# Modular invariant holographic correlators for $\mathcal{N}=4$ SYM with
general gauge group

###### Abstract

We study the stress tensor four-point function for $\mathcal{N}=4$ SYM with
gauge group $G=SU(N)$, $SO(2N+1)$, $SO(2N)$ or $USp(2N)$ at large $N$. When
$G=SU(N)$, the theory is dual to type IIB string theory on $AdS_{5}\times
S^{5}$ with complexified string coupling $\tau_{s}$, while for the other cases
it is dual to the orbifold theory on $AdS_{5}\times S^{5}/\mathbb{Z}_{2}$. In
all cases we use the analytic bootstrap and constraints from localization to
compute 1-loop and higher derivative tree level corrections to the leading
supergravity approximation of the correlator. We give perturbative evidence
that the localization constraint in the large $N$ and finite complexified
coupling $\tau$ limit can be written for each $G$ in terms of Eisenstein
series that are modular invariant in terms of $\tau_{s}\propto\tau$, which
allows us to fix protected terms in the correlator in that limit. In all
cases, we find that the flat space limit of the correlator precisely matches
the type IIB S-matrix. We also find a closed form expression for the $SU(N)$
1-loop Mellin amplitude with supergravity vertices. Finally, we compare our
analytic predictions at large $N$ and finite $\tau$ to bounds from the
numerical bootstrap in the large $N$ regime, and find that they are not
saturated for any $G$ and any $\tau$, which suggests that no physical theory
saturates these bootstrap bounds.

###### Contents

  1. 1 Introduction
  2. 2 $\mathcal{N}=4$ stress-tensor four-point function
    1. 2.1 Basics
    2. 2.2 Strong coupling expansion
    3. 2.3 Constraints from flat space type IIB string theory
  3. 3 Constraints from supersymmetric localization
    1. 3.1 $\mathcal{N}=2^{*}$ sphere free energy at finite $N$
    2. 3.2 $1/N$ expansions
    3. 3.3 Constraining the stress-tensor correlator
  4. 4 Stress-tensor correlator at 1-loop
    1. 4.1 One-loop from tree level
    2. 4.2 Mellin amplitude
    3. 4.3 Flat space limit
    4. 4.4 Fixing 1-loop contact term from localization
    5. 4.5 Extracting CFT data
  5. 5 Comparison to numerical bootstrap
  6. A Matrix model calculations
  7. B Closed form for $R|R$ Mellin amplitude in $SU(N)$
  8. C 1-loop localization integral in Mellin space
  9. D Lorentzian inversion
  10. E Mixed Correlator Bootstrap Setup
    1. E.1 Superconformal blocks
    2. E.2 Free theory
    3. E.3 Short sector in the interacting theory
    4. E.4 Short contributions to reduced correlators

##  1 Introduction

Holographic correlators are correlation functions in a CFTd that are dual to
scattering of particles in string theory or M-theory on AdSd+1 times some
compact manifold. Starting from [1, 2], they have been computed at leading
order tree level[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], higher derivative tree
level [14, 15, 16, 17, 18, 19, 20, 21, 22], and 1-loop [23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 15] in various $d$. In 4d, the simplest maximally
supersymmetric example relates the stress tensor multiplet four-point function
in $\mathcal{N}=4$ $SU(N)$ SYM to scattering of gravitons of type IIB string
theory on $AdS_{5}\times S^{5}$. Since the complexified gauge coupling
$\tau=\frac{\theta}{2\pi}+i\frac{4\pi}{g_{\text{YM}}^{2}}$ transforms under
$SL(2,\mathbb{Z})$ duality invariance, the correlators can be computed at
large $N$ and finite $\tau$ in terms of modular functions like Eisenstein
series, which in the flat space limit precisely match the prediction from the
strongly coupled string theory S-matrix [21, 22], which is also expected to be
modular invariant in terms of the complexified string coupling
$\tau_{s}=\chi+i/g_{s}$. In the small $\mathop{\rm Im}\nolimits\tau$ limit,
these results reproduce the large $N$ and large ’t Hooft coupling
$\lambda\equiv g_{\text{YM}}^{2}N$ expansion, which is dual to weakly coupled
string theory, and is not sensitive to $\mathop{\rm
Re}\nolimits\tau=\mathop{\rm Re}\nolimits\tau_{s}$.

In this paper, we will generalize this $\mathcal{N}=4$ SYM story to general
gauge group $G=SU(N)$, $SO(2N+1)$, $SO(2N)$ or $USp(2N)$. While the $SU(N)$
case is dual to $AdS_{5}\times S^{5}$, the other three groups are dual to the
orbifold theory $AdS_{5}\times S^{5}/\mathbb{Z}_{2}$, so from the bulk
perspective there are two classes of theories: non-orbifold and orbifold. At
tree level supergravity, the correlators for the two cases are identical, but
they start to differ at higher derivative corrections and at loop level. As in
the previously studied $SU(N)$ case, we will fix the protected $R^{4}$ tree
level corrections using the localization constraint derived in [17], which
related an integral of the correlator to derivatives
$\partial_{\tau}\partial_{\bar{\tau}}\partial_{m}^{2}F(m)|_{m=0}$ of the mass
deformed $\mathcal{N}=2^{*}$ free energy. This latter quantity was computed
using localization in terms of a $\text{Rank}(G)$ dimensional matrix model
integral [33], which we computed to any order in large $N$ and finite
$\lambda\sim g_{\text{YM}}^{2}N$ following the $SU(N)$ case considered in
[15]. At large $\lambda$, this computes the perturbative contributions to the
large $N$ finite $\tau$ limit, which we find are consistent with an expansion
in Eisenstein series, as in the $SU(N)$ case. We then use the known type IIB
S-matrix in the flat space limit of the holographic correlator to further
constrain terms at large $N$, which confirms our answer for $R^{4}$ and allows
us to further fix $D^{4}R^{4}$.

One novelty of the general gauge group case relative to $SU(N)$ is that the
transformation under S-duality [34, 35, 36] is more subtle. In general, gauge
groups transform to their Langlands (i.e. GNO) dual under S-duality [37].
While $SU(N)$ and $SO(2N)$ thus transform to themselves with $\tau\to-1/\tau$,
up to discrete subgroups and theta angles that do not effect the quantities we
consider [37, 38, 39],111The precise dual of $SU(N)$ is
$SU(N)/\mathbb{Z}_{N}$, while for $SO(2N)$ it is $SO(2N)/\mathbb{Z}_{2}$ [37].
$SO(2N+1)$ transforms to $USp(2N)$ with $\tau\to-1/(2\tau)$. Consistent with
this non-trivial transformation, we find perturbative evidence that the
$USp(2N)$ localization quantity is expanded in Eisenstein series with argument
$2\tau$, as opposed to $\tau$ for the other cases. If we write these
quantities in terms of $\tau_{s}$, however, then from the AdS/CFT dictionary
$\tau_{s}=2\tau$ for $USp(2N)$ and $\tau_{s}=\tau$ for the other gauge groups,
we find that results in all cases are modular invariant in terms of
$\tau_{s}$. This had to be the case in the flat space limit where the type IIB
S-matrix is modular invariant in terms of $\tau_{s}$, but it is remarkable
that this modularity is inherited by a priori less restricted CFT quantities
dual to string theory on AdS.

We then compute the 1-loop correction coming from supergravity $R$ and $R^{4}$
vertices for both the orbifold and $SU(N)$ cases, using the unitarity cut
method of [40]. This requires an unmixing of double trace operators formed by
two half-BPS operators labeled by their superprimary dimension $p$, e.g. $p=2$
is the stress tensor. In the $SU(N)$ case there are contributions from all
$p\geq 2$, and our answer reproduces previous results [23, 24, 41, 28], except
that we have now been able to resum the double sum Mellin space expression of
[28], see (B.4). For the orbifold case only even $p$ contributes, and the
resulting 1-loop term looks very different from the $SU(N)$ case, yet in the
flat space limit each case differs just by a power of 2 as expected from the
orbifold factor in their bulk duals. Finally, for the 1-loop supergravity term
we fix the unique contact term ambiguity using the previously described
localization constraint, and find that the orbifold case differs from the
$SU(N)$ case by the same factor of 2, which did not have to be the case since
this contact term is subleading in the flat space limit.

One motivation for studying the orbifold theories is that their CFT data at
large $N$ is bigger than the $SU(N)$ case, so if any physical theory saturates
the numerical bootstrap bounds of [42, 43], it would be the orbifold case for
some value of $\tau$. Unfortunately, we find that our large $N$ finite $\tau$
predictions for CFT data do not saturate the bootstrap bounds at large $N$ for
either case, even at the self dual point $\tau=e^{i\pi/3}$ that was
conjectured to saturate the bounds in [43]. Furthermore, we computed bounds
using both the single correlator setup in [42, 43] as well as the mixed
correlator setup in [44] (which included $p=2,3$ half-BPS operators), and
found that the upper bounds were almost indistinguishable at large $N$, even
though the mixed correlator setup does not apply to the orbifold theories
(since they do not contain odd $p$ half-BPS operators), so we would have
expected $SU(N)$ to saturate the mixed bounds and the other cases to saturate
the single correlator bounds.

Looking ahead, it would be nice to compute the other localization constraint
$\partial_{m}^{4}F|_{m=0}$ to all orders in $1/N$ for general gauge group, as
was done for $SU(N)$ in [45], and to try to find the finite $N$ and $\tau$
formulae for both constraints, as has been done for
$\partial_{\tau}\partial_{\bar{\tau}}\partial_{m}^{2}F(m)|_{m=0}$ for the
$SU(N)$ case in [46]. This will involve computing the instanton contributions
to these quantities for general gauge group, which we did not consider in this
paper. Finally, if we eventually want to make contact between numerical
bootstrap and large $N$ analytics, it will probably be necessary to further
constrain the numerical bootstrap by directly imposing the localization
constraints as a function of $\tau$.

