Low energy models of string theory

Poula Tadros and Iiro Vilja

Department of Physics and Astronomy, University of Turku, 20014,
Turku,Finland.

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###### Abstract

String theory is the prime candidate for the theory of everything. However, it
must be defined in ten dimensions to be consistent. To get 4D physics, the 6
other dimensions should be curled up in a small compact manifold, this
procedure is called string compactification. In this review, we will review
different compactification schemes proving that in absence of flux, the
compact manifold must be a Calabi-Yau manifold. Then, we review
compactifications with flux using generalized complex geometry. We then
discuss some applications in cosmology like the swampland project and the
cosmological models derived from it. We then discuss non relativistic string
theories and introduce a toroidal compactifications for such theories.
Finally, we discuss some open questions in the field.

###### Contents

  1. 1 Introduction
  2. 2 Mathematical tools
    1. 2.1 Basic category theory
      1. 2.1.1 Categories and functors
      2. 2.1.2 Diagrams
      3. 2.1.3 Natural transformations
      4. 2.1.4 The duality principle
      5. 2.1.5 Yoneda lemma
    2. 2.2 Manifolds
      1. 2.2.1 De Rham cohomology
      2. 2.2.2 Vector bundles
      3. 2.2.3 Dolbault cohomology
      4. 2.2.4 Integrability
      5. 2.2.5 Chern classes
    3. 2.3 Algebraic geometry
      1. 2.3.1 Singularities and blowups
      2. 2.3.2 Sheafs
      3. 2.3.3 Schemes
      4. 2.3.4 Moduli
      5. 2.3.5 Fibered categories and stacks
    4. 2.4 Cobordisms
  3. 3 String compactifications
    1. 3.1 Compactification of type II string theory on a circle
    2. 3.2 Toroidal compactification of type II superstrings
    3. 3.3 Compactification of type IIB superstrings on $AdS_{5}\times S^{5}$
    4. 3.4 Compactification of heterotic strings on a circle
    5. 3.5 Toroidal compactification of heterotic superstrings
    6. 3.6 Motivation for Calabi-Yau compactifications
  4. 4 Calabi-Yau compactifications
    1. 4.1 Compactifications of bosonic strings as a toy model
    2. 4.2 Type II superstring theories compactifications on Calabi-Yau manifolds
    3. 4.3 Heterotic superstring theories compactifications on Calabi-Yau manifolds
    4. 4.4 Non-supersymmetric string theory compactifications on Calabi-Yau manifolds
  5. 5 Generalized complex geometry
    1. 5.1 G-structures
    2. 5.2 Elements of generalized complex geometry
      1. 5.2.1 The metric and the volume form
      2. 5.2.2 (Twisted) Courant bracket
      3. 5.2.3 Polyforms and pure spinors
      4. 5.2.4 Twisted exterior derivative, generalized complex structures and integrability
    3. 5.3 Generalized Calabi-Yau manifolds
  6. 6 Flux compactifications
    1. 6.1 Fluxes and charges
      1. 6.1.1 Dirac quantization rule
      2. 6.1.2 Types of fluxes
    2. 6.2 Compactifications on twisted tori
      1. 6.2.1 Twisted tori
      2. 6.2.2 The compactification model
    3. 6.3 Compactifications of type II superstrings on Generalized Calabi-Yau manifolds
      1. 6.3.1 Torsion classes
      2. 6.3.2 Type II compactifications with flux
      3. 6.3.3 Generalised geometry formulation
    4. 6.4 Heterotic string theory compactifications with flux
    5. 6.5 Non-supersymmetric string theory compactifications with flux
  7. 7 Applications to cosmology
    1. 7.1 The swampland project
      1. 7.1.1 Introduction
      2. 7.1.2 Overview on the swampland conjectures
      3. 7.1.3 No global symmetries hypothesis
      4. 7.1.4 Cobordism conjecture
      5. 7.1.5 The completeness hypothesis
      6. 7.1.6 The weak gravity conjecture
      7. 7.1.7 The distance conjecture
      8. 7.1.8 AdS distance conjecture
      9. 7.1.9 Non-supersymmetry AdS conjecture
      10. 7.1.10 De Sitter conjecture
    2. 7.2 The possibility of finding De Sitter vacua
      1. 7.2.1 Brane world cosmologies
      2. 7.2.2 Warm inflation
      3. 7.2.3 Dynamical dark energy models
      4. 7.2.4 Other models
  8. 8 Non relativistic string theory
  9. 9 Open problems
    1. 9.1 Moduli stabilization problem
    2. 9.2 The swampland project
  10. 10 Conclusion
  11. 11 References

##  1 Introduction

String theory is currently the most promising candidate for the theory of
everything but it contains a conformal anomaly which means an inconsistency in
the theory. To cancel this anomaly bosonic string theory must be defined on a
26 dimensional spacetime and superstring theories have to be defined on a 10
dimensional spacetime. But we know that we live in 4 dimensions so we have to
reduce the dimensions of string theories spacetimes to 4. The way to do this
is to assume that the ambient spacetime manifold of the string theory is a
product of two manifolds: a 4 dimensional non compact manifold which will
represent the 4 dimensional spacetime we observe, and a higher dimensional
compact manifold (22 dimensional in the case of bosonic string theory and 6
dimensional in case of superstring theories) representing an internal space
which we can not observe because the compact manifold is too small for us to
observe in low energy, this procedure is called a string compactification.

String compactification models usually give rise to massless scalar fields
with no potentials on the 4 dimensional manifolds, these scalars are called
moduli and are problematic from the physical point of view because they can
mediate long range forces which are not observed in nature. To solve this
problem, additional structures are added to the ambient manifold namely
differential forms called fluxes. Fluxes induce potentials to moduli so that
they can not mediate long range forces, this process is called moduli
stabilization. String compactifications on manifolds with fluxes are called
flux compactifications.

On studying string compactification models on a certain manifold, there are
three main steps:

  1. 1.

The study of the possible compactification manifolds and their geometry i.e.
we study which manifold we can compactify the theory on to give a realistic 4
dimensional particle physics theory, and the spectrum of this low energy 4
dimensional model since the geometry of the internal manifold affects the 4
dimensional low energy model.

  2. 2.

The classification of vacua in the compactified model i.e. to study what are
the possible vacua can arise from the theory and see which one makes a
physically viable 4 dimensional model. This step is particularly important if
we want to construct cosmological models from the compactification model as we
will see in chapter 8.

  3. 3.

The study of moduli spaces of the compactification models. This is very
important because the geometry of the moduli space dramatically affects the 4
dimensional model and it also gives hints of which flux should we introduce to
stabilize the moduli.

In this review we focus on step 1, and only get in touch with step 2 in
chapters 6.

In chapter 2, we review some mathematical concepts that will be used
throughout the review. In chapter 3, we introduce some string compactification
models and study the low energy 4 dimensional models arising from them and
their spectra, since there are many compactification models, we introduce the
most generic ones so the reader can follow the same procedure with other
similar models. We study the process for bosonic string theory as a toy model,
type II and heterotic superstring theories. Then, we present some challenges
to the aforementioned models leading to choosing the compactification manifold
to be a Calabi-Yau. In chapter 4, we study compactifications on Calabi-Yau
compactification i.e. string compactifications with the internal manifold to
be Calabi-Yau. There we show that a realistic low energy model from string
compactifications without flux must arise from a Calabi-Yau compactification
in case of heterotic supertring theories, we also present the compactification
schemes of bosonic strings, type II superstring theories and non
supersymmetric string theory. However, as mentioned earlier, this does not
stabilize the moduli in the theory thus, flux must be added. In chapter 5 we
review generalised complex geometry which is the math used in flux
compactifications. In chapter 6, we study different models of flux
compactifications showing how flux can break supersymmetry leading to
realistic low energy models. In chapter 7, we discuss the swampland project
which was designed to test whether a low energy model is consistent with
quantum gravity, in this case string theory, and we present some vacua from
string compactification models motivated by the swampland conjecture
representing possible realistic cosmological models. In chapter 8, we review
non realtivistic string theory, so far there are no compactification models
for non relativistic strings, we construct a toroidal compactification model
in this chapter and directions for future research in the field. In chapter 9,
we present some open problems and what are the directions of research at the
moment.

##  2 Mathematical tools

###  2.1 Basic category theory

####  2.1.1 Categories and functors

Category theory was founded by Eilenberg and Maclane in 1945 [1] as a step to
define natural transformations between mathematical structures. Category
theory can be though of as a unifying scheme that takes a bird’s eye view on
all mathematical structures, for example we can define sets and functions
between them, groups and homomorphisms preserving group structure, topological
spaces and continuous maps (preserving the topological structure), rings and
ring homomorphisms, vector spaces and linear transformations, etc. The main
idea is a structure and a map preserving it thus, we can unify them calling
the structure "objects" and the maps "morphisms" and the whole system of
objects and morphisms as Category.
There are two equivalent ways to properly define Categories. The first uses
the notions of metagraphs and metacategories as presented in [2]. A recent way
defines Categories directly by some axioms. In this review, we use the second
approach.

Definition: A Category $\mathcal{C}$ consists of:

  1. 1.

Objects denoted by capital letters A,B,… collectively denoted by
obj($\mathcal{C}$).

  2. 2.

morphisms between objects denoted by small letters f,g,… . Morphisms between A
and B in the category $\mathcal{C}$ are denoted by $\mathcal{C}(A,B)$ or
Hom(A,B). A is called the domain and B the codomain. The collection of all
morphism in the category $\mathcal{C}$ is denoted by Hom($\mathcal{C}$)

Satisfying the following axioms:

  1. 1.

Composition of morphisms is defined i.e. the map
$\mathcal{C}(A,B)\times\mathcal{C}(B,C)\rightarrow\mathcal{C}(A,C)$ ;
$(f,g)\rightarrow f\circ g$ is defined.

  2. 2.

Associativity of morphisms i.e. $\forall
f\in\mathcal{C}(A,B),g\in\mathcal{C}(B,C)$ and $h\in\mathcal{C}(C,D)$ ;
$(f\circ g)\circ h=f\circ(g\circ h)$.

  3. 3.

The existence of identity morphism to each object i.e. $\forall
A\in\mathcal{C}$ $\exists\ 1_{A}\in\mathcal{C}(A,A)$; $\forall
f\in\mathcal{C}(A,B)$ $f\circ 1_{A}=f=1_{B}\circ f$.

A morphism f from A to B is called invertible if there exists another morphism
g from B to A such that $f\circ g=1_{B}$ and $g\circ f=1_{A}$.

Graphically, we can consider objects as points and morphisms as arrows as
shown in fig.1

CategoryABCf Figure 1: A graphical representation of a Category.

Examples :

  1. 1.

Set: The Category of sets (sets are objects and functions are morphisms).

  2. 2.

Grp: The category of groups (groups are objects and homomorphisms are
morphisms).

  3. 3.

Ab : the category of Abelian groups.

  4. 4.

Top : The category of topological spaces with continuous maps as morphisms.

  5. 5.

$\mathbf{Vect_{k}}$: The category of vector spaces over the field k with
linear transformations as morphisms.

Definition: A category is called small if the collection of objects and the
collection of morphisms form sets. If a category is not small, it is called
large.

Example: Any finite category is small.

Non-Example: The category of sets (Set) is a large category because the
collection of sets is not a set.

Definition: A Monoid is a category with one object.

Example: A group is a monoid (the set is the object and group operation
between elements are morphisms )such that every element has an inverse.

Next we consider mapping between categories.

Definition: A Covariant functor F from a category $\mathcal{C}$ to a category
$\mathcal{A}$ is a mapping assigning objects in $\mathcal{C}$ to objects in
$\mathcal{A}$, and morphisms in $\mathcal{C}$ to morphisms in $\mathcal{A}$,
satisfying the following axioms:

  1. 1.

$F(1_{A})=1_{F(A)}$ for all A in $\mathcal{C}$.

  2. 2.

$\forall f,g\in Hom(C)\ F(fg)=F(f)F(g)$ so called covariant.

Examples:

  1. 1.

The assignment of a fundamental group to topological spaces is considered as a
covariant functor from Top to Ab

  2. 2.

An important example for string theory is that moduli are defined as covariant
functors from the category of schemes to the category of sets as defined
later.

####  2.1.2 Diagrams

Calculations using category theory uses diagrams modelling objects as points
and morphisms as arrows. This method helps writing definitions and theorems in
a more compact way which is easier to handle.

Definition.A diagram is called commutative if composition rules does not
depend on a path in the diagram.

Example: The statement "Diagram 2 commutes" (assuming every arrow is
invertible)is equivalent to the conditions: $f=g^{-1}\circ I\circ h$ , $g\circ
f=I\circ h$ , $h=I^{-1}\circ g\circ f$ , $I=g\circ f\circ h^{-1}$ , $g=I\circ
h\circ f^{-1}$ and the inverse counterparts of these conditions.
This representation is the representation used throughout the review.

ABCDfghI Figure 2: This is an example of a diagram in category theory, this id
labeled in the text by "Diagram 2".

####  2.1.3 Natural transformations

Natural transformations are transformations between functors between two
specified categories. It can be thought of as morphisms between functors
motivating the definition of functor categories. We will not discuss functor
categories in this review.

Definition: Let s and t be functors from the category $\mathcal{C}$ to the
category $\mathcal{A}$, and let C be an arbitrary object in $\mathcal{C}$, and
f be a morphism in $\mathcal{C}$ mapping C to C’. A Natural transformation
$\tau:s\rightarrow t$ is a mapping assigning to each object C in $\mathcal{C}$
an arrow from s(C) to t(C) such that diagram 3 commutes.

s(C)t(C)s(C’)t(C’)$\displaystyle\tau$ct(f)s(f)$\displaystyle\tau c^{\prime}$
Figure 3: This Diagram represents the action of natural transformations if it
commutes. It is refered to in the text by "Diagram 3".

A natural transformation with every object $\tau$c is invertible is called a
natural equivalence.

Example: The most famous example of a natural transformation is a determinant
of matrices with entries in an arbitrary commutative ring, which is a natural
transformation between two functors from the category of commutative rings to
the category of groups.

####  2.1.4 The duality principle

The informal statement of the duality principle is that any result can be
deduced from category theory has a dual statement where all arrows are
reversed i.e. domains and codomains of all morphisms are interchanged.

Definition: For any category $\mathcal{C}$, we can define its opposite
category denoted by $\mathcal{C}^{op}$ consisting of the same objects but all
arrows are reversed. If $\mathcal{C}$ = $\mathcal{C}^{op}$ then it is called
self dual.

Definition: A functor of the form G:$\mathcal{C}^{op}\rightarrow\mathcal{A}$
is called a contravariant functor from $\mathcal{C}$ to $\mathcal{A}$.

It is called contravariant because the composition rule in this case is
$G(f\circ g)=G(g)\circ G(f)$, because all morphisms in $\mathcal{C}$ are
reversed. Note that a functor from $\mathcal{C}^{op}$ to $\mathcal{A}^{op}$ is
covariant because the composition rule is reversed two times.

Example: The assignment of cohomology groups to each topological space is a
contravariant functor form $\mathbf{Top}$ to Ab.

The formal duality principle: For any sentence in the language of category
theory following from axioms of categories, there is a dual statement which
also follows from the same axioms.

####  2.1.5 Yoneda lemma

Yoneda lemma is considered one of the most important results in category
theory. It basically says that we can identify arbitrary objects in arbitrary
categories knowing only their relations with other categories, for example, we
can identify a topological space by knowing how the other topological spaces
are mapped to it continuously.

Definitions:

  1. 1.

Assume that $\mathcal{C}$ is a locally small category. We define the
$Hom_{\mathcal{C}}(-,C)$ as a contravariant functor from a category
$\mathcal{C}$ to Set, sending an object d in $\mathcal{C}$ to the set Hom(D,C)
the set of all morphisms from D to C, and sending morphisms in $\mathcal{C}$
to functions in Set according to the following rule :
If $f:D\rightarrow E$ is a morphism from D to E in $\mathcal{C}$, Then
$Hom_{\mathcal{C}}(f,C)$ is a function from Hom(E,C) to Hom(D,C) as the
composition of the morphism form E to C and f.

  2. 2.

Let F be a covariant functor from $\mathcal{C}$ to $\mathcal{A}$ (locally
small categories). F induces a function

$F_{C,A}:Hom_{\mathcal{C}}(C,A)\rightarrow Hom_{\mathcal{A}(F(C),F(A))}$

F is called faithful if $F_{C,A}$ is injective, and is called full if
$F_{C,A}$ is surjective. If $F_{C,A}$ is bijective then F is called Fully
faithful.

  3. 3.

We denote the category of all functors from $\mathcal{C}$ to $\mathcal{A}$,
with morphisms are natural transformations, by
Func($\mathcal{C}$,$\mathcal{D}$).

  4. 4.

