Paper Title: Entropy, Thermodynamics and the Geometrization of the Language Model.



plain


   
    Entropy, Thermodynamics and the Geometrization of the Language Model
        
    

    Wenzhe Yang
    
    

    
    
    

    
    <https://orcid.org/0000-0002-2456-7453>

    
    



§ ABSTRACT 

In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. 

Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold.








Keywords: Language Model, Moduli Space, Entropy Function, Thermodynamics, Geometrization of Language Model, Boltzmann Manifold.

 




§ INTRODUCTION


In November 2022, OpenAI surprised the world with the release of the highly “intelligent” ChatGPT, which possesses many striking abilities that far exceed any other available systems <cit.>. For example, it can answer questions with a very high accuracy and engage in human-like conversations. It is the first time that a computer performs well enough in human languages that makes people wonder whether it acquires true intelligence. Ever since then, the arm race in the area of Large Language Model (LLM) has become extremely fierce, with the announcements of new LLMs with hundreds of billions of parameters regularly. Now there are abundant Chatbots based on LLMs available in markets, many of which have astonishing new abilities and behaviors <cit.>. Especially, when the number of parameters (e.g., weights) are increased to a threshold, then some surprising behaviors emerges <cit.>, which is extremely intriguing. 

An LLM is a generative model, which means it can practically sample a new sentence given a prompt. More precisely, suppose we are given an arbitrary prompt w_1⋯ w_n, where each w_i is a word, an LLM can compute the probability distribution of the next word, i.e., P(w_n+1|w_1 ⋯ w_n). Next, we sample the next word w_n+1 according to this probability distribution <cit.>. After we obtain w_n+1, we repeat this process and sample w_n+2 according to the probability distribution P(w_n+2|w_1 ⋯ w_n w_n+1) computed by the LLM. We repeat this process until some conditions are met, e.g., the LLM outputs an end-of-sequence symbol.

Let us now briefly review the basic structures of the LLMs based on the transformer architecture, to the extent that will be needed in this paper. For a more detailed and thorough treatment, the readers are referred to the book <cit.>. First, in order for a computer to be able to compute words, words are first embedded into a vector space, which is called the Word-Embedding <cit.>. Suppose 𝒲 is the finite set of all words, a Word-Embedding is an injective map

    ι: 𝒲→ℝ^N,

where the standard inner product ⟨·, ·⟩ on ℝ^N measures the correlations between words. Given an arbitrary vector 𝐯∈ℝ^N, it defines a distribution on 𝒲 via

    P_𝐯(w) = exp( ⟨𝐯, ι(w) ⟩)/∑_w' ∈𝒲exp( ⟨𝐯, ι(w') ⟩),

which is called the Boltzmann distribution in statistical mechanics <cit.>. 


In particular, given a word w ∈𝒲, it also gives us a distribution on 𝒲 via

    P_ι(w)(w') = exp( ⟨ι(w), ι(w') ⟩)/∑_w”∈𝒲exp( ⟨ι(w), ι(w”) ⟩),

the value of which measures the correlations between w and w'. In practice, the embedding ι is chosen such that P_ι(w)(w') is the probability that w' is in an l-neighborhood of w <cit.>. One nice property of this embedding is that generally the embedded vectors {ι(w)| w ∈𝒲} lie in a lower dimensional vector subspace of ℝ^N as these vectors are not independent from each other. A famous linear relation is 

    ι(Queen) - ι(Woman) = ι(King) - ι(Man).

For more details, the readers are referred to the papers <cit.>.

After the Word-Embedding, a sentence w_1⋯ w_n is mapped to a vector

    (ι(w_1),⋯,ι(w_n)) ∈ (ℝ^N)^n.

As the name suggests, a transformer is a transformation

    T_n: (ℝ^N)^n → (ℝ^N)^n,

which is a differentiable often smooth map <cit.>. For the details of the construction of a transformer and some of the ideas behind it, the readers are referred to the paper <cit.> and the book <cit.>. In particular, each transformer depends on internal parameters which are tuned during the training process. Notice that transformers can be nested in the form

    𝒯_n = T_n,k∘⋯∘ T_n,1,

where k is the number of layers and different layers have different parameters. Let Proj be the projection of (ℝ^N)^n to the last ℝ^N in the direct product, hence Proj∘𝒯(ι(w_1),⋯,ι(w_n)) is a vector of ℝ^N, which defines a distribution on 𝒲 via Eq. (<ref>). The training of the transformers on a very large corpse is to make sure that this distribution is a good approximation of P(·|w_1⋯ w_n) <cit.>.

The LLMs have made rapid advance from a numerical and experimental point of view, but there have been very few papers that attempt to understand the properties and behaviors of LLMs from a analytical point of view. More precisely, we now desperately need a theory that can explain these surprising properties and behaviors, which is the main motivation of this paper. 


In this paper, we first use set theory, analysis and probability theory to rigorously define the concept of Causal Language Model (CLM), and show it is equivalent to the Predicative Language Model (PLM), which basically predicts the next word of a prompt. We formulate a generalized distributional hypothesis for a CLM based on functional analysis and topology. We introduce the concept of the moduli space of distributions on words, which is the key metric space to understand the power and properties of a language model.

The development of human language is for human to exchange information efficiently between individuals. In order to analytically study a language model, we must understand how information is quantified during conversations. In this paper, we introduce the entropy function for CLM and PLM, and show how it can play a crucial role in the study of language models. In particular, we analyze the zero points of the entropy function, and show how to use information theory to understand the meaning of them. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. We also formulate an extension conjecture about the properties of AGI using the entropy function.

Next, we use thermodynamics to give an physical interpretation of the language models. In this interpretation, a sentence is a microstate that has a potential energy. All the possible outputs of a prompt define a statistical ensemble, which is distributed according to the Boltzmann distribution <cit.>. We introduce the concept of the partition function, internal energy and Helmholtz free energy for this statistical ensemble associated. Using the Helmholtz free energy, we show how the game of word-predicting can be interpreted as the process of a molecule growing in physics.

Later, we formulate the concept of the geometrization of a language model, and show that the accuracy of a language model is determined by how precise the embedding of the moduli space in a geometrization is. We also define the Word-Embedding with respect to a general manifold with a pairing structure, and introduce the corresponding Boltzmann manifold with respect to it. At last, we show how the current LLMs can be viewed as special cases of the geometrization of language models that use the linear space together with its inner product. Based on our theory, we also pose several important open questions, e.g., which manifold together with a pairing is optimal for the geometrization of language models. Perhaps different languages would need different manifolds and pairings, namely different Boltzmann manifolds in the geometrization. Another important direction is whether the current researches on statistical mechanics and differential geometry can offer new tools in the study of LLMs.


The layout of this paper is as follows. In Section <ref>, we use set theory, analysis and probability theory to rigorously define language models, i.e., CLM and PLM, and we introduce the moduli space of distributions associated with a language model. In Section <ref>, we introduce the entropy function, and show how information theory can be applied to the study of language models. In Section <ref>, we present a thermodynamic interpretation of language models and define statistical functions such as partition function, internal energy and Helmholtz free energy for a language model. In Section <ref>, we formulate the concept of the geometrization of language models, and define the Boltzmann manifold. In Section <ref>, we show that the current LLMs are special cases of the geometrization of language models. In Section <ref>, we conclude this paper and propose several open problems.









§ THE LANGUAGE MODEL AND MODULI SPACE
 



In this section, we will use set theory, analysis and probability theory to give rigorous definitions of language models. We first define what is a general sentence in the sense of set theory <cit.>. Then we introduce the definitions of the General Language Model (GLM), the Causal Language Model (CLM) and the Predicative Language Model (PLM). At last, we define what is called the moduli space of distributions for a PLM, which is the crucial metric space to understand the properties and behaviors of language models.



 §.§ Sentence as a Free Mathematical Sequence?


Suppose 𝒲 is the finite set of all words in a language, the set of all sentences associated with 𝒲 is defined in the same way as in set theory and logic <cit.>. First, let ∅ be the unique sentence with no word, whose length is by definition 0. Let 𝒮_0 be the set that consists of a unique element ∅

    𝒮_0 = {∅}.


The inclusion of ∅ as a sentence will make our analysis simpler from a mathematical point of view. While later we will also see ∅ is a suitable symbol for the end (or even the beginning) of a sentence.

A sentence of length 1 is just a word of 𝒲. Let 𝒮_1 be the set of all sentences of length 1, namely

    𝒮_1 = 𝒲.

Given an positive integer n ≥ 2, let 𝒮_n be the set of all sentences of length n, i.e., an element of 𝒮_n is a sequence

    𝐬_n=w_1 w_2 ⋯ w_n with w_i ∈𝒲.

Here, the words in a sentence do not need to be different from each other, e.g., we allow the case where w_i=w_j, i ≠ j. The set of all sentences of finite lengths is by definition the union of all 𝒮_n

    𝒮 = ⋃_n=0^∞𝒮_n.

Notice that here we allow the length of a sentence to be arbitrarily large. 

In the definition of 𝒮, we have allowed the existences of arbitrary sentences that do not satisfy grammar rules. In this paper, the grammar is instead considered as part of the language model.



The cardinality of 𝒮 is countably infinite, i.e., there exists a bijective map between 𝒮 and ℕ.


Since 𝒲 is a finite set, this lemma is an immediate result of elementary set theory, see the book <cit.> for more details.



Even in this most general setting, we do not consider the case of sentence of infinite length, i.e., infinite sequence, which is very important from both practical and set-theoretic points of view. Practically, human brain has not evolved to understand infinity properly, so it makes no sense to a mortal that a sentence is infinite. Set-theoretically, the set of all infinite sequences has the same cardinality as the real numbers, i.e., its cardinality is ℵ_1 <cit.>. So we do not really want to walk into the deep water of axiomatic set theory.


There is a natural operation defined on 𝒮 that is called concatenation: given a sentence s_m ∈𝒮_m and a sentence s_n ∈𝒮_n, their concatenation is a new sentence s_m s_n that lies in 𝒮_m+n. As a convention, the concatenation between ∅ and s_n is s_n itself, i.e.,

    ∅𝐬_n = 𝐬_n ∅ = 𝐬_n.

This concatenation operation is associative, but not commutative, while the empty sentence ∅ serves as a two-sided identity. So the set 𝒮 has a monoid structure <cit.>.








 §.§ The Causal Language Model


Now we are ready to give a more mathematical and general definition of the language model.


 
Given an arbitrary sentence s ∈𝒮, a General Language Model (GLM) ℒ_G defines a joint probability distribution P_ℒ_G,𝐬 on 𝒮×𝒮, where the probability P_ℒ_G,𝐬(𝐬_1,𝐬_2) measures the likelihood of the concatenation 𝐬_1𝐬𝐬_2 among all possible choices. Namely, P_ℒ_G,𝐬(𝐬_1,𝐬_2) is the probability that there is a sentence 𝐬_1 before 𝐬 and a sentence 𝐬_2 after 𝐬.



Now, we consider the case where the sentence 𝐬_1 on the left side of 𝐬 is ∅, in which case the probability distribution in Definition <ref> defines a distribution on 𝒮. Hence it motivates the following definition.


Given an arbitrary sentence 𝐬∈𝒮, a Causal Language Model (CLM) ℒ defines a probability distribution on 𝒮. Namely, given a prompt 𝐬, we have a distribution P(·|𝐬) such that the probability that ℒ outputs a sentence 𝐬' is P_ℒ(𝐬'|𝐬).


In the definition, we allow the prompt 𝐬 to be ∅, in which case the CLM ℒ defines a probability distribution on 𝒮, i.e., P(𝐬_out|∅). It is called causal because 𝐬' is after 𝐬, so there is a causality between 𝐬 and 𝐬'. From the definition, the input to ℒ can be of arbitrary length, while the output sentence can also be of arbitrary length. In fact, such a definition of CLM is extremely powerful.




Let ℬ(𝒮,ℝ) be the set of all functions with finite absolute summation, i.e.,

    ℬ(𝒮,ℝ) = {f:𝒮→ℝ| ∑_𝐬∈𝒮|f(𝐬)| < ∞}.

The set ℬ(𝒮,ℝ) is a metric space with a metric given by

    ‖ f-g ‖  =  ∑_𝐬∈𝒮 | f(𝐬) - g(𝐬) |.

For more details about this space, the readers are referred to the book <cit.>. Clearly for every 𝐬, the distribution P_ℒ(·|𝐬) is a non-negative function that lies in ℬ(𝒮,ℝ), in fact ‖ f ‖ = 1. Therefore, there exists a map

    ρ_ℒ: 𝒮→ℬ(𝒮,ℝ).


In 1950s, linguists formulated the distributional hypothesis, which basically says that the meaning of words are determined by their contexts <cit.>. We generalize this hypothesis using real analysis.

If ℒ is an intelligent language model, then the map ρ_ℒ is an embedding that is also discrete. Namely, each point ρ_ℒ(𝐬) has a neighborhood that does not contain any other ρ_ℒ(𝐬') where 𝐬≠𝐬'. 

For more details of the meaning of “discrete” in the sense of topology, the readers are referred to the book <cit.>. As a result, the embedding ρ_ℒ induces a metric on 𝒮.



It should be stressed that in the definition of a CLM, we do not require it to have “intelligence” behaviors. For example, for a given prompt 𝐬, the probability distribution P(· | 𝐬) can be a “random” distribution over 𝒮, which can even outputs sentences that are quite pathological. Hence, intelligence is not incorporated in the definition of the language model. Instead, only highly special language models can exhibit intelligence behaviors.


 







 §.§ The Predicative Language Model
 

Now we give the definition of the Predicative Language Model.


Given a sentence 𝐬, a Predicative Language Model (PLM) ℒ_P defines a probability distribution P(·|𝐬) on 𝒲∪∅. Here, P(∅|𝐬) is the probability that ℒ_P outputs nothing and P(w|𝐬) is the probability that ℒ_P outputs the word w ∈𝒲.


We have the following lemma.

The definition of the PLM is equivalent to the definition of the CLM.


If we have a PLM ℒ_P, then by repetitively using it we can generate sentences. More precisely, given an input 𝐬∈𝒮, we can generate a sentence 𝐬'=w_1w_2⋯ w_n until ℒ_P outputs ∅ via

    ℒ_P(𝐬) = w_1,  ⋯,  ℒ_P(𝐬w_1 ⋯ w_n-1) = w_n,  ℒ_P(𝐬w_1 ⋯ w_n) = ∅.

The probability that ℒ_P outputs 𝐬'=w_1⋯ w_n given 𝐬 is 

    P(𝐬'|𝐬)=P(w_1|𝐬) · P(w_2|𝐬w_1) ·…· P(w_n|𝐬w_1 ⋯ w_n-1) ·  P(∅|𝐬𝐬').

In this way, the PLM ℒ_P defines a probability distribution on 𝒮 for any prompt 𝐬, hence it produces a CLM.

We now show the inverse construction from CLM to PLM. Namely, suppose we have a CLM ℒ, we now construct a PLM ℒ_P that reverses the process in the previous paragraph. Given a prompt 𝐬, let us construct a distribution on 𝒲∪∅. First, P(∅|𝐬) is straightforward to define, i.e., P_ℒ(∅|𝐬). For an arbitrary word w, let 
    P(w|𝐬) = ∑_𝐬' ∈𝒮 P_ℒ(w𝐬'|𝐬).

Notice that here the sum is just over all sentences whose first word is w. In this way, we obtain a PLM.



As shown in the proof of this lemma, the empty sentence ∅, which is the unit of the monoid 𝒮, plays the role of ending the sentence outputted by ℒ_P. Hence comes a crucial definition in our treatment of PLM.

 
The space of all possible distributions on 𝒲∪{∅} that come from a PLM ℒ_P is called the moduli space of distributions of ℒ_P, which is expressed as

    ℳ = { P(·| 𝐬)| 𝐬∈𝒮}.


In fact, this moduli space is a metric space. Let ℬ(𝒲∪{∅},ℝ) be the set of all functions with finite absolute summation

    ℬ(𝒲∪{∅},ℝ) = {f:𝒲∪{∅}→ℝ| ∑_w ∈𝒲∪{∅}|f(w)| < ∞}.

The set ℬ(𝒲∪{∅},ℝ) is a metric space with a metric given by

    ‖ f-g ‖  =  ∑_w∈𝒲∪{∅} | f(w) - g(w) |.

For more details, the readers are referred to the book <cit.>. Clearly, for every 𝐬∈𝒮, the distribution P(·|𝐬) ∈ℳ is a non-negative function that lies in ℬ(𝒲∪{∅},ℝ), in fact ‖ P(·|𝐬) ‖ = 1. Therefore, ‖·‖ induces a metric d on the moduli space ℳ

    d: ℳ×ℳ→ℝ.


The moduli space ℳ together with the metric d is the key object to understand a PLM.








§ THE ENTROPY FUNCTION
 


In this section, we introduce the concept of the entropy function and show how it quantifies the information contained in the output for a given prompt. We will study the zero-points, which are also the minimal points of the entropy function, and discuss their crucial properties. We argue that these zero points and the points where the entropy is close to 0 are the key obstacles for a LLM to approximate an intelligent language model. At last, we formulate a conjecture regarding AGI.




 §.§ Information Flow and Entropy


The entropy function for a CLM ℒ is defined just as in statistical mechanics and information theory <cit.>.


Given a prompt s ∈𝒮, the entropy of 𝐬 with respect to a CLM ℒ is given by

    S_ℒ(𝐬)= - ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) log P_ℒ(𝐬'|𝐬),

where the sum is over all possible outputs 𝐬'.

We will see that the entropy function is a key mathematical concept to analyze the behaviors of language models. From the definition, the entropy function is a non-negative function defined on 𝒮

    S_ℒ: 𝒮→ℝ.

As a special example, if the prompt is ∅, i.e., no prompt, the entropy associated with it has a special name which is stated as a definition.

The background entropy of a CLM ℒ is

    S_ℒ(∅) = - ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|∅) log P_ℒ(𝐬'|∅).


The background entropy S_ℒ(∅) is a quantity that gives a zeroth order measurement about how complicated a language is. If S_ℒ(∅) is very large, this intuitively mean that the CLM is quite diverse. While if S_ℒ(∅) is very small, it intuitively means that the CLM is very rigid.


Since the development of language is for individuals to exchange information, so a language model itself must be able to demonstrate this. Here we argue that the definition of the entropy function exactly quantify the information during this process <cit.>. More precisely, given a prompt 𝐬, the entropy S_ℒ(𝐬) measures how vague are the answers to the prompt 𝐬 with respect to this CLM ℒ. If S(𝐬) is very large, this means that the answers can be very diverse. On the other hand, if S(𝐬) is very small, this means the answer to 𝐬 is very rigid. Therefore, after the CLM outputs 𝐬', we can say the information revealed by 𝐬' is S_ℒ(𝐬) <cit.>.


Just like statistical mechanics, we argue that the most important function for a CLM is the entropy function <cit.>.








 §.§ The Zero Points of the Entropy Function






Let us now look at the zero points of S_ℒ. If S_ℒ(𝐬)=0, then it means that the answer to the prompt 𝐬 is unique in the CLM ℒ, i.e., there exists a unique sentence 𝐬' ∈𝒮 such that P_ℒ(𝐬'|𝐬) = 1. Hence there is the interpretation that the CLM ℒ detects that the answer to the prompt 𝐬 is unique, and thus the CLM itself actually contain this information. For example, suppose the prompt 𝐬 is

    𝐬 =

If the entropy function of an intelligent CLM ℒ vanishes at 𝐬, i.e., S_ℒ(𝐬) = 0, then this means there is only one possible output of ℒ for the prompt 𝐬. Of course, an intelligent ℒ that knows about human history will output 

    𝐬' =

with probability 1. It should be noticed that at the point 𝐬, the vanishing of the entropy only implies there is only one possible output to 𝐬. Another CLM ℒ_1 could still have S_ℒ_1(𝐬)=0, but it instead outputs 

    𝐬” = .

We say the CLM ℒ_1 outputs a `wrong' answer because Gauss is a mathematician, so the information given by ℒ_1 is wrong. But `wrong' information is still information, the entropy function itself only quantifies the amount of information, but it cannot detect whether the information contained is right or wrong, which needs to be checked with what happens in the outside world. 

Given a CLM ℒ, a zero point 𝐬 of the entropy function S_ℒ is called a singularity of ℒ.

Hence the conclusion is that a CLM memorize information at its singularities. We argue that the existence of these singularities and the points where the entropy function is very small are the key obstacle for an LLM to approximate an intelligent CLM. 






 §.§ An AGI Conjecture
 


It is hard to define which CLM is Artificial General Intelligence (AGI) mathematically! Our idea is that if a CLM ℒ is an AGI, then it must satisfy certain mathematical properties. In this paper, we formulate a conjecture about this.








 
If a CLM is an AGI, then there exists a finite subset 𝒮_AGI⊂𝒮 such that the entropy function S_ℒ: 𝒮→ℝ is determined by its value on 𝒮_AGI.

The motivation of this conjecture comes from modern mathematics. In modern mathematics, we first give a finite set of definitions and axioms, and then mathematics is about proof, which uses these finite set of rules of prove a result that is not known a priori. The key motivation for Conjecture <ref> is that the value of S_ℒ on 𝒮 is determined using mathematical deductions by its value on the finite set 𝒮_AGI <cit.>. 




§ A THERMODYNAMIC INTERPRETATION OF LANGUAGE MODELS
 

In this section, we will give a thermodynamic interpretation of the CLM defined in Section <ref>. We show how the concepts and properties of the CLM naturally admit physical interpretations. Then we will define thermodynamic functions such as internal energy, Helmholtz free energy on CLM, and show how they help us understand phenomena in language models. At last, we give a physical interpretation to how a sentence “grows” in a PLM.




 §.§ The Boltzmann Distribution


First, let us introduce the Boltzman distribution in statistical mechanics <cit.>. Suppose we are given a physical system that can occupy different microstates, and suppose the energy of the microstate i is E_i. Let k be the Boltzmann constant and T be the temperature, then the probability that the system occupies  microstate i satisfies

    p_i ∝exp( - E_i/kT).

Let the partition function of the system be

    𝒵= ∑_iexp( - E_i/kT),

where the sum is over all possible microstates. Then p_i is just

    p_i = 1/𝒵·exp( - E_i/kT),

which is called the Boltzman distribution. The Boltzmann distribution is the distribution that maximizes the entropy

    S = - ∑_i p_i log p_i

subject to the constraint ∑ p_i =1 and ∑_i p_i E_i is a constant, i.e., the average energy is kept fixed.



 §.§ A Thermodynamic Interpretation
 

We now give a thermodynamic interpretation of CLM. 

In a CLM ℒ, a sentence is a microstate and for every sentence 𝐬∈𝒮, there is an energy E_ℒ(𝐬) associated with it.

The energy of ∅ is called the vacuum energy, and the energy of a word w ∈𝒲 is called its self-energy <cit.>.


As in physics, a word w ∈𝒲 itself can have self-interaction, it has an internal energy E(w), the meaning of this internal energy is that given a random sentence 𝐬, the probability that the first word of 𝐬 is w is proportional to exp(-β E(w) ). Here β is the inverse of temperature.


Given a length-2 sentence w_1w_2, its energy E_ℒ(w_1w_2) is the interaction between the two words w_1 and w_2, and E_ℒ(w_1w_2) is lower if the interaction between w_1 and w_2 is stronger. 

Since a language is essentially a discrete object, a word does not have a continuous degree of freedom. Therefore, there is no kinetic energy for a word, and only potential energy exists when words form a sentence. 

Here comes the definition of the statistical ensemble in a language model.

For a given prompt 𝐬, the statistical ensemble associated with it is 

    {𝐬𝐬'|𝐬' ∈𝒮}.



For later purpose, we introduce another quantity β which is the inverse of temperature <cit.>. For a fixed CLM, the value of β is usually fixed to be 1. 

In a CLM ℒ, for a given prompt 𝐬 the probability that ℒ outputs 𝐬' satisfies

    P_ℒ(𝐬'|𝐬) ∝exp(-β E_ℒ(𝐬𝐬') ),

where E_ℒ(𝐬𝐬') is the energy of the concatenation 𝐬𝐬'.   


More precisely, for a given prompt 𝐬, the probability that ℒ outputs 𝐬' is

    P_ℒ(𝐬'|𝐬) = 1/𝒵_ℒ·exp( - β E_ℒ(𝐬𝐬') ),

where E_ℒ(𝐬𝐬') is the energy of the concatenation 𝐬𝐬' and 𝒵_ℒ is the partition function associated with 𝐬

    𝒵_ℒ(𝐬) = ∑_𝐬' ∈𝒮exp( - β E_ℒ(𝐬𝐬')).

The partition function is in fact a thermodynamic function defined on 𝒮

    𝒵_ℒ:𝒮→ℝ.

 

The average energy of the statistical ensemble associated with 𝐬 is 

    U_ℒ (𝐬) = ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) E_ℒ(𝐬𝐬'),

which is called the internal energy of the ensemble. The internal energy U_ℒ is also a well-defined function on 𝒮

    U_ℒ : 𝒮→ℝ.

But notice that U_ℒ is different from E_ℒ: E_ℒ(𝐬) is the energy of the microstate 𝐬, while U_ℒ(𝐬) is the average energy of the statistical ensemble associated with 𝐬. Recall that the entropy for the statistical ensemble associated to 𝐬 is

    S_ℒ(𝐬)=-  ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) log  P_ℒ(𝐬'|𝐬) ,

which is simplified to

    S_ℒ = β U_ℒ + log𝒵_ℒ.

The Helmholtz free energy is given by

    F_ℒ = U_ℒ - 1/β S_ℒ=- 1/βlog𝒵_ℒ.

The Helmholtz free energy determines whether a sentence 𝐬 is likely to be followed by another sentence or not.







 §.§ A Physical Picture of How a Sentence Grows



Let us now interpret the process of predicting the next word in a PLM using how molecule grows in statistical mechanics. First, because of the equivalence between CLM and PLM, the interpretation in Section <ref> admits immediate generalization to PLM. In this physical picture, a word is view as an atom, and a sentence is a sequence of atoms, whose energy depends on the order of these atoms. Suppose we are given a prompt 𝐬∈𝒮_n, which is composed of n atoms. To predict the next word is to add an atom to 𝐬. For a word w ∈𝒲∪{∅}, the energy of 𝐬w is E(𝐬w), and the internal energy is by definition

    ∑_w ∈𝒲∪{∅} P(w|𝐬) E(𝐬w).

We also have the entropy

    - ∑_w ∈𝒲∪{∅} P(w|𝐬) log P(w|𝐬).

There are two trends that fight each other

    
  * the internal energy (Eq. (<ref>)) wants to be as small as possible,
    
  * the entropy (Eq. (<ref>)) wants to be as large as possible.

As a result, the Helmholtz free energy (Eq. (<ref>)) is minimized, and the resulting distribution is the Boltzmann distribution. After sampling the next word, say w_n+1, we can continue this process with the new sentence 𝐬w_n+1 and sample its next one. This process is highly similar to the process of how a molecule grow in statistical mechanics!


But there remains the question of how to compute the energy of a sentence. A sentence can be very long, hence the computation of its energy is a typical many-body problem. The power of LLM is to approximately compute the interaction energy between the “atoms” in a sentence using machine learning.




§ THE GEOMETRIZATION OF LANGUAGE MODELS
 

In this section, we will introduce the concept of the geometrization of a PLM, which lays the foundation for studying language models using differential geometry. We will also define a more general version of the Word-Embedding, which leads to the concept of a Boltzmann manifold.





 §.§ The Geometrization of PLM


Suppose we have a PLM ℒ_P, whose moduli space of distributions is ℳ (Definition <ref>). Moreover, we have a map

    ρ: 𝒮→ℳ,

where ρ sends a sentence 𝐬∈𝒮 to the distribution P(·| 𝐬) ∈ℳ. Notice that ℳ is a metric space, the details of which can be found in Section <ref>.

Generally, the moduli space ℳ itself is extremely complicated, to study which we need to borrow ideas from both mathematics and physics. The first step is to geometrize this complicated metric space, which “prompts” the following definition.


The ϵ-geometrization of a PLM is a manifold M and a continuous map

    Ψ: ℳ→ M,

where each point of M represents a probability distribution on 𝒲∪∅ that is also computable. Moreover, for every point P(·| 𝐬) of ℳ, the norm of the difference ‖Ψ(P(·| 𝐬))-P(·| 𝐬) ‖ < ϵ(n). Here, the error ϵ(n) can depend on the length of the sentence 𝐬∈𝒮_n.

Notice that the norm ‖·‖ is defined in the metric space ℬ(𝒲∪{∅},ℝ) (Eq. (<ref>)). The composition Ψ∘ρ is a map from 𝒮 to M

    Ψ∘ρ: 𝒮→ M.

The motivation of the geometrization of the moduli space ℳ is that we try to find a finite dimensional manifold M to approximately describe the infinite set ℳ. Then we use the geometry of this manifold to study the bizarre metric space ℳ.




 §.§ The Word-Embedding and the Boltzmann Manifold
 

One way to construct an ϵ-geometrization is via an additional structure on a manifold called the pairing. Suppose M is a manifold, a symmetric pairing is a smooth function 

    ⟨·, ·⟩ : M × M →ℝ,

which maps any two points pt_1, pt_2 ∈ M to a number in ℝ. Here “symmetric” means

    ⟨pt_1, pt_2 ⟩ = ⟨pt_2, pt_1 ⟩.

A Word-Embedding is an injective map

    ι: 𝒲∪{∅}→ M.

Given an arbitrary point pt∈ M, it defines a distribution P_pt on 𝒲∪{∅} via

    P_pt(w) =  exp( - β⟨ pt,ι(w) ⟩)/∑_w' ∈𝒲∪∅exp( - β⟨ pt,ι(w') ⟩),

which is exactly from the Boltzmann distribution <cit.>. Hence every point of M defines a distribution on 𝒲∪{∅} and M is a space of distributions on 𝒲∪{∅}.


The manifold M together with the pairing ⟨·,·⟩ and the Word-Embedding ι is called a Boltzmann manifold. 


Since every point of the Boltzmann manifold M defines a distribution on 𝒲∪{∅}, M is a subspace of  ℬ(𝒲∪{∅},ℝ) (Eq. (<ref>)). Therefore, given two points pt and pt' of M, their distance is

    d(pt,pt') =   ‖ P_pt-P_pt'‖.

This distance function induces a metric g_B on M, which is called the Boltzmann metric, hence M is in fact a Riemannian manifold <cit.>.

Now suppose M gives us an ϵ-geometrization with the Moduli-Embedding Ψ: ℳ→ M. Since 𝒮_n is the direct product 𝒲^n, thus if we believe the map Ψ∘ρ has sufficiently nice properties, then there exists a diagram

    𝒮_n = 𝒲^n @>ι^n>> M^n
    
    @VVρV @VVΛ_nV
    ℳ @>Ψ>> M
      ,

which is commutative up to a small error that depends on n. More precisely, we assume there exists a smooth map Λ_n:M^n → M such that for every 𝐬=w_1w_2⋯ w_n, we have

    Ψ∘ρ (𝐬) = Λ_n( ι(w_1), ⋯, ι(w_n) ).

It means that the map ρ∘Ψ is determined by the map ι and {Λ_n:n ≥ 2 }. The point is that the map Ψ∘ρ is defined on the infinite set 𝒮, which can be highly complicated. While the map ι is defined on a finite set and Λ_n is defined on a smooth manifold, hence can be studied by the modern theory of differential geometry <cit.>.









§ LLM AS A SPECIAL GEOMETRIZATION
 

In this section, we show that the current LLMs based on the transformer architecture are special cases of geometrization in the sense of Section <ref>. We also introduce the fundamental thermodynamic relation of LLMs.




 §.§ The Geometrization and LLM



The Boltzmann manifold used in LLMs is the simplest one: ℝ^N together with the standard inner product ⟨· , ·⟩. In practice, first we obtain a Word-Embedding  <cit.>

    ι: 𝒲∪{∅}→ℝ^N.

Then a point 𝐯 of ℝ^N defines a distribution on 𝒲∪{∅} via 

    P_𝐯(w) = exp(-β⟨𝐯, ι(w)⟩)/∑_w' ∈𝒲∪{∅}exp(-β⟨𝐯, ι(w')⟩),

where β is the inverse of temperature. The Moduli-Embedding is a map

    Φ: ℳ→ℝ^N.

Given a sentence 𝐬∈𝒮, Ψ∘ρ(𝐬) is a vector of ℝ^N, which gives the distribution of the next word of 𝐬 via Eq. (<ref>), i.e.,

    P(w|𝐬) = exp(-β⟨Ψ∘ρ(𝐬), ι(w)⟩)/∑_w' ∈𝒲∪{∅}exp(-β⟨Ψ∘ρ(𝐬), ι(w')⟩).




For every n ≥ 1, there exists a map Λ_n: (ℝ^N)^n →ℝ^N that makes the following diagram commutative up to a small error

    𝒮_n = 𝒲^n @>ι^n>> (ℝ^N)^n
    
    @VVρV @VVΛ_nV
    ℳ @>Ψ>> ℝ^N
    .


Namely, given a sentence 𝐬=w_1⋯ w_n, the distribution of the word after 𝐬 is given by the vector Λ_n(ι(w_1),⋯,ι(w_n)) via Eq. (<ref>). In practice, both ι and Λ_n are determined by data and machine learning <cit.>. Currently, the most popular method to construct the map Λ_n is based on the transformer architecture <cit.>.





 §.§ The Fundamental Thermodynamic Relation of LLMs


Given a geometrization (ℝ^N, ⟨·,·⟩, ι), the properties of w ∈𝒲∪{∅} with respect to this geometrization are determined by the embedding vector ϕ(w) ∈ℝ^N. Borrowing language from physics <cit.>, the space ℝ^N is also called the phase space of a language model. Just as in CLM, in a PLM we also have the entropy function S, the internal energy U and Helmholtz free energy F defined on 𝒮.



Suppose we are given a geometrization (ℝ^N, ⟨·,·⟩, ι) and the maps {Λ_n| n≥ 1 } associated with a PLM. The finite set 𝒮_n is viewed as a subspace of (ℝ^N)^n via the embedding ι^n. Then all thermodynamic functions such as S, U and F defined over 𝒮_n admit smooth extensions to functions over (ℝ^N)^n up to small errors.

Let us now look at the entropy function S as an example. This conjecture implies that there exists a differentiable function S: (ℝ^N)^n →ℝ that makes the following diagram commutative up to a small error

    𝒮_n = 𝒲^n @>ι^n>> (ℝ^N)^n
    
    @VVρV @VVSV
    ℳ @>S>> ℝ.




In fact, from the construction of LLMs using the architecture of transformer, the extensions of thermodynamic functions always exist, but the error cannot be guaranteed to be small.


Suppose the coordinate of (ℝ^N)^n is x_i,j with 1 ≤ i ≤ N and 1 ≤ j ≤ n, then we have the fundamental thermodynamic relation of LLMs

    dU = 1/β dS -∑_i,jf_i,jdx_i,j,

where f_i,j is the generalized force given by

    f_i,j=∂S/∂ x_i,j.




As a function on (ℝ^N)^n, we conjecture that the entropy function S for a LLM is convex. 

Suppose 0 ≤λ≤ 1, given two points 𝐱=(x_i,j) and 𝐱'=(x'_i,j) of (ℝ^N)^n, we have

    S(λ𝐱 + (1 - λ) 𝐱' ) ≥λS( 𝐱) + (1 - λ ) S(𝐱' ).


It has an intuitive interpretation, if you mix the two vectors 𝐱=(x_i,j) and 𝐱'=(x'_i,j), the entropy is always greater than the average. Namely, if you mixed the meanings, it always becomes more vague. 








§ CONCLUSION AND OPEN PROBLEMS
 

In this paper, we have used set theory, analysis and probability theory to rigorously give the definitions of Causal Language Model (CLM) and Predicative Language Model (PLM). We propose a generalization of the distributional hypothesis. We introduce the concept of the moduli space of distributions for a CLM, which is the key metric space to understand the power and properties of a language model. Then we define the entropy function for a CLM, study its properties, and show how its zero points are crucial for a language model to incorporate information and logic. Using the entropy function, we formulate an extension conjecture about AGI.

We have used thermodynamics to show that language models admit thermodynamic interpretations. Namely, a sentence is a microstate and a prompt gives us a statistical ensemble, which is distributed according to the Boltzmann distributions. We also formulate the concept of the geometrization of language models and define the Boltzmann manifold. We show the embedding of the moduli space into a Boltzmann manifold determines how accurate can a geometrization be. At last, we discuss how the current LLMs can be viewed as special cases of geometrization with respect to the linear space together with its inner product.

There are many interesting open questions that deserve further study. Perhaps the most straightforward question is what is the best Boltzmann manifold for the geometrization of a language model. Current LLMs all uses the linear space together with its inner product. But it is likely that different languages would need different Boltzmann manifold for the geometrization. For example, some of the possible choices of Boltzmann manifold with non-trivial topology can be

    ℝ^N × S^M, ℝ^N × (S^1)^M,

where S^k is a k-dimensional sphere. For such manifold, it is very interesting to see whether the non-trivial topology has any effect on the properties of the resulting LLM.

Another important question is whether results in modern physics and mathematics can be applied to study language models. For example, whether statistical physics can help us understand the emergent abilities of LLMs. Whether the results in statistical mechanics and differential geometry can offer a better architecture than the transformer. There are numerous results in the study of many-body interaction systems, it is interesting to see whether these results can have applications to LLMs.







plain





 Paper Title: Persistent Topological Negativity in a High-Temperature Mixed-State.






Department of Physics, University of California, Santa Barbara, CA 93106, USA	


Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106


Department of Physics, University of California, Santa Barbara, CA 93106, USA	

§ ABSTRACT 
We study the entanglement structure of the Greenberger–Horne–Zeilinger (GHZ) state as it thermalizes under a strongly-symmetric quantum channel describing the Metropolis-Hastings dynamics for the d-dimensional classical Ising model at inverse temperature β. This channel outputs the classical Gibbs state when acting on a product state in the computational basis.  When applying this channel to a GHZ state in spatial dimension d>1, the resulting mixed state changes character at the Ising phase transition temperature from being long-range entangled to short-range-entangled as temperature increases. Nevertheless, we show that the topological entanglement negativity of a large region is insensitive to this transition and takes the same value as that of the pure GHZ state at any finite temperature β>0. We establish this result by devising a local operations and classical communication (LOCC) “decoder" that provides matching lower and upper bounds on the negativity in the thermodynamic limit which may be of independent interest. This perspective connects the negativity to an error-correction problem on the (d-1)-dimensional bipartitioning surface and explains the persistent negativity in certain correlated noise models found in previous studies. Numerical results confirm our analysis. 


 Persistent Topological Negativity in a High-Temperature Mixed-State
    Sagar Vijay
    February 2024
===================================================================



Introduction: The nature and fate of quantum entanglement in mixed-states of quantum many-body systems – arising in a description of thermal quantum matter  or after a pure state experiences local decoherence – have been recently investigated in order to understand the universal quantum correlations that can persist in open quantum systems <cit.>. Patterns of long-range-entanglement (LRE) in mixed quantum matter are intricately related to quantum many-body phases which can act as quantum error-correcting codes <cit.>.  Thermal states of quantum many-body systems are also now known to be efficiently classically-preparable at sufficiently high temperatures <cit.>.

The entanglement negativity provides a well-studied diagnostic of bipartite entanglement in a mixed-state, which vanishes if the state can be prepared by from a product state by local operations and classical communication (LOCC) across the entanglement bipartition <cit.>.  While the negativity of a region A grows with the area of its boundary in a generic Gibbs state <cit.>, contributions to the negativity which are independent of coarse-grained details of the bipartitioning surface signal the presence of LRE in a mixed-state and can probe the presence of mixed-state topological quantum order <cit.>.  Diagnostics of non-local mixed-state entanglement, such as the topological entanglement negativity <cit.>, are challenging to study analytically, and the subtle behavior of mixed-state LRE under a finite-depth local quantum channel (FDLC) remains to be fully understood <cit.>.  

In this work, we investigate the fate of mixed-state quantum entanglement under a Metropolis-Hastings <cit.> quantum channel describing the equilibration of a thermal classical system, specifically that of a d-dimensional Ising model at inverse temperature β. When acting on a pure state in the computational basis, this channel simply prepares a classical thermal state of the d-dimensional Ising model. 

The nature of the mixed quantum many-body state produced by the action of this channel on a generic input state is more striking. When the initial state is chosen to be a Greenberger–Horne–Zeilinger (GHZ) state, a symmetric, long-range-entangled ground-state of the ferromagnetic  phase, we show that the resulting quantum many-body mixed state at any finite temperature β > 0 has a negativity which is asymptotically exactly that of the pure GHZ state in the limit that the boundary of the sub-region becomes large. In spatial dimensions d>1, the negativity of a large, contiguous region in this mixed state ρ_β is thus a constant, independent of geometric details of the bipartitioning surface, and insensitive to the thermal phase transition, as summarized in Eq. (<ref>). Nevertheless, the entanglement properties of the mixed-state do change across this transition; we show that ρ_β cannot be expressed as a convex sum of short-range-entangled (SRE) pure states in the ordered phase but is preparable using an FDLC acting on a product initial state in the high-temperature, disordered phase.  This observed behavior is in stark contrast to the negativity of the Gibbs state of a local, quantum many-body Hamiltonian which necessarily vanishes above a finite temperature <cit.>.  

This result is established by using the fact that the entanglement negativity is an entanglement monotone <cit.>, so that the negativity of a subregion A of any state ρ, denoted (ρ), cannot increase under an LOCC operation  across the bipartitioning surface

    ([ρ])≤(ρ).

This property lends a new perspective on mixed-state negativity, which is of possibly broader interest outside of the specific mixed-states that we study. We explicitly construct an LOCC operation that recovers the GHZ state at any finite temperature and for any bipartition in the asymptotic limit that the boundary A becomes large.  We show that for a contiguous bipartition, this channel is the decoder for a (d-1)-dimensional repetition code, which the two parties in the LOCC protocol must use to determine the precise unitary circuit that must be applied to recover the GHZ state.  The well-known success of the (d-1)-dimensional repetition code up to a maximal error strength in dimensions d > 1 is intimately related to the recovery of the GHZ state, and thus the constant entanglement negativity, at any finite temperature. 

We comment on broader utility of this perspective for understanding mixed-state negativity, by demonstrating that this can provide a simple understanding of why the topological entanglement negativity can remain a constant in correlated noise models (e.g. in <cit.>).


Setup: Consider N qubits on a d-dimensional hypercubic lattice, in a state described by the density matrix

    ρ_β≡1/_β∑_σe^-β H|ψ_σ⟩⟨ψ_σ|.

Here, 

    H ≡ -J∑_⟨ i,j⟩Z_iZ_j

is the energy of a classical Ising model on the d-dimensional hypercubic lattice with ferromagnetic (J>0) nearest-neighbor interactions, while _β≡(e^-β H) is the corresponding partition function at inverse temperature β.  Furthermore, the state
|ψ_σ⟩≡ (|σ⟩ + U|σ⟩)/√(2), 
where |σ⟩≡|σ_1,…,σ_N⟩ is a product state in the Pauli Z basis, and U≡∏_jX_j is a unitary transformation which generates the Ising symmetry of Eq. (<ref>).

The mixed-state ρ_β is naturally obtained by starting with a GHZ state |ψ_+⟩ = (|↑⋯⟩ + |↓⋯⟩)/√(2), which is an equal-amplitude superposition of the macroscopically-distinct all-up and all-down qubit configurations, and then applying a channel Φ_β which describes a Metropolis-Hastings algorithm <cit.> in which the energy H is measured, and then a unitary operation X_j is applied on a random qubit with probability min{1, exp(-β Δ E_j)}, where Δ E_j≡ E_new - E_old is the energy difference between the new and old qubit configurations. 

Repeated measurements and feedback, starting from any product state in the Pauli Z basis, will eventually produce a Gibbs ensemble of the d-dimensional classical Ising model at inverse temperature β <cit.>.   We specifically define Φ_β to be  repeated application of the Metropolis-Hastings algorithm so as to produce a thermal steady-state when acting on the all-up state ρ_0≡|↑⋯⟩⟨↑⋯|, i.e.

    Φ_β[ρ_0] = 1/_β∑_σe^-β H|σ⟩⟨σ|.

Each operation that comprises the channel Φ_β manifestly commutes with the Ising symmetry transformation U.  Therefore, U is a strong symmetry of this channel <cit.>, i.e.    UΦ_β[ρ]=Φ_β[Uρ] for any state ρ, so that

    Φ_β[|ψ_+⟩⟨ψ_+|] = 1+U/√(2)Φ_β[ρ_0]1+U/√(2) = ρ_β.


Mixed-State Entanglement:
The state ρ_β witnesses a thermal phase transition with increasing temperature in spatial dimensions d>1 between a ferromagnetically-ordered and a disordered state. At any temperature, the state ρ_β has a strong Ising symmetry Uρ_β = ρ_β.  However, the ferromagnetically-ordered phase is also characterized by long-range-order in local operators which are charged under this symmetry (ρ_βZ_iZ_j) |i-j|→∞0.  These conditions are sufficient to establish  <cit.> that this mixed-state cannot be described as an ensemble of SRE pure-states.  

We may also show that in the disordered phase, ρ_β can be prepared by an FDLC, and is thus an SRE mixed-state [The ability to prepare ρ_β by an FDLC implies that this mixed-state describes an ensemble of SRE pure state.].  In the Supplemental Material <cit.>, we formally show that ρ_β = ℰ_ZZ[|ϕ(β)⟩⟨ϕ(β)|], 
where ℰ_ZZ is a quantum channel that describes maximal dephasing of Z_iZ_j along all nearest-neighbor bonds.  Furthermore, we show <cit.> that the wavefunction |ϕ(β)⟩≡_β^-1/2∑_μe^-β H/2|μ⟩, 
where |μ⟩ denotes a product state in the Pauli X basis, is the exact ground-state of the  Hamiltonian

    H_0(β) = ∑_j[∏_i∈NN(j)e^-β J Z_iZ_j -X_j],

which exhibits a phase transition at a critical value of β=β_c(d) which is in the d-dimensional Ising universality class.  When β<β_c(d),  |ϕ(β)⟩ is the Ising-symmetric ground-state in the disordered phase and can thus be prepared from a product state using a finite-depth local unitary circuit.  Thus in this range of temperatures, ρ_β can be prepared by an FDLC by composing this circuit with the quantum channel ℰ_ZZ.


Conditional Mutual Information:
We now investigate entanglement properties of ρ_β and their relation to the Ising phase transition. The conditional mutual information (CMI) is a measure of tripartite correlations in a state and has been used to study various aspects of entanglement in quantum systems<cit.>, including probing mixed state quantum phases of matter<cit.>. 
Given subsystems A, B, and C, the CMI is defined as

    I(A:C|B)
           =S(AB)+S(BC)-S(B)-S(ABC).

Consider the partitioning shown in Fig. <ref> with r>0, where C is taken to be the complement of AB. Note that for any proper subset A of qubits, ρ_β,A=_A̅(ρ_β)=ρ_β,A^Cl, where ρ_β^Cl=e^-β H/_β is the classical Gibbs state [This in particular shows that for mutual information we have I(A:C)=I^Cl(A:C) provided that r>0, where I^Cl is computed with respect to ρ_β^Cl.]. This shows that the first three terms in Eq. (<ref>) take the same value when computed for ρ_β as for ρ_β^Cl. Moreover, it is straightforward to use  Eq.(<ref>) to show S(ρ_β)=S(ρ_β^Cl)-log(2). Therefore I(A:B|C)=log(2)+I^Cl(A:C|B),
where I^Cl(A:C|B) is the CMI computed for ρ_β^Cl. On the other hand, since B is assumed to be non-empty, there is no interaction term in H with support on both A and C and hence I^Cl(A:C|B)=0 <cit.>, and

    I(A:B|C)=log(2),

independent of r, temperature β and the system size. It is worh mentioning that in contrast, for thermal Gibbs state of local quantum Hamiltonians, CMI decays exponentially with r <cit.>. 


Entanglement Negativity:
We now investigate the entanglement negativity of a subregion A of the mixed-state (<ref>), (ρ_β) ≡ (||ρ_β^T_A||_1 - 1)/2.  We will show that for a region A, 

    (ρ_β) = {[ 1/2 - O(exp(-|∂ A|))                  β>0;                     ;                    0                  β=0 ].

where |∂ A| is the number of qubits on the boundary of A. The second result is easily established. At infinite temperature, ρ_β = 0 = (1 + U)/2^N, so that ρ_β^T_A = ρ_β and (ρ_β=0)=0. 

The first result is determined as follows. The negativity is a convex function <cit.> so that 

    (ρ_β) ≤1/_β∑_σe^β J∑_⟨ i,j⟩σ_iσ_j(|ψ_σ⟩⟨ψ_σ|) = 1/2

for any region A, since |ψ_σ⟩ is related to the GHZ state |ψ_+⟩ by single qubit gates, so that (|ψ_σ⟩⟨ψ_σ|) = (|ψ_+⟩⟨ψ_+|).  The latter is exactly 1/2 for any region A. The negativity is also an entanglement monotone <cit.>, see Eq. (<ref>).  Here, we show that given the state ρ_β for any β > 0 and any region A, there is an LOCC operation  with respect to A and its complement A̅ that recovers the GHZ state with a fidelity that approaches unity exponentially quickly in the size of the boundary of A.  Combining this with (<ref>), allows us to establish Eq. (<ref>).  



We first construct the LOCC channel  for a contiguous bipartition, before generalizing this construction for a non-contiguous subregion A.  The state ρ_β describes the ensemble of pure-states |ψ_σ⟩ drawn with probability e^β J∑_⟨ i,j⟩σ_iσ_j/_β. Given a state from this ensemble, we measure the operators Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .  The resulting pattern of domain walls – an example of which is shown in the left panel of  Fig. <ref> in two spatial dimensions – is consistent with having drawn the state  |ψ_σ⟩ or the state  ∏_j∈ AX_j|ψ_σ⟩ and thus leaves one classical bit's worth of ambiguity in determining the precise state.  Equivalently, this ambiguity corresponds to the two possible ways that the observed domain walls can connect across the bipartition, as shown in Fig. <ref>.   

The preparation of the GHZ state through LOCC thus maps onto a decoding problem in a classical (d-1)-dimensional repetition code living on the interface between A and A.  The classical bit of information is the relative alignment of any two qubits across the bipartition in the Pauli Z basis (e.g. in Fig. <ref>, the classical bit can be taken to be τ_1≡σ_1σ_1).  The measured “syndromes" of the repetition code (e.g. τ_1τ_2 = (σ_1σ_2)(σ_1σ_2)) and the Boltzmann weights for the d-dimensional Ising model are used to perform maximum-likelihood decoding to guess the correct state.  Specifically, let w_1 and w_2 denote the Boltzmann weights corresponding to states (i) |ψ_σ⟩ and (ii) ∏_j∈ AX_j|ψ_σ⟩, respectively. We will guess that the state we have is (i) with probability p ≡ w_1/(w_1+w_2) and (ii) with probability 1-p.  We will subsequently apply the appropriate single-site unitary gates to turn this into the GHZ state.
The LOCC channel is thus 


  * Measure Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .

  * Let |σ⟩ be a product state in the Pauli Z basis which is consistent with the measurement outcomes. Apply the unitary U_1 = ∏_jX_j^(1-σ_j)/2 with probability p, or the unitary U_2 = ∏_j∈ AX_j∏_jX_j^(1-σ_j)/2 with probability 1-p. 


At any finite temperature, the relative probability p/(1-p) = w_1/w_2 scales exponentially in the size of the boundary of A, due to the fact that the energy of an Ising domain wall is extensive in its length.  Thus, the decoder for the repetition code should be successful at any finite temperature, and fail exactly at infinite temperature when w_1 = w_2 = 1/2.  This is consistent with the known behavior of the repetition code with single-qubit bit-flip errors with probability q, for which perfect decoding is possible for any q≤ 1/2. 

We may generalize this channel for the case where A is a non-contiguous region.  In this case, step 1 (above) is replaced by measurements of Z_iZ_j across all pairs of sites entirely within A and entirely within A̅.  As before, only one classical bit's worth of ambiguity thus remains after these measurements, in determining the precise state.  The quantum circuit that takes this state onto the GHZ state is then determined by a decoder for a repetition code with |∂ A| sites.  Thus, decoding the correct quantum circuit should again be successful at any β > 0, as |∂ A| becomes large, regardless of the geometry of the bipartition.  

Consistent with these expectations, we show <cit.> that for any bipartition, the fidelity of the recovered state R(ρ_β) with the GHZ state is exactly

    ℱ((ρ_β),|ψ_+⟩⟨ψ_+|) = 1/2[1 - ⟨tanh(β H_∂)⟩]

where H_∂ denotes terms in the Ising Hamiltonian acting exclusively along bonds that cross the bipartition between A and A and the expectation value is taken in the thermal state for the classical Ising model.  At any finite temperature, the energy H_∂ is negative, and extensively large in the boundary of A, so that ⟨tanh(β H_∂)⟩→ 1 as A becomes large.

Numerical Results: 
To numerically confirm this result, we obtain an alternate expression for the entanglement negativity of any bi-partition <cit.> 

    𝒩_A[ρ_β]=1/4|1-e^2β H_∂[σ]|.

One may now use classical Monte Carlo simulations to compute the thermal average in Eq.(<ref>).
However, care must be taken with sampling the spin configurations, since the expression inside the absolute value could be exponentially large for configurations with an exponentially small Boltzmann weight, in such a way that the overall contribution to the expectation value is O(1). 




The aforementioned issue can be avoided by using a technique known as importance sampling. Let q[σ] denote the probability distribution defined as q[σ_A,σ_]=1/2p[σ_A,σ_]+1/2p[σ_A,_], where p[σ]=e^β J∑_⟨ i,j⟩σ_iσ_j/𝒵_β. To sample from q[σ], one can simply sample from p[σ] and then with 1/2 probability, flip the spins in . We may thus write Eq.(<ref>) as,

    𝒩_A[ρ_β]
           =1/4∑_σ q[σ]p[σ]/q[σ]|1-e^2β H_∂[σ]|
       =1/2|tanh(β H_∂[σ])|_q.

where the final expectation value is taken with respect to the distribution q. The advantage of Eq.(<ref>) compared to Eq.(<ref>) is that the expression inside the expectation value is now bounded by unity everywhere, so that contributions from rare configurations are no longer a concern. 

