Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

The first annual workshop on Learnersourcing: Student-generated Content @
Scale, June 01, 2022, NYC, NY

[ orcid=0000-0001-5249-5162<EMAIL_ADDRESS>]

[ orcid=0000-0001-6502-209X<EMAIL_ADDRESS>]

[ orcid=0000-0001-6829-9449<EMAIL_ADDRESS>]

# Coverage of Course Topics in Learnersourced SQL Exercises

Nea Pirttinen  University of Helsinki, Helsinki, Finland  Arto Hellas  Aalto
University, Espoo, Finland  Juho Leinonen

(2022)

###### Abstract

Learnersourcing is a common task in modern computing classrooms, where it is
used, for example, for the creation of educational resources such as multiple-
choice questions and programming exercises. One less studied type of
learnersourced artefact is SQL exercises. In this work, we explore how well
different SQL topics are covered by learnersourced SQL exercises. Covering
most course topics would allow students to practice the full content of the
course by completing learnersourced exercises. Our results suggest that
learnersourcing can be used to create a large pool of SQL exercises that cover
most of the topics of the course.

###### keywords:

learnersourcing crowdsourcing SQL databases topic coverage

##  1 Introduction

Traditionally, learnersourcing has been used, for example, for creating
multiple-choice questions (MCQs) [1, 2, 3, 4]. In computing classrooms,
learnersourcing is increasingly used for the creation of programming exercises
[5, 6]. In our recent prior work, we introduced a system for crowdsourcing SQL
exercises [7], which to the best of our knowledge is the first such system.

One concrete goal of learnersourcing systems is to create a vast repository of
exercises that could complement or even replace instructor-created exercises.
An important question related to this is whether student-created exercises
cover all the course topics, which would be necessary to allow students to
practice all learning objectives of the course. Prior work has studied the
coverage of course topics of learnersourced MCQs and found that student-
created exercises did cover all course topics [8].

In this work-in-progress paper, we report preliminary results related to how
well learnersourced SQL exercises cover different SQL concepts. Our goal is to
examine whether – by using mostly learnersourced exercises – students will get
to practice the concepts they are expected to learn during the course.

##  2 Tool and Context

###  2.1 SQL Trainer

SQL Trainer is a learnersourcing system for practicing SQL queries [7]. When
using the system, a student is presented with a list of topics in the order of
appearance in the course material. Students can choose to either create
exercises or practice a specific topic. If the student chooses to create an
exercise for a particular topic, the system will ask which database the
student wishes to use (selection is made from a pre-defined list of databases
created by the course instructor). The student is then required to give a name
for the exercise, an exercise description, and a sample solution for that
exercise. Once the exercise is completed, it is added to the pool of exercises
for that topic. If the student chooses to practice a topic, they will be given
a randomly selected exercise that they have not completed yet from the pool of
created exercises.

A more detailed description of SQL Trainer and its usage, as well as the
description of the teacher view, is presented in [7].

###  2.2 Context and Research Questions

For the present study, we use data collected from three introductory database
courses offered by the University of Helsinki, where SQL trainer was used to
support SQL practice. In the courses, SQL trainer had 11 topics (briefly
outlined in Table 1). Approximately 10% of the course grade was based on
completing at least four exercises per course topic and creating at least a
single exercise per course topic. Students were free to complete and create
more exercises than required for the points, however.

Our research questions are as follows:

  * RQ1.

Which topics do students create exercises for?

  * RQ2.

How well do the exercises created by students cover the course topics?

To answer research question 1 and 2, we partially replicate earlier work by
Denny et al. [8] and Purchase et al. [9] by studying to what extent the
exercises that students complete and create cover the course topics. In
particular, we study how many exercises are created by students per course
topic, and how the used SQL concepts match the topics.

##  3 Results and Discussion

Table 1: Course topics. Arranged in the order the topics appear in the cource material. # | Topic
---|---
1 | Select from a table
2 | Filter and order data from a single table
3 | Select from multiple tables
4 | Select from even more tables
5 | Other types of joins
6 | Adding and removing tables
7 | Adding data to a database
8 | Using update and delete
9 | Using functions
10 | Using group by and functions
11 | Functions, group by, having, order

###  3.1 Created Exercises

Table 2: The number of exercises created per topic, 11247 in total. Arranged in the order the topics appear in the course material. Topic | Exercises created
---|---
1 | 1301 (11.6 %)
2 | 1205 (10.7 %)
3 | 1113 (9.9 %)
4 | 1048 (9.3 %)
5 | 666 (5.9 %)
6 | 1063 (9.5 %)
7 | 1047 (9.3 %)
8 | 1037 (9.2 %)
9 | 1028 (9.1 %)
10 | 948 (8.4 %)
11 | 879 (7.8 %)

In total, 1569 students entered the system. Out of these, a total of 1187
created at least one exercise of their own. In total, students created 11247
exercises for 11 different instructor-defined topics. The number of created
exercises for each topic can be found in Table 2 (topic number references
listed in Table 1).

From Table 2, we can see that there is a clear downwards trend in the number
of created exercises per topic. One possible reason for this is that the
latter topics come later in the course and thus fewer students actively
participate at that point in the course. Another potential explanation for the
trend is that the latter topics are related to somewhat more difficult topics
and thus students might not be as inclined to create exercises for those. One
clear outlier is topic 5, which was related to “other types of joins”. Why
this topic elicited fewer learnersourced exercises requires further research.

###  3.2 Topic Coverage

Figure 1: The number of SQL concepts used per topic. Color legend: concept
used in over 80 % of the created exercises in dark blue, over 40 % light blue,
over 10 % yellow, under 10 % gray, and 0 occurrences white.

Figure 1 presents the most common SQL concepts for each topic (topic number
references listed in Table 1). Only concepts that were relatively common
(occurred in over 10% of created exercises in at least a single topic) are
included. In the vast majority of cases, students are able to identify
relevant concepts for the topic, and utilize them when creating an exercise.

Interestingly, based on the SQL concepts used in topic 5, students seem to
majorly prefer creating exercises related to left joins compared to right or
inner joins (which occurred rarely enough that they were not included in the
figure). Similarly, for topic 6, we can see that students are more inclined to
create exercises related to creating tables than dropping tables.

###  3.3 Future Work

Altogether, these preliminary results support the use of learnersourcing for
creating a large exercise pool with a good coverage of course topics. This
provides the opportunity for students to practice a wide variety of SQL
concepts using the learnersourced exercises. In our future work, we are
interested in extending this research to similarly examine how much students
practice the different concepts using the system. In addition, we are keen to
study whether the exercises created by students cover varying levels of
difficulty and if the exercises are effective. Lastly, we are exploring
whether there are differences between demographic groups in to what extent
they participate in learnersourcing activities.

###### Acknowledgements.

We are grateful for the doctoral research grant awarded by Jenny and Antti
Wihuri Foundation to the first author.

## References

  * Denny et al. [2008] P. Denny, J. Hamer, A. Luxton-Reilly, H. Purchase,  Peerwise: students sharing their multiple choice questions,  in: Proceedings of the fourth international workshop on computing education research, 2008, pp. 51–58.
  * Moore et al. [2021] S. Moore, H. A. Nguyen, J. Stamper,  Examining the effects of student participation and performance on the quality of learnersourcing multiple-choice questions,  in: Proceedings of the Eighth ACM Conference on Learning@ Scale, 2021, pp. 209–220.
  * Singh et al. [2021] A. Singh, C. Brooks, Y. Lin, W. Li,  What’s in it for the learners? evidence from a randomized field experiment on learnersourcing questions in a mooc,  in: Proceedings of the Eighth ACM Conference on Learning@ Scale, 2021, pp. 221–233.
  * Khosravi et al. [2019] H. Khosravi, K. Kitto, J. J. Williams,  Ripple: A crowdsourced adaptive platform for recommendation of learning activities.,  Journal of Learning Analytics 6 (2019) 91–105.
  * Pirttinen et al. [2018] N. Pirttinen, V. Kangas, I. Nikkarinen, H. Nygren, J. Leinonen, A. Hellas,  Crowdsourcing programming assignments with crowdsorcerer,  in: Proceedings of the 23rd Annual ACM Conference on Innovation and Technology in Computer Science Education, 2018, pp. 326–331.
  * Denny et al. [2011] P. Denny, A. Luxton-Reilly, E. Tempero, J. Hendrickx,  Codewrite: supporting student-driven practice of java,  in: Proceedings of the 42nd ACM technical symposium on Computer science education, 2011, pp. 471–476.
  * Leinonen et al. [2020] J. Leinonen, N. Pirttinen, A. Hellas,  Crowdsourcing content creation for sql practice,  in: Proceedings of the 2020 ACM Conference on Innovation and Technology in Computer Science Education, 2020, pp. 349–355.
  * Denny et al. [2009] P. Denny, A. Luxton-Reilly, J. Hamer, H. Purchase,  Coverage of course topics in a student generated mcq repository,  ACM SIGCSE Bulletin 41 (2009) 11–15.
  * Purchase et al. [2010] H. Purchase, J. Hamer, P. Denny, A. Luxton-Reilly,  The quality of a peerwise mcq repository,  in: Proceedings of the Twelfth Australasian Conference on Computing Education - Volume 103, ACE ’10, Australian Computer Society, Inc., AUS, 2010, p. 137–146.


# Bootstrapping Mixed MN Correlators in 3D

Stefanos R. Kousvosa,b,c and Andreas Stergioud,e

(May 2022)

###### Abstract

The recent emergence of the modern conformal bootstrap method for the study of
conformal field theories (CFTs) has enabled the revisiting of old problems in
classical critical phenomena described by three-dimensional CFTs. The study of
such CFTs with $O(m)^{n}\rtimes S_{n}$ global symmetry, also known as MN
models, is pursued in this work. Systems of mixed correlators involving scalar
operators in two different representations of the global symmetry group are
considered. Isolated allowed regions are found in parameter space for various
values of $m$ and $n$. These “islands” can be separated into two qualitative
groups: those close to the unitarity bound and those further away. As a by-
product of our analysis generic tensor structures required to bootstrap any
$G^{n}\rtimes S_{n}$ theory with $G$ arbitrary are worked out.

Contents

toc

##  1 Introduction

Since its modern revival, the numerical conformal bootstrap [1] has provided a
powerful and efficient tool for the study of conformal field theories
(CFTs).111For a review see [2] and for a pedagogic introduction see [3]. Its
most remarkable successes have been found in $d=3$ spacetime dimensions, where
it has been used to determine critical exponents and other quantities of
interest in the $O(N)$ family of models, [4] ($N=1$), [5] ($N=2$) and [6]
($N=3$), as well as the supersymmetric Ising ($N=1$) variant [7, 8].

Beyond these widely studied examples, it is desirable to extend applications
to less examined ones. Three-dimensional multi-scalar theories provide ideal
candidates, since they are simultaneously some of the simplest non-trivial
theories one may write down, but at the same time they are also physically
relevant due to their applications to classical critical phenomena. If one
wishes to scan over all the low lying scalar operators as externals in the
theory (which would presumably make results more constraining), the number of
sum rules one needs to consider is (much) larger than the $O(N)$ case. This is
due to the fact that subgroups of $O(N)$ always have (the same or) more
irreducible representations than $O(N)$ in the relevant operator product
expansions (OPEs).222Recently there has been some notable progress regarding
techniques for higher dimensional scans in parameter space; see e.g. [9]. See
also [10] for an example with two externals but many exchanged parameters
scanned over. Additionally, one of the most important assumptions in the
$O(N)$ case, namely $\Delta_{\phi^{\prime}}\geqslant 3$ with $\phi^{\prime}$
the next-to-leading operator in the vector representation of $O(N)$, cannot be
used since in a typical multi-scalar theory the next-to-leading operator in
the defining representation has a strongly relevant dimension. This is because
the vector representation “$\phi^{3}$” operator in a generic multi-scalar
theory cannot be neglected due to the equation of motion as in the $O(N)$
case, which in turn limits the size of the gaps we can impose in this sector.

There are many three-dimensional multi-scalar models to consider. Perhaps the
most well-known one is the cubic model, which is a theory whose global
symmetry group consists of the cubic subgroup $\mathbb{Z}_{2}^{\;\,3}\rtimes
S_{3}$ of $O(3)$. Conformal field theories with cubic and the more general
hypercubic symmetry have been bootstrapped in [11, 12, 13, 14]. Many other
examples to be considered include fixed points already discovered with the
$\varepsilon$ expansion a long time ago,333See for example [15] for a
relatively recent review of known fixed points in the $\varepsilon$ expansion
below four dimensions. as well as a wealth of new $\varepsilon$ expansion
fixed points presented in [16]. With the addition of fermions there are
presumably many fixed points to study with the bootstrap, including
supersymmetric ones [17].