Let $\mathcal{C}$ be a locally small category. Yoneda functor is defined as
$h:\mathcal{C}\rightarrow\mathbf{Func(\mathcal{C}^{op},Set)}$, mapping objects
as $C\rightarrow Hom(-,C)$ and morphisms as a covariant version of definition
1(note that Yoneda functor is covariant).

Lemma(Yoneda): Let $\mathcal{C}$ be a locally small category. For every object
C in $\mathcal{C}$ and for every functor F in Func($\mathcal{C}^{op}$, Set),
there is an isomorphism

$Hom(hC,F)\cong FC,$

where h is the Yoneda functor. Moreover, the isomorphism is natural in the
sense discussed before.

Theorem: The Yoneda functor is fully faithful.

Yoneda’s lemma tells us that all information of an object in a category, say
an unknown manifold we want to study, is entirely embedded in the set of
functors with other objects, in our example if we know how to map other
manifolds to this unknown manifold, then we can determine its structure
entirely.

###  2.2 Manifolds

Here, we assume the reader to be familiar with the definition of real and
complex manifolds, tangent and cotangent spaces, tensors and lie groups. This
section follows the analysis of Ref [3,4].

####  2.2.1 De Rham cohomology

Definition:A differential $\omega$ form is called exact if there exists
another differential form $\Omega$ such that $d\Omega=\omega$, and is called
closed if $d\omega=0$.

It is easy to prove that the vector space of exact forms is a subspace of the
vector space of closed forms. Their quotient is called De Rham cohomology,
denoted by $H^{k}(M)$ (k is the degree of the forms and M is the manifold).

Proposition: If the dimension of the manifold is n then $k>n\implies
H^{k}(M)=0$.

An important property of De Rham cohomology is that we can define
$H^{*}(M)=\bigoplus_{k=0}^{n}H^{k}(M)$ which can be shown to be an
anticommutative graded ring over R. Thus, De Rham cohomology is a
contravariant functor from the category of $C^{\infty}$ manifolds to the
category of anticommutative graded rings.

####  2.2.2 Vector bundles

Definition: A smooth k-dimensional vector bundle is a pair of smooth manifolds
E (called the total space), and M (called the base) with a surjective map
$\pi:E\rightarrow M$ (called the projection), such that

  1. 1.

$\forall p\in M\ \ E_{p}=\pi^{-1}(p)$ (called a fiber of E over p) is a vector
space.

  2. 2.

$\forall p\in M\ \ \exists$ a neighborhood U of p such that U and the product
$\text{U}\times\mathbb{R}^{K}$ are diffeomorphic i.e. there exists a mapping
$\phi:\pi^{-1}(\text{U})\rightarrow\text{U}\times\mathbb{R}^{k}$ such that
$\phi$ is a diffeomorphism and diagram 4 commutes. In this case $\phi$ called
a local trivialization of E,

$\displaystyle\pi^{-1}$(U)UU $\displaystyle\times\
R^{k}$$\pi_{1}$$\displaystyle\varphi$$\displaystyle\pi$ Figure 4: This
diagram’s commutativity represents the condition of local trivialization
existence and it is refered to in the text by "Diagram 4".

  3. 3.

The restriction $\phi|_{E_{p}}:E_{p}\rightarrow\\{p\\}\times\mathbb{R}^{k}$ is
a linear isomorphism.

The idea of defining vector bundle is to associate a vector space to every
point in the manifold, or equivalently, parameterizing a family of vector
spaces by a manifold.

Examples:

  1. 1.

The tangent bundle of a smooth manifold is the disjoint union of all tangent
spaces of every point in the manifold.

  2. 2.

The cotangent bundle of a manifold is the disjoint union of cotangent spaces
of the manifold.

Definition: Let $\pi:E\rightarrow M$ be a vector bundle over M. A section of E
is a map $F:M\rightarrow E$ such that $\pi\circ F=Id_{M}$. A section is called
smooth if F is smooth.

Examples:

  1. 1.

A section over the vector bundle of a manifold is a vector field i.e. the
vector space of derivations on the manifold.

  2. 2.

A section over the cotangent bundle is a covector field i.e. the vector space
of one forms on the manifold.

####  2.2.3 Dolbault cohomology

On a complex manifold, at least locally, we can define coordinates as in real
manifolds, the difference is that coordinated in this case can be divided into
holomorphic coordinates $z_{i}$ and antiholomorphic coordinates $\bar{z}_{i}$,
for example a differential form can locally be written as

$\omega=f(z_{1},z_{2},...,z_{p},\bar{z}_{1},\bar{z}_{2},...,\bar{z}_{q})dz_{1}\wedge
dz_{2}\wedge...\wedge dz_{p}\wedge d\bar{z_{1}}\wedge
d\bar{z}_{2}\wedge...\wedge\bar{z}_{q}.$

This has an important consequence on the cohomology theory, that is the De
Rham complex is now bigraded not just graded as in the real case. Thus, we
have to define a holomorphic and antiholomorphic differentials, cohomologies,
dimensions and so on.

Notations: Let M be a complex mainfold, we denote the bigraded complex vector
bundle of all forms with p holomorphic and q antiholomorphic coordinate
dependence i.e. of bi degree (p,q) by $\Omega^{p,q}$ and
$\Gamma(\Omega^{p,q},M)$ is its space of sections.

We begin with defining adequate differential operators. Beginning with taking
the exterior derivative of a generic differential form and by the chain rule,
we can see that the exterior derivative can be decomposed into holomorphic and
antiholomorphic parts

$d\omega=\partial\omega+\bar{\partial}\omega$

where $\partial$ is the holomorphic part i.e. a differential acting only on
the holomorphic part of $\omega$, and $\bar{\partial}$ is the antiholomorphic
part which leads to the formal definition of Dolbeault operators.

Definition: Let M be a complex manifold, the Dolbeault operators on its charts
are defined as

$\partial:\Gamma(\Omega^{p,q},M)\rightarrow\Gamma(\Omega^{p+1,q},M),$
$\bar{\partial}:\Gamma(\Omega^{p,q},M)\rightarrow\Gamma(\Omega^{p,q+1},M).$

such that each operator is a derivation.

The next step is to define a cohomology with respect to Dolbeault operators,
defining holomorphic and antiholomorphic closed and exact forms is similar to
the real case, so the general cohomology group is defined as the quotient

$H^{p,q}(M)=\frac{ker(\bar{\partial}(\Gamma(\Omega^{p,q},M))}{Im(\bar{\partial}(\Gamma(\Omega^{p,q-1},M)))},$

Similar to Betti numbers, the complex dimensions of Doulbeault cohomology
groups are called Hodge numbers

$h^{p,q}(M)=dim_{C}(H^{p,q}(M)),$

which will be used extensively in chapter 4 and 5 in Calabi-Yau
compactifications.

####  2.2.4 Integrability

Definition: Let M be an almost complex manifold with almost complex structure
J. If the lie bracket of any two holomorphic vector fields is holomorphic,
then J is said to be integrable. An integrable almost complex structure is
said to be a complex structure and an even dimensional manifold M equipped
with the complex structure is called a complex manifold.

Theorem: Let M be an almost complex manifold with almost complex structure J.
J is integrable if and only if the Nijenhuis tensor defined as

$N_{J}(X,Y)=[X,Y]+J[X,JY]+J[JX,Y]-[JX,JY],$

where $[X,Y]$ is the lie bracket of the vector fields X and Y, vanishes for
any two vector fields.

We now define an important type of complex manifolds namely Kahler manifolds.
Firstly, we observe that to any almost complex structure J provided that the
metric g is hermitian, we can associate a hermitian 2-form defined by
$\omega(X,Y)=g(X,JY)$.

Theorem: Let M be a complex manifold with a complex structure J, a hermitian
metric g and an associated hermitian 2-form $\omega$, and let $\nabla$ be the
Levi-Civita connection, then the following statements are equivalent ;

  1. 1.

$\nabla\omega=0$

  2. 2.

$\nabla J=0$

  3. 3.

$d\omega=0$

The manifold satisfying these conditions is called a Kahler manifold.

####  2.2.5 Chern classes

The motivation to define Chern classes is the question of whether or not two
vector bundles on a manifold are isomorphic. Chern classes give a necessary
condition on the existence of such isomorphism i.e. if two bundles do not have
equal Chern classes, they are not isomorphic.

Definition: Let E $\rightarrow$ M be a vector bundle over a manifold M with
connection form $\nabla$ whose curvature form is $R^{\nabla}$. The total Chern
class is defined as

$c(E,t)=\det(I+\frac{i}{2\pi}R^{\nabla})$

The total Chern class can be expanded as

$c(E,t)=\sum_{n=0}^{M}c_{n}(E)$

where $M\leq max(dim(M),rk(E))$ and $c_{n}(E)$ is called the nth Chern class.

In this review, only the first chern class is important and is given by
$c_{1}(E)=\frac{i}{2\pi}\text{t}r(R^{\nabla})$. We can see that the first
chern class has a direct relation to the curvature of the manifold, and also
the canonical bundle. This relation allows to define an interesting subclass
of Kahler manifolds called Calabi-Yau manifolds as follows.

Theorem: Let M be a compact Kahler manifold of complex dimension n. The
following statements are equivalent:

  1. 1.

M has a trivial canonical bundle.

  2. 2.

The holomorphic bundle $TM^{(1,0)}\rightarrow M$ has a vanishing first chern
class.

  3. 3.

M has holonomy SU($n/2$).

  4. 4.

M is Ricci flat.

The manifold satisfying these conditions is called Calabi-Yau manifold.
The theory of Calabi-Yau manifolds is presented in more details in chapter 4.

###  2.3 Algebraic geometry

In this section I will review the basic theory of sheafs and schemes. I assume
the knowledge of algebraic and projective varieties, Noetherian rings,
projective geometry. This section will follow Ref [5,6].

####  2.3.1 Singularities and blowups

We begin by reviewing the concepts of singularities in affine varieties and
how to resolve them.

Definition:Let V be an irreducible variety, a point in V is called singular if
the dimension of its tangent space is greater than the dimension of the
variety.

In case of curves (one dimensional varieties), singular points take the form
of self intersections, cusps or multiples of them.

A way to resolve a singularity i.e. replacing the singular curve with a
regular one, is by blowing up the singularity. This is done by introducing
additional coordinates then express the curve in terms of the new extended
coordinate system which results in a regular curve and an additional
exceptional curve.The procedure is as follows:

Given a curve in the coordinate system $\\{x_{i}\\}$ ; $i=1,2,...,dim(X)$
where X is the ambient space.

  1. 1.

Introduce additional projective coordinates $\\{y_{i}\\}$ and impose the
constraints $x_{i}y_{j}=x_{j}y_{i}$ for $i,j=1,2,...,dim(X)$. This defines a
closed subvariety S$\subset V\times P^{n-1}$

  2. 2.

Define the blowup map $\sigma:S\rightarrow V$ to be the first projection.

  3. 3.

To blowup a singularity $z\in V$ consider the map $\sigma^{-1}:V\rightarrow
S$, the image of every regular point is itself while the image of z is
$z\times P^{n-1}$, and the resulting curve in non singular.

The image of the singular point is mapped to its projectified tangent space
i.e. since the point has infinitely many tangents, every tangent is mapped to
a point in the additional dimensions we introduced, or equivalently each
tangent is mapped to a point in an exceptional curve.

Example: Consider the curve $y^{2}=x^{3}+x^{2}$, the curve has a singularity
(self intersection) at $(0,0)$. To blow up the singularity, firstly we define
a new projective coordinate t, then impose the condition xt=y. Substituting in
the curve we get $x^{2}(t^{2}-x-1)=0$ i.e. two curves: a non singular curve
$t^{2}-x-1=0$ and an exceptional curve $x=0$.
If the resulting curve is again singular, we iterate this procedure until it
is resolved. If it will take infinite number of iterations, it can be resolved
by Newton’s rotating ruler method which is beyond the scope of this review.

Example: Consider the curve $(x^{2}+y^{2}-1)^{2}=0$ which represents two
coincident unit circles whose center is at the origin, each circle is called a
branch of the curve i.e. the curve consists of two coincident branches each
one of them is a unit circle whose center is at the origin. This curve has a
singularity in every point since every point is an intersection point between
the two branches i.e. a self intersection of the curve. This curve’s
singularities can not be resolved by a finite number of blowups because each
singularity requires a blowup and there are infinite number of singularities.
The singularities of this curve can be resolved by Newton’s rotating ruler
method.

####  2.3.2 Sheafs

Definition: Let X be a topological space. Suppose we associate a set
$\mathcal{F}(\text{U})$ (called a section of the presheaf) to every open set U
$\subset$ X , and a map
$\rho^{\text{V}}_{\text{U}}:\mathcal{F}(\text{V})\rightarrow\mathcal{F}(\text{(}U)),$
for every V containing U, satisfying the following:

  1. 1.

$\text{U}=\phi\implies\mathcal{F}(\text{U})$ is a singleton.

  2. 2.

$\forall$ open subset U of X, $\rho^{\text{U}}_{\text{U}}$ is the identity
map.

  3. 3.

$\forall$ open sets $\text{U}\subset\text{V}\subset\text{W}$,
$\rho^{\text{W}}_{\text{U}}=\rho^{\text{V}}_{\text{U}}\circ\rho^{\text{W}}_{\text{V}}$.

This system of sets and maps is called a presheaf, and is denoted by
$\mathcal{F}.$

If all the sets $\mathcal{F}(\text{U})$ are groups, rings, fields,… ,then the
presheaf $\mathcal{F}$ is called a presheaf of groups, rings, fields,… .

Note that the three conditions in the definition are exactly the condition in
the definition of a functor discussed in section 1. This leads to a more
general definition for a presheaf.

Definition: An $\mathcal{S}$-valued presheaf on a small category
$\mathcal{C}$, where $\mathcal{S}$ is a category, is a contravariant functor
from $\mathcal{C}$ to $\mathcal{S}$. We recover the previous definition if we
set $\mathcal{C}=\mathbf{Top(X)}$(The category of open sets in the topological
space X).

Definition: A presheaf $\mathcal{F}$ on a topological space X is a sheaf if
$\forall$ open set U $\subset$ X and $\forall$ open cover of U
($\text{U}=\cup_{\alpha}\text{U}_{\alpha}$), the following conditions are
satisfied:

  1. 1.

Let $s_{1}$ and $s_{2}$ be arbitrary elements of $\mathcal{F}(\text{U})$.
Then,
$\forall\text{U}_{\alpha}\rho^{\text{U}}_{\text{U}_{\alpha}}(s_{1})=\rho^{\text{U}}_{\text{U}_{\alpha}}(s_{2})\implies
s_{1}=s_{2}$.

  2. 2.

Let $S_{\alpha}\in\mathcal{F}(\text{U})$.
$\forall\text{U}_{\alpha},\text{U}_{\beta}\subset\text{U}$,
$\rho^{\text{U}_{\alpha}}_{\text{U}_{\alpha}{\cap}\text{U}_{\beta}}(s_{\alpha})=\rho^{\text{U}_{\beta}}_{\text{U}_{\alpha}\cap\text{U}_{\beta}}(s_{\beta})\implies\exists
s\in\mathcal{F}(\text{U})$; $s_{\alpha}=\rho^{\text{U}}_{\text{U}_{\alpha}}$
for each $\text{U}_{\alpha}$.

Basically a presheaf is a construction to associate data to open sets in a
topological space, a data can be groups so the presheaf is a presheaf of
groups, or rings so the presheaf is a presheaf of rings and so on. A sheaf is
when this data can be "glued" together throughout the topological space, and
can be restricted to smaller sets without distortion i.e. the total data
equals the sum of data restricted in each subset.

Example: An important example of a sheaf is the presheaf of rings on Spec(A)
the topological space of all prime ideals of a ring A. This sheaf is called
the structure sheaf on Spec(A), and is denoted by $\mathcal{O}_{\text{A}}$.

Non-Example: An example of a presheaf that fails to be a sheaf is the presheaf
where $\mathcal{F}(\text{U})$ is the set of constant maps from U to a set M.
This presheaf fails to meet the second condition in the definition of a sheaf.

####  2.3.3 Schemes

Definition:Let X be a topological space, and let $O_{X}$ be a sheaf of rings
on X, A Ringed space is the pair $(X,O_{X})$. In this case the sheaf $O_{X}$
is called the structure sheaf of X. Obviously, ringed spaces form a category.

Definition: A Scheme is a ringed space (X,$O_{X}$) such that every point x has
a neighborhood U (called an affine neighborhood of x)satisfying the condition
that the ringed space (U,$O_{X}|_{\text{U}}$) is isomorphic to Spec(A) for
some ring A.Schemes form a category

Examples:

  1. 1.

Spec(A) itself is a scheme for any ring A.

  2. 2.

Any polynomial with real coefficients determines a scheme in the real
projective space, called a projective hypersurface.

Definition: A scheme X with a morphism from it to another scheme S is called
an S-scheme.