The rest of this paper is organized as follows. In Section 2 we review the
constraints of superconformal symmetry on the stress tensor correlator, as
well as its strong coupling expansion and constraints from type IIB string
theory in the flat space limit. In Section 3, we compute the non-instanton
part of $\partial_{\tau}\partial_{\bar{\tau}}\partial_{m}^{2}F(m)|_{m=0}$ for
general gauge group both at finite $N$ and $\lambda$, and in the large $N$ and
finite $g_{\text{YM}}$ limit (neglecting instantons), which we find is
nontrivially consistent with an expansion in Eisenstein series. In Section 4,
we compute the 1-loop amplitudes with $R$ and $R^{4}$ vertices, check their
flat space limit, fix the 1-loop supergravity contact term ambiguity using the
localization constraint, and extract CFT data. Finally, in Section 5 we
compare this large $N$ analytic data to the numerical bootstrap bounds
recomputed at very high precision, and find that they do not saturate the
bounds. Several technical details are given in various Appendices, and we
include an attached Mathematica notebook with explicit results.

##  2 $\mathcal{N}=4$ stress-tensor four-point function

We begin by reviewing what is already known about the stress tensor multiplet
four-point function. First we discuss general constraints from the
$\mathcal{N}=4$ superconformal group. Then we discuss the two different large
$N$ strong coupling expansions for SYM with general gauge group in Mellin
space: the expansion at finite $\tau$, as well as the more standard ’t Hooft
expansion at large $\lambda\sim g_{\text{YM}}^{2}N$. Lastly, we discuss how
the 10d flat space type IIB string theory S-matrix can be used to constrain
the SYM correlator by taking the flat space limit.

###  2.1 Basics

Let us denote the bottom component of the stress tensor multiplet by $S$. This
operator is a dimension 2 scalar in the ${\bf 20}^{\prime}$ of the
$SU(4)_{R}\cong SO(6)_{R}$, and can thus be represented as a rank-two
traceless symmetric tensor $S_{IJ}(\vec{x})$, with indices $I,J=1,\ldots,6$.
For simplicity we will contract these indices with null polarization vectors
$Y^{I}$, where $Y\cdot Y=0$. We are interested in studying the four-point
function $\langle SSSS\rangle$, which is fixed by conformal and $SU(4)$
symmetry to take the form

$\begin{split}&\langle
S(\vec{x}_{1},Y_{1})S(\vec{x}_{2},Y_{2})S(\vec{x}_{3},Y_{3})S(\vec{x}_{4},Y_{4})\rangle=\frac{Y^{2}_{12}Y^{2}_{34}}{{x}_{12}^{4}{x}_{34}^{4}}\mathcal{S}(U,V;\sigma,\tau)\,,\end{split}$
(2.1)

where we define

$\begin{split}U\equiv\frac{{x}_{12}^{2}{x}_{34}^{2}}{{x}_{13}^{2}{x}_{24}^{2}}\,,\qquad
V\equiv\frac{{x}_{14}^{2}{x}_{23}^{2}}{{x}_{13}^{2}{x}_{24}^{2}}\,,\qquad\sigma\equiv\frac{(Y_{1}\cdot
Y_{3})(Y_{2}\cdot Y_{4})}{(Y_{1}\cdot Y_{2})(Y_{3}\cdot
Y_{4})}\,,\qquad\tau\equiv\frac{(Y_{1}\cdot Y_{4})(Y_{2}\cdot
Y_{3})}{(Y_{1}\cdot Y_{2})(Y_{3}\cdot Y_{4})}\,.\end{split}$ (2.2)

The constraints of superconformal symmetry are given by the Ward identity in
[47], whose solution can be formally solved in two different ways. The first
solution takes the form

$\begin{split}\mathcal{S}(U,V;\sigma,\tau)&=\mathcal{S}_{\text{free}}(U,V;\sigma,\tau)+\Theta(U,V;\sigma,\tau)\mathcal{T}(U,V)\,,\\\
\Theta(U,V;\sigma,\tau)&\equiv\tau+[1-\sigma-\tau]V+\tau[\tau-1-\sigma]U+\sigma[\sigma-1-\tau]UV+\sigma
V^{2}+\sigma\tau U^{2}\,,\end{split}$ (2.3)

where $\mathcal{S}_{\text{free}}(U,V;\sigma,\tau)$ is the free theory
correlator

$\begin{split}\mathcal{S}_{\text{free}}(U,V;\sigma,\tau)=1+U^{2}\sigma^{2}+\frac{U^{2}}{V^{2}}\tau^{2}+\frac{1}{c}\left(U\sigma+\frac{U}{V}\tau+\frac{U^{2}}{V}\sigma\tau\right)\,,\end{split}$
(2.4)

so that all non-trivial interacting information is given by the $R$-symmetry
invariant correlator $\mathcal{T}(U,V)$. The second solution takes the form
[43]

$\begin{split}&\mathcal{S}(U,V;\sigma,\tau)=\Theta(U,V;\sigma,\tau){\mathcal{G}}(U,V)+\Phi_{1}(z,\bar{z};\sigma,\tau)f_{1}(z)+\Phi_{3}(z,\bar{z};\sigma,\tau)f_{2}(z)\\\
&+(\Phi_{1}(\bar{z},z;\sigma,\tau)-\Phi_{2}(U;\sigma,\tau))f_{1}(\bar{z})+(\Phi_{3}(\bar{z},z;\sigma,\tau)-\Phi_{2}(U;\sigma,\tau))f_{2}(\bar{z})+\Phi_{2}(U;\sigma,\tau)f_{3}(z)\,,\\\
\end{split}$ (2.5)

where $\Theta$ is the same as in (2.3) and we define

$\begin{split}U&=z\bar{z}\,,\quad
V=(1-z)(1-\bar{z})\,,\quad\Phi_{1}\equiv\frac{z\bar{z}}{z-\bar{z}}+\frac{z^{2}\bar{z}\tau}{(z-\bar{z})(1-z)}+\frac{z\bar{z}^{2}\sigma}{\bar{z}-z}\,,\quad\Phi_{2}\equiv\sigma
U\,,\\\
\Phi_{3}&\equiv\frac{z(\bar{z}-1)}{z-\bar{z}}+\frac{z^{2}\bar{z}\sigma^{2}}{\bar{z}-z}+\frac{z^{2}\bar{z}\tau^{2}+z^{2}\bar{z}(z-2)\sigma\tau+z(z+\bar{z}-2z\bar{z})\tau}{(z-1)(z-\bar{z})}+\frac{z(z+\bar{z}-\bar{z}^{2})\sigma}{z-\bar{z}}\,.\\\
\end{split}$ (2.6)

If we then equate (2.5) to (2.3), we find that the free theory correlator
fixes the holomorphic functions $f_{j}$ to be

$\begin{split}f_{1}(z)=2+\frac{1}{c}-\frac{1}{z}+\frac{1}{z-1}\,,\qquad
f_{2}(z)=1+\frac{1}{c}-z+\frac{1}{1-z}\,,\qquad
f_{3}(z)=\frac{1}{c}+z+\frac{1}{z}\,,\end{split}$ (2.7)

while the remaining free part contributes to ${\mathcal{G}}(U,V)$, which is
then related to ${\mathcal{T}}(U,V)$ as

$\begin{split}{\mathcal{T}}(U,V)={\mathcal{G}}(U,V)-\left[1+\frac{1}{V^{2}}+\frac{1}{c}\frac{1}{V}\right]\,.\end{split}$
(2.8)

We can expand ${\mathcal{G}}(U,V)$ in terms of long and short multiplets as

$\begin{split}{\mathcal{G}}(U,V)=U^{-2}\sum_{\Delta,\ell}\lambda^{2}_{\Delta,\ell}G_{\Delta+4,\ell}(U,V)+\mathcal{F}^{(0)}_{\text{short}}(z,\bar{z})+c^{-1}\mathcal{F}^{(1)}_{\text{short}}\,,\end{split}$
(2.9)

where $G_{\Delta,\ell}(U,V)$ with scaling dimension $\Delta$ and spin $\ell$
are 4d conformal blocks

$\begin{split}G_{\Delta,\ell}(U,V)&=\frac{z\bar{z}}{z-\bar{z}}(k_{\Delta+\ell}(z)k_{\Delta-\ell-2}(\bar{z})-k_{\Delta+\ell}(\bar{z})k_{\Delta-\ell-2}(z))\,,\\\
k_{h}(z)&\equiv z^{\frac{h}{2}}{}_{2}F_{1}(h/2,h/2,h,z)\,,\end{split}$ (2.10)

while the short multiplets OPE coefficients do not depend on the coupling and
so can be computed from the free theory to give the exact expressions
$\mathcal{F}_{\text{short}}^{(0)}$ and $\mathcal{F}_{\text{short}}^{(1)}$
given in [43]. All non-trivial interacting information in this formulation is
then given by $\Delta$ and $\ell$ for the long multiplets.

###  2.2 Strong coupling expansion

We now restrict our discussion to the case of ${\cal N}=4$ SYM with gauge
groups $SU(N)$, $SO(2N+1)$, $SO(2N)$, or $USp(2N)$. The conformal anomaly is
the dimension of the group divided by four, which for each case gives

$\begin{split}c_{SU(N)}=\frac{N^{2}-1}{4}\,,\qquad
c_{SO(2N)}=\frac{2N^{2}-N}{4}\,,\qquad
c_{SO(2N+1)}=c_{USp(2N)}=\frac{2N^{2}+N}{4}\,.\end{split}$ (2.11)

We will consider two strong coupling limits at large $c$. In the ’t Hooft
limit, we define a ’t Hooft coupling $\lambda$ for each gauge group as

$\begin{split}\lambda_{SU(N)}&\equiv
g_{\text{YM}}^{2}N\,,\qquad\qquad\qquad\,\lambda_{SO(2N)}\equiv
g_{\text{YM}}^{2}\left(N-\frac{1}{4}\right)\,,\\\ \lambda_{SO(2N+1)}&\equiv
g_{\text{YM}}^{2}\left(N+\frac{1}{4}\right)\,,\qquad\lambda_{USp(2N)}\equiv\frac{1}{2}g_{\text{YM}}^{2}\left(N+\frac{1}{4}\right)\,,\end{split}$
(2.12)

and then consider the limit of large $c$ with fixed $\lambda$, and then large
$\lambda$. We will also consider the very strong coupling limit where we keep
the complexified coupling $\tau$ finite, from which we can recover the ’t
Hooft strong coupling limit by expanding in small $\tau$. For either strong
coupling expansion, it is convenient to consider the Mellin transform $M$ of
${\mathcal{T}}$ via