In this work we continue the study of scalar MN models by considering a multi-
correlator study of CFTs with $O(m)^{n}\rtimes S_{n}$ global symmetry in three
dimensions. These models have been analyzed with the $\varepsilon$ expansion
[18, 19, 20, 21, 22, 23, 15] as well as the numerical conformal bootstrap
[24], where a single-correlator study was pursued. Our goal is to find the
minimal set of conditions that allow us to isolate allowed regions in
parameter space at the positions of kinks of the single-correlator bounds
obtained in [24]. As was explored in [24] and also in [25], bounds of $m=2$
theories contain two kinks with potential applications to critical phenomena
for low $n$ and of theoretical interest at larger $n$. There is no proof that
a kink of the type we find in a numerical bootstrap bound must be due to the
presence of an actual CFT. However, we will consider such kinks to be strong
indicators for the presence of actual CFTs. For an in-depth discussion of
experimental realizations of the CFTs we will study and their significance
within field theory we refer the reader to [24].

By considering systems of correlators involving operators in multiple
irreducible representations (irreps) of the global symmetry group, we manage
to find isolated allowed regions (islands) at the positions of kinks of
single-correlator bounds. To get these islands we make assumptions on the
operator spectrum, but we try to keep the number of such assumptions to a
minimum. We find that we typically have to require the presence of a conserved
current and stress-energy tensor and gaps to the next-to-leading spin-1 and
spin-2 operators in the corresponding irreps. Also, if we wish to obtain an
island in the $\Delta_{\phi}$-$\Delta_{S}$ plane,444Where $\phi$ is the order
parameter field and $S$ the scalar singlet. we need to require that the
leading operator in a specific irrep (called $X$ below) saturates the bound
obtained from the single-correlator bootstrap. However, to obtain an island in
the $\Delta_{\phi}$-$\Delta_{X}$ plane, no such assumption is required. We
note that the $\Delta_{X}$ bound is of interest since it displays pronounced
features/kinks in parameter space.

One particularly interesting island is found for the second kink of the
$MN_{20,2}$ theory (see Fig. 11 below). This island is obtained in the
$\Delta_{\phi}\text{-}\Delta_{S}$ plane, where $\phi$ is the defining and $S$
the singlet irrep of $MN_{20,2}$. A notable feature of this island is that the
dimensions $\Delta_{\phi}$ and $\Delta_{S}$ that define it are low, while the
assumptions we make to obtain it force the dimensions of next-to-leading
operators in a variety of irreps to be high.

For the experimentally relevant $MN_{2,2}$ and $MN_{2,3}$ theories we find
islands at the first kinks of the corresponding single-correlator bootstrap
bounds. One would expect the values of $\Delta_{\phi}$ and $\Delta_{S}$ in
these islands to match the results obtained in the $\varepsilon$ expansion.
This is not the case, however, and the potential resolutions are either that
the $\varepsilon$ expansion results in the $\varepsilon\to 1$ limit are not
trustworthy or that the bootstrap kink is due to a theory distinct from that
of the $\varepsilon$ expansion. We note that at large $m$ and large $n$ the
bootstrap results we obtain for $MN_{m,n}$ theories agree very well with the
$\varepsilon$ expansion.

For various values of $m$ and $n$ we initiate a preliminary study of symmetric
tensors $Z^{ab}_{ij}$ of $MN_{m,n}$, similar in spirit to [26]. We find a
family of kinks that in the large $n$ limit converge to the point
$\Delta_{Z}=\frac{1}{2}$ and $\Delta_{Y}=2$, which hints towards a standard
Hubbard–Stratonovich description in this limit. Conversely, in the large $m$
limit, the field $Y$ does not converge to $\Delta_{Y}=2$. We delegate a
careful study of these bounds to future work.

We also work out the required tensor structures (under global symmetry) for
generic groups $G^{n}\rtimes S_{n}$ with $G$ arbitrary. There are at least two
motivations for doing so. First, there exist experimentally relevant cases
among such models. For example, beyond the cubic
($\mathbb{Z}_{2}^{\;\,3}\rtimes S_{3}$), $MN_{2,2}$ ($O(2)^{2}\rtimes S_{2}$)
and $MN_{2,3}$ ($O(2)^{3}\rtimes S_{3}$) models mentioned earlier, there are
also the so called tetragonal theories ($D_{4}{\\!}^{n}\rtimes S_{n}$, with
$D_{4}$ the dihedral group of 8 elements). An incomplete list of references
where the reader may find a number of physically motivated examples is [27,
28, 29, 30, 31]. A second, broader, motivation stems from our pursuit to
better understand the space of three-dimensional CFTs. We hope that our
results will enable a variety of future bootstrap studies.

This paper is organized as follows. In the next section we present a quick
review of MN models as understood from a Lagrangian point of view and describe
our results for global symmetries of the type $G^{n}\rtimes S_{n}$. In section
3 we present our numerical bootstrap results. We conclude in section 4. Four
technical appendices include calculations of crossing equations in a variety
of cases relevant for this work.

##  2 MN and $\boldsymbol{G^{n}\rtimes S_{n}}$ symmetries

###  2.1 Definition and Lagrangian description in $d=4-\varepsilon$

The global symmetry group $MN_{m,n}=O(m)^{n}\rtimes S_{n}$ consists of $n$
copies of the $O(m)$ model that can be permuted with each other. Consider a
scalar field multiplet $\phi^{a}_{i}$, where the upper index labels the copy
and the lower index is an $O(m)$ index. With this field we may write the
multi-scalar $MN_{{m,n}}$-invariant Lagrangian

$\mathscr{L}=\partial_{\mu}\phi^{a}_{i}\hskip
1.0pt\partial^{\mu}\phi^{a}_{i}+m^{2}\hskip
1.0pt\phi^{a}_{i}\phi^{a}_{i}+u\hskip
1.0pt(\phi^{a}_{i}\phi^{a}_{i})^{2}+v\hskip 1.0pt\delta_{abcd}\hskip
1.0pt\phi^{a}_{i}\phi^{b}_{i}\phi^{c}_{j}\phi^{d}_{j}\,,$ (2.1)

where summation over repeated indices is implicit and we keep up to quartic
terms in $\phi$. The field $\phi^{a}_{i}$ transforms in the “defining”
representation of $MN_{m,n}$. In other words, the index $a$ has its values
permuted by $S_{n}$, and $O(m)$ acts as $\phi^{a}_{i}\rightarrow
R_{ij}\phi^{a}_{j}$ with $R$ including proper and improper rotations. If one
were to set $u=0$ in (2.1), then (2.1) would be the Lagrangian of $n$
decoupled $O(m)$ models. Thus, when both $u$ and $v$ are non-zero we obtain a
theory describing $n$ coupled $O(m)$ models. Single-coupling deformations of
the $O(N)$ theory lead to a total of four fixed points [32, 23, 15], which
correspond to the free theory, $n$ coupled $O(m)$ models (i.e. the MN theory),
$n$ decoupled $O(m)$ models, and the $O(mn)$ theory.

The generalization of (2.1) to the $G^{n}\rtimes S_{n}$-symmetric case for $G$
arbitrary is straightforward:

$\mathscr{L}=\partial_{\mu}\phi^{a}_{i}\hskip
1.0pt\partial^{\mu}\phi^{a}_{i}+m^{2}\hskip
1.0pt\phi^{a}_{i}\phi^{a}_{i}+u\hskip
1.0pt(\phi^{a}_{i}\phi^{a}_{i})^{2}+\sum_{r}v_{r}\hskip
1.0pt\delta_{abcd}\hskip 1.0ptT^{r}_{ijkl}\hskip
1.0pt\phi^{a}_{i}\phi^{b}_{j}\phi^{c}_{k}\phi^{d}_{l}\,,$ (2.2)

where $r$ labels the four-index fully symmetric invariant tensors
$T^{r}_{ijkl}$ of $G$, to each of which we associate a coupling $v_{r}$. Only
fields from the same copy of $G$ interact when $u=0$. In tetragonal theories
(see e.g. [15]), where $G=D_{4}$ with $D_{4}$ the dihedral group of eight
elements, we have $r=1,2$.

A minor comment is that instead of viewing (2.1) and (2.2) as $n$ decoupled
$G$-symmetric theories which are consequently coupled via the addition of the
$u$ term, we may also view them as deformations of an $O(mn)$ symmetric theory
with a term that breaks the symmetry down to $G^{n}\rtimes S_{n}$. Both
approaches are fruitful. For example, the authors of [33] study hypercubic
theories, where $G=\mathbb{Z}_{2}$, by performing conformal perturbation
theory around the CFT of $n$ decoupled Ising models, whereas the authors of
[6] discuss the cubic theory, where $G=\mathbb{Z}_{2}$ and $n=3$, by writing
it as a deformation of $O(3)$ and then using conformal perturbation theory.
Both approaches led to different and interesting results.

###  2.2 The $\phi^{a}_{i}\times\phi^{b}_{j}$ OPE

The product of two fields transforming in the defining representation of
$G^{n}\rtimes S_{n}$ is schematically expressed as

$\phi^{a}_{i}\times\phi^{b}_{j}\sim\delta^{ab}\delta_{ij}S+\delta_{ij}X^{ab}+{I_{1}}^{ab}_{ij}+\cdots+{I_{\mathcal{N}}}^{ab}_{ij}+Z^{ab}_{ij}+B^{ab}_{ij}\,.$
(2.3)

Here the lower indices take on values $i,j=1,\ldots,m$ with $m$ the dimension
of the defining irrep of $G$ and the upper indices take on values
$a,b=1,\ldots,n$.555For every value $a_{*}$ of the upper index $a$,
$\phi_{i}^{a_{*}}$ furnishes the defining irrep of the corresponding $G$ in
$G^{n}$. The representations on the right-hand side of (2.3) are to be
understood as follows. The last two representations do not exist if $a=b$,
whereas all other representations exist only if $a=b$. The first
representation, $S$, is the singlet. The representation $X$ is a singlet of
$G^{n}$ and traceless in $S_{n}$, or to rephrase, it is the fundamental
representation of $S_{n}$. The representations $Z$ and $B$ satisfy
$Z^{ab}_{ij}=Z^{ba}_{ji}$ and $B^{ab}_{ij}=-B^{ba}_{ji}$. The remaining irreps
$I_{i},i=1,\ldots,\mathcal{N}$, correspond to non-singlet representations of
$G$. The dimensions of the irreps $(S,X,I_{1},\ldots,I_{\mathcal{N}},Z,B)$ are
$(1,n-1,n\hskip 1.0pt\dim R_{1},\ldots,n\hskip 1.0pt\dim
R_{\mathcal{N}},m^{2}\frac{n(n-1)}{2},m^{2}\frac{n(n-1)}{2})$, where
$R_{1},\ldots,R_{\mathcal{N}}$ are the non-singlet irreps of $G$ that appear
in the $\phi_{i}\times\phi_{j}$ OPE of $G$. An easy mnemonic rule for
obtaining the decomposition in (2.3) is to decompose onto irreps of $G$ when
$a=b$ and to simply symmetrize and antisymmetrize the indices when $a\neq b$.

We may decompose a four point function
$\langle\phi^{a}_{i}\phi^{b}_{j}\phi^{c}_{k}\phi^{d}_{l}\rangle$ into the
irreps that appear in (2.3). Doing this, we find tensor structures of the
global symmetry which in turn determine the sum rules to be eventually studied
numerically. Notably, these tensor structures are projectors.666Assuming they
are normalized properly, which we won’t always do. Nevertheless, by slight
abuse of terminology we will still call them projectors. The explicit
expressions are

$\displaystyle(P^{S})^{abcd}_{ijkl}$
$\displaystyle=\frac{1}{mn}\delta^{ab}\delta^{cd}\delta_{ij}\delta_{kl}\,,$
$\displaystyle(P^{X})^{abcd}_{ijkl}$
$\displaystyle=\frac{1}{m}\delta_{ij}\delta_{kl}\Big{(}\delta^{abcd}-\frac{1}{n}\delta^{ab}\delta^{cd}\Big{)}\,,$
$\displaystyle(P^{I_{1}})^{abcd}_{ijkl}$ $\displaystyle=\delta^{abcd}\hskip
1.0pt{P^{R_{1}}}_{\\!ijkl}\,,$ $\displaystyle\ldots$
$\displaystyle(P^{I_{\mathcal{N}}})^{abcd}_{ijkl}$
$\displaystyle=\delta^{abcd}\hskip 1.0pt{P^{R_{\mathcal{N}}}}_{\\!ijkl}\,,$
$\displaystyle(P^{Z})^{abcd}_{ijkl}$
$\displaystyle=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})\delta_{ik}\delta_{jl}+(\delta^{ad}\delta^{bc}-\delta^{abcd})\delta_{il}\delta_{jk})\,,$
$\displaystyle(P^{B})^{abcd}_{ijkl}$
$\displaystyle=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})\delta_{ik}\delta_{jl}-(\delta^{ad}\delta^{bc}-\delta^{abcd})\delta_{il}\delta_{jk})\,,$
(2.4)

where the tensors
$P^{R_{1}}{}_{\\!ijkl},\ldots,P^{R_{\mathcal{N}}}{}_{\\!ijkl}$ correspond to
the projectors of the irreps $R_{1},\ldots,R_{\mathcal{N}}$ of $G$. Note that
all invariant tensors are expressed in terms of (generalized) Kronecker
deltas, up to the form of the
$P^{R_{1}}{}_{\\!ijkl},\ldots,P^{R_{\mathcal{N}}}{}_{\\!ijkl}$ invariant
tensors.