From the definition we can see that a scheme is the most general construction
we can do projective geometry on. It can be considered as the algebraic
counterpart of a manifold (Manifolds are locally homeomorphic to $R^{n}$, and
a scheme is locally isomorphic to Spec(A)). We define schemes because
Manifolds can not be used in algebraic geometry because here we are dealing
with ratios of polynomials which is far less easier to deal with than smooth
functions.

####  2.3.4 Moduli

Moduli are one of the most important objects in the theory of deformation of
complex structures and consequently string theory. Physically, Moduli are
massless fields with no potential, so they can take any value without
requiring energy but each value can deform the complex structure to give
different physical outcomes. In this review, we will deal with it in a more
mathematical way.

Definition: A Modulus $\mathcal{M}$ is a contravariant functor from the
category of schemes to the category of sets sending every S-scheme to the set
of classes of isomorphic families of S(such a space whose points are
isomorphism classes is called moduli space), and sending every morphism to its
pullback.

Example: Consider the scheme of all annuli, two annuli are conformally
equivalent (isomorphic in the scheme) is the ratio of the outer radius to the
inner radius is equal in both annuli. This can be reformulated into the
statement: for every positive number greater than one, there is an infinite
set of equivalent annuli having the outer to inner radius ratio equal to this
number. Thus, the moduli space for annuli is the set $[1,\infty]$, and the
outer to inner ratio is a modulus.

####  2.3.5 Fibered categories and stacks

###  2.4 Cobordisms

Cobordisms are mathematical objects emerged to answer a question of whether a
given manifold is a boundary of a higher dimensional manifold. This tool will
be useful later when discussing the swampland project as cobordisms are
related to global symmetries in supergravity theories as will discuss later in
chapter 9 in detail. In this section we will briefly introduce cobordisms and
cobordant manifolds and their category theory generalisation.

Definition: Let M and N be two n dimensional smooth, compact manifolds. M and
N are called cobordant if there exists an $n+1$ dimensional manifold with
boundaries W, called the cobordism joining $M$ and $N$, and two embeddings
$i:M\rightarrow\partial W$, and $j:N\rightarrow\partial W$ such that

  1. 1.

$i(M)\cap j(N)=\phi$

  2. 2.

$\partial W=i(M)\cup j(N)$

Figure 5: Here M and N are two dimensional cobordant manifolds due to the
existence of the 3 dimensional manifold W whose boundaries are comprised
entirely of M and N without them intersecting.

Definition: Let M be n dimensional smooth compact manifold, the set of all n
dimensional manifolds cobordant to M is called the cobordism class of M, and
is denoted by [M].

Clearly the set of cobordims classes of n dimensional manifolds equipped with
the disjoint union as an operation is an Abelian group (with the identity is
[$\phi$] and the inverse of each element is the element itself). Thus, the set
of cobordisms classes of n dimensional manifolds is also a category denoted by
$\mathbf{Bord_{n}}$ whose objects are n dimensional manifolds and a morphism
from a manifold $E$ to a manifold $F$ is the $n+1$ dimensional manifolds
defining the cobordism between them.

##  3 String compactifications

In this chapter we review some compactification models of superstring theories
without flux. As type I superstring theory is just type II superstring theory
on an orbifold, it is sufficient to consider only type II and heterotic
superstrings.

###  3.1 Compactification of type II string theory on a circle

The easiest compactification scheme is to compactify on $S^{1}$ leading to a
9D theory. While it is not realistic, it serves as a tool to understand more
sophisticated compactifications. Compactifications on circles and tori are
reviewed in [7-10].

In this scheme the 10D underlying manifold of the theory is written as
$\mathcal{M}_{10}=\mathcal{M}_{9}\times S^{1}$ where $\mathcal{M}_{9}$ is a
non compact 9D manifold.

Since one dimension is compact, its momentum must be quantized. On a circle

$X^{9}(\sigma+l,t)=X^{9}(\sigma,t)+2\pi R\omega$

where $X^{9}$ is the coordinate along the compact dimension, $l$ is the length
of the string, R is the radius of the circle, $\sigma$ and t are the
parameters of the parameter space, and $\omega$ is the winding number i.e. the
homotopy class of the string.

The momentum is then $p_{9}=\frac{k}{R}$ where k is an integer. On decomposing
into left and right moving modes, the momenta are

$p_{L}=\frac{k}{R}+\frac{\omega R}{\alpha^{\prime}}$
$p_{R}=\frac{k}{R}-\frac{\omega R}{\alpha^{\prime}}$

with mass formulae

$M^{2}_{L}=\frac{p_{L}^{2}}{2}+\frac{2}{\alpha^{\prime}}(N_{B}+N_{F}+E_{0})$
$M^{2}_{R}=\frac{p_{R}^{2}}{2}+\frac{2}{\alpha^{\prime}}(\bar{N}_{B}+\bar{N}_{F}+\bar{E}_{0})$

where $N_{B}$ is the left moving bosonic mode number, $N_{F}$ is the left
moving fermionc mode number and $E_{0}$ is a constant, the quantities with
bars are their right moving counterparts. Note that there are no tachyons in
the spectrum.

We now focus on the massless modes. In the original 10D theory, the space is
9D so the symmetry group is $SO(8)$, after the compactification the non
compact space is 8D so the symmetry group must be $SO(7)$ i.e. we need to
decompose the representations of $SO(8)$ into representations of $SO(7)$ to
get the particle content after compactification.

The Neveu Schwartz (NS) sector is decomposed as
$\mathbf{8}_{V}\rightarrow\mathbf{7}\oplus\mathbf{1}$ where $\mathbf{8}_{V}$
is the vector 8D representation of $SO(8)$ and $\mathbf{7}$ is the vector
representation of $SO(7)$. For Ramond (R) sector the decompositions are
$\mathbf{8}_{S}\rightarrow\mathbf{8}$ and
$\mathbf{8}_{C}\rightarrow\mathbf{8}$ where $\mathbf{8}_{S}$ and
$\mathbf{8}_{C}$ are the chiral representations of $SO(8)$ of left and right
movers respectively, and $\mathbf{8}$ is the spinor representation of $SO(7)$.
Note that all information about chirality is lost in the process because no
chirality can be defined in odd dimensions, this indicates that
compactification on a circle can not result in a chiral theory.

Deducing the particle content can be done by taking tensor products of NS and
R sectors to form the full string contents:
\- NS-NS sector:

$(\mathbf{7}\oplus\mathbf{1})\otimes(\mathbf{7}\oplus\mathbf{1})=\mathbf{27}\oplus\mathbf{21}\oplus\mathbf{7}\oplus\mathbf{7}\oplus\mathbf{1}\oplus\mathbf{1}$

i.e. a graviton, a KR field, two vector fields and two scalar fields.
R-NS sector:

$\mathbf{8}\otimes(\mathbf{7}\oplus\mathbf{1})=\mathbf{48}\oplus\mathbf{8}\oplus\mathbf{8}$

i.e. one Rarita- Schwinger field and two spinor fields. and same for NS-R
sector.
R-R sector:
IIA:

$\mathbf{8}\otimes\mathbf{8}\rightarrow(\mathbf{1}\oplus\mathbf{7})\oplus(\mathbf{21}\oplus\mathbf{35})$

i.e. A scalar, one form and a three form.
IIB:

$\mathbf{8}\otimes\mathbf{8}\rightarrow(\mathbf{1})\oplus(\mathbf{7}\oplus\mathbf{21})\oplus(\mathbf{35})$

i.e. a scalar, two form and a four form.
This field contents means that the theory has 32 supercharges ($\mathcal{N}=8$
supergravity in 4D) which is the maximal supersymmetry, we can then conclude
that circle compactifications can not break any supersymmetry. Thus
compactification on circles and spheres without flux can not give a realistic
particle physics theory.

###  3.2 Toroidal compactification of type II superstrings

In this section we study the case of compactification on square tori of
dimension d which are products of d circles. Firstly, we define parameters on
tori. Since a torus is a product of circles and a circle can be considered as
a line segment with its endpoints identified i.e. a quotient space. Therefore,
we can define a torus also as a quotient space as follows:
Firstly, define the equivalence relation $\sim$ on $\mathbb{R}^{d}$ ,where d
is the number of compact dimensions, by

$X^{i}\sim X^{i}+2\pi\text{R}^{i}$

where $\text{R}^{i}$ is the radius of the circle in the i-th compact dimension
comprising the torus and i runs over the compact dimensions.

A torus is defined as the quotient space $\mathbb{R}^{d}/\sim$. In the case
that all $\text{R}^{i}$ are equal the torus is called a square torus.
By this definition we have d compact dimensions, this means that there are d
quantized momenta $\text{P}_{i}=\frac{\text{k}+i}{\text{R}}$ where $k_{i}$ are
integers. Additionally, in this case we have winding number $\omega_{i}$ for
every compactified dimension measuring how many times the string is wrapped on
it, this number coincides with the homology class of a circle.

The Lagrangian for this theory is given by

$S=\frac{1}{2\pi}\int_{0}^{\infty}d^{2}\sigma[\frac{1}{2\alpha^{\prime}}G_{ij}(\gamma^{\mu\nu}\partial_{\mu}X^{i}\partial_{\nu}X^{j})+\frac{1}{\alpha^{\prime}}B_{ij}\partial_{\sigma_{1}}X^{i}\partial_{\sigma_{2}}X^{j}],$

where $G_{ij}$ is a symmetric 2 form (graviton), $B_{ij}$ is an antisymmetric
2 form (KR field), $\sigma_{1}$, $\sigma_{2}$ are the parameters of the
parameter space of the worldsheet, $X^{i}$ are the coordinates on the
worldsheet and $\gamma^{\mu\nu}=diag[1,-1]$.

Since this compactification does not alter any oscillatory mode (because it is
already periodic), it is sufficient to work on the zero modes, i.e. we can
expand the coordinates as follows

$X^{i}=x_{0}^{i}+\dot{x}^{i}\sigma_{1}+\frac{2\pi\text{R}}{\text{l}}\omega^{i}\sigma_{2},$

where l is the length of the string. Here we set $\sigma_{1}$ to be the time
coordinate and $\sigma_{2}$ to be the parameter on the string at a fixed time.
Substituting in the Lagrangian and integrating, we get

$S=\frac{\text{l}}{2\pi}[\frac{1}{2\alpha^{\prime}}G_{ij}(\dot{x}^{i}\dot{x}^{j}-(\frac{2\pi\text{R}}{\text{l}})^{2}\omega^{i}\omega^{j})+\frac{2\pi\text{R}}{\text{l}\alpha^{\prime}}B_{ij}\dot{x}^{i}\omega^{j}]$

The canonical momentum is then given by

$p_{i}=\frac{\text{l}}{2\pi\text{R}}G_{ij}\dot{x}^{j}+B_{ij}\omega^{j},$

But the quantization condition is $p_{i}=\frac{k_{i}}{R}$, equating and
solving for $\dot{x}^{j}$ we get

$\dot{x}^{j}=\frac{2\pi\text{R}}{\text{l}}G^{ij}(\frac{k_{i}}{\text{R}}-K_{ik}\omega^{k})$

substituting back to get the worldsheet coordinates and after some algebra we
get the left and right moving sectors of the momentum

$p_{L,i}=\frac{k_{i}}{\text{R}}+\frac{\text{R}\omega^{j}}{\alpha^{\prime}}(G_{ij}-B_{ij}),$
$p_{\text{R},i}=\frac{k_{i}}{\text{R}}+\frac{\text{R}\omega^{j}}{\alpha^{\prime}}(-G_{ij}-B_{ij}).$

Note that the momenta are no longer parameterized by one integers but two.
Thus, they form a 2D lattice where D is the dimension of the compact torus
used. This lattice is even, self dual and is called Narain lattice.

To prove this, first define the coordinates of the lattice to be
$(p_{L},p_{R})$. In the Lorentzian signature scalar product, the product of
any two points in the lattice is even.
Proof: Let $(p_{L},p_{R})$ and $(p^{\prime}_{L},p^{\prime}_{R})$ be two
arbitrary points in that lattice, the inner product is

$(p_{L},p_{R})\cdot(p_{L},p_{R})=2\sum_{i}[k^{i}\omega^{\prime}_{i}+k^{\prime}_{i}\omega^{i}]$

which is an even integer.

The self dual part is obvious because the theory must be modular invariant
which is a well known symmetry that holds in string theory.

The moduli space of the theory can be deduced from this lattice description
(equivalent lattices correspond to equivalent theories). Narain lattices by
definition are unique up to $SO(d,d)$ transformations, but in this case each
point in the lattice used here is invariant under $SO(d)\times SO(d)$ because
mass states of the theory depends only on the sum of squares of momenta as
will deduced afterwards. Additionally, two lattices are give the same theory
if they are T dual i.e. invariant under T duality group $SO(d,d,Z)$ (Remember
that T duality change the sign of the right movers only so the point in the
lattice will be transformed by an integer value). From this we can say that
the moduli space of the theory is the quotient group

$\frac{SO(d,d)}{SO(d)\times SO(d)\times SO(d,d,Z)}.$

Now we deduce the field content of the theory: From the formulae for momenta
deduced earlier, we can derive the mass formulae for left and right movers to
be

$M_{L}^{2}=\frac{2}{\alpha^{\prime}}(N_{F}+N_{B}+E_{0})+\frac{p^{2}_{L}}{2},$
$M_{R}^{2}=\frac{2}{\alpha^{\prime}}(\bar{N}_{F}+\bar{N}_{B}+E_{0})+\frac{p^{2}_{R}}{2}.$

The field content is derived from the level matching condition as in the usual
string theory. Here we only stress on the fact that the field content has 32
supercharges i.e. $\mathcal{N}=8$ supergravity in 4D, this is because tori are
just products of circles and we saw in the previous section that
compactifications on circles can not break any supersymmetry, thus, the theory
will be maximally supersymmetric.

###  3.3 Compactification of type IIB superstrings on $AdS_{5}\times S^{5}$

Compactification on $AdS_{5}\times S^{5}$ particularly is an interesting model
because it was proven to be exactly dual to $\mathcal{N}=4$ super Yang Mills
theory by two independent methods: the first method was by using AdS/CFT
correspondence, and was shown to be the case by Metsaev and Tseytlin [11], and
the second was by considering a stack of N coincident D3 branes in the limit
$\text{N}\rightarrow\infty$ [12] and studied further in [13]. This means that
the field content and the supersymmetry of the theory is already known. Here
we review the action, equations of motion and some properties.

Firstly, note that the symmetry group is the super lie group $PSU(2,2|4)$
whose super lie algebra has no representations in terms of supermatrices but
in terms of quotients of supermatrices, for more mathematical properties see
[14-16].

The action is unique and consists of two terms: the metric term and the Wess
Zumino term. The metric term is as usual

$S_{1}=-\frac{\text{R}}{4\pi\alpha^{\prime}}\int
d^{2}\sigma[\sqrt{-h}h^{\mu\nu}G_{\mu\nu}],$

where R is the radius of the metric of $AdS_{5}\times S^{5}$, $h^{\mu\nu}$ is
the inverse of the worldsheet metric. It can also be written in terms of a one
form $J_{1}$ defined in [17] as

$S_{1}=\frac{\sqrt{\lambda}}{16\pi}\int str(J_{1}\wedge\star J_{1}),$

where $\lambda=\frac{\text{R}^{4}}{\alpha^{\prime 2}}$, str is the supertrace,
and $\star J_{1}$ is the Hodge dual to $J_{1}$.

The Wess Zumino term is proportional to the only non zero $psu(2,2|4)$
invariant two form namely $str(J_{2}\wedge J_{3})$. Thus, the total action is
given by

$S=\frac{\sqrt{\lambda}}{16\pi}\int str(J_{1}\wedge\star
J_{1})-\frac{\sqrt{\lambda}}{8\pi}\int str(J_{2}\wedge J_{3}).$

The invariance under $PSU(2,2|4)$ induces a conserved current which is given
by

$J=J_{1}+\star J_{3}.$

The equations of motion coincide with the conservation of the dual of the
current namely $d\star J=0$, in terms of $J_{i}$ we get

$d\star J_{1}+dJ_{3}=0.$

The general features of this theory is that its bosonic part coincides with
the standard sigma model on $AdS_{5}\times S^{5}$, the theory has local
$\kappa$ symmetry which is responsible of removing half of the
supersymmetries, that is why this model has $\mathcal{N}=4$ supersymmetry
instead of $\mathcal{N}=8$. The analysis of kappa symmetry is beyond the scope
of this review for more details see [18]. Another feature is that the theory
reduces to the usual type IIB superstring theory in the limit
$\text{R}\rightarrow\infty$. For more detailed analysis of these type of
models see [19-21].

Other compactifications to $AdS$ space exist and are numerous, for example
there are compactifications on orbifolds to $AdS$ like $AdS_{5}\times
S^{5}/Z_{2}$ but this type of models were rules out due to Dirac quantization
rule of charges which will be discussed in chapter 7. Other models include
compactifications on various combinations of circles, tori and other manifolds
are reviewed in [22] and references therein.