In the following we present the numerical results for a 2D system of qubits placed on a L× L square lattice with periodic boundary conditions. 
Fig. <ref> shows the negativity 𝒩_A(ρ_β) when A is taken to be the L×L/2 cylinder. As is clear from Fig. <ref>, in the thermodynamic limit, the negativity is equal to 1/2 even above the Ising critical temperature (marked by the dashed line in Fig.<ref>), as expected from the expression in (<ref>). The fact that the negativity of subregions with extensive boundary is 1/2 irrespective of temperature, means that if one defines the topological negativity as the constant contribution to the negativity as in Ref. <cit.>, one would find that the resulting mixed state has 𝒩_topo=-1/2 even above the critical temperature, despite the fact that for T>T_c – as shown above – ρ_β is an SRE mixed-state. <cit.> (cf. Ref. <cit.>). 

Although the thermal phase transition at T_c has no effect in the negativity of subregions with extensive boundaries, the negativity of finite subregions (while continuous) becomes singular at T=T_c. Fig. <ref> shows the derivative of the negativity of a single site with respect to temperature which shows a singularity at T=T_c, which is consistent with the singularity of the heat capacity at the d=2 classical Ising phase transition, as to be expected from Eq. (<ref>). 




Discussion and Outlook: Using LOCC to recover states with a known pattern of LRE may provide another perspective that can be used to investigate mixed-state LRE under decoherence. For correlated decoherence, this is apparent in specific examples.  The two-dimensional toric code dephased by local operators Z_j+δX_j at each site j, with the vector δ given in Fig. <ref>, was studied in Ref. <cit.>.  Quantum state trajectories under this channel only contain patterns of ℤ_2 charge and flux which are separated by δ.  Therefore, for any dephasing strength, it is possible to use LOCC across the bipartition in Fig. <ref> to deterministically recover the toric code ground-state. The ℤ_2 charge and flux lying on the bipartitioning surface, which cannot be directly measured in the LOCC protocol, can be perfectly inferred from stabilizer measurements in A and A due to the noise correlations.  The negativity for this bipartition and for any dephasing strength <cit.> is thus exactly that of the toric-code ground-state <cit.>. Another example for which the LOCC perspective presented here could be insightful is the study of entanglement structure in the toric code with single-site Y decoherence <cit.>. In this case, while the noise model itself is not correlated, the LOCC protocol can benefit from the correlations that are present in the syndromes due to the extensively many strong symmetries of this channel <cit.>. The detailed study of this model and other correlated noise models is left to a future work. 

We identify other directions implied by our results.  First, it is interesting to consider the fate of mixed-state entanglement under thermalization by genuinely quantum channels which are strongly-symmetric.
More concretely, we may consider thermalizing the GHZ state via a strongly symmetric implementation of the quantum Metropolis-Hastings dynamics<cit.> for the d-dimensional transverse field quantum Ising model. The resulting mixed state would be the projection of the quantum Gibbs state onto the symmetric sector. A similar LOCC protocol to recover the GHZ state can be used to relate the entanglement negativity to the performance of a (d-1)-dimensional repetition code under a coherent correlated noise model. Energetic considerations can be used to argue about the success of the LOCC protocol in turning the resulting mixed state into the GHZ state. This lower bounds the negativity of the resulting mixed-state and may suggest that it 
would have a non-trivial negativity at high temperatures, though this remains to be confirmed. 

Second, while the persistence of topological negativity at high temperatures in our work can be understood as a consequence of the repetition code possessing maximum error threshold, it is interesting to search for settings where a similar line of argument would lead to a different code with a finite threshold. Such a system, would exhibit a finite temperature phase transition in negativity, which may not necessarily coincide with the standard order-disorder thermal phase transition. Furthermore, while we focused on the Metropolis-Hastings channel for the classical Ising model, a possibly fruitful future direction to pursue is to consider other classical systems with different symmetries and study the fate of entanglement in an initially symmetric state (e.g. continues symmetries, higher-form symmetries or non-Abelian symmetries <cit.>). 


Lastly, given that the topological negativity is unable to detect the phase transition in the complexity of the mixed-state ρ_β (from LRE to SRE), it is interesting to look for other quantities that would be able to capture the complexity of the quantum phase transition in our system as well as in more general settings. 

Acknowledgments:
We thank Yimu Bao, Matthew Fisher, Tarun Grover, Tibor Rakovszky, Subhayan Sahu, Shengqi Sang and Yaodong Li for useful discussions. This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 2308708) at UC Santa Barbara. SV is partly supported as an Alfred P. Sloan Research Fellow.








    Supplemental Material





§ ENTANGLEMENT OF Ρ_Β

Here, we show that the state (<ref>) is short-range-entangled (SRE) throughout the disordered phase at sufficiently high temperatures, and in any spatial dimension.  First, observe that this state may be written as

    ρ_β = ∑_μp_μ|ϕ_μ⟩⟨ϕ_μ|

where |ϕ_μ⟩≡ρ_β^1/2|μ⟩/ √(p_μ) and p_μ≡⟨μ|ρ_β|μ⟩, while |μ⟩ is a product state in the Pauli X basis.  Because of the strong symmetry of this state, (ρ_βU) = 1, where U ≡∏_jX_j.  As a result, each pure state appearing in (<ref>) must satisfy ∏_jX_j|ϕ_μ⟩ = |ϕ_μ⟩.  

Furthermore, observe that

    ρ_β^1/2 = 1/√(_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|

It is evident from this expression that [Z_iZ_j,ρ_β^1/2]=0 for any i, j.  As a result, any state |ϕ_μ⟩ appearing in (<ref>) – for which U|ϕ_μ⟩ = |ϕ_μ⟩ – can be obtained by applying single-site unitary gates on the reference state |ϕ_+⟩ = ρ_β^1/2|+⋯ +⟩/√(p_+), where |+⋯ +⟩ is a product state in which X_j=+1 for each qubit:

    |ϕ_μ⟩ = ∏_jZ_j^1-μ_j/2|ϕ_+⟩

Finally, we note that p_μ = ⟨μ|ρ_β|μ⟩ = ⟨+⋯ +|∏_jZ_j^1-μ_j/2ρ_β∏_jZ_j^1-μ_j/2|+⋯ +⟩ = ⟨+⋯ +|ρ_β|+⋯ +⟩ = p_+.  As a result, p_μ = 1/2^N-1 where N is the total number of qubits.  To summarize, we may write (<ref>) as

    ρ_β   = 1/2^N-1∑_μ  s.t.  ∏_jμ_j= +1∏_jZ_j^1-μ_j/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-μ_j/2

We now re-cast the summation by introducing Ising degrees of freedom σ_ij along bonds of the hypercubic lattice, and letting μ_j≡ A_j[σ] ≡∏_i∈ NN(j)σ_ij, where the product is taken over sites that are nearest-neighbor to j.  With periodic boundary conditions, ∏_jA_j[σ] = +1, and as a result, the constraint on the summation ∏_jμ_j = +1 is naturally satisfied by this replacement.  With this replacement, we may write

    ρ_β   = 1/2^N_b∑_σ∏_jZ_j^1-A_j[σ]/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-A_j[σ]/2

Here, N_b = d· N is the total number of bonds on the d-dimensional hypercubic lattice.  The pre-factor ensures the normalization of the density matrix[Another way to derive this pre-factor is to observe that for a given configuration of μ_j satisfying the constraint ∏_jμ_j = 1, there are 2^N_b/2^N-1 different configurations of σ_ij such that μ_j = A_j[σ]. ]. Finally, using the fact that Z_j^1-A_j[σ]/2 = ∏_i∈NN(j)Z_j^1-σ_ij/2, we may write ∏_jZ_j^1-A_j[σ]/2 = ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2, where the product in the second expression is taken over distinct, nearest-neighbor bonds.  As a result,

    ρ_β   = 1/2^N_b∑_σ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2|ϕ_+⟩⟨ϕ_+|∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2

We conclude that ρ_β may be obtained by dephasing Z_iZ_j on each bond of the lattice, with maximal strength, a manifestly finite-depth, local quantum channel.

Finally, we may argue that the state |ϕ_+⟩ is SRE when the system is in a disordered phase.  We do this by first writing

    |ϕ_+⟩ = √(2^N-1)ρ_β^1/2|+⋯ +⟩ = √(2^N-1/_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|+⋯ +⟩ = 1/√(_β)∑_σe^-β H/2|σ⟩

We now observe that (<ref>) is the ground-state of the Hamiltonian

    H = -∑_jQ_j

where

    Q_j≡ -X_j + ∏_i∈NN(j)e^-β J Z_iZ_j


 


§ LOCC OPERATIONS AND RECOVERING THE GHZ STATE


Here, we will show the existence of an LOCC protocol   with respect to bi-partition A and , which transforms state ρ_β back to the GHZ state in the thermodynamic limit. 
The LOCC protocol consists of two steps:


    
  * Measure Z_iZ_j on each link that is entirely in A or in  subregions, finding the outcome μ_ij=± 1. This does not include the links across the boundary, ∂ A.

    
  * For any given set of measurement outcomes μ={μ_i,j}, there exist two possible states which have domain wall configuration consistent with μ. 
     We pick one of the two states based on a coin toss weighted with their respective Boltzmann weight in ρ_β (Eq.(<ref>)) and
      apply a product of Pauli X_i operators to eliminate the domain walls in that assumed configuration.

In the following, we will derive Eq.(<ref>) for the fidelity between the GHZ state, |ψ_+⟩, and the output state (ρ_β).


 §.§ Fidelity


Let {M_μ} be the projective measurements that measure whether there is a domain wall or not on the links which are entirely
in region A or A̅,


    M_μ≡∏_<i,j>∈ A, A̅1+μ_i,jZ_iZ_j/2,

where μ_ij is the measurement outcome of measuring Z_iZ_j and the product only includes next nearest sites which are both in A or both in .
Note that,

    M_μ|ψ_σ⟩ = ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_j|ψ_σ⟩.
 

A set of measurement outcomes μ specifies a certain domain wall configuration in A and , with some domain walls ending on the boundary ∂ A. Let 1∈ A and 1̅∈ be two adjacent sites across the boundary and define τ=± 1 to be a marker for whether or not there is a domain wall across the edge connecting 1 and 1̅, i.e. τ≡σ_1 σ_1̅ (see Fig. <ref>). Note that μ and τ together uniquely specify the global domain wall configuration, because there should be even number of domain walls across each plaquette of the square lattice. In particular if we label the sites along the boundary as shown in Fig.<ref>, we have,

    σ_i σ_ i̅=τ∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Let X_μ,τ be the appropriate product of single qubit Pauli-Xs that corresponds to the domain wall configuration specified by μ and τ. In particular, if M_μ|ψ_σ⟩=|ψ_σ⟩, we have

    X_μ,τ|ψ_σ⟩≡|ψ_+⟩   if σ_1σ_1̅=τ
    ∏_j∈ AX_j|ψ_+⟩   if σ_1σ_1̅≠τ.

Given the measurement outcomes μ, the decoder  applies X_μ,τ with τ=± 1 chosen randomly with probability P_μ,τ. The probability P_μ,τ is chosen to be proportional to the Boltzmann weight of the spin configuration specified by μ and τ and according to Eq.(<ref>) can be written as,

    P_μ,τ = e^β J τ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1/∑_τ'=± 1 e^β J τ' ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Putting everything together, we can compute the average “decoding" fidelity,


    F = ψ_+(ρ_β)ψ_+= ∑_σ,μ,τe^-β H[σ]/𝒵_βP_μ,τ⟨ψ_+|X_μ,τM_μ|ψ_σ⟩^2.


Using Eqs.(<ref>) and (<ref>) this simplifies to


    F = ∑_σ,μ,τe^-β H[σ]/𝒵_β P_μ,τ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_jδ_σ_1σ_ 1̅,τ.

Plugging the expression for P_μ,τ into Eq. (<ref>), we get


    F =   ∑_σ,τe^-β H[σ]/𝒵_βe^β Jτ∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅/∑_τ'=±1 e^β Jτ'∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅δ_σ_1σ_ 1̅,τ
    
         =   ∑_σe^-β H[σ]/𝒵_βe^β J∑_i≥1σ_iσ_ i̅/∑_τ'=±1 e^βτ'σ_1 σ_ 1̅ J∑_i≥1σ_i σ_ i̅
    
         =   1/2∑_σe^-β H[σ]/𝒵_β(1-tanh(β J∑_i≥1σ_i σ_ i̅))
    
         =   1/2[1-⟨tanh(β H_∂)⟩]

where H_∂ is the boundary Hamiltonian between A and A̅. 



 §.§ Mapping to the d-1 repetition code


The problem of choosing which domain wall configuration to assume can be understood as a decoding problem for a (d-1)-dimensional classical repetition code. To see this, note that the decoding task in a classical error correcting code is basically figuring out what error has happened based on a given set of syndromes. In particular the decoding task in the classical repetition code on N bits can be described as follows. Consider an error vector e∈{1,-1}^N such that e_i=-1 if a bit flip has happened on the i'th bit and e_i=1 otherwise. 
The goal of the decoder is to figure out e, knowing only the syndrome values, i.e. e_i e_j for adjacent i and j on the appropriate graph, e.g. the d-dimensional square lattice in the standard d-dimensional repetition code. Moreover, the decoder assumes an underlying noise model, i.e. a random distribution from which the vector e is drawn. For example, in the independent and identically distributed (iid) noise model, it assumes each component e_i is -1 with probability p and 1 with probability 1-p, independent of other components. 

In our setting finding the domain wall configuration is clearly the same task, with e_i defined to be e_i≡σ_i σ_ i̅. Note that we can not know the value of e_i, but we know the value of e_i e_j=μ_i,j μ_i̅,j̅ for adjacent i and j. At low temperatures, the domain walls are rare so most e_is would be 1, and as one increases the temperature more e_i would be -1. The underlying noise model is given by P_μ,τ as described in the previous section. This noise model is not iid, but rather a correlated noise model. However, as we argue in the next section, the repetition code is able to guess e correctly with probability 1 in the thermodynamic limit unless β=0, at which point each e_i is ± 1 compeletly at random. 



§ THRESHOLD FOR THE REPETITION CODE

Here we argue that the repetition code decoding under the noise distribution P_μ,τ is successful with probability 1 for any β>0 in the thermodynamic limit. First we review the argument for maximum error correction threshold in the presence of the iid noise and then we argue that the same should remain true for more generic noise models. We use quantum notation for convenience although the problem we are considering is inherently classical. 

Consider an initial state |Ψ⟩ undergoing some errors, leaving it in the mixed state

    ρ = ∑_e p_e e|Ψ⟩⟨Ψ|e

where e is a product of X bit flips and p_e is the probability of having that error. We rewrite the sum to distinguish between the errors that leave ρ with fewer than N/2 bit flips and errors with higher weights:

    ρ = ∑_e: |e|<N/2p_e e|Ψ⟩⟨Ψ|e + ∑_e:|e|≥ N/2p_e e|Ψ⟩⟨Ψ|e.

The error correction channel, , measures the syndromes, and between two possible error configuration consistent with a syndrome, applies the one with the smaller weight. Therefore it always successfully corrects the errors when there are less than N/2 bit flips, hence,

    (ρ)=   ∑_e: |e|<N/2 p_e (e|Ψ⟩⟨Ψ|e) + ∑_e:|e|≥ N/2p_e  (e|Ψ⟩⟨Ψ|e)
    
        =   ∑_e: |e|<N/2 p_e|Ψ⟩⟨Ψ| + ∑_e:|e|≥ N/2p_e (e|Ψ⟩⟨Ψ|e).


The fidelity of this post-error correction state with the original state is lower bounded as,


    F =   ⟨Ψ|(ρ)|Ψ⟩
    
        =   ∑_e: |e|<N/2 p_e +  ∑_e:|e|≥ N/2p_e ⟨Ψ|(e|Ψ⟩⟨Ψ|e)|Ψ⟩≥∑_e: |e|<N/2 p_e


where the lower bound is because the second term in the expression above is non-negative. Now we will show that this lower bound is equal to one in the thermodynamic limit. First, we rearrange the sum to be over |e|=k, the number of bit flips error, and because  p_k is a binomial distribution,
p_k = Nkp^k(1-p)^N-p.

In the limit of large N, we can approximate the binomial distribution as a Gaussian distribution with mean Np and standard deviation √(Np(1-p)) 
via the central limit theorem. Since N/2 is N/2-Np/√(Np(1-p))=O( √(N)) standard deviations away from the peak of the Gaussian, we see that 

lim_N→∞∑_k=0^N/2-1 p_k = 1

when p<1/2. Since fidelity is upper-bounded by one, the fidelity is indeed equal to one in the thermodynamic limit. Thus, the repetition code will work successfully when p<1/2. 

For the more general case where e is drawn based on the P_μ,τ distribution, the bound in Eq.(<ref>) is still valid. Therefore, to show perfect recovery we only need to show that with probability one, less than half of the edges contain a domain wall. Intuitively, this is true because for β>0, not having a domain wall on an edge is energetically favorable. To make it rigorous, one can define indicator variables I_i=(1-e_i)/2 in terms of which the number of domain walls can be written as k=∑_iI_i. When β >0 it is more probable for I_i to be 0 rather than 1 and hence the expectation value of k/|∂ A| is less than 1/2. Noting that lim_|i-j|→∞Cov(I_i,I_j)=0, the weak law of large numbers for weakly dependent variables<cit.> shows that k/|∂ A| concentrates around its mean, which is less than 1/2 when β>0 in the thermodynamic limit. 



§ NEGATIVITY SPECTRUM

Here we calculate the entanglement negativity between a sub-region A and its complement A̅, in the state ρ_β. 
Let  denote the spin configuration that is related to σ by flipping every spin. Accordingly, the state |ψ_σ⟩ can be written as |ψ_σ⟩=|σ⟩+|⟩/√(√(2)) and Eq. (<ref>) can be rewritten as,


    ρ_β = 1/2𝒵_β∑_σ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅| +|σ⟩⟨σ̅| +|σ̅⟩⟨σ|),

where H[σ]=-J ∑_i,jσ_i σ_j is the classical Ising energy of the spin configuration σ. Taking the partial transpose of ρ_β with respect to sub-region A yields,

    ρ_β^T_A   = 1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,σ_A](|σ_A,σ_A⟩⟨σ_A, σ̅_A| +|σ_A, σ̅_A⟩⟨σ_A,σ_A|)]
       =1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,_A](|σ⟩⟨σ| +|σ⟩⟨σ|)]


where σ_A and σ_A̅ denote the spin configuration σ restericted to regions A and A̅ respectively. Thus, the eigenvalues and eigenstates of ρ_A^T_A are

    λ_σ;±=e^-β H[σ]± e^-β H[σ_A,σ̅_A]/_β, |ψ_σ^(±)⟩≡|σ⟩±|σ⟩/√(2).



We can now calculate the entanglement negativity  from the sum of all the negative eigenvalues. Firstly,
it is only the λ_σ;- eigenvalues which can be negative. 
We can also see that λ_σ;-=λ_σ̅;- from the ℤ_2 symmetry of the Hamiltonian and they correspond to the same eigenvector, up to some phase. 
Moreover, 
note that λ_σ;-=-λ_σ_A,σ̅_A̅;- so for a given spin configuration, either λ_σ;- or λ_σ_A,σ̅_A̅;- would be negative. Thus, we account for the two incidences of double counting with an additional 1/4 and get


    𝒩_A[ρ_β] = 1/4_β∑_σe^-β H[σ] - e^-β H[σ_A,σ_A].


Factoring out e^-β H[σ] we get


    𝒩_A[ρ_β]    = 1/4_β∑_σ e^-β H[σ]1 - e^-β( H[σ_A,σ_A]- H[σ])
       =1/4⟨1 - e^2β H_∂[σ])⟩.






 Paper Title: Relaxation Fluctuations of Correlation Functions: Spin and Random Matrix Models.






pathak.tanay@yukawa.kyoto-u.ac.jp
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics,
 Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan


§ ABSTRACT 
Spectral statistics and correlations are the usual way to study the presence or absence of quantum chaos in quantum systems. We present our investigation on the study of the fluctuation average and variance of certain correlation functions as a diagnostic measure of quantum chaos and to possibly characterize quantum systems based on it. These quantities are related to eigenvector distribution and eigenvector correlation. Using the Random Matrix Theory certain analytical expressions of these quantities, for the Gaussian orthogonal ensemble case, were calculated before. So as a first step we study these quantities for the Gaussian unitary ensemble case numerically, and deduce certain analytical results for the same. We then carry out our investigations in physical system, such as the mixed-field Ising model. For this model, we find that although the eigenvalue statistics follow the behaviour of corresponding random matrices, the fluctuation average and variance of these correlation functions deviate from the expected random matrix theory behaviour. We then turn our focus on the Rosenzweig-Porter model of the Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble types. By using the fluctuation average and variance of these correlations, we identify the three distinct phases of these models: the ergodic, the fractal, and the localized phases. We provide an alternative way to study and distinguish the three phases and firmly establish the use of these correlation fluctuations as an alternative way to characterize quantum chaos.

Relaxation Fluctuations of Correlation Functions:

Spin and Random Matrix Models
    Tanay Pathak  https://orcid.org/0000-0003-0419-2583

    < g r a p h i c s >

    August 5, 2024
=================================================================================

            Report Number: YITP-24-96







§ INTRODUCTION

The dynamics of classical systems have been classified into a well-known hierarchy, the ergodic hierarchy <cit.> which ranges from integrable systems to Bernoulli systems. However, the situation is not quite clear in quantum systems <cit.> and there is no analogous neat categorization and it is not easy to extend the results of the classical phase space to Hilbert space with ease. The signature of quantum chaos is often attributed to the observation that there is a level repulsion amongst the eigenvalues in quantum Hamiltonians whose classical counterpart is chaotic which is called the Bohigas-Giannoni-Schmit (BGS) conjecture <cit.>. This behaviour of eigenvalues and their correlations are well described by the random matrices <cit.>. On the other hand, we have Berry-Tabor conjecture <cit.>, for the quantum system whose classical counterpart is integrable and the eigenvalues in quantum systems do not have any correlation, show level clustering and in the semi-classical limit behaves statistically similar to a random process. Thus, there is an intricate relationship between random matrices as described using Random Matrix Theory (RMT) and quantum chaos <cit.>. Over the years, these properties of quantum chaotic systems have been studied in great detail using various tools, not necessarily related to RMT as well. Some of these include (a non-exhaustive list) the eigenstate thermalization hypothesis (ETH) <cit.>, eigenvalue correlation measures such as the spectral form factor <cit.> and ⟨r|$⟩-value <cit.>, entanglement measures <cit.> and measures utilizing operator growth such as out of time-ordered correlator (OTOC) <cit.> and recently introduced universal operator growth hypothesis and associated Krylov complexity <cit.>, see <cit.> for a detailed review on recent works.

With the motivation to study quantum chaotic systems and possibly to classify them, similar to their classical counterpart, we utilize the ideas put forth in <cit.> regarding the relaxation fluctuations of the correlation function and its intrinsic connection to chaos. The study was motivated by and studied using models having well-defined classical limits which are known to be chaotic such as in quantum maps <cit.>. By studying the quantum correlation functions which can be related to classical correlations and also to the intricate issues of ergodicity and mixing, an attempt was made to classify quantum dynamics similar to their classical counterpart. Of central interest in this study is the correlation function constructed in such a way that they have a well-defined classical limit. In this paper, we will study these quantities with their original formulation and without necessarily considering the classical limits to illuminate properties of certain quantum spin systems, such as the mixed-field Ising model and other models such as the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE) Rosenzweig Porter (RP) models. The mixed field Ising model is a simple model of1/2-spins in a 1D chain with a nearest neighbour interaction and two transverse external fields. The model is known to exhibit both integrable and quantum chaotic limits <cit.>, depending on the strength of the transverse field. The eigenvalues of the model follow the RMT behaviour, yet the model does not have the expected chaotic dynamics, and exhibit weak chaos<cit.>. The RP model is a simple one-parameter model of random matrices that has a rich structure and is well studied in literature <cit.> for its properties. This simple one-parameter model remarkably has a fractal phase (an extended phase) apart from the usually found ergodic (chaotic) and localized (integrable) phases, depending on the value of its parameters. Our findings suggest that even if the eigenvalue statistics follow RMT behaviour it is still possible that the eigenvector correlations do not follow the same, as we will see in the case of the mixed-field Ising model. Such conclusions have also been made for another simple model of quantum chaos, Baker's map <cit.>. Furthermore, for models like the RP model, we show that the average and variance of relaxation fluctuations can used to detect the transition quite well. Our work here also try to advocate the use of eigenvector distribution and their correlations along with usual eigenvalue statistics to quantify quantum chaos.

The outline of the paper is as follows. We start with describing the setup used in Section <ref> and then discuss the results obtained previously for the GOE random matrices. We then briefly use the method to revisit Baker's map in section <ref> and study some of it in some detail. In Section <ref> we then study the mixed field Ising model within the same framework and describe the similarities of its results with the Baker's map. We then study both the GOE and the GUE-RP model in Section (<ref>) and finally end the paper with the conclusion and outlook in Section <ref>.





§ SETUP AND RANDOM MATRIX THEORY

In this section, we first set up the framework. We will mostly review the procedure already outlined in <cit.>. 

We will be interested in the study of Hilbert spaceℋof dimensionD. Let there be two distinct operators on ,P_AandP_B. For our present purpose, we will focus on the case when these operators are projection operatorsP_A + P_B = 𝐈_D,𝐈_Dis theDdimensional identity matrix. The reason for such choice originally was the classical limits of these operators, where classically these correspond to choosing the partitionAandBof relevant phase space𝒫such thatA ∪B= 𝒫 . Consider a unitary evolution operatorUwhich acts on the state in Hilbert space  of dimensionD. The quantity of interest considered in <cit.> is the correlation function 

    C_AB(t)=  (U^t P_A U^-t P_B)/D,

which physically captures the overlap of subspaceAevolved for timetwith the subspaceB. For the case whent →∞,C_AB →(P_A)(P_B)/D^2 = f_Af_B wheref_Aandf_Bdenotes what fraction of total space , sub-spacesAandBare respectively. Using this, we normalize Eq. (<ref>), and hence study the following quantity

    c(t) = C_AB(t)/f_Af_B.

Furthermore, for simplicity, we will choose projectors to be diagonal in the discrete position basis where 

    P_A = ∑_n^f_AD|n⟩⟨n|,

andP_B = 𝐈_D - P_ASinceU(or for that matterH) has a discrete and finite spectrum, the correlation given by Eq. (<ref>) is a sum of purely oscillatory terms, hence giving rise to fluctuation in correlation function even after equilibrium has set in. We will mainly be interested in the study of the average and the variance of these fluctuations denoted byandσ_c^2respectively. We can representc(t)in terms of eigenfunctions of the evolution operatorU, denoted byψ_kwith the following eigenvalue equation 


    U |ψ_k⟩ = e^i E_k|ψ_k⟩   k = 1, ⋯ D,

whereE_kis the eigenenergy. The correlation function can then be written as 

    c(t)   = 1/D f_A(1-f_B)∑_k_1,k_2 ∑_n_1=1 ^D f_A∑_n_2= D f_A+1^D e^i t (E_k_2-E_k_1
       ×⟨n_2|ψ_k_1|⟨%s|%s⟩⟩ψ_k_1|n_2⟨n_1|ψ_k_2|⟨%s|%s⟩⟩ψ_k_2|n_2,

where we have used the fact thatf_A +f_B = 1.
The time average of the above quantity can be simply taken by substitutingk_1 = k_2. Denotingf_Aasfwe get following after simplification

    = 1/D f(1-f)∑_k =0^D ∑_n_1=1 ^D f∑_n_2= D f + 1^D 
        |⟨n_1|ψ_k|⟨%s|%s⟩⟩n_2|ψ_k|^2.

Thus,is a measure of the distribution of eigenvectors in the two sub-spacesAandB. Another feature of the average of correlations is that they capture the amount of spread of eigenstates in the subspaceAandB. If they are equally distributed then the average tends towards the classical time average and is equal to 1. This progression of results towards classically results means the result in the largeNlimit (noticeħ_effis defined to be equal to1/Nfor such systems and largeNimpliesħ_eff →0). To capture this feature it is important to look at the scaling ofwithN.  In <cit.>
it was suggested via various different models that< 1, which implies the reluctance to participate equally in both partitions. However, there is still a possibility of≥1if there are symmetries present in the systems for e.g. the parity symmetry which forces the wavefunctions to be essentially identical in the two subspacesAandB. Although in the absence of any symmetries. 
Similarly the variance ofc(t)can be written as follows after simplification

    σ_c^2 = 2/D^2f^2(1-f)^2∑_k_2>k_1∑_n=1^f D-1|⟨n|ψ_k_1|⟨%s|%s⟩⟩ψ_k_2|n|^4,

which implies that the variance is a measure of correlations between distinct pairs of eigenvectors. 

Using the properties, that the real eigenvectors of GOE matrices are uniformly distributed over aD-dimensional sphere and none of the eigenvectors is preferred, it was analytically derived in <cit.> that the average and variance of the fluctuations are given by 

    ⟨c_avg|_⟩GOE   = N/N+2∼ 1- 2/N, 
    ⟨σ_c^2|_⟩GOE   = 3/N^2,

where⟨·|$⟩ denotes the ensemble average of the quantities.
It is important to note that the above results are independent of the fraction f. We have plotted the above results in Fig. (<ref>) along with the numerical results obtained using the GOE matrices. We have also plotted the results for the GUE matrices. For numerical purposes for both GOE and GUE type random matrices, we have taken f= 1/2 and system sizes N= 128 (2048), 256 (1024), 512 (512), 1024(128), 2048(64) , 4096(32), where the numbers in the bracket denote the number of samples. Using the detailed numerical analysis, as outlined in Appendix <ref>, we deduce the following results for GUE matrices

    ⟨|_⟩GUE   = N/N+1∼ 1- 1/N,
    ⟨σ_c^2|_⟩GUE   = 2/N^2 + 𝒪(1/N^2).

These are also plotted along with the GUE results in Fig. (<ref>) ( the dashed black lines). These behavior of ⟨|$⟩ and⟨σ_c^2|$⟩ are universal, in the sense that they are independent of the partition f used. 
 


§ BAKER'S MAP: A REVISIT

As a brief overview of the setup described in the previous section, we consider a simple quantum model exhibiting chaos, the quantum Baker's map <cit.>. Classically the map is defined via the following recursion corresponding to the phase space points (q_n,p_n)

    (q_n+1, p_n+2)= 
        (2 q_n, p_n/2)              0< q_n < 1/2,
    (2q_n-1, (p_n+1)/2)     1/2 < q_n < 1.

The quantized version of the map corresponds to the following unitary operator U, in position representation, with Hilbert space dimension N

    B = G_N^-1([  G_N / 2        0;        0 G_N / 2, ])

where G_N denotes the discrete Fourier transform and is given as 
(G_N)_l,m = 1/√(N)exp(-2 π i (l+1/2)(m+1/2)).
The classical map can be shown to be chaotic and mixing with the Lyapunov exponent equal to ln(2), which is also equal to the Lyapunov exponent obtained using OTOC in the quantized Baker's map too <cit.>. We remark that we can form various kinds of generalized Baker's map similar to Eq. (<ref>), by taking into account the different partition schemes in the classical version, e.g. Eq. (<ref>) is isomorphic to  (1/2,1/2) Bernoulli process <cit.>. As another example, we can consider a classical map with Bernoulli scheme (2/3,1/3) <cit.> given as follows

    (q_n+1, p_n+2)= 
        ( 3q_n/2, 2 p_n/3)              0< q_n≤2/3,
    (3q_n-2, (p_n+2)/3)   2/3 < q_n < 1.

The corresponding unitary operator is then given by 

    B = G_N^-1([ G_2 N / 3         0;         0  G_N / 3. ])

The (1/2,1/2) scheme also has a parity symmetry with the parity operator given by P = -G_N^2. The eigenvalues of the unitary operator U are given by e^iθ, where θ is the eigenangle, θ∈ [0,2 π] which are uniformly distributed on a circle. The spacing distribution of eigenangles for the chaotic operator follows the GOE distribution. In Fig. (<ref>) we show the eigenangle spacing distribution of Baker's map with Bernoulli scheme (1/2,1/2) after symmetry resolution and unfolding. We observe quite a good agreement with the GOE spacing distribution, which is given by the red solid line in Fig. (<ref>). We now study the c_avg and σ_c^2 of the fluctuations for various schemes of Baker's map. 











The Fig (Bottom). (<ref>) suggest the following trend of  and σ_c^2 with system size N

    = 1- α/N^β, 
    σ_c^2   = α_σ/N^β_σ.

For various schemes, we obtain the following values of the best-fit parameter for =

    (2/3,1/3)    : α = 1.07 , β = .78 , 
    
        (1/4,3/4)    : α = .37 , β = .98 , 
    
        (1/3,2/3)    : α = .94 , β = 1.02.


Similarly, for various schemes, we obtain the following values of the best-fit parameter for σ_c^2

    (2/3,1/3)    : α_c = 2.84 , β_c = 1.97, 
    
        (1/2,1/2)    : α_c = 4.94, β_c = 1.97, 
    
        (1/4,3/4)    : α_c = 2.39 , β_c = 1.95, 
    
        (1/3,2/3)    : α_c = 3.24 , β_c = 1.97.


We observe that for σ_c^2 for the Baker's map corresponding to the (1/2,1/2) scheme, the slope is the same as other schemes, β_σ≈ -2, yet it has an offset from other schemes and stands markedly different than other curves which nearly overlap on top of each other. Similar results reporting deviation of (1/2,1/2) scheme from (2/3,1/3) schemes were also reported in <cit.> and a possible reason suggested for such deviation is the presence of a large number of scarred states in Baker's map for this particular scheme <cit.>. We merely report these results of Baker's map as an exercise to understand the machinery of the method described. However, it is important to further understand the origin of these results in greater detail.
 


§ MIXED-FIELD ISING MODEL

We now consider a simple spin chain model that exhibits quantum chaos depending on the values of its parameters, the mixed field Ising model. The model is given as follows 

    H = - ∑_i=1^N-1σ_i^zσ_i+1^z - ∑_i=1^N-1 ( h_xσ_i^x+ h_zσ_i^z),

where σ_i^x and σ_i^y are the Pauli matrices at site i, defined as 

    σ_k^i= 𝐈_2^⊗(i-1)⊗σ_k^i⊗𝐈_2^⊗(N-i)

for k ∈{x,y,z}. We consider open boundary conditions, with N sites with the Hilbert space of dimension d = 2^N.

The model is known to exhibit both chaotic and integrable regimes <cit.> depending on the strength of longitudinal and transverse fields as follows

    (h_x,h_z)=    
          (-1.05,0.5),   Chaotic
    
         (-1,0).       Integrable

Since the model has parity symmetry hence we look for these properties by focusing on symmetry resolves blocks, so as to eradicate the effect of symmetry on these quantities. Once the symmetries are resolved we then carry out our analysis.



In Fig. (<ref>) we look at the variation of c(t) with time for the two regimes. For practical purposes, and to bypass the need to further resolve the extra symmetry arising for the integrable case, unlike Eq. (<ref>), we take the parameters to be (h_x,h_z) = (-1,0.001). We observe that for the integrable regime, the fluctuations are larger than those of the chaotic regime, and there is a clear distinction between the two. 



The integrable and chaotic regimes are quite evident from the scaling of  and σ_c_+ with N. For the integrable case, the results in Fig. (<ref>), suggest the scaling behaviour of the following form 

    ()_+ = 1- α_+/ N ^β_+,

with fitting parameters: α = 0.058, γ = 0.12. The slope of the best-fit line in Fig (Top).(<ref>) is really small ∼ -0.12. In fact, we practically observe that for larger N the value saturates implying that as N increases  does not approach unity and is constant. This tendency of  to not scale with N is a sign of the integrability of the system. We also explore the variance and observe the following scaling behaviour

    σ^2_+ = κ_+/ N ^δ_+,

with best-fit parameters κ_+ = 0.46, δ_+= 1.54. Unlike the behaviour  for the variance, we see that the scaling of variance with N. It has been observed in <cit.> as well that for integrable systems σ_c^2 can scale with N but the coefficient is non-universal with a strong dependence on partition f. 




For the chaotic regime, the results for the deviation of average relaxation from unity are shown in Fig (Top). (<ref>) for both the parity sectors. We observe the following power-law-like behaviour of average fluctuation with N 

    ()_± = 1- α_±/ N ^β_±,

where ± denotes the +1 and -1 parity sectors respectively with α_±≈ 1, β_±≈7/10, which deviates from the RMT predictions as discussed in Section <ref>, where α= 2 and β =1. This implies that in some sense, the Ising model is less chaotic as compared to other models of quantum chaos which follow the RMT behaviour closely. We have already observed similar features in the case of the quantum Baker's map. These results hint towards a hierarchy amongst systems exhibiting quantum chaos and suggest that not all systems are as chaotic as others. Furthermore, it shows that eigenvalue statistics are not enough to characterize the dynamics of the quantum systems, as it is possible that although the eigenvalue statistics of these systems follow RMT behaviour closely, the correlation of eigenvectors as discussed in the present context is not in agreement with the corresponding RMT results. We would also like to remark that spin 1/2 systems are the most quantum systems we can have and thus they have peculiar features within the context of quantum chaos as has been reported before in various studies using other measures as well <cit.>. 




§ ROSENZWEIG-PORTER MODEL

To elucidate the features of integrable to chaotic transition in more detail we now study a random matrix model, the Rosenzweig-Porter (RP) model <cit.>. The model is given by 

    H = A + 1/N^γ/2 B ,

where A is a diagonal matrix with its elements chosen identically and independently from a standard normal distribution and B is a GOE matrix, whose elements are drawn from a normal distribution with zero mean (for both diagonal and off-diagonal elements) and variance ⟨B_ii|^⟩2= 1 and ⟨B_ij|^⟩2 = 1/2 for i ≠ j respectively. Depending on the value of the free parameter γ, the model has the following three phases

    γ = 
         [0,1]     Ergodic ,
    
        [1,2]      Fractal ,
    
        [2,∞)   Localized .


These phases are characterized using the fractal dimensions which are defined using the inverse participation ratio (IPR). For a particular state ψ_k, we define the inverse participation ratio  in a basis ϕ_i <cit.> as follows

    IPR = ∑_i| ⟨ϕ_i|ψ_k||⟩^4.

The scaling of IPR is given as follows

    IPR∼ N^-D,

where D is called the fractal dimensions. For the ergodic phase when the states are uniformly delocalized over all the available states D= 1, however, for the localized phase, it is D= 0 as the states are localized over a single state. The fractal phase is interesting as D lies between 0 and 1 implying the states are delocalized over the basis, but not fully. 



 Paper Title: Constraints on the $γ$-parameter for the vacuum solution of Cotton gravity with geodesics and shadows.







	 ednaldobarrosjr@gmail.com
Faculdade de Física, Universidade Federal do Pará, Campus Universitário de Tucuruí, CEP: 68464-000, Tucuruí, Pará, Brazil

     
    tarcisojunior17@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill

	 fslobo@ciencias.ulisboa.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal
Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     esialg@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill
Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil

   drubiera@ucm.es
Departamento de Física Téorica and IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain

      
        fc53497@alunos.fc.ul.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     henriquefisica2017@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill



	





§ ABSTRACT 

We consider a recently introduced extension of General Relativity dubbed as Cotton gravity (CG), based on the use of the Cotton tensor, to estimate the size of a new constant γ appearing within a spherically symmetric, vacuum solution of the theory. Taking into account its non-asymptotically flat character, we use the inferred size of the central brightness depression of the supermassive object at the heart of the Milky Way galaxy (Sgr A*) by the Event Horizon Telescope to constrain at 2σ the CG parameter as γ M ≈ 3.5 × 10^-12. We study the potential observational consequences from the smallness of such a value using exact and numerical expressions for the deflection angle, optical images from optically and geometrically thin accretion disks, isoradials, and instability scales (Lyapunov index) of nearly bound geodesics associated to photon rings. Our results point towards the impossibility to distinguish between these two geometries using current and foreseeable techniques in the field of interferometric detection of optical sources.






Constraints on the γ-parameter for the vacuum solution of Cotton gravity with geodesics and shadows

    Henrique A. Vieira
    August 5, 2024
====================================================================================================







§ INTRODUCTION



Despite the undeniable success of Einstein's General Relativity (GR) in explaining a large number of gravitational phenomena in different regimes <cit.>, there are several drawbacks of the theory. On the one hand, the accelerated expansion of the Universe, as found by Type-Ia supernovae observations  <cit.>, can only be accounted for, in the frame of GR, via a hypothetical fluid known as dark energy <cit.>, whose properties can be assimilated to that of a cosmological constant term <cit.> yet there is no direct proof of its existence. Similarly, the explanation of galactic rotation curves demands more than 90% of the total mass of most galaxies to be dominated by a non-baryonic form of matter dubbed as dark matter <cit.>, which we have been unable to detect in any terrestrial observatory yet. On more theoretical grounds, the problems with the unavoidable development of space-time singularities inside black holes and in the early cosmological evolution and the loss of predictability they bring about <cit.>, the Hawking process and the associated information loss paradox <cit.>, and the development of a framework unifying gravity and quantum mechanics (quantum gravity <cit.>), are challenges that still haunt the gravitational community.

In response to these challenges, a large avenue of research has emerged in the last few decades arranged around alternative approaches to GR, generally dubbed as modified theories of gravity (see e.g. <cit.> for some reviews). Within this context, recently Harada presented a theory called “Cotton Gravity" (CG) <cit.>. In this proposal, the effects describing gravity are attributed to the Cotton tensor, and the corresponding field equations contain third-order derivatives. The main interest in this proposal comes from the fact that any solution of the Einstein equations, with or without a cosmological constant term, also satisfies the field equations of CG. Furthermore, the cosmological constant itself emerges as an integration constant out of the field equations. In addition, in <cit.> Harada used the post-Newtonian limit of CG to solve the field equations numerically, interpreting the rotation curves of several galaxies without employing dark matter sources. Note that the possibility that dark matter may be geometric effects of modified theories of gravity has been explored in the literature <cit.>.


Regarding new solutions within CG, the first non-trivial one with static, spherical symmetry was found in <cit.>, which extends the Schwarzschild solution via a new parameter γ. Other non-trivial solutions were found in <cit.>. On the other hand, in <cit.> it was emphasized that the Codazzi tensor can be derived from the equations of CG, which implies that Harada's formulation can be understood as a modification of the GR equations due to the presence of this residual tensor. This way, with the introduction of the Codazzi tensor, it became possible to express CG through a new formulation, which allowed for the representation  of conformally flat cosmological solutions without a vacuum in CG <cit.>. 

However, there is ongoing debate about the viability of CG as a potential extension of GR. Some researchers see CG as promising for addressing GR's limitations, while others doubt its theoretical consistency <cit.>. Further research is needed to determine whether CG can replace or complement GR, or if it will remain a theoretical curiosity.


In order to set constraints in the new parameter γ of CG, one can resort to the gravitational lensing of light. This is the deflection of light trajectories as they pass close to massive bodies (stars, black holes, galaxies, etc) and can act like a natural telescope by amplifying the source brightness and even creating multiple images. 
Recent developments related to this effect can be divided into two groups:  microlensing (i.e. the measurement of the collective magnification of various images) and strong lensing <cit.>. For the sake of this work we are (mostly) interested in the strong field regime, given the fact that it offers the possibility of testing GR beyond the weak-field limit tests carried out so far. In such a limit,  Bozza <cit.> proposed a formalism to calculate the deflection angle for every spherically symmetric metric in terms of a logarithmic expansion. Such a formalism has been very influential in advancing analytical methods to compare theoretical predictions with real images <cit.>, including alternative spherically symmetric metrics <cit.>.


The black hole shadow and photon ring heuristics are two such examples of visual phenomena caused by strong gravitational lensing <cit.>. The former consists in the dark silhouette of a black hole that is projected in the celestial plane, with a critical boundary separating backwards-traced geodesics that are scattered towards infinity from those that are captured by a black hole <cit.>. The latter appears as an infinite series of discrete and stacked, exponentially-damped in luminosity, lensed images of the emission surrounding a black hole, under the assumption of an optically thin and non-spherical accretion disk <cit.>. The recent progress in high-frequency very-long-baseline interferometry (VLBI) provides a direct way of comparing such heuristics with actual images, by the Event Horizon Telescope (EHT) Collaboration, of the super-heated overflow of the plasma around the supermassive central objects of M87* <cit.> and Sgr A* <cit.>. In turn, this provides an unprecedented opportunity to perform tests of new gravitational Physics, which has been put to good use very recently, see e.g.  <cit.>.

The main aim of this work is therefore to make use of this opportunity to study the optical appearances of the CG black hole above. In particular, we shall employ the reported bounds on Sgr A*'s shadow by the EHT Collaboration to constrain the CG black hole's parameter space. Next, we generate images of this geometry within this restricted parameter space when illuminated by an optically and geometrically thin accretion disk, modeled by three semi-analytic emission profiles developed by Gralla-Lupsasca-Marrone (GLM) <cit.>, and compare its optical appearance with that of a Schwarzschild black hole. Furthermore, we study the Lyapunov exponents of nearly bound orbits to elaborate on the differences between both geometries. In this work we consider units G=c=1. 

This paper is outlined in the following manner: In Sec. <ref>, we briefly present the field equations of CG and the static, spherically symmetric metric used throughout this work. In Sec. <ref>, we obtain the null geodesic equation and the effective potential of the CG black hole, using the above-mentioned spherically symmetric metric.
Section <ref> addresses gravitational lensing in CG, where we first derive the exact analytical expression for the deflection angle using elliptic integrals, followed by an approximate expression for the weak field regime.
In Sec. <ref>, we generate the images of the Cotton black hole using the restricted γ-space parameter obtained in the previous sections,  and qualitatively compare its appearance to that of a Schwarzschild black hole of the same mass parameter.
In Sec. <ref>, we perform an analysis of isoradial curves in the Cotton and Schwarzschild space-times, to complement the black hole images presented in the previous section. In Sec. <ref>, we explore the Lyapunov exponent of nearly bound orbits, which is a quantity that measures the instability scale of these orbits.
Finally, in Sec. <ref>, we summarize our results and conclude.
 



§ A SPHERICALLY SYMMETRIC SOLUTION OF COTTON GRAVITY




Recently, as mentioned above, a generalization of the gravitational field equations of GR, dubbed as “Cotton Gravity" (CG) <cit.>, was proposed. The field equations read as follows

    C_αμν=κ^2∇_βΘ_βαμν^β,

where κ^2=8π, is  Newton's constant in suitable units, ∇_β is the usual covariant derivative  built with the Christoffel symbols of the metric, C_αμν  is the Cotton tensor defined as

    C_αμν≡∇_μR_να-∇_νR_αμ-1/6(g_αν∂_μ-g_αμ∂_ν)R,

where R_μν stands for the Ricci tensor and R  is the Ricci scalar, and finally the energy-momentum tensor  Θ_βαμν is defined as

    Θ_βαμν   ≡   1/2(g_βμΘ_αν-g_αμΘ_βν-g_βνΘ_αμ+g_ανΘ_βμ) 
        
          -1/6(g_βμg_αν-g_αμg_βν)Θ ,

with Θ_βμ=g^ανΘ_βαμν and Θ=g^βμΘ_βμ. 

The most remarkable feature of CG gravity is that every solution of the Einstein equations with a cosmological constant term G_μν+Λ g_μν=κ^2Θ_μν satisfies Eq. (<ref>) <cit.>, while the opposite is not true. In contrast to GR, however, the cosmological constant in CG is determined by integration of the field equations. Furthermore, the conservation law (∇_βΘ_μν^β=0) is a natural consequence of the theory, just like in GR, as can be verified by contracting Eq. (<ref>) by g^αν to yield

    g^ανC_αμν=κ^2∇_βΘ_μν^β=0.


For the sake of this work we use the exact vacuum solution obtained by Harada in <cit.>. Similarly to GR, where R_μν=0, this solution is obtained in Cotton Gravity when C_αμν=0 is considered. Such a solution belongs to the static, spherically symmetric class of metrics with line element

    ds^2=-A(r)dt^2+dr^2/A(r)+C(r)(dθ^2+sin^2θ dϕ^2).

whose metric function reads as <cit.>

    A(r)=1-2M/r-Λ/3r^2+γ r  ,       C(r)=r^2  .

This is the first non-trivial solution found in CG theory, which contains a new parameter γ, not present in the Schwarzschild case of the GR, and which breaks the asymptotically flat character of the space-time. Moreover, the cosmological constant is obtained naturally as an integration constant no matter the value of the cosmological constant Λ, which we take to be vanishing in this work. We draw attention to the fact that solutions containing a linear γ r term have been obtained in other theoretical contexts (e.g. <cit.>).






§ GEODESICS AND EFFECTIVE POTENTIAL





 §.§ Gravitational lensing


In this section, we obtain the null geodesic equation and the effective potential of the CG black hole, using the spherically symmetric line element (<ref>). To this end, we consider the point-particle Lagrangian density given by

    L=1/2g_μνẋ^μẋ^ν=ϵ,

where ẋ^μ is the four-velocity of the photon,  ϵ=(-1,0,1) indicates the time-like, null, or space-like character of the geodesic, respectively, and an overdot denotes the derivative with respect to the affine parameter λ. Taking into account the metric (<ref>) yields the following form for the Lagrangian

    L=1/2[-A(r)ṫ^2+ṙ^2/A(r)+C(r)θ̇^2+C(r) sin^2 θϕ̇^2]=ϵ .



Given  the spherically symmetric character of the space-time, we can consider the motion in the equatorial plane, θ=π/2, without any loss of generality. In addition, since the Lagrangian (<ref>) does not directly depend on t and ϕ, the Euler-Lagrange equations imply the conservation of two quantities:


    E   =A(r)ṫ,
     
    L   =C(r)ϕ̇.,


where E is the energy of the particle and L the angular momentum, respectively (per unit mass). Since we are interested in null geodesics, ϵ=0, Eq. (<ref>) with the conserved quantities (<ref>) and (<ref>), leads to

    ṙ^2=1/b^2-V_ eff(r).

where we have conveniently re-absorbed a factor L in the affine parameter λ, and introduced the impact parameter b ≡ L/E. In this equation, the effective potential reads as

    V_ eff(r)=A(r)/C(r).

which for the  metric function considered in this work, Eq. (<ref>), reads

    V_ eff(r)=1/r^2-2 M/r^3+γ/r.

The maximum of the effective potential determines the region of unstable orbits, dubbed as the photon sphere, whose radius is denoted by r=r_p, and the corresponding impact parameter as the critical impact parameter, dubbed by b_c. The latter separates those orbits that are captured by the black hole, b<b_c, from those that are scattered back to asymptotic infinity, b>b_c. Those that have b=b_c may linger indefinitely at the photon sphere, turning several times around the black hole. To determine these quantities, we note that circular light orbits obey ṙ=0 and r̈=0 simultaneously. Under these conditions, r_p and b_c are expressed for the metric function (<ref>) as


    r_p   =√(1+6 γ  M)-1/γ,
    
        b_c   =3√(6)  M/√((1+ √(1+6 γ   M)+6 γ   M √(1+6 γ  M)+9 γ  M)),


which for γ=0 it recovers the critical b_c=3√(3)M and the photon sphere radius r_p=3M of the Schwarzschild geometry.

With the expressions (<ref>) and (<ref>) we can develop the equation of motion which describes the trajectory of the photon as

    (dr/dϕ)^2=C^2(r)(1/b^2-V_ eff).

These are the general expressions for a given spherically symmetric metric. 





 §.§ Shadow's radius and constraints to the γ parameter



The shadow of a black hole is a silhouette, a  region marked by a strong decrease in the observed luminosity, projected onto the celestial plane of a distant observer. This projection arises from photons originating from a background source, which are absorbed due to its intense gravity. To calculate the angular size and the shadow of the black hole, we adopt the formalism described by <cit.>. In this approach, they consider a static observer located at (r_0, θ_0=π/2), projecting an angle θ relative to the radial coordinate r, as given by

    tanθ=√(C(r)/A^-1(r))dϕ/dr|_r=r_0.

The shadow of the black hole, understood as the region which borders (on the observer's plane image) the critical curve, is thus constructed by the light rays emitted from the observer back to the source.

To proceed with our derivation, we conveniently re-write Eq.(<ref>) into

    (dr/dϕ)^2=C(r)/A^-1(r)(C(r)/A(r)E^2/L^2-1).

A light ray with impact parameter b>b_c will be scattered off the black hole at a radial distance r_0>r_m, while  the photon sphere (i.e. the locus of unstable bound geodesics, r_p) can be found via the condition dr/dϕ|_r_p=0  which allows to express the impact parameter b as

    1/b^2=A(r_p)/C(r_p).

Now, by introducing the definition

    h^2(r)=C(r)/A(r),

we can rewrite Eq. (<ref>) as

    (dr/dϕ)^2=C(r)/A^-1(r)(h^2(r)/h^2(r_p)-1).


Substituting Eq.(<ref>) into Eq.(<ref>), and after proper manipulations, allows to find the expression

    sin^2θ=h^2(r_p)/h^2(r_0),

which yields the angular size of the black hole's shadow θ_s as

    sin^2θ_s=b_c^2/h^2(r_0)=b_c^2A(r_0)/C(r_0).

Using the expressions worked out above, for CG the angular shadow's size reads as 

    sin^2θ_s   =   2/r_0^2γ^2 (γ +8γ^2M)(-1+6M√((1+6γ M))
       +   √((1+6γ M))-9γ M )(1-2M/r_0+γ r_0).








Similarly, we can obtain an expression for the shadow radius as seen by a far-away observer, a regime in which we approximate sin(θ) ∼θ and θ_s=b_c/r_0
which gives us the following relation (assuming θ=π/2)

    r_s=r_p√(g_tt(r_0)/g_tt(r_p)).

For CG this reads as

    r_s=(√(a)-1)/γ√((√(a)-1)[r_0(γ r_0+1)-2M]/r_0(1-√(a)+4γ M)) ,

where a=1+6γ M. In Fig. <ref> we depict the behaviour of the shadow's radius in CG with the parameter γ, considering the black hole distance and mass estimates reported in Table <ref>.