$\begin{split}{\mathcal{T}}(U,V)=\int_{-i\infty}^{i\infty}\frac{ds\,dt}{(4\pi
i)^{2}}U^{\frac{s}{2}}V^{\frac{t}{2}-2}\Gamma\bigg{[}2-\frac{s}{2}\bigg{]}^{2}\Gamma\bigg{[}2-\frac{t}{2}\bigg{]}^{2}\Gamma\bigg{[}2-\frac{u}{2}\bigg{]}^{2}M(s,t)\,,\end{split}$
(2.13)

where $u\equiv 4-s-t$. Crossing symmetry $M(s,t)=M(t,s)=M(s,u)$ and the
analytic properties of the Mellin amplitude then restrict $M(s,t)$ to have a
$1/c$ and $1/\lambda$ expansion of the form (for a detailed description, see
[15])

$\displaystyle
M=\frac{1}{c}\Big{[}8M^{R}+\frac{B_{0}^{R^{4}}M^{0}}{\lambda^{\frac{3}{2}}}+\frac{1}{\lambda^{\frac{5}{2}}}[B_{2}^{D^{4}R^{4}}M^{2}+B_{0}^{D^{4}R^{4}}M^{0}]$
$\displaystyle\qquad\qquad+\frac{1}{\lambda^{3}}[B_{3}^{D^{6}R^{4}}M^{3}+B_{2}^{D^{6}R^{4}}M^{2}+B_{0}^{D^{6}R^{4}}M^{0}]+O(\lambda^{-\frac{7}{2}})\Big{]}$
$\displaystyle+\frac{1}{c^{2}}\Big{[}\lambda^{\frac{1}{2}}\bar{B}_{0}^{R^{4}}M^{0}+[M^{{R}|{R}}+\bar{B}^{{R}|{R}}_{0}M^{0}]+\frac{1}{\lambda}[\bar{B}_{3}^{D^{6}R^{4}}M^{3}+\bar{B}_{2}^{D^{6}R^{4}}M^{2}+\bar{B}_{0}^{D^{6}R^{4}}M^{0}]+O(\lambda^{-\frac{3}{2}})\Big{]}$
$\displaystyle+\frac{1}{c^{3}}\left[\lambda^{\frac{3}{2}}\bar{\bar{B}}_{2}^{D^{4}R^{4}}M^{2}+\lambda\left[\bar{\bar{B}}_{3}^{D^{6}R^{4}}M^{3}+\bar{\bar{B}}_{2}^{D^{6}R^{4}}M^{2}+\bar{\bar{B}}_{0}^{D^{6}R^{4}}M^{0}\right]+O(\lambda^{\frac{1}{2}})\right]$
$\displaystyle+\frac{1}{c^{4}}\left[\lambda^{3}\left[\bar{\bar{\bar{B}}}_{3}^{D^{6}R^{4}}M^{3}+\bar{\bar{\bar{B}}}_{2}^{D^{6}R^{4}}M^{2}+\bar{\bar{\bar{B}}}_{0}^{D^{6}R^{4}}M^{0}\right]+O(\lambda^{\frac{5}{2}})\right]+O(c^{-5})\,,$
(2.14)

where the $B$’s are numerical coefficients that cannot be fixed from symmetry
alone. Here, $A|B$ refers to a term that receives contributions from a 1-loop
Witten diagrams with $A,B$ vertices.222These terms can also receive
contributions from other Witten diagrams that contribute at the same order,
such as cubic supergravity vertices for $R|R$. Abstractly, each Mellin
amplitude is simply an allowed solution to the analytic bootstrap constraints
at the given order in $c$ and $\lambda$. Terms at order $1/c^{g+1}$ correspond
in the flat space limit to genus-$g$ corrections to the type IIB S-matrix in
the small $g_{s}$ expansion. On AdS5, these terms receive contributions from
$l$-loop Witten diagrams with $l\leq g$. The leading order term is tree-level
supergravity [48, 49]

$\begin{split}M^{R}=&\frac{1}{(s-2)(t-2)(u-2)}\,,\end{split}$ (2.15)

whose coefficient is fixed by requiring that the unprotected R-symmetry
singlet of dimension two that appears in the conformal block decomposition of
the free part ${{\mathcal{S}}}_{\text{free}}$ in (2.3) is not present in the
full correlator [2]. The $M^{n}$ terms in (2.14) arise from contact Witten
diagrams with vertices of the form $D^{2n}R^{4}$ and are degree $n$ crossing
symmetric polynomials [49]

$\begin{split}M^{0}=1\,\qquad M^{2}=s^{2}+t^{2}+u^{2}\,,\qquad
M^{3}=stu\,.\end{split}$ (2.16)

The 1-loop term ${M}^{{R}|{R}}$ arises from a loop Witten diagram with two $R$
vertices and scales as $c^{-2}$, and so includes a constant contact term
ambiguity. We will discuss this 1-loop term more in Section 4, along with
other 1-loop terms ${M}^{{R}|{R^{4}}}$ and ${M}^{{R^{4}}|{R^{4}}}$ that scale
as $c^{-2}\lambda^{-\frac{5}{2}}$ and $c^{-2}\lambda^{-3}$, respectively, and
so include several contact term ambiguities. The analytic bootstrap similarly
restricts the $1/c$ and finite $\tau$ expansion to be

$\displaystyle M$
$\displaystyle=\frac{8}{c}M^{R}+\frac{1}{c^{\frac{7}{4}}}{\tilde{B}}_{0}^{R^{4}}(\tau,\bar{\tau})M^{0}+\frac{1}{c^{2}}[M^{{R}|{R}}+\overline{B}^{{R}|{R}}_{0}M^{0}]+\frac{1}{c^{\frac{9}{4}}}[{\tilde{B}}_{2}^{D^{4}R^{4}}(\tau,\bar{\tau})M^{2}+{\tilde{B}}_{0}^{D^{4}R^{4}}(\tau,\bar{\tau})M^{0}]$
$\displaystyle\quad+\frac{1}{c^{\frac{5}{2}}}[{\tilde{B}}_{3}^{D^{6}R^{4}}(\tau,\bar{\tau})M^{3}+{\tilde{B}}_{2}^{D^{6}R^{4}}(\tau,\bar{\tau})M^{2}+{\tilde{B}}_{0}^{D^{6}R^{4}}(\tau,\bar{\tau})M^{0}]+O(c^{-\frac{11}{4}})\,,$
(2.17)

where the ${\tilde{B}}$’s now depend on $\tau$ except the $R|R$ terms, which
are the same as in the ’t Hooft expansion. Next, we will discuss how some of
these coefficients can be fixed by comparing to type IIB string theory in the
flat space limit.

###  2.3 Constraints from flat space type IIB string theory

The type IIB string theory S-matrix is restricted by supersymmetry to be
proportional to a single function $f(s,t)$

$\begin{split}{\cal A}(s,t)={\mathcal{A}}_{R}(s,t)f(s,t)\,,\end{split}$ (2.18)

where $s,t,u=-s-t$ are 10d Mandelstam variables and ${\mathcal{A}}_{R}$ is the
tree-level four-point supergravity amplitude given in e.g. [50]. We can expand
at small string length $\ell_{s}$ to get

$\begin{split}f(s,t)=1+\ell^{6}_{s}{f}_{R^{4}}+\ell^{8}_{s}{f}_{R|R}+\ell^{10}_{s}{f}_{D^{4}R^{4}}+\ell^{12}_{s}{f}_{D^{6}R^{4}}+\ell_{s}^{14}{f}_{R|R^{4}}+\dots\,,\end{split}$
(2.19)

where each coefficient depends non-trivially on the complexified string
coupling $\tau_{s}=\chi_{s}+i/g_{s}$. The lowest few corrections are
protected, and so can be computed in terms of modular functions as [51, 52,
53, 54]

$\begin{split}f_{R^{4}}&=\frac{stu}{64}g_{s}^{\frac{3}{2}}E({\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})\,,\\\
f_{D^{4}R^{4}}&=\frac{stu(s^{2}+t^{2}+u^{2})}{2^{11}}g_{s}^{\frac{5}{2}}E({\scriptstyle\frac{5}{2}},\tau_{s},\bar{\tau}_{s})\,,\\\
f_{D^{6}R^{4}}&=\frac{3(stu)^{2}}{2^{12}}g_{s}^{3}{\mathcal{E}}(3,{\scriptstyle\frac{3}{2}},{\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})\,.\end{split}$
(2.20)

The definition of
${\mathcal{E}}(3,{\scriptstyle\frac{3}{2}},{\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})$
is given in [55, 22] and will not be used in this work, while the other
modular functions are Eisenstein series $E(r,\tau_{s},\bar{\tau}_{s})$, which
can be expanded at small $g_{s}$ as

$\begin{split}E(r,\tau_{s},\bar{\tau}_{s})={}&\frac{2\zeta(2r)}{g_{s}^{r}}+2\sqrt{\pi}g_{s}^{r-1}\frac{\Gamma(r-\frac{1}{2})}{\Gamma(r)}\zeta(2r-1)\\\
&+\frac{2\pi^{r}}{\Gamma(r)\sqrt{g_{s}}}\sum_{k\neq 0}\left\lvert
k\right\rvert^{r-{\scriptstyle\frac{1}{2}}}\sigma_{1-2r}(|k|)\,K_{r-\frac{1}{2}}(2\pi
g_{s}^{-1}\left\lvert k\right\rvert)\,e^{2\pi ik\chi_{s}}\,,\end{split}$
(2.21)

where the divisor sum $\sigma_{p}(k)$ is defined as
$\sigma_{p}(k)=\sum_{d>0,{d|k}}d^{p}$, and $K_{r-\frac{1}{2}}$ is the Bessel
function of second kind. Note that each Eisenstein series includes only two
perturbative in $g_{s}$ terms, which implies that in the small $g_{s}$
expansion $f_{R^{4}}$ and $f_{D^{4}R^{4}}$ get corrections at only genus 1 and
2, respectively. The modular function
${\mathcal{E}}(3,{\scriptstyle\frac{3}{2}},{\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})$
only has four perturbative terms [55, 22], so $f_{D^{6}R^{4}}$ is only
corrected up to genus 3. The small $\ell_{s}$ expansion in 10d maps to the
large $c\sim N^{2}$ expansion in SYM for the various gauge groups according to
the dictionary [56, 57, 58]:

$\displaystyle SU(N):$
$\displaystyle\qquad\frac{L^{4}}{\ell^{4}_{s}}={g_{\text{YM}}^{2}N}=\lambda\,,\qquad$
$\displaystyle\tau_{s}=\tau\,,$ (2.22) $\displaystyle SO(2N):$
$\displaystyle\qquad\frac{L^{4}}{\ell^{4}_{s}}=2{g_{\text{YM}}^{2}\Big{(}N-\frac{1}{4}\Big{)}}=2\lambda\,,\qquad$
$\displaystyle\tau_{s}=\tau\,,$ $\displaystyle SO(2N+1):$
$\displaystyle\qquad\frac{L^{4}}{\ell^{4}_{s}}=2{g_{\text{YM}}^{2}\Big{(}N+\frac{1}{4}\Big{)}}=2\lambda\,,\qquad$
$\displaystyle\tau_{s}=\tau\,,$ $\displaystyle USp(2N):$
$\displaystyle\qquad\frac{L^{4}}{\ell^{4}_{s}}={g_{\text{YM}}^{2}\Big{(}N+\frac{1}{4}\Big{)}}=2\lambda\,,\qquad$
$\displaystyle\tau_{s}=2\tau\,.$

The flat space limit formula [59, 60, 17, 45] then relates the Mellin
amplitude to the 10d amplitude defined in (2.19) as

$\begin{split}f(s,t)=\frac{stu}{32}\lim_{L/\ell_{s}\to\infty}L^{6}c\int_{\kappa-i\infty}^{\kappa+i\infty}\frac{d\alpha}{2\pi
i}\,e^{\alpha}\alpha^{-6}{M}\left(\frac{L^{2}}{2\alpha}s,\frac{L^{2}}{2\alpha}t\right)\,,\end{split}$
(2.23)

where the dependence on the different SYM gauge groups comes from the
relationship between $L$, $\ell_{s}$, and the CFT parameters $c$ and $\lambda$
or $\tau$ in (2.22). We can apply this to $M^{R^{4}}$ in the finite $\tau$
expansion (2.17) to fix its coefficient to be

$\begin{split}{\tilde{B}}_{0}^{R^{4}}(\tau,\bar{\tau})&=\frac{15}{4\sqrt{2\pi^{3}}\mathfrak{o}^{\frac{3}{4}}}E({\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})\,,\end{split}$
(2.24)

where recall that the orbifold theories ($SO(2N+1)$, $SO(2N)$ and $USp(2N)$)
differ from $SU(N)$ by a power of 2 due to the different AdS/CFT dictionary
(2.22), which we denote using the orbifold factor $\mathfrak{o}$, which is 1
for $SU(N)$ and 2 for the orbifold theories. Also, $\tau_{s}=\tau$ for all
theories except $USp(2N)$ with $\tau_{s}=2\tau$. The ’t Hooft expansion can
then be recovered by looking at the perturbative terms in the Eisenstein
series and writing $g_{\text{YM}}$ in terms of $\lambda$ and $c$ to get

$\begin{split}{B}_{0}^{R^{4}}&=\frac{120\zeta(3)}{\mathfrak{o}^{\frac{3}{2}}}\,,\qquad\bar{B}_{0}^{R^{4}}=\frac{5}{8\mathfrak{o}^{\frac{1}{2}}}\,.\end{split}$
(2.25)

The flat space constraint is not sufficient to completely fix the other Mellin
amplitudes, but can be used together with other methods that we will now
describe.

##  3 Constraints from supersymmetric localization

There are two known constraints on the stress tensor correlator coming from
supersymmetric localization. The first relates mixed $\tau$ and hypermultiplet
mass $m$ derivatives of the ${\cal N}=2^{*}$ sphere free energy $F=-\log Z$ to
the integrated correlator in Mellin space as [17, 45]

$\begin{split}c^{2}I_{2}[M(s,t)]=&\frac{c}{8}\frac{\partial_{m}^{2}\partial_{\tau}\partial_{\bar{\tau}}F}{\partial_{\tau}\partial_{\bar{\tau}}F}\Big{|}_{m=0}\,,\\\
\end{split}$ (3.1)

where the integral is defined as

$\begin{split}I_{2}[M(s,t)]&\equiv-\frac{1}{8}\int_{-i\infty}^{i\infty}\frac{ds\,dt}{(2\pi
i)^{2}}\Gamma\Big{[}2-\frac{s}{2}\Big{]}\Gamma\Big{[}\frac{s}{2}\Big{]}\Gamma\Big{[}2-\frac{t}{2}\Big{]}\Gamma\Big{[}\frac{t}{2}\Big{]}\Gamma\Big{[}2-\frac{s}{2}-\frac{t}{2}\Big{]}\Gamma\Big{[}\frac{s+t}{2}\Big{]}M(s,t)\,.\end{split}$
(3.2)

The second constraint relates $\partial_{m}^{4}F\big{|}_{m=0}$ to a different
integral of the correlator whose explicit form is given in [45], but which we
will not consider in detail in this paper. The $\mathcal{N}=2^{*}$ sphere
partition function for general gauge group $G$ was computed using
supersymmetric localization in [33] to get

$\begin{split}Z(m)=\frac{1}{|W|}\int
d^{r}a|Z_{\text{inst}}(m,\tau,a)|^{2}e^{-\frac{8\pi^{2}N}{\lambda}(a,a)}\frac{1}{H(m)^{r}}\prod_{\alpha\in\Delta}\frac{\alpha(a)H(\alpha(a))}{H(\alpha(a)+m)}\,,\end{split}$
(3.3)

where $r$ denotes the rank of $G$, $W$ is the Weyl group, $(a,a)$ is the
Killing form for the choice of basis,333The Killing form is given by
$\frac{1}{2T_{r}}\mathop{\rm tr}\nolimits_{r}$, where $\mathop{\rm
tr}\nolimits_{r}$ is the usual trace in the representation $r$, and $T_{r}$ is
the Dynkin index. $\Delta$ is the set of roots, and $H(z)$ is a product of two
Barnes G-functions, namely $H(z)=e^{-(1+\gamma)z^{2}}G(1+iz)G(1-iz)$. The
quantity $\left\lvert Z_{\text{inst}}(m,\tau,a)\right\rvert^{2}$ represents
the contribution to the localized partition function coming from instantons
located at the North and South poles of $S^{4}$ [61, 62, 63, 64]. In this
section, we will first compute the relevant derivatives of $Z(m)$ for
$G=SU(N),SO(2N),SO(2N+1),USp(2N)$ at finite $N$ and $\lambda$ but neglecting
the instanton term. Then we will expand to all orders in $1/c$ and large
$\lambda$, where instantons would not contribute anyway because they are non-
perturbative in the ’t Hooft limit [65]. We will then write $\lambda$ in terms
of $c$ and $\tau$ to compute the perturbative contributions to the large $c$
finite $\tau$ limit, which can be conjecturally completed to finite $\tau$
Eisenstein series for each gauge group, as was proven in the $SU(N)$ case in
[21, 22]. Finally, we will plug these results into the constraint (3.1) to fix
the $R^{4}$ term in the correlator, which matches the results from the flat
space limit, and to fix the $D^{4}R^{4}$ term by also using the flat space
limit.

###  3.1 $\mathcal{N}=2^{*}$ sphere free energy at finite $N$

If we neglect the instanton term in (3.3), then we can compute $Z(m)$ in a
small $m$ expansion for finite $N$ and $\lambda$ for any gauge group using the
method of orthogonal polynomials [66]. For instance, at $m=0$ the free energy
for each gauge group is shown in Appendix A to be444Since $\left\lvert
Z_{\text{inst}}(0,\tau,a)\right\rvert^{2}=1$, this result also holds in
general.

$\begin{split}F(0)=-4c\log
g_{\text{YM}}+\text{$g_{\text{YM}}$-independent}\,,\end{split}$ (3.4)

where recall that $c$ is defined differently for each gauge group. We can then
simplify the quantity on the RHS of (3.1) to be

$\begin{split}\mathcal{F}\equiv-\frac{1}{16g_{\text{YM}}^{4}}\partial_{m}^{2}\partial_{g_{\text{YM}}^{-2}}^{2}F\big{|}_{m=0}\,,\end{split}$
(3.5)

where $\theta$ does not appear because we have ignored instantons. We can then
take the mass derivatives of the localized partition function in (3.3) for
each gauge group to get

$\begin{split}\mathcal{F}_{SU(N)}&=\frac{g_{\text{YM}}^{-4}}{16}\partial_{g_{\text{YM}}^{-2}}^{2}{\sum_{i\neq
j}\langle K^{\prime}(a_{i}-a_{j})\rangle}\,,\\\
\mathcal{F}_{SO(2N)}&=\frac{g_{\text{YM}}^{-4}}{16}\partial_{g_{\text{YM}}^{-2}}^{2}{\sum_{i\neq
j}\Big{[}\langle K^{\prime}(a_{i}-a_{j})\rangle+\langle
K^{\prime}(a_{i}+a_{j})\rangle\Big{]}}\,,\\\
\mathcal{F}_{SO(2N+1)}&=\frac{g_{\text{YM}}^{-4}}{16}\partial_{g_{\text{YM}}^{-2}}^{2}\Bigg{[}\sum_{i}\langle
K^{\prime}(a_{i})\rangle+{\sum_{i\neq j}\Big{[}\langle
K^{\prime}(a_{i}-a_{j})\rangle+\langle
K^{\prime}(a_{i}+a_{j})\rangle\Big{]}}\Bigg{]}\,,\\\
\mathcal{F}_{USp(2N)}&=\frac{g_{\text{YM}}^{-4}}{16}\partial_{g_{\text{YM}}^{-2}}^{2}\Bigg{[}\sum_{i}\langle
K^{\prime}(2a_{i})\rangle+{\sum_{i\neq j}\Big{[}\langle
K^{\prime}(a_{i}-a_{j})\rangle+\langle
K^{\prime}(a_{i}+a_{j})\rangle\Big{]}}\Bigg{]}\,,\\\ \end{split}$ (3.6)

where the expectation values are taken in terms of the zero mass partition
function, and $K(z)\equiv-\frac{H^{\prime}(z)}{H(z)}$ can be conveniently
written in terms of its Fourier transform