###  3.4 Compactification of heterotic strings on a circle

In this section we review the compactification of heterotic superstrings on a
circle without Wilson lines, compactifications with Wilson lines are presented
in [9].
Heterotic strings are combination of bosonic left movers and superstrings as
right movers. Thus, we repeat the analysis in section 3.2 with the adequate
changes and show that using heterotic string theory can eliminate half of
supercharges in the same compactification model.

Once again, compactification on a circle means that one coordinate must
satisfy

$X^{9}(\sigma+\text{l},t)=X^{9}(\sigma,t)+2\pi R\omega,$

where $X^{9}$ is the compact dimension, $\sigma$ and $\tau$ are the parameters
parameterizing the string’s worldsheet, l is the length of the string, R is
the radius of the circle and $\omega$ is the winding number.

The corresponding momentum is quantized as in type II case with the difference
that the left moving part is bosonic and is 26 dimensional. Thus, the left
moving momentum consists of two parts: 10D part similar to the superstring
momentum discussed before, and an additional internal 16D momentum denoted by
$P_{int}=\sqrt{\frac{2}{\alpha^{\prime}}}P$ where the
$\sqrt{\frac{2}{\alpha^{\prime}}}$ factor is added for future convenience.

The momenta of the 10D part are given by

$p_{R}=\frac{k}{R}+\frac{R\omega}{\alpha^{\prime}},$
$p_{L}=\frac{k}{R}-\frac{R\omega}{\alpha^{\prime}},$

where $p_{R}$ and $p_{L}$is the right and the left moving momenta respectively
and the square of the mass is given by

$M^{2}_{R}=\frac{p^{2}_{R}}{2}+\frac{2}{\alpha^{\prime}}(\bar{N_{B}}+\bar{N_{F}}+E_{0}),$
$M^{2}_{L}=\frac{p^{2}_{L}}{2}+\frac{P^{2}_{int}}{2}+\frac{2}{\alpha^{\prime}}(N_{B}-1)=\frac{p^{2}_{L}}{2}+\frac{2}{\alpha^{\prime}}(N_{B}+P-1),$

where $M_{R}$ and $M_{L}$ are the right and left movers’ mass respectively.
Note that in the case of bosonic strings the momentum shift is known to be -1
which is the reason why bosonic string spectrum has tachyons, in cases of
heterotic $E_{8}\times E_{8}$ and SO(32) strings tachyons do not exist. Also,
the left mover’s mass has a contribution from the internal momentum but no
fermionic contribution as it is a bosonic sector.

We now derive the particle content and prove that this theory has half the
number of supercharges of its type II counterpart. The decomposition of
$SO(8)$ for right movers is done as before namely,
$\mathbf{8}_{V}\rightarrow\mathbf{7}\oplus\mathbf{1}$ and
$\mathbf{8}_{C}\rightarrow\mathbf{8}$. As before, chirality is lost i.e. it is
not possible to obtain chiral theories from this compactification scheme. For
the left movers we do not have Ramond and Neveu-Schwartz sectors, the lowest
energy particle states are as follows:

  1. 1.

$\alpha_{-1}^{i}|0>$ where $i-1,2,...,10$ are the 10D states propagating with
the right moving superstring sector and is decomposed as
$\alpha_{-1}^{i}|0>\rightarrow\mathbf{7}\oplus\mathbf{1}$.

  2. 2.

$\alpha^{I}_{-1}|0>$ where $I=11,12,...,26$ are the 16D states in the internal
space of the bosonic sector which decompose as $\alpha^{I}_{-1}|0>\rightarrow
1$, as they are in a different space so no gauge group affects them.

  3. 3.

$|P>_{I}$ which are the on shell momentum states (remember that bosonic string
states are specified by two parameters: energy and momentum), these states
also decomposed as $|P>_{I}\rightarrow\mathbf{1}$ as it is in the internal
space.

To get the whole particle content of the heterotic theory, we take tensor
products of each superstring state with each bosonic string state

$\mathbf{8}_{V}\otimes\alpha^{i}_{-1}|0>\rightarrow(\mathbf{7}\oplus\mathbf{1})\otimes(\mathbf{7}\oplus\mathbf{1})=\mathbf{27}\oplus\mathbf{21}\oplus\mathbf{7}\oplus\mathbf{7}\oplus\mathbf{1}\oplus\mathbf{1}$

i.e. a metric, a KR field, two vectors , a scalar and a dilaton.

$\mathbf{8}_{V}\otimes\alpha_{-1}^{I}|0>\rightarrow(\mathbf{7}\oplus\mathbf{1})\otimes\mathbf{1}=\mathbf{7}\oplus\mathbf{1}$

i.e. a spinor and a scalar.

$\mathbf{8}_{V}\otimes|P>_{I}\rightarrow(\mathbf{7}\oplus\mathbf{1})\times\mathbf{1}=\mathbf{7}\oplus\mathbf{1}$

i.e. a vector and a scalar.

$\mathbf{8}_{C}\otimes\alpha_{-1}^{i}|0>\rightarrow\mathbf{8}\otimes(\mathbf{7}\oplus\mathbf{1})=\mathbf{48}\oplus\mathbf{8}\oplus\mathbf{8}$

i.e. a Rarita-Schwinger field (gravitino) and two spinors.

$\mathbf{8}_{C}\otimes\alpha^{I}_{-1}|0>\rightarrow\mathbf{8}\otimes\mathbf{1}=\mathbf{8}$

i.e. a spinor.

$\mathbf{8}_{C}\otimes|P>_{I}\rightarrow\mathbf{8}\otimes\mathbf{1}=\mathbf{8}$

i.e. a spinor.

This particle content is a 9D supergravity with 16 supercharges so in 4D we
get $\mathcal{N}=4$ supergravity, this is half the supersymmetry of type II on
a circle due to the fact that bosonic states never decomposes to spinors, this
results in decreasing the number of spinors, so the number of supercharges by
half.

###  3.5 Toroidal compactification of heterotic superstrings

We generalize the previous section to tori (products of circles). As in the
case of type II superstrings we consider square d dimensional tori.
The action of the theory is a mixture between actions on the 10D space of
superstrings and the internal bosonic 16D space and is given by

$S=\int
d^{2}\sigma[G_{IJ}\eta^{\mu\nu}-B_{IJ}\epsilon^{\mu\nu}]\partial_{\mu}X^{I}\partial_{\nu}X^{J},$

where $I,J$ are indices in the internal space, $\mu,\nu$ are indices on the
superstring space, $\epsilon^{\mu\nu}$ is the totally antisymmetric tensor in
the superstring space, $\eta^{\mu\nu}$ is the inverse of the Minkowski metric
in the superstring space, $\sigma_{1}$ and $\sigma_{2}$ are the parameters in
the parameter space and the derivatives are on superstring space coordinates.

Following similar calculations to section 3.3, the momenta are given by

$P_{I}=\sqrt{\frac{2}{\alpha^{\prime}}}P_{I}$
$p_{L,i}=\frac{k_{i}}{R}+\frac{R\omega^{j}}{\alpha^{\prime}}(G_{ij}-B_{ij}),$
$p_{R,i}=\frac{k_{i}}{R}+\frac{R\omega^{j}}{\alpha^{\prime}}(-G_{ij}-B_{ij}),$

where $P_{I}$ is the internal momentum in the 16D internal space of the left
movers and the index I runs over this space and $\omega^{i}$ are the winding
numbers.
These results are similar to that of type II but with extra internal momentum
for the bosonic extra 16D internal space. The extra momentum is a part of the
left moving sector i.e. the quantized left moving momentum space is $16+d$
dimensional not d dimensional as the right moving momentum space.

The next step is to deduce the moduli space of the theory. The momenta form a
$d(d+16)$ Narain lattice with coordinates represents left and right moving
momenta (so the dimension of the lattice is the product of dimensionalities of
the two momentum spaces).

As the mass spectrum only depends on the magnitude of vectors in the lattice,
the lattice is unique up to $SO(16+d,d)$ rotations, but left and right moving
momenta rotate in the groups $SO(16+d)$ and $SO(d)$ respectively i.e. each
point is invariant under the group $SO(16+d)\times SO(d)$. Considering also
the T duality group $SO(16-d,d,Z)$, we get the moduli space

$\frac{SO(16+d,d)}{SO(16_{d})\times SO(d)\times SO(16-d,d,Z)}.$

Finally, the mass spectrum is given by

$M_{L}^{2}=\frac{2}{\alpha^{\prime}}(P_{I}P^{I}+N_{B}-1)+\frac{p^{2}_{L}}{2},$
$M_{R}^{2}=\frac{2}{\alpha^{\prime}}(\bar{N}_{F}+\bar{N}_{B}+E_{0})+\frac{p^{2}_{R}}{2},$

here the index I is being summed over.

The particle content of the theory is deduced similar to the last section, it
gives a 9D supergravity with 16 supercharges i.e. $\mathcal{N}=4$, 4D
supergravity which is half of the number of supercharges of type II toroidal
compactification.

It is worth mentioning that the resulting theory is not chiral (no chiral
fermions can be found in the spectrum) because on decomposing group
representations, chiral representations with different chiralities decompose
to the same lower dimensional group, so the chirality is lost, this happens in
circle and toroidal compactifications because only rotation groups are used.

###  3.6 Motivation for Calabi-Yau compactifications

In all the previous compactifications, there are some problems:

  1. 1.

There are too many supercharges; a realistic model of particle physics must be
at most $\mathcal{N}=1$ supersymmetric in 4D which is not true in circle,
toroidal compactifications and compactifications to $AdS$ spaces discussed
before.

  2. 2.

The models discussed before can never produce chiral fermions which were
observed in nature (this can also be done in some orbifold compactifications
[23-25] but it is beyond the scope of the review).

To solve these problems we must choose another suitable manifold to compactify
on. In their renowned paper [26] Candelas, Horowitz, Strominger and Witten
showed that this manifold must be a Calabi-Yau manifold.

The paper studies heterotic string compactification on Calabi-Yau manifolds.
They proved that $\mathcal{N}=1$ 4D supersymmetry requires the manifold to
have $SU(3)$ holonomy, and it must be Ricci flat to eliminate the conformal
anomaly as we will discuss in chapter 5.

Other results show that Calabi-Yau manifold compactifications can produce
chiral fermions and generations of particles similar to the standard model
(the number of generations is equal to half of the manifold’s Euler
characteristic). Thus, it is the best candidate to compactify on without
turning on any flux.

Note that in any of the previous compactification schemes turning on flux can
break supersymmetry and introduce chiral fermions which gives us a realistic
models without dealing with complicated manifolds like Calabi-Yau manifolds,
flux compactifications are discussed in detail in chapter 7.

##  4 Calabi-Yau compactifications

In this chapter we show why compactifications on Calabi-Yau manifolds without
flux give the most phenomenologically plausible low energy particle physics
theories than compactifications any other manifolds (also without flux). We
also discuss the problems this scheme faced resulting in the introduction of
flux compactifications and the use of new mathematics namely, generalised
complex geometry.

We study the aforementioned topics in the cases of type II superstrings (type
I theories can be viewed as type II theories on an orbifold), heterotic
superstrings and heterotic non supersymmetric string theories. Then, we
briefly review some explicit models on specific Calabi-Yau manifolds.

###  4.1 Compactifications of bosonic strings as a toy model

In this section we present the technique used in [27] and use it on bosonic
string theory as a toy model so that we can quote its results in the later
sections.
We begin with the Polyakov action in D dimensions describing a string
propagating on an ambient manifold $\mathcal{M}$:

$S=-\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{\gamma}[(\gamma^{\alpha\beta}G_{MN}(X)+i\epsilon^{\alpha\beta}B_{MN}(X))\partial_{\alpha}X^{M}\partial_{\beta}X^{N}+\alpha^{\prime}\Phi\text{R}],$

where $M,N=1,2,...,D$, $\alpha^{\prime}$ is the Regge slope,
$\alpha,\beta=1,2$, $d^{2}\sigma=d\sigma^{1}d\sigma^{2}$ such that
$\sigma^{1}$ and $\sigma^{2}$ are the parameters used to parameterize the
world sheet, $\gamma^{\alpha\beta}$ is the induced metric on the world sheet,
$\gamma=\det{\gamma^{\alpha\beta}}$, $G_{MN}$ is the metric of the ambient
manifold, $\epsilon^{\alpha\beta}$ is the totally antisymmetric tensor i.e.
the permutation symbol, $B_{MN}$ is an antisymmetric tensor field i.e. Kalb-
Ramond field, $\partial_{\alpha}=\frac{\partial}{\partial\sigma^{\alpha}}$,
$X^{M}$ are the coordinates on the ambient manifold, $\Phi$ is a scalar field
i.e. the dilaton and R is the Ricci scalar of the world sheet.

In this chapter we use complex coordinates as parameters i.e. $\sigma^{1}=z$
and $\sigma^{2}=\bar{z}$ because they are more convenient in the later
transformation to superstring models. In this case we use the notations
$\partial_{1}=\partial=\partial_{z}$ and
$\partial_{2}=\bar{\partial}=\partial_{\bar{z}}$. We also use the conformal
gauge $\gamma^{\alpha\beta}=e^{-2\eta}\delta^{\alpha\beta}$, where $\eta$ is a
function of the parameters $\sigma^{1}$ and $\sigma^{2}$. In this case
$R=-4\partial\bar{\partial}\eta e^{-2\eta}$.
The action in this gauge reads

$S=-\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}ze^{(d-2)\eta}[(G_{MN}(X)+iB_{MN}(X))\partial_{\alpha}X^{M}\partial_{\beta}X^{N}+\alpha^{\prime}\Phi(-4\partial\bar{\partial}\eta)],$
(1)

where $d$ is the dimension of the target space, but since in string theory the
target space is the world sheet i.e. 2 dimensional therefore, the exponential
factor cancels from the action.

For this theory to be conformally invariant i.e. conformal anomaly free, the
expectation value of the trace of the energy momentum tensor must vanish.
Calculating the mentioned quantity we get

$<T^{\rho}_{\rho}>=(-\frac{1}{2\alpha^{\prime}}\beta^{G}_{MN}G^{\alpha\beta}-\frac{i}{2\alpha^{\prime}}\beta^{B}_{MN}\epsilon^{\alpha\beta})\partial_{\alpha}X^{M}\partial_{\beta}X^{N}+2\beta^{\Phi}\partial\bar{\partial}\eta
e^{-2\eta},$

where $\rho=1,2$, $G^{\alpha\beta}$ is the pulled back metric to the world
sheet, and the beta functions $\beta^{G}$, $\beta^{B}$ and $\beta^{\Phi}$ are
given by

$\beta^{G}_{MN}=\alpha^{\prime}(R_{MN}+2\nabla_{M}\nabla_{N}\Phi-\frac{1}{4}H_{MPQ}H_{N}^{PQ})+O(\alpha^{\prime
2}),$ (2)
$\beta^{B}_{MN}=\alpha^{\prime}(-\frac{1}{2}\nabla^{P}H_{PMN}+\nabla^{P}\Phi
H_{PMN})+O(\alpha^{\prime 2}),$ (3)
$\beta^{\Phi}=\alpha^{\prime}(\frac{D-26}{6\alpha^{\prime}}-\frac{1}{2}\nabla^{2}\Phi+\nabla_{M}\Phi\nabla^{M}\Phi-\frac{1}{24}H_{MNP}H^{MNP})+O(\alpha^{\prime
2}),$ (4)

where $H=dB$ is the field strength of the field $B$, $\nabla_{M}$ is the
covariant derivative, $D$ is the dimensions of the ambient manifold and
$R_{MN}$ is the Ricci tensor of the ambient manifold. For the details of the
calculations see [28-31].

We require that $<T^{\rho}_{\rho}>=0$ i.e. all the beta functions must be zero
for the theory to be anomaly free. Turning off all flux i.e. $H=0$ we get the
requirements

$R_{MN}+2\nabla_{M}\nabla_{N}\Phi=0,$ (5)
$\frac{D-26}{6\alpha^{\prime}}-\frac{1}{2}\nabla^{2}\Phi+\nabla_{M}\Phi\nabla^{M}\Phi=0.$
(6)

So far we were using a general D dimensional manifold as an ambient manifold
for the theory and we need to compactify all the extra dimensions to get a 4
dimensional particle physics theory i.e. the ambient manifold must be
decomposed as $\mathcal{M}_{D}=\mathcal{M}_{4}\times\mathcal{M}_{D-4}$, where
$\mathcal{M}_{4}$ is a 4 dimensional manifold called the external manifold
which we take to be Minkowskian, and $\mathcal{M}_{D-4}$ is a (D-4)
dimensional manifold called the internal manifold, and we require that the
internal manifold to be compact. Also, we impose the condition
$\nabla^{2}\Phi=0$ otherwise we will have a fifth long range force which is
not observed in nature. This implies that $\nabla_{M}\Phi$ is a constant and
since the internal manifold is compact $\Phi$ must attain a maximum on
$\mathcal{M}_{D-4}$ i.e. $\nabla_{M}\Phi=0$ at this point so on the whole
manifold. Using these requirements in equations 4 and 5, we get

$R_{ab}=0,$ $D=26$

where $a,b$ run over the internal manifold’s coordinated, this means that the
internal manifold is Ricci flat and the spacetime dimensions must be 26.