We now aim to restrict the parameter space of γ by assessing the compatibility of the CG geometry's corresponding shadow radius r_s with the EHT's inferred shadow size of Sgr A* using the data above. This is so because, as argued in <cit.>, it is reported that the observed shadow size of Sgr A* can be inferred by utilizing the size of the bright emission ring as its proxy, subject to a calibration factor that multiplicatively accounts for the theoretical and observational uncertainties. Such a correlation becomes possible thanks to priors on Sgr A*'s mass-to-distance ratio obtained via orbital dynamics measurements of the stars closest to the galactic center (i.e. dubbed S-stars), and which are reported in Table <ref>.  These measurements allow the EHT collaboration to estimate the fractional deviation δ between Sgr A*'s inferred shadow size r_sh and that of a Schwarzschild black hole r_sh,Sch= 3 √(3) M, with angular radius ω_g = M/D  via the formula

    r_sh/M=(δ +1) 3 √(3) ,

which, under the assumption of a normal distribution of the estimation uncertainties, yields the following shadow size bounds <cit.>

    4.55 ≲ r_sh/M ≲ 5.22  ,

at 1σ deviation and

    4.21 ≲ r_sh/M ≲ 5.56  ,

at 2σ deviation.



We note that, to quantify the theoretical biases in the calibration factor utilized in the inference of the shadow's size of Sgr* A, the EHT collaboration performs simulations of black hole images to study a wide range of effects concerning the accretion flow properties and the underlying space-time. In this regard, the collaboration assumes the Kerr metric by default, although they also cover black hole metrics that either deviate from the Kerr space-time or describe alternative geometries, such as wormholes or boson stars. Because their analysis does not include the Cotton geometry, by performing tests with the above constraints we introduce a theoretical bias in our own analysis, since we assume these results to be valid for the CG spacetime. The extent to which the Cotton geometry could modify the above fractional deviation, and the constraints stemming from it, could be evaluated by performing a similar analysis to that by the EHT collaboration. However, such an effort lies well beyond the purpose of this work. 

To perform tests on the CG metric with its shadow radius, we compare the evolution of r_s as a function of the logarithm of γ M, in Fig. <ref>. We restrict our analysis to positive values of γ, which are required to describe the large-distance gravitational acceleration. 
Because we are dealing with a non-asymptotically flat space-time, one must take into account that its asymptotic Arnowitt-Deser-Misner (ADM) mass <cit.> is undefined for an observer at radial infinity and cannot be associated with the mass parameter appearing in the metric.
To circumvent this issue, we shall take a similar approach to <cit.> and report the the shadow radius in units of the local mass M, based on the reasoning that the stellar orbital dynamics of the S group of stars probe this quantity, since they are influenced by the local gravity of Sgr A*. 
Additionally, since the CG black hole's shadow radius depends explicitly on the observer's radial position, recall Eq.(<ref>), we compute the shadow radius considering an observer located at r_O ∼ 8000  kpc <cit.>. 




Upon inspection of Fig. <ref>, we verify that r_sh starts to increase around values of γ M ∼ 10^-12. Below this value, the CG black hole shadow radius coincides with that of a Schwarzschild black hole. We note that this is qualitatively the same behavior that is evidenced in <cit.> for the Rindler metric. Actually, the contribution of γ to the shadow radius becomes significant when the term γ r ≫ 1, while the upper 2σ bound in Eq.(<ref>) becomes saturated at a value of approximately γ M ≃ 3.525 × 10^-12.
For the values of γ M predicted by galaxy radial acceleration rotation curves in <cit.>, or the recent constraints obtained by analyzing CG's influence on CMB power spectrum and large scale structures <cit.> (i.e. both estimating γ M ∼ 10^-17), the shadow radius of the Cotton black hole lies well within the constraints imposed by Sgr A*'s shadow bounds, though there is no appreciable difference between it and its Schwarzschild counterpart. This result is not surprising, given that γ encodes large-distance gravitational effects within the CG framework. 
We have also found that the Cotton black hole shadow radius remains within the 2σ bounds for γ M ∼ 10^20. 
Such values describe extremely large and unphysical gravitational acceleration effects, exceeding any other current estimates by several orders of magnitude. For this reason, we have opted for excluding them from the γ parameter range in Fig. <ref>.










§ GRAVITATIONAL LENSING



This section deals with the phenomenon of gravitational lensing for CG. First, we will obtain the exact analytical expression for the deflection angle in terms of elliptic integrals. Then we will calculate an approximate expression valid for the weak field regime.





 §.§ Exact deflection angle 


To proceed with our derivation, it is convenient to make the following variable change in the radial coordinate u = 1/r in Eq. (<ref>). This allows us to re-write the photon equation of motion as

    ( du/d ϕ)^2 = 2 M G(u),

where the function G(u) reads, for CG, as

    G(u) =u^3 - u^2/2M  - u γ/2M + 1/2Mb^2.


The general expression for the deflection angle at a finite distance has been worked out in <cit.> as

    α = √(1/2M)( ∫_u_R^u_0du/√(G(u))+∫_u_S^u_0du/√(G(u))) + Ψ_R - Ψ_S,

where u_0 = 1/r_0 is the inverse of the distance of closest approach, while u_R = 1/r_R and u_S are the inverse of the radius of the observer and the source, respectively, and the indices refer to the position of the observer (R) and the source (S). 
The function Ψ encodes a correction term accounting for the finite distance, and is given by

    Ψ = arcsin( bu √(A(u))) .

Note that if we take the limit of  γ→ 0, and consider also that observer and source are located at infinity we will have

    α_SC = √(2/M)∫_0^u_0du/√(G(u)) - π,

which is the well-known deflection angle formula for the Schwarzschild solution.  This change in the definition of the deflection angle is necessary because the Cotton metric is not asymptotically flat. Moving on, the integrals of Eq. (<ref>) can be solved by rewriting the function G(u) as

    G(u) = (u-u_1)(u-u_2)(u-u_3),

where u_1,u_2 and u_3 are the roots of the cubic polynomial. We can check that u_0 = 1/r_0 is one of the roots of this polynomial. We have chosen (for reasons that will become clear later) u_2 = 1/r_0. Now we can write

    u^3 -u^2(u_1 +u_3 +1/r_0) +u (u_1 u_3 +u_1/r_0 + u_3/r_0) 
       
    
         - u_1 u_3/r_0  = u_3 - u^2/2M - u γ/2M + 1/2Mb^2.

After comparing the coefficients in Eq. (<ref>), we get

    u_1 +u_3 + 1/r_0 =   1/2M, 
        u_1u_3 + u_1/r_0 + u_3/r_0 =  
            - γ/2M.

Solving the above system we find that u_1 and u_3 are  given by

    u_1 = -2 M
       +r_0 ∓√(-(12 M^2 -8 γ  M r_0^2-4 M r_0-r_0^2))/4 M r_0 , 
       u_3 = -2 M+r_0 ±√(- (12 M^2-8 γ  M r_0^2-4 M
       r_0-r_0^2))/4 M r_0,

respectively. Note that we have two different solutions, one of which is u_1> u_3 and another that u_3> u_1.
Since our goal is that these roots fulfill the condition u_3> u_2> u_1, we choose the negative sign in the expression u_1 and the positive sign in the expression u_3. Now both integrals of Eq. (<ref>) can be put into the form

    I_1(u^') =  ∫_u^'^u_2du/√((u-u_1)(u-u_2)(u-u_3))

with u_3>u_2>u^'>u_3 and u^' being any finite distance. The result of the above integral is 

    I(u^') = 2/√(u_3-u_1) F(δ,k^2),

where F(δ,k^2) is an incomplete first-order elliptic integral of the parameters

    δ   =   arcsin√((u_3-u_1)(u_2-u^')/(u_2-u_1)(u_3-u^')), 
    
        k^2    =   u_2-u_1/u_3-u_1.



 To get the result of the integrals given in (<ref>) we just need evaluate I(u_R)+I(u_S). The complete result will also depend on the correction given by the functions Ψ(u_R) - Ψ(u_S). 
For the Cotton metric, the function Ψ is 

    Ψ(u) = sin ^-1(b u √(1-2 M u+γ/u)),

while we have for the parameters

    δ(u^') = sin ^-1{ 2 √(2)[ 
        (M r_0 (1-u^' r_0) √(r_0^2 c)/√(r_0^2 c)  +6 M  r_0- r_0^2) ×
    ×(√( r_0^2 c)-2 M
        r_0 (2 R  r_0+1)+ r_0^2)^-1]^1/2},












where c=-12 M^2+4 M r_0 (2 γ  r_0+1)+r_0^2

    k^2= √( r_0^2-12 M^2+4 M r_0 (2 γ 
      r_0+1))+6 M -r_0/2√(r_0^2 -12 M^2+4 M
       r_0 (2 γ  r_0+1)).

which closes all the expressions we needed to characterize the deflection angle in CG. It is also interesting to re-write these expressions in terms of the impact parameter by inverting its definition, in CG, as 

    r_0    =   b^2 γ/3+2/3 b √(b^2 γ ^2+3)×
          ×cos[1/3cos^-1((2 b^4 γ ^3 +9 b^2 γ
       -54 M)/2 b (b^2 γ^2+3)^3/2)].









For the sake of the graphical representation of the deflection angle we prefer to use the variable b^' = 1-b_c/b. This variable is limited to a value between 0 ≤ b^'≤ 1, where zero represents the strong field limit and one the weak field one. In Figs. <ref> and <ref> we show the deflection angle as a function of the variable b^' for values of γ in the range from 10^-16 to 10^-9. We see that α decreases with increasing γ and that there is no significant change in the values of α from γ <10^-13. If we increase the value of the parameter γ beyond the order of magnitude of 10^-9, the deflection angle begins to give negative values. Observationally, this value diverges for b^' =0 and tends to zero for b^' =1. We expect our theoretical result to reproduce this behaviour.
In other words, for the phenomenon of light deflection, Cotton's parameter is limited to values smaller than γ = 10^-9.










§ IMAGES OF BLACK HOLES WITH THIN ACCRETION DISKS



Having restricted the γ-space parameter, we now proceed to generate the images of the Cotton black hole (by pushing the value saturating the upper 2σ bound of the Sgr A* shadow) and qualitatively compare its appearance to that of a Schwarzschild black hole of the same mass parameter. We consider the setting of a black hole illuminated by an optically (i.e. fully transparent to its own radiation) and geometrically thin accretion disk, when observed at both a face-on inclination (i.e. θ_o = 0^∘) and at a nearly edge-on inclination (i.e. θ_o = 80^∘).  

We begin our considerations from Eq. (<ref>), which we conveniently re-write in the form

    dϕ/dr=±b/C(r)√(1/1-b^2V_ eff),

that describes the variation in a photon's azimuthal angle as a function of its radial coordinate. By numerically ray-tracing the previous equation for a bunch of trajectories within some range of the impact parameter b, we compute the null geodesics from an observer's screen backwards towards their point of origin. Figure <ref> displays the geodesics in the Schwarzschild space-time (left panel) for M=1, and in the CG space-time (right panel), the latter with a value of γ M = 3.525 × 10^-12. Upon close observation of the figure, one highlights the absence of any significant qualitative differences between the near-horizon lensing properties of the two geometries, under the considered parameter values. In addition to larger distances, the effects of γ become more noticeable at much higher orders of magnitude than those considered here, as is discussed in Section <ref> below. 





To simulate the emission from an accretion disk we resort to the semi-analytic models provided by Gralla, Lupsasca and Marrone in Ref. <cit.>, which describe the radial emission profile of an accretion disk based on a Standard Unbound (SU) Johnson distribution function given by

    I(r;γ,μ,σ)=e^-1/2[γ+arcsinh(r-μ/σ)]^2/√((r-μ)^2+σ^2),

where the γ, μ and σ parameters modify the skewness, peak location and kurtosis of the distribution, respectively. Such models are designed to reproduce the most prominent image features of time-averaged General Relativistic Magneto-Hydrodynamic (GRMHD) simulations of a thick accretion flow around a Kerr black hole, providing a less computationally taxing alternative towards the generation of black hole images <cit.>.
These models describe an infinitesimally thin, axisymmetric, radially dependent emission profile, assumed to be monochromatic in the disk's frame of reference and optically thin at the emitting frequency. In the absence of absorption, the relativistic radiative transfer equation <cit.> implies that the specific intensity I_ν/ν^3 is conserved along a light ray such that I_ν_o=g^3 I_ν_e, where the subscripts o and e denote the observer and emitter frames, respectively. The quantity g is defined as the redshift factor g=ν_o/ν_e = √(A(r)/A(r_∞)). Since the disk is geometrically thin, a light ray gains additional brightness for each time it intersects the accretion disk while on its way to a distant observer (i.e. every n half-orbit). Thus, considering a monochromatic emitter I_ν_e≡ I(r), the integrated (bolometric) intensity I=∫ I_ν_o d ν_o in the observer's plane is computed according to

    I_o(r)=∑_n(A(r)/A(r_o))^2.I(r)|_r=r_n.



For the purpose of generating black hole images we restrict our analysis up to the n=2 photon ring, as higher-order photon rings will provide a negligible contribution to the net luminosity (as shall be explained in the next section). We shall suitably adapt the three models introduced in <cit.> to a spherically symmetric scenario, to simulate the accretion flow around the Schwarzschild and Cotton black holes, with the parameters

    GLM1   :   γ=-3/2  ,   μ=0   ,      σ=M/2 , 
    GLM2   :   γ=0   ,     μ=0   ,      σ=M/2 , 
    GLM3   :   γ=-2  ,    μ=17M/3  ,  σ=M/4 .

whose corresponding emission profile is depicted in Fig. <ref>.



The GLM1 and GLM2 models are utilized to simulate an accretion disk with an emission profile extending all the way towards the event horizon, being distinguished by the radial emission decay near this region. In contrast, the GLM3 model represents an accretion disk with its emission peak located near the radius of the innermost stable circular orbits (r_ISCO), which is 6M for a Schwarzschild black hole. The latter allows us to separate the individual luminosity contributions of the n=1 and n=2 rings from the direct emission, thereby facilitating the comparison of these features between the Schwarzschild and Cotton black holes, whereas in the former models the photon rings appear as a discrete set of boosts of luminosity superimposed on top of the disk's direct emission <cit.>. In our simulations, the observed luminosity is normalized with respect to its total value, to ensure both geometries are compared in identical settings. We resort to the Geodesic Rays and Visualization of IntensiTY profiles (GRAVITYp) ray-tracing code, developed by some of us.



By inspection of Fig. <ref>, we report no appreciable visual differences between a Schwarzschild and a Cotton black hole of the same mass, with γ M ≃ 3.525 × 10^-12, regardless of the emission conditions. At the considered distance, the Cotton black hole's corresponding event horizon radius, photon sphere radius and critical impact parameter, are nearly indistinguishable from those of a Schwarzschild black hole of the same mass M,  differing only by a factor of the same order of magnitude of γ. Since for a non-asymptotically flat geometry the shadow radius does not coincide with the critical impact parameter, the geometries appear identical. This similarity persists onto the photon ring(s), where the gravitational lensing features play a significant role. In the literature, however,  various black holes with a similar critical impact parameter have been found to be distinguished by the properties of their corresponding photon rings <cit.>. These differences occur when, within the parameter space compatibility with Sgr A* shadow radius, the additional parameter in a given geometry modifies significantly the effective potential (i.e. its shape, location, concavity) seen by photons on their approach to a black hole, from the canonical Schwarzschild case. For some geometries, however, the constraints to the parameter space are too restrictive, to the point that there are virtually no changes to the effective potential, which is our case. Finally, we point out that although Eq. (<ref>) suggests the observed intensity depends on the observer's radial coordinate, for a non-asymptotically flat space-time, there are no visual differences between the Cotton and Schwarzschild black holes since the intensities are normalized with respect to their total value. Without normalizing the intensities, the observed intensities of the accretion disk emission in the Cotton spacetime would decrease, as in this case A(r) →∞ for an observer at infinity.





§ ISORADIALS



In this section, we perform an analysis of isoradial curves in the Cotton and Schwarzschild space-times, to complement the black hole images presented in the previous section. We solve the isoradial equatorial curves corresponding to the n=0 emission, following the analytical approximation provided by <cit.>, which states that the arrival point of a light ray may be mapped onto an observer's plane with Cartesian coordinates (X',Y') as 


    [ X'; Y' ]
        = R/√(A(R))√(1-(1-x)^2)[  cos(φ); sin(φ). ]

Here, R denotes a given radial distance from the origin, while φ corresponds to the polar angle in the observer's plane[For a better understanding of this coordinate representation, the reader is referred to Fig. 7 of <cit.>.]. The value of x may be computed via the following approximation


    x    =    A(R)y {1
     -e/100(1-A(R))y[ ln(1-y/2) +y/2] 
     
           +(315 I_1^2 -135 I_2 + 180 A^2(R) +120 I_1 +32)/720y^2
        
                   + (I_1/2 + 1/6)y
     },









with y being expressed by the trigonometric relation between the polar angle in the observer's plane and the observer's inclination angle θ_o via

    y=1+sin(φ)/√(sin^2(φ)+cot^2(θ_o)),

and where the quantities I_1 and I_2 are calculated as

    I_1=∫_0^1 P(u)   du  , 
    
     I_2=∫_0^1 P^2(u)   du,

with P(u)=-4u^2A(u) + A(R), and where the variable change r=R/u was considered to obtain A(u).

By solving the above equations for all polar angles, for a light ray with its emission point located at a constant r=R, one obtains the isoradial curves of the n=0 emission. In Fig. <ref>, we display isoradial curves for a Schwarzschild black hole and for a Cotton black hole with γ M ≃ 3.525 × 10^-12, as seen by an observer with inclination angle θ_o = 80^∘. 



This result emphasizes the similarity in the horizon-scale lensing phenomena between these two geometries, as seen by the complete overlap of the isoradial curve of both space-times; something which is in agreement with the analysis in the previous section. 




§ THEORETICAL LYAPUNOV EXPONENT




In this section, we explore the Lyapunov exponent of nearly bound orbits, a quantity that measures the instability scale of such orbits. Since these instabilities take place near the photon sphere, the Lyapunov exponent tracks a local effect and is therefore independent of the observer's distance to the black hole. 

We begin our analysis by considering the effect of a small radial perturbation near the photon sphere (i.e. r=δ r + r_p  , δ r ≪ r_p) and apply it to Eq. (<ref>). We perform a second-order Taylor series expansion of the effective potential V_ eff around r_p

    V_ eff(r)≃ V_ eff(r_p)+V'_ eff(r_p)δ r+1/2V”_ eff(r_p)(δ r)^2,

where the first term of the expansion corresponds to the critical impact parameter b_c and the second term disappears, since unstable circular null geodesics correspond to critical maxima of the effective potential. The prime notation above denotes a derivative with respect to the radial coordinate. Substituting this result into Eq. (<ref>), leads to the following expression 

    (d δ r/dϕ)^2 = -1/2C^2(r_p)V”_ eff(r_p)(δ r)^2.

By writing the differential equation in the form

    πd δ r/dϕ = λδ r,

whose solutions are of an exponential form, δ r_n=δ r_0e^λ n, we obtain the Lyapunov exponent λ which quantifies the instability scale of a small perturbation. Computing the second derivative of the effective potential and utilizing the critical curve equation C'(r_p)A(r_p)-C(r_p)A'(r_p)=0 <cit.>, yields the following expression for the theoretical Lyapunov exponent

    λ=π(AC”-A”C/2)^1/2.


For the Schwarzschild geometry γ = π, indicating that an initial perturbation grows (in radial distance) by a factor e^γ with each half-orbit around the black hole. For the Cotton geometry, the Lyapunov exponent reads

    λ=π(1+γ r_p)^1/2.

This result supports the qualitative analysis done in the previous sections. The λ exponent of the CG black hole, differs from the Schwarzschild black hole by an order of magnitude proportional to γ.  Consequently, despite the exponential behaviour driven by the Lyapunov exponent, the deviations between these two geometries are captured by photon rings of a similarly order, requiring many half-orbits around the black hole before becoming significantly different. The gravitational lensing causing the n=1 and n=2 photon rings is far from being sensitive to the contribution of γ. Thus, one cannot hope to be able to distinguish between the Schwarzschild and CG geometries via their n=1 and n=2 ring features (i.e. relative intensities, relative widths), which are the ones that could be within reach of future interferometric detectors <cit.>.





§ CONCLUSION


In this work we investigate the phenomenon of gravitational lensing and its associated observables within an extension of GR recently proposed in the literature and dubbed as Cotton gravity. We estimated the size of the new parameter γ of Cotton gravity and which arises with a newly found spherically symmetric vacuum (Schwarzschild-like) solution, by making a numerical comparison with bound on the shadow's size of the supermassive black hole at the center of our galaxy (Sgr A*) reported by the Event Horizon Telescope Collaboration. Such a bound  is saturated, for CG, for values as small as γ M = 3.525 × 10^-12 (at 2σ in such bounds), where we took into account some subtleties associated to the non-asymptotic flatness of CG geometry to identifying the mass parameter of it with the one of the Schwarzschild case. 

To study the impact of this result in geodesic effects, we first computed the exact expression for the deflection angle in the form of elliptic integrals. There are two important considerations about this result. Firstly, in order to obtain the result of the Schwarzschild solution, one must first assume γ=0 and additionally u_R and u_S=0 (which corresponds to the consideration of infinite distances). This is because the result for Schwarzschild is usually determined using the definition for infinite distances, which is not possible in the case of Cotton gravity given its non-asymptotically flat character. Secondly, the deflection angle can be written as a function of the impact parameter b, though under an explicit cumbersome form. Furthermore, we computed the deflection angle using a Taylor series approximation to solve the integrals, which yields a result which is valid for the weak field regime where M/b ≪ 1. We have considered a linear approximation for the parameter γ in addition to the series with respect to the mass (we expect that γ has in reality a much smaller order of magnitude than mass and distances), and checked the consistence of the analytical approximation with the numerical integration of the deflection angle. Overall, we find that the deflection angle remains unchanged for values of the CG parameter γ < 10^-14.


The smallness of such a parameter of CG drove us to consider the chances of distinguishing between Schwarzschild and CG black holes using observables associated to gravitational lensing. To this end, we simulated the appearance of a CG black hole, with such a value of the γ parameter, surrounded by a thin accretion disk, modelled by three semi-analytic radial intensity profiles. We found the constraints placed on γ by Sgr A*'s shadow bounds to be too tight for any significant visual differences to appear. In this regard, we note that although in CG the observed intensity depends on the observer's radial distance, our images do not show any difference in this regard due to having normalized intensity profiles. 

In looking for further settings to search for the presence of any differences, we studied the isoradial curves of a CG black hole against those of a Schwarzschild black hole, utilizing a generalization of the Poutanen formula. We found that there is a full overlap of the isoradial curves between the CG and Schwarzschild geometries, confirming the absence of any visual differences in this regard. We also considered the theoretical Lyapunov exponent for a Cotton gravity black hole, associated to the instability scale of photon rings. We derived its exact expression by perturbing the photon equation of motion about the photon sphere radius. The CG theoretical Lyapunov exponent was found to differ from Schwarzschild's by a factor proportional to CG's γ parameter, and thus photon rings do not help in discriminating between both geometries, at least for those rings (n=1 and n=2) that could be expected to be measured in the last decades via interferometric means.

To conclude, the analysis carried out in this work points that the CG space-time recently found in the literature is very unlikely to be distinguishable from the expectations of the canonical Schwarzschild black holes with the devices we have at this moment and in the foreseeable near future.



MER thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Brazil, for partial financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
FSNL acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT) Scientific Employment Stimulus contract with reference CEECINST/00032/2018, and funding through the research grants UIDB/04434/2020, UIDP/04434/2020 and PTDC/FIS-AST/0054/2021.
DRG is supported by the Spanish Agencia Estatal de Investigación Grant No. PID2022-138607NB-I00, funded by MCIN/AEI/10.13039/501100011033, UE (European Union), ERDF (European Regional  Development Fund) A way of making Europe.


999


















Will:2014kxa
C. M. Will,

Living Rev. Rel. 17 (2014) 4.


SupernovaSearchTeam:1998fmf
A. G. Riess et al. [Supernova Search Team],
Astron. J. 116  (1998) 1009.


SupernovaCosmologyProject:1998vns
S. Perlmutter et al. [Supernova Cosmology Project],
Astrophys. J. 517 (1999) 565.


Copeland:2006wr
E. J. Copeland, M. Sami and S. Tsujikawa,

Int. J. Mod. Phys. D 15, 1753-1936 (2006).




Carroll:2000fy
S. M. Carroll,

Living Rev. Rel. 4 (2001), 1.



Senovilla:2022vlr
J. M. M. Senovilla,

Gen. Rel. Grav. 54 (2022)  151.


Salucci:2007tm
P. Salucci, A. Lapi, C. Tonini, G. Gentile, I. Yegorova and U. Klein,

Mon. Not. Roy. Astron. Soc. 378 (2007), 41-47.



Hawking:2005kf
S. W. Hawking,

Phys. Rev. D 72 (2005) 084013.

QG
C. Rovelli, “Quantum Gravity", Cambridge University Press (Cambridge Monographs on Mathematical Physics, 2004.).


Clifton:2011jh
T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis,
Phys. Rept. 513  (2012) 1.


Capozziello:2011et
S. Capozziello and M. De Laurentis,
Phys. Rept. 509 (2011) 167.



Harada:2021bte
J. Harada,
Phys. Rev. D 103   (2021) L121502.


Harada:2022edl
J. Harada,
Phys. Rev. D 106 (2022) 064044.


Capozziello:2004us
S. Capozziello, V. F. Cardone, S. Carloni and A. Troisi,

Phys. Lett. A 326 (2004), 292-296.




Capozziello:2006ph
S. Capozziello, V. F. Cardone and A. Troisi,

Mon. Not. Roy. Astron. Soc. 375 (2007), 1423-1440.




Boehmer:2007kx
C. G. Boehmer, T. Harko and F. S. N. Lobo,

Astropart. Phys. 29 (2008), 386-392.




Boehmer:2007glt
C. G. Boehmer, T. Harko and F. S. N. Lobo,

JCAP 03 (2008), 024.




Capozziello:2012qt
S. Capozziello, T. Harko, T. S. Koivisto, F. S. N. Lobo and G. J. Olmo,

JCAP 07 (2013), 024.




Capozziello:2013yha
S. Capozziello, T. Harko, T. S. Koivisto, F. S. N. Lobo and G. J. Olmo,

Astropart. Phys. 50-52 (2013), 65-75.




Sussman:2023eep
R. A. Sussman and S. Najera,
[arXiv:2312.02115 [gr-qc]].


Gogberashvili:2023wed
M. Gogberashvili and A. Girgvliani,
Class. Quant. Grav. 41  no.2, 025010 (2024).

Mantica:2022flg
C. A. Mantica and L. G. Molinari,

Gen. Rel. Grav. 55 (2023) no.4, 62


Sussman:2023wiw
R. A. Sussman and S. Najera,
[arXiv:2311.06744 [gr-qc]].


Clement:2023tyx
G. Clément and K. Nouicer,

[arXiv:2312.17662 [gr-qc]].




Sussman:2024iwk
R. A. Sussman, C. A. Mantica, L. G. Molinari and S. Nájera,

[arXiv:2401.10479 [gr-qc]].



Clement:2024pjl
G. Clément and K. Nouicer,

[arXiv:2401.16008 [gr-qc]].



Sussman:2024qsg
R. A. Sussman, C. A. Mantica, L. G. Molinari and S. Nájera,

[arXiv:2402.01992 [gr-qc]].


Congdon 
Arthur B. Congdon, and 
Charles R. Keeton, “Principles of Gravitational Lensing: Light Deflection as a Probe of Astrophysics and Cosmology", Springer,  (2018).


Bozza:2002zj
V. Bozza,
Phys. Rev. D 66 (2002) 103001.





Bozza:2012by
V. Bozza and L. Mancini,
Astrophys. J. 753 (2012) 56.



Pietroni:2022cur
S. Pietroni and V. Bozza,
JCAP 12 (2022),¡ 018.



Tsukamoto:2020iez
N. Tsukamoto,
Phys. Rev. D 102 (2020) 104029.



Tsukamoto:2020bjm
N. Tsukamoto,
Phys. Rev. D 103 (2021) 024033.



Tsukamoto:2021fsz
N. Tsukamoto,
Phys. Rev. D 104 (2021) 124016.




Zhang:2022nnj
J. Zhang and Y. Xie,
Eur. Phys. J. C 82 (2022)  471.



Gralla:2019xty
S. E. Gralla, D. E. Holz and R. M. Wald,
Phys. Rev. D 100 (2019) 024018.


Falcke:1999pj
H. Falcke, F. Melia and E. Agol,
Astrophys. J. Lett. 528 (2000) L13.



Cunha:2018acu
P. V. P. Cunha and C. A. R. Herdeiro,
Gen. Rel. Grav. 50 (2018) 42.



Gralla:2019drh
S. E. Gralla and A. Lupsasca,
Phys. Rev. D 101 (2020) 044031.



Johnson:2019ljv
M. D. Johnson,  et al.
Sci. Adv. 6 (2020)  eaaz1310.



EventHorizonTelescope:2019dse
K. Akiyama et al. [Event Horizon Telescope],
Astrophys. J. Lett. 875 (2019) L1.






EventHorizonTelescope:2022wkp
K. Akiyama et al. [Event Horizon Telescope],
Astrophys. J. Lett. 930 (2022) 12.


Vagnozzi:2022moj
S. Vagnozzi, et al.
Class. Quant. Grav. 40 (2023) 165007.



Gralla:2020srx
S. E. Gralla, A. Lupsasca and D. P. Marrone,
Phys. Rev. D 102 (2020) 124004.


Mannheim:1988dj
P. D. Mannheim and D. Kazanas,

Astrophys. J. 342 (1989) 635.

Ghosh:2015cva
S. G. Ghosh, L. Tannukij and P. Wongjun,

Eur. Phys. J. C 76 (2016) 119.

Soroushfar:2015wqa
S. Soroushfar, R. Saffari, J. Kunz and C. Lämmerzahl,

Phys. Rev. D 92 (2015) 044010.

Kiselev:2002dx
V. V. Kiselev,

Class. Quant. Grav. 20 (2003) 1198.


Grumiller:2010bz
D. Grumiller,

Phys. Rev. Lett. 105 (2010), 211303
[erratum: Phys. Rev. Lett. 106 (2011).



Perlick:2021aok
V. Perlick and O. Y. Tsupko,
Phys. Rept. 947 (2022) 1.



Do:2019txf
T. Do, et al.
Science 365 (2019) 664.



EventHorizonTelescope:2022xqj
K. Akiyama et al. [Event Horizon Telescope],
Astrophys. J. Lett. 930 (2022) L17.



Arnowitt:1962hi
R. L. Arnowitt, S. Deser and C. W. Misner,
Gen. Rel. Grav. 40 (2008) 1997.


Xia:2024tps
P. Xia, D. Zhang, X. Ren, B. Wang and Y. C. Ong,
[arXiv:2405.07209 [astro-ph.CO]].


Ishihara:2016vdc
A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura and H. Asada,
Phys. Rev. D 94 (2016) 084015.


Gold:2020iql
R. Gold, et al.

Astrophys. J. 897 (2020) 148.


daSilva:2023jxa
L. F. D. da Silva, F. S. N. Lobo, G. J. Olmo and D. Rubiera-Garcia,
Phys. Rev. D 108 (2023) 084055.



Claros:2024atw
J. Claros and E. Gallo,
[arXiv:2403.18543 [gr-qc]].



Claudel:2000yi
C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis,
J. Math. Phys. 42 (2001) 818.

Ayzenberg:2023hfw
D. Ayzenberg, et al.

[arXiv:2312.02130 [astro-ph.HE]].












 Paper Title: Aspects of Stability, Rigidity and Unitarity in String Vacua.



















 



6062


fancy




[LE, RO]

[RE]
[LO]





@twoside 
    
    
    
    -2.0



























































minitoctocrightmargin2.55em plus 1fil

@mtc@verse@verse
@endmtc@verse@verse
@verse#1@mtc@verse#1-1.5ex -1
        @verse@endmtc@verse




shapes,arrows



compat=newest
plot coordinates/math parser=false
plotmarks





    dirac/.style=
            mark=triangle*,
            mark options=scale=2,
            ycomb,
            scatter,
            visualization depends on=y/abs(y)-1 ,
            scatter/@pre marker code/.code=[rotate=90*,yshift=-2pt]
        



plain
THMTheorem[section]
Lem[THM]Lemma
Prop[THM]Proposition
Not[THM]Notation


definition
DEFDefinition[section]
ConjConjecture[section]
EXMPExample[section]

remark
RemaRemark[section]
*NoteNote
CaseCase




calc



theorems

[auto counter, number within=section]Thm[2][]
    enhanced,
    colframe=black,
    colback=white,
    colbacktitle=white,
    coltitle=black,
    attach boxed title to top left=xshift=5mm,yshift*=-/2,
    fonttitle=,
    title=Theorem : #2,#1





definition
DefDefinition[section]
[outerlinewidth=0pt, innerlinewidth=0.5pt, middlelinewidth=1pt, middlelinecolor=white]Def








[
    spaceabove = 15pt, spacebelow=15pt,  
    headfont   = ,  
    headformat = ▾ Example 
,
    headpunct  = ,    
    notefont   = ,           
    notebraces = — ,           
    bodyfont   = =0em =0em 1.05, 
    preheadhook   = *4,  
    postheadspace = 0pt,              
    prefoothook   = 
 ]examplestyle

[style=examplestyle, numberwithin=section]Exmp



    thmbox/.style=
            enhanced,
            breakable,
            sharp corners=all,
            fonttitle=,
            fontupper=,
            top=0mm,
            bottom=0mm,
            right=0mm,
            colback=white,
            colframe=white,
            colbacktitle=white,
            coltitle=black,
            attach boxed title to top left,
            boxed title style=empty, size=minimal, bottom=1.5mm,
            overlay unbroken =
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)|-
                    (frame.south east)–(frame.north east);,
            overlay first=
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);
                    (frame.north east)–(frame.south east);,
            overlay middle=
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);
                    (frame.north east)–(frame.south east);,
            overlay last=
                    ([xshift=3.5mm]frame.north west)|-
                    (frame.south east)–(frame.north east);,
        ,
    S/.style=thmbox,
            overlay unbroken =
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
            overlay first=
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
            overlay middle=
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
            overlay last=
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
        ,
    L/.style=thmbox,
            overlay unbroken =
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)|-([xshift=15mm]frame.south west);,
            overlay first=
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
            overlay middle=
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
            overlay last=
                    ([xshift=3.5mm]frame.north west)|-([xshift=15mm]frame.south west);,
        ,
    LQ/.style=thmbox,
            overlay unbroken =
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)|-([xshift=10cm]frame.south west);
                    [anchor=east] at (frame.south east) □;,
            overlay first=
                    (title.south west)–(title.south east);
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
            overlay middle=
                    ([xshift=3.5mm]frame.north west)–([xshift=3.5mm]frame.south west);,
            overlay last=
                    ([xshift=3.5mm]frame.north west)|-([xshift=10cm]frame.south west);
                    [anchor=east] at (frame.south east) □;,
        ,


[]LEMLemmathmbox,Stheo
[]ProofProofthmbox,LQtheo











5.25cm







patterns
shapes.geometric

decorations.markings
arrows
fit


babel













black







@twoside @̧page
            empty
            @twocolumn

























[slg]symbolslistsyisygSymbols




long-sc-short




A****** 



Sname=, description=***, type=symbolslist







eqaed
    
    matrix
      pretableaumatrix/.style=
        ampersand replacement=&,
        matrix of math nodes,
        outer sep=1mm,
        inner sep=0mm,
        anchor=center,
        row sep=between borders,-,
        column sep=between borders,-,
        dottedentry/.style=densely dotted,
        dashedentry/.style=densely dashed,
        spaceentry/.style=draw=none,execute at begin node=,
      ,
      pretableaunode/.style=
        font=,
        draw=gray,
        sharp corners,
        rectangle,
        anchor=base,
        text height=3.75mm,
        text depth=1.25mm,
        minimum height=5mm,
        minimum width=5mm,
        inner sep=0mm,
        outer sep=0mm,
        doublewidth/.style=minimum width=10mm,
        footnotesize/.style=font=,
        scriptsize/.style=font=,
      ,
      tableaumatrix/.style=
        pretableaumatrix,
        every node/.append style=
          pretableaunode,
        ,
      ,
      medtableaumatrix/.style=
        pretableaumatrix,
        every node/.append style=
          pretableaunode,
          font=,
          text height=2.75mm,
          text depth=.75mm,
          minimum height=3.5mm,
          minimum width=3.5mm
        ,
      ,
      smalltableaumatrix/.style=
        pretableaumatrix,
        every node/.append style=
          pretableaunode,
          font=,
          text height=1.85mm,
          text depth=.15mm,
          minimum height=2.5mm,
          minimum width=2.5mm,
        ,
      ,
      tinytableaumatrix/.style=
        pretableaumatrix,
        every node/.append style=
          pretableaunode,
          font=,
          text height=1.25mm,
          text depth=.15mm,
          minimum height=1.75mm,
          minimum width=1.75mm
        ,
      ,
      tableau/.style=
        baseline=-1.25mm,
        every matrix/.style=tableaumatrix,
      ,
      medtableau/.style=
        baseline=-1.25mm,
        every matrix/.style=medtableaumatrix,
      ,
      smalltableau/.style=
        baseline=-1.25mm,
        every matrix/.style=smalltableaumatrix,
      ,
      preshapetableaumatrix/.style=
        pretableaumatrix,
        execute at end cell=,
        every node/.append style=
          draw=black,
          anchor=base,
          inner sep=0mm,
          outer sep=0mm,
        ,
        shadedentry/.style=fill=gray,
        darkshadedentry/.style=fill=darkgray,
      ,
      medshapetableaumatrix/.style=
        preshapetableaumatrix,
        every node/.append style=
          text height=2.75mm,
          text depth=.75mm,
          minimum height=3.5mm,
          minimum width=3.5mm
        ,
      ,
      shapetableaumatrix/.style=
        ampersand replacement=&,
        matrix of math nodes,
        outer sep=0mm,
        inner sep=0mm,
        anchor=base,
        row sep=between borders,-,
        column sep=between borders,-,
        execute at begin cell=,
        every node/.append style=draw,anchor=base,text height=1mm,text depth=.5mm,minimum size=1.5mm,inner sep=0mm,outer sep=0mm,
      ,
      shapetableau/.style=
        every matrix/.style=shapetableaumatrix,
      ,
      topalign/.style=
        every matrix/.append style=name=maintableau,anchor=maintableau-1-1.base,
        baseline,
      ,
    empty
    
    Università degli Studi di Torino 
     Scuola di Dottorato
    
    
    
    
        < g r a p h i c s >
    
    
     Aspects of Stability, Rigidity and Unitarity in String Vacua
     Giorgio Leonearabic
    
    Università degli Studi di Torino 
     Scuola di Dottorato
    
     Dottorato in Fisica
    
     Aspects of Stability, Rigidity and Unitarity in String Vacua
     Giorgio Leone
     PhD advisor: Carlo Angelantonj=0.9cm
    = 615pt
    =40pt
    
    nopatch=itemempty
    
    
    "Wovon man nicht sprechen kann, darüber muss man schweigen." 
    
    "Whereof one cannot speak, thereof one must be silent." 
    L. Wittgenstein, Tractus Logico-Philosophicus, (1921). 
    empty1.45
    
    CHAPTER: ACKNOWLEDGEMENTS
    empty
    
    This thesis could not have been conceived without my advisor, prof. Carlo Angelantonj, to whom I am grateful in so many ways that the present few lines cannot encode to the full extent. First, I am very thankful and honoured for his support in my abilities and ideas even when I felt I did not deserve it. I am grateful for his kindness and availability in everything I have ever needed starting from the first weeks of the pandemic and ending with these last days of my PhD journey. I really enjoyed all the stimulating discussions we had in these years which also allowed me to learn a lot of physics. I want to thank him also for all the time and patience he spent reading and correcting my notes, papers and the present manuscript, and for his suggestions that have surely improved my writing ability. I am also very grateful to him for teaching me how to be a professional scientist and behave towards the scientific community. I will carry all these invaluable experiences with me in the future. 
       
    
     I would like to thank Prof. Augusto Sagnotti for his comments and interest in my work, which allowed me to continue my research activity. His support has an invaluable meaning to me. I would like to thank Prof. Ralph Blumenhagen for his feedback on the present manuscript and the stimulating discussions during my time in Munich and the conference in Corfou that improved my work.
    
     I would like to thank my collaborators, Prof. Emilian Dudas, Cezar Condeescu and Prof. Ioannis Florakis, for the stimulating discussions that expanded my knowledge and opened new perspectives on the topics I have been working on.
    
     A special thank is deserved to my collaborator and friend, Ivano Basile, who taught me a lot of new physics I did not know of and opened my eyes to new worlds to be explored, as well as for his kindness and availability even outside physics. His passion and enthusiasm have been stimulating and inspirational in my research activity.
    
     During these years I had the opportunity to be involved in interesting conversations, from which my work undoubtedly benefited, with many brilliant colleagues and senior scientists including Prof. Steve Abel, Roberta Angius, Prof. Ignatios Antoniadis, Veronica Collazuol, Niccolò Cribiori, Arun Debray, Matilda Delgado, Prof. Keith Dienes, Markus Dierigl, Bernardo Fraiman, Mariana Grana, Matteo Inglese, Christian Kneissl, Cameron Krulewski, Prof. Carlo Maccaferri, Andriana Makridou, Prof. Marco Meineri, Miguel Montero, Luca Nutricati, Natalia Pacheco-Tallaj, Hector Parra De Freitas, Prof. Herve Partouche, Alessandro Pini, Salvatore Raucci, Flavio Tonioni, Prof. Nicolaos Toumbas, Irene Valenzuela and Matthew Yu.
    
     I would like to thank also my PhD colleagues and friends with whom I have shared the office and everyday life in these years: Andrea Arduino, Valerio Belocchi, Gloria Bertolotti, Dripto Biswas, Niccolò Brizio, Andrea Bulgarelli, Elia Cellini, Valerio Belocchi, Elia de Sabbata, Jakub Kryss, Luca Orusa, Federica Rossi, Alberto Ruffino, Francesco Sarandrea, Antonio Smecca and Paolo Vallarino. I am honoured to have shared my intellectual space with them.
    
     I would like to thank my mother, my father and my brother who have always supported me, even during my tough times.
    
     I am extremely grateful to Claudio Arnetoli, who helped me go through difficult times and grow as a human being and scientist. 
    
     I would like to thank my old friends Cecilia Belletti, Fabio Buccoliero, Bo De Baene, Stefania Frand-Genisot, Darix Gribaudo, Paolo Racca, Federico Russo and Isabella Sanna who have always been there for me even though our paths took different directions.
    
     Finally, I would like to thank my partner Micol, for her support in these years. I am very grateful for bearing the burden of my anxiety in the challenges that I had to face, still giving me all the strength and motivation I needed to overcome them and grow as a human being.
    
    empty=0.9cm
    = 615pt
    =40pt
    
    
    [0]Résumé–Abstractabstract1.45§ ABSTRACT  In this Thesis we investigate properties of stability and unitarity of the string landscape in ten and lower dimensions. The dissertation explores these aspects by intertwining a detailed analysis of string vacua, with and without supersymmetry, with a bottom-up study driven by unitarity. In particular, in Chapter <ref> the possibility of formulating a necessary and sufficient condition for the classical stability of non-supersymmetric string vacua is discussed, emphasising the examples in ten and nine dimensions. In Chapter <ref>, new solutions are presented in six dimensions both for non-supersymmetric BSB vacua and those with minimal 𝒩=(1,0) supersymmetry, arising from a non-trivial cancellation of the R-R tadpoles. In particular, solutions for the T^4/ℤ_6 supersymmetric and BSB orientifolds are discussed, including a non-trivial non-supersymmetric vacuum that cannot be further deformed, a property absent in its supersymmetric counterpart. In addition, the insertion of probe strings charged with respect to R-R 2-forms is discussed in both cases, along with the unitarity conditions for the theory living on the world-volume to be adapted to these scenarios. Finally, Chapter <ref> discusses the role played by a new kind of global anomaly, arising from the inconsistency of effective field theories under topology change. A systematic analysis of six-dimensional supergravity theories with SU(2) gauge group and one tensor multiplet and U(1) gauge group with no tensor multiplets is discussed. Novel constraints are found and their cancellation in string vacua is verified. The latter discussion involves compactifications of supersymmetric heterotic theories whose related anomaly is guaranteed to cancel by a known theorem in the literature. In addition to supersymmetric heterotic theories, we have investigated the SO(16)×SO(16) heterotic theory compactified on the T^4/ℤ_6 orbifold and the Gepner orientifold with no tensor multiples, showing how such anomalies are cancelled. Even though expected, this result provides a non-trivial check for the consistency of these vacua, which was not guaranteed to hold  a priori by any theorem known in the literature. 
    
    
    
    
    roman  =0.9cm
    = 615pt
    =40pt
    
    
    
    
    
    
    
     -0.7cm
    
    
    
    
    CHAPTER: INTRODUCTION
    1.45 -20pt
    
    =0.9cm
    
    
    
    
    § INTRODUCTION
     String theory  <cit.> has provided a successful framework in which gravity, matter and gauge interactions admit a consistent unified quantum description. Despite its enormous success and despite the impressive progress in the last decades, many questions are still unanswered, especially when trying to make contact with low-energy phenomenology. Indeed, string theory is characterised by universal dynamics in the trans-Planckian regime <cit.> constraining the behaviour of scattering amplitudes at the Planck scale, which, however, cannot be directly tested by nowadays experiments. Thus, with the state of the art of our technology, making contact with the current phenomenology requires studying the theory in the low energy regime, aiming to incorporate the well-established standard model in a proper limit. However, the infra-red (IR) physics admits manifold manifestations and the task turns out to be challenging both technically and conceptually. In particular, the need to provide a stringy description of supersymmetry breaking has dramatic consequences on the phenomenology, and on the structure and dynamics of the theory itself. In the last forty years, many ways of breaking supersymmetry have been realised in heterotic string vacua <cit.>, through non-trivial magnetic fields in open strings <cit.>, via type 0 <cit.> and type II orientifolds with a non-trivial action on the compact directions <cit.>, with orientifold planes having positive tension and R-R charges, both in ten <cit.> and lower dimensions <cit.>, and via a combination of all these effects <cit.>. However, in most of these models, tachyonic instabilities arise, implying that the background cannot be trusted even classically, since, after tachyon condensation, it is expected to decay into a new, at least metastable, vacuum. This process is relatively under control when the tachyon is associated to the instability of (anti)brane configurations in simple set-ups <cit.>, while only in recent years some progress has been made in understanding the situation when the tachyon belongs to a ten-dimensional heterotic spectrum <cit.>[A previous study showing similar features has been discussed in <cit.> for the type IIB superstring theory at finite temperature.], and yet a systematic analysis is lacking. Such features are characteristic of tachyonic theories and thus one may wonder whether they underlie an intrinsic property which may distinguish them from non-tachyonic ones.
    
     In closed-string vacua, the interplay between the ultra-violet (UV) and IR degrees of freedom, guaranteed by the presence of modular invariance, allows one to address this issue by looking at the massive excitations of the string spectrum and thus at its global behaviour. A first attempt in this direction was carried out in <cit.> and led to the development of the so-called  asymptotic supersymmetry, according to which the IR finiteness of the cosmological constant implies an overall cancellation between on-shell bosonic and fermionic degrees of freedom. A few years later, a refined analysis involving the Rankin-Selberg-Zagier transform <cit.> revealed that the sum of the degrees of freedom within a given cut-off is proportional to the vacuum energy up to oscillating terms depending on the non-trivial zeros of the Riemann zeta function <cit.>. This outcome extends the result of <cit.>, since it gives an expression for the sum over the massive states for a finite value of the cut-off, but fails to uncover the origin of classical stability. Indeed, although very appealing from a mathematical perspective, it is not clear how to adapt such theorems to the cases in which the integrand grows exponentially at the cusp and admits a non-vanishing Fourier zero mode, the relevant situation for tachyonic vacua[If the integrand function grows exponentially but admits a vanishing zero mode, there is the possibility of regulating the partition function by subtracting the exponentially divergent piece, according to a conjecture by Zagier. In this case the vacuum admits unphysical tachyons and an explicit realisation of Zagier's conjecture has been shown in <cit.>.]. 
    
     A different route was followed in <cit.>, where it was observed that, in a tachyonic-free vacuum, bosons and fermions follow an oscillating pattern as the mass increases[The same observation has been made in <cit.> where the oscillations between baryons and mesons were traced back to an underlying modular invariance.], called  misaligned supersymmetry. These oscillations have been described by the large-mass behaviour of the partition function once the mass levels are continued to real values, by defining the so-called  sector-averaged sum. Indeed, such a continuation interpolates among the  physical masses of the tower associated to each character of the conformal field theory (CFT), thus allowing to realise a mutual, if partial, cancellation when bosonic and fermionic characters are added together. In <cit.>, the cancellation was proven to hold for the leading term and was conjectured to take place also for the sub-leading ones if and only if tachyons are absent in the spectrum. Moreover, the complete cancellation was conjectured to be reflected in the observed oscillations, which would then provide a necessary and sufficient condition for classical stability. These conclusions have been supported by the analysis in <cit.>, where a complete cancellation was shown to occur for the SO(16) ×SO(16) heterotic string in D=10<cit.>, with bosons and fermions following the conjectured oscillatory pattern. Although a significant step forward was taken in <cit.>, proving Dienes' conjecture calls for a better understanding of the features of tachyonic vacua. Indeed, simple numerical computations (see also <cit.> for previous analyses) actually reveal a  slightly more complicated story. For instance, oscillations have been observed in all the non-supersymmetric heterotic theories in ten dimensions, as well as for the tachyonic region of the moduli space for a Scherk-Schwarz compactification of the heterotic and type IIB superstring in D=9. These models provide counter-examples to the presence of oscillations as a necessary and sufficient condition for the absence of tachyons, but still leave the possibility of ascribing the latter to a complete cancellation of the large-mass behaviour of the string spectrum. The issue was finally settled in <cit.>, where it was understood that oscillations are reflected in the presence of fermions, which also determines the cancellation of the leading growth of the sector-averaged sum. Furthermore, it was shown that a complete cancellation takes place only when the spectrum has no tachyonic instabilities, thus proving Dienes' conjecture[It can be shown that the behaviour of the sector-averaged sum is dictated by the deepest tachyon, consistently with the observation in <cit.>.]. In view of these considerations, the oscillations are not reflected in the vanishing of the sector-averaged sum, and thus one needs to differentiate the two concepts, which were originally thought to be linked. However, since the former is trivially a consequence of the presence of fermions, it is more appropriate to define  misaligned supersymmetry as the vanishing of the sector-averaged sum. 
    
     It is natural to try to extend the analysis to the study of classical stability for orientifold vacua <cit.>. However, in such a case, the interplay between the IR and UV properties of the spectrum, previously guaranteed by modular invariance, is lost, and thus the physical meaning of the  orientifolded sector-averaged sums might not encode tachyonic instabilities. Indeed, the modular group maps different descriptions of the same amplitude, while one-loop vacuum amplitudes are mapped to the tree-level propagation of unoriented closed-string states between boundaries or crosscaps (or D-branes or O-planes) by an S or P transformation. This connection, as first noted in <cit.>, implies that the sector-averaged sums for orientifold amplitudes grow exponentially if and only if a tachyonic character is present in the dual description <cit.>. As a consequence, from such analysis, one can only draw either a necessary condition for the absence of tachyons in the closed-string spectrum or a sufficient criterion for the classical stability of the vacuum. Still, it fails to provide a necessary and sufficient condition, since a non-trivial action of the orientifold projection implies the associated sector-averaged sums are non-vanishing although the tachyon is projected away. It would then be required to realise a mutual cancellation between sector-averaged sum on different Riemann surfaces, which cannot take place. Therefore, misaligned supersymmetry cannot be considered the right tool to uncover the reason behind the classical stability of orientifold vacua, and new ideas and technologies are needed to finally address the issue.   
    
     Although the vast majority of non-supersymmetric models contain tachyonic instabilities, a few allowed solutions are classically stable, as it happens, in ten dimensions, for the aforementioned heterotic SO(16) ×SO(16) model, the type 0'B superstring <cit.> and the Sugimoto model <cit.>. Among the 10D tachyon-free vacua, the Sugimoto model <cit.> stands out since the tree-level closed-string amplitudes are supersymmetric, as reflected in the presence of the massless ten-dimensional gravitino, while bosons and fermions transform into different representations of the  USp(32) gauge group, thus breaking supersymmetry. Although the simultaneous presence of the massless gravitino and of non-supersymmetric matter seems at first sight puzzling, a consistent coupling is achieved since supersymmetry is realised non-linearly <cit.> on the open-string sector, with a massless goldstino playing the role of the Volkov-Akulov field <cit.>. Furthermore, in such a setting, the breaking of supersymmetry must be regarded as taking place at the string scale, since there is no order parameter that may recover supersymmetry. Microscopically, such configuration is realised through the presence of orientifold planes with positive tension and R-R charge, O9_+, which calls for the introduction of anti-branes, D9, to avoid anomalies. As a result, the R-R tadpole of the non-dynamical ten-form vanishes, but the uncancelled NS-NS dilaton tadpole contributes to the scalar potential. Such corrections, however, identify a new vacuum in which the background geometry is modified  <cit.>, but for which a full-fledged string theory analysis is missing, since it is not known how to quantise the theory on the new background nor how to describe it in the wrong vacuum <cit.>.  This feature is not peculiar to the Sugimoto model but also applies to the other non-supersymmetric vacua that are classically stable. In this scenario, one can thus, at most, develop a low energy effective field theory analysis that can be trusted up to the order of the dilaton tadpole. Although not exhaustive, such approach has been extremely successful in enriching our insights on the geometry characterising these vacua, leading to interesting features such as spontaneous compactification on a space with boundaries or a distinctive cosmological evolution <cit.>.
    