We now give a set of explicit examples to clear up any confusion.

####  2.2.1 Example 1: hypercubic theories

We start with hypercubic theories, which have been bootstrapped in [11, 12,
13, 14]. In this example, $G=\mathbb{Z}_{2}$ and so the lower indices
($i,j,k,l$) may be dropped. Also, in the OPE between two operators charged
under $\mathbb{Z}_{2}$ only the singlet representation appears, thus we have
no “$I$” representations. Hence, the projectors are

$\displaystyle\begin{split}(P^{S})^{abcd}&=\frac{1}{n}\delta^{ab}\delta^{cd}\,,\\\
(P^{X})^{abcd}&=\delta^{abcd}-\frac{1}{n}\delta^{ab}\delta^{cd}\,,\\\
(P^{Z})^{abcd}&=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})+(\delta^{ad}\delta^{bc}-\delta^{abcd}))\,,\\\
(P^{B})^{abcd}&=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})-(\delta^{ad}\delta^{bc}-\delta^{abcd}))\,,\end{split}$
(2.5)

where the irreps $(S,X,Z,B)$ have dimensions
$(1,n-1,\frac{n(n-1)}{2},\frac{n(n-1)}{2})$, in agreement with the dimensions
stated above when $m=1$.

####  2.2.2 Example 2: MN theories

Next, we consider the MN theories studied in [24] and [25]. In this example we
have $G=O(m)$. Since the OPE between two vectors of $O(m)$ exchanges, beyond
the singlet, the two-index traceless symmetric $T$ and antisymmetric $A$
irreps, we have $R_{1}=T$ and $R_{2}=A$. For these irreps we know that
$P^{R_{1}}{}_{\\!ijkl}=\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{m}\delta_{ij}\delta_{kl})$
and
$P^{R_{2}}{}_{\\!ijkl}=\frac{1}{2}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$.
We thus obtain

$\displaystyle\begin{split}(P^{S})^{abcd}_{ijkl}&=\frac{1}{mn}\delta^{ab}\delta^{cd}\delta_{ij}\delta_{kl}\,,\\\
(P^{X})^{abcd}_{ijkl}&=\frac{1}{m}\delta_{ij}\delta_{kl}\Big{(}\delta^{abcd}-\frac{1}{n}\delta^{ab}\delta^{cd}\Big{)}\,,\\\
(P^{I_{1}})^{abcd}_{ijkl}&=\delta^{abcd}\hskip
1.0pt{P^{R_{1}}}_{\\!ijkl}\,,\\\
(P^{I_{2}})^{abcd}_{ijkl}&=\delta^{abcd}\hskip
1.0pt{P^{R_{2}}}_{\\!ijkl}\,,\\\
(P^{Z})^{abcd}_{ijkl}&=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})\delta_{ik}\delta_{jl}+(\delta^{ad}\delta^{bc}-\delta^{abcd})\delta_{il}\delta_{jk})\,,\\\
(P^{B})^{abcd}_{ijkl}&=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})\delta_{ik}\delta_{jl}-(\delta^{ad}\delta^{bc}-\delta^{abcd})\delta_{il}\delta_{jk})\,,\end{split}$
(2.6)

where the representations $(S,X,I_{1},I_{2},Z,B)$ have dimensions

$(1,n-1,n\frac{(m-1)(m+2)}{2},n\frac{m(m-1)}{2},m^{2}\frac{n(n-1)}{2},m^{2}\frac{n(n-1)}{2})\,,$
(2.7)

in agreement with the general formulas given earlier and the computations of
[24]. In what follows we will rename $I_{1}$ and $I_{2}$ as $Y$ and $A$ to
keep the same names as [24].

####  2.2.3 Example 3: Tetragonal theories and their generalizations

We may also study a generalization of the tetragonal theories bootstrapped in
[24]. In this example we have $G=\mathbb{Z}_{2}^{\;\,m}\rtimes S_{m}$. Thus,
similarly,

$\displaystyle\begin{split}(P^{S})^{abcd}_{ijkl}&=\frac{1}{mn}\delta^{ab}\delta^{cd}\delta_{ij}\delta_{kl}\,,\\\
(P^{X})^{abcd}_{ijkl}&=\frac{1}{m}\delta_{ij}\delta_{kl}\Big{(}\delta^{abcd}-\frac{1}{n}\delta^{ab}\delta^{cd}\Big{)}\,,\\\
(P^{I_{1}})^{abcd}_{ijkl}&=\delta^{abcd}\Big{(}\delta_{ijkl}-\frac{1}{m}\delta_{ij}\delta_{kl}\Big{)}\,,\\\
(P^{I_{2}})^{abcd}_{ijkl}&=\delta^{abcd}\hskip
1.0pt\tfrac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-2\delta_{ijkl})\,,\\\
(P^{I_{3}})^{abcd}_{ijkl}&=\delta^{abcd}\hskip
1.0pt\tfrac{1}{2}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})\,,\\\
(P^{Z})^{abcd}_{ijkl}&=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})\delta_{ik}\delta_{jl}+(\delta^{ad}\delta^{bc}-\delta^{abcd})\delta_{il}\delta_{jk})\,,\\\
(P^{B})^{abcd}_{ijkl}&=\tfrac{1}{2}((\delta^{ac}\delta^{bd}-\delta^{abcd})\delta_{ik}\delta_{jl}-(\delta^{ad}\delta^{bc}-\delta^{abcd})\delta_{il}\delta_{jk})\,,\end{split}$
(2.8)

where now the irreps $(S,X,I_{1},I_{2},I_{3},Z,B)$ have dimensions

$(1,n-1,n(m-1),n\frac{m(m-1)}{2},n\frac{m(m-1)}{2},m^{2}\frac{n(n-1)}{2},m^{2}\frac{n(n-1)}{2})\,.$
(2.9)

These indeed agree with and generalize to arbitrary $m$ the results of [24].

###  2.3 The $\phi^{a}_{i}\times X^{bc}$ OPE

In our mixed correlator system we will need to consider the OPE of $\phi$ with
$X$. A first observation is that the decomposition of this OPE does not depend
on $G$ because $X$ is a singlet of $G$. We also know that $\phi\times X$
should exchange $\phi$, as dictated by self-consistency of the three-point
function, i.e. calculating the three point function with different OPEs should
give the same result. Lastly, the existence of the defining representation
($\phi$) on the right hand side of the OPE implies the existence also of an
antisymmetric representation ($\bar{A}$) on the right-hand side, see e.g. [34,
Chapter 4.3]. If one now counts the dimensions of these two irreps ($\phi$ and
$\bar{A}$), their sum turns out equal to the product of the dimensions on the
left-hand side of the OPE. Thus, the OPE is fully decomposed. Schematically,
the decomposition looks like

$\phi^{a}_{i}X^{bc}\sim(P^{\phi})^{abcdef}\phi^{d}_{i}X^{ef}+(P^{\bar{A}})^{abcdef}\phi^{d}_{i}X^{ef}\,,$
(2.10)

where

$(P^{\phi})^{abcdef}=\delta_{abcdef}-\frac{1}{n}(\delta^{bc}\delta^{adef}+\delta^{ef}\delta^{abcd})+\frac{1}{n^{2}}\delta^{ad}\delta^{bc}\delta^{ef}$
(2.11)

and

$(P^{\bar{A}})^{abcdef}=-\delta^{abcdef}+\frac{n-1}{n}\delta^{ad}\delta^{bcef}+\frac{1}{n}(\delta^{bc}\delta^{adef}+\delta^{ef}\delta^{abcd})-\frac{1}{n}\delta^{ad}\delta^{bc}\delta^{ef}\,.$
(2.12)

It is a straightforward exercise to contract the tensors quoted above in
(2.10) and obtain the expected representation dimensions. Note that (2.11) and
(2.12) are the projectors we need in order to obtain the sum rules from the
corresponding crossing equations. An additional check one can perform is test
the above decomposition for some discrete group $G$ in which the
decompositions are known, and then remembering that the formula is
$G$-independent one has proved the result.

###  2.4 The $X^{ab}\times X^{cd}$ OPE

It is useful to remember at this point that the $X$ representation is simply
the fundamental representation of $S_{n}$, and so its OPE decomposition is
already known from e.g. [35]; see also the later works [11] and [12]. Hence,
both projectors and crossing equation sum rules are known. Note that the
fundamental of $S_{n}$ is usually written with one index. For the explicit map
between the one- and two- index notations see [34].

###  2.5 The $n\geqslant 4$ $Z^{ab}_{ij}\times Z^{cd}_{kl}$ OPE

In this work we will also consider the $\langle ZZZZ\rangle$ single correlator
bootstrap for values of $n\geqslant 4$ (and $m\geqslant 2$). This is because
we can then probe various large parameter limits, such as the
$m\rightarrow\infty$ and $n\rightarrow\infty$ limits.777For $m\geqslant 2$ and
$n\geqslant 4$ the sum rules take their most general form, valid for any such
$m$ and $n$. In Appendix Appendix B. $\boldsymbol{Z\times Z}$ single
correlator sum rules we additionally work out the $n=2$ case, which is
special. Probing such limits may give hints about the existence or not of
possible perturbative expansions of theories satisfying these single
correlator constraints. One such example could be a theory where the order
parameter field transforms in the $Z$ irrep. A similar study was performed for
traceless tensors of $O(n)$ in [26] and adjoints of $SU(n)$ in [36], where
theories with order parameter fields in these representations were analysed.
The reader may skip this subsection on a first reading since it is rather
technical, and return on a second read-through.

For $n\geqslant 4$ one finds 21 irreducible representations on the right-hand
side of the $n\geqslant 4$ $Z^{ab}_{ij}\times Z^{cd}_{kl}$ OPE.888Note that
for $n=3$ there are fewer irreps. Also, the invariant tensor
$\delta_{abcdefgh}$ becomes expressible in terms of tensors with fewer
indices, see e.g. [13]. We omit an analysis of the $n=3$ case in the present
work since we are interested in the large parameter limits. We present the
projectors of the 21 irreps in Appendix Appendix C. $\boldsymbol{\langle
ZZZZ\rangle}$ projectors for $\boldsymbol{n\geqslant 4}$ and $\boldsymbol{G}$
arbitrary, where we also explain the generalization to arbitrary group $G$.
When all indices $a$, $b$, $c$ and $d$ are different, we have three exchanged
irreps. It is convenient to define
$T^{ab,cd}_{ij,kl}=Z^{ab}_{ij}Z^{cd}_{kl}+Z^{cd}_{kl}Z^{ab}_{ij}$ and
$\bar{T}^{ab,cd}_{ij,kl}=Z^{ab}_{ij}Z^{cd}_{kl}-Z^{cd}_{kl}Z^{ab}_{ij}$, which
obviously satisfy $T^{ab,cd}_{ij,kl}=T^{cd,ab}_{kl,ij}$ and
$\bar{T}^{ab,cd}_{ij,kl}=-\bar{T}^{cd,ab}_{kl,ij}$. With this in mind, we have

$Z^{ab}_{ij}\times Z^{cd}_{kl}\sim
T^{ab,cd}_{ij,kl}+\bar{T}^{ab,cd}_{ij,kl}\,.$ (2.13)

The right-hand side may now be further decomposed by symmetrizing and
antisymmetrizing indices,

$\displaystyle\begin{split}Z^{ab}_{ij}\times
Z^{cd}_{kl}&\sim(T^{ab,cd}_{ij,kl}+T^{ac,bd}_{ik,jl}+T^{ad,bc}_{il,jk})+(T^{ab,cd}_{ij,kl}-T^{ac,bd}_{ik,jl})+(T^{ab,cd}_{ij,kl}-T^{ad,bc}_{il,jk})\\\
&\quad+(\bar{T}^{ab,cd}_{ij,kl}+\bar{T}^{ac,bd}_{ik,jl})+(\bar{T}^{ab,cd}_{ij,kl}+\bar{T}^{ad,bc}_{il,jk})+(\bar{T}^{ab,cd}_{ij,kl}-\bar{T}^{ac,bd}_{ik,jl})+(\bar{T}^{ab,cd}_{ij,kl}-\bar{T}^{ad,bc}_{il,jk})\,,\end{split}$
(2.14)

which may also be rewritten as

$\displaystyle\begin{split}Z^{ab}_{ij}\times Z^{cd}_{kl}&\sim To\hskip
0.3pttS^{abcd}_{ijkl}+BB^{ad,bc}_{il,jk}+BB^{ac,bd}_{ik,jl}\\\
&\quad+BZ^{ad,bc}_{il,jk}+BZ^{ac,bd}_{ik,jl}+ZB^{ad,bc}_{il,jk}+ZB^{ac,bd}_{ik,jl}\,,\end{split}$
(2.15)

where $To\hskip 0.3pttS$ is totally symmetric in all pairs of
indices,999Remember that permuting $a$ with $b$ means we must permute $i$ with
$j$, hence we do not permute indices, but pairs of them. whereas for $BB$ and
$BZ$ we have the relations
$BB^{ac,bd}_{ik,jl}=-BB^{ca,bd}_{ki,jl}=-BB^{ac,db}_{ik,lj}=BB^{bd,ac}_{jl,ik}$
and
$BZ^{ac,bd}_{ik,jl}=-BZ^{ca,bd}_{ki,jl}=+BZ^{ac,db}_{ik,lj}=-BZ^{bd,ac}_{jl,ik}$.
Lastly, notice that $BZ$ and $ZB$ are the same representation since
$BZ^{ab,cd}_{ij,kl}=ZB^{cd,ab}_{kl,ij}$.