Since bosonic string theory has no fermions in its spectrum therefore, it has
no spinor fields so the internal manifold may not be Calabi-Yau. However,
bosonic string theory can not be used to construct a realistic model of
particle physics at low enegies but used just as a toy model for calculations
needed later as we will see in the next few sections.

###  4.2 Type II superstring theories compactifications on Calabi-Yau
manifolds

In this section we will repeat the analysis in the previous section in the
case of type II syperstring theories.
The first step is to introduce superpartners for the bosonic variables. We
define the Rarita-Schwiger field $\psi^{M}$ to be the superpartner of $G_{MN}$
called a gravitino, and $\chi^{M}$ to be the superpartner of $X^{M}$ called
the gaugino.

The next step is to define an action which is invariant under local
supersymmetry transformations. To do this step we need to define new
variables, first of all the introduction of a fermionic variables implies the
introduction of fermionic coordinates on the world sheet, we denote them by
$\theta$ and $\bar{\theta}$. Fermionic coordinates are Grassmannian numbers
i.e. they satisfy $\theta^{2}=\bar{\theta}^{2}=0$, the theory of Grassmannian
numbers is beyond the scope of this review, for more details see [32].
Accordingly, we define a new derivative operators as follows

$D=\partial_{\theta}+\theta\partial_{z}.$
$\bar{D}=\partial_{\bar{\theta}}+\bar{\theta}\partial_{\bar{z}}.$

where $\partial_{\theta}=\frac{\partial}{\partial\theta}$ and
$\partial_{\bar{\theta}}=\frac{\partial}{\partial\bar{\theta}}$.
We decompose the superfields according to Grassmannian numbers theory as
follows:

$X^{M}(z,\bar{z},\theta,\bar{\theta})=X^{M}(z,\bar{z})+\theta\chi^{M}(z)+\bar{\theta}\bar{\chi^{M}}(\bar{z}),$
$G_{MN}(z,\bar{z},\theta,\bar{\theta})=G_{MN}+\theta\psi^{M}(z)+\bar{\theta}\bar{\psi}(\bar{z}).$

Noting that in the complex coordinates and their Grassmannian counterparts all
gamma matrices vanish except $\gamma^{z}_{\theta\theta}$ and
$\gamma^{\bar{z}}_{\bar{\theta}\bar{\theta}}$, and both are equal to the
identity matrix. We can use the action (1) directly with the following
transformations

$X^{M}\rightarrow X^{M}(z,\bar{z},\theta,\bar{\theta}),$ $G_{MN}\rightarrow
G_{MN}(z,\bar{z},\theta,\bar{\theta}),$ $\partial\rightarrow D,\ \ \ \ \ \ \ \
\ \bar{\partial}\rightarrow\bar{D},$
$\eta\rightarrow\eta(z,\bar{z}.\theta,\bar{\theta}),$ $d^{2}z\rightarrow
d^{2}zd^{2}\theta,$

where $d^{2}\theta=d\theta d\bar{\theta}$.
The action then reads

$S=-\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}zd^{2}\theta[(G_{MN}(z,\bar{z},\theta,\bar{\theta})+iB_{MN}(z,\bar{z},\theta,\bar{\theta})))DX^{M}(z,\bar{z},\theta,\bar{\theta}))\bar{D}X^{N}(z,\bar{z},\theta,\bar{\theta}))$
$+\alpha^{\prime}\Phi(-4D\bar{D}\eta(z,\bar{z},\theta,\bar{\theta})))].$ (7)

This action is clearly invariant under supersymmetry transformations. For the
explicit form of the transformations (without flux) see [30].

With similar calculations to the previous section, the expectation value of
the energy momentum tensor is given by

$<T^{\rho}_{\rho}>=(-\frac{1}{2\alpha^{\prime}}\beta^{G}_{MN}G^{\alpha\beta}-\frac{i}{2\alpha^{\prime}}\beta^{B}_{MN}\epsilon^{\alpha\beta})DX^{M}\bar{D}X^{N}+2\beta^{\Phi}D\bar{D}\eta
e^{-2\eta},$

where the beta function are given by

$\beta^{G}_{MN}=\alpha^{\prime}(R_{MN}+2\nabla_{M}\nabla_{N}\Phi-\frac{1}{4}H_{MPQ}H_{N}^{PQ})+O(\alpha^{\prime
2}),$ (8)
$\beta^{B}_{MN}=\alpha^{\prime}(-\frac{1}{2}\nabla^{P}H_{PMN}+\nabla^{P}\Phi
H_{PMN})+O(\alpha^{\prime 2}),$ (9)
$\beta^{\Phi}=\alpha^{\prime}(\frac{D-10}{4\alpha^{\prime}}-\frac{1}{2}\nabla^{2}\Phi+\nabla_{M}\Phi\nabla^{M}\Phi-\frac{1}{24}H_{MNP}H^{MNP})+O(\alpha^{\prime
2}),$ (10)

where $H=dB$ is the field strength of the field $B$, $\nabla_{M}$ is the
covariant derivative but replacing the ordinary derivative with the previously
introduced supersymmetric derivative, $D$ is the dimensions of the ambient
manifold and $R_{MN}$ is the Ricci tensor of the ambient manifold.

Similar to the previous section we turn off all fluxes and require that all
the beta functions to vanish. This gives us the conditions

$D=10,$ $R_{MN}=0,$

i.e. the theory is anomaly free only in 10 spacetime dimensions, and the
ambient manifold is Ricci flat.

The next step is to assume that the ambient manifold is of the form
$\mathcal{M}_{4}\times\mathcal{M}_{6}$, where $\mathcal{M}_{4}$ is Minkowski
space. The condition on the internal space is

$R_{mn}=0,$

where $m,n=1,2,...,6$.

On the other hand the supersymmetry transformation for $\psi^{M}$ can be used
to derive another condition on the internal manifold as follows:
The transformation is

$\delta\psi^{M}=\nabla_{M}\epsilon,$

where $\epsilon$ is the supersymmetry parameter and a similar condition for
$\bar{\psi}$, we continue the analysis with $\psi$ only and it should be
understood that there are similar equations for $\bar{\psi}$.

Since the action is supersymmetric therefore,
$\delta\psi^{M}=\delta\bar{\psi}=0$. Thus

$\nabla_{M}\epsilon=0.$ (11)

And since the ambient manifold was decomposed into internal and external
manifolds, the spinor $\epsilon$ must also be decomposed to an internal part
and an external part

$\epsilon=\xi\otimes\rho.$

where $\xi$ is a spinor in the external Minkowsli space and $\rho$ is a spinor
in the internal manifold.
Substituting in eq.(11) and noting that the covariant derivative vanish on the
Minkowski space, we get the condition

$\nabla_{m}\rho=0.$

This means that there exist two covariantly constant spinors on the internal
manifold(one from $\psi$ and one from $\bar{\psi}$). Thus, The internal
manifold is a 6D complex manifold with two covariantly constant spinors which
is Ricci scalar i.e. a Calabi Yau manifold.
The problem with this result is that there are two covariantly constant
spinors on the manifold, this means that the resulted particle physics theory
in 4D will be $\mathcal{N}=2$ supersymmetric which is not plausible
phenomenologically as discussed before in chapter 4.

Another concern is that Calabi Yau manifolds have large moduli spaces which
are translated into the spectrum as massless fields with no potential. This
means that the vacuum expectation value of these fields can change without any
energy cost i.e. freely, this kind of fields was not observed in nature
moreover, the masses of particles and values of coupling constants in the 4D
theory depend on the vacuum expectation value of the moduli. Thus, we need to
fix moduli values, this process is called moduli stabilization and require
some flux to be turned on motivating new type of compactifications called flux
compactifications which will be discussed in chapter 7.

###  4.3 Heterotic superstring theories compactifications on Calabi-Yau
manifolds

In this section we repeat the procedure in the previous section in the case of
heterotic superstring theories.

The superpartners are to be introduced similar to the previous section but in
this case we introduce one Grassmannian coordinate $\theta$, that is because
in heterotic string theory only the left movers are supersymmetric, we also
use complex coordinates. To define an appropriate action we begin with
defining new supersymmetric values similar to the previous section. We define
a derivative operator

$D=\partial_{\theta}+\theta\partial_{z},$

and supersymmetric variables as follows:

$X^{M}(z,\bar{z},\theta)=X^{M}(z,\bar{z})+\theta\chi^{M}(z),$
$G_{MN}(z,\bar{z},\theta,\bar{\theta})=G_{MN}+\theta\psi^{M}(z).$

The action is defined using (1) with the following transformations

$X^{M}\rightarrow X^{M}(z,\bar{z},\theta,\bar{(}\theta)),$ $G_{MN}\rightarrow
G_{MN}(z,\bar{z},\theta),$ $\partial\rightarrow D,$
$\eta\rightarrow\eta(z,\bar{z},\theta),$ $d^{2}z\rightarrow d^{2}zd\theta,$

The action then reads

$S=-\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}zd\theta[(G_{MN}(z,\bar{z},\theta)+iB_{MN}(z,\bar{z},\theta)))DX^{M}(z,\bar{z},\theta))\bar{\partial}X^{N}(z,\bar{z},\theta))$
$+\alpha^{\prime}\Phi(-4D\bar{\partial}\eta(z,\bar{z},\theta)))].$ (12)

The expectation value of the energy momentum tensor is given by

$<T^{\rho}_{\rho}>=(-\frac{1}{2\alpha^{\prime}}\beta^{G}_{MN}G^{\alpha\beta}-\frac{i}{2\alpha^{\prime}}\beta^{B}_{MN}\epsilon^{\alpha\beta})DX^{M}\bar{\partial}X^{N}+2\beta^{\Phi}D\bar{\partial}\eta
e^{-2\eta},$

where the beta function are similar to the beta functions in equations 8,9,10
but with only half of the derivatives are supersymmetric. Equating them to
zero we get

$D=10,$ $R_{MN}=0.$

i.e. the theory only anomaly free in 10 spacetime dimensions and the ambient
manifold is again Ricci flat.
Decomposing the manifold into an external and internal part
$\mathcal{M}=\mathcal{M}_{4}\times\mathcal{M}_{6}$, where $\mathcal{M}_{4}$ is
a Minkowski space and $\mathcal{M}_{6}$ is compact, the condition on the
internal manifold is

$R_{mn}=0.$

The transformation is

$\delta\psi^{M}=\nabla_{M}\epsilon,$

where $\epsilon$ is the supersymmetry parameter.

The action is supersymmetric therefore, $\delta\psi^{M}=0$. Thus

$\nabla_{M}\epsilon=0.$ (13)

The ambient manifold was decomposed into internal and external manifolds thus,
the spinor $\epsilon$ must also be decomposed to an internal part and an
external part

$\epsilon=\xi\otimes\rho.$

where $\xi$ is a spinor in the external Minkowsli space and $\rho$ is a spinor
in the internal manifold.
Substituting in eq.(13) we get the condition

$\nabla_{m}\rho=0.$

On the contrary of the previous section, here there exists only one
covariantly constant spinors on the internal manifold. Thus, The internal
manifold is a Calabi-Yau manifold.

This compactification scheme gives $\mathcal{N}=1$ supersymmetric 4D theory
which is phenomenologically plausible but the issue of moduli is still a
problem for this scheme as well.

###  4.4 Non-supersymmetric string theory compactifications on Calabi-Yau
manifolds

Non supersymmetric string theories are theories resulting from supersymmetry
breaking of heterotic superstring theories, the most famous non supersymmetric
string theory is the $SO(16)\times SO(16)$ which is the theory resulted from
supersymmetry breaking of $E_{8}\times E_{8}$ heterotic string theory, or
equivalently its T-dual the $SO(32)$ heterotic string theory. The details of
non supersymmetric string theories are beyond the scope of this review, for
more details see [33].

We will not repeat the procedure again, instead we describe how to get non
supersymmetric string theories compactifications from heterotic string
theories compactifications. The relation between the two type of theories’
compactifications is shown in fig.5 below. To complete our description we
state how to break supersymmetry in heterotic string theories, then the rest
will be similar to the previous two sections.

To break supersymmetry in heterotic string theories we use certain topological
twists, the theory of topological twists is beyond the scope of this review
for more details see [34]. In the upcoming discussion, we follow [35]:
Firstly, we divide the fermionic and bosonic variables into to two categories:
the variables in the non supersymmetric theory and will be labeled by the
subscript $A$ and variables not in the non supersymmetric theory which are
labeled by the subscript $X$ if they belong to the $E_{8}\times E_{8}$ theory,
or $Y$ if they belong to the $SO(32)$ theory. This categorization is displayed
in figure 1 in Ref [35].
Now we define the supersymmetry breaking twists as follows:
From $E_{8}\times E_{8}$ theory we apply the twists

$X^{M}_{A}\rightarrow X^{M}_{A},\ \ \ \ \ \ \ X^{M}_{X}\rightarrow-X^{M}_{X},$
$\chi_{A}^{M}\rightarrow-\chi_{A}^{M},\ \ \ \ \ \ \
\chi_{X}^{M}\rightarrow\chi_{X}^{M}.$

Here we assumed that all gauginos have positive chirality.
From $SO(32)$ theory we apply the twists

$X^{M}_{A}\rightarrow X^{M}_{A},\ \ \ \ \ \ \ X^{M}_{Y}\rightarrow-X^{M}_{Y},$
$\chi_{A}^{M}\rightarrow-\chi_{A}^{M},\ \ \ \ \ \ \
\chi_{Y}^{M}\rightarrow\chi_{Y}^{M}.$

Here we assumed that all gauginos have negative chirality.

Applying these twists to the heterotic string action, we get a non
supersymmetric string theory. Then we calculate the expectation value of the
energy momentum tensor, equate the beta functions to zero, decompose the
ambient manifold then deduce a condition on the internal manifold. This
procedure has a caveat that after the twist there are no preserved spinors on
the internal manifolds. Thus, we can not derive any restrictive conclusion on
it unless we consider manifolds with line bundles, this indeed gives the
restriction we need to conclude that the manifold must be Calabi-Yau, but
since the line bundles are considered as background flux (because they are on
the manifold itself not derived from the theory), twists and flux
compactifications will be discussed in more detail in chapter 7.

##  5 Generalized complex geometry

Generalized complex geometry was introduced by Hitchin [36] and Gualtieri’s
PhD thesis [37]. The main motivation is to give a geometric meaning to the
$B$-field in a rigorous way. In this chapter we review the basic theory
following [38].

###  5.1 G-structures

Firstly, we set up the notation used. Throughout this chapter let $M$ be a
compact manifold with local patches $U_{\alpha}$, $E$ be a bundle over $M$
with fiber $F$ and section $s$, $TM$ be the tangent bundle over $M$, $T^{*}M$
be the cotangent bundle over $M$.

As discussed in chapter 2, the bundle $E$ is locally a product $M\times F$ but
globally it can be a much more complicated structure. Therefore, we must
define transformations to see how does the fiber transform from one patch to
other neighbouring patches, in this way we may be able to examine global
properties of the bundle. These transformations are called transition
functions, and they play a key role in defining G-structures as we will see.
It is easy to see that the set of transition functions with the composition of
functions for a group, this group is called the structure group. The other key
concept to define a G-structure is the tangent frame bundle.

Definition: The tangent frame bundle denoted by $FM$ associated to $TM$ is a
bundle over M whose fiber at each point $p\in M$ is the set of ordered basis
of the tangent space of $M$ at this point $T_{P}M$.

Let’s describe this bundle in a more practical way, consider two different
patches $U_{\alpha}$ and $U_{\beta}$ which are nieghbourhoods of a point $p\in
M$ with local trivializations ($p,\epsilon_{\alpha}$) and
($p,\epsilon_{\beta}$) respectively, where $\epsilon_{\alpha}$ and
$\epsilon_{\beta}$ are the local basis for $T_{p}M$ on the patch
$U_{\alpha}$($\beta$), physicists call them local frames of reference. Now
suppose there is an overlap between the two patches, then the relations
between the bases on the two patches are

$\epsilon_{\alpha}=\epsilon_{\beta}t_{\alpha\beta},$

where $t_{\alpha\beta}$ are the transition function from $U_{\alpha}$ to
$U_{\beta}$.
But from differential geometry we know that the relations between basis on
different patches are given by

$\epsilon_{\alpha}=\frac{\partial x}{\partial\bar{x}}\epsilon_{\beta},$

where $x=$($x_{1},x_{2},...,x_{n}$) such that $x_{i}$ are the normal
coordinates over $U_{\alpha}$, and $\bar{x}$ is the same for $U_{\beta}$. Thus
$\frac{\partial x}{\partial\bar{x}}$ is the Jacobian matrix that transforms
one set of normal coordinated to the other.