     A similar situation can be realised in lower dimensions, where however the presence of orientifold planes of different dimensionality provides richer scenarios. For instance, in D=6, the standard orientifold projection can be dressed with an involution, σ, acting on the twisted two-cycles, which trades the standard O5_- planes of the supersymmetric orientifolds <cit.> with O5_+, with positive tension and charge[This is not the only combination that can be realised in D=6, since the trivial and σ involutions are still compatible with the lower dimensional version of the Sugimoto vacuum involving O9_+ and O5_+ planes and its variation with O9_+ and O5_-, respectively. However, these cases are straightforward generalisations of the results discussed in this thesis and we will not comment on these possibilities any further.]. Therefore one is forced to introduce D5 branes to cancel the R-R tadpoles of the non-dynamical six-form, giving rise to a non-linear realisation of supersymmetry <cit.>, in a similar way as the 10D case. Such configuration realises what in literature is known as Brane Supersymmetry Breaking (BSB), whose simplest setting was first discussed in <cit.>. Here, the world-sheet parity operator  is dressed with the involution σ and the ℤ_2 action arising from the point group of the orbifold compactification T^4/ℤ_2. As a result, the fixed loci of the orientifold action identify O9_- and sixteen O5_+ planes whose contributions to the tadpole of the non-dynamical R-R forms are cancelled by placing D5 branes on a single ℤ_2 fixed point. Such a vacuum is anomaly-free but leaves behind a non-vanishing scalar potential generated by the untwisted NS-NS tadpoles. However, although this is the simplest choice, this is not the only possible one, and one still has the freedom to distribute the D5 branes among all the fixed points <cit.>. This induces a non-trivial cancellation of the D9 branes twisted charge against that of D5 ones, with a corresponding reduction of the open-string moduli that only allows deformations of the model via complete brane recombination <cit.>. Furthermore, uncancelled twisted NS-NS tadpoles are present and determine a richer potential, including contributions from twisted scalars. The latter may lead to a very interesting dynamics since non-vanishing vevs would lead to a spontaneous resolution of the orbifold singularities.
    
     If for the T^4/ℤ_2 BSB orientifold placing D5 branes on  all the fixed points is optional, and indeed such configuration is connected to the standard solution in <cit.> via brane recombination, for the BSB orientifold built on the T^4/ℤ_4 orbifold it is mandatory. The crucial difference resides in the fact that now O5_+ planes carry twisted charges for the non-dynamical R-R six-forms to be cancelled by those of D-branes. Hence, the model in such a case is  truly rigid since D-branes cannot be recombined altogether, without leading to an anomalous model <cit.>. Fractional orientifold planes may seem exotic but are present even in more standard supersymmetric realisations <cit.> involving the ℤ_3 and ℤ_6 point groups or the four-dimensional Z orientifold <cit.>. The T^4/ℤ_3 orbifold does not admit a BSB variant, while the BSB version of the  T^4/ℤ_6  was discussed in <cit.>, and still involves fractional O9_- and O5_+ planes. Similarly to the T^4/ℤ_4 case, the branes configuration guarantees the cancellation of the twisted R-R tadpoles, but, since there is only one ℤ_6 fixed point, the main properties of this vacuum follow closely those of the T^4/ℤ_2 model. Nevertheless, such a model admits a solution with rigid branes, with a scalar potential comprising blown-up moduli, potentially leading to a very interesting dynamics. 
    
     The low-energy effective action for these BSB vacua, with and without additional contributions to the scalar potential, admits a geometric interpretation <cit.>. Indeed, the gauge couplings can be expressed as a combination of scalars, J, which determines the metric of the moduli space in the tensor branch, similarly to what happens for the 𝒩=(1,0) supersymmetric vacua in D=6<cit.>. In the latter case, however, the J form[In F-theory vacua the J-form plays the role of the Kähler form <cit.>, but we will refer to it as such even outside the realm of F-theory.]  is related by supersymmetry to the Wess-Zumino counterterms induced by the generalised Green-Schwarz-Sagnotti (GSS) mechanism <cit.>.  Therefore, with 𝒩=(1,0) supersymmetry, the gauge couplings are linked to the data of the R-R tadpoles, which determine the structure of the reducible anomaly polynomial. However, this connection is only present in supersymmetric settings, since the real  stringy origin of the gauge couplings is tied to the NS-NS tadpoles, and thus one should  a priori use these data to characterise the J-form. This is manifest when supersymmetry is broken  à la BSB, so that the naïve construction of the J form in terms of the 't-Hooft anomaly coefficients induces ghost-like couplings for scalars and gauge fields in the low energy action <cit.>, spoiling unitarity.  This does not imply that the Kähler form cannot be defined, and indeed an explicit solution that preserves unitarity is obtained using the structure of the NS-NS tadpoles, thus solving the puzzle raised in <cit.>. Of course, this observation is consistent with the construction of the Kähler form for supersymmetric vacua, where R-R and NS-NS tadpoles are equal, but allows to extend the definition also to cases when supersymmetry is non-linearly realised. 
    
     Once the J-form is found, the unitarity conditions arising from the introduction of probe string defects, demanded by the completeness hypothesis <cit.>, can be analysed quantitatively adapting to the BSB context the results of <cit.>. These models correspond to  bona fide string theory vacua, and thus are expected to be consistent, so that these unitarity bounds have to be interpreted as a consistency check, in the same spirit as the cancellation of gauge and gravitational anomalies. The original formulation of <cit.> requires the presence of 𝒩=(0,4) supersymmetry on the defect, but the validity of the completeness hypothesis should go beyond the realm of supersymmetry and can be adapted to BSB vacua, since the microscopic theory living on the defect is known. This point of view is also instrumental in extending the conditions of <cit.> to non-supersymmetric set-ups, completing the task initiated in <cit.>. Actually, a generalisation of the latter is also required for supersymmetric models, since in both cases defects admitting an interpretation in terms of instantons of the gauge theory of the corresponding higher dimensional brane realise a Kač-Moody algebra on both left and right moving sectors of the CFT in the IR. Therefore, the precise formulation of the unitarity conditions of such  instanton defects would require a detailed knowledge of the CFT realised by the charged non-chiral bosons, which is notoriously difficult to achieve. 
    
     From a bottom-up perspective, the generalisation of the unitarity constraints of string defects might play a non-trivial role in constraining the landscape of the consistent effective field theories (EFT) that can be coupled to gravity. However, to make such conditions relevant for the swampland program <cit.> (see <cit.> for reviews), one ought to develop an independent low-energy argument through which they can be formulated. In 𝒩=(1,0) supergravity, the values of the central charges c_L,   c_R of the CFT in the IR can be deduced from the knowledge of the 't-Hooft coefficient of the SU(2) R-symmetry, since c_R simply follows from the properties of the 𝒩=(0,4) supersymmetry algebra in D=2. Therefore, the unitarity bounds translate into a correct identification, in the anomaly polynomial, of the R-symmetry that in <cit.> has been associated with the maximal subgroup of the orthogonal group of the non-compact space transverse to the defect. However, such an identification does not apply to instanton defects, and thus one needs to extend the analysis to spot it correctly[When supersymmetry is broken, there is no R-symmetry to be realised so that the argument should be rephrased completely.] (determining the central charge realised on the right moving sector of the CFT). Despite the difficulties, unitarity conditions from probe defects supporting the identifications of <cit.> and from anomaly cancellation of the bulk theory are very effective in sharpening the supersymmetric landscape  <cit.>, even beyond the 6D case <cit.>. Further constraints can be deduced for theories with at least one tensor multiplet through the  null charge conjecture <cit.>, which requires the existence of a defect whose charge Q satisfies Q · Q=0. The latter implies that every consistent realisation of an EFT in D=6 should arise from an internal theory admitting a description that is either geometric or dual to a geometric one.  
    
     Although these unitarity constraints have been extensively discussed, recently it has been understood that the full extent of the consistency conditions arising from unitarity has not yet been harnessed. Indeed, a new perspective on the meaning of anomalies paved the way for the exploration of new consistency conditions, suited for vacua both with and without supersymmetry. Traditionally, anomalies can be traced to the presence of ambiguities in the definition of the partition functions of chiral fields[For chiral p-form fields even the action is ill-defined.]. However, the set of theorems stated and proved by Dai and Freed <cit.>, and further extensions <cit.>, allowed to resolve such ambiguities by interpreting chiral fields living on X as a boundary mode of a non-chiral gapped one living in one dimension higher, Y, such that ∂ Y=X. For a generic quantum field theory (QFT), the allowed spaces Y can be properly restricted <cit.>, but in a theory of quantum gravity, topology is allowed to change, so that Y can be arbitrarily chosen among the spaces preserving the structure of X. In this sense, the anomaly is not related anymore to the failure of gauge invariance but to the dependence of the theory on the arbitrary manifold Y. The latter is encoded in a topological quantum field theory, denoted for clear reasons the  anomaly theory, defined on all the  closed spaces Y_cl obtained by glueing two different choices for Y. The anomaly theory evaluated on all the possible manifolds Y_cl identifies the so-called   Dai-Freed anomalies<cit.>, which comprise local anomalies <cit.> whenever Y_cl is himself a boundary and global ones <cit.> when Y_cl is a mapping torus, but they also include more general anomalies arising from spacetime topology change. Computing the anomaly theory on every closed manifold seems still a daunting task. However, this is simplified by taking advantage of the Dai-Freed theorems. Indeed, it has been shown that, for two manifolds that are equal up to a boundary, the anomaly theory differs only through the local anomaly. Thus a vanishing local anomaly allows to reorganise all closed spaces Y_cl in terms of bordism classes preserving the structure of X, for which the anomaly theory is a topological invariant. The set of bordism classes forms a discrete abelian group Ω^X_str_D+1 under the disjoint union, and thus it is enough to show that the theory of interest is anomaly-free if the anomaly theory vanishes on the generators of Ω^X_str_D+1.
    
     In six-dimensional string vacua, local anomalies are cancelled via the GSS mechanism, and thus we shall require the extended 7D closed manifolds to preserve the associated Bianchi identity. Demanding that the Bianchi identity be preserved at the level of integer cohomology identifies the  twisted string structure used to organise bordism classes. Specifically in six dimensions, the chiral 2-form fields entering the GSS mechanism belong to the gravity and tensor multiplets, so that multiple Bianchi identities are to be satisfied at the level of differential forms. Whether all the Bianchi identities have to be uplifted to integer cohomology is the subject of an open debate, since in <cit.> it was shown that satisfying only one Bianchi identity at the level of integer cohomology implies anomaly cancellation for known heterotic string theory vacua. In this thesis, following <cit.>, we shall make the minimal choice of satisfying at least one of the Bianchi identities, while still requiring the freedom to choose consistently all the ones present at the level of de Rham cohomology. The shape of the bordism groups associated to this choice is not known in the mathematical literature, and therefore a systematic analysis of Dai-Freed anomalies for string theory vacua along the lines of <cit.> cannot be performed[A different route has been taken in <cit.>, where using the technology of the equivariant topological modular forms <cit.> they have been able to show the absence of such anomalies for all the heterotic supersymmetric string vacua, generalising the 2D case of <cit.>.]. Nevertheless, it is still possible to perform the analysis on those backgrounds on which the anomaly theory can be exactly computed[There could still be the possibility to have a bordism group generated by a background on which our computational power is lost.], aiming on the one hand to verify its cancellation for string vacua and on the other hand to exclude inconsistent low energy effective theories <cit.>. Such non-trivial backgrounds are the Lens spaces L_p^7, obtained as the quotient S^7/ℤ_p, satisfying at least one Bianchi identity, for which the anomaly theory of fermions is well known <cit.>. Furthermore, theories admitting chiral 2-forms contribute through the so-called  quadratic refinement, reproducing locally the GSS term and defined through a characteristic equation linked to the cohomology pairing on the closed D+1 manifold.  A priori every solution to the latter is a consistent quadratic refinement, and up to now it is not known how to deduce such information from an independent path. This piece of information is crucial for the anomaly cancellation in D=6 supersymmetric theories <cit.>, and in <cit.> we considered all the possible solutions on L_p^7 for which only a specific choice leads to a vanishing anomaly theory. In particular, the shape of the quadratic refinement for spin bordism <cit.> turns out not to be suited for the consistency of the EFTs, and thus for preserving unitarity. Put differently, since the quadratic refinement characterises the partition function of the chiral 2-forms, requiring a specific choice of the latter identifies specific classes of chiral forms, excluding all the other vacua characterised by different choices. In <cit.>, 𝒩=(1,0) supergravity theories in six dimensions with at most one tensor multiplet were studied for abelian and simply-laced gauge groups, for which theories admitting non-chiral 2-forms are free of Lens space anomalies, while the same result does not hold for those theories admitting chiral forms unless a specific choice for the quadratic refinement is made. In particular, up to this condition that still requires an explanation, infinite families with arbitrary U(1) charges and no tensor multiplets turn out to be excluded. Or, phrased differently, Dai-Freed anomalies for a U(1) gauge group restrict the possible chiral 2-forms to a very specific choice whose features are still unclear. On the other hand, the anomaly cancellation was verified for explicit string constructions involving the compactification of the supersymmetric heterotic theories on K3 orbifolds and the non-supersymmetric SO(16) ×SO(16) heterotic vacuum on the same compactification spaces, as well as the Gepner orientifold with no tensor multiplets. The choice for the quadratic refinement in <cit.> is not suited to guarantee the consistency of these string vacua.  
    
    
    
    § OUTLINE
     In this thesis, we will review the results in <cit.> and <cit.>, by discussing the classical stability of non-supersymmetric vacua, the rigidity of brane configuration in D=6 orientifolds and unitarity conditions from probe strings and Dai-Freed anomalies, intertwined with a bottom-up analysis for 6D supergravities. For each aspect, we shall begin with a general discussion before addressing particular examples in which these features are concretely realised. To this end, the present dissertation is divided into four chapters, organised as follows:
    
    
      Chapter <ref> contains a discussion on the classical stability of non-supersymmetric string vacua in ten and nine dimensions, focusing on the possibility of distinguishing the two cases. In particular, we will show that it is possible to identify a necessary and sufficient condition for the absence of tachyons for closed-string vacua, while a similar criterion cannot be formulated when orientifold projections are performed.  
       
      	
      Chapter <ref> explores new solutions for six-dimensional vacua in cases where supersymmetry is realised both linearly and non-linearly, discussing local anomalies and unitarity constraints from string probes. Particular emphasis will be placed on the T^4/ℤ_6 orientifolds.
       
      	
      Chapter <ref> analyses the role played by Dai-Freed anomalies in six-dimensional string vacua, with and without supersymmetry, and 𝒩=(1,0) supergravity with at most one tensor multiplet. In particular, we shall discuss, as examples, supergravities admitting an SU(2) or U(1) gauge group with one and zero tensor multiplets. Gepner orientifolds are then addressed, along with the SO(16) ×SO(16) heterotic string compactified on the T^4/ℤ_6 orbifold.
       
      	
      Chapter <ref> summarises our results and provides possible extensions and outlook.
    
    
    CHAPTER: CLASSICAL STABILITY: MISALIGNED SUPERSYMMETRY
    1.45
    
    § ROOTS OF MISALIGNED SUPERSYMMETRY: THE SECTOR-AVERAGED SUM
    
    
    Stability is one of the main issues to be faced when trying to characterise the dynamics and the features of a non-supersymmetric string theory vacuum. Indeed, for models admitting space-time supersymmetry, the GSO  projection <cit.> eliminates the tachyon from the string spectrum and the vacuum-to-vacuum amplitudes on a generic Riemann surface vanish. As a result, tachyonic instabilities and quantum corrections to the background geometry are absent at each order in perturbation theory, thus implying that these models are perturbatively stable. However, when space-time supersymmetry is absent or broken, nothing prevents tachyons and quantum corrections from being present, and indeed characterising the stability of non-supersymmetric string vacua is still the object of intense activity. Even though understanding the final fate of these vacua is still an open problem, a significant step forward has been taken in recent years in finding a property that protects the classical stability[Here and in the rest of the thesis classical stability is used as a synonym for the absence of tachyons in the tee-level spectrum.] of the vacuum, known in the literature as  misaligned supersymmetry.
    
     Misaligned supersymmetry was first introduced by Keith Dienes for closed-string vacua in <cit.> and relies on the interplay between the UV and IR regimes implied by modular invariance. The key observation in <cit.> was to address the question of classical stability by looking at the large-mass behaviour of the string spectrum, which for closed strings is encoded in the one-loop partition function computed on a Riemann surface with the topology of a torus. We will limit ourselves to points of the moduli space where the vacuum is described by a rational conformal field theory (RCFT), which admits a finite number of conformal primaries M, tensored with the CFT given by D-2 non-compact free bosons. In general, the characters which identify the holomorphic and anti-holomorphic sectors are different but, for convenience, we shall use the same symbol  χ to label both. Hence, the contributions from the holomorphic sector will be denoted by χ_a and those from the anti-holomorphic sectors by χ̅_a, while non-compact bosons always contribute with suitable powers of the Dedekind eta functions. As a result, we can write the torus partition function as
    
        𝒵   =∫_ℱ dμ 𝒯
           = ∫_ℱ dμ 1/( √(τ_2)η̅ η)^D-2∑_a,bχ̅_a  𝒩_ab χ_b 
        	 ,
    
    where τ=τ_1 + i τ_2 is the complex structure modulus of the world-sheet torus, ℱ is the fundamental domain of SL (2;ℤ) and dμ = τ_2^-2  dτ_1   dτ_2 is the modular invariant measure. The so-called GSO matrix 𝒩_ab enforces the GSO projection and it is to be invariant under the modular transformations affecting the holomorphic and anti-holomorphic characters. To simplify the notation even further, we define the so-called  pseudo-characters, χ̌_a = η^2-D χ_a, which, strictly speaking, do not correspond anymore to the characters of the RCFT, since they carry the non-trivial modular weight 1-D/2. As a consequence, the action of the modular group is dressed with additional phases and powers of τ, identifying the pseudo-characters as  vector-valued modular forms
        χ̌_a (γ·τ ) = (cτ + d )^1-D/2  M(γ )_ab χ̌_b (τ ) ,
    
    where γ is a generic element of SL (2;ℤ), 1-D/2 is the weight of the modular forms, M(γ ) is a unitary matrix, including a non-trivial multiplier system, representing the action of γ on the χ̌s, and a sum over the repeated index b is understood.
    
     The pseudo-characters χ̌_a,  from now on simply referred to as the characters χ_a with the check omitted, admit power series expansions in the  nomeq=e^2π i τ of the form
    
        χ_a (q) = ∑_n=0^∞ d_a (n) q^H_a + n ,
    
    with 
    
        H_a = h_a - c/24
    
    expressed in terms of the conformal weight h_a and the central charge c of the full theory, including the D-2 non-compact bosons. The variable τ corresponds to the Teichmüller parameter describing the complex structure of the torus.
    
    
     The content of the physical spectrum, encoded in (<ref>), can be directly extracted by imposing the level-matching condition on the q-expansion of the characters, which in the large-mass limit simply reduces to  
    
        d(n) ≡∑_a,b𝒩_ab d̅_a (n+ H_b -H̅_a)   d_b (n)   .
    
    Therefore the behaviour of (<ref>) depends on the expressions for the degeneracies d_a(n) of the characters χ_a which is dictated by the Rademacher  exact formula  <cit.> for D >2[The formula is conjectured to hold also for the zero weight D=2 case, in which a refined estimate for the error determining the convergence of the expression is believed to exist <cit.>.]
        d_a (n) = 2π ∑_ℓ=1^∞∑_b  |   H_b <0  Q^(ℓ , n)_ab  f_b (ℓ , n)   ,
    
    adapted for the vector-valued modular forms <cit.>, through the known  Circle Method<cit.>. In the expression reported above, |H_b| ≤ 1[One can easily modify the previous expression whenever |H_b| ≤ h, with h a positive integer. In this case, the sum over b includes all Fourier modes d_b (m) which have m+H_b <0 for which we have to replace |H_b|→ |m+H_b| into (<ref>).] and we have defined the  generalised Kloosterman sum
        Q ^(ℓ , n)_ab = i^1-D/2 ∑_p=0 (p,ℓ )=1^ℓ -1 e^2π i/ℓ( H_b p' - p (n+H_a ))  (M^-1_ℓ ,p)_ab  ,
    
    where the generic modular transformation is parametrised as 
    
        γ_p,ℓ = [      - p' 1 + pp'/ℓ;        -ℓ         p ] ,
    
    with p' fixed by the condition γ_p,ℓ∈SL (2;ℤ). Furthermore, the behaviour of the degeneracy (<ref>) is dictated by 
    
        f_b (ℓ , n) = 2π d_b (0)/ℓ (|H_b|/n+H_a)^D/4  I_D/2( 4π/ℓ√( |H_b | (n+ H_a ))) ,
    
    which includes the modified Bessel function of the first kind I_α (x), admitting the asymptotic expansion 
    
        I_α (x) ∼e^x/√(2π x) ( 1- 4α^2-1/8x + …) .
     Therefore to deduce the large-mass behaviour of the spectrum, it is enough to plug (<ref>) into (<ref>) and take the limit n ≫ |H_a|. However, one subtlety has to be kept in mind: the connection between the asymptotic growth of states and the absence of tachyons cannot be formulated in terms of the physical d (n). A comparison between them requires the continuation of the variable n to the reals, thus "filling" the leftover mass levels. As a result, this implies the introduction of the continuous enveloping functions Φ_a (n), which reproduce the d_a (n) for the special values of n associated with the actual masses <cit.>. Using these functions, we can define the  sector-averaged sum as introduced in <cit.>
        ⟨ d (n) ⟩ ( 𝒯) ≡∑_a,b𝒩_ab Φ̅_a (n+ H_b -H̅_a) Φ_b (n ) ,
    
    where the enveloping functions behave asymptotically
    
        Φ_a (n) ∼∑_b   |  H_b <0d_b (0)/√(2) |H_b|^(d-1)/4/n^(d+1)/4    [ Q^(1,n)_ab e^4π√(|H_b| n) + Q^(2,n)_ab/√(2)  e^2π√(|H_b| n). 
         
           . + Q^(3,n)_ab/√(3)   e^4π/3√(|H_b| n) +…] ,
    
    with the first two terms dictated by the S and P modular transformations 
    
        Q_ab^(1,n) = i^1-d/2  S_ab  ,
        		
           Q_ab^(2,n) = i^1-d/2  (-1)^n P_ab  ,
        		
                   ⋮
     With the definition above, it has been possible to show in <cit.> that without tachyons the leading term of the expression vanishes and it has been conjectured that actually a complete cancellation among the enveloping functions would entail the stability of the vacuum. To prove the conjecture, hence, it is required to extend the analysis to the sub-leading orders. However, the study of the sub-dominant contributions to ⟨ d (n) ⟩ ( 𝒯) is a bit more involved and we shall employ a refinement of the expression (<ref>) along the lines of <cit.>. To this end, we notice that the Qs are periodic functions of n, with period ℓ, 
    
        Q^(ℓ,n+ℓ m)_ab = Q^(ℓ,n)_ab ,
    
    for any integer m.  Moreover, since the number of physical states (<ref>) depends on the product of the Q functions from the holomorphic and anti-holomorphic characters,  it is convenient to decompose the degrees of freedom into classes organised by the common periodicity v_ℓ,ℓ̅ = ℓ ℓ̅/gcd (ℓ,ℓ̅) of the two Qs and associate to each class its own enveloping function Φ_a (n , { w} ). As a result, we can write the  refined sector-averaged sum
    
        ⟨ d (n ) ⟩  ( 𝒯)    = ∑_a,b∑_{w  }𝒩_ab Φ̅_a (n, { w  + H_b - H̅_a } ) Φ_b (n, { w  } )
        		
           ≡∑_a,b∑_c,d   H_c, H̅_d <0∑_ℓ,ℓ̅=1^∞∑_w=0^v_ℓ,ℓ̅-1𝒩_ab  Q^(ℓ,w)_bc Q̅^(ℓ̅,  w+H_b - H̅_a)_ad  f_c (ℓ,n ) f̅_d (ℓ̅, n)  .
    
    It can be noticed that the refinement does not affect the leading growth so that the considerations made in <cit.> still apply.
    
     The definition of the sector-averaged sums can be extended to the vacuum-to-vacuum amplitudes relevant for orientifold vacua. These are given by the remaining (un)oriented Riemann surfaces with boundaries and cross-caps and vanishing Euler characteristic <cit.>: the Klein bottle, the annulus and the Möbius strip. We can write them in a compact way  as  
    
        𝒵_I=   ∑_a=0^M-1Z_I^a  χ_a = ∑_a=0^M-1Z_I^a ∑_n=0^∞ d_a(n) q^n+H_a ,
      
    with I=1,2,3 labels the Klein bottle, annulus and Moebius strip amplitudes, respectively, while the Z_I^a are suitable integers. In (<ref>), χ_a are the  real characters depending on the modulus τ of the double covering torus whose real part is fixed. For 𝒦 and 𝒜 the Teichmüller parameter is purely imaginary (τ= 2 i τ_2 and τ = iτ_2/2, respectively) so that the characters defined in (<ref>) are real, whereas for the Möbius strip surface the modulus of the covering torus is  1/2 + i τ_2/2, and one needs to introduce the extra phase e^-i π H_a to make them real <cit.>. This extra phase will not play any role in the following discussion and thus, with an abuse of notation, we shall omit it. These amplitudes describe the one-loop propagation of closed and open strings, which in 𝒦 and ℳ flip their orientation. It is conventional to refer to this description as the  loop-channel which involves a  vertical proper-time<cit.>. Similarly, by exchanging the length σ^1 of the string with the proper time σ^0 on the world-sheet through the S or P transformation[Actually, for the Möbius strip this inversion of proper time is realised via the P=T^1/2 S T^2 S T^1/2 transformation <cit.>.], we can describe the free propagation of closed strings between boundaries ( i.e. D-branes) and cross-caps ( i.e. O-planes) <cit.> via the  tree-level amplitudes 
    
        𝒵_I=  ∑_a=0^M-1Z_I^a  χ_a= ∑_a=0^M-1Z_I^a ∑_n=0^∞ d_a(n) q^n+H_a ,
       
    where q=e^-2 πℓ is written in terms of the new horizontal proper time ℓ. For the Möbius amplitudes it carries an extra minus sign, which originates from the non-vanishing real part of τ. As a result, the coefficients  Z_I^a and Z_I^a entering the eqs. (<ref>) and (<ref>) are not independent but are connected to each other by the i^1-D/2S or i^1-D/2P modular transformations. 
    
     From the expressions (<ref>) and (<ref>), it is straightforward to extract the large mass behaviour in the two channels encoded in the  sector-averaged sums associated to the corresponding amplitudes
    
        ⟨ d(n) ⟩ ( 𝒵_I  ) =  ∑_a=0^M-1 Z_I^a  Φ_a(n)  ,     ⟨ d(n) ⟩ ( 𝒵_I  ) =  ∑_a=0^M-1 Z_I^a  Φ_a(n )  .
       
    As in the closed-string case, 
    
        Φ_a (n) = ∑_ℓ =1^∞∑_w=0^ℓ -1 Φ_a (n,w ) ,
    
    are the enveloping functions associated with the degeneracies of the real characters, once the continuation of n to the reals and the refinement on the spectrum are employed. Employing the same refinement used for closed-string vacua, we can define the refined enveloping functions Φ_a (n, w) for the Klein bottle and annulus amplitudes as 
    
        Φ(n, w )= ∑_b   |    H_b < 0∑_ℓ=1^∞∑_w=0^ℓ-1 Q^(ℓ,w)_a b f_b(ℓ, n)   .
     
    In the Möbius strip amplitude, they involve the new Kloosterman sums
    
        Q̂_ab^( ℓ,w)= (-1)^w Q^(ℓ,w)_a b ,
       
    where the alternating sign is clearly associated to the fixed real part of the Teichmüller parameter. The presence of (-1)^w affects their periodicity in w, which is now ℓ̂= lcm(2,ℓ). This little change has remarkable consequences for the property of the sector-averaged sum associated with the Möbius strip amplitudes both in the direct and transverse channel, and reflects itself in the need of using the P transformations to connect the two channels, in the same way as the S transformation does for the Klein bottle and the annulus amplitudes.  
    
     Now that we have defined the sector-averaged sums for all relevant amplitudes, we can explore their properties and their links to the presence of tachyons in the spectrum. This will allow us, on one hand, to clarify the role of oscillations and also to prove Dienes' conjecture for oriented closed-string vacua. On the other, we shall only establish sufficient conditions for the classical stability of the vacuum, as well as a necessary criterion for the absence of closed-string tachyons, for orientifolds.  
                
    
    
    
    § MISALIGNED SUPERSYMMETRY FOR CLOSED STRING VACUA
    
    
    Let us now dig into the study of the properties of the refined sector-averaged sum for closed-string vacua. The leading order of (<ref>) can be straightforwardly deduced from the expression for the Rademacher expansion (<ref>) and the asymptotic behaviour of the modified Bessel function (<ref>). As can be extracted from (<ref>), the leading contribution is dictated by the deepest tachyon, with the smallest conformal weight, included in χ_0, associated with the ubiquitous NS vacuum which plays the role of the identity in the RCFT. As a result, the leading growth of each character is universal and reads
    
        d_a (n) ∼ e^4π√(c n/24) + … ,
    
    in agreement with the Cardy formula <cit.>. Plugging (<ref>) into the definition of the sector-averaged sum (<ref>) allows one to identify the leading contribution to the growth of states degeneracies     
    
        ⟨ d (n) ⟩ ( 𝒯) ∼  e^4π√(n)( √(c_L/24)+√( c_R /24)) ∑_a,b𝒩_ab = e^C_tot √(n)  ∑_a,b𝒩_ab  ,
    
    determined by the SL (2;ℂ) invariant vacuum, where we have defined the  total central charge[This reflects the terminology used in literature since the work of <cit.>. Nevertheless, it is an abuse of notation since the total central charge should correspond to c_L+c_R.]C_tot= √(c_L/24)+√( c_R /24).
    However, whether or not the asymptotic growth is controlled by C_tot depends entirely on the particular properties of 𝒩, since the GSO matrix must yield a modular invariant partition function. Consequently, we can relate the entry for the NS-NS vacuum to the sum over all the characters appearing in the partition function,
    
        𝒩_00 = 1/M ∑_a,b𝒩_ab
    
    since S_a0=i^D/2-1/√(M), for a RCFT with all the M characters resolved. Moreover, the entries 𝒩_ab can only be ± 1 or zero, which yields the following two scenarios:
    
      *  If the spectrum  does contain the deepest tachyon,  i.e.𝒩_00=1, one has 
    	
        ∑_a,b𝒩_ab≠ 0     and    ⟨ d (n) ⟩ ( 𝒯) ∼ e^C_tot √(n) .
    
      *  If the spectrum  does not contain the deepest tachyon,  i.e.𝒩_00=0,  then
    	
        ∑_a,b𝒩_ab = 0     and    ⟨ d (n) ⟩ ( 𝒯) ∼ e^C_eff √(n)
    
    	with an  effective central chargeC_eff < C_tot. 
    
    The condition 𝒩_00=0 clearly requires that fermions be present in the spectrum and, in particular, the number of bosonic sectors must equal the number of fermionic ones. On the contrary, in the absence of extended symmetries, namely when each character appears only once in  (<ref>), the condition 𝒩_00=1 implies that the spectrum only contains bosonic excitations. These observations were already contained in <cit.> and, in fact, represent its main result. However, it is important to stress that C_eff < C_tot does  not imply classical stability since, although the leading tachyon must indeed be absent, the condition 𝒩_00 =0 does not automatically exclude the possibility that other tachyons be present, as happens in most non-supersymmetric theories. This also implies that the mere presence of oscillations, which have been observed in  all non-supersymmetric theories with space-time fermions, has to be ascribed to the presence of the latter states and thus cannot be considered as a necessary and sufficient condition for classical stability.   
    
     On the other hand, the full extent of this simple analysis can lead to a further consideration. Indeed, as things stand now, the presence of fermions is not enough of course to guarantee the absence of  all tachyons but it is only a necessary condition. Thus, on a perturbative level, requiring the vacuum to be at least classically stable unavoidably introduces fermions, justifying their presence in a phenomenologically meaningful universe.      
    
     We can now turn to the analysis of the sub-leading contributions. This is where the refinement becomes effective and useful, since taking advantage of the expression in (<ref>), we can arrange the mass-levels in sub-classes  w=k_ℓ + ℓ r, determined by the periodicity of the generalised Kloosterman sum (<ref>) reported in (<ref>)
        ⟨ d (n) ⟩ ( 𝒯)= ∑_a,b∑_c,d   H_c, H̅_d <0∑_ℓ,ℓ̅=1^∞∑_k_ℓ=0^ℓ-1∑_r=0^ℓ̅/gcd (ℓ,ℓ̅)-1   𝒩_ab  Q^(ℓ,k_ℓ )_bc Q̅^(ℓ̅,  k_ℓ + r ℓ + H_b - H̅_a)_ad 
           × f_c (ℓ,n ) f̅_d (ℓ̅, n)   ,
    
    where we have used
    
        ∑_w=0^v_ℓ,ℓ̅ -1 = ∑_k_ℓ =0^ℓ-1∑_r=0^ℓ̅/gcd (ℓ,ℓ̅)-1  .
    
    This is the expression from which we can extract our main results. We have to distinguish the two cases ℓ = ℓ̅ and ℓ≠ℓ̅. In the latter case, if ℓ does not divide ℓ̅, there is no contribution to the sector-averaged sum, since
    
        ∑_r=0^ℓ̅/gcd (ℓ,ℓ̅)-1Q̅^(ℓ̅,  k_ℓ + r ℓ + H_b - H̅_a)_ad =0 .
    
    This property follows from 
    
        ∑_r=0^ℓ̅/gcd (ℓ,ℓ̅)-1Q̅^(ℓ̅,  k + r ℓ )_ad   = ∑_r=0^ℓ̅/gcd (ℓ,ℓ̅)-1∑_p̅=0 (p̅,ℓ̅)=1^ℓ̅-1 e^-2 π i/ℓ̅ (p̅ ( k + r ℓ)+  p̅ ' H̅_d ) ( M^-1_ℓ̅, p̅)_ad
           =∑_p̅=0  (p̅, ℓ̅)=1^ℓ̅-1  e^-2 π i/ℓ̅ (p̅ k +  p̅ ' H̅_d ) ( M^-1_ℓ̅, p̅)_ad 1-e^-2π i p̅ℓ̅/gcd (ℓ,ℓ̅)/1-e^- 2π i p̅ℓ/ℓ̅
           =0  ,
    
    since ℓ̅ is an integer multiple of gcd (ℓ, ℓ̅). Similarly, if ℓ̅= m ℓ, for some integer m, eq. (<ref>) still holds since p̅ and ℓ̅ must be co-prime, and thus
    
        ∑_r=0^m-1 e^-2 π i p̅ r/m = 0 .
    
    The case ℓ=ℓ̅ is a bit more involved and, as we shall see, discriminates between tachyonic and non-tachyonic string vacua. The Q functions have now the same periodicity, which implies that in eq. (<ref>) the sum over r is trivial, since nothing depends on r, and
    
        ⟨ d (n) ⟩ ( 𝒯 )    = ∑_a,b∑_c,d   H_c, H̅_d <0∑_ℓ =1^∞∑_ p=0 (p, ℓ )=1^ℓ -1∑_p̅=0 (p̅,  ℓ )=1^ℓ -1𝒩_ab ( M^-1_ℓ ,  p)_bc ( M^-1_ℓ , p̅)_ad^*
        		 
               × e^-2π i/ℓ  [ (p-p̅ )  H_b - (p' H_c - p̅ ' H̅_d )] 
        		f_c (ℓ,n ) f̅_d ( ℓ , n)  ∑_k_ℓ=0^ℓ-1 e^- 2 π i/ℓ (p-p̅ ) k_ℓ .
    
    The sum over k_ℓ imposes the condition p=p̅ which identifies the holomorphic and anti-holomor­phic modular transformations. For large n, using the modular invariance of the GSO matrix 𝒩_ab,  and taking into account that in string theory tachyons have -1≤H̅_a , H_b <0, one arrives at the final result
    
        ⟨ d(n) ⟩ ( 𝒯 ) = ∑_ℓ=1^∞∑_a, bH̅_a = H_b <0 ℓ φ (ℓ )  𝒩_ab   f_b (ℓ,n ) f̅_a ( ℓ , n)   ,
    
    where φ (ℓ) is a function counting the number of co-prime integers with ℓ, the  so-called Euler totient function.    The exponential growth of the sector-averaged sum is thus directly linked to the presence of tachyons in the physical string spectrum. In fact,  if physical tachyons, with H_b = H̅_a <0, are present in the string spectrum, then
    
        ⟨ d(n) ⟩ ( 𝒯 ) ∼∑_a,b  H_b = H̅_a <0𝒩_ab   d_b (0)  d̅ _a (0) |H_b|^(d-1)/2/2  n^(d+1)/2 ∑_ℓ=1^∞ φ (ℓ)   e^8π/ℓ√(|H_b|  n) .
    
    The exponential growth is then dictated by the conformal weight H_b of the  deepest physical tachyon,  C_eff = 8 π√(|H_b|). Instead, in the absence of on-shell tachyons the sector-averaged sum ⟨ d (n) ⟩ ( 𝒯 ) vanishes, and thus
    
        C_eff =0 .
     All in all, thanks to modular invariance and employing the clever refinement used in <cit.>, we have proved that the asymptotic growth rate of the sector-averaged sum is dictated by the mass of the  lightest states, whether tachyonic or massless,
    
        C_eff = 4 π√(|α' m^2_lightest |)≤ C_tot ,
    
    and thus the  necessary and sufficient condition for classical stability is the vanishing of both the sector-averaged sum and the effective central charge, as conjectured by Dienes in <cit.>. Notice that the deepest tachyon also determines the  effective Hagedorn temperature T_eff of strings at finite temperature <cit.>, and thus links it to C_eff. 
    
     The analysis is fully general and applies to any vacuum of oriented closed strings. It extends the discussion of <cit.>, which heavily relies on the representation of the characters in terms of eta quotients of special type, which is a rather restrictive requirement, not met by most non-supersymmetric string vacua. Obviously, when both representations are possible the two results coincide.
    
    
    
    § MISALIGNED SUPERSYMMETRY FOR ORIENTIFOLD VACUA
    
    
    For orientifold vacua, the conclusion reached in the previous Section cannot hold since modular invariance is no longer a property of the vacuum-to-vacuum amplitudes. This means that the large-mass behaviour encoded in the sector-averaged sum does not reflect anymore the presence of physical tachyons in the tree-level spectrum. Nevertheless, one may naïvely expect that since the closed-string UV/IR mixing translates into the connection between dual descriptions in orientifold amplitudes, the sector-averaged sum may encode the presence of tachyonic characters in the dual channel. This expectation is going to be confirmed in the following analysis, in which the properties of the generalised Kloosterman sums (<ref>), valid for the Klein bottle and annulus amplitudes, and the (<ref>), valid for the Möbius strip amplitude, play a key role.    
    
     Indeed, the properties of the Kloosterman sum drastically simplify the expressions of the enveloping functions. In fact, as shown previously, 
    
        ∑_w=0^ℓ-1  Q^( ℓ , w )_ab   = 	∑_w=0^ℓ-1∑_ p=0 ( p, ℓ )=1^ℓ -1 e^-2 π i/ℓ ( p w +  p '  H_b ) ( M^-1_ℓ , p)_ab ,
        		
           = ∑_ p=0 ( p, ℓ )=1^ℓ -1  e^-2 π i/ℓ   p '  H_b  ( M^-1_ℓ , p)_ab∑_w=0^ℓ-1 e^-2 π i/ℓ  p w  ,
        		
           = ∑_ p=0 ( p, ℓ )=1^ℓ -1  e^-2 π i/ℓ   p '  H_b  ( M^-1_ℓ , p)_abℓδ_p,0  ,
    
    which vanishes identically unless ℓ=1, since only in this case p=0 is coprime with ℓ. Therefore, the only contribution to the growth of the sector-averaged sum comes from ℓ=1, 
    
        Φ_a (n) = ∑_b  |  H_b <0  Q^ (1, 0)_ab  f_b (1,n ) =   i^1-D/2 ∑_b  |  H_b <0 S_ab  f_b (1,n) ,
    
    and, in the last equality, we have used the explicit formula (<ref>) for Q^ (1, 0)_ab. Notice that this expression only involves the S modular transformation that relates the 1-loop amplitudes, 𝒦 and 𝒜, to their dual counterparts,  𝒦 and 𝒜, describing the tree-level propagation of closed strings between pairs of O-planes and D-branes.
    
     The Möbius strip amplitudes depend on the modified Kloosterman sum, which also vanishes unless ℓ=2. In fact,
    
        ∑_w=0^ℓ̂-1Q̂^( ℓ , w )_ab   = ∑_w=0^ℓ̂-1 (-1)^w  ∑_p=0  (p,ℓ)=1^ℓ-1  e^- 2π i/ℓ (pw + p' H_b ) ( M^-1_ℓ , p)_ab
           =∑_ p=0 ( p, ℓ )=1^ℓ -1  e^-2 π i/ℓ   p '  H_b  ( M^-1_ℓ , p)_ab∑_r=0^ℓ̂/gcd(ℓ̂,ℓ)-1∑_k=0^ℓ-1 e^-2 π i/ℓ ( p- ℓ/2) (k+ ℓ r) 
           = ∑_ p=0 ( p, ℓ )=1^ℓ -1  e^-2 π i/ℓ   p '  H_b  ( M^-1_ℓ , p)_ab∑_r=0^ℓ̂/gcd(ℓ̂,ℓ)-1  e^-2 π i  ( p- ℓ/2)  rℓ δ_p,ℓ/2 ,
    
    where in the second line we have written w=k+ℓ r and converted the sum over w into the sums over k and r. Clearly, the only non-vanishing term is for ℓ=2 and p=1, so that
    
        ∑_w=0^1Q̂_ab^(2,w) = 2 i^1-D/2  (T^1/2 S T^2 S T^1/2 )_ab = 2 i^1-D/2  P_ab ,
    
    and we recognise the modular transformation P which connects the dual amplitudes ℳ and ℳ, describing the orientifold projection of the open-string spectrum and the propagation of closed-string states between a boundary and a cross-cap, respectively. 
    
    
     With these considerations, we can now state the main result of this Section, which extends misaligned supersymmetry of oriented closed strings to the case of orientifold constructions. In fact, eq. (<ref>)  implies
    
        ⟨ d(n) ⟩ (𝒵_I  )  = i^1-D/2 ∑_a=0^M-1∑_b   |  H_b <0 Z_I^a  S_ab  f_b (1 , n ) =   ∑_b  |  H_b <0 Z_I^b    f_b(1, n)  ,
    
    for the Klein bottle and annulus amplitudes, I=1,2, while eq. (<ref>) implies that
    
        ⟨ d(n) ⟩ (ℳ )  =2 i^1-D/2 ∑_a=0^M-1∑_b   |  H_b <0 Z^a_3   P_ab  f_b (2 , n ) = 2  ∑_b  |  H_b <0 Z_3^b    f_b(2, n)  ,
    
    for the Möbius strip amplitude. In both cases, the sector-averaged sum associated to the loop-channel amplitudes grows exponentially if and only if a closed-string tachyons propagates in the tree-level channel. Using the explicit expression (<ref>), one can write
    
        ⟨ d(n) ⟩ (𝒵_I  ) =  √(ℓ_I 2)∑_b  |  H_b <0 Z_I^b    d_b(0)    | H_b  |^(d-1)/4/n^(d+1)/4  e^4π/ℓ_I √( | H_b  |   n) ,
    
    with ℓ_I =1 (ℓ_I =2) for I= 1, 2 (I=3). Similarly, the sector-averaged sum associated to the tree-level amplitudes reads
    
        ⟨ d(n) ⟩  ( 𝒵_I  )     = ℓ_I  ∑_a  |  H_a <0 Z_I^a  f_a(ℓ_I,n)
        		
           =  √(ℓ_I2)∑_a  |  H_a <0 Z_I^a    d_a(0)    | H_a  |^(d-1)/4/n^(d+1)/4  e^4π/ℓ_I √( | H_a  |   n) ,
    
    and grows exponentially if and only if a tachyonic character is present in the associated loop-channel amplitudes. If no tachyons are present in the tree-level (loop) channel, the sector-averaged sum ⟨ d(n) ⟩ (𝒵_I  ) (⟨ d(n) ⟩  ( 𝒵_I  )) vanishes identically. This means that ⟨ d(n) ⟩ (𝒵_I) carries information about the coupling of closed-string tachyons to D-branes or orientifold planes, while ⟨ d(n) ⟩  ( 𝒵_I ) encodes the presence of tachyonic characters in the one-loop partition functions. Therefore the latter must be considered for the study of the classical stability of the vacuum. This result clarifies the physical meaning of the sector-averaged sum for orientifold vacua and thus allows to extend the notion of misaligned supersymmetry to this context, matching our expectations. 
    
     Notice that such considerations only provide either a necessary condition for the absence of closed-string tachyons or a sufficient criterion for the stability of the vacuum. Indeed, closed-string tachyons could be present in the spectrum even though their coupling to D-branes or O-planes vanishes, which implies that ⟨ d(n) ⟩ (𝒵_I  ) =0 cannot guarantee the classical stability of the vacuum. Furthermore, albeit the vanishing of the sector-averaged sum of the parent closed oriented string and of the ⟨ d(n) ⟩  ( 𝒵_I  )s automatically guarantees the classical stability of the orientifold vacua, it is not true in general that a ⟨ d(n) ⟩  ( 𝒵_I  ) ≠ 0 implies that physical tachyons are present in the spectrum. In fact, tachyons could be projected away by Ω, and when this happens each individual sector-averaged sum can experience a non-vanishing exponential growth. Although one would naïvely expect that upon summing the contributions from all amplitudes 𝒯 and 𝒵_I this exponential growth should disappear, this cannot occur. In fact, the sector-averaged sum (<ref>) associated to the torus amplitude contains infinite contributions with ℓ≥ 1, which clearly cannot be cancelled by ⟨ d(n) ⟩ ( 𝒦), which has only the leading term, ℓ=1.
    
     A similar story goes on for the open-string sector. Because of the properties (<ref>) and (<ref>) of the generalised Kloosterman sums, only one term contributes to the sector-averaged sum, and has ℓ=1 for the transverse annulus but ℓ =2 for the transverse Möbius strip amplitude, which yields different growth rates so that they can never cancel. 
    
     This suggests that the sector-averaged sums associated with different Riemann surfaces cannot be directly compared. This is not surprising since they compute different properties of the vacuum. On the one hand, ⟨ d(n) ⟩ (𝒯) controls the asymptotic growth of physical degrees of freedom and therefore it is tied to the spectrum of oriented closed strings. On the other hand, the quantities ⟨ d(n) ⟩ (𝒵_I) carry no global information on the spectrum, but rather to the one-point couplings of closed-string states to D-branes or O-planes. 
    
      Therefore, we can only conclude that it is enough for the classical stability of the vacuum to satisfy ⟨ d(n) ⟩ (𝒯)=0 and ⟨ d(n) ⟩ (𝒵_I)=0 . Actually, because the mere role of the Klein bottle and Möbius strip amplitudes is to enforce the orientifold projection on 𝒯 and 𝒜, the simultaneous vanishing of the sector-averaged sums associated to the torus and to the tree-level channel annulus amplitudes is a sufficient condition for the classical stability of the orientifold vacuum. On the contrary, no information can be drawn from the vanishing of ⟨ d(n) ⟩ ( 𝒦) and ⟨ d(n) ⟩ ( ℳ), since tachyons could be associated to oriented strings. 
    
     All in all, these considerations show that the sector-averaged sums have nothing to say for vacua in which classical stability is a consequence of a non-trivial action of the orientifold projection. Some concrete examples will be now provided through which the general discussions of the previous Sections can be shown at work.
    
    
    
    § AN EXAMPLE IN TEN DIMENSIONS
    
    
    A simple arena where to test our results is given by ten-dimensional string vacua with no space-time supersymmetry. Indeed, in ten dimensions the number of consistent constructions realised in string theory is finite 
    <cit.> and thus a systematic analysis can be carried out. In all these cases, the spectrum is organised according to the conjugacy classes of the ten-dimensional little group SO(8) combined with themselves or with characters describing the representations of the gauge group for the heterotic string. Furthermore, such characters are dressed with the Dedekind η functions from the non-compact bosons  <cit.> giving rise to irrational CFT. Still, following <cit.>, we can overcome this problem by defining the pseudo-characters 
    
        (O_2n , V_2n , S_2n , C_2n ) →(O_2n/η^8, V_2n/η^8 , S_2n/η^8 , C_2n/η^8) ,
    
    and including suitable phases in the modular transformations 
    
        T   :     (O_2n , V_2n , S_2n , C_2n ) → e^- i π (n+8)/12   (O_2n , - V_2n , e^i π n/4  S_2n , e^i π n/4  C_2n )  ,
    
    and 
    
        S :  [ O_2n; V_2n; S_2n; C_2n ]→τ^-4 1/2[      1      1      1      1;      1      1     -1     -1;      1     -1   i^-n - i^-n;      1     -1 - i^-n   i^-n ] [ O_2n; V_2n; S_2n; C_2n ] .
     All these theories can be studied at once by noticing that their partition functions can be compactly written as
    
        𝒯_A = ∑_a,b=0^3 χ̅^A_a 𝒩_ab χ_b  ,
    
    where χ = (O_8 , V_8 , S_8 , C_8) denotes the left-moving characters, which are common to all ten-dimen­sional non-supersymmetric theories, and χ̅_a^A denotes the right-moving characters, which depend on the specific model A. Thus, once the characters are identified, the partition functions are determined by the choice of the GSO matrices that read
    
        𝒩_het = [  0  1  0  0;  1  0  0  0;  0  0 -1  0;  0  0  0 -1 ] ,
    
    for the heterotic theories, while 
    
        𝒩_0A = [ 1 0 0 0; 0 1 0 0; 0 0 0 1; 0 0 1 0 ] ,
        	    𝒩_0B = [ 1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ] ,
    
    for the type 0A and type 0B theories and 
    
        𝒩_IIA = [  0  0  0  0;  0  1  0 -1;  0 -1  0  1;  0  0  0  0 ] ,
        	    𝒩_IIB = [  0  0  0  0;  0  1 -1  0;  0 -1  1  0;  0  0  0  0 ] ,
    
    for the type IIA and type IIB theories.  
    
     The type IIB, 0A and 0B superstring vacua are left-right symmetric on the world-sheet and thus can be modded out by the world-sheet parity Ω<cit.>, possibly combined with extra symmetries. 
    Following the rules of the orientifold construction (see, for instance, <cit.>), the Klein bottle projection for the type IIB and type 0A superstrings is unique <cit.>, while four independent choices are allowed for the type 0B <cit.>.
    
     The analysis for the ten-dimensional heterotic models has been carried out in great detail in <cit.> reproducing the results of <cit.> for the SO(16) ×SO(16) model. The vacua with 𝒩=(1,1) world-sheet supersymmetry and their related orientifolds have been discussed in <cit.> reproducing the results of <cit.> for the type 0'B superstring. The analysis for the type 0A superstring in <cit.> is somewhat quick and schematic, and thus we take advantage of this opportunity to provide a more detailed discussion.
    
     The type 0A superstring  <cit.> combines the left and right moving SO(8) characters via the GSO matrix (<ref>). In this case, the spectrum is purely bosonic, and thus the tachyon related to the identity of the algebra is present. In addition, the massless states comprise the dilaton, the metric, the Kalb-Ramond field, two RR 1-forms and two RR 3-form fields.
    
     Compared to the type 0B case, the only changes occur for the bosonic content in the RR sector, and thus play no role in the computation of the sector-averaged sum associated with the torus amplitude with the one reported in <cit.> for the type 0B superstring,
    
        ⟨ d(n) ⟩ ( 𝒯 ) = ∑_ℓ=1^∞φ(ℓ) e^8 π/ℓ√(n/2)/(2n)^11/2 ,
     
    where the exponential growth is dictated by the NS vacuum associated to the tachyon with mass H_0=-1/2.
    
     As already mentioned, the type 0A superstring is left-right symmetric, but admits only one type of orientifold projection <cit.>, resulting in the Klein bottle amplitude 
    
        𝒦= O_8 + V_8   .
         
    Under the action of the S modular transformation, the tree-level propagation of unoriented closed-string states is encoded in the transverse channel 
    
        𝒦̃= 2^5 { O_8 + V_8 }  .
    
    The structure of the amplitude is not affected when moving from the one-loop and tree-level channel and  vice versa. Furthermore, in both cases, there is a tachyonic character, which shows that the closed-string tachyon is kept by the orientifold projection[It eliminates the Kalb-Ramond field and one copy of the RR 1-form and 3-form.] and couples to orientifold planes. 
    Although there are only NS NS tadpoles, and thus the introduction of the open sector is not mandatory, we shall be a bit more general and include D-branes into the game to exploit the properties of the sector-averaged sums associated to each amplitude. Since in the transverse channel only the O_8 and V_8 characters may appear, the only available choice for the transverse annulus amplitude is given by
    
        𝒜̃= 2^-5{ (n_B-n_F)^2 O_8 +  (n_B+n_F)^2 V_8}  ,
     
    where we have introduced the stack of branes n_B and n_F, leading in the direct channel to 
    
        𝒜= (n_B^2+n_F^2) (O_8 + V_8)   -   2   n_B   n_F (S_8 + C_8)   .
      
    Aside for an overall sign ε, we can compute the Möbius strip amplitude in the transverse channel as 
    
        ℳ̃= 2 ε{ (n_B-n_F) Ô_8 +  (n_B+n_F) V̂_8}  ,
     
    that after a P transformation gives 
    
        ℳ= - ε{ (n_B-n_F) Ô_8 -  (n_B+n_F) V̂_8}  .
     
    Therefore we can read the massless open-string spectrum according to the choice of ε. If ε=1, the open sector provides a gauge boson in the adjoint of USp(n_B) ×USp(n_F) gauge group, with a tachyon transforming into the (
    
    ,1)+(1,&
    ) representations and left and right fermion transforming in the bi-fundamental representation. If ε=-1 the gauge group becomes a product of orthogonal ones SO(n_B) ×SO(n_F) with a tachyon in  (&
    ,1)+(1,
    
    ) and still with fermions transforming in the bi-fundamental representations.  
    
     The sector-averaged sums associated to the transverse amplitudes come straightforwardly 
    
        [                     ⟨ d(n) ⟩ (𝒦 )= e^4 π√(n/2)/  (2 n)^11/4  ,;                                                               ;       ⟨ d(n) ⟩ (𝒜 )= e^4 π√(n/2)/(2n)^11/4 ( n_B -n_F  )^2   ,;                                                               ; ⟨ d(n) ⟩ (ℳ )= - εe^2 π√(n/2)/  2^9/4 n^11/4 ( n_B -n_F  )   , ]
    
    reflecting the coupling of the closed-string tachyon to boundaries and cross-caps. Since all the terms are non-vanishing, this is enough to conclude that the closed-string tachyon survives the orientifold projection. Indeed, this can be seen from the one-loop Klein bottle amplitude and is reflected into the computation of the sector-averaged sum associated to the transverse channel
    
        ⟨ d(n) ⟩  ( 𝒦 ) =e^4 π√(n/2)/ (2 n)^11/4 .
    