Next we must consider the representations where one or more of the copy (i.e.
upper) indices are equal in $Z^{ab}_{ij}\times Z^{cd}_{kl}$. These are rather
simple: if a pair of copy indices are equal we decompose onto irreps of that
copy of $G$, whereas if two copy indices are different we simply symmetrize
and antisymmetrize. Let us give some examples. Consider $a=c$ and $b\neq d$.
We have

$\displaystyle Z^{ab}_{ij}\times
Z^{cd}_{kl}\sim(Z^{ab}_{ij}Z^{cd}_{kl}+Z^{ad}_{il}Z^{cb}_{kj})+(Z^{ab}_{ij}Z^{cd}_{kl}-Z^{ad}_{il}Z^{cb}_{kj})\sim
RZ^{ac,bd}_{ik,jl}+RB^{ac,bd}_{ik,jl}\,,$ (2.16)

where the part $R$ simply signifies that $a=c$ but is not an irrep. To
decompose $R$ onto irreps we must decompose the indices $i$ and $k$ onto $G$
irreps and then subtract the trace of copies from the singlet of $G$:

$RZ^{ac,bd}_{ik,jl}\sim\Big{(}RZ^{ac,bd}_{ik,jl}+RZ^{ac,bd}_{ki,jl}-\frac{2}{m}\delta_{ik}RZ^{ac,bd}_{mm,jl}\Big{)}+(RZ^{ac,bd}_{ik,jl}-RZ^{ac,bd}_{ki,jl})+\frac{2}{m}\delta_{ik}RZ^{ac,bd}_{mm,jl}\,,$
(2.17)

and

$\displaystyle\begin{split}\delta_{ik}RZ^{ac,bd}_{mm,jl}\sim\delta_{ik}\Big{(}RZ^{ac,bd}_{mm,jl}-\frac{1}{n-2}\delta_{ac}RZ^{ee,bd}_{mm,jl}\Big{)}+\frac{1}{n-2}\delta_{ik}\delta_{ac}RZ^{ee,bd}_{mm,jl}\sim
XZ^{ac,bd}_{ik,jl}+SZ^{ac,bd}_{ik,jl}\,,\end{split}$ (2.18)

where we have absorbed factors of Kronecker deltas in the representations in
order to simplify the notation. Combining all this information together we
find

$\displaystyle\begin{split}RZ^{ac,bd}_{ik,jl}&\sim
YZ^{ac,bd}_{ik,jl}+AZ^{ac,bd}_{ik,jl}+XZ^{ac,bd}_{ik,jl}+SZ^{ac,bd}_{ik,jl}\,,\end{split}$
(2.19)

which we remind the reader is for $a=c$ and $b\neq d$. For the full expression
one needs to consider all possible combinations.

The last case we must consider is when we have pairs of copy indices equal,
e.g. $a=c$ and $b=d$. We must again decompose onto irreps of $G$. Let us start
by first decomposing $j$ and $l$ onto $G$ irreps:

$Z^{ab}_{ij}\times Z^{cd}_{kl}\sim
RR^{ac,bd}_{ik,jl}\sim\Big{(}RR^{ac,bd}_{ik,jl}+RR^{ac,bd}_{ik,lj}-\frac{2}{m}\delta_{jl}RR^{ac,bd}_{ik,mm}\Big{)}+(RR^{ac,bd}_{ik,jl}-RR^{ac,bd}_{ik,lj})+\frac{2}{m}\delta_{jl}RR^{ac,bd}_{ik,mm}\,.$
(2.20)

This is rewritten as

$Z^{ab}_{ij}\times Z^{cd}_{kl}\sim
RY^{ac,bd}_{ik,jl}+RA^{ac,bd}_{ik,jl}+\frac{2}{m}\delta_{jl}RR^{ac,bd}_{ik,mm}\,,$
(2.21)

where we use $RR^{ac,bd}_{ik,jl}$ to denote an object that is not an irrep but
has $a=c$ and $b=d$. We will now focus on the last term, which is the most
complicated. We decompose the $i$ and $k$ indices onto $G$ irreps,

$RR^{ac,bd}_{ik,mm}\sim\Big{(}RR^{ac,bd}_{ik,mm}+RR^{ac,bd}_{ki,mm}-\frac{2}{m}\delta_{ik}RR^{ac,bd}_{nn,mm}\Big{)}+(RR^{ac,bd}_{ik,mm}-RR^{ac,bd}_{ki,mm})+\frac{2}{m}\delta_{ik}RR^{ac,bd}_{nn,mm}\,,$
(2.22)

which is equivalent to

$RR^{ac,bd}_{ik,mm}\sim
YR^{ac,bd}_{ik,mm}+AR^{ac,bd}_{ik,mm}+\frac{2}{m}\delta_{ik}RR^{ac,bd}_{nn,mm}\,.$
(2.23)

The last term is again the most complicated.101010Since, for example, we know
$YR^{ac,bd}_{ik,mm}\sim
YR^{ac,bd}_{ik,mm}-\delta^{bd}YR^{ac,ee}_{ik,mm}/(n-1)+\delta^{bd}YR^{ac,ee}_{ik,mm}/(n-1)$
or stated in terms of the irreps $YR^{ac,bd}_{ik,mm}\sim
YX^{ac,bd}_{ik,jl}+YS^{ac,bd}_{ik,jl}$ where again we have absorbed Kronecker
deltas and numerical factors in the definitions of the irreps for simplicity.
Defining $T^{ac,bd}=RR^{ac,bd}_{nn,mm}$, we subtract all possible traces to
construct irreducible objects:

$\displaystyle\begin{split}T^{ac,bd}&\sim
T^{ac,bd}-\frac{1}{n-2}(\delta_{bd}T^{ac,ee}+\delta_{ac}T^{ee,bd})+\frac{\delta_{ac}\delta_{bd}}{(n-1)(n-2)}T^{ee,ff}\\\
&\quad+\frac{\delta_{bd}}{n-2}\Big{(}T^{ac,ee}-\frac{\delta_{ac}}{n}T^{ee,ff}\Big{)}+\frac{\delta_{ac}}{n-2}\Big{(}T^{ee,bd}-\frac{\delta_{bd}}{n}T^{ee,ff}\Big{)}\\\
&\quad-\delta_{ac}\delta_{bd}\Big{(}\frac{1}{(n-1)(n-2)}-\frac{2}{n(n-2)}\Big{)}T^{ee,ff}\\\
&\sim\bar{X}^{ac,bd}+XS^{ac,bd}+SX^{ac,bd}+\delta_{ac}\delta_{bd}S\,,\end{split}$
(2.24)

where again we have absorbed various factors in the definitions of the irreps.
The other irreps follow rather straightforwardly, hence we omit a detailed
analysis. In conclusion we find the 21 irreps ($S$, $XS$, $\bar{X}$, $XY$,
$XA$, $SY$, $SA$, $YY$, $AA$, $YA$, $BB$, $To\hskip 0.3pttS$, $XB$, $XZ$,
$SB$, $SZ$, $YB$, $YZ$, $AB$, $AZ$, $BZ$) with respective dimensions ($S:1$,
$XS:n-1$, $\bar{X}:n(n-3)/2$, $XY:n(n-2)(m-1)(m+2)/2$, $XA:n(n-2)(m-1)m/2$,
$SY:n(m-1)(m+2)/2$, $SA:n(m-1)m/2$, $YY:n(n-1)(m+2)^{2}(m-1)^{2}/8$,
$AA:n(n-1)(m-1)^{2}m^{2}/8$, $YA:n(n-1)(m-1)^{2}m(m+2)/4$,
$BB:2m^{4}n(n-1)(n-2)(n-3)/(4!)$, $To\hskip
0.3pttS:m^{4}n(n-1)(n-2)(n-3)/(4!)$, $XB:(n-3)m^{2}n(n-1)/2$,
$XZ:(n-3)m^{2}n(n-1)/2$, $SB:m^{2}n(n-1)/2$, $SZ:m^{2}n(n-1)/2$,
$YB:(m+2)(m-1)m^{2}n^{2}(n-1)/4$, $YZ:(m+2)(m-1)m^{2}n^{2}(n-1)/4$,
$AB:(m-1)m^{3}n^{2}(n-1)/4$, $AZ:(m-1)m^{3}n^{2}(n-1)/4$,
$BZ:3m^{4}n(n-1)(n-2)(n-3)/(4!)$).

##  3 Numerical results

With the group theory outlined above and the sum rules given in the appendices
we are ready to start obtaining numerical results. We start by probing the
system of four point correlators obtained by considering all combinations of
external operators $\phi^{a}_{i}$ and $X^{bc}$. We then conclude the chapter
by presenting bounds pertaining to four point functions of symmetric tensors
$Z^{ab}_{ij}$.

###  3.1 Islands close to the unitarity bound

In this section we study the portion of parameter space “close” to the
unitarity bound. By this we refer to values of the order parameter scaling
dimensions roughly around $0.5\text{-}0.53$; these correspond to typical
values found in fixed points of multi-scalar theories. However, this does not
necessarily mean that the theories we find are due to the fixed point of some
multi-scalar Lagrangian description. For $n$ sufficiently large we find kinks,
and corresponding islands, that converge to the expected values, that is
$\Delta_{\phi}=\Delta_{\phi}^{O(m)}$ and
$\Delta_{X}=\Delta_{S}^{O(m)}$.111111Whereas for the scalar singlet one has
and $\Delta_{S}=d-\Delta_{S}^{O(m)}$, although we do not present results for
this in the present work. Here $\Delta_{\phi}^{O(m)}$ is the scaling dimension
of the order parameter field and $\Delta_{S}^{O(m)}$ is the scaling dimension
of the lowest lying scalar singlet, both in the theory of $n$ decoupled $O(m)$
models. For smaller values of $n$, which also correspond to the
phenomenologically interesting values, it is not clear whether the kinks (and
their islands) correspond to the ordinary $d=4-\varepsilon$ fixed points or
are some new hitherto unknown CFTs.

We separate our results into two groups depending on how we obtain the
islands. In the case where the islands are found in the
$\Delta_{\phi}$-$\Delta_{X}$ plane of parameter space, we impose gaps on
certain sectors mainly guided by the extremal functional method [37,
38].121212One can also try to justify the gaps from the large $n$ point of
view. For example, we know that the leading $X^{ab}$ operator has scaling
dimension $\Delta_{X}=\Delta_{S}^{O(m)}+O(1/n)$, so we expect the subleading
operator to have dimension either
$\Delta_{X^{\prime}}=\Delta_{S^{\prime}}^{O(m)}+O(1/n)$ or
$\Delta_{X^{\prime}}=2\Delta_{S}^{O(m)}+O(1/n)$ or
$\Delta_{X^{\prime}}=\Delta_{S}^{O(m)}+2+O(1/n)$ (remember
$X^{ab}\sim(\delta^{abcd}-\frac{1}{n}\delta^{ab}\delta^{cd})\phi^{c}_{i}\phi^{d}_{i}$
in the weakly coupled limit). All these satisfy $\Delta_{X^{\prime}}>3$ at
$n\rightarrow\infty$. The main issue with this line of reasoning though is
that we do not have the explicit corrections to subleading order in $1/n$.
Since these gaps are motivated by the extremal functional, their choice is not
rigorous. Nevertheless, we believe that even in absence of rigor our islands
are sufficient to motivate that something particularly interesting takes place
in the corresponding region of parameter space. We also present islands in the
$\Delta_{\phi}$-$\Delta_{S}$ plane, specifically in the phenomenologically
interesting cases $O(2)^{2}\rtimes S_{2}$ and $O(2)^{3}\rtimes S_{3}$. For
these cases, in addition to assumptions described above, we demand the
saturation of certain exclusion bounds. More specifically, we demand that the
exclusion bounds in the $X$ sector131313These are the exclusion plots that
display kinks/changes of slope. are saturated.