Equating the last two equations we see that the group transition functions
i.e. the structure group can be represented as a group of linear
transformations i.e. the structure group is $GL(n,\mathbb{R})$. On the other
hand the basis $\epsilon_{\alpha}$ can also be considered as a matrix (as each
element for example $\epsilon_{1}$ is an element of the basis i.e. a vector).
Thus, $\epsilon_{\alpha}\in GL(n,\mathbb{R})$. Then, collectively the fiber of
the frame tangent bundle is $GL(n,\mathbb{R})$. This gives an equivalent
definition for the tangent frame bundle.

Equivalent definition: A tangent frame bundle is the principal bundle whose
fiber is the structure group.

We are interested in the cases where there exists an invariant structure over
the manifold $M$, i.e. the same on all patches, this implies that the
transition functions that change this invariant structure must be excluded
leaving a subgroup G of the old structure group as the reduced structure group
or equivalently, a reduced tangent frame bundle over $M$.

Definition:A manifold $M$ has a G-structure if it is possible to reduce its
structure group to the group G, and the invariant used to do so is called the
G-invariant.

Example: Suppose we have a globally defined linear transformation

$J:TM\rightarrow TM,$

such that

$J^{2}=-Id,$

i.e. an almost complex structure. In this case every matrix preserving this
structure is a block matrix of two diagonal blocks each one is an
$\frac{n}{2}\times\frac{n}{2}$ complex matrices. This implies that the group
of all matrices preserving $J$ is isomorphic to $GL(\frac{n}{2},\mathbb{C})$
i.e. an almost complex manifold always has a $GL(\frac{n}{2},\mathbb{C})$
structure.

###  5.2 Elements of generalized complex geometry

The procedure is to replace the tangent bundle with the sum of the tangent and
the cotangent bundle $TM\rightarrow TM\oplus T^{*}M$ called the generalized
tangent bundle, and repeat everything in complex geometry with it.

We define a generalized tangent vector $\mathbb{X}$ as a sum of a tangent
vector and a one form

$\mathbb{X}=X\oplus\xi,$

where $X\in TM$ is called the vector part, and $\xi\in T^{*}M$ is called the
one form part.

####  5.2.1 The metric and the volume form

On the generalized tangent bundle we can define a metric in a natural way,
unlike the tangent bundle, as follows: given two generalized vectors
$\mathbb{X}=X\oplus\xi$ and $\mathbb{Y}=Y\oplus\eta$, we define the canonical
metric by

$\mathcal{I}(\mathbb{X},\mathbb{Y})=\frac{1}{2}(\xi(Y)+\eta(X)).$

It is easy to see that this metric has a signature of $(n,n)$, and is defined
globally on the manifold thus, it reduces the structure group to $O(n,n)$.
Additionally, this metric comes with a natural volume form defined as

$vol_{\mathcal{I}}=\frac{1}{(d!)^{2}}\epsilon^{i_{1}i_{1},,,i_{n}}\partial_{x^{i_{1}}}\wedge...\wedge\partial_{x^{i_{n}}}\wedge\epsilon_{i_{1}i_{1},,,i_{n}}dx^{i_{1}}\wedge...\wedge
dx^{i_{n}},$

where $\epsilon^{i_{1}...i_{n}}$ and $\epsilon_{i_{1}...i_{n}}$ are totally
antisymmetric permutation symbols and
$\partial_{x^{i_{j}}}=\frac{\partial}{\partial x^{i_{j}}}$.
Since there are two multiplied permutation symbols in the volume form
therefore, the volume form does not depend on the orientation thus, the
structure group is reduced once more to the transformations that do not change
orientation i.e. $SO(n,n)$. This means that the generalized tangent bundle
comes naturally with an $SO(n,n)$ structure on its base manifold.
The generators for $SO(n,n)$ are

$\begin{bmatrix}A\ \ \ \ \ \ \ \ \ 0\\\ \ \ 0\ \ \ (A^{-1})^{T}\end{bmatrix},\
\ \ \ \ \ \ \ \ e^{B}=\begin{bmatrix}Id\ \ \ 0\\\ B\ \ \ Id\end{bmatrix},\ \ \
\ \ \ \ \ \ e^{\beta}=\begin{bmatrix}Id\ \ \ \beta\\\ 0\ \ \ Id\end{bmatrix},$

where $A$ is the structure group of the tangent bundle, $B$ is a two form,
$\beta$ is a two vector.
it is easy to see that the action of $e^{B}$ is

$e^{B}(X\oplus\xi)=X+(\xi-B\cdot X),$

where $\cdot$ is the interior product i.e. contraction. Similarly for
$e^{\beta}$

$e^{\beta}(X\oplus\xi)=(X-\xi\cdot\beta)+\xi.$

The actions of $e^{B}$ and $e^{\beta}$ are usually called twists.

####  5.2.2 (Twisted) Courant bracket

The Courant bracket in generalized complex geometry is the counterpart of the
Lie bracket in differential geometry, it is defined as follows

Definition: Let $\mathbb{X}=X\oplus\xi$ and $\mathbb{Y}=Y\oplus\eta$ be two
generalized vector fields, their Courant bracket is given by

$[\mathbb{X},\mathbb{Y}]_{C}=[X,Y]+\mathcal{L}_{X}\eta-\mathcal{L}_{Y}\xi-\frac{1}{2}d(\eta\cdot
X-\xi\cdot Y),$

where $[X,Y]$ is the Lie bracket of $X$ and $Y$, $\mathcal{L}_{X}$ is the Lie
derivative with respect to $X$, $d$ is the exterior derivative and $\eta\cdot
X$ is their contraction. This operation is well defined but it fails as a
generalization of a Lie bracket because it is not a Lie bracket itself because
it does not satisfy the Bianchi identity. To fix that we define the twisted
Courant bracket by a closed three form $H$ as follows

$[\mathbb{X},\mathbb{Y}]_{H}=[\mathbb{X},\mathbb{Y}]_{C}+(H\cdot X)\cdot Y.$

It is called "twisted" because it satisfies

$[e^{B}\mathbb{X},e^{B}\mathbb{Y}]_{H-dB}=e^{B}[\mathbb{X},\mathbb{Y}]_{H}.$

This is indeed a Lie bracket unless $H=0$ it would be a Courant bracket.

Physically, we interpret $B$ as the B-field in string compactifications and
$H$ to be its field strength i.e. a flux. Thus, if we turn on fluxes on
compactifying we can use use generalized complex geometry to give a geometric
meaning to the flux $H$ as the twist in the internal manifold and for the
B-field as the action of the coordinated due to this twist, in the language of
field theories $H$ is a physical field and $B$ is its potential.

####  5.2.3 Polyforms and pure spinors

Definition: A polyform $\phi$ is the formal sum of forms of different
dimensions.
The interior product of polyforms and generalized tangent vectors is defined
as

$\mathbb{X}\cdot\phi=X\cdot\phi+\xi\wedge\phi,$

where $\mathbb{X}=X\oplus\xi$.
Now let’s compute the action of an anticommutator of two generalized tangent
vectors on a polyform, the result is

$\\{\mathbb{X},\mathbb{Y}\\}\cdot\phi=2\mathcal{I}(\mathbb{X},\mathbb{Y})\phi.$

This is interesting because it means that generalized tangent vectors form a
Clifford algebra (act as gamma matrices in quantum field theory language).
Thus, polyforms transform in the spin representation of $Spin(n,n)$, we can
use this to define different kinds of spinors like Majorana-Weyl spinors with
a rigorous geometrical meaning but this is beyond the scope of this review.

Another interesting fact is that polyforms come with a natural bilinear form
called Mukai pairing defined as follows:
Let $\phi_{1}$ and $\phi_{2}$ be two polyforms, their Mukai pairing is

$<\phi_{1},\phi_{2}>=\phi_{1}\wedge\sigma(\phi_{2})|_{top},$

where $\sigma$ is the operator reversing all indices of the polyform and
$|_{top}$ denotes the projection of the top form i.e. the form with dimension
n. This pairing is equivalent to the charge conjugation matrix in classical or
quantum field theory.
Note that Mukai pairing is invariant under twisting i.e.

$<e^{B}\phi_{1},e^{B}\phi_{2}>=<\phi_{1},\phi_{2}>.$

So far we dealt with general polyforms and shown that they transform as
spinors, now we define a special class of polyforms called pure spinors.
Firstly we need to define the null space of a polyform which is a similar to
null spaces in linear algebra.

Definition: Let $\phi$ be a polyform, the null space of $\phi$ is the
subbundle

$L_{\phi}=\\{\mathbb{X};\mathbb{X}\cdot\phi=0\\},$

i.e. all generalized tangent vectors annihilating $\phi$.

Definition: A subbundle $L$ is called isotropic if for every two generalized
tangent vectors $\mathbb{X}$ and $\mathbb{Y}$ in L

$\mathcal{I}(\mathbb{X},\mathbb{Y})=0.$

Definition: We say that $L$ is maximally isotropic if its rank equals half the
rank of the generalized tangent bundle.

Definition: A polyform $\phi$ is called a pure spinor if its null space is
maximally isotropic.

A pure spinor and its isotropic null space induce a decomposition of the space
of all polyforms as follows

$\Lambda^{*}T^{*}M\otimes\mathbb{C}=\bigoplus_{-n/2\leq k\leq n/2}U_{k},$

where

$U_{k}=\Lambda^{n/2-k}L\cdot\phi,$

where we denote $\Lambda^{*}T^{*}M$ to be the space of all polyforms and
$\Lambda^{n/2-k}$ to be the space of all $n/2-k$ dimensional sections of
$L\cdot\phi$, this means that $U_{k}$ is what we get be applying an
antisymmetric product of $n/2-k$ generalized tangent vectors of $L$ on $\phi$.
This decomposition is called a filtration.

####  5.2.4 Twisted exterior derivative, generalized complex structures and
integrability

Now we define the analogue of the exterior derivative in generalized complex
geometry i.e. the twisted exterior derivative.
Given a polyform $\phi$, its twisted exterior derivative is given by

$d_{H}\phi=d\phi+H\wedge\phi,$

which satisfies

$d^{2}_{H}=0.$

Definition: A pure spinor $\phi$ is called a generalized Calabi Yau a la
Hitchin if

$d_{H}\phi=0,$

i.e. it is closed under twisted exterior derivative.
Now we introduce generalized complex structures.

Definition:A generalized almost complex structure is a map

$\mathcal{J}:TM\oplus T^{*}M\rightarrow TM\oplus T^{*}M,$

satisfying

$\mathcal{J}^{2}=-Id,$

and that the metric is Hermitian with respect to $\mathcal{J}$ i.e.

$\mathcal{I}(\mathcal{J}\mathbb{X},\mathcal{J}\mathbb{Y})=\mathcal{I}(\mathbb{X},\mathbb{Y})$

for every generalized tangent vectors $\mathbb{X}$ and $\mathbb{Y}$.
The previous two conditions reduce the structure group to $U(n/2,n/2)$. Note
that we can define a generalized almost complex structure $\mathcal{J}$ from
an almost complex structure $J$ as follows:

$\mathcal{J}=\begin{bmatrix}-J\ \ \ \ \ \ 0\\\ 0\ \ \ \ \ \
J^{T}\end{bmatrix}.$

This structure decomposes the bundle into two, each of which corresponds to an
eigenvalue of $\mathcal{J}$ namely i and -i, the bundle corresponds to i is
called $L_{\mathcal{J}}$ and the other is $\bar{L}_{\mathcal{J}}$. The
integrability of such structure can be studies using two different but
equivalent approaches, the first straightforward one is analogous to the
integrability of complex structures studied in chapter 2:

Definition: The generalized almost complex structure $\mathcal{J}$ is
integrable if $L_{\mathcal{J}}$ is involutive under twisted Courant bracket
i.e.

$\mathbb{X},\mathbb{Y}\in\Gamma(L_{\mathcal{J}})\implies[\mathbb{X},\mathbb{Y}]_{H}\in\Gamma(L_{\mathcal{J}}).$

The other way is to firstly notice that we can associate a complex pure spinor
with isotropic null space to every generalized almost complex structure, and
this null space is the i eigenbundle of the generalized complex bundle. we
denote this associated spinor by $\phi_{\mathcal{J}}$.

Theorem: An almost complex generalized bundle is integrable if and only if its
associated spinor satisfies

$d_{H}\phi_{\mathcal{J}}=\mathbb{X}\cdot\phi_{\mathcal{J}},$

for some generalized tangent vector $\mathbb{X}$.

This can be used as an equivalent definition of integrability. As in the case
of complex geometry, an integrable generalized almost complex bundle is called
a generalized complex bundle.

###  5.3 Generalized Calabi-Yau manifolds

In this section the aim is to define the generalized Calabi-Yau geometry which
is of utmost importance in flux compactifications as we will see the next
chapter.

As we have seen in the previous chapter the existence of a globally defined
spinor on the internal manifold i.e. some preserved supersymmetry, forces the
geometry to be Calabi-Yau i.e. to have $SU(n/2)$ holonomy group on its
patches. In the G-structure language we say that supersymmetry forces an
$SU(n/2)$ structure on the internal manifold. Here we define the generalized
version of this situation. We begin by defining the $U(n/2)\times U(n/2)$
structure as follows:

Definition: A $U(n/2)\times U(n/2)$ structure consists of two generalized
almost complex structures $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ such that
they commute and the metric $-\mathcal{I}\mathcal{J}_{1}\mathcal{J}_{2}$ is
positive definite. This structure is called H twisted generalized Kahler
structure if the two generalized almost complex structures are H integrable.

Theorem: The generalized tangent bundle has a $U(n/2)\times U(n/2)$ structure
if and only if there exists pure spinor bundles $\Psi_{1}$ and $\Psi_{2}$ such
that they satisfy:

  1. 1.

$\Psi_{2}\in\Gamma(U_{0})$, where $U_{i}$ is a filtration associated to
$\Psi_{1}$. Equivalently, $\Psi_{1}\in\Gamma(V_{0})$, where $V_{i}$ is a
filtration associated to $\Psi_{2}$.

  2. 2.

The metric on the generalized complex structures to which $\Psi_{1}$ and
$\Psi_{2}$ are associated is positive definite.

The obstruction of introducing globally defined spinors on this structure is
the overall factor ambiguity of $\psi_{1}$ and $\Psi_{2}$, to remove this
ambiguity we reduce the structure group to $SU(n/2)\times SU(n/2)$ be imposing
the normalization condition

$<\Psi_{1},\bar{\Psi_{1}}>=<\Psi_{2},\bar{\Psi_{2}}>=0,$

where $<.>$ is the Mukai pairing and $\bar{\Psi_{i}}$ is the complex conjugate
of $\Psi_{i}$. In this case $\Psi_{1}$ and $\Psi_{2}$ are globally defined,
physically this means that there is some preserved supersymmetry in the
resulting theory. We can now define the generalize Calabi-Yau geometry which
will be used in the next chapter.

Definition: A generalized Calabi-Yau geometry a la Gualtieri is an
$SU(n/2)\times SU(n/2)$ structure such that

$d_{H}\Psi_{1}=0,$

and

$d_{H}\Psi_{2}=0.$

This geometry is also called twisted generalised Calabi-Yau geometry.

This will be used to model the compactification manifold of string theory with
flux, this is remarkable because it allows us to describe fields and fluxes in
string theory in a purely geometric way as we will see in the next chapter.

##  6 Flux compactifications

In the previous chapters we saw that string compactifications have some
problems like non broken supersymmetry and the fact that these types of
compactifications give rise to moduli which can be interpreted as a long range
force carriers. To deal with these problems, we have to turn on some fluxes to
give potentials to the moduli in order to stabilize them. In this chapter we
discuss some models of flux compactifications. Since this topic is very
diverse with many models being proposed, we only discuss the most generic
model in each string theory type and refer to other models.

###  6.1 Fluxes and charges

By a flux we mean a differential form defined on the manifold in question.
Fluxes representing potentials introduced in the theory whose field strength
satisfy a certain Bianchi identity as we will see. We present the different
types of potential forms (i.e. fluxes) below.

To define a charge, consider an n-form flux $F_{\mu_{1}\mu_{2}...\mu_{n}}$,
this represents the field strength of an (n-1) potential. We define the charge
of the mentioned flux by

$Q=\int_{\partial\Sigma}F_{\mu_{1}\mu_{2}...\mu_{n-1}0},$

where $\Sigma$ is the manifold the flux is defined on, and $\partial\Sigma$ is
its boundary defined when setting the nth coordinate to a value, without loss
of generality we can set it to zero.