$\begin{split}K^{\prime}(z)=-\int_{0}^{\infty}d\omega\frac{2\omega[\cos(2\omega
z)-1]}{\sinh^{2}\omega}\,.\end{split}$ (3.7)

As shown in Appendix A, the zero mass partition functions are Gaussian matrix
models, so we can compute the expectation values of the exponential (3.7) in
these ensembles using orthogonal polynomials following [67] to get

$\begin{split}\mathcal{F}_{SU(N)}&=-\int_{0}^{\infty}\frac{\omega
g_{\text{YM}}^{-4}}{8\sinh^{2}\omega}\partial_{g_{\text{YM}}^{-2}}^{2}\sum_{i,j=1}^{N}e^{\frac{-\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\Bigg{[}L_{i-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{j-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad-(-1)^{i-j}L_{i-1}^{j-i}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{j-1}^{i-j}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\Bigg{]}\,,\\\
\mathcal{F}_{SO(2N)}&=-\int_{0}^{\infty}\frac{\omega
g_{\text{YM}}^{-4}}{4\sinh^{2}\omega}\partial_{g_{\text{YM}}^{-2}}^{2}\sum_{i,j=1}^{N}e^{\frac{-\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\Bigg{[}L_{2(i-1)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{2(j-1)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
L_{2(i-1)}^{2(j-i)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{2(j-1)}^{2(i-j)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\Bigg{]}\,,\\\
\mathcal{F}_{SO(2N+1)}&=-\int_{0}^{\infty}\frac{\omega
g_{\text{YM}}^{-4}}{4\sinh^{2}\omega}\partial_{g_{\text{YM}}^{-2}}^{2}\Bigg{(}e^{\frac{-\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\sum_{i,j=1}^{N}\Bigg{[}L_{2i-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{2j-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\\\
&\qquad-
L_{2i-1}^{2(j-i)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{2j-1}^{2(i-j)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\Bigg{]}+e^{\frac{-\omega^{2}g_{\text{YM}}^{2}}{8\pi^{2}}}\sum_{i=1}^{N}L_{2i-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\Bigg{)}\,,\\\
\mathcal{F}_{USp(2N)}&=-\int_{0}^{\infty}\frac{\omega
g_{\text{YM}}^{-4}}{4\sinh^{2}\omega}\partial_{g_{\text{YM}}^{-2}}^{2}\Bigg{(}e^{\frac{-\omega^{2}g_{\text{YM}}^{2}}{8\pi^{2}}}\sum_{i,j=1}^{N}\Bigg{[}L_{2i-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{8\pi^{2}}}\right)L_{2j-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{8\pi^{2}}}\right)\\\
&\qquad-
L_{2i-1}^{2(j-i)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{8\pi^{2}}}\right)L_{2j-1}^{2(i-j)}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{8\pi^{2}}}\right)\Bigg{]}+e^{\frac{-\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\sum_{i=1}^{N}L_{2i-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{2\pi^{2}}}\right)\Bigg{)}\,,\\\
\end{split}$ (3.8)

where $L_{i}^{j}(z)$ are generalized Laguerre polynomials, and the $SU(N)$
case was already computed in [15]. For $SO(2N+1)$, the result holds for $N>1$,
while for $SO(3)$ we must rescale the quantity in parentheses by
$g_{\text{YM}}\to\sqrt{2}g_{\text{YM}}$, where the difference comes from the
discontinuity of the standard Killing form $(a,a)$, i.e. of the Dynkin index,
for $SO(N)$ with $N=3$ and $N>3$. One can check that these formulae satisfy
the Lie algebra isomorphisms $SU(2)\cong SO(3)\cong USp(2)$, $SU(4)\cong
SO(6)$, $SU(2)\times SU(2)\cong SO(4)$, and $SO(5)\cong USp(4)$.

###  3.2 $1/N$ expansions

We can now take the large $N$ limit of these finite $N$ results. From the
$SU(N)$ case considered in [15], which was computed to all orders in $1/N$
using topological recursion for $U(N)$ Gaussian matrix models [68, 69], we
expect that the finite $\lambda$ answer should be written in terms of Bessel
functions. Since the topological recursion relations have not yet been derived
for the other gauge groups, we will instead directly expand the finite $N$ and
$\lambda$ result in (3.8). We do this by first considering the weak coupling
expansion, where we define the weak coupling limit ’t Hooft coupling for each
gauge group as

$\begin{split}\lambda^{SU(N)}_{w}&=Ng_{\text{YM}}^{2}\,,\qquad\qquad\quad\;\,\lambda^{SO(2N)}_{w}=2Ng_{\text{YM}}^{2}\,,\\\
\lambda^{SO(2N+1)}_{w}&=(2N+1)g_{\text{YM}}^{2}\,,\qquad\lambda^{USp(2N)}_{w}=\frac{1}{2}(2N+1)g_{\text{YM}}^{2}\,.\end{split}$
(3.9)

We then perform the sums over $i,j$ in (3.8) at each order in $\lambda_{w}$,
take the large $N$ expansion, and finally resum $\lambda_{w}$ at each order in
$1/N$. To order $O(N^{0})$ we found

$\begin{split}&{\mathcal{F}}_{SU(N)}=N^{2}\int_{0}^{\infty}d\omega\,\omega\frac{J_{1}(\frac{\sqrt{\lambda_{w}}}{\pi}\omega)^{2}-J_{2}(\frac{\sqrt{\lambda_{w}}}{\pi}\omega)^{2}}{4\sinh^{2}\omega}+O(N^{0})\,,\\\
\end{split}$ (3.10)

and

$\begin{split}&\mathcal{F}_{SO(2N)}=\int_{0}^{\infty}\frac{d\omega}{\sinh^{2}\omega}\Bigg{[}\frac{N^{2}}{2}\Big{(}\frac{\left(\lambda_{w}\omega^{2}-4\pi^{2}\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}}{\lambda_{w}\omega}-\omega
J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}+\frac{4\pi
J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)}{\sqrt{\lambda_{w}}}\Big{)}\\\
&+\frac{N\omega}{16\pi}\left(2\pi
J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}+\sqrt{\lambda_{w}}w\left(1-4J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)-2\sqrt{\lambda_{w}}\omega
J_{1}\left({\scriptstyle\frac{2\omega\sqrt{\lambda_{w}}}{\pi}}\right)\right)\Bigg{]}+O(N^{0})\,,\\\
\end{split}$ (3.11)

and

$\begin{split}&\mathcal{F}_{SO(2N+1)}=\int_{0}^{\infty}\frac{d\omega}{\sinh^{2}\omega}\Bigg{[}\frac{N^{2}}{2}\Big{(}\frac{\left(\lambda_{w}\omega^{2}-4\pi^{2}\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}}{\lambda_{w}\omega}-\omega
J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}+\frac{4\pi
J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)}{\sqrt{\lambda_{w}}}\Big{)}\\\
&-\frac{N}{16\pi\lambda_{w}\omega}\Big{(}2\lambda_{w}^{3/2}\omega^{3}J_{1}\left({\scriptstyle\frac{2\omega\sqrt{\lambda_{w}}}{\pi}}\right)+8\pi\lambda_{w}\omega^{2}J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}-{{\lambda^{\frac{3}{2}}_{w}}\omega^{3}J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)}\\\
&+4\sqrt{\lambda_{w}}\omega\left(\lambda_{w}\omega^{2}-8\pi^{2}\right)J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)+2\left(16\pi^{3}-5\pi\lambda_{w}\omega^{2}\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}\Big{)}\Bigg{]}+O(N^{0})\,,\end{split}$
(3.12)

and

$\begin{split}&\mathcal{F}_{USp(2N)}=\int_{0}^{\infty}\frac{d\omega}{\sinh^{2}\omega}\Bigg{[}\frac{N^{2}}{2}\Big{(}\frac{\left(\lambda_{w}\omega^{2}-4\pi^{2}\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}}{\lambda_{w}\omega}-\omega
J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}+\frac{4\pi
J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)}{\sqrt{\lambda_{w}}}\Big{)}\\\
&-\frac{N}{16\pi\lambda_{w}\omega}\Big{(}\lambda_{w}^{3/2}\omega^{3}J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)+2\lambda_{w}^{3/2}\omega^{3}J_{1}\left({\scriptstyle\frac{2\omega\sqrt{\lambda_{w}}}{\pi}}\right)+8\pi\lambda_{w}\omega^{2}J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}-4{{\lambda^{\frac{3}{2}}_{w}}\omega^{3}J_{1}\left({\scriptstyle\frac{2\omega\sqrt{\lambda_{w}}}{\pi}}\right)}\\\
&+4\sqrt{\lambda_{w}}\omega\left(\lambda_{w}\omega^{2}-8\pi^{2}\right)J_{0}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)+2\left(16\pi^{3}-5\pi\lambda_{w}\omega^{2}\right)J_{1}\left({\scriptstyle\frac{\omega\sqrt{\lambda_{w}}}{\pi}}\right)^{2}\Big{)}\Bigg{]}+O(N^{0})\,,\end{split}$
(3.13)

where $\lambda_{w}$ is defined in each case in (3.9). Note that the $SU(N)$
case only includes even powers of $N$ and pairs of Bessel functions, while the
other cases include all powers of $N$ and single Bessel functions. This
pattern continues at subleading orders in $1/N$, which are given in the
attached Mathematica file. For $SU(N)$, these results match those computed
from topological recursion in [15].