Obtaining islands by demanding saturation of exclusion bounds (in our case the
$X$ bound) is somewhat morally similar to the extremal functional method where
one may again demand saturation of a bound to then find an approximate
spectrum that corresponds to it. However, we believe that our method minimizes
the probability that a zero of the extremal function is spurious. This is
because in our approach we do not try to find approximate zeroes of the
functional, but instead show that the positions of these zeroes cannot be
excluded by the bootstrap algorithm (whereas the parameter space surrounding
them can be). Also, in conjunction with additional assumptions we can provide
a minimum and maximum value for $\Delta_{\phi}$. Lastly, we note that we have
found these islands to depend smoothly on the precise position of the $X$
bound. In other words, if $\Delta_{X}$ were to change a little there would be
no major changes in the corresponding $\Delta_{\phi}$-$\Delta_{S}$ plane
island. We confirmed this behavior while working on [14], albeit for a
slightly different symmetry, namely $\mathbb{Z}_{2}^{\;\,n}\rtimes S_{n}$.
Another way to think of our method for obtaining islands in the
$\Delta_{\phi}$-$\Delta_{S}$ plane, is that we apply the usual bootstrap
algorithm, but to a specific slice of parameter space. This slice is precisely
the one that maximizes the scaling dimension $\Delta_{X}$.

We note that for our numerical calculations in the present work we use qboot
[39]141414We also encourage the reader to see [40], which automates the
derivation of sum rules, although we did not use it for the present paper
since our groups of interest were not supported. and SDPB [41, 42]. Some of
the spectrum calculations were performed with PyCFTBoot [43].

####  3.1.1 Islands in the $\Delta_{\phi}\text{-}\Delta_{X}$ plane of
parameter space

In Figs. 2 and 4 we plot the islands corresponding respectively to the kinks
in Figs. 1 and 3. Note that we do not present an island in the
$\Delta_{\phi}$-$\Delta_{X}$ plane for $n=2$, even though it is
phenomenologically interesting. This is because up to $\Lambda=60$ in qboot
the allowed region obtained is very large. For Fig. 2 and Fig. 1 we see good
agreement with the perturbative expectation.151515SRK thanks Bernardo Zan for
pointing out aspects of the large $n$ description. That is, as
$n\rightarrow\infty$ we expect $\Delta_{X}\rightarrow{\Delta_{S}}^{O(m)}$,
hence for Fig. 2 in particular we have
$\Delta_{X}\sim{\Delta_{S}}^{O(2)}=1.50946(22)$ and
$\Delta_{\phi}\sim{\Delta_{\phi}}^{O(2)}=0.519050(40)$ [5]. The interested
reader is referred to [44] for the large $n$ limit in the case
$G=\mathbb{Z}_{2}$ or to [45] for a more recent reference. For Figs. 4 and 3
the situation is more complicated. We cannot make any conclusive statements
with regards to the theory captured by this island.

Fig. 1: Single correlator $MN_{2,100}$ exclusion bound. The cross is the
$O(2)$ model [5]. The line corresponds to the maximum allowed scaling
dimension of the first scalar $X$ operator as a function of the scaling
dimension of the order parameter operator. In qboot we used $\Lambda=45$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=20$. Fig. 2: $MN_{2,100}$ island corresponding to the kink in
Fig. 1 obtained using the mixed $\phi\text{-}X$ system of correlators. The
cross is the $O(2)$ model [5]. The island assumes that the second scalar $X$
and spin-1 $A$ operators have scaling dimensions that satisfy $\Delta\geqslant
3.0$, and the first spin-$2$ singlet after the stress tensor has a dimension
that satisfies $\Delta\geqslant 4.0$. The first spin-1 $A$ operator satisfies
$\Delta_{A}=2.0$ since it is the conserved vector of $O(2)$, and the first
spin-2 $S$ operator satisfies $\Delta_{T_{\mu\nu}}=3.0$. We also fixed the
ratio of the OPE coefficients with which the stress tensor appears,
$\lambda_{\phi\phi
T_{\mu\nu}}/\lambda_{XXT_{\mu\nu}}=\Delta_{\phi}/\Delta_{X}$. Lastly, we
imposed $\Delta_{\phi^{\prime}}\geqslant 1.0$ for the second operator in the
defining representation. In qboot we used $\Lambda=27$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$. Fig. 3: Single-correlator $MN_{2,3}$ exclusion bound. The
line corresponds to the maximum allowed scaling dimension of the first scalar
$X$ operator as a function of the scaling dimension of the order parameter
operator. In qboot we used $\Lambda=45$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=20$. Fig. 4: $MN_{2,3}$ island corresponding to the kink in Fig.
3 obtained using the mixed $\phi$-$X$ system of correlators. The island
assumes that the second scalar $X$ and spin-1 $A$ operators have scaling
dimensions that satisfy $\Delta\geqslant 3.0$ and the first spin-$2$ singlet
after the stress tensor has a dimension that satisfies $\Delta\geqslant 4.0$.
The first spin-1 $A$ operator satisfies $\Delta_{A}=2.0$ since it is the
conserved vector of $O(2)$, and the first spin-$2$ $S$ operator satisfies
$\Delta_{T_{\mu\nu}}=3.0$. We also fixed the ratio of the OPE coefficients
with which the stress tensor appears, $\lambda_{\phi\phi
T_{\mu\nu}}/\lambda_{XXT_{\mu\nu}}=\Delta_{\phi}/\Delta_{X}$. Lastly, we
imposed $\Delta_{\phi^{\prime}}\geqslant 1.0$ for the next-to-leading operator
in the defining representation. To obtain this figure with qboot [39] we used
$\Lambda=27$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$.

####  3.1.2 Islands in the $\Delta_{\phi}\text{-}\Delta_{S}$ plane of
parameter space

The motivation behind studying the islands in the $\Delta_{\phi}$-$\Delta_{S}$
plane of parameter space is twofold. On the one hand, the
$\Delta_{\phi}$-$\Delta_{S}$ plane of parameter space is the most immediately
relevant one in terms of critical exponents since $\nu=1/(3-\Delta_{S})$ and
$\beta=\Delta_{\phi}/(3-\Delta_{S})$. On the other, this way (i.e. by
demanding saturation of the $X$ sector exclusion bound) we can study
specifically the theory that creates the kink by excluding the rest of
parameter space. We do again emphasize that this approach is not rigorous in
the usual sense of the term as used in the bootstrap.

With these considerations in mind, in Figs. 5 and 9 we obtain islands
pertaining to the theories saturating the corresponding $MN_{{2,3}}$ and
$MN_{{2,2}}$ $X$ sector bounds. Note that in Fig. 6 we present again the
$MN_{2,3}$ island but with much larger gaps in order to demonstrate that as
our gaps approach the values predicted by the extremal functional method the
islands become more smooth. Lastly, in Fig. 7 we explicitly show the overlap
of these two figures.

Additionally, note that the rightmost tip of the left blue island in Fig. 9
extends to the $\Delta_{\phi}$ of the kink of Fig. 8. Thus, the putative
theory that lives at that kink is allowed under the assumptions mentioned in
the caption of Fig. 9. Given that this is a marginal case, however, further
numerical work with stronger numerics and more refined methods is required to
provide further clarity.

Fig. 5: $MN_{2,3}$ island corresponding to the theory saturating Fig. 3
obtained using the mixed $\phi\text{-}X$ system of correlators. To obtain the
island we imposed $\Delta_{S^{\prime}}\geqslant 3.0$, $\Delta_{A}=2.0$ ($O(2)$
conserved vector), $\Delta_{A^{\prime}}\geqslant 3.0$,
$\Delta_{X^{\prime}}\geqslant 3.0$ and $\Delta_{\phi^{\prime}}\geqslant 1.0$.
Lastly, we assumed that the first scalar $X$ operator saturates the bound of
Fig. 3. To obtain this figure with qboot we used $\Lambda=27$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$. Fig. 6: $MN_{2,3}$ island corresponding to the theory
saturating Fig. 3 obtained using the mixed $\phi\text{-}X$ system of
correlators. To obtain the island we imposed $\Delta_{S^{\prime}}\geqslant
3.5$, $\Delta_{A}=2.0$ ($O(2)$ conserved vector),
$\Delta_{A^{\prime}}\geqslant 3.9$, $\Delta_{X^{\prime}}\geqslant 3.0$ and
$\Delta_{\phi^{\prime}}\geqslant 1.5$. Lastly, we assumed that the first $X$
operator saturates the bound of Fig. 3. To obtain this figure with qboot we
used $\Lambda=20$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$. Fig. 7: $MN_{2,3}$ islands of the last two figures, namely
Fig. 5 and Fig. 6, in a combined plot. We emphasize that the blue island is
computed at a lower $\Lambda$, and that is why it appears larger on the left
despite the stronger spectrum assumptions used to obtain it compared to the
green island. The stronger spectrum assumptions have an effect on the right
part of the island. Fig. 8: Mixed correlator ($\phi\text{-}X$) $MN_{2,2}$
exclusion bound. The line corresponds to the maximum allowed scaling dimension
of the first scalar $X$ operator as a function of the scaling dimension of the
order parameter operator. Here we used the mixed- instead of the single-
correlator system, since in this particular case the sum rules are simpler and
hence numerically cheaper, than the $n\geqslant 3$ cases. To obtain this
figure with qboot we used $\Lambda=45$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=20$.  Fig. 9: $MN_{2,2}$ island corresponding to the theory
saturating Fig. 8 obtained using the mixed $\phi$-$X$ system of correlators.
To obtain the island we imposed $\Delta_{S^{\prime}}\geqslant 3.0$,
$\Delta_{A}=2.0$ ($O(2)$ conserved vector), $\Delta_{A^{\prime}}\geqslant
3.0$, $\Delta_{X^{\prime}}\geqslant 3.0$ and $\Delta_{\phi^{\prime}}\geqslant
1.0$. Lastly, we assumed that the first scalar $X$ operator saturates the
bound of Fig. 8. In qboot we used $\Lambda=35$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$. We also display (in blue) two narrow islands that correspond
to the allowed region that remains if we choose to saturate the $\Delta_{X}$
bound at $\Lambda=35$ instead of $\Lambda=45$ (Fig. 8 is obtained at
$\Lambda=45$).

###  3.2 Islands further away from the unitarity bound

In [24] more kinks further away from the unitarity bound were observed, such
as the one in Fig. 10. These were subsequently studied in [25]. In this work
we probe them using two different strategies. The first strategy is to
bootstrap them directly, i.e. to demand that the exclusion plot with the kink
is saturated and impose assumptions inspired by the extremal functional method
to obtain an isolated allowed region. The second strategy is to try and study
the kinks indirectly, based on the observation made in [25] that the spectrum
at these kinks contained operators in other representations very close to the
unitarity bound, namely fields in the $Y$ and $Z$ irreps. Hence, since the
numerical bootstrap tends to be stronger closer to the unitarity bound, we can
bootstrap these fields instead of the initial $\phi$ field we were studying.
We only give a brief example of this strategy in this work, and leave a more
complete treatment to future work.

An interesting observation about the second kinks is that, to our knowledge,
they have appeared in all works studying scalars involving the group $S_{n}$.
More specifically they have appeared in [11] ($S_{n}\times\mathbb{Z}_{2}$),
[34] ($\mathbb{Z}_{2}^{\;\,n}\rtimes S_{n}$) and [25] ($O(m)^{n}\rtimes
S_{n}$). Notably, they always appear in what we call the $X$ sector in this
work. Second kinks have also appeared in other theories [46, 47, 48, 26, 49,
36].

Note that in Fig. 11 we were still able to obtain an island even if we imposed
$\Delta_{S^{\prime}}\geqslant 5.0$. This is not allowed, though, for unitary
CFTs in 3D, as can be seen from [50, Fig. 3].161616We thank Alessandro Vichi
for bringing this to our attention In other words, the bootstrap seems to be
completely insensitive to that operator. This is not the first time we observe
this phenomenon. For example when working on [51], where we had concrete large
$n$ predictions to compare to, the bootstrap seemed to be completely
insensitive to the second scalar singlet at the antichiral fixed point for
$m=2$ and $n=10$, even though all other CFT data agreed exceptionally well
with the perturbative predictions. We believe that sensitivity to the second
scalar singlet should be restored once $S$ is included as an external
operator.

Fig. 10: Single-correlator $MN_{20,2}$ exclusion bound. The line corresponds
to the maximum allowed scaling dimension of the first scalar $X$ operator as a
function of the scaling dimension of the order parameter operator. In qboot we
used $\Lambda=45$,
$\ell=\\{0,\ldots,60,63,64,66,67,73,74,77,78,81,82,85,86,89,90,93,94,97,98\\}$
and $\nu_{\max}=26$. Fig. 11: $MN_{20,2}$ island in the $\phi$-$S$ plane of
parameter space. To obtain this figure we assumed that the first spin-two
singlet is the stress-energy tensor and that $X$ saturates the bound of Fig.
10. We further imposed $\Delta_{S^{\prime}}\geqslant 3.0$,
$\Delta_{T_{\mu\nu}^{\prime}}\geqslant 6.0$ (next-to-leading spin-two
singlet), $\Delta_{X^{\prime}}\geqslant 10.0$,
$\Delta_{\phi^{\prime}}\geqslant 1.0$. In qboot we used $\Lambda=35$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$.