    The open-string tachyon is also unoriented, implying non-vanishing contributions to the sector-averaged sums of both transverse annulus and Möbius strip amplitudes  
    
        [ ⟨ d(n) ⟩ (𝒜  )= e^4 π√(n/2)/(2 n)^11/4 ( n_B^2 + n_F^2  )   ,;                                                              ;  ⟨ d(n) ⟩ (ℳ  )= εe^2 π√(n/2)/ 2^9/4 n^11/4 ( n_B -n_F  )   . ]
    
    Choosing ε=-1 and the Chan-Paton factors to be n_B=1 and n_F=0, the tachyon is projected away from the open spectrum. Note, however, that there is no way to provide a mutual cancellation between the associated sector-averaged sums, since the powers and the coefficients characterising the expressions in (<ref>) are different.	
    
    
    
    § AN EXAMPLE IN D=9
    
    
    Aside from the ten-dimensional examples described so far, there are other ways to build non-supersymmetric string vacua. A notable option is breaking supersymmetry also via continuous deformations, employing the Scherk-Schwarz mechanism <cit.>, or its T-dual version: M-theory breaking <cit.>. Focusing on the simplest scenario with the internal manifold described by a circle S^1(R), one can realise both constructions as freely acting orbifolds <cit.>, where the supersymmetry breaking generator g is combined with a suitable shift δ along the compact direction. Explicitly, this means that g = (-1)^F, with F the space-time fermion number, while δ acts as δ: y → y+π R for the Scherk-Schwarz case and δ:  y_L,R→ y_L,R±α'/2 Rπ for the M-theory breaking. The latter action has a natural interpretation in terms of the T-dual variable ỹ = y_L-y_R implying that the two oriented closed-string vacua are equivalent. However, new physics emerges in the orientifold case, since T-duality changes the dimensionality of the orientifold planes and D-branes, as well as the Ω projection, providing for the M-theory breaking construction an additional projection which eliminates the tachyon <cit.>.  Therefore, we shall focus on the latter[For a parallel discussion of misaligned supersymmetry for the Scherk-Schwarz and M-theory breaking compactification see <cit.>.] by first discussing the role of misaligned supersymmetry for the oriented spectrum and then for the two orientifold projections. 
    
     Let us start from the torus partition function obtained from a freely acting orbifold of the type IIB superstring generated by (-1)^Fδ
        𝒯_M   = 12 |V_8 - S_8|^2  ∑_m,nΛ_m,n (R) + 12 |V_8 + S_8 |^2  ∑_m,n (-1)^n  Λ_m,n (R)
        		
            + 12 |O_8 - C_8|^2  ∑_m,nΛ_m+1/2,n (R) + 12 |O_8 + C_8|^2  ∑_m,n (-1)^n Λ_m+1/2,n (R)   ,
    
    where the action of δ on the winding modes and its modular completion are shown. The Kaluza-Klein momenta and windings associated to the compact direction contribute with the standard Narain lattice
    
        Λ_m,n = q^α'/4( m/R + nR/α ')^2/η q̅^α'/4( m/R - nR/α ')^2/η̅ .
    
    In the R→ 0 limit, the orbifold action is trivialised and one recovers the supersymmetric  IIB theory, while for generic values of the radius R supersymmetry is spontaneously broken and the gravitini acquire a mass m≃ R. The excitations of the NS-NS vacuum in | O_8|^2 survive the GSO projection in the twisted sector, and the lightest state has mass
    
        m^2_|O_8|^2 =-1/2 α' +1 /4( 1/2 R)^2 .
    
    This scalar is then massive for small values of R, but becomes tachyonic above the critical radius R_c = √(α') / 2 √(2). As the radius increases, more and more states become tachyonic, and it is clear that these models represent an ideal ground to study the realisation of  misaligned supersymmetry in string theory[See for instance <cit.> for a first study of  misaligned supersymmetry in the context of the T-dual Scherk-Schwarz reductions.]. 
    
     To illustrate the analysis of Section <ref> on the degeneracies of states, we need to select rational values for R^2/α ' = s/t ∈ℚ since, in this case, the Narain lattice reduces to a RCFT 
    
        ∑_m,nΛ_m,n (R) →∑_a=0^2st-1λ_a  λ̅_a l ,
    
    with the 2st characters defined as
    
        λ_a (q) = ∑_m q^st ( m + a/2st)^2/η (q) .
    
    Note that λ_0 and λ_st are real, while λ_a and λ_2st-a, a=1,… , st-1,  form conjugate pairs. 
    The λs have conformal weight h_a = a^2/4st, and thus H_a=a^2/4st-1/24, with conjugate pairs carrying the same weight.  Notice that H_a <0 for a < √(2s) and p=0, so that the associated characters can describe tachyonic states. In eq. (<ref>) the anti-holomorphic characters have index a l, where l = rt + s v, with the integers r and v satisfying the relation 
    
        rt - vs =1   ,
    
    and the label a l is defined modulo 2st. 
    As shown in <cit.>, these characters are eigenstates of the shift operator δ only for even t, but must be broken into sub-characters for odd t. For simplicity, here we shall restrict the discussion  to the even-t case, where
    
        δ :  λ_a → (-1)^a/2s λ_a ,
    
    and we  also take s=1, so that the condition (<ref>) can be easily solved by r=0 and v=-1, for any t[Other choices for s and t yield equivalent results.]. In this setting, the action of the modular group on these characters is encoded in the T and S matrices
    
        T_a b = e^i π( a^2/2t-1/12) δ_a b ,    
        	S_a b = e^2 π i a b/2t/√(2t) .
     
    The new characters that enter the partition function are thus <cit.>
        {χ_α}_α=0^8t-1 = (O_8 , V_8 ,  S_8 ,  C_8 ) ⊗{λ_a }_a=0^2t-1 .
    
    where the χ_α have shifted conformal weights  H_2tp+a = a^2/4t - 1/2δ_p,0, with p=0,… ,3. In terms of the characters (<ref>), the torus amplitude associated to the (-1)^F  δ orbifold of the type IIB superstring reads
    
        𝒯_M   = ∑_a=0^1  (  | χ_2t+at |^2 +  | χ_4t+at |^2  )
        		
           + ∑_b=1^t/2-1(  χ_2t+2bχ̅_2t-2b+ χ_4t+2bχ̅_4t-2b.
        		
                       .
        		+ χ_2t+t+2b χ̅_2t-t-2b + χ_4t+t+2b χ̅_4t-t-2b)
        		
           -∑_b=0^t/2-1(  χ_2t+2b+1χ̅_4t-2b-1+ χ_4t+2b+1χ̅_2t-2b-1.
        		
                       .
        		+ χ_2t+t+2b+1 χ̅_4t-t-2b-1 + χ_4t+t+2b+1 χ̅_2t-t-2b-1)
        		
           +  ∑_a=0^1 (  | χ_2a+1/2 t |^2 +  | χ_6t+2a+1/2t |^2  )
        		
           + ∑_b=1^t/2-1(  χ_t/2-2bχ̅_t/2+2b+ χ_6t+t/2-2bχ̅_6t+t/2+2b.
        		
                       .
        		+ χ_3t/2-2b χ̅_-t/2+2b + χ_6t+3t/2-2b χ̅_6t-t/2+2b)
        		
           - ∑_b=0^t/2-1(  χ_t/2-2b-1χ̅_t/2+2b+1+ χ_6t+t/2-2b-1χ̅_6t+t/2+2b+1.
        		
                       .
        		+ χ_3t/2-2b-1 χ̅_-t/2+2b+1 + χ_6t+3t/2-2b-1 χ̅_6t-t/2+2b+1)   .
     Given the expression for the torus partition function, we can proceed by computing the sector-averaged sum, which reads, for t < 8,   
    
        ⟨ d(n) ⟩ ( 𝒯)=  (8-t/16 )^4 1/ n^5 ∑_ℓ=1^∞ φ (ℓ)   e^2π/ℓ√((8-t)  n) ,
     
    while it vanishes for t ≥ 8, as expected. The Scherk-Schwarz reduction at this level can be obtained via a T-duality operation, in which the radius of compactification is sent to R →α'/R. This means that the sector-averaged sum for this vacuum is related to (<ref>) by exchanging s ↔ t. We can therefore define the effective central charge in terms of the radius R
        C_eff = 
        		2 π√(8-α'/R^2 )   for   R^2 > α'1/8 ,
        		
        
        		0    for    R^2 ≤α'1/8  ,
     
    where the limits R → 0 and R →∞ recover the results for the type IIB and type 0B superstrings respectively, as shown in figure <ref> once the radius is continued to real values. Recalling that the sector-averaged sum is tied to the number of degrees of freedom, the behaviour of (<ref>) suggests that the M-theory breaking compactification does not correspond to a continuous deformation which, as such, should imply that the number of degrees of freedom does not change with the radius R. It would be interesting to make this observation more quantitative, by deforming the partition function whose free energy can be shown to depend quadratically on C_eff<cit.>. This would characterise the phase transition as a first order and it would be interesting to look for a holographic dual in AdS_3 for which, in the large central charge limit, the Hagedorn-like phase transitions of the CFT are mapped to Hawking-Page transitions of the gravity theory, dominated by the entropy of BTZ black holes <cit.>. However, such a study would require a clearer picture on the meaning of the averaging procedure in this context, and lies beyond the scope of this thesis. 
    
     The amplitude described in (<ref>) is invariant under world-sheet parity, and can be orientifolded by adding the Klein bottle amplitude that, for the standard choice of Ω, reads
    
        𝒦_M   =  ∑_a=0^1(  χ_a t+ 2t - χ_at+4t)
        		+ ∑_a=0^1( χ_2a+1/2 t  - χ_2a+1/2t+ 6 t )  .
    
    Notice that aside from the first two terms associated to the V_8 and S_8 characters dressed with the KK and winding excitations, two extra contributions appear associated to O_8 and C_8.  In the transverse channel
    
        𝒦_M=  2^5 2/√(t)∑_b=0^t/2-1{χ_2t+4b- χ_4t+4b+2} ,
    
    so that this vacuum involves pairs of O9_- and O9_- planes which do not carry a net RR charge, as can be seen from the fact that the characters χ_4t+4b+2= S_8  λ_4b+2 are massive, m^2∝(4b+2)^2/4t = 1/4 R^2  (4 b+2)^2. In the formal limit R→ 0 they become massless, the O9_- planes decouple and a net RR tadpole emerges from (<ref>). Taking this limiting case into account,  we shall also impose the RR tadpoles associated to the χ_4t+4b+2 characters.
    
     Following <cit.>, we add (n_1,n_2) stacks of branes and (n_3,n_4) stacks of antibranes so that
    
        𝒜_M= 2^-52/√(t)∑_b=0^t/2-1   { ( n_1 + n_2 + n_3 + n_4  )^2  χ_2t + 4 b  +  ( n_1 - n_2 + n_3 - n_4  )^2 χ_2t+ 2 + 4b .  
        		
           .  -   ( n_1 + n_2 - n_3 - n_4  )^2 χ_4t +4b-  ( n_1 - n_2 - n_3 + n_4  )^2  χ_4t+4b+2} ,
    
    and
    
        ℳ_M= - 2 2/√(t)∑_b=0^t/2-1   { ( n_1 + n_2 + n_3 + n_4  )  χ̂_2t + 4 b   -    ( n_1 - n_2 - n_3 + n_4  ) χ̂_4t+4b+2}.
     The tadpole conditions 
    
        n_1 + n_2 + n_3 + n_4 = 32 ,      n_1 + n_2 - n_3 - n_4 = 0 ,
    
    associated to the massless characters χ_2t and χ_4t, together with the “massive” tadpole n_1 - n_2 - n_3 + n_4 = 32 for χ_4t+4b+2, admit the unique solution
    
        n_1 = n_4 =16 ,      n_2 = n_3 =0 .
    
    The open spectrum can be then deduced by employing the S and P modular transformations, yielding 
    
        𝒜_M= ∑_a=0^1{    ( n_1^2+n_2^2 + n_3^2+n_4^2 )   (   χ_2t + 2 a /2 t  -χ_4t + 2 a /2 t )     
        		
            + 2  ( n_1 n_2 + n_3 n_4  )   (   χ_2t + 2 a +1/2 t  -χ_4t + 2 a +1 /2 t )     
        		
            + 2  ( n_1 n_3 + n_2 n_4  )   (   χ_2 a /2 t  -χ_6t + 2 a  /2 t )     
        		
            +  2  ( n_2 n_3 + n_1 n_4  )   (  χ_2 a+1 /2 t  - χ_6t+2 a+1 /2 t ) } ,
    
    and 
    
        ℳ_M= -  ∑_a=0^1   { ( n_1 + n_2 + n_3 + n_4 )  χ̂_2t+2 a /2 t  -   ( n_1 - n_2 - n_3 + n_4 ) (-1)^a χ̂_4t+2 a /2 t} .
     At low energy, this M-theory breaking orientifold describes a graviton, the dilaton and a RR two-form from the closed-string sector together with an SO(16) ×SO(16) gauge group coupled to a left-handed fermion in the adjoint representation. For large values of the compactification radius,  i.e. for t<8 a real closed-string tachyon and an open-string one in the (16,16) representation emerge. 
    
     The amplitudes can be shown to be compatible with those of <cit.> once the value R^2 = α '/t is chosen, and the momentum and winding sums are written in terms of the λ characters. 
    
     We can now study the classical stability for this M-theory breaking vacuum by computing the associated sector-averaged sums. An inspection of the direct channel amplitudes gives ⟨ d(n) ⟩ ( 𝒵_I  )=0, reflecting the absence of closed-string tachyons that freely propagate between D-branes or O-planes. This, however, does not guarantee the classical stability of the whole construction, since a tachyon can be present in 𝒯_M for special values of t. A completely different story regards the sector-averaged sum associated with the transverse channel. 
    In fact, from 𝒦 one finds
    
        ⟨ d(n) ⟩  ( 𝒦_M  )= 2 1/√(2) n^5/2 ( 8-t/16 )^2 e^π√((8-t)  n) ,
    
    for t<8, while it vanishes if t ≥ 8. This behaviour is consistent with the fact that the closed-string spectrum is tachyon-free for t ≥ 8, while a tachyon is present when t<8.
    
     The sector-averaged sum from the transverse annulus amplitude has a similar behaviour
    
        ⟨ d(n) ⟩ ( 𝒜_M)=  2 1/√(2) n^5/2 2 n_1 n_4   ( 8-t/16 )^2 e^π√((8-t)  n)  ,
    
    reflecting the presence of a tachyon in the bi-fundamental representation (16, 16) when t<8. When t ≥ 8, ⟨ d(n) ⟩ ( 𝒜_M)=0 since all would-be tachyons are actually massive. Finally, the sector-averaged sum of the transverse Moebius strip amplitude vanishes identically, consistently with the fact that the open-string tachyon, when present, is oriented and valued in the bi-fundamental representations of SO(16) ×SO(16). Therefore, in this case, the vanishing of the sector-averaged sum ⟨ d (n) ⟩ (𝒜̃_M) is a necessary and sufficient condition for the classical stability of the open-string spectrum, while for the stability of the closed-string sector we must require that both ⟨ d (n)⟩ (𝒯_M) and  ⟨ d (n)⟩ (𝒦̃_M) vanish. 
    
     As discussed at the beginning of this Section, the Klein bottle of eq. (<ref>) is not the only orientifold projection compatible with (<ref>). Indeed, in <cit.> it was shown that other options for Ω are possible, and in particular one can choose to project away the closed-string tachyon by using 
    
        𝒦'_M =  ∑_a=0^1(  χ_a t+ 2t - χ_at+4t)
        	- ∑_a=0^1( χ_2a+1/2 t  - χ_2a+1/2t+ 6 t )   .
    
    This change of sign affects the geometry of the orientifold planes, since now
    
        𝒦'_M=  2^5 2/√(t)∑_b=0^t-1{χ_2t+4b+2- χ_4t+4b} ,
    
    which reveals the presence of O9_- and O9_+ planes which have a net RR charge but vanishing tension. The annulus amplitude is still given by eqs. (<ref>) and (<ref>), while the tree-level channel Möbius strip amplitude reads
    
        ℳ'_M= - 2 2/√(t)∑_b=0^t-1   { ( n_1 - n_2 + n_3 - n_4  )  χ̂_2t + 4 b+2   -    ( n_1 + n_2 - n_3 - n_4  ) χ̂_4t+4b}.
      
    The RR tadpole conditions
    
        n_1 + n_2 - n_3 - n_4 =32  ,      n_1 - n_2 - n_3 + n_4 =0
    
    are clearly incompatible with the NS-NS tadpole n_1 + n_2 + n_3 + n_4 =0 since the O-planes are tensionless. Upon a P modular transformation, we get  
    
        ℳ'_M= -  ∑_a=0^1   { ( n_1 - n_2 + n_3 - n_4  ) (-1)^a χ̂_2t+2 a /2 t -  ( n_1 + n_2 - n_3 - n_4  ) χ̂_4t+2 a /2 t} .
     The light spectrum from the closed strings comprises a graviton, the dilaton and a RR two-form, and no extra light states emerge at special values of the compactification radius, since the would-be tachyon is removed by the new orientifold projection. From (<ref>) and (<ref>) we read the gauge group SO(n_1) ×USp(n_2) ×SO(n_3) ×USp(n_4). Left-handed fermions and tachyons transform in the representations (
    
    ,1;1,1)+(1,
    
    ;1,1)+(1,1;&
    ,1)+(1,1;1,&
    ) and (
    ,1;
    ,1) + (1,
    ;1,
    ), respectively. When t<8 additional tachyons in the representations (
    ,1;1,
    ) + (1,
    ;
    ,1) appear. All open-string ta­chyons can be eliminated by taking the minimal solution n_1=n_2=16 and n_3=n_4=0 of the RR tadpoles, but we shall focus on the more general set-up allowing the simultaneous presence of branes and anti-branes to fully analyse the behaviour of the sector-averaged sums. 
    It is possible to show that, choosing the value R^2 = α '/t in the model of <cit.> one obtains the amplitudes presented in this section.
    
    
     As in the case of the standard M-theory breaking, ⟨ d(n) ⟩ (𝒵'_I) ≡ 0 for every loop amplitude, since no closed-string tachyon propagates between D-branes or O-planes. The sector-averaged sums associated with the tree-level channels are, instead, slightly modified and read 
    
        ⟨ d(n) ⟩ ( 𝒦'_M )= - 2 1/√(2) n^5/2 ( 8-t/16 )^2 e^π√((8-t)  n) ,
    
    and 
    
        ⟨ d(n) ⟩ ( 𝒜'_M )=   1/√(2) n^5/2   { 2 (  n_1 n_4 + n_2 n_3  )   2   ( 8-t/16 )^2 e^π√( (8-t)   n).
         
           . + 2 (  n_1 n_3 + n_2 n_4  ) 1/4   e^4 π√(n/2)},
    
    while ⟨ d(n) ⟩ ( ℳ'_M)=  0 since, also in this case, open-string tachyons are oriented. Even though the only difference with the standard M-theory breaking analysis is in the minus sign in eq. (<ref>), the physical interpretation is  deeply different. It tells us that the tachyon that appears in the spectrum when t < 8 is actually projected out. However, eq. (<ref>) fails to cancel the sector-averaged sum  (<ref>) of the torus because of the appearance of the sub-leading contributions, ℓ >1.
    
     To conclude,  eq. (<ref>) reflects the presence of tachyons in the spectrum, as long as anti-branes are present. Also in this case, since all tachyons transform in bi-fundamental representations, a necessary and sufficient condition for the stability of the open-string sector is the vanishing of  ⟨ d(n) ⟩ ( 𝒜'_M ). 
    
    
    
    
    
    CHAPTER: RIGIDITY: NEW SOLUTIONS FOR BSB AND SUSY K3 ORIENTIFOLDS
    
    
    § ORIENTIFOLD VACUA IN SIX DIMENSIONS
    
    
    Orientifold vacua in six dimensions have received much attention, since their dynamics is highly constrained by anomaly cancellation, and thus are an ideal playground for model building and for studying supersymmetry breaking. In particular, we will focus on orientifolds obtained from the orbifold limits of K3 manifolds, T^4/ℤ_N, with N=2,3,4,6<cit.>, on which the nonlinear sigma model is exactly solvable, and thus we can use an exact world-sheet description to investigate these vacua. In particular, in D=6 the standard world-sheet parity operator can be dressed with the orbifold action with fixed loci given by orientifold planes with different dimensionalities, namely O9_- and O5_-. One must thus introduce D9 and D5 branes to avoid local anomalies <cit.>. This configuration preserves 𝒩=(1,0) supersymmetry, and the spectrum is organised in suitable super multiplets. However, this choice of the orientifold action is also compatible with the presence of O9_+ and O5_+ planes requiring D9 and D5 branes, which thus provide a compactification of the Sugimoto model, where, as in the ten-dimensional case, supersymmetry is broken in the full open-string sector. Actually, this is not the only way to break supersymmetry, since it is possible to dress the orientifold projection with an involution that flips the twisted 2-cycles of K3 which implies the simultaneous presence of orientifold planes with positive and negative tension and charge[The case of a non-vanishing background B-field <cit.> will not be presented in this thesis.]. The possible configurations in this scenario are given by O9_+ and O5_- or, alternatively, O9_- and O5_+ planes, requiring the introduction of D9 and D5 or D9 and D5 branes, respectively. These scenarios are completely equivalent and are known in the literature as  Brane Supersymmetry Breaking (BSB) <cit.> and will be the subject (with its supersymmetric counterpart) of this Chapter[We will focus on the configuration with O9_- and O5_+ planes. The other aforementioned choices are straightforward.]. The original construction <cit.> was based on the T^4/ℤ_2 orbifold, while generalisations to the ℤ_4 and ℤ_6 cases[The BSB version of the T^4/ℤ_3 cannot be realised since there are no O5 planes identified by the orientifold action.] have been recently discussed in <cit.>. These models are particularly interesting, since they do not have tachyonic instabilities and admit a tree-level supersymmetric closed-string sector coupled to non-supersymmetric open strings. 
    The consistency of the vacuum is guaranteed since supersymmetry is, actually, non-linearly realised <cit.>, with the massless goldstino in the singlet of the gauge group playing the role of the Volkov-Akulov field <cit.>. Its presence, together with the non-vanishing dilaton tadpole, is hence crucial in writing down a consistent interaction term for the massless gravitino, which is coupled to a conserved current. In this setting supersymmetry is broken in the open-string sector, but such a breaking has to be thought of as occurring at the string scale, since no order parameter is present which may restore the symmetry.  
    
     In <cit.>, the simplest realisation of such class of vacua was presented, where, for instance, the cancellation of R-R tadpoles was achieved by placing D5 branes on a single ℤ_2 fixed point. However, this is not the only option at our disposal and one can still choose to place D5 branes on all the fixed points. In this case, the charges of D9 branes can be cancelled against those of D5 branes, following an observation made in <cit.> in a different setting. Such a mechanism may lead to non-trivial solutions, since the D9 and D5 branes involved are fractional as one can see from the particle interpretation of the vacuum-to-vacuum orientifold amplitudes.    
    On the other hand, this way of cancelling twisted charges may leave behind additional uncancelled NS-NS tadpoles, thus providing new contributions to the scalar potential. Therefore in addition to the usual dilaton contribution from the untwisted tadpoles, new terms arise involving blown up moduli, which could be responsible for an eventual spontaneous resolution of orbifold singularities, in case the dynamics forces them to get non-trivial vevs. Furthermore, when BSB orientifolds are built from higher order point group orbifolds, O-planes become fractional, as reflected in the absence of open-string moduli that would allow partial recombination of branes leaving the charge of orientifold planes uncancelled. As explained in <cit.>, this is manifest for the BSB orientifold built upon the T^4/ℤ_4 orbifold, for which the presence of fractional O5_+ planes located on the four ℤ_4 fixed points enforces the introduction of four stacks of D5 branes on the top of them and local charge cancellation forbids D5 branes to be moved away. A similar situation applies for the T^4/ℤ_6 orbifold in which both O9_- and O5_+ planes are fractional but since the ℤ_6 action identifies only one fixed point, such feature is less manifest. Nevertheless, a solution without open-string moduli is also available in this case and will be discussed in the present Chapter.
    
     Although this way of cancelling D-brane twisted charges seems to be specific to BSB vacua, a similar situation applies to supersymmetric vacua as well[For instance, the observation pointed out in <cit.> was made for D=4 supersymmetric ℤ_6 orientifolds.]. Indeed, also for the supersymmetric T^4/ℤ_4 and T^4/ℤ_6 orientifolds D9 and D5 branes are fractional, and one can choose to cancel the R-R charges of D9 branes against those of the D5 ones, once the latter are placed over all the fixed points. As a result, we obtain a finite number of non-trivial solutions to the R-R tadpole conditions that determine different  bona fide string vacua. However, when 𝒩=(1,0) space-time supersymmetry is present, orientifold planes are fractional only for the ℤ_6[Also in the T^4/ℤ_3 orientifold O9_- planes are fractional, but we will not discuss the features of this vacuum.] point group, for which the previous considerations apply. In the rest of this Chapter, we will present a parallel discussion of the BSB and supersymmetric vacua built from the T^4/ℤ_6 orbifold and we will enlighten the features just described. 
    
     For simplicity, we consider a factorised T^4 = T^2 × T^2, with complex coordinates (z_1,z_2). The ℤ_6 group acts as a 60-degree rotation on each T^2, which is then a symmetry for a torus with fixed complex structure U= e^π i /3. Compatibility with supersymmetry requires an equal or opposite rotation on the two T^2s. Our choice is
    
        g:   (z_1 , z_2) →  (e^2π i/6 z_1 , e^-2π i /6 z_2) ,
    
    where g is the generator of ℤ_6. 
    
     In orbifold compactifications, the structure of fixed points plays a crucial role, since it gives the multiplicities of the twisted states and determines the structure of the associated Hilbert spaces. The action of g has a single fixed point with coordinates ζ=(0,0), which is also fixed under the action of g^5. The elements g^2 and g^4 form a ℤ_3 subgroup. Aside from the origin, eight of the nine fixed points arrange in four ℤ_6 doublets, and the associated Hilbert spaces only support ℤ_3 projectors. Similarly, g^3 generates a ℤ_2 subgroup and the fifteen fixed points different from the origin, arrange into five ℤ_6 triplets, with only a ℤ_2 projection.  
    
     This structure of fixed points is reflected in the various terms of the torus partition function, which is well known and is explicitly written in <cit.>. Keeping only the massless states, we can write the partition function in terms of the SO(4)-characters associated to the little group of the six-dimensional Minkowski space, yielding
    
        𝒯_IIB =    | V_4 - 2   S_4  |^2 + 2  | 2   O_4 -  C_4  |^2  
        
           +    | 2   O_4 -  C_4  |^2 +   | 2   O_4 -  C_4  |^2  
        
           + (1+4)[   | 2   O_4 -  C_4  |^2 +   | 2   O_4 -  C_4  |^2   ]
        
           +(1+5)   | 2   O_4 -  C_4  |^2   .
    
    The first line corresponds to the contribution from the untwisted sector, the second one encodes the contribution from the unique ℤ_6 fixed point from the g and g^5-twisted sectors, while in the third the extra multiplicity 4 takes into account the four ℤ_3 doublets form the g^2 and g^4-twisted sectors. Similarly for the last line associated to the g^3-twisted sector, where the 5 encodes the five ℤ_2 triplets. This partition function yields the (unique) massless spectrum with 𝒩=(2,0) supersymmetry, with one gravity multiplet and twenty-one tensor multiplets. 
    
     Starting from this oriented closed-string sector, one can implement the two different orientifold projections giving rise to the supersymmetric and BSB vacua.
    
    
     Although in eq. (<ref>) all the massless characters appear left-right symmetric, when the full massive excitations are considered this parity symmetry only survives for the first contribution in the untwisted sector and the last one associated with the g^3-twisted sector. Indeed, the geometric action of the orbifold combines the g^k and the g^N-k twisted characters, resulting into a charge conjugate modular invariant partition function. This means that the orientifold projection yields the Klein bottle amplitude with massless characters
    
        𝒦^(σ) =  ( V_4- 2   S_4  ) + (-1)^σ (1+5)   ( 2   O_4 - C_4  ) ,
    
    where we have introduced the involution σ, which acts trivially on the g^3-twisted cycles in the supersymmetric case, σ=0, while it flips the parity of these cycles in the BSB scenario σ=1. In the supersymmetric case, this means that the orientifold projection selects a 𝒩=(1,0) hyper multiplet on each ℤ_2 fixed point, while for the BSB case the projection selects a tensor mulitplet. As a result, we have one tensor multiplet and two hyper multiplets arising from the untwisted sector, one tensor multiplet and one hyper multiplet from the g and g^5 twisted sectors, five tensor multiplets and five hyper multiplets from the g^2 and g^4 twisted sectors, while the g^3 twisted sector yields six hyper multiplets in the supersymmetric case or six tensor multiplets for the BSB construction, as a consequence of the non-trivial involution σ. To summarise, the choice σ=0 provides fourteen hyper multiplets and seven tensor multiplets, while σ=1 leads to thirteen tensor multiplets and eight hyper multiplets.
    
     The nature of orientifold planes is encoded in the transverse channel Klein bottle amplitude, which reads
    
        𝒦̃^(σ) =   2^-5/6 (-2^5 √(v) - (-1)^σ2^5/√(v))^2  ( V_4- 2   S_4 ) 
        
            + 2^-5/6 ( - 2^5 √(v) + (-1)^σ2^5/√(v))^2   ( 2   O_4-  C_4 + 2   O_4-  C_4  ) 
        
           + 2^-4/9[ 8·  q_2,t^2 + (q_2,t -(-1)^σ 3 q_2,t )^2 ] 
         ( 2   O_4-  C_4)
         
            + 2^-4/9[ 8· q_4,t^2 + (q_4,t -(-1)^σ 3 q_4,t )^2 ] 
         ( 2   O_4-  C_4)   .
    
    In the first line, we have presented the contribution to the untwisted tadpoles. It  explicitly shows that the choice of the involution determines which orientifold planes are considered: σ=0 reflects the presence of O-planes carrying negative tension and charge O9_- and O5_-, while σ=1 requires orientifold planes with opposite tension and charge, namely O9_- and O5_+. The last two lines encode the contribution from the g^2 and g^4-twisted sectors, and reflect the geometry of the configuration of orientifold planes. Indeed, the O9_- planes fill all space-time and are thus charged under the twisted R-R forms located on all the ℤ_3 fixed points. The O5_+ planes are instead placed at the ℤ_2 fixed points, and thus only the one located at the origin (which corresponds to the unique ℤ_6 fixed point) carries a non-trivial charge. The g^2,4-twisted charges of O9_- planes are given by q_2,t = -(-1)^σ q_4,t= (-1)^σ 8, while the O5_∓ planes are coupled to the six-form through  ∓ 3 q_α,t, following the usual formula <cit.>
        q =  q^9√(# fixed points/# occupied fixed points) .
    
    To cancel irreducible anomalies <cit.>, one must introduce D9 and D5 branes with the appropriate charges required to cancel the R-R tadpoles. In general, the Chan-Paton factors do feel the action of the orbifold group, so that each element of the point group g^α identifies the corresponding brane images, and we label with N_α ( D_α) the action of g^α on D9 (D5 or D5) branes. Furthermore, since the ℤ_6 orbifold has only one fixed point, we introduce a single stack of D5 or D5 branes on it. Consequently, the massless contribution to the transverse channel of the annulus amplitude reads 
    
        𝒜̃^(σ) =    2^-5/6{ V_4 ( N_0 √(v) + D_0/√(v))^2   - 2   S_4 ( N_0 √(v) + (-1)^σD_0/√(v))^2  
           + 4  O_4  ( N_0 √(v) - D_0/√(v))^2   - 2   C_4 ( N_0 √(v) -(-1)^σD_0/√(v))^2  
           + 2^2  ∑_α=1,5[ 2   O_4 (N_α - D_α)^2  - C_4 (N_α - (-1)^σ D_α )^2 ]
        
           + 4/3∑_α =2,4[ (2   O_4 - C_4  )   8  N_α^2    + 2   O_4 (N_α - 3   D_α)^2 - C_4 (N_α -(-1)^σ 3  D_α)^2  ] 
        
            +  [ (2   O_4 - C_4)  15   N_3^2    + 2   O_4 (N_3 - 4  D_3 )^2    - C_4 (N_3 -(-1)^σ 4  D_3 )^2  ] } ,
    
    while the massless terms in the transverse channel of Möbius strip amplitude can be computed as geometric means of 𝒦̃ and 𝒜̃, as required by compatibility
    
        ℳ̃^(σ) = - 1/3   {V̂_4 ( √(v) +(-1)^σ1/√(v)) ( N_0 √(v) + D_0/√(v)) 
        
           + 4  Ô_4( √(v) -(-1)^σ1/√(v))( N_0 √(v) - D_0/√(v)) 
        
           - 2  Ŝ_4 ( √(v) +(-1)^σ1/√(v)) ( N_0 √(v) +(-1)^σD_0/√(v)) 
        
            - 2  Ĉ_4 ( √(v) -(-1)^σ1/√(v))( N_0 √(v) -(-1)^σD_0/√(v)) 
        
            - 2^-3/3∑_α=2,4[ 8   q_α,t  N_α  (2  Ô_4 - Ĉ_4  ) .
        
                      + 2 Ô_4   (N_α - 3  D_α) (1-(-1)^σ 3 ) q_α,t
           .            - Ĉ_4   (N_α - (-1)^σ 3  D_α) (1-(-1)^σ 3) q_α,t]  .
    
    Given all the amplitudes (<ref>), (<ref>) and (<ref>), one can straightforwardly deduce the R-R tadpole conditions which then read
    
        N_0 = 32 ,      D_0 = 32 ,
    
    from the untwisted sector, 
    
        N_1,5 -(-1)^σ D_1,5 = 0 ,
    
    from the g and g^5 twisted sectors,
    
        N_2 =-(-1)^σ N_4 =- (-1)^σ  8 ,       N_2,4 -(-1)^σ 3 D_2,4 = ±(3-(-1)^σ )   8 ,
    
    from the g^2 and g^4 twisted sectors, and, finally,
    
        N_3 -(-1)^σ 4 D_3 =0 ,      N_3 =0 ,
    
    from the g^3 twisted sector.
    
     Note that for σ=0 bosonic and fermionic degrees of freedom in the eqs. (<ref>) and (<ref>) have the same tadpoles, consistently with the presence of supersymmetry in this vacuum, while they are different for the choice σ=1, when supersymmetry is broken. This means that, for the supersymmetric case, satisfying R-R tadpoles necessarily implies the cancellation of the NS-NS ones as well. However, in the BSB case, one is left with an uncancelled tension that contributes to the scalar potential. In general, we will choose to satisfy the twisted R-R tadpole conditions via a mutual cancellation between the twisted charges of D9 and D5 branes, which schematically implies 
    
        V (ϕ , ξ ) = (D_0 + 32)  e^-ϕ + e^-ϕ[ (N_1-D_1)   (ξ_1 + ξ_5 ) + (N_2-3   D_2)    (ξ_2 + ξ_4 ) + O (ξ^2 )] ,
    
    where the field ϕ corresponds to the ten-dimensional dilaton while ξ_α correspond to massless scalars localised on the ℤ_6 and ℤ_3 fixed points. Such scalars are massless and they could play a non-trivial role in the dynamics since a non-vanishing vev would imply a spontaneous blown-up of the orbifold singularity. 
    
     In eq. (<ref>), the first and last terms are always present since they are fixed by the solution to the corresponding R-R tadpoles, D_0+32=64 and N_2-3   D_2=-16, while the second term could be present or not according to our choice to cancel the g(g^5)-twisted R-R tadpoles. 
    
    
    
    
     §.§ The T4/Z6 BSB orientifold
    
    
    In this case σ=1 and the involution acts non-trivially, calling for the presence of D5 branes. Furthermore, the choice of σ also affects the structure of the Chan-Paton labels. In particular, in our setting the Chan-Paton factors are real and one can decompose N_α and D_α into labels that are eigenvalues of the orbifold action
    
        N_β = ∑_γ=0^5 e^2 π i βγ/6  n_γ ,
            
        D_(k) , β = ∑_γ=0^5 e^2 π i βγ/6  d_(k) , γ .
    
    The solution to the tadpole conditions is not unique and reads
    
        n_0 = 8+2a ,    n_1 = n_5 = 4+a ,    n_2 = n_4 = 4-a ,    n_3 =8-2a ,
        
            d_0 = 8-2a ,    d_1 = d_5 = 4-a ,    d_2 = d_4 = 4+a ,    d_3 =8+2a ,
    
    with a=0,…, 4. This parameter controls the way the twisted charges are cancelled between D9 and D5 branes.
    Performing S and P modular transformations on (<ref>) and (<ref>), we can extract the direct channel amplitudes
    
        𝒜_99 =    (n_0^2 + 2 n_1 n_5 + 2 n_2 n_4 + n_3^2 ) ( V_4 - 2   S_4 ) 
        
           + 2 (n_0 n_5 + n_1 n_4 + n_2 n_3 ) (2   O_4 - C_4 ) 
        
           + 2  (n_0 n_1 + n_5 n_2 + n_4 n_3 ) (2   O_4 - C_4 ) ,
    
    and
    
        ℳ_9 = - (n_0 + n_3 ) ( V̂_4 - 2  Ŝ_4 ) ,
    
    for open strings stretched between D9 branes,
    
        𝒜_5̅5̅ =    (d_0^2 + 2 d_1 d_5 + 2 d_2 d_4 + d_3^2 ) ( V_4 - 2   S_4 )  
        
           + 2 (d_0 d_5 + d_1 d_4 + d_2 d_3 ) (2   O_4 - C_4 )  
        
           + 2  (d_0 d_1 + d_5 d_2 + d_4 d_3 ) (2   O_4 - C_4 )  ,
    
    and
    
        ℳ_5̅ = (d_0 + d_3 ) ( V̂_4 + 2  Ŝ_4) ,
    
    for open strings stretched between D5 branes, and
    
        𝒜_95̅ =    2 (n_0 d_0 + n_1 d_5 + n_2 d_4 + n_3 d_3 + n_4 d_2 + n_5 d_1 )(- S_4)
        
           + 2 (n_0 d_5 + n_1 d_4 + n_2 d_3 + n_3 d_2 + n_4 d_1 + n_5 d_0) O_4
        
           +2 (n_0 d_1 + n_5 d_2 + n_4 d_3 + n_3 d_4 + n_2 d_5 + n_1 d_0) O_4  ,
    
    for open strings stretched between D9s and D5s. Supersymmetry is thus broken on the D5 branes, since the gauge bosons and would-be gauginos come in different representations in (𝒜_5̅5̅ + ℳ_5̅)/2, while in 𝒜_95̅ a different GSO projection determines the presence of a left-handed symplectic Majorana-Weyl fermion and two reals scalars in different representations of the gauge group. The Chan-Paton gauge group is then
    
        G_CP =   SO (8+2a ) ×U (4+a) ×U (4-a) ×SO (8-2a ) |_D9
           ×USp (8-2a ) ×U (4-a) ×U (4+a) ×USp (8+2a ) |_D5 .
    
    The charged matter can be readily extracted from the previous partition functions, and comprises vectors in the adjoint representation of G_CP together with a left-handed Majorana-Weyl fermions in the representations
    
        _9_1 + ( 
        ×
        )_9_2 +  (
        ×
        )_9_3 +
        
        
        _9_4  +
        
        _5̅_1 + (
        ×
        )_5̅_2 +  ( 
        ×
        )_5̅_3 + 
        
        _5̅_4 ,
    
    four scalars and a right-handed Majorana-Weyl fermion (i.e. a full hyper multiplet) in the representations
    
        (
        _9_1 , 
        _9_2) +
        (
        _9_2 , 
        _9_3) +
        (
        _9_3 ,  
        _9_4)  +
        (
        _5̅_1 , 
        _5̅_2)+
        (
        _5̅_2 , 
        _5̅_3)+
        (
        _5̅_3 ,  
        _5̅_4)
         ,
    
    a left-handed symplectic Majorana-Weyl fermion in the representations 
    
        (
        _9_1 ,  
        _5̅_1) +
        (
        _9_2 , 
        _5̅_2) +
        (
        _9_3 , 
        _5̅_3) +
        (
        _9_4 , 
        _5̅_4) +
        (
        _9_2 ,  
        _5̅_2) +
        (
        _9_3 ,  
        _5̅_3) ,
    
    and two scalars in the representations
    
        (
        _9_1 ,  
        _5̅_2) +
        (
        _9_2 , 
        _5̅_3) +
        (
        _9_3 ,  
        _5̅_4) +
        (
        _9_4 , 
        _5̅_3) +
        (
        _9_2 ,  
        _5̅_1) +
        (
        _9_3 ,  
        _5̅_2) 
          .
    
    To lighten the notation, the representation _p_i refers to the fundamental of the i-th factor in the gauge group on the Dp (anti-)brane, as ordered in (<ref>), and similarly for the antisymmetric representations. 
    
     For a=4, this vacuum configuration is rigid since branes cannot be recombined further and moved into the bulk. Indeed, although scalars in bi-fundamental representations are always present, moving branes into the bulk would lead to an anomalous model, since tadpoles cannot be satisfied anymore. A more detailed analysis would require the computation of the F and D terms, which however lies beyond the scope of this Thesis. However, other solutions allow a deformation to take place. For instance, one can consider the example where D9 and D5 branes are recombined to 
    
        G_CP = SO (4) |_D9 bulk×USp (4) |_D5 bulk×SO (4)^2 |_D9×USp (4)^2 |_D5 ,
    
    with left-handed Majorana-Weyl fermions in the antisymmetric representation of all group factors, four scalars in 
    
        (10, 1; 1, 1; 1,1) + (1, 6; 1, 1; 1,1)   ,
     
    a right-handed Majorana-Weyl fermion in 
    
        (10, 1; 1, 1; 1,1) + (1, 10; 1, 1; 1,1)   ,
    
    a left-handed symplectic Majorana-Weyl fermion in 
    
        (1, 1; 4, 1; 4,1) + (1, 1; 1, 4; 1,4)   ,
    
    two scalars and a left-handed symplectic Majorana-Weyl fermion in 
    
        (4, 1; 1, 1; 4,1)
        + (4, 1; 1, 1; 1,4)+ (1, 4; 4, 1; 1,1)
        + (1, 4; 1, 4; 1,1)   ,
     
    and in six copies of 
    
        (4, 4; 1, 1; 1,1)   .
     
    The branes on the fixed points cannot be further deformed, unless the scalars associated with strings stretching between the bulk branes and those on the fixed points are turned on. This would imply a recombination of the various branes, although one cannot bring everything in the bulk, since this would leave the g^2 and g^4 twisted R-R charge of the O-planes un-matched. As before, the analysis of the allowed higgsing requires a detailed knowledge of the F and D terms and of the allowed magnetisations of this model, which are beyond the scope of this study. 
    
    
    
    
     §.§ The T4/Z6 supersymmetric orientifold
    
    
    The choice of σ=0 implies instead a "trivial" action on the twisted two-cycles which induces O9_- and O5_- planes requiring D9 and D5 branes to cancel anomalies. The action of the orbifold group on the Chan-Paton labels is summarised in
    
        N_β = e^π i β /6∑_γ=0^5 e^2 π i βγ/6  n_γ ,
            
        D_ β = e^π i β /6∑_γ=0^5 e^2 π i βγ/6  d_ γ ,
    
    The extra phase implies that the Chan-Paton labels are all complex, leading to unitary gauge groups. The general solution of the R-R tadpoles is 
    
        n_0=n_5= 4+a   ,    n_1=n_4= 8   ,    n_2=n_3= 4-a   ,
            
           d_0=d_5= 4+a   ,    d_1=d_4= 8   ,    d_2=d_3= 4-a  ,
    
    with gauge group 
    
        G_CP= U(4+a) ×U(8) ×U(4-a) |_D9×U(4+a) ×U(8) ×U(4-a) |_D5  .
    
    The parameter a=0,…, 4 depends on how the twisted R-R charges are cancelled between D9 and D5 branes.
    The contribution of the massless states to the open-string amplitudes reads 
    
        𝒜_99=    (2 n_0    n_5 + 2 n_1   n_4 + 2 n_2   n_3) ( V_4 - 2   S_4 ) 
            
           +  ( 2 n_0   n_4+ 2 n_1   n_3 + n_2^2 + n_5^2) (2   O_4 - C_4 )
            
           + (n_0^2 +  2 n_1   n_5 + 2 n_2   n_4 + n_3^2 ) (2   O_4 - C_4 )
    
    and
    
        ℳ_9=  - (  n_2 + n_5) (2  Ô_4 - Ĉ_4 )- (n_0 + n_3) (2  Ô_4 - Ĉ_4 )
    
    for open strings stretched between D9 branes, 
    
        𝒜_5 5=    (2 d_0    d_5 + 2 d_1   d_4 + 2 d_2   d_3) ( V_4 - 2   S_4 )  
            
           + ( 2 d_0   d_4+ 2 d_1   d_3 + d_2^2 + d_5^2)  (2   O_4 - C_4 )
            
           + (d_0^2 +  2 d_1   d_5 + 2 d_2   d_4 + d_3^2 ) (2   O_4 - C_4 )
    
    and
    
        ℳ_5= - (  d_2 + d_5) (2  Ô_4 - Ĉ_4 ) - (d_0 + d_3) (2  Ô_4 - Ĉ_4 )
    
    for open strings stretched between D5 branes, and 
    
        𝒜_95=     (n_0   d_5  + n_1   d_4 + n_2   d_3 + n_3   d_2 + n_4   d_1 + n_5   d_0) (2   O_4 - C_4 )
    
    for open strings between D9 and D5 branes. The light spectrum is supersymmetric and comprises a vector multiplet in the adjoint representation of the gauge group (<ref>), one hyper multiplet in the representations
    
        ( 
        
        , 1, 1;1, 1, 1)  + ( 
        , 
        ,1; 1, 1, 1)  + (1,  
        ,
        ; 1, 1, 1) + ( 1, 1, 
        
        ; 1, 1, 1)   ,
                
           ( 1, 1, 1;
        
        , 1, 1)  + ( 1, 1, 1; 
        , 
        ,1)  + (1, 1, 1; 1,  
        ,
        ) + ( 1, 1, 1; 1, 1, 
        
        )   ,
         
           (
        ,1,1; 
        ,1 , 1) + (1,
        ,1;1, 
        , 1) + (1,1,
        ;1,1, 
        )    ,
    
    from open strings stretched between D9/D9, D5/D5 and D9/D5 branes, respectively.
    In this supersymmetric vacuum, the tadpole conditions are weaker than in its BSB counterpart, since a recombination of branes is always possible, even with the minimal choice a=4. Indeed, in such a case, one can move branes in the bulk yielding the gauge group
    
        G_CP = USp (4) |_D9 bulk×USp (4) |_D5 bulk×U (4) |_D9×U (4) |_D5 ,
    
    whose light spectrum can be deduced by setting n_0=n_2=0 in the amplitudes of eqs.(<ref>), (<ref>), (<ref>),   (<ref>), (<ref>) and by adding the contribution from the bulk branes. All in all, the spectrum comprises a vector multiplet in the adjoint of (<ref>), an hyper multiplet in the representations
    
        (6, 1; 1; 1) + (1, 6;  1; 1)  +  (1, 1; 4 ; 4̅) + (4, 1; 1; 4) + (1, 4; 4 ; 1)    ,
    
    and in three copies of 
    
        (4, 4; 1 ; 1)   .
    
    
    § ANOMALY POLYNOMIALS AND THE KÄHLER J FORM
    
    
    Tadpoles in orientifold constructions are deeply tied to the structure of the low-energy effective action. In particular, the R-R tadpoles determine the charge through which D-branes and O-planes are coupled to R-R fixed forms, including the topological Chern-Simons terms entering the Green-Schwarz-Sagnotti mechanism <cit.>. All these terms are captured in the expansion of the Wess-Zumino coupling for D-branes  <cit.> and O-planes <cit.>
        S_Dp  = - ∫_M_ptr  C ∧ e^i F_p∧√(Â(R))  ,
    
    and 
    
        S_Op_± = - ± 2^p-4 ∫_M_p C ∧√(L̂ (R/4 ))  ,
    
    where C= ∑_p C_(p) collects the various p-form fields, R and F_p describe the curvature of the tangent and Dp-brane gauge bundle, Â(R) corresponds to the A-roof genus and L̂ (R  ) the Hirzebruch polynomial. From eqs. (<ref>) and (<ref>), one can directly read the R-R tadpoles from the coefficients multiplying the non-dynamical 6-forms and the topological coupling dictated by the Chern-Simons term   
    
        S_GS = ∫Ω_αβ  C_2^α∧ X_4^β  ,
     
    where  Ω_αβ is an SO (1,n_T) invariant metric with mostly minus signature, and n_T is the number of tensor multiplets. The term X_4^α is a polynomial in the curvatures of the tangent and gauge bundles  
    
        X_4^α= 1/2 ( a^α tr  R^2 + ∑_i b_i^α/λ_i tr  F_i^2  )   ,
    
    where the 't Hooft coefficients a^α and b_i^α depend on the charges of Dp branes, q^Dp, and Op planes, q^Op, 
    
        b_i^α = ℓ_α/2 √(N) λ_i   q^D_i_α ,      
        a^α = ℓ_α/2 √(N) ∑_p q^Op_α ,
    
    as follows from eqs. (<ref>) and (<ref>). 
    In the  expression above, λ is the group theoretic factor tied to the highest root and ℓ_α is a geometric factor given by
    
        ℓ   = (1,1; 1/√(2) sin (π/2)1_16 σ) ,    for   N=2 ,
        
        ℓ   = (1,1; 1/sin (2 π /6 ) 1_9) ,    for   N=3 ,
        
        ℓ   = (1,1; 1/sin (π /4 ) 1_4 ; 1/√(2) sin (2π/4)1_10 σ) ,    for   N=4 ,
        
        ℓ   = (1,1; 1/sin ( π /6 ) ; 1/sin (2π/6) 1_5 ; 1/√(2) sin (3π/6)1_6 σ) ,    for   N=6 ,
    
    with the coefficients sin (aπ/N) clearly related to the number of fixed points in the various twisted sectors, and 1_d a 
    d-dimensional vector whose entries are all equal to one. For the BSB N=6 orientifold the charges are given by
    
        q^O9_-   = ( -4,-4 ; 0; - 1_5 ; 0_6 ) , 
        
        q^D9_1   = (1,1; 1; 1_5 ; 1_6 ) , 
        
        q^D9_2   = (2,2; 1 ; - 1_5 ; -2 1_6 ) , 
         
        q^D9_3   = (2,2; -1 ;  - 1_5 ;  2 1_6 ) , 
        
        q^D9_4   = (1,1; -1;  1_5 ; - 1_6 )  ,
            
        q^O5_+^i   = ( 14,-14 ; 0 ; -3 δ_5^1 ; 0_6  )  ,  
        
        q^D5_1   = (-1,1;1;  3 δ_5^1 ; 4 δ^1_6 ) , 
         
        q^D5_2   = (-2,2; 1; - 3 δ_5^1 ; -8 δ^1_6 )  , 
         
        q^D5_3   = (-2,2;-1;  -3 δ_5^1 ; 8 δ^1_6 ) , 
         
        q^D5_4   = (-1,1; -1; 3  δ_5^1 ; - 4 δ^1_6 )  ,
    
    while for the supersymmetric N=6 orientifold the charges are
    
        q^O9_-   = ( -4,-4 ; 0;  1_5 ) , 
        
        q^D9_1   = (2,2; √(3); 1_5  ) , 
        
        q^D9_2   = (2,2; 0 ; - 21_5  ) , 
         
        q^D9_3   = (2,2; - √(3) ;   1_5 ) , 
            
         q^O5_-^i   = ( -14,-14 ; 0 ; -3 δ_5^1  )  , 
        
        q^D5_1   = (2,-2;- √(3);  3 δ_5^1 ) , 
         
        q^D5_2   = (2,-2; 0; - 6 δ_5^1  )  , 
         
        q^D5_3   = (2,-2; √(3);  3 δ_5^1  ) .
    
    Here 0_d denotes a d-dimensional null vector and δ_d^i is a d-dimensional vector whose only non-vanishing component is {δ^i_d}_j = δ^i_j.
    
     In eq. (<ref>), we have chosen to write the topological coupling using a general basis, although from the expansion of the terms (<ref>) and (<ref>) it appears in diagonal form. This choice has the advantage of reflecting the geometry of D-branes and O-planes <cit.> but obscures some features reflecting other properties of these vacua. Indeed, it is always possible to find a basis where the self-duality of the lattice is manifest, which follows <cit.> from the identification of the dyonic strings as gauge instantons of charges b_i for the gauge group G_i<cit.> and a gravitational instanton with charge a. Although this requirement plays a non-trivial role in constraining the landscape of low-energy effective theories and will be discussed in great detail in the next Chapter, in what follows we shall rely on the diagonal basis for the anomaly polynomial.
    
     The cancellation of R-R tadpoles <cit.> implies the vanishing of all irreducible anomalies and allows to factorise the anomaly polynomial as   
    
        I_8 = 12 Ω_αβ X_4^α∧ X_4^β  ,
     
    which is then cancelled by the contribution from (<ref>), via the celebrated Green-Schwarz-Sagnotti mechanism <cit.>. In fact, requiring a non-trivial transformation of the 2-forms
    
        δ C_2^α= ω_2^(1)^α  ,
    
    provides the counter-terms that cancel the reducible anomaly, where ω_2^(1)^α is tied to the Chern-Simons form ω_3^α and X_4^α via the descent equations
    
        δω_3^α= d  ω_2^(1)^α
           d  H^α = X_4^α  ,
    
    with H^α= d   C_2^α - ω_3^α.
    In our case, the anomaly polynomial for the BSB ℤ_6 orientifold is given by
    
        I_8    =1192 (  tr F_9,1^2 + 2tr F_9,2^2 + 2 tr F_9,3^2 + tr F_9,4^2 . 
            