The next step is to compute the strong coupling expansion by looking at large
’t Hooft coupling. For this we want to use the ’t Hooft coupling that is
naturally related to the AdS radius, which is the $\lambda$ defined for each
gauge group in (2.12).555For $SU(N)$, the strong and weak coupling definitions
coincide, but for the other cases they differ. To do the large $\lambda$
expansion, we use the Mellin-Barnes formulae for Bessel functions:

$\begin{split}J_{\mu}(x)J_{\nu}(x)&=\frac{1}{2\pi i}\int_{c-\infty
i}^{c+\infty
i}ds\frac{\Gamma(-s)\Gamma(2s+\mu+\nu+1)\left(\frac{1}{2}x\right)^{\mu+\nu+2s}}{\Gamma(s+\mu+1)\Gamma(s+\nu+1)\Gamma(s+\mu+\nu+1)}\,,\\\
J_{\mu}(x)&=\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty
i}ds\frac{\Gamma(-s)\left(\frac{1}{2}x\right)^{\mu+2s}}{\Gamma(s+\mu+1)}\,,\\\
\end{split}$ (3.14)

perform the $\omega$ integrals using

$\begin{split}\int_{0}^{\infty}d\omega\
\frac{\omega^{a}}{\sinh^{2}\omega}=\frac{1}{2^{a-1}}\Gamma(a+1)\zeta(a)\,,\end{split}$
(3.15)

and finally close the $s$ contours to the left (closing to the right would
give the weak coupling expansion). For $SU(N)$ we get

$\begin{split}\mathcal{F}_{SU(N)}&=c\left(\frac{1}{4}-\frac{3\zeta(3)}{\lambda^{3/2}}+\frac{45\zeta(5)}{4\lambda^{5/2}}+O(\lambda^{-\frac{7}{2}})\right)+\Big{(}-\frac{\sqrt{\lambda}}{64}+\frac{1}{16}+O(\lambda^{-\frac{3}{2}})\Big{)}\\\
&\qquad+\frac{1}{c}\Big{(}\frac{{\lambda^{\frac{3}{2}}}}{24576}+O(\lambda^{\frac{1}{2}})\Big{)}+O(c^{-2})\,,\\\
\end{split}$ (3.16)

where we wrote $N$ in terms of $c$ using (2.11), which is where the difference
between $U(N)$ and $SU(N)$ finally enters, and these results recover those in
[15]. For $SO(2N)$, $SO(2N+1)$ and $USp(2N)$, after using the different
definitions of $c$ in (2.11) and $\lambda$ in (2.12) for each case, we find
essentially the same answer

$\begin{split}\mathcal{F}_{orb}&=c\left(\frac{1}{4}-\frac{3\zeta(3)}{2\sqrt{2}\lambda^{3/2}}+\frac{45\zeta(5)}{16\sqrt{2}\lambda^{5/2}}+O(\lambda^{-\frac{7}{2}})\right)+\Big{(}-\frac{\sqrt{\lambda}}{64\sqrt{2}}+\frac{1}{128}+O(\lambda^{-\frac{3}{2}})\Big{)}\\\
&\qquad+\frac{1}{c}\Big{(}\frac{{\lambda^{\frac{3}{2}}}}{24576\sqrt{2}}+O(\lambda^{\frac{1}{2}})\Big{)}+O(c^{-2})\,,\end{split}$
(3.17)

which involved nontrivial cancellations so that other powers of $c$ and
$\lambda$ would not appear. For all gauge groups we only find half integer
powers of $\lambda$, unlike $\partial_{m}^{4}F\big{|}_{m=0}$ as computed for
$SU(N)$ in [45], which included both integer and half integer powers of
$\lambda$.

Finally, we consider the large $c$ and finite $\tau$ limit. We can compute the
terms which are perturbative in $g_{\text{YM}}$ from the ’t Hooft limit by
simply expressing $\lambda$ in terms of $c$ and $g_{\text{YM}}$ using the
different definitions for each gauge group, and then reexpanding at large $c$.
By expanding (3.16) and (3.17) to several more orders in this manner using the
expressions in the attached Mathematica notebook, we found that they are
nontrivially consistent with the Eisenstein series expansions

$\begin{split}&\mathcal{F}_{SU(N)}=\frac{c}{4}+\frac{1}{16}-c^{\frac{1}{4}}\frac{3E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{32\sqrt{2}\pi^{\frac{3}{2}}}+\frac{1}{c^{\frac{1}{4}}}\frac{45E({\scriptstyle\frac{5}{2}},\tau,\bar{\tau})}{1024\sqrt{2}\pi^{\frac{5}{2}}}\\\
&\;+\frac{1}{c^{\frac{3}{4}}}\Bigg{[}\frac{4725E({\scriptstyle\frac{7}{2}},\tau,\bar{\tau})}{262144\sqrt{2}\pi^{\frac{7}{2}}}-\frac{423E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{65536\sqrt{2}\pi^{\frac{3}{2}}}\Bigg{]}+\frac{1}{c^{\frac{5}{4}}}\Bigg{[}\frac{99225E({\scriptstyle\frac{9}{2}},\tau,\bar{\tau})}{4194304\sqrt{2}\pi^{\frac{9}{2}}}-\frac{4005E({\scriptstyle\frac{5}{2}},\tau,\bar{\tau})}{1048576\sqrt{2}\pi^{\frac{5}{2}}}\Bigg{]}\\\
&\;+\frac{1}{c^{\frac{7}{4}}}\Bigg{[}\frac{245581875E({\scriptstyle\frac{11}{2}},\tau,\bar{\tau})}{4294967296\sqrt{2}\pi^{\frac{11}{2}}}-\frac{6440175E({\scriptstyle\frac{7}{2}},\tau,\bar{\tau})}{1073741824\sqrt{2}\pi^{\frac{7}{2}}}+\frac{93303E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{134217728\sqrt{2}\pi^{\frac{3}{2}}}\Bigg{]}+O(c^{-\frac{9}{4}})\,,\\\
\end{split}$ (3.18)

and

$\displaystyle\mathcal{F}_{orb}=\frac{c}{4}+\frac{1}{128}-c^{\frac{1}{4}}\frac{3E({\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})}{64\cdot{2}^{\frac{1}{4}}\pi^{\frac{3}{2}}}+\frac{1}{c^{\frac{1}{4}}}\frac{45E({\scriptstyle\frac{5}{2}},\tau_{s},\bar{\tau}_{s})}{2048\cdot
2^{\frac{3}{4}}\pi^{\frac{5}{2}}}$ (3.19)
$\displaystyle+\frac{1}{c^{\frac{3}{4}}}\Bigg{[}\frac{4725E({\scriptstyle\frac{7}{2}},\tau_{s},\bar{\tau}_{s})}{1048576\cdot
2^{\frac{1}{4}}\pi^{\frac{7}{2}}}-\frac{207E({\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})}{262144\cdot
2^{\frac{1}{4}}\pi^{\frac{3}{2}}}\Bigg{]}+\frac{1}{c^{\frac{5}{4}}}\Bigg{[}\frac{99225E({\scriptstyle\frac{9}{2}},\tau_{s},\bar{\tau}_{s})}{16777216\cdot
2^{\frac{3}{4}}\pi^{\frac{9}{2}}}-\frac{4545E({\scriptstyle\frac{5}{2}},\tau_{s},\bar{\tau}_{s})}{4194304\cdot
2^{\frac{3}{4}}\pi^{\frac{5}{2}}}\Bigg{]}$
$\displaystyle+\frac{1}{c^{\frac{7}{4}}}\Bigg{[}\frac{245581875E({\scriptstyle\frac{11}{2}},\tau_{s},\bar{\tau}_{s})}{34359738368\cdot
2^{\frac{1}{4}}\pi^{\frac{11}{2}}}-\frac{11656575E({\scriptstyle\frac{7}{2}},\tau_{s},\bar{\tau}_{s})}{8589934592\cdot
2^{\frac{1}{4}}\pi^{\frac{7}{2}}}+\frac{55503E({\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})}{1073741824\cdot
2^{\frac{1}{4}}\pi^{\frac{3}{2}}}\Bigg{]}+O(c^{-\frac{9}{4}})\,,$

where we used the perturbative terms in the Eisenstein expansion (2.21), and
recall that $\tau_{s}=\tau$ for all gauge groups except $USp(2N)$ where
$\tau_{s}=2\tau$. Note that there are only two perturbative terms in the
Eisenstein series expansion, which corresponds to the fact that there is a
finite number of perturbative terms at each order in $1/c$. For $SU(N)$, this
recovers the result that was rigorously derived in [21, 22].

###  3.3 Constraining the stress-tensor correlator

We can now input the localization results for
$\partial_{m}^{2}\partial_{\tau}\partial_{\bar{\tau}}F\big{|}_{m=0}$ into the
integrated constraint (3.1) to constrain the stress tensor correlator as
expanded at large $N$ in (2.14) and (2.17). For the degree $n$ polynomial
Mellin amplitudes $M^{n}$ in (2.16) as well as the tree level supergravity
amplitude $M^{R}$ in (2.15), we can compute the integrals in (3.2) using
Barnes’ Lemma

$\begin{split}\int_{-i\infty}^{i\infty}\frac{ds}{2\pi
i}\Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)=\frac{\Gamma(a+c)\Gamma(b+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)}\,,\end{split}$
(3.20)

to get [17, 45]

$\begin{split}I_{2}[M^{R}]&=\frac{1}{32}\,,\quad\qquad
I_{2}[M^{0}]=-\frac{1}{40}\,,\qquad I_{2}[M^{2}]=-\frac{2}{35}\,,\qquad
I_{2}[M^{3}]=-\frac{4}{35}\,.\\\ \end{split}$ (3.21)

The localization constraint (3.1) then fixes the $R^{4}$ coefficients to the
values in (2.24) and (2.25) that were independently fixed from the flat space
limit, which is a check of AdS/CFT at genus-1 for general gauge group,
generalizing the $SU(N)$ case first shown in [17]. We can then combine this
localization constraint with the flat space limit constraint to rigorously fix
the $D^{4}R^{4}$ coefficients in the ’t Hooft limit to be

$\begin{split}{B}_{2}^{D^{4}R^{4}}&=-\frac{1}{3}{B}_{0}^{D^{4}R^{4}}=\frac{630\zeta(5)}{\mathfrak{o}^{\frac{5}{2}}}\,,\qquad\bar{\bar{B}}_{2}^{D^{4}R^{4}}=-\frac{1}{3}\bar{\bar{B}}_{0}^{D^{4}R^{4}}=-\frac{7}{3072\mathfrak{o}^{\frac{1}{2}}}\,.\end{split}$
(3.22)

We can also use the conjectured localization expressions in (3.18) and (3.19)
for the large $c$ and finite $\tau$ limit to get

$\begin{split}{\tilde{B}}_{2}^{D^{4}R^{4}}(\tau_{s},\bar{\tau}_{s})&=-\frac{1}{3}{\tilde{B}}_{0}^{D^{4}R^{4}}(\tau_{s},\bar{\tau}_{s})=\frac{315}{128\sqrt{2\pi^{5}}\mathfrak{o}^{\frac{5}{4}}}E({\scriptstyle\frac{5}{2}},\tau_{s},\bar{\tau}_{s})\,.\end{split}$
(3.23)