####  6.1.1 Dirac quantization rule

In string theory charges due to a flux are defined on D branes are represented
by currents through the strings attached to these branes. Thus, be charge
conservation the strings must form a closed loop or the branes must be wrapped
on closed cycles, this must be satisfied by all the flux in the
compactification scheme to be viable, this condition leads to Dirac
quantization condition

$\int_{\Sigma_{n}}F_{n}=2\pi N,$

where $F_{n}$ is an n-form flux, $\Sigma_{n}$ is an n cycle and $N$ is a
natural number. For more details on different approaches to derive this
condition see [39].

The most important consequence (of Dirac quantization condition is that it
rules out some compactifications for example the compactification on
$AdS\times S^{5}/Z_{2}$, to solve this issue we have to introduce orientifolds
with fractional charges to satisfy the conditions, these type of
compactifications are beyond the scope of this review, for more details and
how orientifolds of different dimensions are introduced see [40].

####  6.1.2 Types of fluxes

In this section we present the types of fluxes derived from string theory
action directly i.e. not introduced by hand on the manifold.

Remember that superstring theories have two sectors: Neveau Schwarz (NS)
sector and Ramond (R) sector each of which has its own potential forms. We
focus here on the massless section of both sectors, in NS massless sector
there is the metric tensor, the Kalb Ramond field ($B$) and the dilaton
($\Phi$), the metric can not be considered as a flux because it is a property
of the manifold so the fluxes here are $B$ and $\Phi$, we will discuss the
dilaton flux later.
we define the field strength form of $B$ as follows

$H=dB,$

which satisfies the Bianchi identitiy $dH=0$.

For the massless R sector we have different potentials according to the
theory, here we use the democratic formulation [41]. In this sector (called RR
sector because it is derived from R sector of left and right movers) we can
define three types of field strength forms

  1. 1.

$F_{n}=dC_{n-1}+m\text{e}^{B},$

  2. 2.

$F_{n}^{(10)}=dC-H\wedge C+m\text{e}^{B},$

  3. 3.

$F^{(10)}=\sum_{n}F_{n}^{(10)},$

where $C$ in both cases is the potential with the right rank, and m is the
mass parameter $m=F_{0}$. The Bianchi identity of the flux are given by

  1. 1.

$dF_{n}=0,$

  2. 2.

$dF^{(10)}_{n}=0,$

  3. 3.

$dF^{(10)}=H\wedge F^{(10)},$

with another condition imposed by the democratic formulation

$F^{(10)}_{n}=(-1)^{[n/2]}\star F^{(10)}_{10-n},$

where $[n/2]$ is the integer part of $n/2$.

Finally, we address the dilaton which is a scalar field i.e. a zero form, in
theories allowing zero forms flux in their RR sector, the dilaton is combined
with the form to form one flux for example in type IIA superstring theory,
there is an RR zero form $C_{0}$ then the total flux will be
$\tau=C_{0}+ie^{\Phi}$. In theories who do not allow zero form potentials the
dilaton will represent the topological term in the action i.e. the coupling
constant for gravitational interactions as usual in string theory.

###  6.2 Compactifications on twisted tori

We saw in chapter 3 that toroidal compactifications results in $\mathcal{N}=8$
supesrymmetric low energy models which is too much supersymmetry for a
realistic model and the fact that a torus can not induce chiral fermions.
These two problems can be solved by introducing an additional geometric
structure to the torus, namely a twist, which is equivalent to turning on a
flux in the string model. In this section we briefly review the resulting low
energy model and the challenges it faces.

####  6.2.1 Twisted tori

We begin by defining a twisted torus mathematically as follows: Let $M$ be a d
dimensional manifold with coordinates $x^{i}$, we equip the manifold $M$ with
a basis of non vanishing one form $\sigma_{m}$ i.e. a rank n tensor $T$ on $M$
can be written as
$T=T_{\mu_{1}\mu_{2}...\mu_{n}}\sigma^{1}\sigma^{2}...\sigma^{n}$ where
$T_{\mu_{1}\mu_{2}...\mu_{n}}$ are the tensor’s components and
$\mu_{i}=1,2,...,d\ \forall i\in\\{1,2,...,n\\}$. This manifold with the one
form basis is called a twisted d dimensional torus by a group $G$ if

  1. 1.

The coordinated $x^{i}$ satisfy periodicity conditions: $x^{i}$ is identified
to $x^{i}+2\pi R^{i}$ where $R^{i}$ is the radius of the torus in the i-th
direction (as each direction is a circle as the case of ordinary torus).

  2. 2.

the form basis satisfy the structure equation

$d\sigma_{m}+\frac{1}{2}f^{m}_{np}\sigma^{n}\wedge\sigma^{p}=0,$

where the quantities $f^{m}_{np}$ are the structure constant of a lie group
$G$ representing the twisting as will be explained later, these quantities
satisfy the Jacobi identity

$f^{q}_{[mn}f^{t}_{p]q}=0.$

where $[mnq]$ is a commutator of indices.

To describe twisted tori are we have to note that a torus has two classes of
non contractible closed curves, and a vector field defined on the torus can be
transformed along the curves in a well defined manner. The main idea of
twisting is to transform a vector field defined on the torus by the action of
the group $G$ wherever it is transformed along a non contractible closed
curve, it is similar to holonomy but for finite (not infinitesimal) non
contractible curves, a simple example is shown in figure 6 below.

Figure 6: A simple example of a twisted torus when the twisting group $G$ is
the rotation group, a vector field defined on this twisted torus will be
rotated each time it revolves along a non contractible curve i.e. one rotation
alone the curve will be equivalent to an action of the rotation group.

Note that when the twisting group $G$ is the trivial group, all the structure
constants are zero and we recover the ordinary torus.
From this definition we can easily see that in the context of string theory, a
twisted torus can be thought of as a torus with flux turned on [42], and from
this we study the compactification on this geometry as flux compactification
on a torus.

####  6.2.2 The compactification model

To perform a compactification of bosonic string theory on these type of tori,
we use generalised geometry discussed in chapter 6 to get an $O(d,d)$
invariant theory where $d$ is the dimension of the twisted torus.

Firstly, we define the Lie algebra of symmetries of the theory as

$[Z_{m},Z_{n}]=f^{p}_{mn}Z_{p}-K_{mnp}X^{p},$
$[X^{m},Z_{n}]=-f^{m}_{np}X^{p},$ $[X^{m},X^{n}]=0,$

where $X^{n}$ are the generators of the gauge transformations of the flux,
$Z_{n}$ are the generators of the diffeomorphisms on the twisted torus,
$f^{n}_{mp}$ are the structure constants of the underlying Lie group of the
twisted torus, and $K_{nmp}$ are constant coefficients satisfying the
integrability conditions $K_{t[mn}f^{t}_{pq]}=0$.
To define a generalised geometry we define the generalised vector

$T_{A}=(Z_{m},X^{m}),$

where $A=1,2,...,2d$, which is an $O(d,d)$ vector as it satisfies

$[T_{A},T_{B}]=f_{AB}^{C}T_{C}.$

Secondly, we write the Lagrangian as

$\mathcal{L}=R\star\mathbb{I}+\frac{1}{2}\star d\phi\wedge
d\phi-\frac{\text{e}^{a\phi}}{2}\star F\wedge F,$

where $R$ is the Ricci scalar of the manifold
$M_{d+4}=\mathcal{M}_{4}\times\mathbb{T}^{d}$ such that $\mathbb{T}^{d}$ is
the d dimensional twisted torus, $\mathbb{I}$ is the identity form, $\phi$ is
the dilaton and $F=dB$ where $B$ is the Kalb Ramond field.
Now we separate the internal space i.e. the twisted torus from the internal
Minkowski space and write the metric as

$ds=g_{\mu\nu}dy^{\mu}dy^{\nu}+g_{mn}dx^{m}dx^{n},$

where $g_{\mu\nu}$ is the Minkowski metric with $\mu,\nu=1,2,3,4$, $y^{\mu}$
are the coordinates on the Minkowski space, $g_{mn}$ is the internal metric
where $m,n=1,2,...,d$.

In this formalism we note that the external component of the Kalb Ramond field
$B_{\mu\nu}$ transforms as a two form, the internal components $B_{mn}$
transforms also as a two form and $B_{m}$ transforms as a d vector
additionally, there is another d vector $A^{m}$ representing the connection
form on the torus. From these fields we can construct the generalised metric

$\mathcal{M}^{AB}=\begin{bmatrix}g^{mn}&-B_{mn}g^{np}\\\
-B_{mp}g^{np}&g_{mn}+g^{pq}B_{mp}B_{nq}\end{bmatrix},$

along with the generalised vectors

$\mathcal{A}^{A}=\begin{bmatrix}A^{m}\\\ B_{m}\end{bmatrix},$

representing the internal vector due to the connection on the generalised
geometry and

$\mathcal{F}^{A}=\begin{bmatrix}F^{m}\\\ G_{m}-B_{mn}F^{n}\end{bmatrix},$

where $G_{m}$ is defined in [46], this generalised vector representing the
flux.

Using all these variables we can write the Lagrangian of the compactified
theory as

$\mathcal{L}=\text{e}^{\phi}(R\star\mathbb{I}+\star d\phi\wedge
d\phi+\frac{1}{2}\star G_{3}\wedge G_{3}+\frac{1}{4}L_{AC}L_{BD}\star
D\mathcal{M}^{AB}\wedge D\mathcal{M}^{CD}$
$-\frac{1}{2}L_{AC}L_{BD}\mathcal{M}^{AB}\star\mathcal{F}^{C}\wedge\mathcal{F}^{D}-\frac{1}{12}\mathcal{M}^{AD}\mathcal{M}^{BE}\mathcal{M}^{CF}t_{ABC}t_{DEF}$
$+\frac{1}{4}\mathcal{M}^{AD}L^{BE}L^{CF}t_{ABC}t_{DEF}),$

where $G_{3}$, $D\mathcal{M}^{AB}$ and $L^{AB}$ are defined in [45].

This models bosonic string theory on twisted tori, for supersymmetric
counterparts see[43-47], the resulting theories contains chiral fermions
solving the problem in toroidal compactifications without flux however, the
supersymmetric models give $\mathcal{N}=4$ supersymmetric low energy models
after adding flux which is an improvement compared to the $\mathcal{N}=8$
theories without flux, but still has too much supersymmetry to be a realistic
model. For models with additional flux are introduced see [42].

###  6.3 Compactifications of type II superstrings on Generalized Calabi-Yau
manifolds

In this section we review compactifications of types IIA and IIB superstring
theories with flux on manifolds with $SU(3)$ structure, we focus on flux
compactifications leaving $\mathcal{N}=1$ unbroken supersymmetry, and prove
that any consistent such theory is equivalent to a compactification on
generalised Calabi-Yau manifold reviewed in chapter 6.

####  6.3.1 Torsion classes

In this section we define torsion classes for manifolds with $\text{SU}(3)$
structure, as fluxes leading to $\mathcal{N}=1$ supersymmetric low energy
effective models are completely specified by the manifold’s torsion classes
[48].
From basic differential geometry, we know that to any manifold we can define a
torsion tensor. On manifolds with $\text{SU}(3)$ structure, the torsion tensor
$T^{m}_{np}\in\Lambda^{1}\otimes(\text{su}(3)\oplus\overline{\text{su}(3)})$
can be decomposed into tensors of different representations as follows

$T^{m}_{np}=(\mathbf{1}\oplus\mathbf{1})\otimes(\mathbf{8}\oplus\mathbf{8})\otimes(\mathbf{6}\oplus\mathbf{\bar{6}})\otimes
2(\mathbf{3}\oplus\mathbf{\bar{3}}),$

where the bar represents the adjoint representation. We call the first term
i.e. $\mathbf{1}\oplus\mathbf{1}$ the first torsion class of the manifold,
obviously it represents a complex scalar and is denoted by $W_{1}$. The second
term is a complex $(1,1)$ form, it is called the second torsion class and is
denoted by $W_{2}$. The third term is a real formal sum of forms $(2,1)+(1,2)$
and is denoted by $W_{3}$. The last term is in fact two copies of a
representation so is two classes, one is a real one vector and one is a real
one form, the former is called the fourth torsion class, denoted by $W_{4}$,
and the later is called the fifth torsion class denoted by $W_{5}$.

For a manifold with both complex structure $J$ and a symplectic structure
$\Omega$ we can write [49]

$dJ=\frac{1}{2}\text{Im}(\bar{W_{1}}\Omega)+W_{4}\wedge J+W_{3},$
$d\Omega=W_{1}J^{2}+W_{2}\wedge J+\bar{W_{5}}\wedge\Omega.$

Having written the differentials of the complex and symplectic structures in
terms of torsion classes, we can classify manifolds with $\text{SU}(3)$
structures only by knowing their torsion classes as in [49], this will be used
in the next section to specify the types of manifolds used in the
compactifications we are considering.

####  6.3.2 Type II compactifications with flux

Firstly, we answer the question: why do we consider only manifolds with
$\text{SU}(3)$ structure? The answer is that we require that the manifold is
Calabi-Yau if all the fluxes vanish to match the results in chapter 5. Thus,
an $\text{SU}(n)$ structure is required and we take $n=3$ so the manifold will
be 6 dimensional.

Now, we write the supersymmetry transformations and derive the $\mathcal{N}=1$
solutions. For type II superstrings, the supersymmetry transformations are

$\delta\psi_{M}=(\nabla_{M}+\frac{1}{8}H_{MNP}\Gamma^{NP}\Gamma_{11}+\frac{e^{\Phi}}{16}\sum_{n=0}^{9}\frac{1}{n!}F_{N_{1}N_{2}...N_{n}}\Gamma^{N_{1}N_{2}...N_{n}}\Gamma_{M}\Gamma_{11}^{[\frac{n}{2}]}\sigma^{1})\epsilon,$
$\delta\lambda=(\Gamma^{N}\partial_{N}\phi+\frac{1}{2}\Gamma^{MPQ}H_{MPQ}\Gamma_{11}+\frac{1}{8}\text{e}^{\phi}\sum_{n=0}^{9}\frac{(-1)^{n}}{n!}(5-n)F_{P_{1}P_{2}...P_{N}}\Gamma^{P_{1}P_{2}...P_{N}}\Gamma_{11}^{[n/2]}\sigma^{1})\epsilon$

for type IIA and

$\delta\psi_{M}=(\nabla_{M}-\frac{1}{8}H_{MNP}\Gamma^{NP}\sigma^{3}+\frac{e^{\Phi}}{16}[\sum_{n\
odd}\frac{1}{n!}F_{N_{1}N_{2}...N_{n}}\Gamma^{N_{1}N_{2}...N_{n}}\Gamma_{M}\sigma^{1}+$
$\sum_{n\
even}\frac{i}{n!}F_{N_{1}N_{2}...N_{n}}\Gamma^{N_{1}N_{2}...N_{n}}\Gamma_{M}\sigma^{2}])\epsilon,$

$\delta\lambda=(\Gamma^{N}\partial_{N}\phi-\frac{1}{2}\Gamma^{MPQ}H_{MPQ}\sigma^{1}+\frac{\text{e}^{\phi}}{8}[\sum_{n\
odd}\frac{(-1)^{n}}{n!}(5-n)F_{P_{1}P_{2}...P_{N}}\Gamma^{P_{1}P_{2}...P_{N}}\sigma^{1}+$
$i\sum_{n\
odd}\frac{(-1)^{n}}{n!}(5-n)F_{P_{1}P_{2}...P_{N}}\Gamma^{P_{1}P_{2}...P_{N}}\sigma^{2}])\epsilon$

for type IIB, where $\psi_{M}$ are the two gravitinos, $\lambda$ is a vector
containing the two dilatinos, $\epsilon$ is the supersymmetry parameter i.e.
supersymmetry spinor, $\Gamma$ are the Dirac matrices in different dimensions,
$F$ are the RR fluxes, $H$ are the NS fluxes, $\phi$ is the dilaton, $\nabla$
is the covariant derivative, $\sigma^{i}$ are the Pauli matrices. Note that
when all fluxes vanish we get the results in chapter 5.

As explained before, we can split the supersymmetry Weyl spinors into internal
and external parts as follows

$\epsilon^{1}=\xi^{1}_{+}\otimes\eta_{+}+\xi^{1}_{-}\otimes\eta_{-},$
$\epsilon^{2}=\xi^{2}_{+}\otimes\eta_{-}+\xi^{2}_{-}\otimes\eta_{+},$

for type IIA, and

$\epsilon^{1}=\xi^{1}_{+}\otimes\eta_{+}+\xi^{1}_{-}\otimes\eta_{-},$
$\epsilon^{2}=\xi^{2}_{+}\otimes\eta_{+}+\xi^{2}_{-}\otimes\eta_{-},$

for type IIB, where $\xi_{1,2}$ are the 4D spinors, $\eta^{1,2}$ are the 6D
spinors and the subscripts $+$ and $-$ label the chirality of the spinors.