###  3.3 The $n=2$ $Z^{ab}_{ij}\times Z^{cd}_{kl}$ single correlator bootstrap

In Fig. 12 we display the bound for the maximum allowed scaling dimension for
the first operator in the $SY$ representation as a function of $\Delta_{Z}$.
For this plot we use the sum rules from Appendix Appendix B.
$\boldsymbol{Z\times Z}$ single correlator sum rules. There seems to be a kink
roughly around the position of the second $SY$ operator and first $Z$ operator
of the corresponding $MN_{{100,2}}$ theory studied in [25]. Beyond this
similarity at the level of certain scaling dimensions we do not know if this
kink is due to the same theory or some other theory. This is in part due to
the fact that [25, Fig. 4] converges very slowly with $\Lambda$.

Fig. 12: Single correlator $MN_{100,2}$ exclusion bound using the $\langle
ZZZZ\rangle$ crossing equation. The line corresponds to the maximum allowed
scaling dimension of the first $SY$ (remember $SY=Y$) operator as a function
of the scaling dimension of the order parameter operator. To obtain this
figure with qboot we used $\Lambda=20$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$.

###  3.4 The $n\geqslant 4$ $Z^{ab}_{ij}\times Z^{cd}_{kl}$ single correlator
bootstrap

With the expressions for the sum rules of a four-point function of
$Z^{ab}_{ij}$ operators worked out for generic $m\geqslant 2$ and $n\geqslant
4$, it is interesting to study the behavior of the bounds in various parameter
limits. To this end, in Fig. 13 we probe the large $m$ limit, whereas in Fig.
14 we probe the large $n$ limit. We find that the most interesting exclusion
bound is the one in the $SY$ sector, which displays a sharp kink for all
values of $m$ and $n$ tested. In the large $n$ limit the kink converges to the
point $(\Delta_{SY}=2$, $\Delta_{\phi}=1$) which hints at a
“standard”171717For the “non-standard” large $n$ description see the
discussion in section 3.1.1 large $n$ Hubbard–Stratonovich description. On the
other hand, it is not clear what the theory converges to in the large $m$
limit.

Another interesting thing to look at is the exclusion bound for the first
operator transforming in the $BB$ representation. Representations that are
antisymmetric in two pairs of indices tend to require higher powers of the
field or derivatives in order to be written down and not vanish
identically;181818We thank Alessandro Vichi for pointing us in this direction.
see related discussions in [49] and [36]. We note that this is true for
theories where the fields may be written as polynomials of other fields, i.e.
which have a weakly coupled description. However, we do expect intuitively
that some qualitative features may carry over to the strong coupling limit.
This is indeed what we observe in Fig. 15 where we find a kink that has (very
roughly) $\Delta_{BB}\sim 2\Delta_{Z}+2\sim 4\Delta_{Z}$.

Let us note that we have checked that our bounds do not change if we assume
that the external $Z$ operator appears in its OPE with itself. We checked this
by looking at the $\Delta_{SZ}\sim\Delta_{Z}$ exclusion bound, for e.g. $m=5$
and $n=5$ and did not see any difference in our bounds (up to a vertical
precision of $10^{-6}$ that we checked). More explicitly, we checked that if
$Z$ is exchanged in the $Z\times Z$ OPE, then the exclusion bound on the
second exchanged operator in this representation is identical to the exclusion
bound of the first exchanged operator in the $Z$ irrep if one assumes that the
external operator is not exchanged. Additionally, we checked that for e.g.
$m=1000$ and $n=4$ the corresponding $\Delta_{SY}$ exclusion bound in Fig. 13
remains unchanged even after adding the assumption $\Delta_{SZ}\geqslant 1.0$.
The question of whether or not $Z$ should appear in its OPE with itself is
important, since if it does not, that would exclude a cubic (in powers of the
order parameter field) term in a possible Hamiltonian/Lagrangian description
of the theory. This is because in this case the $Z$ field would transform
under an additional $\mathbb{Z}_{2}$ symmetry.

Fig. 13: Single correlator MN exclusion bound using the $n\geqslant 4$
$\langle ZZZZ\rangle$ crossing equation. We display the behavior for $n=4$
fixed and for increasing $m$ ($m=10$ green, $m=100$ red and $m=1000$ blue).
The line corresponds to the maximum allowed scaling dimension of the first
$SY$ (remember $SY=Y$) operator as a function of the scaling dimension of the
order parameter operator. To obtain this figure with qboot we used
$\Lambda=15$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$. Fig. 14: Single correlator MN exclusion bound using the
$n\geqslant 4$ $\langle ZZZZ\rangle$ crossing equation. We display the
behavior for $m=10$ fixed and for $n=4$ (red) and $n=10$ (blue). The line
corresponds to the maximum allowed scaling dimension of the first $SY$
(remember $SY=Y$) operator as a function of the scaling dimension of the order
parameter operator. To obtain this figure with qboot we used $\Lambda=15$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$. Fig. 15: Single correlator MN exclusion bound using the
$n=10$ and $m=2$ $\langle ZZZZ\rangle$ crossing equation. The line corresponds
to the maximum allowed scaling dimension of the first $BB$ operator in the
$Z\times Z$ OPE. To obtain this figure with qboot we used $\Lambda=15$,
$\ell=\\{0,\ldots,50,55,56,59,60,64,65,69,70,74,75,79,80,84,85,89,90\\}$ and
$\nu_{\max}=25$.

##  4 Summary and conclusion

We have studied kinks and islands that arise when three-dimensional CFTs with
$O(m)^{n}\rtimes S_{n}$ global symmetry are analyzed with the numerical
bootstrap both “close” and “far” from the unitarity bound. With regards to the
analysis close to the unitarity bound, we found that in the large number of
copies limit the kinks converge to the expected position from perturbative
calculations. However, when we examine theories with a small number of copies
we do not agree with the $\varepsilon$ expansion predictions. This has two
interpretations. The first interpretation is that the theories we find are
those of the $\varepsilon$ expansion, but for some reason the actual
perturbative predictions are very inaccurate. The second is that the kinks are
due to another theory that converges to the same point when the number of
copies is taken to be large. Note that this is precisely the same qualitative
behavior observed in the case of hypercubic theories.

In addition to the above, we have worked out the tensor structures for
problems including groups of the form $G^{n}\rtimes S_{n}$ with $G$ and $n$
arbitrary. To this end, as an application of these results we bootstrapped the
four-point function of symmetric tensors $Z^{ab}_{ij}$ of $O(m)^{n}\rtimes
S_{n}$. Although we found various interesting features, we left a more
detailed analysis of these theories for future work.

There are various future directions that stem from the present work. One is to
bootstrap a mixed system of correlators involving $\phi^{a}_{i}$ and
$Z^{bc}_{jk}$, this would give us a better handle on theories like the ones in
Figs. 11 and 12. Also, having worked out the tensor structures for
$G^{n}\rtimes S_{n}$ it would be interesting to bootstrap the theories
consisting of $n$ copies of the $m$ state Potts models and see if we can find
any interesting features; see e.g. [52].

Acknowledgments

We thank Mocho Go for assistance regarding the implementation of OPE
coefficient relations and definitions in qboot. We are grateful to Alessandro
Vichi for useful discussions and suggestions and to Johan Henriksson for
reading through the manuscript and providing useful comments. Additionally, we
thank two anonymous referees for their careful reading and constructive
comments that helped improve this manuscript. Numerical computations in this
paper were run on the Crete Center for Quantum Complexity and Nanotechnology.
This research used resources provided by the Los Alamos National Laboratory
Institutional Computing Program, which is supported by the U.S. Department of
Energy National Nuclear Security Administration under Contract No.
89233218CNA000001. This research used resources of the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.

The research work of SRK was supported by the Hellenic Foundation for Research
and Innovation (HFRI) under the HFRI PhD Fellowship grant (Fellowship Number:
1026). The research work of SRK also received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement no. 758903).

Research presented in this article was supported by the Laboratory Directed
Research and Development program of Los Alamos National Laboratory under
project number 20180709PRD1. AS is funded by the Royal Society under the grant
“Advancing the Conformal Bootstrap Program in Three and Four Dimensions”.

## Appendix A. $\boldsymbol{\phi\text{-}X}$ system sum rules

In this appendix we collect the sum rules that result from each crossing
equation in the system of $\phi$-$X$ mixed correlator equations. We define
$F_{\pm,\Delta,l}^{ij,kl}$ as in [53] and we also use $F^{\pm}$ and $F_{\pm}$
interchangeably.

###  A.1 $\langle\phi\phi\phi\phi\rangle$ crossing equation

The single correlator sum rules have already appeared in [24] and [25]. We
quote them below for completeness:

$\displaystyle\sum_{S^{+}}c_{O}^{2}\begin{pmatrix}0\\\ F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ 0\\\ 0\\\ F^{+}_{\Delta,\hskip 1.0pt\ell}\\\
0\end{pmatrix}+\sum_{X^{+}}c_{O}^{2}\begin{pmatrix}0\\\ -F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ n\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ 0\\\ -\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ n\hskip 1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{Y^{+}}c_{O}^{2}\begin{pmatrix}0\\\ 0\\\
\frac{m-2}{2}\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ m\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ 0\\\ \frac{m-2}{2}\hskip
1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{Z^{+}}c_{O}^{2}\begin{pmatrix}F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ \frac{1}{2}\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\
-\frac{1}{2}\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\
-F^{-}_{\Delta,\hskip 1.0pt\ell}\\\ -\frac{1}{2}\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ +\frac{1}{2}\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\end{pmatrix}$ $\displaystyle\hskip
184.9429pt+\sum_{A^{-}}c_{O}^{2}\begin{pmatrix}0\\\ 0\\\ -\frac{1}{2}\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ F^{-}_{\Delta,\hskip 1.0pt\ell}\\\
0\\\ \frac{1}{2}\hskip 1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{B^{-}}c_{O}^{2}\begin{pmatrix}F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ -\frac{1}{2}\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\
\frac{1}{2}\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\
-F^{-}_{\Delta,\hskip 1.0pt\ell}\\\ \frac{1}{2}\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ -\frac{1}{2}\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\
0\\\ 0\\\ 0\\\ 0\end{pmatrix}.$ (A.1)

###  A.2 $\langle\phi X\phi X\rangle$ crossing equation

Using the projectors (2.11) and (2.12) we find the sum rules

$\sum_{O}\lambda_{\phi XO_{\bar{A}}}^{2}F_{\mp,\Delta,l}^{\phi X,\phi X}=0\,,$
(A.2)

and

$\sum_{O}\lambda_{\phi XO_{\phi}}^{2}F_{\mp,\Delta,l}^{\phi X,\phi X}=0\,.$
(A.3)

###  A.3 $\langle\phi\phi XX\rangle$ crossing equation

For the crossing equation from the four-point function $\langle\phi\phi
XX\rangle$ we also need the projectors corresponding to the decomposition of
$\langle\phi^{a}_{i}\phi^{b}_{j}X^{cd}X^{ef}\rangle$ onto irreps. To find
these we need to know the irreps the $\phi\times\phi$ and $X\times X$ OPEs
have in common. These are the $X$ and $S$ irreps. The corresponding projectors
turn out to be [13, 34]

$P^{S}_{abcdef}=\delta_{ab}\Big{(}\delta_{cdef}-\frac{1}{n}\delta_{cd}\delta_{ef}\Big{)}$
(A.4)

and

$P^{X}_{abcdef}=\delta_{abcdef}-\frac{1}{n}\Big{(}\delta_{ab}\delta_{cdef}+\delta_{cd}\delta_{abef}+\delta_{ef}\delta_{abcd}\Big{)}+\frac{2}{n^{2}}\delta_{ab}\delta_{cd}\delta_{ef}\,.$
(A.5)

We thus obtain the sum rules

$\sum_{O}(\lambda_{\phi\phi
O_{S+}}\lambda_{XXO_{S+}}F_{\mp,\Delta,l}^{\phi\phi,XX}\pm(-1)^{l}\frac{1}{n}\lambda_{\phi
XO_{y}}^{2}F_{\mp,\Delta,l}^{X\phi,\phi X}\pm(-1)^{l}\frac{1}{n}\lambda_{\phi
XO_{\bar{A}}}^{2}F_{\mp,\Delta,l}^{X\phi,\phi X})=0$ (A.6)

and

$\sum_{O}(\lambda_{\phi\phi
O_{X+}}\lambda_{XXO_{X+}}F_{\mp,\Delta,l}^{\phi\phi,XX}\pm(-1)^{l}\lambda_{\phi
XO_{y}}^{2}F_{\mp,\Delta,l}^{X\phi,\phi X}\mp(-1)^{l}\lambda_{\phi
XO_{\bar{A}}}^{2}F_{\mp,\Delta,l}^{X\phi,\phi X})=0\,.$ (A.7)

###  A.4 $\langle XXXX\rangle$ crossing equation

For the $\langle XXXX\rangle$ crossing equation the sum rules have been
already computed in the literature, albeit in a slightly different context,
see e.g. [11] and [12]. We quote

$\displaystyle\sum_{S^{+}}c_{O}^{2}\begin{pmatrix}0\\\ F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ F^{+}_{\Delta,\hskip 1.0pt\ell}\\\
0\end{pmatrix}+\sum_{X^{+}}c_{O}^{2}\begin{pmatrix}0\\\ 0\\\
-\frac{4}{n+1}\hskip 1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ \hskip
1.0ptF^{-}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{E^{+}}c_{O}^{2}\begin{pmatrix}\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ \frac{2(n-1)}{n}\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ -\frac{(n+1)(n-2)}{n(n-1)}\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ -\frac{n+1}{2(n-1)}\hskip
1.0ptF^{-}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{\bar{S}^{-}}c_{O}^{2}\begin{pmatrix}F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ 0\\\ F^{+}_{\Delta,\hskip 1.0pt\ell}\\\
0\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\end{pmatrix}.$ (A.8)

####  A.4.1 The $n=3$ case

For $n=3$ the sum rules become

$\displaystyle\sum_{S^{+}}c_{O}^{2}\begin{pmatrix}0\\\ F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ F^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{X^{+}}c_{O}^{2}\begin{pmatrix}F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ 0\\\ -2\hskip 1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{\bar{S}^{-}}c_{O}^{2}\begin{pmatrix}F^{-}_{\Delta,\hskip
1.0pt\ell}\\\ -\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\
F^{+}_{\Delta,\hskip 1.0pt\ell}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix},$

which are equivalent to those of $O(2)$.