For both the $R^{4}$ and $D^{4}R^{4}$ cases, we observe that the finite $\tau$
localization results for $SU(N)$ (3.18) and the orbifold cases (3.19) differ
by a factor of $2^{\frac{l}{8}}$, where $l$ is the $\ell_{s}$ scaling of the
corresponding term in the type IIB S-matrix (2.19). This is the same factor
that appeared in the constraint from the flat space limit formula (2.23),
which arises from the different AdS/CFT dictionary (2.22) for each case, and
is ultimately explained by the orbifold factor in the bulk duals. This
suggests that for all tree level terms666Since $c$ is the only expansion
parameter at finite $\tau$, from the bulk perspective there is in general no
difference between tree and loop level. At low orders, however, one can
distinguish between tree and loop terms by the different power of $c$ that
multiply them., the difference between the orbifold and non-orbifold cases
will simply be this orbifold factor, as was similarly observed for M-theory
duals in [30, 31]. For $SU(N)$, the $D^{6}R^{4}$ coefficients in (2.24) and
(2.25) were fixed in [45, 22] using the localization constraint from
$\partial_{m}^{4}F\big{|}_{m=0}$ combined with the
$\partial_{m}^{2}\partial_{\tau}\partial_{\bar{\tau}}F\big{|}_{m=0}$
constraint and the flat space limit. We can then divide by $2^{\frac{l}{8}}$
for $l=12$ to conjecturally obtain the orbifold results from the known $SU(N)$
results as

$\begin{split}{\tilde{B}}_{3}^{D^{6}R^{4}}(\tau_{s},\bar{\tau}_{s})&=-4{\tilde{B}}_{2}^{D^{6}R^{4}}(\tau_{s},\bar{\tau}_{s})=-\frac{1}{4}{\tilde{B}}_{0}^{D^{6}R^{4}}(\tau_{s},\bar{\tau}_{s})=\frac{945{\mathcal{E}}(3,{\scriptstyle\frac{3}{2}},{\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})}{64\pi^{3}\mathfrak{o}^{\frac{3}{2}}}\,.\end{split}$
(3.24)

The ’t Hooft limit coefficients in (2.24) can then be extracted from the
perturbative part of the modular function as given in [45, 22] to get

$\begin{split}{{{B}}}_{3}^{D^{6}R^{4}}&=-4{{{B}}}_{2}^{D^{6}R^{4}}=-\frac{1}{4}{{{B}}}_{0}^{D^{6}R^{4}}=\frac{5040\zeta(3)^{2}}{\mathfrak{o}^{3}}\,,\\\
{{{\bar{B}}}}_{3}^{D^{6}R^{4}}&=-4{{{\bar{B}}}}_{2}^{D^{6}R^{4}}=-\frac{1}{4}{{{\bar{B}}}}_{0}^{D^{6}R^{4}}=\frac{105\zeta(3)}{4\mathfrak{o}^{2}}\,,\\\
{\bar{{\bar{B}}}}_{3}^{D^{6}R^{4}}&=-4{\bar{{\bar{B}}}}_{2}^{D^{6}R^{4}}=-\frac{1}{4}{\bar{{\bar{B}}}}_{0}^{D^{6}R^{4}}=\frac{21}{256\mathfrak{o}}\,,\\\
\bar{\bar{{\bar{B}}}}_{3}^{D^{6}R^{4}}&=-4\bar{\bar{{\bar{B}}}}_{2}^{D^{6}R^{4}}=-\frac{1}{4}\bar{\bar{{\bar{B}}}}_{0}^{D^{6}R^{4}}=\frac{1}{221184}\,.\end{split}$
(3.25)

We will explore more constraints from localization and the flat space limit in
the next section, where we consider 1-loop terms.

##  4 Stress-tensor correlator at 1-loop

We will now compute the 1-loop terms with $R$ and $R^{4}$ vertices in the
large $c$ expansion for the $SU(N)$ and other gauge groups, dual to
$AdS_{5}\times S^{5}$ and $AdS_{5}\times S^{5}/\mathbb{Z}_{2}$, respectively.
Since the results for the ’t Hooft expansion can be trivially obtained from
the finite $\tau$ expansion, as discussed in previous sections, we will only
show the finite $\tau$ expansion in this section. In particular the $R|R$,
$R|R^{4}$, and $R^{4}|R^{4}$ terms appear at orders $c^{-2}$,
$c^{-\frac{11}{4}}$, and $c^{-\frac{7}{2}}$, respectively. We obtain the
1-loop terms by first computing the 1-loop double-discontinuity (DD) from tree
and GFFT data, and then using it to write the entire correlator in Mellin
space using crossing symmetry up to contact term ambiguities. We then compare
the correlators for both theories to the relevant 1-loop corrections to the
10d S-matrix in the flat space limit, and find a precise match for 1-loop
amplitudes on both $AdS_{5}\times S^{5}$ and $AdS_{5}\times
S^{5}/\mathbb{Z}_{2}$. For $R|R$, we use the localization constraints from the
previous section to fix the unique contact term ambiguity for each theory, and
find that they are simply related by the orbifold factor. Finally, we extract
low-lying CFT data using two methods: the Lorentzian inversion integral
applied to the DD that does not apply to certain low values of spin [70, 71],
and a projection method applied to the entire Mellin amplitude [1, 72]. Each
method agrees in general, while only the projection method can be used to
extract the low spin data that is affected by the contact term ambiguities,
which in some cases we fixed using localization.

###  4.1 One-loop from tree level

Since the short multiplets are all $1/c$ exact, only long multiplets will
appear at 1-loop. These long multiplets are double trace operators that can be
written as
$S_{p}\partial_{\mu_{1}}\dots\partial_{\mu_{\ell}}(\partial^{2})^{n}S_{p}$,
where $S_{p}$ is the $\Delta=p$ bottom component of the single trace half-BPS
multiplet dual to the $p$th lowest KK mode, so that the stress tensor is
$S_{2}$. The long multiplets thus have spin $\ell$ and twist
$t\equiv\Delta-\ell=2p+2n$ for integer $n$ so for $t\geq 2$ there are $t/2$
such degenerate operators due to the different ways of adding $p$ and $n$ to
get the same twist, which we label using the degeneracy label $I$. We can
expand the CFT data of these long operators as

$\begin{split}\Delta_{t,\ell,I}&=t+\ell+c^{-1}\gamma^{R}_{t,\ell,I}+c^{-\frac{7}{4}}\gamma^{R^{4}}_{t,\ell,I}+c^{-2}\gamma^{R|R}_{t,\ell,I}+\dots\,,\\\
(\lambda_{p,t,\ell,I})^{2}&=(\lambda^{(0)}_{p,t,\ell,I})^{2}+c^{-1}(\lambda^{R}_{p,t,\ell,I})^{2}+c^{-\frac{7}{4}}(\lambda^{R^{4}}_{p,t,\ell,I})^{2}+c^{-2}(\lambda^{R|R}_{p,t,\ell,I})^{2}+\dots\,,\\\
\end{split}$ (4.1)

where $\lambda_{p,t,\ell,I}$ is the OPE coefficient in $S_{p}\times S_{p}$. At
GFFT and tree level, the only difference between the $AdS_{5}\times S^{5}$ and
$AdS_{5}\times S^{5}/\mathbb{Z}_{2}$ theories is that the orbifold projects
out all long multiplets constructed from $S_{p}$ with odd $p$, while at loop
level the values of the CFT data themselves will differ. We can then apply
this large $c$ expansion to the block expansion in (2.9) of the position space
correlator ${\mathcal{G}}(U,V)$ to get $R|R$ at order $c^{-2}$:

$\begin{split}{\mathcal{G}}^{R|R}=&\sum_{t=4,6,\dots}\sum_{\ell\in\text{Even}}\Big{[}\frac{1}{8}\langle(\lambda^{(0)}_{t,\ell})^{2}(\gamma^{R}_{t,\ell})^{2}\rangle(\log^{2}U+4\log
U\partial_{t}^{\text{no-log}}+4(\partial_{t}^{\text{no-log}})^{2})\\\
&+\frac{1}{2}\langle(\lambda^{R})^{2}_{t,\ell}\gamma^{R}_{t,\ell}\rangle(\log
U+2\partial_{t}^{\text{no-log}})\\\
&+\frac{1}{2}\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{{R}|{R}}_{t,\ell}\rangle(\log
U+2\partial_{t}^{\text{no-
log}})+\langle(\lambda^{{R}|{R}}_{t,\ell})^{2}\rangle\Big{]}U^{-2}G_{t+\ell+4,\ell}(U,V)\,,\end{split}$
(4.2)

where we suppressed the $p=2$ subscripts for simplicity, $\langle\rangle$
denotes the sum over the degeneracy label $I$ in (4.1), and
$\partial_{t}^{\text{no-log}}G_{t+\ell+4,\ell}$ means that after taking the
derivative we consider the term that does not include a $\log U$, since the
terms with a log have already been written separately. The expression for
${\mathcal{G}}^{R^{4}|R^{4}}$ at order $c^{-\frac{7}{2}}$ is identical except
we replace $R\to R^{4}$ and the sum for the long multiplets is now restricted
to $\ell=0$, since tree $R^{4}$ only contributes to $\ell=0$ CFT data. The
expression for ${\mathcal{G}}^{R|R^{4}}$ at order $c^{-\frac{11}{4}}$ is
similar except we replace the $\frac{1}{8}$ in the first line by $\frac{1}{4}$
since the vertices are different.