           .              - tr F_5̅,1^2  - 2tr F_5̅,2^2 - 2 tr F_5̅,3^2 - tr F_5̅,4^2  )^2 
            
           - 1192 ( -8 tr R^2 + tr F_9,1^2 + 2 tr F_9,2^2 + 2 tr F_9,3^2 +  tr F_9,4^2  .
            
           .               +  tr F_5̅,1^2 + 2 tr F_5̅,2^2 + 2 tr F_5̅,3^2 +  tr F_5̅,4^2  )^2  
             
             - 148 (   tr F_9,1^2 + tr F_9,2^2 -  tr F_9,3^2 -  tr F_9,4^2 
            .
            
           .               + tr F_5̅,1^2 + tr F_5̅,2^2 - tr F_5̅,3^2-  tr F_5̅,4^2     )^2
            
            - 1144 [  ( - 4 tr R^2 + tr F_9,1^2 - tr F_9,2^2 -  tr F_9,3^2 + tr F_9,4^2  
            . .
            
           .               + 3  ( tr F_5̅,1^2 - tr F_5̅,2^2 -  tr F_5̅,3^2 +  tr F_5̅,4^2  )  )^2   
            
            - 1144 (  -tr R^2 + tr F_9,1^2 - tr F_9,2^2 - tr F_9,3^2 + tr F_9,4^2  )^2 
            
            -1284 ( tr F_9,1^2 - 2 tr F_9,2^2 + 2tr F_9,3^2 - tr F_9,4^2 
            . 
            
            -196 ( tr F_5̅,1^2 - 2 tr F_5̅,2^2 + 2tr F_5̅,3^2 -  tr F_5̅,4^2  )^2  
            
              -15284 ( tr F_9,1^2 - 2 tr F_9,2^2 + 2tr F_9,3^2 - tr F_9,4^2  )^2   ,
    
    while for the supersymmetric one it reads
    
        I_8    =148 ( -4 tr R^2 + tr F_9,1^2 + tr F_9,2^2 + tr F_9,3^2 + tr F_5,1^2 + tr F_5,2^2 + tr F_5,3^2  )^2 
           - 148 ( tr F_9,1^2 + tr F_9,2^2 + tr F_9,3^2 - tr F_5,1^2 - tr F_5,2^2 - tr F_5,3^2  )^2   
          
              - 116 (   tr F_9,1^2 - tr F_9,3^2 - tr F_5,1^2 + tr F_5,3^2   )^2 
          
            - 1144 ( - 2 tr R^2 + tr F_9,1^2 - 2  tr F_9,2^2 +  tr F_9,3^2 - 3  ( tr F_5,1^2 -2  tr F_5,2^2 +  tr F_5,3^2  )  )^2  
          
            - 472 ( tr R^2 + tr F_9,1^2 - 2  tr F_9,2^2 + tr F_9,3^2   )^2    .
    
    In both expressions, we notice that the presence of fractional orientifold planes is reflected in the appearance of the first Pontryagin class (roughly tr R^2) in the last two lines of eq (<ref>) for the supersymmetric case and in the seventh and ninth lines of eq. (<ref>), which are associated with the coupling to the twisted R-R forms.
    
     All in all, there is a deep connection between the R-R tadpoles, encoding the charges of O-planes and D-branes, and the  Wess-Zumino coupling in the low-energy effective action, which is encoded in the expression (<ref>) for the 't Hooft coefficients. For vacua with 𝒩=(1,0) supersymmetry, these in turn also determine the kinetic term for the gauge fields <cit.>
        12  J_α b_i^α tr  F_i∧⋆  F_i   .
    
    The J_α[Following <cit.>, the symbol J_α is used to identify the vector v^r of <cit.>. J plays the role of the Kähler form determining the geometry of the scalar manifold.] function depends on the n_T scalars τ in the tensor multiplets, which parametrise the coset SO (1 , n_T)/SO (n_T ). Positivity of the kinetic term for the scalar fields and the gauge vectors then requires <cit.>
        J· J >0    and     J· b_i >0  ,
    
    where the inner product is taken with Ω_αβ. 
    
     Although the supersymmetry invariance of the low energy effective action requires that the gauge kinetic function be given in terms of the anomaly vector b_i, its origin in string theory is different. In orientifold vacua, the gauge fields originate from the open-string sector, and it should thus be possible to extract the coupling (<ref>) from the tadpoles. Indeed, turning on a background magnetic field H_i on the D-branes, and expanding the J function as J_α∼τ_α +  O(τ^2),  eq. (<ref>) becomes schematically
    
        τ_α b_i^α  H_i^2.
    
    This equation identifies the coefficients b_i^α with the tadpoles of the scalars in the tensor multiplets, i.e. the one-point functions of the τ fields with the various D-branes. Now, while the anti-self-dual tensors C_2^- originate from the R-R sector, their scalar super partners are of NS-NS type, and therefore the b_i^α in (<ref>) and (<ref>) are actually related to the NS-NS tadpoles, and not to the R-R ones. Clearly, supersymmetry identifies them, and this is the reason why the same anomaly vectors enter both in eqs. (<ref>) and (<ref>). 
    
     This observation suggests that some care is needed when dealing with BSB vacua. In fact, in this class of orientifolds, R-R and NS-NS tadpoles are, in general, no longer related by supersymmetry, since for the D5 branes the tension and charges are not equal. The kinetic terms for the gauge fields should then involve new coefficients b̃_i^α,
    
        12  J ·b̃_i  tr  F_i ∧⋆ F_i ,
    
    which can be extracted from the NS-NS tadpoles and are related to the "tension" of branes t^D_i via
    
        b̃_i^α = ℓ_α/2 √(N) λ_i   t^D_i_α .
     
    Now t^D_i= q^D_i for D-branes, but t^D_i=-q^D_i for anti-branes, and thus they differ from the anomaly vectors by a sign flip in the entries associated with the gauge factors living on the antibranes. This is fully compatible with supersymmetry, since on the D5 branes (or, in general, in the non-supersymmetric sector) supersymmetry is realised non-linearly, which implies that each term in the corresponding Lagrangian is invariant by itself <cit.>. 
    
     Replacing the coefficients b_i with b̃_i in the eq. (<ref>) allows to define a J function also for the T^4/ℤ_N BSB orientifolds <cit.>, since the conditions 
    
        J· J > 0     and     J·b̃_i >0
    
    are now compatible. This solves the puzzle on the non-existence of the J form in <cit.>, where the original condition (<ref>) led to the presence of ghosts.  If, following <cit.>, we also impose the positivity of the Gauss-Bonnet term, J· a <0, the components of the J vector must satisfy J_0 > - J_1 and J_1 <0, where, for simplicity, we have set to zero the remaining entries
    
        J=  (J_0 , - |J_1 |; 0_4 ; 0_10 ) ,      J_0 > | J_1| .
    
    
    § DEFECTS AND (NEW) UNITARITY CONSTRAINTS
    
    
    In six dimensions the term described in (<ref>) arises form the coupling of the R-R 2-forms C_2^α to D9 and D5 or D5 branes and allows to cancel the local anomaly of the theory via an anomalous transformation of C_2^α. However, the latter fields admit also a SO(1,n_T) invariant coupling
    
        S_2D = - Q^α Ω_αβ∫ C_2^β ,
    
    involving the presence of one-dimensional defect for this six-dimensional vacuum, which in our case consist of D1 branes localised on the internal manifolds or D5' branes wrapping the entire T^4/ℤ_N space[Clearly, one can also have D1, D5' and magnetised D5' branes. In these cases, the discussion would follow a similar pattern.]. However, the non-trivial behaviour of C_2^α under local gauge and Lorentz transformations, as required by the Green-Schwarz-Sagnotti mechanism, induces an anomaly inflow on the world-volume of the defect which, in a consistent vacuum, must be cancelled by the anomalous contribution of its two-dimensional degrees of freedom <cit.>
        I_4     = Q· X_4 +12 Q· Q  χ (N) 
        
           = 12( Q· a  tr  R^2 +Q· Q χ (N) +∑_i Q· b_i/λ_i tr  F_i^2 )= - I_Inflow  .
    
    The pontryagin class in the above equation encodes the curvature of the tangent bundle of the whole six-dimensional space-time which can be decomposed in terms of the curvature of the tangent bundle of the 2D world-sheet and its SO(4)-normal bundle N via
    
        tr  R^2 = -12 p_1 (T_2 ) + c_2 (u) + c_2 (v)  ,     and  χ (N) = c_2 (v) - c_2 (u) ,
    
    where we have used the decomposition SO (1,5) = SO (1,1) ×SU (2)_u ×SU (2)_v and the definition of the Euler class  χ(N) of the normal bundle. Thus, the anomaly polynomial can be written as
    
        I_4 = - 112 3 Q· a   p_1 (T_2)  + 12 (Q· Q + Q · a )  c_2 (v)  + 12 (Q· Q - Q · a)  c_2 (u)  +12∑_i Q· b_i/λ_i tr F_i^2 .
    
    When the theory on the defect flows to the IR, we can interpret the coefficient of p_1 (T_2) as the difference of the central charges from the left and right movers, c_R - c_L =  6 Q· a. In supersymmetric vacua, the correct identification of the SU(2)_R R-symmetry would allow to completely determine the c_R, taking advantage of the properties of the 𝒩=(0,4) superconformal algebra which implies c_R=3 k_R, with k_R the 't Hooft coefficient of the R-symmetry. However, such identification is typically difficult to determine. In <cit.>, SU(2)_u is identified with the R-symmetry, but such an assumption cannot hold universally. In fact, both left and right moving fields can transform non-trivially under SU (2)_u, and thus both would contribute to the coefficient of c_2 (u). However, according to 𝒩=(0,4) supersymmetry only right movers should be allowed to transform under the true R-symmetry. As shown in <cit.>, it seems that SU (2)_u can be identified with the R-symmetry only if a single D1 brane is moved away from the orbifold fixed points, for which the result in <cit.> is thus reproduced. In all other cases, c_2 (u) also receives contributions from left-moving excitations, and it is not known in literature how to get access to this piece of information purely from anomaly inflow considerations. 
    
     For BSB orientifolds the situation is even worse. In fact, the D1 and D5' defects do not, in general, enjoy 𝒩 = (0,4) supersymmetry on their world-volume. This is due to the presence of anti-branes, which flip the chirality of the excitations of the strings stretched between the D1 or D5' and the D5 branes, thus explicitly breaking supersymmetry. This means that the notion of R-symmetry is lost, and thus the relation c_R= 3 k_R does not hold.  
    
     Nevertheless we can take advantage of the fact that when flowing to the IR fixed point, only the massless states survive, so that knowing the light spectrum of the UV degrees of freedom living on the defects allows to determine the central charges of the the left and right moving sectors of corresponding two-dimensional CFT. Such a top-down approach allows then to accumulate empirical data on the 2D CFT on the D1 and D5' branes, instrumental for deriving refined consistency conditions in a sought-after bottom-up approach. 
    
     We can therefore discuss the general features of the defects in our class of vacua with D9 and D5 (anti-)branes. In this case, the D1 branes located on a ℤ_N fixed point carry the same gauge group of D9 branes <cit.>, while the gauge group is orthogonal if they are moved in the bulk. The Chan-Paton group of D5' branes is equal in structure to that of D5 or D5 branes sitting on a ℤ_N fixed point, although tadpole conditions may force some of the group factors to be absent on the D5 branes. Open strings stretched between D1 and D9 branes have ND boundary conditions along the eight transverse directions, and therefore the light excitations involve left-handed fermions which are singlets of the 𝒩 = (0,4) superalgebra. Open strings stretched between D1 and D5 branes have DD boundary conditions along the internal space, so that the light excitations now comprise a full right-moving super multiplet (two right-moving scalars and Majorana-Weyl fermions) and a left-moving Majorana-Weyl fermion and a pair of left-moving chiral bosons. Open strings stretched between D1 and D5 branes give rise to the same fields but with flipped chirality, which is, indeed, incompatible with 𝒩 = (0,4) supersymmetry, as D1 and  D5 branes would preserve super-charges of opposite chirality. 
    
     For the interactions of the D5 ' branes with D9 and D5 branes we have a full super multiplet, together with a left-moving fermion in the 95' sector, while the 55' sector only involves a left-moving fermion. Trading D5 with D5 branes implies a chirality flip for the fermionic singlet in the 5̅ 5' sector. The presence of massless scalars in the 51 and 95' strings suggests that D5' (D1) branes can be interpreted as (anti-) instantons of D9 (D5 or D5) branes. This identification is further supported by the fact that the twisted charges of the putative instanton brane and of the physical brane match. In these cases, the charge vector Q can be expressed in terms of the anomaly vectors b_i associated with the gauge groups of the D9 (D5 or D5) branes.
    
     In general, the defects are charged under the gauge group of the vacuum. In the IR, the theory decouples from the bulk, and the D9 and D5 or D5 gauge groups become global symmetries realised on the defect CFT as Kač-Moody algebras in both left and right moving sectors[Actually, this only occurs for the Kač-Moody algebra associated with the gauge group admitting the defect as an instanton.]. This is reflected in the sign of the Kač-Moody level k_i, which can be extracted from the coefficient of tr  F_i^2 in the anomaly polynomial I_4,  and in its contribution to the central charge. 
    
     For instance, for the gauge group G_9,i on the D9 branes with dual Coxeter number h_i^∨, the algebra on the D1 defect is realised at level k_i ≥ 0 and thus
    
        c_i = k_i  dim  G_9,i/k_i + h_i^∨ ,
    
    since the Kač-Moody algebra is realised by a left-moving fermion, which is a singlet of the 𝒩 = (0,4) superalgebra. For the gauge group G_5̅ ,i or G_5,i on the D5 or D5 branes, the level of the Kač-Moody algebra can be positive or negative. An analogous feature also emerges on the D5' branes, where the levels of the D5 branes are positive, while those associated with the D9 group can be positive or negative. Similarly, when D5 branes replace the D5 ones, the level of the associated Kač-Moody algebra is negative.
    
     Upon removing the contribution of the centre of mass (CM) degrees of freedom (four non-chiral scalars and four right-moving Majorana-Weyl fermions) which decouple in the IR, a natural generalisation of the KSV unitarity constraints is 
    
        ∑_i |  k_i>0k_i  dim  G_i/k_i + h_i^∨≤ c_L-4_CM    and    ∑_i |  k_i<0|k_i|  dim  G_i/|k_i| + h_i^∨≤ c_R-6_CM ,
    
    both for the left and right-moving sectors. However, in the UV, charged (non-chiral) scalar fields are present on the defect, and their role in the realisation of the Kač-Moody algebras in the IR is unclear. These scalar fields are non-compact and their dynamics in the IR is difficult to determine. They could be free and generate an abelian algebra or describe an independent interacting sector of the theory. In both cases, they are expected to contribute to the left and right central charges and, if this happens, the inequalities (<ref>) cannot be saturated. Indeed, this is the case in all string constructions where the defects admit an instanton interpretation.
    
     As a final comment, notice that it is always possible to introduce D1 branes in the bulk of the compactification orbifold, which only couple to the two-forms C_2^0 and C_2^1 from the untwisted sector. In the decompactification limit, one is expected to recover the ten-dimensional type I superstring with a defect coupled to the (non-chiral) R-R two-form C_2 existing in D=10. Since in D=6C_2^0 and C_2^1 are the two chiral components of C_2, the D1 brane in the bulk must carry the same charge, Q^0 = ± Q^1, thus satisfying the null-charge conjecture, Q· Q =0<cit.>. We expect that this is the case for any six-dimensional vacuum with at least one tensor multiplet. The existence of null-charge branes allowed to put in the swampland various models which pass the KSV constraints, but do not admit a string or F-theory construction <cit.>.
    
     We can then now move to the discussion of the defects introduced in the examples analysed in this Chapter, namely the supersymmetric and BSB orientifolds of T^4/ℤ_6.
    
    
    
     §.§ Examples on T4/Z6
    
    
    The spectrum characterising the world-sheet theory of the D1 and D5' defects can be deduced, following <cit.>, by performing an orbifold projection of the toroidal construction, as done for instance in <cit.>. We will present only the light spectrum and refer to the Appendix of <cit.> for the explicit construction of the amplitudes.
    
    
    
    
      §.§.§ D1 branes in the T4/Z6 BSB orientifold
    
    
    We start our analysis by considering D1 branes sitting on the ℤ_6 fixed point of the T^4 /ℤ_6 orientifold discussed in Section <ref>. Such D1 branes support the Chan-Paton gauge group SO(r_1)  ×U(r_2)  ×U(r_3) ×SO(r_4), reproducing the structure of that of the D9 branes in the same vacuum. The light excitations are summarised in table <ref>, where the representations of the various fields with respect to the SO (4) ∼SU (2)_u ×SU (2)_v group, transverse to the D1 world-volume, are given.  The second line in the table includes the contribution of the centre of mass degrees of freedom which are expected to decouple in the IR. From the microscopic data in the table <ref>, we can compute the anomaly polynomial,
     
        I_4   =12  ( -(r_1+  r_4)  trR^2  - χ(N)  ((r_1-r_2)^2 + (r_4-r_3)^2+ (r_2-r_3)^2  ) .
        		
                + r_12 tr F_9,1^2 + r_42 tr F_9,4^2 + r_2  tr F_9,2^2 + r_3  tr F_9,3^2
        		
                 + (r_1-r_2)  tr F_5̅,1^2 + (r_4-r_3)  tr F_5̅,4^2 
          
           .      - (r_1+r_3-2r_2)  tr F_5̅,2^2 - (r_4+r_2-2 r_3)  tr F_5̅,3^2  )   ,
    
       which cancels the bulk inflow if the charge vector is
       
        Q   = r_1   b_5 + r_2   b_6 + r_3   b_7 + r_4   b_8
         
           = (    r_1+2r_2 + 2r_3  + r_42 √(6),  -   r_1+2r_2 + 2r_3  + r_42 √(6);-r_1-r_2+r_3+r_4√(6); .
        	
           .     -r_1+r_2+ r_3- r_4√(2) ; 0_4; -r_1+2 r_2-2r_3 + r_4√(3); 0_5  )  .
    
    Notice that this solution for Q guarantees the positivity of the tension of the defect, Q· J >0, with the Kähler form J given in eq. (<ref>). From (<ref>) we also extract the levels of the Kač-Moody algebras in the IR,
    
        k_1= r_1    ,
                 
            k_5= r_1 - r_2   ,
                    k_2=r_2   , 
                
            k_6=2 r_2 - r_1 - r_3   ,
                    k_3=r_3   , 
                
            k_7= 2r_3 - r_2 - r_4   ,
                    k_4=r_4   ,
            
            k_8=r_4-r_3  .
    
    Notice that, according to our choice of the Chan-Paton labels, k_i may be positive or negative, implying the associated the Kač-Moody algebra to be realised on the left or right-moving sector of the CFT. 
    
    
     From the solution (<ref>), we read that the D1 branes can be interpreted as instantons of the gauge theories on the D5 branes, since Q ∝ b_5,6,7,8. This observation is supported by the presence of moduli in the bi-fundamental representations (  
    _1_1, 
    _5̅_1 ) + ( 
    _1_2, 
    _5̅_2 )+ (  
    _1_3, 
    _5̅_3  ) + ( 
    _1_4, 
    _5̅_4  ). 
    
     Finally, one can check that the constraints (<ref>) are satisfied with c_L and c_R computed from table <ref>, once the degrees of freedom of the centre of mass are removed. As expected, this implies that the two-dimensional CFT is unitary and this string construction is consistent. As an example, in the a=0 vacuum, where the D5 support a USp (8) ×U(4) ×U(4) ×USp(8) gauge group and D9 branes a SO (8) ×U(4) ×U(4) ×SO(8) gauge group on the fixed point, the simple choice r_1=1 and r_2=r_3=r_4=0 gives
    
        ∑_i   |   k_i ≥ 0k_i  dim G_i /k_i + h_i^∨=4 + 6< c_L -4_CM = 28 , 
            
           ∑_i   |   k_i <0|k_i|  dim G_i /|k_i| + h_i^∨ = 4 < c_R -6_CM = 20   ,
    
    where the centre of mass degrees of freedom have been removed from the counting of c_L,R. 
    
     One has the option of moving the D1 branes away from fixed points. As in <cit.>, this turns out to be the configuration which satisfies the minimal constraints of <cit.>. The gauge group on the defect is SO (r) and the microscopic degrees of freedom are listed in table <ref>, while the D1-D5 sector is absent, since the open strings with these boundary conditions are now massive. Also in this case, the second line in the table contains the centre of mass degrees of freedom, associated with the singlet in the decomposition of the twofold symmetric representation of orthogonal groups. The anomaly polynomial now reads
    
        I_4= 12 ( -2 r  tr R^2 + r2tr F_9,1^2 + r tr F_9,2^2 + r  tr F_9,3^2 + r2tr F_9,4^2   )   ,
    
    which cancels the inflow of the bulk theory if the charge vector is
    
        Q =  ( r,-r ; 0_12 )   .
    
    As expected, the D1 branes couple only to the two-forms from the untwisted sector, and therefore satisfy the null-charge condition <cit.>. For r=1, i.e. for a single D1 brane, the left and right central charges
    
        c_L= 20 +4_CM  ,
            
        c_R=  6 +6_CM  ,
    
    agree with the those given in <cit.>. We would like to stress, though, that for the case of an arbitrary stack of D1 branes, r>1, the expressions of c_L,R in <cit.> seem no-longer to be correct. 
    
     Upon reading the coefficients k_1=k_2=k_3=k_4=1 from I_4, and taking into account that the Kač-Moody algebra is realised only in the left-moving sector, the unitarity constraint reads
    
        ∑_i   |   k_i ≥ 0k_i  dim G_i /k_i + h_i^∨= 16 < c_L-4_CM =20   ,
    
    and is clearly satisfied.
    
    
    
    
      §.§.§ D5' branes in the T4/Z6 BSB orientifold
    
    
    
    We can now consider the other kind of defects for this vacuum given by D5' branes wrapping the compactification space, whose associated Chan-Paton gauge group is  USp(r_1) ×U(r_2)  ×U(r_3) ×Usp(r_4). The spectrum of light excitations is summarised in table <ref>. Clearly, in general, the D5' branes probe all ℤ_N fixed points, which include the origin where  D5 branes are placed. The second line contains the contribution of the centre of mass degrees of freedom of the defects, associated to the various singlets of the Chan-Paton gauge group. The anomaly polynomial is 
       
        I_4   = 12  ( (r_1+r_4) trR^2  - χ(N)  ((r_1-r_2)^2 + (r_4-r_3)^2+ (r_2-r_3)^2  ) .
        		
                 -  r_12 tr F_5̅,1^2 - r_42 tr F_5̅,4^2 - r_2  tr F_5̅,2^2 - r_3  tr F_5̅,3^2
        		
                 + (r_2-r_1)  tr F_9,1^2 + (r_3-r_4)  tr F_9,4^2 
          
           .      + (r_1+r_3-2r_2)  tr F_9,2^2 + (r_4+r_2-2r_3)  tr F_9,3^2 )   ,
    
       which cancels the anomaly inflow if
       
        Q    =   ( r_1+2r_2+2r_3+r_42√(6),r_1+2r_2+2r_3+r_42√(6); r_1-r_4+r_2-r_3√(6) ; r_1-r_2-r_3+r_43√(2) ; r_1-r_2-r_3+r_431_4;  .
        	
           .     r_1-2 r_2+2 r_3-r_44 √(3) ; r_1-2 r_2+2 r_3-r_441_5  )   .
    
    From this expression, we can see that Q ∝ b_1,2,3,4, implying that the D5' branes are naturally interpreted as instantons of the D9 ones. Indeed, the light spectrum involves scalars in the corresponding bi-fundamental representations. 
    
     The solution (<ref>), together with the expression (<ref>) for the Kähler J-form, guarantees that the D5' branes have positive tension as well, Q· J >0. Moreover, from eq. (<ref>) we read
    
        k_1= 2(r_2-r_1)   ,
             
              k_5= -r_1   ,
                k_2 = r_1 + r_3 -2 r_2   , 
             
            k_6 = -r_2   , 
                     k_3=r_4+r_2-2r_3   , 
             
            k_7=-r_3   ,
                    k_4=2(r_3-r_4)   , 
            
            k_8=- r_4    .
    
    The unitarity constraints are satisfied both in the left and right-moving CFTs. For instance, still for a=0, r_1=1 and r_2=r_3=r_4=0 and one finds
    
        ∑_i   |   k_i ≥ 0k_i  dim G_i /k_i + h_i^∨=  4 < c_L-4_CM =20   ,
            
           ∑_i   |   k_i <0|k_i|  dim G_i /|k_i| + h_i^∨=  7 + 4 = 11 < c_R - 6_CM= 28   .
    
    
      §.§.§ D1 branes in the T4/Z6 supersymmetric orientifold
    
    
    Analogous considerations can be made when supersymmetry is present. In such a case, the spectrum is organised according to the 𝒩=(0,4) superalgebra representations described in the Section <ref>. We can start by analysing the D1 brane defects.  In such a case the gauge group is given by U(r_1)×U(r_2)×U(r_3) and gives rise to the light spectrum summarised in table <ref>. The anomaly polynomial reads
        
        I_4= 12     (-(r_1+  r_3 ) trR^2   - χ(N)  ((r_1-r_2)^2 + (r_3-r_2)^2  ) .
        		
            + r_1  tr F_9,1^2 + r_2  tr F_9,2^2 + r_3  tr F_9,3^2
        		
           .  - (r_1-r_2)  tr F_5,1^2 - (2 r_2 -r_1-r_3)  tr F_5,2^2 - (r_3-r_2)  tr F_5,3^2  )   ,
    
    which cancels the bulk inflow if the charge vector is
        
        Q=  ( r_1+r_2+r_3√(6),- r_1+r_2+r_3√(6);r_3-r_1√(2); -r_1-2 r_2+r_3√(2); 0_4  )    .
    
    From the anomaly polynomial, it is possible to read the level of the Kač-Moody algebras
    
        k_1= r_1   ,
            
            k_4= r_2-r_1   ,             k_2= r_2   , 
            
            k_5= r_1+ r_3 - 2 r_2   , 
                    k_3= r_3   , 
           
             k_6= r_2-r_3   .
    
    The minimal choice r_1=1,   r_2=r_3=0 induces a realisation of the algebra both on the left and right moving sectors, which can be shown to satisfy the unitarity constraints,
    
        ∑_i   |   k_i ≥ 0k_i  dim G_i /k_i + h_i^∨=4 + 8=12  < c_L-4_CM=32    ,
            
           ∑_i   |   k_i <0|k_i|  dim G_i /|k_i| + h_i^∨=4  < c_R-6_CM=24   ,
    
    where we have taken a=0 for simplicity.
        
    
      §.§.§ D5' branes in the T4/Z6 supersymmetric orientifold
    
    
    Finally, we can discuss D5' branes for such vacuum, which also support a U(r_1)×U(r_2)×U(r_3) gauge group. The light spectrum is summarised in table <ref>
    and gives the anomaly polynomial 
        
        I_4= 12     ( -(r_1+r_3)  trR^2  - χ(N)  ((r_1-r_2)^2 + (r_2-r_3)^2  ) .
        		
            + (r_2-r_1)  tr F_9,1^2 + (r_1+r_3-2 r_2)  tr F_9,2^2 + (r_2-r_3)  tr F_9,3^2  
        		
           . + r_1  tr F_5,1^2 + r_2  tr F_5,2^2 + r_3  tr F_5,3^2   )   ,
    
    which is cancelled by the bulk inflow if the charge vector is
    
        Q    =  ( r_1 + r_2 + r_3√(6), r_1 + r_2 + r_3√(6); r_1-r_3√(2) ; r_1  - 2 r_2+ r_33√(2) ; -r_1 + 2 r_2-r_331_4  )   .
    
    The structure of the anomaly polynomial entails the levels of the Kač-Moody algebras to be
    
        k_1= r_2-r_1   , 
            
            k_4= r_1   , 
                k_2= r_1+ r_3 - 2 r_2   ,
        
            k_5= r_2   ,
                k_3= r_2-r_3   ,
        
            k_6= r_3   .
    
    As before, we can thus evaluate the anomaly constraints again for the choice a=0 and r_1=1   , r_2=r_3=0
        ∑_i   |   k_i ≥ 0k_i  dim G_i /k_i + h_i^∨=4  < c_L-4_CM=32    ,
            
           ∑_i   |   k_i <0|k_i|  dim G_i /|k_i| + h_i^∨=4 + 8=12  < c_R-6_CM=24   .
    
    
    CHAPTER: UNITARITY: DAI-FREED ANOMALIES
    
    
    § A MODERN VIEW ON ANOMALIES
    
    
    The discussion in the previous Chapter makes it clear that the implications of unitarity have not been fully exploited, yet despite the intense activity that has been devoted to the subject since the '70s <cit.> (see e.g.<cit.> for reviews). Indeed, the analysis of the previous Chapter shows how admitting the possibility that a one-dimensional defect be present in the spectrum entails further constraints on the low-energy effective theory, beyond the simple requirement of anomaly cancellation of the bulk theory. The introduction of this kind of defect, which is natural from a string theory perspective, since it simply reflects the coupling to the ubiquitous p-form fields, has been recently linked to the conjectured existence of a unique theory of quantum gravity <cit.>, resulting from the interconnected web of dualities among the known string theory constructions <cit.>. This extended notion of duality requires the knowledge of all processes allowed in quantum gravity (even at the non-perturbative level) and of all possible defects, as suggested by the so-called  completeness hypothesis<cit.>. Combining the latter with unitarity leads to non-trivial requirements, which have been nevertheless shown to hold in F-theory vacua <cit.>, in D=10<cit.> and in D=6 perturbative string theory constructions <cit.>. One can thus approach the issue of describing the shape of the string landscape by adopting a complementary bottom-up view as part of the  swampland program<cit.>, in which unitarity combined with the completeness hypothesis implies further consistency conditions to be satisfied by low-energy theories admitting a UV completion in quantum gravity, along the lines of <cit.>. In this perspective, one can combine unitarity with other general properties of the string landscape to deduce additional constraints on the string landscape. Requiring that topology change be part of the processes admitted by a theory of quantum gravity entails additional constraints <cit.>. This picture emerges naturally in the current understanding of anomalies pioneered by the theorems on the Dirac operator by Dai and Freed <cit.>. 
    
     Traditionally, anomalies can be described as the violation of gauge invariance of the one-loop effective action under gauge transformations that are either connected (local anomalies) or disconnected (global anomalies) to the identity <cit.>. Both can be ultimately traced back to the ambiguities occurring in the definitions of the partition function of chiral fermionic and bosonic fields. The first can be described by the descent procedure and are encoded in the anomaly density I_D+2, which is a D+2-dimensional polynomial in the curvature of the tangent and gauge bundle <cit.>, while the others are described by building a mapping torus X × [0,1]/ ∼ where the equivalence relation identifies (x,0) with (g(x),1), with g the considered gauge transformation. The mapping torus thus depends on the homotopy classes of the gauge group <cit.> whose associated anomaly vanishes if the topological phase ϕ_T of the kinetic operator O_T on the mapping torus <cit.> does. Recently, it has been shown <cit.> that these ambiguities can be clarified by describing the chiral fields on a given spacetime X as a boundary mode of a gapped Dirac spinor or a non-chiral p+1-form living in a D+1-dimensional space Y such that ∂ Y=X with suitable elliptic boundary conditions[These results have been obtained working in Euclidean signature. The case of Minkowski's signature has been covered in <cit.>.]<cit.>, required in order for O_T to be self-adjoint on X. Schematically, we can write the partition function of these chiral fields living on X=∂ Y as 
    
        𝒵(L, Y)=  | 𝒵_T(X)  | e^-2 π i ϕ_T(Y)
    
    where |𝒵_T(X)| corresponds to the absolute value of the partition function of the chiral field living on the boundary once we have imposed the elliptic boundary conditions L. Note that, contrary to what happens in standard treatments of anomalies, the definition (<ref>) has a well-defined phase, at the cost however of a non-trivial dependence  a priori on the extension Y, that should preserve the structure characterising X.
    The space Y has no physical meaning, and therefore for a physically meaningful theory, the partition function should not depend on the choice of the extension. The dependence on the latter is all contained in the phase ϕ_T(Y) and can be quantified by taking the ratio of the partition function on two different extensions 
        𝒜_T(Y ∪Y̅')= 1/2 π ilog𝒵(L, Y')/𝒵(L, Y)
    
    where Y ∪Y̅' ≡ Y_cl is a closed manifold obtained by glueing Y and Y' with the opposite orientation along the common boundary. The resulting partition function 𝒜_T(Y_cl) describes an invertible topological field theory[An invertible topological field theory is quantum field theory whose Hilbert space on a closed manifold is one-dimensional <cit.>.]<cit.>, known in literature as the  anomaly theory. This nomenclature is justified by the fact that the usual treatment of anomalies is reproduced by specific choices of the closed manifold Y_cl. Indeed, whenever it is a boundary of a D+2 dimensional space Y_cl= ∂ Z the Atiyah-Patodi-Singer (APS) <cit.> theorem implies
    
    𝒜_T(∂ Z) = Index(O_T) - ∫_Z I_D+2  ,
    
    
    where Index(O_T) is the index of the operator O_T and it is an integer number. Similarly, the usual global anomalies are reproduced if we choose Y_cl to be a mapping torus. However, these two examples do not exhaust  a priori all the possible D+1 dimensional manifolds and therefore different choices of the manifolds Y_cl may be new sources of anomalies, the so-called  Dai-Freed anomalies <cit.>. It is then crucial to determine the set of manifolds that are to be considered. For a generic quantum field theory, constraints can restrict the choice of the closed manifolds Y_cl<cit.>, but for a theory of quantum gravity, topology is allowed to change and thus  all the manifolds should be considered. Luckily, our task can be simplified further, since it can be shown that if two closed manifolds are equal up to a boundary the difference between the anomaly theories on the two spaces is encoded in the local anomaly. Thus if the latter vanishes, the anomaly depends on the equivalence classes of spaces defined up to a boundary. This space actually forms an abelian group under the disjoint union known as the  bordism groupΩ_D+1^X_str. Therefore to cancel anomalies it is enough to evaluate 𝒜_T on the generator of this group.
    
     In 6D (super)gravity, local anomalies are cancelled by employing the celebrated Green-Schwarz-Sagnotti mechanism, introduced in the previous Chapter, which requires the Bianchi identity  
    
        d H_3^α = X_4^α =  a^α/4  p_1(R) - ∑_i b_i^α/λ_i  c_2(F_i)   ,
     
    to be satisfied where we have used the definition the characteristic classes[For the abelian case studied in <ref>, the second Chern class should be replaced by -c_1^2/2.]1/4 p_1(R)= -1/2tr R^2, c_2(F)= 1/2tr F^2 in (<ref>). This means that, in the procedure of extending manifolds, one must preserve the structure inherited by the Bianchi identity, the so-called  twisted-string structure<cit.> (or string-G, for a given gauge group G), in addition to the spin structure with a principal bundle[In certain settings the spin structure can be modified to a pin^±, spin^c or spin^ℤ_4 structure. This is crucial to find the correct consistency conditions, for instance when reducing M-theory over non-orientable manifolds where I-fold defects emerge <cit.> and certain topological symmetries are broken <cit.>.] valid for the case of gauge theories with fermions. Turning off any gauge contribution reduces to the ordinary string structure, yielding the bordism group Ω_7^string = 0. This shows that there are no purely gravitational Dai-Freed anomalies in six dimensions when the Green-Schwarz-Sagnotti mechanism cancels the local anomaly. In the presence of gauge fields, twisted string structures can instead generate novel anomalies <cit.>. If such an anomaly is found it may still be possible to cancel it with a “topological” version of the Green-Schwarz-Sagnotti mechanism <cit.>, but we will not consider this subtle possibility in the present discussion.
    
    
     The right-hand side of (<ref>), which we dub the Bianchi class, is thus trivial in de Rham cohomology. However, one should require the Bianchi class to be trivial in integral cohomology, which can include torsional classes undetected by differential forms. This was originally argued in <cit.>, and (at least for perturbative heterotic strings) follows from Dai-Freed anomaly cancellation on the world-sheet <cit.>.  However, specifically for theories in D=6, there is an additional subtlety to be taken into account. These settings allow for the presence of many chiral 2-forms which lead to additional Bianchi identities. As discussed in <cit.>, it seems that the correct requirement to impose on admissible anomaly backgrounds is to satisfy at least one identity (at the level of integral cohomology). This turns out to eliminate all anomalies in string theory examples, and can produce non-trivial bordism groups. Imposing multiple identities simultaneously generically kills all independent characteristic classes. Therefore in the following examples, we shall focus on backgrounds satisfying at least one Bianchi identity but spaces trivialising all the available ones have been discussed in <cit.>.
    
     Nevertheless, even imposing one Bianchi identity, the computation of these bordism groups is a complicated mathematical task that has not been fully addressed yet. Therefore, we cannot push until the end the proof of the cancellation of Dai-Freed anomalies for string theory vacua but, we can analyse backgrounds satisfying the Bianchi identity on which the anomaly theory can be computed. These backgrounds are Lens spaces, given by the quotient of the seven-sphere with a discrete group, L_p^7=S^7/ℤ_p<cit.>. On the one hand, this allows to verify, without knowing the exact shape of the Ω_7^string-G, the cancellation of anomalies for (non-)supersymmetric heterotic theories and for the Gepner orientifold with no tensor multiplets. On the other hand, it allows to seek constraints for low-energy theories from a bottom-up perspective.
    
     For our discussion it is therefore crucial to specify which is the anomaly theory for the fields of interest. In the case of 6D non-supersymmetric heterotic theories the fields contributing to the anomaly are simply Majorana-Weyl fermions and the 2-form fields, while if supersymmetry is present there is an additional contribution coming from the gravitini. In the following, we are going to specify the structure of 𝒜_T for each of these fields and their values on the Lens spaces that are needed for the rest of the Chapter.
    
    
    
     §.§ Anomaly: fermions
    
    
    Let us start by discussing the contribution of fermionic fields to the anomaly theory. This is the context in which the Dai-Freed theorems were originally formulated <cit.>. The partition function for a chiral fermion is obtained from that of a massive Dirac spinor defined on the extended space Y by imposing the set L of boundary conditions which localises a chiral fermion of a given chirality on the boundary X<cit.>
        𝒵_1/2,+ (L, Y  )=  | Pf ( D_D^+(X)  )  | e^-2 i πη_D (Y)
    
    where the Pfaffian of the Dirac operator is the regularised product of the positive eigenvalues[When the Dirac operator in the bulk Y has zero modes, the boundary conditions should be properly modified (see for instance <cit.> for details). However for the analysis of anomalies such subtleties play no role and we shall skip them.]. The phase given by η (Y) is the so-called  eta invariant<cit.> and corresponds to the regularised sum of the signs of the eigenvalues of the Dirac operator. From the general considerations above, because of the glueing properties of the η-invariant, the anomaly theory is nothing but the phase of (<ref>)
        𝒜_D(Y_cl)= η_D(Y_cl)   ,
    
    that, in the case Y_cl=∂ Z, reduces to <cit.>
        Index^D= η_D(∂ Z) + ∫_Z  ( Â(R)  ch_R(F)  )  .
     
    Here ch_R(F)=tr_R  e^i F is the Chern character for the associated gauge bundle in the representation R.
    
     
     In a similar fashion one can write the partition function of chiral gravitini. However, one must take into account that a massive Rarita-Schwinger field in D+1 dimensions localises a chiral Rarita-Schwinger field and a Weyl fermion of opposite chirality on the D-dimensional boundary[We are assuming D to be even in these arguments.]<cit.>, and that the D-dimensional Rarita-Schwinger operator identifies the degrees of freedom of the propagating gravitino and a Weyl fermion. As a result, the structure of the partition function for the D-dimensional gravitino field is dictated by
    	
        𝒵_grav,+ (L, Y  )=  | Pf ( D_RS^+(X)  )  | / | Pf ( D_D^+(X)  )  |^2   e^-2 i πη_RS (Y)+ 4 i πη_D(Y)    ,
    
    where the operator D_RS^+(X) is the Rarita-Schwinger operator defined on the product between the spin and the tangent bundle.  The anomaly is thus described by the combination of eta invariants of the Dirac and Rarita-Schwinger operators <cit.>
        η_grav(Y_cl)= η_RS(Y_cl) - 2  η_D(Y_cl)   ,
    
    where the phase η_RS(Y_cl) is related to the density index via the Atiyah-Patodi-Singer (APS) index theorem <cit.>
        Index^RS= η_RS(∂ Z) + ∫_Z   ( Â(R)   ( ch(2 R) - 1)  )   ,
     
    with ch(2 R)= tr  e^ 2 i R.
    
     On Lens spaces the values of the η-invariants are known and we will use their expression on L_p^7 to compute explicitly the anomaly theory for fermions. To do so, we can embed the ℤ_p line bundle into the gauge group bundle, implying the decomposition of a given representation in terms of the ℤ_p charge q. As a result, the contribution to the anomaly theory of fermions transforming in a given representation of the gauge group comes from η-invariants carrying non-trivial ℤ_p charges associated to the representation itself <cit.>
        η_D_q(L^7_p)     = -  p^4 + 10p^2 - 11 - 30 p^2 q^2 - 60 pq + 60 pq^3 - 30 q^4 + 60 q^2/720p  , 
        η_RS_0(L^7_p)     = -  7p^4 - 170p^2 + 163/720p  ,
     
    where for our case we only need the uncharged η-invariant for the Rarita-Schwinger field[When discussing the consistency of type IIB supergravity under the full S-duality group <cit.> charged η-invariants for the Rarita-Schwinger operator are relevant, but it will not be of interest for us.].
    
     The discussion presented so far describes the contribution to the anomaly theory of the fermionic fields. This, however, is not the end of the story, since string theory and (super)gravity include additional relevant fields in their spectra: the (chiral) p-form fields.
    
    
    
    
    
    
     §.§ Anomaly: bosons
    
    
    In string theory, p-form fields are ubiquitous and may give a non-trivial contribution to the anomaly theory. However, although the action and partition function of non-chiral forms are well-understood and under control, there are difficulties in the definition of similar actions for chiral forms, which prevents one from writing down consistent partition functions straightforwardly. Furthermore, the standard techniques that describe abelian gauge fields (whether chiral or not) in terms of differential forms miss all restrictions arising from a consistent coupling to matter in topologically non-trivial backgrounds. The formalism that naturally incorporates the holonomy of the U(1) gauge bundle encoding a non-trivial topology describes p-form fields as classes in a (generalised) differential cohomology Ȟ^*(X), known as Cheeger-Simons <cit.> or, equivalently, Deligne cohomology <cit.>. Following <cit.>, we can translate such a piece of information in terms of cochains, according to which an abelian p-form gauge fields is given by
    
        𝐀̌= ( A, N, F)  ∈ Ȟ^p(X)  ,
    
    where A ∈ C^p(X, ℝ) (the space of real p-cochains) is the gauge connection, N ∈ C^p+1(X, ℤ) defines the  characteristic class and F ∈ C^p+1(X, ℝ) is the associated field strength. The non-trivial topological data are encoded in the characteristic integer cohomology element [N] ∈ H^p+1(X,ℤ), which carries torsional information invisible to de Rham cohomology curvature classes [F] ∈ H_dR^p+1(X). The two are connected via to the co-boundary of A
        N= F - δ A   ,
    
    which however hides a gauge redundancy <cit.>
        A → A + δ a + n   ,      	N → N - δ n   ,
    
    with a ∈ C^p-1(X,ℝ) and n ∈ C^p(X, ℤ). This constrains the structure of the action and of the partition function, which should make sense at the level of differential cohomology, and, thus must be invariant under the transformations in (<ref>). 
    
     The situation however is substantially different depending on whether the p-form uplifted in differential cohomology is or is not chiral. For non-chiral forms, requiring that the action be invariant under the gauge transformations (<ref>) when 𝐀̌ is electrically coupled to 𝐁̌∈Ȟ^p+1(X) and magnetically to 𝐂̌∈Ȟ^D-p+1(X) implies the presence of an additional SPT phase <cit.> (and local counter-terms) defined on Y.  As a result, one can write the partition function for the non-chiral p-form fields as
     
        𝒵_𝐀̌ (Y,L)= e^2 π i (-1)^D-p ( 𝐂̌, 𝐁̌ )_Y ∫ [ D 𝐀̌ ] e^-S[𝐀̌,X]  ,
    
    where we have introduced the cohomology pairing <cit.>
        ( 𝐀̌_1, 𝐀̌_2  )_Y= ∫_Y A_𝐀̌_1 *𝐀̌_2  ,
     
    dictated by the product between differential characters[The present definition uses the notion of the  cup product<cit.> and the homotopy cochain q(ω_1,ω_2) acting on differential forms ω_1 and ω_2 according to	
    	
        ω_1 ∧ω_2 - ω_1 ∪ω_2 = q(δω_1, ω_2)+ (-1)^p_1 q(ω_1, δω_2) + δ q( ω_1,  ω_2)   .
    ]
        N_𝐀̌_1 *𝐀̌_2= N_𝐀̌_1∪ N_𝐀̌_2  ,
        	   
            F_𝐀̌_1 *𝐀̌_2= F_𝐀̌_1∧ F_𝐀̌_2  ,
        	   
           A_𝐀̌_1 *𝐀̌_2= A_𝐀̌_1∪ F_𝐀̌_2 + (-1)^p_1+1 F_𝐀̌_1∪ A_𝐀̌_2 + q(F_𝐀̌_1, F_𝐀̌_2 )   .
    
    The definition in (<ref>) holds only when the electrically coupled field 𝐁̌ is topologically trivial on X. Although such requirement seems rather restrictive, it actually corresponds to the unique available choice, since whenever 𝐁̌ is non-trivial the partition function vanishes (see <cit.> for details). Furthermore, in writing the expressions above we have extended the definitions of the background fields to 𝐂̌∈Ȟ^D-p+1(Y) and 𝐁̌∈Ȟ^D-p+1(Y), for which the topological triviality must hold only on the boundary X.
     
      The anomaly is thus given by
    
        𝒜_𝐀̌(Y_cl)= (-1)^D-p ( 𝐂̌, 𝐁̌ )_Y_cl  .
    
    In our context, the background fields are the combination of the first Pontyagin and the second Chern classes entering eq. (<ref>). On Lens spaces, the fields are flat and thus the cohomology pairing depends only on the integer flux. However, requiring the twisted string structure to hold implies that at least one of the two fields is also topologically trivial, thus yielding a vanishing pairing.  
    
     If the p-forms are instead chiral, we can describe the partition function of the p-form field as a boundary mode of a massive p+1-form field, similarly to what has been done for chiral fermions. The resulting expression is quite cumbersome and can be found explicitly in <cit.> wich also provides a detailed derivation. For our purpose it is enough to notice that the expression for the anomaly theory of chiral p-forms coupled to a background field Č∈Ȟ^p+1(Y) is given by 
    
        𝒜_𝐀̌(Y_cl)= 𝒬_Y_cl(Č) ≡𝒬_Y_cl(Č) - 𝒬_Y_cl(0)   ,
    
    where we have introduced the (inhomogeneous)  quadratic refinement𝒬<cit.> of the cohomology pairing, defined via the characteristic equation (see e.g.<cit.> for an application in a physical context)
    
        𝒬_W( Ǎ_1+ Ǎ_2 ) -  𝒬_W( Ǎ_1 ) -  𝒬_W(  Ǎ_2 ) + 𝒬_W ( 0 )=  ( Ǎ_1,  Ǎ_2  )_W  ,
     
    on a probe manifold W. In general, the choice of the quadratic refinement is dictated by a general solution of the characteristic equation, subject to the constraint that whenever Y_cl= ∂ Z
        Q_∂ Z(Č)= ∫_Z {1/2 ( w + F_Č )^2 - 1/8L̂ (R  )}  ,
    
    where L̂ (R  ) is the Hirzebruch polynomial that has already appeared in eq. (<ref>). In eq. (<ref>), we have introduced an integral lift w of the  Wu class defined as the characteristic element of the cup product pairing <cit.>, corresponding in six dimensions to w=- p_1/4. In our case the background field Č is given by the Chern character č, whose curvature and characteristic classes are simply given by the second Chern class c_2(F) as an element of de Rham and integer cohomology, respectively <cit.>. In general, these requirements do not admit a unique solution and up to now it is unclear how to choose the right quadratic refinement systematically. Moreover, for the settings relevant for our discussion, chiral fields also take values in a lattice of signature (1,n_T), depending on the number n_T of tensor multiplets. Hence, the previous considerations have to be slightly modified. Specifically, chiral forms arise as boundary modes of Ȟ^4(Y) ⊗Λ<cit.>, where Λ is the SO(1,n_T) lattice determined by the structure of local anomalies (<ref>). In this scenario, the lattice-valued quadratic refinement on Y_cl= ∂ Z reads
    
        Q_∂ Z^Λ(č)= ∫_Z {1/2 ( a_0 w +  b_0 F_č )^2 -  1/2∑_i=1^n_T ( a_i w +  b_i F_č )^2 + n_T-1/8 L̂ (R  ) }  .
    
    Notice that (<ref>) can be equivalently interpreted as the contribution of one antiself-dual and n_T self-dual fields, as befits the traditional additive view on anomalies for chiral p-form fields.
    
     Determining the quadratic refinement on a generic manifold Y can be cumbersome and is uniquely fixed only for null-bordant manifolds, as described in <cit.> for Spin-bordism. In the settings at stake, we do not know if the twisted string bordism group vanishes, nor if the relevant backgrounds actually lie in the trivial class, so 𝒬_Y could  a priori be different. Nevertheless, we can exploit the fact that in the purely gravitational case Ω_7^string=0 to constrain 𝒬_Y(0). Applying the APS index theorem to (<ref>), this gravitational contribution can be cast in the form[Strictly speaking, since the Bianchi class is either p_14 or 34 p_1, it is not obvious that the expression of 𝒬(0) is unique as in the case of the standard string structure trivialising p_12.]
        𝒬_∂ Z ( 0)= ∫_Z  ( 1/2 ( a/4 p_1   )^2- 1/8L̂ (R  )  ) = -  7(35 a^2-3) /8 η_D(∂ Z) +(a^2-1) /8 η_grav(∂ Z)   ,
    
    
    where a is the relevant coefficient in the Bianchi classes in (<ref>) and η_grav is defined in (<ref>). Then, one can use the formulae in (<ref>) to evaluate this expression on Lens spaces. However, turning on the gauge fields spoils the preceding observations and one can only consider a generic solution of the characteristic equation. On Lens  spaces, a general solution to the characteristic equation can be explicitly computed and can be parametrised by an integer m=0, …, 2p-1 as <cit.>
        𝒬(Ǎ )= -  a(a+m)/2p  ,
    
    since the cohomology pairing on Lens spaces is
    	
        ( Ǎ, 𝐁̌ )_L_p^7= -  a b/p  ,
      
    where [ N_Ǎ ]= a y and [ N_𝐁̌ ]= b y, with y a generator of the integral cohomology group.
    
     In top-down examples, the choice of the quadratic refinement may not be arbitrary. For instance, in <cit.> it has been argued that F-theory selects a single possibility for ℤ_3 gauge groups, although a general rule to determine 𝒬 is not known. A complete analysis that describes the combination of quadratic refinements allowing to cancel the anomaly can be achieved in the simple setting of SU(2) models, but in general one would need some restrictions on the possible quadratic refinements. 
    
    
     Now that we have specified all the ingredients of our anomaly theory, we can proceed to evaluate Dai-Freed anomalies for supergravity and string theory vacua. In <cit.>, the analysis has been carried out for supergravity with at most one tensor multiplet and simply-laced gauge groups. In the following, we shall only discuss the simple setting for the SU(2) with n_T=1 and the abelian U(1) gauge group without tensor multiplets, since the cases for all the other gauge groups have only increasing technical difficulties, without any additional conceptual novelty. We shall then discuss the compactification of the non-supersymmetric SO(16) ×SO(16) heterotic theory on the T^4/ℤ_6 orbifold and the Gepner orientifold with no tensor multiples, following the analysis in <cit.>.  
    
    
    
    § DAI-FREED ANOMALIES FOR SU(2) IN 6D SUPERGRAVITY
    
    
    In this Section we shall focus on six-dimensional supergravity with one tensor multiplet and SU(2) as a gauge group. In such a setting, the contribution to the anomaly theory involves 3 (the adjoint representation) vector multiplets, d_T[In the following the multiplicities of the various representations R will be denoted d_R.]  (d_F) hyper multiplets transforming into the trivial (fundamental) representation, the gravity multiplet and the neutral tensor multiplet. The latter two comprise, aside from the usual contribution coming from chiral fermions, 2-form fields, which, as described in the previous Section, play a non-trivial role in the structure of the anomaly theory. The fermionic contribution is already highly constrained by local anomalies. Indeed, the cancellation of the irreducible purely gravitational piece of the anomaly polynomial implies that the number of trivial and fundamental hyper multiplets satisfy   
    
        d_T +  2 d_F =247   .
         
    Notice that here we are looking at constraints for  would-be  UV complete theories with SU(2) as the full gauge group. If SU(2) is embedded into a larger gauge group, eq. (<ref>) still holds by defining d_T as a signed integer, where the negative contribution arises from the vector multiplets transforming into the adjoint representation of the turned-off gauge groups. Such a case would imply a much larger family to consider, and the control over these theories is lost. For the purpose of uncovering the role played by Dai-Freed anomalies in sharpening the string landscape or testing particular (top-down) examples, this consideration can be ignored. Therefore, taking into account the cancellation of the irreducible gravitational anomalies, the reducible anomaly polynomial takes the form
    
        I_8 =  ( tr R^2  )^2- 1/24tr R^2  tr F^2  ( d_F-4 ) + 1/48 ( tr F^2  )^2   ( d_F - 16  ) ,
    
    where we used the decomposition tr F^4= 1/2 ( tr F^2  )^2 valid for SU(2). In order to cancel the reducible piece of the anomaly, we shall impose the factorisation (<ref>), where however the implementation of p-form fields as elements of differential cohomology requires that the 't Hooft coefficients be integer-valued and embedded into a self-dual lattice <cit.>. With one tensor multiplet, such embedding is possible in two ways, depending on whether the metric Ω_αβ is either diagonal or off-diagonal. This translates into constraints on the allowed values of d_T and d_F. For instance, the off-diagonal case,
    	
        Ω=	[ 0 1; 1 0 ]  ,
    
    allows a factorisation of the anomaly polynomial if d_F=4 + 12 s, which yields
    	
        I_8 = 1/4 ( 2 tr R^2 -  tr F^2   )   ( 2 tr R^2 +  (1 - s)  tr F^2   ) .
     