###  A.5 $n=2$ sum rules

With the exception of the single correlator sum rules, all the above sum rules
are drastically simplified when $n=2$, this is because the irrep $X$ becomes
one dimensional then.191919Hence, all OPEs involving $X$ only exchange one
irrep. We thus have

$\sum_{S^{+}}\lambda_{XXO_{S}}^{2}F^{-XX,XX}_{\Delta,l}=0\,,$ (A.9)

$\sum_{\phi^{\pm}}\lambda_{OXO_{\phi}}^{2}F^{-OX,OX}_{\Delta,l}=0\,,$ (A.10)

$\sum_{S^{+}}\lambda_{OOO_{S}}\lambda_{XXO_{S}}F^{\mp
OO,XX}_{\Delta,l}\pm\sum_{\phi^{\pm}}(-1)^{l}\lambda_{OXO_{\phi}}^{2}F^{\mp
XO,OX}_{\Delta,l}=0\,,$ (A.11)

in addition to the single correlator sum rules which remain as defined
earlier.

## Appendix B. $\boldsymbol{Z\times Z}$ single correlator sum rules

The $n=2$ $Z\times Z$ bootstrap the sum rules are simplified drastically by
the observation that $Z^{ab}_{ij}$ and $Z^{cd}_{kl}$ necessarily have ($a=c$
and $b=d$) or ($a=d$ and $b=c$). Thus if we assume that the field
$Z^{ab}_{ij}$ has, lets say, $a=1$ then by definition $b=2$. Hence, the
projectors are

$\displaystyle\begin{split}{P^{S}}_{ijklmnop}&=P^{S}_{ikmo}P^{S}_{jlnp}\,,\\\
{P^{SY}}_{ijklmnop}&=P^{S}_{ikmo}P^{Y}_{jlnp}+P^{Y}_{ikmo}P^{S}_{jlnp}\,,\\\
{P^{SA}}_{ijklmnop}&=P^{S}_{ikmo}P^{A}_{jlnp}+P^{A}_{ikmo}P^{S}_{jlnp}\,,\\\
{P^{YA}}_{ijklmnop}&=P^{Y}_{ikmo}P^{A}_{jlnp}+P^{A}_{ikmo}P^{Y}_{jlnp}\,,\\\
{P^{YY}}_{ijklmnop}&=P^{Y}_{ikmo}P^{Y}_{jlnp}\,,\\\
{P^{AA}}_{ijklmnop}&=P^{A}_{ikmo}P^{A}_{jlnp}\,,\end{split}$ (B.1)

i.e. the products of projectors of $O(m)$. By $S$ we denote the scalar
singlet, by $A$ the antisymmetric and by $Y$ the rank-two traceless symmetric
irreps of $O(m)$. We obtain the following sum rules

$\displaystyle\sum_{S^{+}}c_{O}^{2}\begin{pmatrix}0\\\ \hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ 0\\\ 0\\\ 0\\\ \hskip
1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{SY^{+}}c_{O}^{2}\begin{pmatrix}0\\\ -2\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ \hskip 1.0ptF^{-}_{\Delta,\hskip
1.0pt\ell}\\\ 2\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ -2\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ \frac{m-4}{2}\hskip
1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{YY^{+}}c_{O}^{2}\begin{pmatrix}\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ \hskip 1.0ptF^{-}_{\Delta,\hskip
1.0pt\ell}\\\ -\hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ (m-2)\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ (m+2)\hskip 1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\\\ \frac{2-m-m^{2}}{2}\hskip 1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{AA^{+}}c_{O}^{2}\begin{pmatrix}\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ 0\\\ 0\\\ 0\\\ -m\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ 0\end{pmatrix}$ $\displaystyle\hskip
184.9429pt+\sum_{SA^{-}}c_{O}^{2}\begin{pmatrix}0\\\ 0\\\ -\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ 0\\\ -2\hskip
1.0ptF^{+}_{\Delta,\hskip 1.0pt\ell}\\\ \frac{m}{2}\hskip
1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}+\sum_{YA^{-}}c_{O}^{2}\begin{pmatrix}2\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ -m^{2}\hskip 1.0ptF^{-}_{\Delta,\hskip
1.0pt\ell}\\\ \hskip 1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ m\hskip
1.0ptF^{-}_{\Delta,\hskip 1.0pt\ell}\\\ 2\hskip 1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\\\ \frac{m^{2}-m}{2}\hskip 1.0ptF^{+}_{\Delta,\hskip
1.0pt\ell}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\ 0\\\
0\end{pmatrix}.$ (B.2)

## Appendix C. $\boldsymbol{\langle ZZZZ\rangle}$ projectors for
$\boldsymbol{n\geqslant 4}$ and $\boldsymbol{G}$ arbitrary

The projectors that correspond to the $\langle
Z^{ab}_{ij}Z^{cd}_{kl}Z^{ef}_{mn}Z^{gh}_{op}\rangle$ correlator are rather
extended 16 index objects. In order to simplify their presentation we will
introduce “Blocks”, not to be confused with conformal blocks. With the use of
Blocks the projectors can be presented in somewhat compact form. Also, we will
make no distinction between upper and lower indices. It will be implicitly
assumed that indices $a$-$h$ label the copy of $G$, and the indices $i$-$p$
are $G$ indices. This setup is not the most general possible with respect to
$G$ indices, but we use it for simplicity of demonstration. We hope that the
generalization to arbitrary indices of $G$ will become obvious from our
presentation. There are three groups of representations that appear in the
$Z^{ab}_{ij}\times Z^{cd}_{kl}$ OPE. These are:

Group I: Representations with $a$ = $c$ or $d$ OR $b$ = $c$ or $d$.

Group II: Representations with $a$ = $c$ or $d$ AND $b$ = $d$ or $c$.

Group III: Representations with $a$, $b$, $c$ and $d$ all different.

To reiterate, we have Group I: two pairs of indices equal, Group II: one pair
of indices equal, Group III: no pairs of indices equal. We also remind the
reader that the operators $Z^{ab}_{ij}$ have $a\neq b$.

###  C.1 Group I representations

Let us start by recalling the projectors of the $G$ irreps (we take $G=O(m)$
in order to be explicit, but the generalization to any $G$ is trivial). We
have $P_{g_{1}}^{ijkl}=\frac{1}{m}\delta^{ij}\delta^{kl}$,
$P_{g_{2}}^{ijkl}=\frac{1}{2}(\delta^{ik}\delta^{jl}-\delta^{il}\delta^{jk})$
and
$P_{g_{3}}^{ijkl}=\frac{1}{2}(\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}-\frac{2}{m}\delta^{ij}\delta^{kl})$.
Next we define the following useful tensors:

$\displaystyle\begin{split}{R_{1}}_{abcdefgh}&=(\delta_{ac}\delta_{bd}-\delta_{acbd})(\delta_{eg}\delta_{fh}-\delta_{egfh})\,,\\\
{R_{2}}_{abcdefgh}&=\delta_{aceg}\delta_{bd}\delta_{fh}-\delta_{abcdeg}\delta_{fh}-\delta_{afcheg}\delta_{bd}+\delta_{abcdefgh}\,,\\\
{R_{3}}_{abcdefgh}&=\delta_{aceg}\delta_{bdfh}-\delta_{abcdefgh}\,.\end{split}$
(C.1)

With these tensors in hand we can now define the Blocks. We denote these with
“$B$” for short. They are

$\displaystyle\begin{split}{B_{S}}^{abcdefgh}_{ijklmnop}&=P_{g_{1}}^{ikmo}P_{g_{1}}^{jlnp}{R_{1}}_{abcdefgh}\,,\\\
{B_{XS}}^{abcdefgh}_{ijklmnop}&=P_{g_{1}}^{ikmo}P_{g_{1}}^{jlnp}\Big{(}{R_{2}}_{abcdefgh}-\frac{1}{n}{R_{1}}_{abcdefgh}\Big{)}\,,\\\
{B_{XY}}^{abcdefgh}_{ijklmnop}&=P_{g_{1}}^{ikmo}P_{g_{3}}^{jlnp}\Big{(}{R_{3}}_{abcdefgh}-\frac{1}{n-1}{R_{2}}_{badcfehg}\Big{)}\,,\\\
{B_{XA}}^{abcdefgh}_{ijklmnop}&=P_{g_{1}}^{ikmo}P_{g_{2}}^{jlnp}\Big{(}{R_{3}}_{abcdefgh}-\frac{1}{n-1}{R_{2}}_{badcfehg}\Big{)}\,,\\\
{B_{SY}}^{abcdefgh}_{ijklmnop}&=P_{g_{1}}^{ikmo}P_{g_{3}}^{jlnp}{R_{2}}_{badcfehg}\,,\\\
{B_{SA}}^{abcdefgh}_{ijklmnop}&=P_{g_{1}}^{ikmo}P_{g_{2}}^{jlnp}{R_{2}}_{badcfehg}\,,\\\
{B_{YY}}^{abcdefgh}_{ijklmnop}&=P_{g_{3}}^{ikmo}P_{g_{3}}^{jlnp}{R_{3}}_{badcfehg}\,,\\\
{B_{AA}}^{abcdefgh}_{ijklmnop}&=P_{g_{2}}^{ikmo}P_{g_{2}}^{jlnp}{R_{3}}_{badcfehg}\,,\\\
{B_{Y\\!A}}^{abcdefgh}_{ijklmnop}&=P_{g_{3}}^{ikmo}P_{g_{2}}^{jlnp}{R_{3}}_{badcfehg}\,,\\\
{B_{\bar{X}}}^{abcdefgh}_{ijklmnop}&=P_{g_{1}}^{ikmo}P_{g_{1}}^{jlnp}\Big{(}{R_{3}}_{badcfehg}-\frac{1}{n-2}({R_{2}}_{abcdefgh}+{R_{2}}_{badcfehg})\\\
&\hskip
170.71652pt+\frac{1}{(n-1)(n-2)}{R_{1}}_{abcdefgh}\Big{)}\,.\end{split}$ (C.2)

To finally obtain the projectors from the Blocks we must perform
symmetrizations, which are the same for all Blocks, hence we write $B_{g}$,
where $g$ is one of the irreps considered above. These symmetrizations are

$\displaystyle\begin{split}{{{B_{g}}^{\prime}}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}&={{{B_{g}}}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}+{{{B_{g}}}\hskip
1.0pt}^{abcdefhg}_{ijklmnpo}\,,\\\ {{{B_{g}}^{\prime\prime}}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}&={{{B_{g}}^{\prime}}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}+{{{B_{g}}^{\prime}}\hskip
1.0pt}^{abcdfegh}_{ijklnmop}\,,\\\ {{B_{g}}^{\prime\prime\prime}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}&={{B_{g}}^{\prime\prime}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}+{{B_{g}}^{\prime\prime}\hskip
1.0pt}^{abdcef,gh}_{ijlkmnop}\,,\\\ {{P_{g}}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}&={{{B_{g}}^{\prime\prime\prime}}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}+{{B_{g}}^{\prime\prime\prime}}{\hskip
1.0pt}^{bacdefgh}_{jiklmnop}\,,\end{split}$ (C.3)

where the last line in (C.3) corresponds to the final expression for the
projector in irrep $g$. Note that the above symmetrizations are necessary in
order to take into account the symmetry $Z^{ab}_{ij}=Z^{ba}_{ji}$. To apply
these equations to a different group $G$ one simply needs to replace the
expressions $P_{g_{1}}$, $P_{g_{2}}$ and $P_{g_{3}}$ with those of their group
of choice.