This setup in general gives $\mathcal{N}=2$ supersymmetry as discussed in
chapter 5, here we can adjust the fluxes so that the two spinors are not
independent i.e. we require the 4D spinors to be proportional then solve for
the vacuum solutions using the adjusted flux, in practice we impose the
condition that the 4D spinors are proportional then solve for the fluxes
representing the required vacua.
The imposed conditions are

$\xi^{1}=a\xi,$ $\xi^{2}=b\xi,$

where $a$ and $b$ are complex functions. This means that the two 4D spinors
are proportional to a common spinor and thus the resulting vacuum will
preserve one globally defined spinor i.e. $\mathcal{N}=1$ supersymmetric.

Substituting in the supersymmetry transformations and equate to zero, we get a
set of equations relating possible fluxes with the torsion classes presented
in the previous section, after rather tedious calculations [50] we get the set
of $\mathcal{N}=1$ solutions in terms of torsion classes as following tables
taken from [48]

Figure 7: The set of all $\mathcal{N}$=1 solutions of type IIA superstring
theory. The columns represent the solutions and the rows represent the fluxes
in each representation. Figure 8: $\mathcal{N}$=1 solutions for type IIB
superstring theory with the last column is displayed separately

$2abW_{3}=\text{e}^{\phi}(a^{2}+b^{2})\star_{6}F_{3}^{(6)},$

$(a^{2}-b^{2})W_{3}=-(a^{2}+b^{2})\star_{6}H_{3}^{(6)},$ (*)

$2abH_{3}^{(6)}=-\text{e}^{\phi}(a^{2}-b^{2})F_{3}^{(6)},$

where $\star_{6}$ is the 6 dimensional Hodge star operation and the
superscript represents the representation the flux in,

$\text{e}^{\phi}F_{3}^{(\bar{3})}=\frac{-4iab(a^{2}+b^{2})}{a^{4}-2ia^{3}b+2iab^{3}+b^{4}}\bar{\partial}a,$
$\text{e}^{\phi}F_{5}^{(\bar{3})}=\frac{-4iab(a^{2}-b^{2})}{a^{4}-2ia^{3}b+2iab^{3}+b^{4}}\bar{\partial}a,$
$H_{3}^{(\bar{3})}=\frac{-2i(a^{2}+b^{2})(a^{2}-b^{2})}{a^{4}-2ia^{3}b+2iab^{3}+b^{4}}\bar{\partial}a,$

$W_{4}=\frac{2(a^{2}-b^{2})^{2}}{a^{4}-2ia^{3}b+2iab^{3}+b^{4}}\bar{\partial}a,$
(**)

$\bar{W}_{5}=\frac{2(a^{4}-4a^{2}b^{2}+b^{4})}{a^{4}-2ia^{3}b+2iab^{3}+b^{4}}\bar{\partial}a,$
$\bar{\partial}A=\frac{4(ab)^{2}}{a^{4}-2ia^{3}b+2iab^{3}+b^{4}}\bar{\partial}a,$
$\bar{\partial}\phi=\frac{2(a^{2}+b^{2})^{2}}{a^{4}-2ia^{3}b+2iab^{3}+b^{4}}\bar{\partial}a.$

These represent all possible $\mathcal{N}=1$ supersymmetric low energy models
from type II superstring theory.

####  6.3.3 Generalised geometry formulation

In this section we derive all the previous results using generalised geometry
reviewed in chapter 5, we prove that all the solutions can be encoded in 4
equations and are compactifications on twisted generalised Calabi-Yau
manifolds.

Here we have a complex structure $J$ and a symplectic structure $\Omega$ on a
generalised manifold with $\text{SU}(3)\oplus\text{SU}(3)$ structure. In this
case we can define two no where vanishing bispinors [51,52]

$\Phi_{+}=\eta_{+}\otimes\eta_{+}^{\dagger},$
$\Phi_{-}=\eta_{+}\otimes\eta_{-}^{\dagger}.$

On the generalised manifold we can consider the two bispinors as two
components of one generalised bispinor i.e. one spinor is conserved so the
solutions have $\mathcal{N}=1$ supersymmetry.
Now we substitute in the supersymmetry transformations and equate to zero (the
full calculations are done in [53]), the resulting equations are: For type IIA

$\text{e}^{-2A+\phi}(d+H\wedge)(\text{e}^{2A-\phi}{a\bar{b}\Phi_{+}})=0,$
$\text{e}^{-2A+\phi}(d+H\wedge)(\text{e}^{2A-\phi}{ab\Phi_{-}})=dA\wedge\bar{a}\bar{b}\bar{\Phi_{-}}-\frac{\text{e}^{\phi}}{16}[(|a|^{2}-|b|^{2})F_{IIA+}-i(|a|^{2}+|b|^{2})\star
F_{IIA-}],$

where $F_{IIA\pm}=F_{0}\pm F_{2}+F_{4}\pm F_{6}$.
The first equation tells us that any compactification manifold must be twisted
generalised Calabi-Yau and the second equation gives information about the
type of twisted generalised Calabi-Yau used for each solution i.e. the
equation used to tell us for each choice of flux which manifold must be used.
For type IIB

$\text{e}^{-2A+\phi}(d-H\wedge)(\text{e}^{2A-\phi}{ab\Phi_{-}})=0,$
$\text{e}^{-2A+\phi}(d+H\wedge)(\text{e}^{2A-\phi}{ab\Phi_{-}})=dA\wedge\bar{a}b\bar{\Phi_{+}}+\frac{\text{e}^{\phi}}{16}[(|a|^{2}-|b|^{2})F_{IIB+}-i(|a|^{2}+|b|^{2})\star
F_{IIB-}],$

where $F_{IIB\pm}=F_{1}\pm F_{3}+F_{5}$.
The first equation says that the manifold is twisted generalised Calabi-Yau
and the second gives the information about classifications of the solutions.

###  6.4 Heterotic string theory compactifications with flux

As shown in chapter 4, low energy models of heterotic string theories are
already $\mathcal{N}=1$ supersymmetric but it is important to consider flux
compactifications because the moduli problem in heterotic string theories are
more severe than type II theories, this is due to the back reaction on the
geometry when introducing new objects; in type II theories back reactions are
always controlled and can be dealt with perturbatively, in heterotic string
theories this is not always the case. Thus, we need to introduce flux to
stabilize the moduli in a way that control all possible changes in the
geometry.

In this section we review flux compactifications of a generic heterotic
superstring theory with only $H$ flux turned on.
Following the analysis of [54], we begin by writing down the supersymmetry
transformations and equating them to zero to get

$(\nabla_{M}+\frac{1}{8}H_{MNP}\Gamma^{NP})\epsilon=0,$
$(\Gamma^{M}\partial_{M}\phi)\epsilon=0,$ $\Gamma^{MN}F_{MN}\epsilon=0,$

the third equation is trivially satisfied in our case as only $H$ flux are on.
Decomposing $\epsilon$ into 4D and 6D parts

$\epsilon=\xi\otimes\eta+\bar{\xi}\otimes\bar{\eta},$

where $\xi$ is the 4D part, $\eta$ is the 6D part,
$\bar{\xi}=(B^{(4)}\xi)^{*}$ and $\bar{\eta}=(B^{(6)}\eta)^{*}$, the first two
equations imply the existence of a complex and symplectic forms as follows

$J_{m}^{n}=-i\eta^{\dagger}\Gamma_{m}^{n}\eta,$
$\Omega_{mnp}=\bar{\eta}^{\dagger}\Gamma_{mnp}\eta.$

Substituting in the equations, using the Bianchi identity for the flux, we get

$H_{mnp}=3J_{[m}^{r}\nabla_{|}r|J_{np]},$

where the commutator is over $m,n,p$ only and

$H_{mnp}J^{np}=-4J_{m}^{p}\partial_{p}\phi.$

The second equation tells us that the $H$ flux is not primitive unless the
dilaton is a constant but in this case we must impose the condition $H=0$ to
get a supergravity solution[55,56], if we require that the flux is non zero we
must have a non constant dilaton so a non primitive flux. In this case we
decompose the flux into primitive and non primitive parts as follows

$H_{mnp}=H^{P}_{mnp}+H^{NP}_{mnp}=(H_{mnp}-\frac{3}{4}J_{[mn}H_{p]rs}J^{rs})+\frac{3}{4}J_{[mn}H_{p]rs}J^{rs},$

applying the first supersymmetry transformation we get

$H^{NP}=-\star(d\phi\wedge J).$

Substituting back into the transformations we get the two equations

$H=\star\text{e}^{2\phi}d(\text{e}^{-2\phi}J),$
$d(\frac{1}{2}\text{e}^{-\phi}J\wedge\text{e}^{-\phi}J)=0.$

Performing the conformal transformation $\bar{J}=\text{e}^{-\phi}J$, the
second equation will be

$d(\bar{J}\wedge\bar{J})=0.$

This defines a geometry called the conformally balanced geometry. There is an
important special case where $H^{P}=0$, the first condition gives $d\bar{J}=0$
which is a geometry related to the Kahler geometry by a conformal
transformation so called conformally Kahler geometry.

To summarize, If the flux has a non zero primitive part a consistent low
energy model with non zero $H$ flux must be due to a compactification on a
conformally balanced manifold, while if the primitive part is zero the
compactification manifold must be conformally Kahler.

###  6.5 Non-supersymmetric string theory compactifications with flux

In this section we review flux compactifications of
$\text{SO}(16)\times\text{SO}(16)$ heterotic non supersymmetric string theory,
this theory is particularly important because it is the only non relativistic
string theory with no tachyons [57].

It was shown that the low energy effective models of the
$\text{SO}(16)\times\text{(SO)(16)}$ heterotic non supersymmetric string
theory are identical to those of $E_{8}\times E_{8}$ and $\text{SO}(32)$
heterotic supersymmetric string theories[36,58]. Thus compactifications of
this theory can be deduced directly from supersymmetric heterotic theories
compactifications i.e. if we want to compactify with certain flux, we can do
the compactification on the supersymmetric counterpart then apply a
supersymmetry breaking twist as explained in figure 5.

##  7 Applications to cosmology

Having discussed string compactifications, now we present some applications to
cosmology. In this chapter we discuss two important applications: the
swampland project and the cosmological constant problem and the attempts to
solve it by trying to find De Sitter vacua from low energy effective models of
string theory.

###  7.1 The swampland project

####  7.1.1 Introduction

The swampland program was proposed by Vafa [59] to answer an important
question in effective field theory: Can every effective field theory be
embedded in a consistent high energy theory of quantum gravity? or in string
theory, Does every low energy models come from some consistent string
compactification?
The answer to this question is no. There are some constraints that tell us
whether the low energy model is a result of a consistent string
compactification (more generally, a low energy model of consistent quantum
gravity theory), these constraints are called the swampland conjectures.
Effective field theories satisfy the swampland conjectures i.e. can be lifted
to a consistent quantum gravity theory are said to be in the landscape,
otherwise the theory is said to be in the swampland.

The swampland conjectures are in fact very restrictive and may be broken for
some theories at high energies. Thus, the swampland conjectures, also called
the swampland constraints, form a cone as drawn in figure 9, that is because
the constraints become stronger and more restrictive as the energy scale goes
higher. For comprehensive review of the project see [60]. The next figure
illustrates nicely the idea of the swampland project.

In the next section we review the swampland conjectures one by one and explain
what are their impact on the physics of string compactifications as well as
the mathematical requirements for the theory to be consistent.

Figure 9: the horizontal plane represents the space of all low energy
effective field theories, and the vertical axis representing the energy scale.
For the theories in the swampland, the effective field theory description
breaks down at certain energy scale and the high energy theory is not
consistent with quantum gravity. For theories in the landscape many theories
also break down at certain point if it breaks the swampland constraints but
all consistent theories with quantum gravity must be in the landscape.

####  7.1.2 Overview on the swampland conjectures

The swampland conjectures can be divided into three groups displayed as rows
in figure 10 below.

  1. 1.

The first row i.e. the "no global symmetry" hypothesis, the cobordism
conjecture and the completeness hypothesis, basically tell us that no global
symmetries are allowed in a quantum gravity theory, gravity breaks all global
symmetries.

  2. 2.

The second row i.e. the weak gravity conjecture and the distance conjecture
give constraints on the spectrum of the low energy model and describe hoe the
effective field theory breaks down for some theories at high energy scales.

  3. 3.

The third row i.e. the non supersymmetry AdS conjecture, the De Sitter
conjecture and the AdS distance conjecture dive constraints on the possible
consistent stable vacua allowed for the low energy effective model to be
consistent with quantum gravity.

Figure 10: A schematic diagram showing the relations between the swampland
conjectures.

Now we study the conjectures one by one.

####  7.1.3 No global symmetries hypothesis

The hypothesis [61-64] states that no global symmetries are allowed in any
theory of quantum gravity.
Another equivalent statement: Any symmetry in a consistent quantum gravity
theory must be either broken or gauged.

No global symmetries hypothesis is the most widely accepted among the
swampland conjectures due to strong evidences from physics point of view
especially cosmological arguments however, there is no mathematical proof for
it yet.
We motivate the hypothesis by considering the contrapositive argument i.e.
what will go wrong if there are global symmetries in a theory of quantum
gravity?

  1. 1.

The first motivation comes from black hole physics: Assume the existence of a
global symmetry, then by Noether theorem we must have a global charge. Thus, a
black hole can be charged under this symmetry. Upon decay the black hole will
lose mass but not the global charge because Hawking radiation do not carry
global charge, this will form a stable remnant which is not physical by
itself. Moreover, since particles of any group representation carry the global
charge, we can form a black hole easily as the remnant is very light, doing
that we end up with an infinite number of stable black hole remnants which is
of course non physical [61-65].

  2. 2.

Another motivation from black hole physics arguing that due to no hair
theorem, a black hole can be completely specified by its mass, angular
momentum and gauge symmetry charge. Thus, a global charge can not be measured
i.e. has infinite uncertainty, this means that the black hole has infinite
entropy and that is also non physical [66].

Now we show that string theory satisfy this hypothesis as follows: It was
proven that there can not be a global symmetry in the target space of a
perturbative string theory even if we begin with a global symmetry in the
parameter space [67].

Here we give an outline of the proof: Consider a global symmetry in the
parameter space, the world sheet charge is given by

$Q=\frac{1}{2\pi i}\oint(j_{z}dz-j_{\bar{z}}d\bar{z}),$

where $j_{z}$ and $j_{\bar{z}}$ are the currents of the global symmetry. The
vertex operators for the currents are given by

$j_{z}\bar{\partial}X^{\mu}\text{e}^{ikX},$ $\partial
X^{\mu}j_{\bar{z}}\text{e}^{ikX}.$

These operators correspond to massless gauge vector particles on the world
sheet. Thus, a global symmetry in the parameter space is translated into a
gauge symmetry on the world sheet which is the physical space according to
string theory.

####  7.1.4 Cobordism conjecture

The conjecture states that in a consistent N dimensional quantum gravity
theory compactified on a D dimensional internal manifold, all cobordism
classes must vanish [68]. To motivate the conjecture, we ask what will go
wrong if a cobordism class is non vanishing?
The answer is that we will get a global charge so a global symmetry. The
global charge can be constructed as follows: Assume that we have a non
vanishing cobordism class with cardinality M i.e. the compactification
manifolds of different elements of the class are cobordant, then we have M
higher dimensional manifolds each is the cobordism of an element of the class
each of which is separate from the other completely, if we label these
cobordism manifolds this label will be a global charge as no manifold from a
certain class can change its class i.e. the label will be the same whatever
interactions or fluxes are added.

An important implication to the cobordism conjecture is that it implies that
every consistent compactification manifold of a quantum gravity theory is a
boundary of some higher dimensional manifold. So we can think of the quantum
gravity theory as this higher dimensional manifold, and every possible low
energy effective field theory is a boundary of this manifold. This is a
powerful statement because after compactification all possible effective field
theories will be present but separated by domain walls.

Figure 11: According to the cobordism conjecture every possible low energy
model of a consistent quantum gravity theory is realised in the 4D space but
separated by domain walls, in the case of a manifold with no boundary, the
effective field theory space ends with the domain wall [69].

Before concluding the section we give two important notes:

  1. 1.

In mathematics literature, this topic is studied but using bordism not
cobordism but the two approaches are exactly equivalent due to Yoneda lemma
discussed in chapter 2(considering bordisms/cobordisms in the categorial
sense). The two approaches are dual and can be transformed to each other at
any point during any proof or argument, in this review we use cobordisms.

  2. 2.

In the case of flux compactifications where there are structures on the
compactification manifolds, different types of cobordisms are defined and the
cobordism conjecture apply for every type separately, in some cases we are
required to add certain structures to some compactfications to satisfy the
conjecture, this topic is beyond the scope of this review.

####  7.1.5 The completeness hypothesis