    This factorisation determines the structure of the twisted string bordism group implied by the Bianchi identities
    	
        d  H_1= 12 p_1 + c_2
        			  , 
            d  H_2= 12 p_1 +(s-1) c_2   .
     We now build SU(2) bundles on L_p^7 backgrounds by including the defining line bundle of ℤ_p into SU(2). Following <cit.>, there are two possibilities to describe such inclusion. Indeed, we can embed ℤ_p as a subgroup of U(1) into k diagonal blocks in the fundamental representation of SU(2), which however only allows two values for k, namely k=0,1. For k=0 the gauge bundle is trivial and one only probes the gravitational anomaly via the uncharged η invariants (<ref>)𝒜(L_p^7) = -  p^4 + 11p^2 - 12/3p  .
    
    Notice that in this case there is no quadratic refinement in general, but the only allowed backgrounds are ordinary string manifolds with p_12 = 0. For Lens spaces this requires p=2, and L_2^7 = ℝP^7 is the real projective 7-space. The resulting gravitational anomaly
    
    𝒜(ℝP^7)    = -8 
        
       = 0   mod  1
        
       ≡_1 0
    
    
    consistently with the fact that it is a bordism invariant of Ω_7^string = 0. Therefore, in the following analysis of n_T=1 models, we will not consider trivial gauge bundles. Here and in the following we shall use the short-hand notation ≡_p to indicate the equivalence modulo p.
    
     For k=1, we are embedding the ℤ_p subgroup into the Cartan subalgebra of SU(2), and we can use the eigenvalues for the Cartan generator to deduce how the contribution of charged fermions decomposes into suitable combinations of η_q. The eta invariants η_q are known and thus the anomaly for k=1 is given by
    	
        𝒜 (L_p^7 )    = d_F ( η_1 + η_-1 )-  ( η_2 + η_-2+η_0  ) + (d_T+1) η_0- η_grav
           ≡_1 d_F (p^2-1) - 4p^4 + 40/12p  .
     The Bianchi classes are now encoded in (<ref>), and one must evaluate them on Lens spaces by using the known fact that p_1/4=y generates the integer cohomology group H^4(L_p^7, ℤ) = ℤ_p, and the second Chern class is given by c_2(L_p^7)=-k y. This means that eq.(<ref>) evaluates to (2-k)y and (2-k(s-1))y. For k=1, The only Bianchi class that can be trivialised on a Lens space is (3-s)y. Thus, setting s = 3 + m p for some integer m the Lens background has the appropriate twisted string structure, and one finds
    
    𝒜(L^7_p) ≡_1 -  (p-1)   p   (p + 1)/3  ,
    
    
    which also vanishes because the numerator is always divisible by three.
    
     For the diagonal lattice, in which the bilinear form reads
    	
        Ω=	[  1  0;  0 -1 ]  ,
    
    the number of fundamentals allowing an integral factorisation  of the anomaly polynomial has to be of the form
    	
        d_F=10 + 12 s   .
    
    For these values it is easy to see that the polynomial factorises according to 
    	
        I_8 = 1/8{ ( 3 tr R^2 - s tr F^2   )^2 -   ( tr R^2 +  (1 - s)  tr F^2   )^2 }.
     A complete study of Dai-Freed anomalies of such theories would require knowing the proper SU(2)-twisted string bordism group, where the twisted string structure is spelt out by the Bianchi identities
    	
        d  H_1= 34 p_1 + s  c_2   ,
        			
            d  H_2= 14 p_1 + (s-1) c_2   .
     
    Following the analysis above, we restrict to Lens spaces L_p^7 which provide particularly simple families of candidate backgrounds to work with, even though we do not know if these exhaust all the nontrivial bordism representatives. 
    	
     We build SU(2) bundles on L_p^7 backgrounds by including the defining line bundle of ℤ_p into SU(2) according to the possible embeddings described before for k=0,1. For k=0, the anomaly theory reduces to the purely vanishing gravitational anomaly
    𝒜(L_p^7) = -  p^4 + 11p^2 - 12/3p + 𝒬_+(0) - 𝒬_-(0) = 0   .
    
    For k=1, charged fermions decompose into representations with different ℤ_p charges, for which the eta invariants η_q are easily calculated. The anomaly for k=1 is thus given by
    	
        𝒜 (L_p^7 ) ≡_1 d_F (p^2-1) - 4p^4 + 40/12p + 𝒬_+ - 𝒬_-   ,
    
    where we have added to (<ref>) the contribution of the quadratic refinement for the two chiral 2-forms that on Lens spaces is given by (<ref>).
    
     We can now proceed to find the Lens spaces which satisfy the twisted string structure on which we are going to compute the relevant η-invariants and the quadratic refinement. For k=1 the Bianchi classes are (3-s)y and (2-s)y and one can show that for any Lens space trivialising one of the Bianchi classes, the fermionic anomaly is always an integer multiple of 1/p. This means that there always exists an appropriate choice of m_± in (<ref>) that trivialises the anomaly theory. This is crucial, since the heterotic and F-theory landscapes include models in this family.  However, the choice of the quadratic refinement constrains the dynamics of chiral p-form fields in a very specific way, making other realisations of chiral forms inconsistent. In particular, if one uses the suitable combination of η invariants reproducing locally the term (<ref>) along the lines of <cit.>, the anomaly cannot be cancelled.
    
     Finally, there is a family of models that admits both the diagonal and off-diagonal embedding given by d_F=16+24s. For this family, we can evaluate the anomaly theory in the same way as it has been done for the off-diagonal model and indeed the anomaly theory can be shown to be given by (<ref>).
    
     To summarise, 6D supergravity models with fermions in both trivial and fundamental representations give rise either to a vanishing anomaly if the 2-form fields are non-chiral in generalised differential cohomology, or to an anomaly that a specific choice of the quadratic refinement can cancel. As shown in <cit.>, the discussion follows a similar pattern and leads to the same conclusions if the set-up is modified by considering U(1), SU(n), Spin(2n), E_7 and E_8 gauge groups with hyper multiplets in fundamental, adjoint, symmetric and anti-symmetric representations, still under the assumption of the presence of one tensor multiplet. We can perform a similar analysis even for the case of no tensor multiplets that can be studied by considering the sample of models given by the infinite families of <cit.>.
    
    
    
    § DAI-FREED ANOMALIES FOR U(1) IN 6D SUPERGRAVITY
    
    
    In this Section we are going to focus on U(1) models with no tensor multiplets, that are known to give rise to different infinite families satisfying all the known swampland criteria but for which no string or F-theory realisation is known <cit.>. Without tensor multiplets there is only one possible embedding of the anomaly which unavoidably entails chiral 2-forms. This means that the choice of the quadratic refinement is crucial in order to address the consistency of these theories, and indeed we are going to argue that the anomaly theory is eventually non-trivial for large charges, thus excluding all but a finite number of these models. Actually, it is always possible to choose a quadratic refinement that cancels the anomaly but, to date, we are still lacking a top-down mathematical principle selecting a suitable 𝒬[It can be also argued that the presence of families with arbitrary large charge increase the effective gauge coupling g_eff = gQ, which in D=6 has dimensions of length, and it can be observed to violate perturbatively the (upper bound to the) quantum gravity cutoff Λ_QG determined by the magnetic weak gravity conjecture <cit.>. Intuitively, this may suggest that large charges are obstructed in weakly coupled gauge theories when quantum gravity is involved.].
    
     As we mentioned, without tensor multiplets there is only one possible lattice embedding with a=3, and the factorised anomaly polynomial is
    
    	I_8   = 12 ( 34 p_1 + b2 c_1^2   )^2   .
        
    In <cit.> a family of charges q,r,q+r admitting such factorisation was found, with fixed multiplicities
    
        d_q= 54   ,      d_r= 54   ,      d_q+r= 54   .
    
    Here q,r are integers determining the anomaly coefficient b=- 6( q^2 +r q + r^2).
    
     The anomaly theory for these models is described by
    
        𝒜 (Y  ) =    27 ( η_q + η_-q- 2 η_0) +  27 ( η_r + η_-r- 2 η_0)
         
           +  27 ( η_q+r + η_-q-r- 2 η_0) -b(b+2m)/8p  ,
    
    where we have considered the general form of the quadratic refinement in terms of an integer parameter m[Notice that the gravitational anomaly is cancels out because of the definition of the quadratic refinement and the triviality of the corresponding string bordism group.]. Using the expressions contained in (<ref>), we can compute the value of (<ref>) for Lens spaces L_p^7 with p a divisor of | 3-3( q^2 +r q + r^2)|, which trivialises the Bianchi class. For general m and p, one has
    
    𝒜(L_p^7) ≡_1 β (p^2+3m-18)/2p  ,
    
    
    where β≡ -b/6 = q^2+q r+r^2 is the only combination of charges that appears. This is also true for the Bianchi class, which is 3(1-β)y. For p=3 one has 
    
    𝒜(L_3^7) ≡_1 β (m-3)/2  ,
    
    
    which vanishes either for even β (q and r even) or m = 3 (mod 2). For general β, p=β-1 remains a valid choice, and we now study this case for large charge. We restrict to choices of quadratic refinement such that m is bounded (mod 2p), and show that all but finitely many theories are excluded with this assumption unless m=6. For this family of models the choice of <cit.> amounts to m = p^2-2, which for p even is bounded mod 2p for p = β-1 ≫ 1. For p odd, m ≡_2p p-2 = β - 3 with β even, so for this choice m is not bounded and a separate analysis is needed.
    
     To begin with, notice that if (q,r) →∞ in ℝ^2 then β→ +∞. Indeed,
    
    
        2β - |(q,r)|^2 = (q+r)^2 ≥ 0  ⟹ β≥1/2  |(q,r)|^2 → +∞  .
    
    
    For m=6 the anomaly always vanishes. Assuming that m ≠ 6 and bounded (taken between 0 and 2p-1), letting p = β-1 one has
    
    𝒜(L_β-1^7)    β≫ 1∼β (β^2-β+3(m-6)) + 3(m-6)/2β + 3(m-6)/2β^2
       ≡β  N(β) + N_0/2β + N_0/2β^2  ,
    
    
    where for our purposes we only need use that N(β) ∈ℤ[β] is a polynomial with integer coefficients and that N_0 ∈ℤ is an integer. The first term cannot be integer for large enough β, since the numerator cannot be a multiple of β, for β > N_0. The second term is subleading for large β, and thus cannot compensate the fractional part since the series is asymptotic. Furthermore, having packaged the divergent part in the first term, all other terms resum to a finite result, namely the limit of (<ref>) (as a rational number rather than mod 1) with its divergent contribution subtracted. All in all, in order to cancel the anomaly on L_β-1^7 one necessarily needs m=6, whereas for p=3 only even charges survive unless m=3 mod 2. If m does not depend on the choice of background, one can save at most even charges, if any.
    
     In order to conclude our analysis for the choice of quadratic refinement of <cit.>, for even β≡ 2z and p=β-1, the parameter m = p^2-2 ≡_2p p-2 = 2z-3 is not bounded for z ≫ 1. The anomaly simplifies to
    
    𝒜(L_2z-1^7) ≡_1 2z^2+z-13/2z-1≡_1 -  12/2z-1  ,
    
    
    which once more shows that the anomaly is eventually non-vanishing.
    
     The result just described is confirmed by a numerical scan. Indeed, fixing the quadratic refinement to the choice m=p^2-2, which corresponds to the definition in <cit.>, only 6 models out of 441 are anomaly-free[There are additional models for which the analysis cannot be performed, since they would formally yield p=0 as valid backgrounds. These correspond to (q,r)={ (-1,0), (0,-1), (1,0), (0,1), (1,-1), (-1,1) }.]. The charges are allowed to vary with |q|≤ 10 and |r|≤ 10, but the non-anomalous models are bounded by |q|=2 and |r|=2, as listed in table <ref>.
      To summarise our findings, without a mechanism indicating which specific quadratic refinements are selected in string or F-theory, we cannot place these theories in the swampland with certainty. Näively one can expect to exclude most of them, since only a very specific choice for each background cancels all the anomalies that we have investigated. It would be interesting to further explore this issue, trying to deduce a top-down criterion to select the allowed quadratic refinement(s). Nonetheless, we now know that, unless m = 6, no choice with m bounded can cancel the anomaly on L_β-1^7 for infinitely many charges, while the anomaly on L_3^7 only saves even charges unless m=3 mod 2.
    
    
    
    § THE GEPNER ORIENTIFOLD WITH NO TENSOR MULTIPLETS
    
    
    The importance and role played by the quadratic refinement can be further investigated from the allowed top-down string theory realisations. Gepner <cit.> has shown that consistent superstring world-sheet theories in a background with SU(n) holonomy can be realised in terms of tensor products of 𝒩=2 superconformal minimal models, corresponding to special points of the moduli space of a compactification on Calabi-Yau manifolds CY_n. Focusing on type IIB strings compactified in D=6 on CY_2, the chiral spectrum is unique and comprises the gravity multiplet and twenty-one tensor multiplets with 𝒩=(2,0) spacetime supersymmetry. The orientifold algorithm <cit.> then allows to halve the number of supercharges to 𝒩=(1,0), but the light spectra depend on the point of the moduli space in which the model is constructed. In the following, we will focus on the orientifold projection of the Gepner model with 81 characters worked out in <cit.>, comprising ten hyper multiplets in the antisymmetric representation of the Chan-Paton gauge group SO(8) from the open-string sector and twenty-one neutral hyper multiplets from the closed-string sector. Our interest in this model is motivated by the absence of tensor multiplets, which highlights the single chiral 2-form contained in the gravity multiplet.
    
     The full anomaly theory for a generic closed manifold Y contains a single quadratic refinement, and takes the form
    
        𝒜 ( Y  )= (10-1)  η_28 (Y) + 21  η_0 (Y) - η_grav (Y) + 𝒬(Y)   .
    
    On boundaries Y=∂ Z, (<ref>) reproduces the anomaly polynomial
    	
        𝒜 ( ∂ Z  )    ≡_1 - ∫_Z I_8 = - 9/8∫_Z  ( tr R^2 - tr F^2 )^2   ,
    
    from which one reads the Bianchi identity
    
        d H= 3  ( 14 p_1 + c_2  ) = X_4   .
    
    The Green-Schwarz-Sagnotti term cancelling (<ref>) arises from the quadratic refinement 𝒬_Y ( 3 č)=𝒬_Y( 3 č) + 𝒬_Y(0), contributing to the gravitational anomaly as
    
        𝒬_Y ( 0)= ∫_Z {1/2 ( a/4 p_1   )^2- 1/8 L } = -  7(35 a^2-3) /8 η_D(Y) +(a^2-1) /8 η_grav(Y)   ,
    
    
    where a=3 in the present case. Thus the only unknown contribution is 𝒬_Y( 3 č), which solves (<ref>) and can be parametrised as in (<ref>) on Lens spaces. To evaluate the Bianchi identity, we need to specify the second Chern class for Spin(2n) bundles, which corresponds to c_2(L_p^7)=- 2k y<cit.>, with k=0,1,2 parametrising the allowed bundles. Eq. (<ref>) singles out Lens spaces L_p^7 with  p=3, p = 1 - 2 k or p = 3(1 - 2k). Thus, p=3   , 9 are allowed, depending on the value of k. In these cases, the anomaly in (<ref>) evaluates to 
    
        𝒜 (L_p^7)    = 9 (k (2 k - 1)(η_1+ η_-2) + 4 k (4 - 2 k)( η_1+ η_-1)  + 𝒬_Y( 3 č)
            
                + ( 18 (k (2 k - 1) + 4 k (4 - 2 k))  28 · 9 + 21 - 7 · 39 )η_0
            
            = 3k/p( m + 3 (p^2 - 2kp + p - 2) ) 
            
           ≡_1 3k/p  (m - 6)   ,
    
    where the quadratic refinement is parametrised as in (<ref>),
    
        𝒬_Y( 3 č)= - -6 k(-6k+m)/2p  ,     with  m=0,…, 2p-1   mod  2p   .
    
    Choosing p=3, valid for all k, manifestly trivialises (<ref>), while for k=2 and p=9 one finds
    
        𝒜 (L_9^7) ≡ _1 2/3  m   ,
    
    which cancels only for m=0 mod 3. This is not the value of m one finds for the choice in <cit.>, corresponding to m=2. All in all, we have shown that the anomaly vanishes in this model for a specific choice of quadratic refinement for all the allowed k and p, although the rationale behind this choice remains obscure.
     
    
    
    
    
    § NON-SUPERSYMMETRIC HETEROTIC MODELS
    
    	
    In the previous Sections we have discussed the role played by Dai-Freed anomalies as a consistency swampland criterion for six-dimensional 𝒩=(1,0) supergravity theories with simply laced and abelian gauge groups. The reason why one could call this a swampland condition lies in the origin of Dai-Freed anomalies, which require spacetime topology change. It is natural to think of this as an intrinsically quantum-gravitational effect, whereby cancellation of such anomalies may be unnecessary if the theory is not coupled to gravity. 
    
     The analysis performed in <cit.> of supergravity theories in D=6 involve K3 compactification in the orbifold limit of the heterotic SO(32)<cit.> and E_8 × E_8<cit.> theories, for which it has been shown that anomalies are cancelled or there exists a specific choice of the quadratic refinement to guarantee it. This result is neither new nor surprising, since <cit.> has shown that anomalies of this kind are always absent in heterotic constructions when supersymmetry is present. 
    
     However, an analogous general result is not available for non-supersymmetric vacua[To our knowledge, neither for supersymmetric and non-supersymmetric orientifold vacua a similar result is available. Hence, also the possibility to cancel anomalies for the Gepner orientifold was not guaranteed.] and thus there is no a priori guarantee that Dai-Freed anomalies would cancel for these theories as well. Nevertheless, the cancellation of Dai-freed anomalies is connected to the consistency under topology change and thus makes no reference to supersymmetry. Hence, it is important to check their absence in non-supersymmetric settings. This program has been initiated in <cit.>, where ten-dimensional tachyon-free models have been shown to be free of Dai-Freed anomalies. This also implies that any smooth geometric compactification thereof is anomaly-free, but, in principle, lower-dimensional vacua may arise from different constructions. In light of these considerations, the purpose of this section is to verify that Dai-Freed anomalies on Lens spaces cancel for certain non-supersymmetric heterotic orbifolds in six dimensions. For concreteness, we shall discuss SU(2) anomalies on Lens spaces for the SO(16)×SO(16) heterotic model[The global form of the gauge group is not in fact SO(16) ×SO(16), but this subtlety will be immaterial for our considerations.] compactified on K3 in its orbifold limits. For simplicity we shall study orbifolds <cit.> whose point group P is a discrete subgroup of the SU(2) holonomy of K3, and particular emphasis will be put on the T^4/ℤ_6 orbifold realisation[The complete analysis can be found in <cit.>.]. In addition, in the following we shall also restrict our attention to the standard embedding, in which the gauge connection on one of the SO(16) factors is identified with the spin connection, so that the orbifold action on the gauge world-sheet fermions and on those on space-time are identical. Since in the settings at stake supersymmetry is absent, these restrictions are not imposed by general principles. Rather, they are chosen for convenience and it is easy to relax these requirements to investigate other corners of the landscape. With this setup, the point group action on the world-sheet fermions is dictated by the shift vectors
    
        v_st=1/6 (1,-1)   ,     	v_gauge= 1/6 (0^6,1,-1) ⊗ (0^8)   ,
            
    for spacetime and gauge degrees of freedom respectively, acting on world-sheet fermions and bosons via
    
        g ·ψ^i_R= e^2 π i v_gauge^iψ^i_R   ,    g ·ψ^i_L = e^2 π i v_st^iψ^i_L   ,      and     	g · z^i= e^2 π i v_st^i z^i   ,
    
    where ψ^i_L,R denotes the complex combinations of Majorana spinors determined by the complex structure on the i-th torus T^2 parametrised by z_i. With this choice of shift vectors, the point group yields a complex action on world-sheet fermions and bosons, preventing the gauge group enhancement U(1) →SU(2) from taking place, as in the case of ℤ_2. This means that the breaking of the gauge group is maximal and reflects a simple K3 compactification <cit.>, 
    
        SO(16) ×SO(16) →SO(12) ×SU(2) ×U(1) ×SO(16)   .
     
    Furthermore, the SO(4) characters are no longer eigenvectors of the orbifold action, and indeed the corresponding partition function can only be expressed in terms of SU(2) and U(1) characters. However, there is no “algorithmic” procedure to determine the level of the affine U(1) algebra, and thus the characters appearing in the partition function, which means that we can only determine the SU(2) representations of the string states from the the q-expansion of the modular blocks, leaving the U(1) charges undetermined. This difficulty can be ignored if our aim is simply to study Dai-Freed anomalies for the SU(2) gauge group, in which the U(1) factor in (<ref>) is turned off. In any case, a more complete discussion would require computing the relevant twisted string bordism group.
    
     Following the partition function reported in <cit.>, it is possible to the deduce the light spectrum, which gives the gauge boson in the adjoint of SO(12) ×SU(2) ×SO(16) ×U(1), the graviton, the non-chiral Kalb-Ramond field and the dilaton in the singlet of the gauge group, eight scalars in the representation (1,1,1), four scalars in the representation (12,2,1), a doublet of left-handed fermions in the representation  (32_s,2,1) ⊕ (1,1,128_s) ⊕ (1,2,16) and a doublet of right-handed fermions in the representation (32_c,1,1) ⊕ (12,1,16). From the twisted sectors we have instead two hundred and sixteen scalars in the representation  (1,1,1), thirty-six scalars in the representation  (12,2,1), nine doublets of left-handed fermions in the representation  (1,2,16) and nine doublets of right-handed fermions in the representation (32_c,1,1).
    
     Form the light spectrum the anomaly theory can be straightforwardly deduced, yielding 
    
        𝒜_fermions ( Y  )   = η^D_(32_c,1,1) ( Y  )+ η^D_(12,1,16) ( Y  ) + 9  η^D_(32_c,1,1) ( Y  )
        		
           - η^D_(1,1,128_s) ( Y  )- η^D_(32_s,2,1) ( Y  )- η^D_(1,2,16) ( Y  )- 9  η^D_(1,2,16) ( Y  )   ,
     
    which locally reproduces the corresponding factorised anomaly polynomial 
        I_8    = 1/2 ( 2  tr R^2  -  12 tr F_1^2  -  tr F_2^2 - 12 tr F_3^2    )  ( tr F_3^2 - 2  tr F_1^2  + 8  tr F_2^2   )
         
           = 1/2  X_4^1 ∧ X_4^2  ,
    
    as required by the Green-Schwarz-Sagnotti mechanism. The above factorisation thus tells us that the bosonic piece of the anomaly arises from a non-chiral 2-form field and it is encoded in the cohomology pairing (<ref>)
        𝒜_B-fields ( Y  )= ( 𝐀̌^1, 𝐀̌^2)_Y   ,
     
    where 𝐀̌^i=(  N^i, A^i, X_4^i ) are the Cheeger-Simons characters with [N^i]= [ X_4^i ]_ℤ and [X_4^i]= [ X_4^i ]_dR<cit.>. Thus, the complete expression for the anomaly theory on a general background is given by
    
        𝒜 (Y  )= 𝒜_fermions ( Y  ) -  ( 𝐀̌^1, 𝐀̌^2)_Y   .
    
    If the i=1 character is topologically trivial,  i.e.[X_4^1]_ℤ =0, the coupling reduces to the known Chern-Simons coupling A^1 ∧ X_4^2, and thus, when Y=∂ Z, this contribution reproduces the Green-Schwarz-sagnotti term X_4^1 ∧ X_4^2. Requiring that the differential characters 𝐀̌^i be topologically trivial entails a twisted string structure on Y, since at the level of de Rham cohomology one obtains the Bianchi identities
    
        X_4^1= 12 p_1 + 12 c_2(F_1)+ c_2(F_2) + 12 c_2(F_3)   ,
        		
           X_4^2= 4    c_2(F_1) - 16   c_2(F_2)  -2   c_2(F_3)   .
    
    One can choose to trivialise either class at the integral level, but of course the consistency of the model should not depend on this choice. The corresponding twisted string bordism groups ought to classify equivalent anomaly backgrounds.
    
     We now study anomalies on Lens spaces, turning off all the groups aside from the first SU(2)[Turning on only the second SU(2) makes the second Bianchi identity trivial. However, this forbids the Lens space as a valid background for any p, as will be explained in the following.]. In this case L_p^7 trivialises X_4^2 for p=2,4,8,16, since the Pontryagin and Chern classes are 1/2 p_1= 2y and c_2=-y with y a generator of degree-four cohomology. On these backgrounds, the contribution from (<ref>) vanishes, as in the off-diagonal supersymmetric models, and thus the anomaly is simply given in terms of the net number of fundamentals for the first SU(2), d_F=12 · 16 as in (<ref>). As a result,
    
        𝒜 ( L_p^7  ) = 12 · 16  η̃_1  = 12 · 16  p^2-1/12 p≡_1 0   ,
    
    for our specific values of p.
    It is worth noting that the available backgrounds are the only ones for which the anomaly vanishes, while it would not have been the case for any other value of p.
    
    
    
    
    
    
    CHAPTER: CONCLUSION
     
    
    § SUMMARY AND OUTLOOK
     In the present dissertation we have analysed various aspects of the string landscape. Particular emphasis has been devoted to the study of tachyonic instabilities and the dynamics of (non-)supersymmetric string vacua, along with unitarity conditions induced by the completeness hypothesis and the consistency under topology change. This latter issue has been addressed both by looking at the consistency of the low-energy supergravity models and by studying explicit examples of string theory vacua, even without supersymmetry.
    
     In Chapter <ref>, we have addressed the absence of tachyons, of non-supersymmetric string vacua in various dimensions in connection with the large-mass behaviour of the net number of degrees of freedom encoded in the sector-averaged sums associated with one-loop amplitudes. When restricting to the closed-string case, we have shown that the latter depends on the mass of the lightest state, independently of whether it is a tachyon or massless particle, and is strictly smaller than C_tot in the presence of fermions. This result allowed us to show that the necessary and sufficient condition for classical stability is C_eff =0 (and ⟨ d (n) ⟩ (𝒯)), as conjectured in <cit.>. 
    
     The analysis performed in this context is based on the analytic continuation of the degrees of freedom obtained crucially continuing the physical discrete masses to real values. This approach is different from those used in  <cit.> and <cit.> which, using number theoretic methods, allowed to connect the behaviour of the signed sum over the physical spectrum with the vacuum energy, assumed to be finite. It would be interesting to connect misaligned supersymmetry and the latter approach.
    
     For orientifold vacua, the situation changes dramatically. In this scenario, the exponential growth of the number of degrees of freedom of the one-loop partition functions is related to infrared divergences of the tree-level amplitudes, and  vice versa. This is not surprising, since this UV/IR interplay descends from the dual interpretation of the Klein bottle, annulus and Möbius strip amplitudes as loop or tree-level diagrams. This allowed us to interpret misaligned supersymmetry associated with the loop channel as the decoupling of closed-string tachyons from D-branes and O-planes. Hence, it provides a  necessary, but not sufficient, condition for their absence from the physical spectrum. Similarly, the vanishing of the sector-averaged sums associated with both the torus and transverse annulus amplitudes is a  sufficient, but not necessary, condition for the absence of tachyons, since the Klein bottle and Möbius strip amplitudes only implement the orientifold projection. However, such a condition is not necessary, since tachyons might be removed through a non-trivial orientifold projection. In these cases, misaligned supersymmetry would require non-trivial cancellations among amplitudes with different topologies. However, the sector-averaged sums for the torus and the tree-level 𝒦, 𝒜 and ℳ amplitudes are built on CFTs on different Riemann surfaces, and thus carry different information. As a result, it appears difficult, if not impossible, to compare them. 
    
     In this sense,  misaligned supersymmetry cannot uncover the reason behind the IR finiteness of the vacuum energy for general orientifold vacua, and the question is still open. New ideas and technologies need to be developed to fully address this issue.
    
     In Chapter <ref>, we have focused our attention on a class of classically stable vacua in six dimensions, in which supersymmetry is realised non-linearly through the insertion of orientifold planes of positive tension and charge. In particular, for K3 orientifolds, the orientifold action is dressed with an involution that introduces O5_+ planes, and thus requires  D5 branes for tadpole cancellation. In this set-up, known as  Brane Supersymmetry Breaking, we have explicitly presented the orientifold of T^4/ℤ_6 orbifold admitting a non-trivial solution of the tadpole conditions, which forbids further recombination of branes. The situation is actually different in the supersymmetric case, where, regardless of the choice of the specific solution to R-R tadpole conditions, it is always possible to deform the model by moving brane into the bulk, as shown for instance for the T^4/ℤ_6 example.
    
     Furthermore, we have discussed the implications of the introduction of string defects coupled to the R-R 2-forms of the bulk super multiplets. Such defects couple to a combination of tensor-multiplet scalars, denoted as J, in such a way that the tension of the defect and the contributions to the energy-momentum tensor from the kinetic terms of scalars and gauge fields are positive, thus avoiding the presence of ghosts.
    Although for the supersymmetric case the existence of such J-form has been shown since the '90s <cit.>, doubts on the possibility of extending its definition to BSB vacua were raised in the literature <cit.>. In the present thesis, we have uncovered its stringy origin, determining its correct expression in the case of BSB vacua. Once the J form has been defined, it is possible to couple defects with positive tension that are also required to support a microscopic unitary theory. This translates into non-trivial conditions for the central charges of the Kač-Moody algebras realising the D-branes gauge groups. In general, the algebra is realised on either the left or right-moving sector, but whenever the defects are instantons of a gauge group of the bulk theory, the algebra can be realised on both, implying additional constraints. The bottom-up analysis known up to now in the literature <cit.> only covers the first case when one defect is considered, and there is no model-independent study describing instanton strings or situations in which more coincident defects are involved. However, since for orientifold vacua the microscopic theory living on string defects is known, it has been possible to evaluate these conditions explicitly, and we have found a correlation between the presence of scalars in the string/five brane-amplitudes and the interpretation of the defects as the corresponding gauge instantons. However, up to now, there is no knowledge on how to extend such results from a bottom-up perspective, which would provide  sharper unitarity conditions on string probes. 
    
     As shown in <cit.>, non-trivial "rigid" solutions of the tadpole conditions, implying the presence of fewer open-string moduli that forbid brane recombination are also present in the 4Dℤ_2 ×ℤ_2 orientifold with discrete torsion <cit.>. Therefore a straightforward extension would be seeking similar solutions for the other orientifolds in D=4, for which it would be interesting to study the consistency of probe defects along the lines of <cit.>.
    
    
     In the last part of the dissertation, Chapter <ref>, we have continued to explore the implications of unitarity by addressing the cancellation of generalised global anomalies arising from topology change, known as  Dai-Freed anomalies. We have adopted a twofold approach to address this issue based on the interplay between a bottom-up study of the consistency of 6D supergravity theories with a minimal amount of supersymmetry and a top-down analysis of known string models. In particular, we have explicitly presented the case of supergravity theories with an SU(2) gauge group and one tensor multiplet and the case of a U(1) gauge group with no tensor multiples. In the former case, we have found that the anomaly theory changes according to the nature of the 2-form fields. If the latter is trivial the anomaly theory is trivialised on all the tested backgrounds satisfying the twisted-string structure dictated by the Bianchi identity associated with the Green-Schwarz-Sagnotti mechanism. If the 2-forms are chiral the anomaly theory depends on the choice of the quadratic refinement describing the partition function of such fields, which cannot be determined  a priori. A similar conclusion is reached for the abelian case with no tensor multiples, for which we have seen that Dai-Freed anomalies exclude infinite families of vacua, up to a specific choice of the quadratic refinement. Extending this analysis to theories with more tensor multiplets, and understanding the details of the twisted string structure,  i.e. whether the bordism groups depend on the choice of Bianchi identity to trivialise, are interesting challenges to be pursued in the future. On the other hand, we have studied how such anomalies are cancelled for string vacua, including the non-supersymmetric heterotic theory with an SO(16) ×SO(16) gauge group compactified on the T^4/ℤ_6 orbifold and a Gepner orientifold with no tensor multiplets that goes beyond the validity of the result of <cit.>, which guarantees their vanishing for heterotic supersymmetric theories. 
    
    
     A natural development of the discussion in the last Chapter would be an investigation of the anomaly inflow on defects at the Dai-Freed level. Indeed, even in a bottom-up approach, exploiting the completeness principle <cit.> arising from cobordism triviality <cit.> and holography <cit.> can predict the existence of novel non-perturbative defects in the theory (see e.g.<cit.> for a detailed analysis in type IIB supergravity). In turn, the consistency of anomaly inflow on these defects can restrict the theory even with lower amounts of supersymmetry, as exemplified in <cit.>. Another promising avenue is the study of equivariant topological modular forms <cit.>, a generalised cohomology theory believed to encode deformation classes of 2D SQFTs (and thus of heterotic world-sheets) <cit.> according to the Stolz-Teichner conjecture <cit.>. As already mentioned, this approach was fruitfully employed in <cit.> to exclude all anomalies in supersymmetric heterotic strings, but an equivariant version <cit.> could apply to non-supersymmetric settings and to refined invariants taking into account gauge charges. This approach could exclude further models from the perturbative heterotic landscape.
    
    
    
     
      
      
       
        
        
      
      
    
      
      
      
    
      
    
    
    
    
    tocchapterBibliography100Green:2012oqa
    M. B. Green, J. H. Schwarz, and E. Witten,
      http://dx.doi.org/10.1017/CBO9781139248563Superstring Theory
      Vol. 1: 25th Anniversary Edition.
     Cambridge Monographs on Mathematical Physics. Cambridge University
      Press, 11, 2012.
    
    Green:2012pqa
    M. B. Green, J. H. Schwarz, and E. Witten,
      http://dx.doi.org/10.1017/CBO9781139248570Superstring Theory
      Vol. 2: 25th Anniversary Edition.
     Cambridge Monographs on Mathematical Physics. Cambridge University
      Press, 11, 2012.
    
    Polchinski:1998rq
    J. Polchinski, http://dx.doi.org/10.1017/CBO9780511816079String
      theory. Vol. 1: An introduction to the bosonic string.
     Cambridge Monographs on Mathematical Physics. Cambridge University
      Press, 12, 2007.
    
    Polchinski:1998rr
    J. Polchinski, http://dx.doi.org/10.1017/CBO9780511618123String
      theory. Vol. 2: Superstring theory and beyond.
     Cambridge Monographs on Mathematical Physics. Cambridge University
      Press, 12, 2007.
    
    Becker:2006dvp
    K. Becker, M. Becker, and J. H. Schwarz,
      http://dx.doi.org/10.1017/CBO9780511816086String theory and
      M-theory: A modern introduction.
     Cambridge University Press, 12, 2006.
    
    Blumenhagen:2013fgp
    R. Blumenhagen, D. Lüst, and S. Theisen,
      http://dx.doi.org/10.1007/978-3-642-29497-6Basic concepts of
      string theory.
     Theoretical and Mathematical Physics. Springer, Heidelberg, Germany,
      2013.
    
    Kiritsis:2019npv
    E. Kiritsis, String Theory in a Nutshell: Second Edition.
     Princeton University Press, USA, 4, 2019.
    
    Johnson:2023onr
    C. V. Johnson, http://dx.doi.org/10.1017/9781009401371D-Branes.
     Cambridge Monographs on Mathematical Physics. 2023.
    
    Gross:1987ar
    D. J. Gross and P. F. Mende, “String Theory Beyond the Planck Scale,”http://dx.doi.org/10.1016/0550-3213(88)90390-2 Nucl. Phys. B 303 (1988) 407–454.
    
    Gross:1987kza
    D. J. Gross and P. F. Mende, “The High-Energy Behavior of String Scattering
      Amplitudes,”http://dx.doi.org/10.1016/0370-2693(87)90355-8
      Phys. Lett. B 197 (1987) 129–134.
    
    Mende:1989wt
    P. F. Mende and H. Ooguri, “Borel Summation of String Theory for Planck Scale
      Scattering,”http://dx.doi.org/10.1016/0550-3213(90)90202-O
      Nucl. Phys. B 339 (1990) 641–662.
    
    Rohm:1983aq
    R. Rohm, “Spontaneous Supersymmetry Breaking in Supersymmetric String
      Theories,”http://dx.doi.org/10.1016/0550-3213(84)90007-5 Nucl.
      Phys. B 237 (1984) 553–572.
    
    Kounnas:1988ye
    C. Kounnas and M. Porrati, “Spontaneous Supersymmetry Breaking in String
      Theory,”http://dx.doi.org/10.1016/0550-3213(88)90153-8 Nucl.
      Phys. B 310 (1988) 355–370.
    
    Ferrara:1988jx
    S. Ferrara, C. Kounnas, M. Porrati, and F. Zwirner, “Superstrings with
      Spontaneously Broken Supersymmetry and their Effective Theories,”http://dx.doi.org/10.1016/0550-3213(89)90048-5 Nucl. Phys. B 318 (1989) 75–105.
    
    Antoniadis:1988jn
    I. Antoniadis, C. Bachas, D. C. Lewellen, and T. N. Tomaras, “On
      Supersymmetry Breaking in Superstrings,”http://dx.doi.org/10.1016/0370-2693(88)90679-X Phys. Lett. B 207 (1988) 441–446.
    
    Kounnas:1989dk
    C. Kounnas and B. Rostand, “Coordinate Dependent Compactifications and
      Discrete Symmetries,”http://dx.doi.org/10.1016/0550-3213(90)90543-M Nucl. Phys. B 341 (1990) 641–665.
    
    Antoniadis:1990ew
    I. Antoniadis, “A Possible new dimension at a few TeV,”http://dx.doi.org/10.1016/0370-2693(90)90617-F Phys. Lett. B 246 (1990) 377–384.
    
    Kiritsis:1997ca
    E. Kiritsis and C. Kounnas, “Perturbative and nonperturbative partial
      supersymmetry breaking: N=4 -> N=2 ->
      N=1,”http://dx.doi.org/10.1016/S0550-3213(97)00430-6 Nucl.
      Phys. B 503 (1997) 117–156,
      http://arxiv.org/abs/hep-th/9703059 arXiv:hep-th/9703059.
    
    Bachas:1995ik
    C. Bachas, “A Way to break supersymmetry,”  (1995) ,
      http://arxiv.org/abs/hep-th/9503030 arXiv:hep-th/9503030.
    
    Berkooz:1996km
    M. Berkooz, M. R. Douglas, and R. G. Leigh, “Branes intersecting at
      angles,”http://dx.doi.org/10.1016/S0550-3213(96)00452-X Nucl.
      Phys. B 480 (1996) 265–278,
      http://arxiv.org/abs/hep-th/9606139 arXiv:hep-th/9606139.
    
    Angelantonj:2000hi
    C. Angelantonj, I. Antoniadis, E. Dudas, and A. Sagnotti, “Type I strings on
      magnetized orbifolds and brane transmutation,”http://dx.doi.org/10.1016/S0370-2693(00)00907-2 Phys. Lett. B 489 (2000) 223–232,
      http://arxiv.org/abs/hep-th/0007090 arXiv:hep-th/0007090.
    
    Blumenhagen:2000wh
    R. Blumenhagen, L. Goerlich, B. Kors, and D. Lust, “Noncommutative
      compactifications of type I strings on tori with magnetic background flux,”http://dx.doi.org/10.1088/1126-6708/2000/10/006 JHEP
      10 (2000) 006, http://arxiv.org/abs/hep-th/0007024
      arXiv:hep-th/0007024.
    
    Aldazabal:2000cn
    G. Aldazabal, S. Franco, L. E. Ibanez, R. Rabadan, and A. M. Uranga,
      “Intersecting brane worlds,”http://dx.doi.org/10.1088/1126-6708/2001/02/047 JHEP
      02 (2001) 047, http://arxiv.org/abs/hep-ph/0011132
      arXiv:hep-ph/0011132.
    
    Cvetic:2001tj
    M. Cvetic, G. Shiu, and A. M. Uranga, “Three family supersymmetric standard -
      like models from intersecting brane worlds,”http://dx.doi.org/10.1103/PhysRevLett.87.201801 Phys. Rev. Lett. 87 (2001) 201801,
      http://arxiv.org/abs/hep-th/0107143 arXiv:hep-th/0107143.
    
    MarchesanoBuznego:2003axu
    F. G. Marchesano Buznego, “Intersecting D-brane models,”http://arxiv.org/abs/hep-th/0307252 arXiv:hep-th/0307252.
    
    Larosa:2003mz
    M. Larosa and G. Pradisi, “Magnetized four-dimensional
      Z2 x Z2 orientifolds,”http://dx.doi.org/10.1016/S0550-3213(03)00551-0 Nucl. Phys. B 667 (2003) 261–309,
      http://arxiv.org/abs/hep-th/0305224 arXiv:hep-th/0305224.
    
    Blumenhagen:2006ci
    R. Blumenhagen, B. Kors, D. Lust, and S. Stieberger, “Four-dimensional String
      Compactifications with D-Branes, Orientifolds and Fluxes,”http://dx.doi.org/10.1016/j.physrep.2007.04.003 Phys. Rept. 445 (2007) 1–193,
      http://arxiv.org/abs/hep-th/0610327 arXiv:hep-th/0610327.
    
    Sagnotti:1995ga
    A. Sagnotti, “Some properties of open string theories,” in International Workshop on Supersymmetry and Unification of Fundamental
      Interactions (SUSY 95), pp. 473–484.
     9, 1995.
    http://arxiv.org/abs/hep-th/9509080
      arXiv:hep-th/9509080.
    
    Angelantonj:1998gj
    C. Angelantonj, “Nontachyonic open descendants of the 0B string theory,”http://dx.doi.org/10.1016/S0370-2693(98)01430-0 Phys. Lett. B 444 (1998) 309–317,
      http://arxiv.org/abs/hep-th/9810214 arXiv:hep-th/9810214.
    
    Blum:1997gw
    J. D. Blum and K. R. Dienes, “Strong/weak coupling duality relations for
      nonsupersymmetric string theories,”http://dx.doi.org/10.1016/S0550-3213(97)00803-1 Nucl. Phys. B 516 (1998) 83–159,
      http://arxiv.org/abs/hep-th/9707160 arXiv:hep-th/9707160.
    
    Antoniadis:1998ki
    I. Antoniadis, E. Dudas, and A. Sagnotti, “Supersymmetry breaking, open
      strings and M theory,”http://dx.doi.org/10.1016/S0550-3213(98)00806-2 Nucl. Phys. B 544 (1999) 469–502,
      http://arxiv.org/abs/hep-th/9807011 arXiv:hep-th/9807011.
    
    Antoniadis:1998ep
    I. Antoniadis, G. D'Appollonio, E. Dudas, and A. Sagnotti, “Partial breaking
      of supersymmetry, open strings and M theory,”http://dx.doi.org/10.1016/S0550-3213(99)00232-1 Nucl. Phys. B 553 (1999) 133–154,
      http://arxiv.org/abs/hep-th/9812118 arXiv:hep-th/9812118.
    
    Angelantonj:1999gm
    C. Angelantonj, I. Antoniadis, and K. Forger, “Nonsupersymmetric type I
      strings with zero vacuum energy,”http://dx.doi.org/10.1016/S0550-3213(99)00344-2 Nucl. Phys. B 555 (1999) 116–134,
      http://arxiv.org/abs/hep-th/9904092 arXiv:hep-th/9904092.
    
    Sugimoto:1999tx
    S. Sugimoto, “Anomaly cancellations in type I
      D9-anti D9 system and the USp(32)
      string theory,”http://dx.doi.org/10.1143/PTP.102.685 Prog.
      Theor. Phys. 102 (1999) 685–699,
      http://arxiv.org/abs/hep-th/9905159 arXiv:hep-th/9905159.
    
    Antoniadis:1999xk
    I. Antoniadis, E. Dudas, and A. Sagnotti, “Brane supersymmetry breaking,”http://dx.doi.org/10.1016/S0370-2693(99)01023-0 Phys. Lett. B 464 (1999) 38–45,
      http://arxiv.org/abs/hep-th/9908023 arXiv:hep-th/9908023.
    
    Aldazabal:1999jr
    G. Aldazabal and A. M. Uranga, “Tachyon free nonsupersymmetric type IIB
      orientifolds via brane - anti-brane systems,”http://dx.doi.org/10.1088/1126-6708/1999/10/024 JHEP
      10 (1999) 024, http://arxiv.org/abs/hep-th/9908072
      arXiv:hep-th/9908072.
    
    Angelantonj:1999ms
    C. Angelantonj, I. Antoniadis, G. D'Appollonio, E. Dudas, and A. Sagnotti,
      “Type I vacua with brane supersymmetry breaking,”http://dx.doi.org/10.1016/S0550-3213(00)00052-3 Nucl. Phys. B 572 (2000) 36–70,
      http://arxiv.org/abs/hep-th/9911081 arXiv:hep-th/9911081.
    
    Angelantonj:2000xf
    C. Angelantonj, R. Blumenhagen, and M. R. Gaberdiel, “Asymmetric
      orientifolds, brane supersymmetry breaking and nonBPS branes,”http://dx.doi.org/10.1016/S0550-3213(00)00518-6 Nucl. Phys. B 589 (2000) 545–576,
      http://arxiv.org/abs/hep-th/0006033 arXiv:hep-th/0006033.
    
    Angelantonj:2024iwi
    C. Angelantonj, C. Condeescu, E. Dudas, and G. Leone, “Rigid vacua with Brane
      Supersymmetry Breaking,”http://dx.doi.org/10.1007/JHEP04(2024)103 JHEP 04
      (2024) 103, http://arxiv.org/abs/2403.02392
      arXiv:2403.02392 [hep-th].
    
    Angelantonj:2003hr
    C. Angelantonj and I. Antoniadis, “Suppressing the cosmological constant in
      nonsupersymmetric type I strings,”http://dx.doi.org/10.1016/j.nuclphysb.2003.09.047 Nucl. Phys. B 676 (2004) 129–148,
      http://arxiv.org/abs/hep-th/0307254 arXiv:hep-th/0307254.
    
    Angelantonj:2005hs
    C. Angelantonj, M. Cardella, and N. Irges, “Scherk-Schwarz breaking and
      intersecting branes,”http://dx.doi.org/10.1016/j.nuclphysb.2005.07.025 Nucl. Phys. B 725 (2005) 115–154,
      http://arxiv.org/abs/hep-th/0503179 arXiv:hep-th/0503179.
    
    Sen:1998sm
    A. Sen, “Tachyon condensation on the brane anti-brane system,”http://dx.doi.org/10.1088/1126-6708/1998/08/012 JHEP
      08 (1998) 012, http://arxiv.org/abs/hep-th/9805170
      arXiv:hep-th/9805170.
    
    Sen:1999nx
    A. Sen and B. Zwiebach, “Tachyon condensation in string field theory,”http://dx.doi.org/10.1088/1126-6708/2000/03/002 JHEP
      03 (2000) 002, http://arxiv.org/abs/hep-th/9912249
      arXiv:hep-th/9912249.
    
    Sen:2002in
    A. Sen, “Tachyon matter,”http://dx.doi.org/10.1088/1126-6708/2002/07/065 JHEP
      07 (2002) 065, http://arxiv.org/abs/hep-th/0203265
      arXiv:hep-th/0203265.
    
    Sen:2004nf
    A. Sen, “Tachyon dynamics in open string theory,”http://dx.doi.org/10.1142/S0217751X0502519X Int. J. Mod. Phys. A 20 (2005) 5513–5656,
      http://arxiv.org/abs/hep-th/0410103 arXiv:hep-th/0410103.
    
    Schnabl:2005gv
    M. Schnabl, “Analytic solution for tachyon condensation in open string field
      theory,”http://dx.doi.org/10.4310/ATMP.2006.v10.n4.a1 Adv.
      Theor. Math. Phys. 10 no. 4, (2006) 433–501,
      http://arxiv.org/abs/hep-th/0511286 arXiv:hep-th/0511286.
    
    Hellerman:2007zz
    S. Hellerman and I. Swanson, “A Stable vacuum of the tachyonic
      E(8) string,”http://arxiv.org/abs/0710.1628 arXiv:0710.1628 [hep-th].
    
    Kaidi:2020jla
    J. Kaidi, “Stable Vacua for Tachyonic Strings,”http://dx.doi.org/10.1103/PhysRevD.103.106026 Phys. Rev. D 103 no. 10, (2021) 106026,
      http://arxiv.org/abs/2010.10521 arXiv:2010.10521
      [hep-th].
    
    Kaidi:2023tqo
    J. Kaidi, K. Ohmori, Y. Tachikawa, and K. Yonekura, “Nonsupersymmetric
      Heterotic Branes,”http://dx.doi.org/10.1103/PhysRevLett.131.121601 Phys. Rev.
      Lett. 131 no. 12, (2023) 121601,
      http://arxiv.org/abs/2303.17623 arXiv:2303.17623
      [hep-th].
    
    Antoniadis:1991kh
    I. Antoniadis and C. Kounnas, “Superstring phase transition at high
      temperature,”http://dx.doi.org/10.1016/0370-2693(91)90442-S
      Phys. Lett. B 261 (1991) 369–378.
    
    Kutasov:1990sv
    D. Kutasov and N. Seiberg, “Number of degrees of freedom, density of states
      and tachyons in string theory and CFT,”http://dx.doi.org/10.1016/0550-3213(91)90426-X Nucl. Phys. B 358 (1991) 600–618.
    
    Rank
    R. A. Rankin, “Contributions to the theory of Ramanujan's functions τ
      (n) and similar arithmetic functions, I,” Proc. Cambridge Philos.
      Soc. 35 (1939) 351–372.
    
    Selb
    A. Selberg, “ Bemerkugen über eine Dirichletsche Reihe, die mit der Theorie
      der Modulformen nahe verbunden ist,” Arch. Math. Naturvid.
      43 (1940) 47–50.
    
    Zagier2
    D. B. Zagier, “The Rankin-Selberg method for automorphic functions which are
      not of rapid decay,” J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981) 415–437.
    
    Angelantonj:2010ic
    C. Angelantonj, M. Cardella, S. Elitzur, and E. Rabinovici, “Vacuum
      stability, string density of states and the Riemann zeta function,”http://dx.doi.org/10.1007/JHEP02(2011)024 JHEP 02
      (2011) 024, http://arxiv.org/abs/1012.5091 arXiv:1012.5091
      [hep-th].
    
    Dienes:1994np
    K. R. Dienes, “Modular invariance, finiteness, and misaligned supersymmetry:
      New constraints on the numbers of physical string states,”http://dx.doi.org/10.1016/0550-3213(94)90153-8 Nucl. Phys. B 429 (1994) 533–588,
      http://arxiv.org/abs/hep-th/9402006 arXiv:hep-th/9402006.
    
    Cudell:1992bi
    J.-R. Cudell and K. R. Dienes, “Are there oscillations in the baryon/meson
      ratio?,”http://dx.doi.org/10.1103/PhysRevLett.69.1324 Phys.
      Rev. Lett. 69 (1992) 1324–1327,
      http://arxiv.org/abs/hep-ph/9207242 arXiv:hep-ph/9207242.
      [Erratum: Phys.Rev.Lett. 69, 2311 (1992)].
    
    Cribiori:2020sct
    N. Cribiori, S. Parameswaran, F. Tonioni, and T. Wrase, “Misaligned
      Supersymmetry and Open Strings,”http://dx.doi.org/10.1007/JHEP04(2021)099 JHEP 04
      (2021) 099, http://arxiv.org/abs/2012.04677
      arXiv:2012.04677 [hep-th].
    
    Dixon:1986iz
    L. J. Dixon and J. A. Harvey, “String Theories in Ten-Dimensions Without
      Space-Time Supersymmetry,”http://dx.doi.org/10.1016/0550-3213(86)90619-X Nucl. Phys. B 274 (1986) 93–105.
    
    Alvarez-Gaume:1986ghj
    L. Alvarez-Gaume, P. H. Ginsparg, G. W. Moore, and C. Vafa, “An
      O(16) x O(16) Heterotic
      String,”http://dx.doi.org/10.1016/0370-2693(86)91524-8 Phys.
      Lett. B 171 (1986) 155–162.
    
    Faraggi:2020fwg
    A. E. Faraggi, V. G. Matyas, and B. Percival, “Type 0
      Z2 x Z2 heterotic string
      orbifolds and misaligned supersymmetry,”http://dx.doi.org/10.1142/S0217751X21501748 Int. J. Mod. Phys. A 36 no. 24, (2021) 2150174,
      http://arxiv.org/abs/2010.06637 arXiv:2010.06637
      [hep-th].
    
    Sara
    S. Bonansea, “Properties of Non-Supersymmetric Heterotic Vacua,” msc
      thesis, Università degli Studi di Torino, 2016.
    
    Angelantonj:2023egh
    C. Angelantonj, I. Florakis, and G. Leone, “Tachyons and misaligned
      supersymmetry in closed string vacua,”http://dx.doi.org/10.1007/JHEP06(2023)174 JHEP 06
      (2023) 174, http://arxiv.org/abs/2301.13702
      arXiv:2301.13702 [hep-th].
    
    Dienes:2012dc
    K. R. Dienes, M. Lennek, and M. Sharma, “Strings at Finite Temperature:
      Wilson Lines, Free Energies, and the Thermal Landscape,”http://dx.doi.org/10.1103/PhysRevD.86.066007 Phys. Rev. D 86 (2012) 066007,
      http://arxiv.org/abs/1205.5752 arXiv:1205.5752 [hep-th].
    
    Sagnotti:1987tw
    A. Sagnotti, “Open Strings and their Symmetry Groups,” in NATO
      Advanced Summer Institute on Nonperturbative Quantum Field Theory (Cargese
      Summer Institute).
     9, 1987.
    http://arxiv.org/abs/hep-th/0208020
      arXiv:hep-th/0208020.
    
    Pradisi:1988xd
    G. Pradisi and A. Sagnotti, “Open String Orbifolds,”http://dx.doi.org/10.1016/0370-2693(89)91369-5 Phys. Lett. B 216 (1989) 59–67.
    
    Horava:1989vt
    P. Horava, “Strings on World Sheet Orbifolds,”http://dx.doi.org/10.1016/0550-3213(89)90279-4 Nucl. Phys. B 327 (1989) 461–484.
    
    Bianchi:1990yu
    M. Bianchi and A. Sagnotti, “On the systematics of open string theories,”http://dx.doi.org/10.1016/0370-2693(90)91894-H Phys. Lett. B 247 (1990) 517–524.
    