###  C.2 Group II representations

For Group II representations the steps are very similar with the ones
described for Group I. We must simply define some new tensors. These are

$\displaystyle{RR_{1}}_{abcdefgh}$
$\displaystyle=\delta_{aceg}(\delta_{bf}\delta_{dh}-\delta_{bfdh})-(\delta_{acegbf}\delta_{dh}-\delta_{acegbfdh})-(\delta_{acegdh}\delta_{bf}-\delta_{acegbfdh})$
(C.4) $\displaystyle{RR_{2}}_{abcdefgh}$
$\displaystyle=\delta_{ac}\delta_{eg}(\delta_{bf}\delta_{dh}-\delta_{bfdh})-\delta_{eg}(\delta_{acbf}\delta_{dh}+\delta_{acdh}\delta_{bf}-2\delta_{acbfdh})\,,$
$\displaystyle\quad-\delta_{ac}(\delta_{egbf}\delta_{dh}+\delta_{egdh}\delta_{bf}-2\delta_{egbfdh})+(\delta_{acbfeg}\delta_{dh}+\delta_{acdheg}\delta_{bf}-2\delta_{acegbfdh})$
$\displaystyle\quad+(\delta_{acbf}\delta_{egdh}+\delta_{acdh}\delta_{bfeg}-2\delta_{acegbfdh})\,,$

whereas the Blocks for Group II representations are

$\displaystyle{B_{XB}}^{abcdefgh}_{ijklmnop}$
$\displaystyle=({RR_{1}}_{abcdefgh}-\frac{1}{n-2}{RR_{2}}_{abcdefgh})P_{g_{1}}^{ikmo}\delta_{jn}\delta_{lp}\,,$
(C.5)
$\displaystyle\quad-({RR_{1}}_{abcdehgf}-\frac{1}{n-2}{RR_{2}}_{abcdehgf})P_{g_{1}}^{ikmo}\delta_{jp}\delta_{ln}\,,$
$\displaystyle{B_{XZ}}^{abcdefgh}_{ijklmnop}$
$\displaystyle=({RR_{1}}_{abcdefgh}-\frac{1}{n-2}{RR_{2}}_{abcdefgh})P_{g_{1}}^{ikmo}\delta_{jn}\delta_{lp}\,,$
$\displaystyle\quad+({RR_{1}}_{abcdehgf}-\frac{1}{n-2}{RR_{2}}_{abcdehgf})P_{g_{1}}^{ikmo}\delta_{jp}\delta_{ln}\,,$
$\displaystyle{B_{SB}}^{abcdefgh}_{ijklmnop}$
$\displaystyle={RR_{2}}_{abcdefgh}P_{g_{1}}^{ikmo}\delta_{jn}\delta_{lp}-{RR_{2}}_{abcdehgf}P_{g_{1}}^{ikmo}\delta_{jp}\delta_{ln}\,,$
$\displaystyle{B_{SZ}}^{abcdefgh}_{ijklmnop}$
$\displaystyle={RR_{2}}_{abcdefgh}P_{g_{1}}^{ikmo}\delta_{jn}\delta_{lp}+{RR_{2}}_{abcdehgf}P_{g_{1}}^{ikmo}\delta_{jp}\delta_{ln}\,,$
$\displaystyle{B_{YB}}^{abcdefgh}_{ijklmnop}$
$\displaystyle={RR_{1}}_{abcdefgh}P_{g_{3}}^{ikmo}\delta_{jn}\delta_{lp}-{RR_{1}}_{abcdehgf}P_{g_{3}}^{ikmo}\delta_{jp}\delta_{ln}\,,$
$\displaystyle{B_{YZ}}^{abcdefgh}_{ijklmnop}$
$\displaystyle={RR_{1}}_{abcdefgh}P_{g_{3}}^{ikmo}\delta_{jn}\delta_{lp}+{RR_{1}}_{abcdehgf}P_{g_{3}}^{ikmo}\delta_{jp}\delta_{ln}\,,$
$\displaystyle{B_{AB}}^{abcdefgh}_{ijklmnop}$
$\displaystyle={RR_{1}}_{abcdefgh}P_{g_{2}}^{ikmo}\delta_{jn}\delta_{lp}-{RR_{1}}_{abcdehgf}P_{g_{2}}^{ikmo}\delta_{jp}\delta_{ln}\,,$
$\displaystyle{B_{AZ}}^{abcdefgh}_{ijklmnop}$
$\displaystyle={RR_{1}}_{abcdefgh}P_{g_{2}}^{ikmo}\delta_{jn}\delta_{lp}+{RR_{1}}_{abcdehgf}P_{g_{2}}^{ikmo}\delta_{jp}\delta_{ln}\,.$

We may now apply (C.3) and obtain the final expression for the projectors.

###  C.3 Group III representations

For the last group of representations we need to define one more tensor
structure:

$\displaystyle{R_{4}}^{abcdefgh}_{ijklmnop}$
$\displaystyle=(\delta_{ae}\delta_{bf}\delta_{cg}\delta_{dh}-\delta_{cg}\delta_{dh}\delta_{abef}-\delta_{bf}\delta_{dh}\delta_{aceg}-\delta_{bf}\delta_{cg}\delta_{adeh}-\delta_{ae}\delta_{dh}\delta_{bcfg}$
(C.6)
$\displaystyle\quad-\delta_{ae}\delta_{cg}\delta_{bdfh}-\delta_{ae}\delta_{bf}\delta_{cdgh}+\delta_{aedh}\delta_{bcfg}+\delta_{aecg}\delta_{bdfh}+\delta_{aebf}\delta_{cdgh}$
$\displaystyle\quad+2(\delta_{dh}\delta_{abcefg}+\delta_{cg}\delta_{abdefh}+\delta_{bf}\delta_{adcehg}+\delta_{ae}\delta_{dbchfg})-6\delta_{aecgbfdh})\delta_{im}\delta_{ko}\delta_{jn}\delta_{lp}\,,$

leading to the Blocks

$\displaystyle{B_{BB}\hskip 1.0pt}^{abcdefgh}_{ijklmnop}$
$\displaystyle={R_{4}}^{abcdefgh}_{ijklmnop}-{R_{4}}^{abcdgfeh}_{ijklonmp}-{R_{4}}^{abcdehgf}_{ijklmpon}+{R_{4}}^{abcdghef}_{ijklopmn}\,,$
(C.7) $\displaystyle{B_{BZ}\hskip 1.0pt}^{abcdefgh}_{ijklmnop}$
$\displaystyle={R_{4}}^{abcdefgh}_{ijklmnop}-{R_{4}}^{abcdgfeh}_{ijklonmp}+{R_{4}}^{abcdehgf}_{ijklmpon}-{R_{4}}^{abcdghef}_{ijklopmn}\,,$
$\displaystyle{B_{To\hskip 0.3pttS}\hskip 1.0pt}^{abcdefgh}_{ijklmnop}$
$\displaystyle={R_{4}}^{abcdefgh}_{ijklmnop}+{R_{4}}^{abcdefhg}_{ijklmnpo}\,.$

For the representations $BB$ and $BZ$ we may simply use (C.3) to obtain the
projectors. On the other hand, $To\hskip 0.3pttS$ needs a slightly more
elaborate formula, which is due to our explicit choice of Block. We have

$\displaystyle{{{B_{To\hskip 0.3pttS}}^{\prime}}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}$ $\displaystyle={B_{To\hskip 0.3pttS}\hskip
1.0pt}^{abcdefgh}_{ijklmnop}+{B_{To\hskip 0.3pttS}\hskip
1.0pt}^{abcdegfh}_{ijklmonp}+{B_{To\hskip 0.3pttS}\hskip
1.0pt}^{abcdehgf}_{ijklmpon}\,,$ (C.8) $\displaystyle{{{P_{To\hskip
0.3pttS}}}\hskip 1.0pt}^{abcdefgh}_{ijklmnop}$ $\displaystyle={{{B_{To\hskip
0.3pttS}}^{\prime}}\hskip 1.0pt}^{abcdefgh}_{ijklmnop}+{{{B_{To\hskip
0.3pttS}}^{\prime}}\hskip 1.0pt}^{abcdfegh}_{ijklnmop}+{{{B_{To\hskip
0.3pttS}}^{\prime}}\hskip 1.0pt}^{abcdgfeh}_{ijklonmp}+{{{B_{To\hskip
0.3pttS}}^{\prime}}\hskip 1.0pt}^{abcdhfge}_{ijklpnom}\,.$

###  C.4 Comments

The above tensors can be derived if we know the symmetry properties of each
irrep (symmetric, antisymmetric, traceless, trace, …), by writing down a
tensor with the same symmetry and then contracting with a tensor that enforces
the constraints $a\neq b$, $c\neq d$, $e\neq f$ and $g\neq h$. Such a tensor
can be found in a straightforward way. First, define a reduced Kronecker delta
$\delta_{ab}^{r}=\delta_{ab}-\delta_{abr}$, which is the same as the usual
Kronecker delta, but equal to zero if the indices are equal to a value $r$.
Now we can define
$P^{abcd}_{a^{\prime}b^{\prime}c^{\prime}d^{\prime}}=\delta_{aa^{\prime}}^{b}\delta_{bb^{\prime}}\delta_{cc^{\prime}}^{d}\delta_{dd^{\prime}}$
which is simplified as
$P^{abcd}_{a^{\prime}b^{\prime}c^{\prime}d^{\prime}}=(\delta_{aa^{\prime}}\delta_{bb^{\prime}}-\delta_{aa^{\prime}bb^{\prime}})(\delta_{cc^{\prime}}\delta_{dd^{\prime}}-\delta_{cc^{\prime}dd^{\prime}})$
by plugging in the definition of the reduced Kronecker delta. Now suppose that
we had found a tensor
$T^{a^{\prime}b^{\prime}c^{\prime}d^{\prime}e^{\prime}f^{\prime}g^{\prime}h^{\prime}}$
with all the required symmetry properties to describe some irrep. We could
turn it into a projector using the following equation:

$T^{abcdefgh}=P^{abcd}_{a^{\prime}b^{\prime}c^{\prime}d^{\prime}}P^{efgh}_{e^{\prime}f^{\prime}g^{\prime}h^{\prime}}T^{a^{\prime}b^{\prime}c^{\prime}d^{\prime}e^{\prime}f^{\prime}g^{\prime}h^{\prime}}\,,$
(C.9)

where the left-hand side is our projector. One can also define generalized
reduced Kronecker deltas in order to impose more complicated constraints. For
example, let us consider a tensor with four indices which must take different
values and is totally symmetric (this is relevant for the $To\hskip 0.3pttS$
irrep). We may start with a tensor
$T^{a^{\prime}b^{\prime}c^{\prime}d^{\prime}}$ that is simply totally
symmetric but with arbitrary index values. The constraint can then be imposed
by contracting with
$P^{abcd}_{a^{\prime}b^{\prime}c^{\prime}d^{\prime}}=\delta_{aa^{\prime}}^{bcd}\delta_{bb^{\prime}}^{cd}\delta_{cc^{\prime}}^{d}\delta_{dd^{\prime}}$.
Note that
$\delta^{cd}_{bb^{\prime}}=\delta_{bb^{\prime}}-\delta_{bb^{\prime}c}-\delta_{bb^{\prime}d}+\delta_{bb^{\prime}dc}$.
But the utility of this formalism is now clear: if one wishes to evaluate a
relation such as
$\delta_{bb^{\prime}}^{cd}\delta_{cc^{\prime}}^{d}\delta_{dd^{\prime}}$, the
last term in $\delta^{cd}_{bb^{\prime}}$, namely $\delta_{bb^{\prime}dc}$, can
be dropped since it gives zero when contracted with
$\delta_{cc^{\prime}}^{d}\delta_{dd^{\prime}}=(\delta_{cc^{\prime}}\delta_{dd^{\prime}}-\delta_{cc^{\prime}dd^{\prime}})$.
Thus, in conclusion, as long as we calculate expressions like
$\delta_{aa^{\prime}}^{bcd}\delta_{bb^{\prime}}^{cd}\delta_{cc^{\prime}}^{d}\delta_{dd^{\prime}}$
from right to left, it is sufficient to take
$\delta_{bb^{\prime}}^{a_{1}a_{2}\ldots
a_{n}}=\delta_{bb^{\prime}}-\sum_{i=0}^{n}\delta_{bb^{\prime}a_{i}}$. This
allows us to algorithmically impose constraints on tensors.

## Appendix D. $\boldsymbol{\langle ZZZZ\rangle}$ sum rules for
$\boldsymbol{n\geqslant 4}$

Due to the rather extended size of the sum rules derived from the crossing
equation corresponding to $\langle ZZZZ\rangle$ for $n\geqslant 4$ we attach
them in an auxiliary file.

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