    Bianchi:1990tb
    M. Bianchi and A. Sagnotti, “Twist symmetry and open string Wilson lines,”http://dx.doi.org/10.1016/0550-3213(91)90271-X Nucl. Phys. B 361 (1991) 519–538.
    
    Bianchi:1991eu
    M. Bianchi, G. Pradisi, and A. Sagnotti, “Toroidal compactification and
      symmetry breaking in open string theories,”http://dx.doi.org/10.1016/0550-3213(92)90129-Y Nucl. Phys. B 376 (1992) 365–386.
    
    Dudas:2000bn
    E. Dudas, “Theory and phenomenology of type I strings and M theory,”http://dx.doi.org/10.1088/0264-9381/17/22/201 Class. Quant.
      Grav. 17 (2000) R41–R116,
      http://arxiv.org/abs/hep-ph/0006190 arXiv:hep-ph/0006190.
    
    Angelantonj:2002ct
    C. Angelantonj and A. Sagnotti, “Open strings,”http://dx.doi.org/10.1016/S0370-1573(02)00273-9 Phys. Rept. 371 (2002) 1–150,
      http://arxiv.org/abs/hep-th/0204089 arXiv:hep-th/0204089.
      [Erratum: Phys.Rept. 376, 407 (2003)].
    
    Israel:2007nj
    D. Israel and V. Niarchos, “Tree-Level Stability Without Spacetime Fermions:
      Novel Examples in String Theory,”http://dx.doi.org/10.1088/1126-6708/2007/07/065 JHEP
      07 (2007) 065, http://arxiv.org/abs/0705.2140
      arXiv:0705.2140 [hep-th].
    
    Niarchos:2000kw
    V. Niarchos, “Density of states and tachyons in open and closed string
      theory,”http://dx.doi.org/10.1088/1126-6708/2001/06/048 JHEP 06 (2001) 048,
      http://arxiv.org/abs/hep-th/0010154 arXiv:hep-th/0010154.
    
    Leone:2023qfd
    G. Leone, “Tachyons and Misaligned Supersymmetry in orientifold vacua,”http://dx.doi.org/10.1007/JHEP11(2023)066 JHEP 11
      (2023) 066, http://arxiv.org/abs/2308.09757
      arXiv:2308.09757 [hep-th].
    
    Dudas:2000nv
    E. Dudas and J. Mourad, “Consistent gravitino couplings in nonsupersymmetric
      strings,”http://dx.doi.org/10.1016/S0370-2693(01)00777-8 Phys.
      Lett. B 514 (2001) 173–182,
      http://arxiv.org/abs/hep-th/0012071 arXiv:hep-th/0012071.
    
    Volkov:1972jx
    D. V. Volkov and V. P. Akulov, “Possible universal neutrino interaction,” JETP Lett. 16 (1972) 438–440.
    
    Fischler:1986ci
    W. Fischler and L. Susskind, “Dilaton Tadpoles, String Condensates and Scale
      Invariance,”http://dx.doi.org/10.1016/0370-2693(86)91425-5
      Phys. Lett. B 171 (1986) 383–389.
    
    Fischler:1986tb
    W. Fischler and L. Susskind, “Dilaton Tadpoles, String Condensates and Scale
      Invariance. 2.,”http://dx.doi.org/10.1016/0370-2693(86)90514-9
      Phys. Lett. B 173 (1986) 262–264.
    
    Dudas:2004nd
    E. Dudas, G. Pradisi, M. Nicolosi, and A. Sagnotti, “On tadpoles and vacuum
      redefinitions in string theory,”http://dx.doi.org/10.1016/j.nuclphysb.2004.11.028 Nucl. Phys. B 708 (2005) 3–44,
      http://arxiv.org/abs/hep-th/0410101 arXiv:hep-th/0410101.
    
    Keller:2007nd
    C. A. Keller, “Brane backreactions and the Fischler-Susskind mechanism in
      conformal field theory,”http://dx.doi.org/10.1088/1126-6708/2007/12/046 JHEP
      12 (2007) 046, http://arxiv.org/abs/0709.1076
      arXiv:0709.1076 [hep-th].
    
    Kitazawa:2008hv
    N. Kitazawa, “Tadpole Resummations in String Theory,”http://dx.doi.org/10.1016/j.physletb.2008.01.028 Phys. Lett. B 660 (2008) 415–421,
      http://arxiv.org/abs/0801.1702 arXiv:0801.1702 [hep-th].
    
    Pius:2014gza
    R. Pius, A. Rudra, and A. Sen, “String Perturbation Theory Around Dynamically
      Shifted Vacuum,”http://dx.doi.org/10.1007/JHEP10(2014)070
      JHEP 10 (2014) 070,
      http://arxiv.org/abs/1404.6254 arXiv:1404.6254 [hep-th].
    
    Dudas:2000ff
    E. Dudas and J. Mourad, “Brane solutions in strings with broken supersymmetry
      and dilaton tadpoles,”http://dx.doi.org/10.1016/S0370-2693(00)00734-6 Phys. Lett. B 486 (2000) 172–178,
      http://arxiv.org/abs/hep-th/0004165 arXiv:hep-th/0004165.
    
    DeWolfe:2001nz
    O. DeWolfe, D. Z. Freedman, S. S. Gubser, G. T. Horowitz, and I. Mitra,
      “Stability of AdS(p) x M(q)
      compactifications without supersymmetry,”http://dx.doi.org/10.1103/PhysRevD.65.064033 Phys. Rev. D 65 (2002) 064033,
      http://arxiv.org/abs/hep-th/0105047 arXiv:hep-th/0105047.
    
    Gubser:2001zr
    S. S. Gubser and I. Mitra, “Some interesting violations of the
      Breitenlohner-Freedman bound,”http://dx.doi.org/10.1088/1126-6708/2002/07/044 JHEP
      07 (2002) 044, http://arxiv.org/abs/hep-th/0108239
      arXiv:hep-th/0108239.
    
    Dudas:2002dg
    E. Dudas, J. Mourad, and C. Timirgaziu, “Time and space dependent backgrounds
      from nonsupersymmetric strings,”http://dx.doi.org/10.1016/S0550-3213(03)00248-7 Nucl. Phys. B 660 (2003) 3–24,
      http://arxiv.org/abs/hep-th/0209176 arXiv:hep-th/0209176.
    
    Dudas:2010gi
    E. Dudas, N. Kitazawa, and A. Sagnotti, “On Climbing Scalars in String
      Theory,”http://dx.doi.org/10.1016/j.physletb.2010.09.040 Phys.
      Lett. B 694 (2011) 80–88,
      http://arxiv.org/abs/1009.0874 arXiv:1009.0874 [hep-th].
    
    Mourad:2016xbk
    J. Mourad and A. Sagnotti, “AdS Vacua from Dilaton Tadpoles and Form
      Fluxes,”http://dx.doi.org/10.1016/j.physletb.2017.02.053 Phys.
      Lett. B 768 (2017) 92–96,
      http://arxiv.org/abs/1612.08566 arXiv:1612.08566
      [hep-th].
    
    Basile:2018irz
    I. Basile, J. Mourad, and A. Sagnotti, “On Classical Stability with Broken
      Supersymmetry,”http://dx.doi.org/10.1007/JHEP01(2019)174 JHEP 01 (2019) 174, http://arxiv.org/abs/1811.11448
      arXiv:1811.11448 [hep-th].
    
    Antonelli:2019nar
    R. Antonelli and I. Basile, “Brane annihilation in non-supersymmetric
      strings,”http://dx.doi.org/10.1007/JHEP11(2019)021 JHEP 11 (2019) 021, http://arxiv.org/abs/1908.04352
      arXiv:1908.04352 [hep-th].
    
    Basile:2020mpt
    I. Basile and S. Lanza, “de Sitter in non-supersymmetric string theories:
      no-go theorems and brane-worlds,”http://dx.doi.org/10.1007/JHEP10(2020)108 JHEP 10
      (2020) 108, http://arxiv.org/abs/2007.13757
      arXiv:2007.13757 [hep-th].
    
    Basile:2021krk
    I. Basile and A. Platania, “String tension between de Sitter vacua and
      curvature corrections,”http://dx.doi.org/10.1103/PhysRevD.104.L121901 Phys. Rev. D 104 no. 12, (2021) L121901,
      http://arxiv.org/abs/2103.06276 arXiv:2103.06276
      [hep-th].
    
    Mourad:2021qwf
    J. Mourad and A. Sagnotti, “On warped string vacuum profiles and cosmologies.
      Part I. Supersymmetric strings,”http://dx.doi.org/10.1007/JHEP12(2021)137 JHEP 12
      (2021) 137, http://arxiv.org/abs/2109.06852
      arXiv:2109.06852 [hep-th].
    
    Mourad:2021roa
    J. Mourad and A. Sagnotti, “On warped string vacuum profiles and cosmologies.
      Part II. Non-supersymmetric strings,”http://dx.doi.org/10.1007/JHEP12(2021)138 JHEP 12
      (2021) 138, http://arxiv.org/abs/2109.12328
      arXiv:2109.12328 [hep-th].
    
    Raucci:2022bjw
    S. Raucci, “On new vacua of non-supersymmetric strings,”http://dx.doi.org/10.1016/j.physletb.2022.137663 Phys. Lett. B 837 (2023) 137663,
      http://arxiv.org/abs/2209.06537 arXiv:2209.06537
      [hep-th].
    
    Baykara:2022cwj
    Z. K. Baykara, D. Robbins, and S. Sethi, “Non-Supersymmetric AdS from String
      Theory,”http://arxiv.org/abs/2212.02557 arXiv:2212.02557
      [hep-th].
    
    Raucci:2022jgw
    S. Raucci, “On codimension-one vacua and string theory,”http://dx.doi.org/10.1016/j.nuclphysb.2022.116002 Nucl. Phys. B 985 (2022) 116002,
      http://arxiv.org/abs/2206.06399 arXiv:2206.06399
      [hep-th].
    
    Mourad:2022loy
    J. Mourad and A. Sagnotti, “A 4D IIB flux vacuum and supersymmetry breaking.
      Part I. Fermionic spectrum,”http://dx.doi.org/10.1007/JHEP08(2022)301 JHEP 08
      (2022) 301, http://arxiv.org/abs/2206.03340
      arXiv:2206.03340 [hep-th].
    
    Gimon:1996ay
    E. G. Gimon and C. V. Johnson, “K3 orientifolds,”http://dx.doi.org/10.1016/0550-3213(96)00356-2 Nucl. Phys. B 477 (1996) 715–745,
      http://arxiv.org/abs/hep-th/9604129 arXiv:hep-th/9604129.
    
    Dabholkar:1996pc
    A. Dabholkar and J. Park, “Strings on orientifolds,”http://dx.doi.org/10.1016/0550-3213(96)00395-1 Nucl. Phys. B 477 (1996) 701–714,
      http://arxiv.org/abs/hep-th/9604178 arXiv:hep-th/9604178.
    
    Pradisi:2001yv
    G. Pradisi and F. Riccioni, “Geometric couplings and brane supersymmetry
      breaking,”http://dx.doi.org/10.1016/S0550-3213(01)00441-2
      Nucl. Phys. B 615 (2001) 33–60,
      http://arxiv.org/abs/hep-th/0107090 arXiv:hep-th/0107090.
    
    Angelantonj:2011hs
    C. Angelantonj, C. Condeescu, E. Dudas, and G. Pradisi, “Non-perturbative
      transitions among intersecting-brane vacua,”http://dx.doi.org/10.1007/JHEP07(2011)123 JHEP 07
      (2011) 123, http://arxiv.org/abs/1105.3465 arXiv:1105.3465
      [hep-th].
    
    Angelantonj:1996uy
    C. Angelantonj, M. Bianchi, G. Pradisi, A. Sagnotti, and Y. S. Stanev,
      “Chiral asymmetry in four-dimensional open string vacua,”http://dx.doi.org/10.1016/0370-2693(96)00869-6 Phys. Lett. B 385 (1996) 96–102,
      http://arxiv.org/abs/hep-th/9606169 arXiv:hep-th/9606169.
    
    Romans:1986er
    L. J. Romans, “Selfduality for Interacting Fields: Covariant Field Equations
      for Six-dimensional Chiral Supergravities,”http://dx.doi.org/10.1016/0550-3213(86)90016-7 Nucl. Phys. B 276 (1986) 71.
    
    Sagnotti:1992qw
    A. Sagnotti, “A Note on the Green-Schwarz mechanism in open string
      theories,”http://dx.doi.org/10.1016/0370-2693(92)90682-T Phys.
      Lett. B 294 (1992) 196–203,
      http://arxiv.org/abs/hep-th/9210127 arXiv:hep-th/9210127.
    
    Ferrara:1997gh
    S. Ferrara, F. Riccioni, and A. Sagnotti, “Tensor and vector multiplets in
      six-dimensional supergravity,”http://dx.doi.org/10.1016/S0550-3213(97)00837-7 Nucl. Phys. B 519 (1998) 115–140,
      http://arxiv.org/abs/hep-th/9711059 arXiv:hep-th/9711059.
    
    Riccioni:1998th
    F. Riccioni and A. Sagnotti, “Consistent and covariant anomalies in
      six-dimensional supergravity,”http://dx.doi.org/10.1016/S0370-2693(98)00846-6 Phys. Lett. B 436 (1998) 298–305,
      http://arxiv.org/abs/hep-th/9806129 arXiv:hep-th/9806129.
    
    Kim:2019vuc
    H.-C. Kim, G. Shiu, and C. Vafa, “Branes and the Swampland,”http://dx.doi.org/10.1103/PhysRevD.100.066006 Phys. Rev. D 100 no. 6, (2019) 066006,
      http://arxiv.org/abs/1905.08261 arXiv:1905.08261
      [hep-th].
    
    Green:1984sg
    M. B. Green and J. H. Schwarz, “Anomaly Cancellation in Supersymmetric
      D=10 Gauge Theory and Superstring Theory,”http://dx.doi.org/10.1016/0370-2693(84)91565-X Phys. Lett. B 149 (1984) 117–122.
    
    Green:1984bx
    M. B. Green, J. H. Schwarz, and P. C. West, “Anomaly Free Chiral Theories in
      Six-Dimensions,”http://dx.doi.org/10.1016/0550-3213(85)90222-6
      Nucl. Phys. B 254 (1985) 327–348.
    
    Angelantonj:2020pyr
    C. Angelantonj, Q. Bonnefoy, C. Condeescu, and E. Dudas, “String Defects,
      Supersymmetry and the Swampland,”http://dx.doi.org/10.1007/JHEP11(2020)125 JHEP 11
      (2020) 125, http://arxiv.org/abs/2007.12722
      arXiv:2007.12722 [hep-th].
    
    Polchinski:2003bq
    J. Polchinski, “Monopoles, duality, and string theory,”http://dx.doi.org/10.1142/S0217751X0401866X Int. J. Mod. Phys. A 19S1 (2004) 145–156,
      http://arxiv.org/abs/hep-th/0304042 arXiv:hep-th/0304042.
    
    Banks:2010zn
    T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and
      Gravity,”http://dx.doi.org/10.1103/PhysRevD.83.084019 Phys.
      Rev. D 83 (2011) 084019,
    http://arxiv.org/abs/1011.5120 arXiv:1011.5120 [hep-th].
    
    
    Vafa:2005ui
    C. Vafa, “The String landscape and the swampland,”http://arxiv.org/abs/hep-th/0509212 arXiv:hep-th/0509212.
    
    Brennan:2017rbf
    T. D. Brennan, F. Carta, and C. Vafa, “The String Landscape, the Swampland,
      and the Missing Corner,”http://dx.doi.org/10.22323/1.305.0015
      PoS TASI2017 (2017) 015,
      http://arxiv.org/abs/1711.00864 arXiv:1711.00864
      [hep-th].
    
    Palti:2019pca
    E. Palti, “The Swampland: Introduction and Review,”http://dx.doi.org/10.1002/prop.201900037 Fortsch. Phys. 67 no. 6, (2019) 1900037,
      http://arxiv.org/abs/1903.06239 arXiv:1903.06239
      [hep-th].
    
    vanBeest:2021lhn
    M. van Beest, J. Calderón-Infante, D. Mirfendereski, and I. Valenzuela,
      “Lectures on the Swampland Program in String Compactifications,”http://dx.doi.org/10.1016/j.physrep.2022.09.002 Phys. Rept. 989 (2022) 1–50,
      http://arxiv.org/abs/2102.01111 arXiv:2102.01111
      [hep-th].
    
    Agmon:2022thq
    N. B. Agmon, A. Bedroya, M. J. Kang, and C. Vafa, “Lectures on the string
      landscape and the Swampland,”http://arxiv.org/abs/2212.06187 arXiv:2212.06187
      [hep-th].
    
    Kumar:2010ru
    V. Kumar, D. R. Morrison, and W. Taylor, “Global aspects of the space of
      6D N=1 supergravities,”http://dx.doi.org/10.1007/JHEP11(2010)118 JHEP 11
      (2010) 118, http://arxiv.org/abs/1008.1062 arXiv:1008.1062
      [hep-th].
    
    Park:2011wv
    D. S. Park and W. Taylor, “Constraints on 6D Supergravity Theories with
      Abelian Gauge Symmetry,”http://dx.doi.org/10.1007/JHEP01(2012)141 JHEP 01
      (2012) 141, http://arxiv.org/abs/1110.5916 arXiv:1110.5916
      [hep-th].
    
    Taylor:2018khc
    W. Taylor and A. P. Turner, “An infinite swampland of
      U(1) charge spectra in 6D supergravity
      theories,”http://dx.doi.org/10.1007/JHEP06(2018)010 JHEP 06 (2018) 010, http://arxiv.org/abs/1803.04447
      arXiv:1803.04447 [hep-th].
    
    Lee:2019skh
    S.-J. Lee and T. Weigand, “Swampland Bounds on the Abelian Gauge Sector,”http://dx.doi.org/10.1103/PhysRevD.100.026015 Phys. Rev. D 100 no. 2, (2019) 026015,
      http://arxiv.org/abs/1905.13213 arXiv:1905.13213
      [hep-th].
    
    Kim:2019ths
    H.-C. Kim, H.-C. Tarazi, and C. Vafa, “Four-dimensional
      N=4 SYM theory and the swampland,”http://dx.doi.org/10.1103/PhysRevD.102.026003 Phys. Rev. D 102 no. 2, (2020) 026003,
      http://arxiv.org/abs/1912.06144 arXiv:1912.06144
      [hep-th].
    
    Katz:2020ewz
    S. Katz, H.-C. Kim, H.-C. Tarazi, and C. Vafa, “Swampland Constraints on 5D
      N=1 Supergravity,”http://dx.doi.org/10.1007/JHEP07(2020)080 JHEP 07
      (2020) 080, http://arxiv.org/abs/2004.14401
      arXiv:2004.14401 [hep-th].
    
    Tarazi:2021duw
    H.-C. Tarazi and C. Vafa, “On The Finiteness of 6D Supergravity Landscape,”http://arxiv.org/abs/2106.10839 arXiv:2106.10839
      [hep-th].
    
    Morrison:2021wuv
    D. R. Morrison and W. Taylor, “Charge completeness and the massless charge
      lattice in F-theory models of supergravity,”http://dx.doi.org/10.1007/JHEP12(2021)040 JHEP 12
      (2021) 040, http://arxiv.org/abs/2108.02309
      arXiv:2108.02309 [hep-th].
    
    Martucci:2022krl
    L. Martucci, N. Risso, and T. Weigand, “Quantum gravity bounds on
      N=1 effective theories in four
      dimensions,”http://dx.doi.org/10.1007/JHEP03(2023)197 JHEP 03 (2023) 197, http://arxiv.org/abs/2210.10797
      arXiv:2210.10797 [hep-th].
    
    Baykara:2023plc
    Z. K. Baykara, Y. Hamada, H.-C. Tarazi, and C. Vafa, “On the String Landscape
      Without Hypermultiplets,”http://arxiv.org/abs/2309.15152
      arXiv:2309.15152 [hep-th].
    
    Hayashi:2023hqa
    H. Hayashi, H.-C. Kim, and M. Kim, “Spectra of BPS Strings in 6D Supergravity
      and the Swampland,”http://arxiv.org/abs/2310.12219
      arXiv:2310.12219 [hep-th].
    
    Montero:2020icj
    M. Montero and C. Vafa, “Cobordism Conjecture, Anomalies, and the String
      Lamppost Principle,”http://dx.doi.org/10.1007/JHEP01(2021)063
      JHEP 01 (2021) 063,
      http://arxiv.org/abs/2008.11729 arXiv:2008.11729
      [hep-th].
    
    Hamada:2021bbz
    Y. Hamada and C. Vafa, “8D supergravity, reconstruction of internal geometry
      and the Swampland,”http://dx.doi.org/10.1007/JHEP06(2021)178
      JHEP 06 (2021) 178,
      http://arxiv.org/abs/2104.05724 arXiv:2104.05724
      [hep-th].
    
    Bedroya:2021fbu
    A. Bedroya, Y. Hamada, M. Montero, and C. Vafa, “Compactness of brane moduli
      and the String Lamppost Principle in D>6,”http://dx.doi.org/10.1007/JHEP02(2022)082 JHEP 02
      (2022) 082, http://arxiv.org/abs/2110.10157
      arXiv:2110.10157 [hep-th].
    
    Freed:1986hv
    D. S. Freed, “On determinant line bundles,” Conf. Proc. C
      8607214 (1986) 189–238.
    
    Dai:1994kq
    X.-z. Dai and D. S. Freed, “Eta invariants and determinant lines,”http://dx.doi.org/10.1063/1.530747 J. Math. Phys. 35
      (1994) 5155–5194, http://arxiv.org/abs/hep-th/9405012
      arXiv:hep-th/9405012. [Erratum: J.Math.Phys. 42, 2343–2344 (2001)].
    
    Freed:2016rqq
    D. S. Freed and M. J. Hopkins, “Reflection positivity and invertible
      topological phases,”http://dx.doi.org/10.2140/gt.2021.25.1165
      Geom. Topol. 25 (2021) 1165–1330,
      http://arxiv.org/abs/1604.06527 arXiv:1604.06527
      [hep-th].
    
    Yonekura:2018ufj
    K. Yonekura, “On the cobordism classification of symmetry protected
      topological phases,”http://dx.doi.org/10.1007/s00220-019-03439-y Commun. Math. Phys. 368 no. 3, (2019) 1121–1173,
      http://arxiv.org/abs/1803.10796 arXiv:1803.10796
      [hep-th].
    
    Witten:2015aba
    E. Witten, “Fermion Path Integrals And Topological Phases,”http://dx.doi.org/10.1103/RevModPhys.88.035001 Rev. Mod. Phys. 88 no. 3, (2016) 035001,
      http://arxiv.org/abs/1508.04715 arXiv:1508.04715
      [cond-mat.mes-hall].
    
    Garcia-Etxebarria:2018ajm
    I. García-Etxebarria and M. Montero, “Dai-Freed anomalies in particle
      physics,”http://dx.doi.org/10.1007/JHEP08(2019)003 JHEP 08 (2019) 003, http://arxiv.org/abs/1808.00009
      arXiv:1808.00009 [hep-th].
    
    Alvarez-Gaume:1983ict
    L. Alvarez-Gaume and P. H. Ginsparg, “The Topological Meaning of Nonabelian
      Anomalies,”http://dx.doi.org/10.1016/0550-3213(84)90487-5
      Nucl. Phys. B 243 (1984) 449–474.
    
    Alvarez-Gaume:1983ihn
    L. Alvarez-Gaume and E. Witten, “Gravitational Anomalies,”http://dx.doi.org/10.1016/0550-3213(84)90066-X Nucl. Phys. B 234 (1984) 269.
    
    Alvarez-Gaume:1984zlq
    L. Alvarez-Gaume and P. H. Ginsparg, “The Structure of Gauge and
      Gravitational Anomalies,”http://dx.doi.org/10.1016/0003-4916(85)90087-9 Annals Phys. 161 (1985) 423. [Erratum: Annals Phys. 171, 233 (1986)].
    
    Alvarez-Gaume:1984zst
    L. Alvarez-Gaume, S. Della Pietra, and G. W. Moore, “Anomalies and Odd
      Dimensions,”http://dx.doi.org/10.1016/0003-4916(85)90383-5
      Annals Phys. 163 (1985) 288.
    
    Alvarez-Gaume:2022aak
    L. Alvarez-Gaume and M. A. Vazquez-Mozo, “Anomalies and the Green-Schwarz
      Mechanism,”http://arxiv.org/abs/2211.06467
      arXiv:2211.06467 [hep-th].
    
    Witten:1982fp
    E. Witten, “An SU(2) Anomaly,”http://dx.doi.org/10.1016/0370-2693(82)90728-6 Phys. Lett. B 117 (1982) 324–328.
    
    Witten:1985xe
    E. Witten, “Gloal gravitational anomalies,”http://dx.doi.org/10.1007/BF01212448 Commun. Math. Phys. 100 (1985) 197.
    
    Witten:1985mj
    E. Witten, “Global anomalies in string theory,” in Symposium on
      Anomalies, Geometry, Topology.
     6, 1985.
    
    Witten:2019bou
    E. Witten and K. Yonekura, “Anomaly Inflow and the η-Invariant,” in
      The Shoucheng Zhang Memorial Workshop.
     9, 2019.
    http://arxiv.org/abs/1909.08775 arXiv:1909.08775
      [hep-th].
    
    Basile:2023zng
    I. Basile and G. Leone, “Anomaly constraints for heterotic strings and
      supergravity in six dimensions,”http://arxiv.org/abs/2310.20480 arXiv:2310.20480
      [hep-th].
    
    Basile:2023knk
    I. Basile, A. Debray, M. Delgado, and M. Montero, “Global anomalies &
      bordism of non-supersymmetric strings,”http://dx.doi.org/10.1007/JHEP02(2024)092 JHEP 02
      (2024) 092, http://arxiv.org/abs/2310.06895
      arXiv:2310.06895 [hep-th].
    
    Tachikawa:2021mvw
    Y. Tachikawa, “Topological modular forms and the absence of a heterotic
      global anomaly,”http://dx.doi.org/10.1093/ptep/ptab060 PTEP 2022 no. 4, (2022) 04A107,
      http://arxiv.org/abs/2103.12211 arXiv:2103.12211
      [hep-th].
    
    hopkins
    M. J. Hopkins, “Topological modular forms, the Witten genus, and the theorem
      of the cube,” in  Proceedings of the International Congress of
      Mathematicians, S. D. Chatterji, ed., pp. 554–565.
     Birkhäuser Basel, Basel, 1995.
    
    Tachikawa:2021mby
    Y. Tachikawa and M. Yamashita, “Topological Modular Forms and the Absence of
      All Heterotic Global Anomalies,”http://dx.doi.org/10.1007/s00220-023-04761-2 Commun. Math. Phys. 402 no. 2, (2023) 1585–1620,
      http://arxiv.org/abs/2108.13542 arXiv:2108.13542
      [hep-th]. [Erratum: Commun.Math.Phys. 402, 2131 (2023)].
    
    Debray:2021vob
    A. Debray, M. Dierigl, J. J. Heckman, and M. Montero, “The anomaly that was
      not meant IIB,”http://arxiv.org/abs/2107.14227
      arXiv:2107.14227 [hep-th].
    
    Debray:2023yrs
    A. Debray, M. Dierigl, J. J. Heckman, and M. Montero, “The Chronicles of
      IIBordia: Dualities, Bordisms, and the Swampland,”http://arxiv.org/abs/2302.00007 arXiv:2302.00007
      [hep-th].
    
    Dierigl:2022zll
    M. Dierigl, P.-K. Oehlmann, and T. Schimannek, “The discrete Green-Schwarz
      mechanism in 6D F-theory and elliptic genera of non-critical strings,”http://dx.doi.org/10.1007/JHEP03(2023)090 JHEP 03
      (2023) 090, http://arxiv.org/abs/2212.04503
      arXiv:2212.04503 [hep-th].
    
    Hsieh:2020jpj
    C.-T. Hsieh, Y. Tachikawa, and K. Yonekura, “Anomaly Inflow and
      p-Form Gauge Theories,”http://dx.doi.org/10.1007/s00220-022-04333-w Commun. Math. Phys. 391 no. 2, (2022) 495–608,
      http://arxiv.org/abs/2003.11550 arXiv:2003.11550
      [hep-th].
    
    Gliozzi:1976jf
    F. Gliozzi, J. Scherk, and D. I. Olive, “Supergravity and the Spinor Dual
      Model,”http://dx.doi.org/10.1016/0370-2693(76)90183-0 Phys.
      Lett. B 65 (1976) 282–286.
    
    Gliozzi:1976qd
    F. Gliozzi, J. Scherk, and D. I. Olive, “Supersymmetry, Supergravity Theories
      and the Dual Spinor Model,”http://dx.doi.org/10.1016/0550-3213(77)90206-1 Nucl. Phys. B 122 (1977) 253–290.
    
    Rademacher:1937a
    H. Rademacher, “A Convergent Series for the Partition Function p(n),” Proc. Natl. Acad. Sci. 23 (1937) 78.
    
    Rademacher:1937b
    H. Rademacher, “On the Partition Function p(n),” Proc. Lond. Math.
      Soc. 43 (1937) 241.
    
    Rademacher:1938
    H. Rademacher, “ On the Fourier Coefficients of Certain Modular Forms of
      Positive Dimension,” Ann. Math. 39 (1938) 433.
    
    Dijkgraaf:2000fq
    R. Dijkgraaf, J. M. Maldacena, G. W. Moore, and E. P. Verlinde, “A Black hole
      Farey tail,”http://arxiv.org/abs/hep-th/0005003
      arXiv:hep-th/0005003.
    
    Manschot:2007ha
    J. Manschot and G. W. Moore, “A Modern Farey Tail,”http://dx.doi.org/10.4310/CNTP.2010.v4.n1.a3 Commun. Num. Theor.
      Phys. 4 (2010) 103–159,
      http://arxiv.org/abs/0712.0573 arXiv:0712.0573 [hep-th].
    
    Hardy
    G. H. Hardy and S. Ramanujan, “Asymptotic Formulaæ in Combinatory
      Analysis,” Proc. Lond. Math. Soc. 17 (1918) 75.
    
    Kani:1989im
    I. Kani and C. Vafa, “Asymptotic Mass Degeneracies in Conformal Field
      Theories,”http://dx.doi.org/10.1007/BF02096934 Commun. Math.
      Phys. 130 (1990) 529–580.
    
    Cribiori:2021gbf
    N. Cribiori, D. Lust, and M. Scalisi, “The gravitino and the swampland,”http://dx.doi.org/10.1007/JHEP06(2021)071 JHEP 06
      (2021) 071, http://arxiv.org/abs/2104.08288
      arXiv:2104.08288 [hep-th].
    
    farkas2012riemann
    H. Farkas and I. Kra,  Riemann Surfaces.
     Graduate Texts in Mathematics. Springer New York, 2012.
    <https://books.google.it/books?id=6pXuBwAAQBAJ>.
    
    Cardy:1981fd
    J. L. Cardy, “Duality and the Theta Parameter in Abelian Lattice Models,”http://dx.doi.org/10.1016/0550-3213(82)90464-3 Nucl. Phys. B 205 (1982) 17–26.
    
    Kawai:1986vd
    H. Kawai, D. C. Lewellen, and S. H. H. Tye, “Classification of Closed
      Fermionic String Models,”http://dx.doi.org/10.1103/PhysRevD.34.3794 Phys. Rev. D 34 (1986) 3794.
    
    Antoniadis:1986rn
    I. Antoniadis, C. P. Bachas, and C. Kounnas, “Four-Dimensional
      Superstrings,”http://dx.doi.org/10.1016/0550-3213(87)90372-5
      Nucl. Phys. B 289 (1987) 87.
    
    Horava:2007hg
    P. Horava and C. A. Keeler, “M-Theory Through the Looking Glass: Tachyon
      Condensation in the E(8) Heterotic String,”http://dx.doi.org/10.1103/PhysRevD.77.066013 Phys. Rev. D 77 (2008) 066013,
      http://arxiv.org/abs/0709.3296 arXiv:0709.3296 [hep-th].
    
    Kaidi:2019pzj
    J. Kaidi, J. Parra-Martinez, and Y. Tachikawa, “Classification of String
      Theories via Topological Phases,”http://dx.doi.org/10.1103/PhysRevLett.124.121601 Phys. Rev.
      Lett. 124 no. 12, (2020) 121601,
      http://arxiv.org/abs/1908.04805 arXiv:1908.04805
      [hep-th].
    
    Kaidi:2019tyf
    J. Kaidi, J. Parra-Martinez, Y. Tachikawa, and A. Debray, “Topological
      Superconductors on Superstring Worldsheets,”http://dx.doi.org/10.21468/SciPostPhys.9.1.010 SciPost Phys. 9 (2020) 10, http://arxiv.org/abs/1911.11780
      arXiv:1911.11780 [hep-th].
    
    BoyleSmith:2023xkd
    P. Boyle Smith, Y.-H. Lin, Y. Tachikawa, and Y. Zheng, “Classification of
      chiral fermionic CFTs of central charge ≤ 16,”http://arxiv.org/abs/2303.16917 arXiv:2303.16917
      [hep-th].
    
    Sagnotti:1996qj
    A. Sagnotti, “Surprises in open string perturbation theory,”http://dx.doi.org/10.1016/S0920-5632(97)00344-7 Nucl. Phys. B
      Proc. Suppl. 56 (1997) 332–343,
      http://arxiv.org/abs/hep-th/9702093 arXiv:hep-th/9702093.
    
    Cribiori:2021txm
    N. Cribiori, S. Parameswaran, F. Tonioni, and T. Wrase, “Modular invariance,
      misalignment and finiteness in non-supersymmetric strings,”http://dx.doi.org/10.1007/JHEP01(2022)127 JHEP 01
      (2022) 127, http://arxiv.org/abs/2110.11973
      arXiv:2110.11973 [hep-th].
    
    Scherk:1979zr
    J. Scherk and J. H. Schwarz, “How to Get Masses from Extra Dimensions,”http://dx.doi.org/10.1016/0550-3213(79)90592-3 Nucl. Phys. B 153 (1979) 61–88.
    
    Ferrara:1987es
    S. Ferrara, C. Kounnas, and M. Porrati, “Superstring Solutions With
      Spontaneously Broken Four-dimensional Supersymmetry,”http://dx.doi.org/10.1016/0550-3213(88)90639-6 Nucl. Phys. B 304 (1988) 500–512.
    
    Dudas:2000sn
    E. Dudas and J. Mourad, “D-branes in nontachyonic 0B orientifolds,”http://dx.doi.org/10.1016/S0550-3213(00)00781-1 Nucl. Phys. B 598 (2001) 189–224,
      http://arxiv.org/abs/hep-th/0010179 arXiv:hep-th/0010179.
    
    Dudas:2003wp
    E. Dudas, J. Mourad, and C. Timirgaziu, “On cosmologically induced
      hierarchies in string theory,”http://dx.doi.org/10.1088/1475-7516/2004/03/005 JCAP
      03 (2004) 005, http://arxiv.org/abs/hep-th/0309057
      arXiv:hep-th/0309057.
    
    Abel:2015oxa
    S. Abel, K. R. Dienes, and E. Mavroudi, “Towards a nonsupersymmetric string
      phenomenology,”http://dx.doi.org/10.1103/PhysRevD.91.126014
      Phys. Rev. D 91 no. 12, (2015) 126014,
      http://arxiv.org/abs/1502.03087 arXiv:1502.03087
      [hep-th].
    
    Keller:2011xi
    C. A. Keller, “Phase transitions in symmetric orbifold CFTs and
      universality,”http://dx.doi.org/10.1007/JHEP03(2011)114 JHEP 03 (2011) 114, http://arxiv.org/abs/1101.4937
      arXiv:1101.4937 [hep-th].
    
    Dixon:1985jw
    L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten, “Strings on Orbifolds,”http://dx.doi.org/10.1016/0550-3213(85)90593-0 Nucl. Phys. B 261 (1985) 678–686.
    
    Dixon:1986jc
    L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten, “Strings on Orbifolds.
      2.,”http://dx.doi.org/10.1016/0550-3213(86)90287-7 Nucl. Phys.
      B 274 (1986) 285–314.
    
    Aspinwall:1996mn
    P. S. Aspinwall, “K3 surfaces and string duality,” in Theoretical
      Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields,
      Strings, and Duality, pp. 421–540.
     11, 1996.
    http://arxiv.org/abs/hep-th/9611137
      arXiv:hep-th/9611137.
    
    Gimon:1996rq
    E. G. Gimon and J. Polchinski, “Consistency conditions for orientifolds and
      D-manifolds,”http://dx.doi.org/10.1103/PhysRevD.54.1667 Phys.
      Rev. D 54 (1996) 1667–1676,
      http://arxiv.org/abs/hep-th/9601038 arXiv:hep-th/9601038.
    
    Angelantonj:1999jh
    C. Angelantonj, “Comments on open string orbifolds with a nonvanishing
      B_ab,”http://dx.doi.org/10.1016/S0550-3213(99)00662-8
      Nucl. Phys. B 566 (2000) 126–150,
      http://arxiv.org/abs/hep-th/9908064 arXiv:hep-th/9908064.
    
    Kakushadze:1998bw
    Z. Kakushadze, G. Shiu, and S. H. H. Tye, “Type IIB orientifolds with NS-NS
      antisymmetric tensor backgrounds,”http://dx.doi.org/10.1103/PhysRevD.58.086001 Phys. Rev. D 58 (1998) 086001,
      http://arxiv.org/abs/hep-th/9803141 arXiv:hep-th/9803141.
    
    Angelantonj:1999xf
    C. Angelantonj and R. Blumenhagen, “Discrete deformations in type I vacua,”http://dx.doi.org/10.1016/S0370-2693(99)01480-X Phys. Lett. B 473 (2000) 86–93,
      http://arxiv.org/abs/hep-th/9911190 arXiv:hep-th/9911190.
    
    Aldazabal:1999nu
    G. Aldazabal, D. Badagnani, L. E. Ibanez, and A. M. Uranga, “Tadpole versus
      anomaly cancellation in D=4 D=6 compact IIB
      orientifolds,”http://dx.doi.org/10.1088/1126-6708/1999/06/031
      JHEP 06 (1999) 031,
      http://arxiv.org/abs/hep-th/9904071 arXiv:hep-th/9904071.
    
    Bianchi:2000de
    M. Bianchi and J. F. Morales, “Anomalies & tadpoles,”http://dx.doi.org/10.1088/1126-6708/2000/03/030 JHEP
      03 (2000) 030, http://arxiv.org/abs/hep-th/0002149
      arXiv:hep-th/0002149.
    
    Douglas:1995bn
    M. R. Douglas, “Branes within branes,” NATO Sci. Ser. C
      520 (1999) 267–275, http://arxiv.org/abs/hep-th/9512077
      arXiv:hep-th/9512077.
    
    Green:1996dd
    M. B. Green, J. A. Harvey, and G. W. Moore, “I-brane inflow and anomalous
      couplings on d-branes,”http://dx.doi.org/10.1088/0264-9381/14/1/008 Class. Quant. Grav. 14 (1997) 47–52,
      http://arxiv.org/abs/hep-th/9605033 arXiv:hep-th/9605033.
    
    Morales:1998ux
    J. F. Morales, C. A. Scrucca, and M. Serone, “Anomalous couplings for
      D-branes and O-planes,”http://dx.doi.org/10.1016/S0550-3213(99)00217-5 Nucl. Phys. B 552 (1999) 291–315,
      http://arxiv.org/abs/hep-th/9812071 arXiv:hep-th/9812071.
    
    Stefanski:1998he
    B. Stefanski, Jr., “Gravitational couplings of D-branes and O-planes,”http://dx.doi.org/10.1016/S0550-3213(99)00147-9 Nucl. Phys. B 548 (1999) 275–290,
      http://arxiv.org/abs/hep-th/9812088 arXiv:hep-th/9812088.
    
    Seiberg:2011dr
    N. Seiberg and W. Taylor, “Charge Lattices and Consistency of 6D
      Supergravity,”http://dx.doi.org/10.1007/JHEP06(2011)001 JHEP 06 (2011) 001, http://arxiv.org/abs/1103.0019
      arXiv:1103.0019 [hep-th].
    
    Duff:1996cf
    M. J. Duff, H. Lu, and C. N. Pope, “Heterotic phase transitions and
      singularities of the gauge dyonic string,”http://dx.doi.org/10.1016/0370-2693(96)00420-0 Phys. Lett. B 378 (1996) 101–106,
      http://arxiv.org/abs/hep-th/9603037 arXiv:hep-th/9603037.
    
    Dudas:2001wd
    E. Dudas, J. Mourad, and A. Sagnotti, “Charged and uncharged D-branes in
      various string theories,”http://dx.doi.org/10.1016/S0550-3213(01)00552-1 Nucl. Phys. B 620 (2002) 109–151,
      http://arxiv.org/abs/hep-th/0107081 arXiv:hep-th/0107081.
    
    Adler:1969gk
    S. L. Adler, “Axial vector vertex in spinor electrodynamics,”http://dx.doi.org/10.1103/PhysRev.177.2426 Phys. Rev.
      177 (1969) 2426–2438.
    
    Bell:1969ts
    J. S. Bell and R. Jackiw, “A PCAC puzzle: π^0 →γγ in the
      σ model,”http://dx.doi.org/10.1007/BF02823296 Nuovo
      Cim. A 60 (1969) 47–61.
    
    Fujikawa:2004cx
    K. Fujikawa and H. Suzuki,
      http://dx.doi.org/10.1093/acprof:oso/9780198529132.001.0001Path
      integrals and quantum anomalies.
     2004.
    
    Bilal:2008qx
    A. Bilal, “Lectures on Anomalies,”http://arxiv.org/abs/0802.0634 arXiv:0802.0634 [hep-th].
    
    Harvey:2005it
    J. A. Harvey, “TASI 2003 lectures on anomalies,” 9, 2005.
    http://arxiv.org/abs/hep-th/0509097
      arXiv:hep-th/0509097.
    
    10.1093/acprof:oso/9780198507628.001.0001
    R. A. Bertlmann,
      http://dx.doi.org/10.1093/acprof:oso/9780198507628.001.0001Anomalies in Quantum Field Theory.
     Oxford University Press, 11, 2000.
    <https://doi.org/10.1093/acprof:oso/9780198507628.001.0001>.
    
    McNamara:2019rup
    J. McNamara and C. Vafa, “Cobordism Classes and the Swampland,”http://arxiv.org/abs/1909.10355 arXiv:1909.10355
      [hep-th].
    
    Witten:1995ex
    E. Witten, “String theory dynamics in various dimensions,”http://dx.doi.org/10.1016/0550-3213(95)00158-O Nucl. Phys. B 443 (1995) 85–126,
      http://arxiv.org/abs/hep-th/9503124 arXiv:hep-th/9503124.
    
    Monnier:2018nfs
    S. Monnier and G. W. Moore, “Remarks on the Green–Schwarz Terms
      of Six-Dimensional Supergravity Theories,”http://dx.doi.org/10.1007/s00220-019-03341-7 Commun. Math. Phys. 372 no. 3, (2019) 963–1025,
      http://arxiv.org/abs/1808.01334 arXiv:1808.01334
      [hep-th].
    
    Lee:2022spd
    Y. Lee and K. Yonekura, “Global anomalies in 8D supergravity,”http://dx.doi.org/10.1007/JHEP07(2022)125 JHEP 07
      (2022) 125, http://arxiv.org/abs/2203.12631
      arXiv:2203.12631 [hep-th].
    
    MR1867354
    A. Hatcher,  Algebraic topology.
     Cambridge University Press, Cambridge, 2002.
    
    Yonekura:2016wuc
    K. Yonekura, “Dai-Freed theorem and topological phases of matter,”http://dx.doi.org/10.1007/JHEP09(2016)022 JHEP 09
      (2016) 022, http://arxiv.org/abs/1607.01873
      arXiv:1607.01873 [hep-th].
    
    Fukaya:2019qlf
    H. Fukaya, M. Furuta, S. Matsuo, T. Onogi, S. Yamaguchi, and M. Yamashita,
      “The Atiyah–Patodi–Singer Index and Domain-Wall
      Fermion Dirac Operators,”http://dx.doi.org/10.1007/s00220-020-03806-0 Commun. Math. Phys. 380 no. 3, (2020) 1295–1311,
      http://arxiv.org/abs/1910.01987 arXiv:1910.01987
      [math.DG].
    
    Witten:2018lgb
    E. Witten, “A note on boundary conditions in Euclidean gravity,”http://dx.doi.org/10.1142/S0129055X21400043 Rev. Math. Phys. 33 no. 10, (2021) 2140004,
      http://arxiv.org/abs/1805.11559 arXiv:1805.11559
      [hep-th].
    
    Freed:2014iua
    D. S. Freed, “Anomalies and Invertible Field Theories,”http://dx.doi.org/10.1090/pspum/088/01462 Proc. Symp. Pure Math. 88 (2014) 25–46,
      http://arxiv.org/abs/1404.7224 arXiv:1404.7224 [hep-th].
    
    Hopkins:2002rd
    M. J. Hopkins and I. M. Singer, “Quadratic functions in geometry, topology,
      and M theory,” J. Diff. Geom. 70 no. 3, (2005) 329–452,
      http://arxiv.org/abs/math/0211216 arXiv:math/0211216.
    
    Atiyah:1975jf
    M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and
      Riemannian Geometry 1,”http://dx.doi.org/10.1017/S0305004100049410 Math. Proc. Cambridge
      Phil. Soc. 77 (1975) 43.
    
    Tachikawa:2024ucm
    Y. Tachikawa and H. Y. Zhang, “On a ℤ_3-valued discrete
      topological term in 10d heterotic string theories,”http://arxiv.org/abs/2403.08861 arXiv:2403.08861
      [hep-th].
    
    McNamaraThesis
    J. McNamara, The Kinematics of Quantum Gravity.
     PhD thesis, Harvard University, 2022.
    http://arxiv.org/abs/https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37372201
      https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37372201.
    
    Dierigl:2022reg
    M. Dierigl, J. J. Heckman, M. Montero, and E. Torres, “IIB string theory
      explored: Reflection 7-branes,”http://dx.doi.org/10.1103/PhysRevD.107.086015 Phys. Rev. D 107 no. 8, (2023) 086015,
      http://arxiv.org/abs/2212.05077 arXiv:2212.05077
      [hep-th].
    
    Dierigl:2023jdp
    M. Dierigl, J. J. Heckman, M. Montero, and E. Torres, “R7-branes as charge
      conjugation operators,”http://dx.doi.org/10.1103/PhysRevD.109.046004 Phys. Rev. D 109 no. 4, (2024) 046004,
      http://arxiv.org/abs/2305.05689 arXiv:2305.05689
      [hep-th].
    
    Yonekura:2024bvh
    K. Yonekura, “Constraints on the topology of Type IIB string theory,”http://arxiv.org/abs/2403.04127 arXiv:2403.04127
      [hep-th].
    
    10.1007/BFb0075216
    J. Cheeger and J. Simons, “Differential characters and geometric invariants,”
      in  Geometry and Topology, pp. 50–80.
     Springer Berlin Heidelberg, Berlin, Heidelberg, 1985.
    
    Deligne
    P. Deligne, “Théorie de hodge, ii,” Publications Mathématiques de
      l'Institut des Hautes Études Scientifiques 40 (1971) 5–57.
    
    Freed:2006yc
    D. S. Freed, G. W. Moore, and G. Segal, “Heisenberg Groups and Noncommutative
      Fluxes,”http://dx.doi.org/10.1016/j.aop.2006.07.014 Annals
      Phys. 322 (2007) 236–285,
      http://arxiv.org/abs/hep-th/0605200 arXiv:hep-th/0605200.
    
    Freed:2006ya
    D. S. Freed, G. W. Moore, and G. Segal, “The Uncertainty of Fluxes,”http://dx.doi.org/10.1007/s00220-006-0181-3 Commun. Math. Phys. 271 (2007) 247–274,
      http://arxiv.org/abs/hep-th/0605198 arXiv:hep-th/0605198.
    
    Freed:2000ta
    D. S. Freed, “Dirac charge quantization and generalized differential
      cohomology,” 11, 2000.
    http://arxiv.org/abs/hep-th/0011220
      arXiv:hep-th/0011220.
    
    Baez:1995xq
    J. C. Baez and J. Dolan, “Higher dimensional algebra and topological quantum
      field theory,”http://dx.doi.org/10.1063/1.531236 J. Math.
      Phys. 36 (1995) 6073–6105,
      http://arxiv.org/abs/q-alg/9503002 arXiv:q-alg/9503002.
    
    Freed:2007vy
    D. S. Freed, M. J. Hopkins, and C. Teleman, “Consistent orientation of moduli
      spaces,”http://arxiv.org/abs/0711.1909 arXiv:0711.1909
      [math.AT].
    
    Lurie:2009keu
    J. Lurie, “On the Classification of Topological Field Theories,”http://arxiv.org/abs/0905.0465 arXiv:0905.0465 [math.CT].
    
    Schommer-Pries:2017sdd
    C. Schommer-Pries, “Invertible Topological Field Theories,”http://arxiv.org/abs/1712.08029 arXiv:1712.08029
      [math.AT].
    
    Michelangelo
    M. Tartaglia, “Self-dual fields in 6D Supergravity,” msc thesis,
      Ludwig-Maximilians-Universitaet München, 2023.
    
    Arkani-Hamed:2006emk
    N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, “The String landscape,
      black holes and gravity as the weakest force,”http://dx.doi.org/10.1088/1126-6708/2007/06/060 JHEP
      06 (2007) 060, http://arxiv.org/abs/hep-th/0601001
      arXiv:hep-th/0601001.
    
    Harlow:2022ich
    D. Harlow, B. Heidenreich, M. Reece, and T. Rudelius, “Weak gravity
      conjecture,”http://dx.doi.org/10.1103/RevModPhys.95.035003
      Rev. Mod. Phys. 95 no. 3, (2023) 035003,
      http://arxiv.org/abs/2201.08380 arXiv:2201.08380
      [hep-th].
    
    Gepner:1987qi
    D. Gepner, “Space-Time Supersymmetry in Compactified String Theory and
      Superconformal Models,”http://dx.doi.org/10.1016/0550-3213(88)90397-5 Nucl. Phys. B 296 (1988) 757.
    
    Angelantonj:1996mw
    C. Angelantonj, M. Bianchi, G. Pradisi, A. Sagnotti, and Y. S. Stanev,
      “Comments on Gepner models and type I vacua in string theory,”http://dx.doi.org/10.1016/0370-2693(96)01124-0 Phys. Lett. B 387 (1996) 743–749,
      http://arxiv.org/abs/hep-th/9607229 arXiv:hep-th/9607229.
    
    Honecker:2006qz
    G. Honecker and M. Trapletti, “Merging Heterotic Orbifolds and K3
      Compactifications with Line Bundles,”http://dx.doi.org/10.1088/1126-6708/2007/01/051 JHEP
      01 (2007) 051, http://arxiv.org/abs/hep-th/0612030
      arXiv:hep-th/0612030.
    
    Walton
    M. A. Walton, “Heterotic string on the simplest calabi-yau manifold and its
      orbifold limits,”http://dx.doi.org/10.1103/PhysRevD.37.377
      Phys. Rev. D 37 (Jan, 1988) 377–390.
      <https://link.aps.org/doi/10.1103/PhysRevD.37.377>.
    
    Vafa:1994rv
    C. Vafa and E. Witten, “On orbifolds with discrete torsion,”http://dx.doi.org/10.1016/0393-0440(94)00048-9 J. Geom. Phys. 15 (1995) 189–214,
      http://arxiv.org/abs/hep-th/9409188 arXiv:hep-th/9409188.
    
    Heidenreich:2020pkc
    B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius, and
      I. Valenzuela, “Chern-Weil global symmetries and how quantum gravity avoids
      them,”http://dx.doi.org/10.1007/JHEP11(2021)053 JHEP 11 (2021) 053, http://arxiv.org/abs/2012.00009
      arXiv:2012.00009 [hep-th].
    
    Heidenreich:2021xpr
    B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius, and
      I. Valenzuela, “Non-invertible global symmetries and completeness of the
      spectrum,”http://dx.doi.org/10.1007/JHEP09(2021)203 JHEP 09 (2021) 203, http://arxiv.org/abs/2104.07036
      arXiv:2104.07036 [hep-th].
    
    McNamara:2021cuo
    J. McNamara, “Gravitational Solitons and Completeness,”http://arxiv.org/abs/2108.02228 arXiv:2108.02228
      [hep-th].
    
    Harlow:2018tng
    D. Harlow and H. Ooguri, “Symmetries in quantum field theory and quantum
      gravity,”http://dx.doi.org/10.1007/s00220-021-04040-y Commun.
      Math. Phys. 383 no. 3, (2021) 1669–1804,
      http://arxiv.org/abs/1810.05338 arXiv:1810.05338
      [hep-th].
    
    douglas2014topological
    C. Douglas, J. Francis, A. Henriques, and M. Hill,  Topological Modular
      Forms.
     Mathematical Surveys and Monographs. American Mathematical Society,
      2014.
    <https://books.google.de/books?id=n0W9BQAAQBAJ>.
    
    Gukov:2018iiq
    S. Gukov, D. Pei, P. Putrov, and C. Vafa, “4-manifolds and topological
      modular forms,”http://dx.doi.org/10.1007/JHEP05(2021)084 JHEP 05 (2021) 084, http://arxiv.org/abs/1811.07884
      arXiv:1811.07884 [hep-th].
    
    Gaiotto:2019asa
    D. Gaiotto, T. Johnson-Freyd, and E. Witten, “A Note On Some Minimally
      Supersymmetric Models In Two Dimensions,”http://arxiv.org/abs/1902.10249 arXiv:1902.10249
      [hep-th].
    
    Stolz:2011zj
    S. Stolz and P. Teichner, “Supersymmetric field theories and generalized
      cohomology,”http://arxiv.org/abs/1108.0189
      arXiv:1108.0189 [math.AT].
    
    Chua_2022
    D. Chua, “C2-equivariant topological modular forms,”http://dx.doi.org/10.1007/s40062-021-00297-1 Journal of Homotopy
      and Related Structures 17 no. 1, (1, 2022) 23–75.
    
    gepner2023equivariant
    D. Gepner and L. Meier, “On equivariant topological modular forms,” 2023.
    
    empty