As shown in [40], the entire 1-loop term up to the contact term ambiguities
described in Section 2.2 can in fact be constructed from the $\log^{2}U$ terms
shown above, which are written in terms of GFFT and tree data, since under
$1\leftrightarrow 3$ crossing

$\begin{split}V^{2}{\mathcal{G}}(U,V)-U^{2}{\mathcal{G}}(V,U)+(U-V)\frac{1}{c}+U^{2}-V^{2}=0\,,\end{split}$
(4.3)

these are related to $\log^{2}V$ terms that are the only contributions at this
order to the DD, which can be used to reconstruct the full 1-loop correlator
as shown in [70, 71]. A subtlety is that the $I$ sum
$\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\gamma^{B}_{t,\ell}\rangle$
for 1-loop vertices $A,B$ is what appears in the $\log^{2}U$ term, whereas the
different sums $\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\rangle$
and $\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{B}_{t,\ell}\rangle$ are what
appear at tree level. As shown in Appendix A of [41], one can compute
$\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\gamma^{B}_{t,\ell}\rangle$
from GFFT $\langle S_{p}S_{p}S_{q}S_{q}\rangle$ and tree level $\langle
S_{2}S_{2}S_{p}S_{p}\rangle$ data as

$\begin{split}\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\gamma^{B}_{t,\ell}\rangle=\sum_{p=2}^{t/2}\frac{\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{A}_{t,\ell}\rangle\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{B}_{t,\ell}\rangle}{{\langle(\lambda^{(0)}_{p,t,\ell})^{2}\rangle}}\,,\end{split}$
(4.4)

where we summed over each $p$ where a given twist $t$ long multiplet appears.
For $SU(N)$, this includes all integer $p$, while for the orbifold cases it
only includes even $p$. The GFFT average OPE coefficients were extracted from
$\langle S_{p}S_{p}S_{p}S_{p}\rangle$ in e.g. [73]777We include a factor of
$1/p^{2}$ that was missing from [73].

$\begin{split}&\langle(\lambda^{(0)}_{p,t,\ell})^{2}\rangle\equiv\sum_{I=1}^{t/2}(\lambda^{(0)}_{p,t,\ell,I})^{2}=\frac{24(\ell+1)\left(\frac{t}{2}-2\right)!\left(\frac{t}{2}!\right)^{2}(\ell+t+2)\left(\ell+\frac{t}{2}-1\right)!}{\left(\frac{t}{2}+2\right)!t!p^{2}(p+1)(p-2)!((p-1)!)^{3}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times\frac{\left(\left(\ell+\frac{t}{2}+1\right)!\right)^{2}\left(p+\frac{t}{2}\right)!\left(\ell+p+\frac{t}{2}+1\right)!}{\left(\ell+\frac{t}{2}+3\right)!(2\ell+t+2)!\left(\frac{t}{2}-p\right)!\left(\ell-p+\frac{t}{2}+1\right)!}\,,\end{split}$
(4.5)

while the analogous
$\langle\lambda^{(0)}_{p,t,\ell}\lambda^{(0)}_{q,t,\ell}\rangle$ vanishes for
$p\neq q$ since $\langle S_{p}S_{p}S_{q}S_{q}\rangle$ is trivial in GFFT. The
supergravity average anomalous dimensions were extracted from $\langle
S_{2}S_{2}S_{p}S_{p}\rangle$ in [73] to get in our conventions

$\begin{split}\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{R}_{t,\ell}\rangle=\frac{\pi(-1)^{p+1}\Gamma\left(\ell+\frac{t}{2}+2\right)\left(-p+\frac{t}{2}+1\right)_{p}\Gamma\left(p+\frac{t}{2}+1\right)}{2^{2\ell+2t+1}\Gamma(p-1)\Gamma(p)\Gamma\left(\frac{t+1}{2}\right)\Gamma\left(\ell+\frac{t}{2}+\frac{3}{2}\right)}\,,\end{split}$
(4.6)

which is the same for orbifold and non-orbifold. The $R^{4}$ expression
depends on the theory, and was extracted for the $SU(N)$ theory in the ’t
Hooft limit in [41]. As discussed in Section 2.3, the flat space limit implies
that for $R^{4}$ the finite $\tau$ results can be recovered from the ’t Hooft
limit by simply promoting $\zeta(3)$ to
$E({\scriptstyle\frac{3}{2}},\tau_{s}.\bar{\tau}_{s})$ using (2.21), while the
difference between $AdS_{5}\times S^{5}$ and $AdS_{5}\times
S^{5}/\mathbb{Z}_{2}$ theories is only due to the orbifold factor. From the
results in [41] we can thus obtain

$\begin{split}\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{R^{4}}_{t,\ell}\rangle=\frac{E({\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})(-1)^{p+1}(p)_{4}(t-2)t(t+2)^{2}\Gamma\left(\frac{t}{2}+4\right)\Gamma\left(p+\frac{t}{2}+2\right)\Gamma\left(\frac{t}{2}+2\right)\delta_{\ell,0}}{\mathfrak{o}^{\frac{3}{4}}\sqrt{2\pi}2^{2t+12}(p-2)!\Gamma(p+4)\Gamma\left(\frac{t+3}{2}\right)\Gamma\left(\frac{t+5}{2}\right)\Gamma\left(\frac{t}{2}-p+1\right)}\,.\end{split}$
(4.7)

We can then use this data to perform the $p,t,\ell$ sums for the $\log^{2}U$
term in (4.4) in a small $z$ expansion as

$\begin{split}{\mathcal{G}}^{A|B}\Big{|}_{\log^{2}U}=z^{2}h_{A|B}^{(2)}(\bar{z})+z^{3}h_{A|B}^{(3)}(\bar{z})+\cdots\,.\end{split}$
(4.8)

The $z$-slices take the same form for orbifold and non-orbifold but differ
depending on the vertices $A,B$ as

$\begin{split}h^{(n)}_{R|R}(\bar{z})&=\frac{1}{\bar{z}^{n+2}}\left(p_{1}^{(n)}(\bar{z})\text{Li}_{2}(\bar{z})+p_{2}^{(n)}(\bar{z})\log(1-\bar{z})^{2}+p_{3}^{(n)}(\bar{z})\log(1-\bar{z})+p_{4}^{(n)}(\bar{z})\right)\,,\\\
h^{(n)}_{A|R^{4}}(\bar{z})&=\frac{1}{\bar{z}^{n+3}}\left(p_{1}^{(n+1)}(\bar{z})\log(1-\bar{z})+p_{2}^{(n+1)}(\bar{z})\right)\,,\end{split}$
(4.9)

where $A$ can be $R$ or $R^{4}$, and $p_{i}^{(n)}(\bar{z})$ are certain
polynomials of degree $n$.

###  4.2 Mellin amplitude

We can now complete the position space DD to the entire correlator using
crossing symmetry in Mellin space following [28, 30]. To convert to Mellin
space, we recognize that the $\log^{2}(z)\log^{2}(1-\bar{z})$ terms in (4.8)
emerge from simultaneous poles in the Mellin amplitude in $s$ and $t$. Taking
residues in (2.13) we see that such poles contribute to the correlator as

$M=\sum\limits_{m,n=2}^{\infty}\frac{c_{mn}}{(s-2m)(t-2m)}\ \rightarrow\
{\mathcal{T}}\big{|}_{\log^{2}(z)\log^{2}(1-\bar{z})}=\sum\limits_{m,n=2}^{\infty}\frac{c_{mn}\Gamma(m+n)^{2}}{16\Gamma(m-1)^{2}\Gamma(n-1)^{2}}U^{m}V^{n-2}\,.$
(4.10)

We can now solve for the $c_{mn}$ by comparing (4.8) to (4.10) in a double
expansion around $z=0$ and $\bar{z}=1$. After crossing symmetrizing we check
that the Mellin amplitude reproduces the full $\log^{2}(z)$ contribution (4.8)
which implies that there are no further poles in $s$ and, by crossing
symmetry, neither in $t$ or $u$. We carried out this procedure for each theory
and found a similar structure for each case. For $M^{R|R}$ we got

$\begin{split}&M^{R|R}(s,t)=\sum_{m,n=2}^{\infty}\Bigg{[}\frac{c_{mn}}{(s-2m)(t-2n)}+\frac{c_{mn}}{(t-2m)(u-2n)}+\frac{c_{mn}}{(u-2m)(s-2n)}-b_{mn}\Bigg{]}+C\,.\end{split}$
(4.11)

For $SU(N)$, the coefficients $c_{mn}=c_{nm}$ were computed in [28] as

$\displaystyle c^{SU(N)}_{mn}={}$
$\displaystyle\frac{(m-1)^{2}m^{2}}{5(m+n-1)}+\frac{2(m-1)^{2}\left(3m^{2}-6m+8\right)}{5(m+n-2)}-\frac{9m^{4}-54m^{3}+123m^{2}-126m+44}{5(m+n-3)}$
$\displaystyle-\frac{4\left(m^{2}-4m+9\right)(m-2)^{2}}{5(m+n-4)}+\frac{6(m-3)^{2}(m-2)^{2}}{5(m+n-5)}\,.$
(4.12)

These coefficients diverge in the large $m\sim n$ limit, so in [15] the
$b_{mn}$ was chosen to regulate the sum in this limit, while the constant $C$
was then chosen so that the Mellin amplitude matches the position space
correlator of [24], which gave

$\begin{split}b^{SU(N)}_{mn}=\frac{9mn}{2(m+n)^{3}}\,,\qquad
C^{SU(N)}=-\frac{39}{16}-\frac{13}{8}\pi^{2}+9\zeta(3)\,.\end{split}$ (4.13)

As shown in [24], this convention for the 1-loop Mellin amplitude gives CFT
data that is analytic in spin including spin zero, while other conventions
would change the constant contact term and so would give different results for
spin zero. Since we anyway add the contact term ambiguity $\bar{B}_{0}^{R|R}$
in (2.14), the value of $C$ here is a matter of convenience. In Appendix B we
explain how this Mellin amplitude can be summed to give a closed form
expression.

For the orbifold cases, we found that the $c_{mn}$ are related to those of
$SU(N)$ as

$\begin{split}c^{orb}_{mn}=\frac{1}{2}c^{SU(N)}_{mn}+d_{mn}\,,\end{split}$
(4.14)

where the new term $d_{mn}$ takes the form

$\begin{split}d_{mn}={}&\frac{d^{(1)}_{mn,m+n-1}\,(m+n-1)+d^{(2)}_{mn,m+n-1}\,{}_{3}F_{2}\left(1,1-m,1-n;\frac{3}{2},2-m-n;1\right)}{768(m-1)^{2}(2m+1)(n-1)^{2}(2n+1)(m+n-1)}\,,\end{split}$
(4.15)

and we define the polynomials

$\begin{split}d^{(1)}_{p,q}={}&-32p^{5}\left(q^{2}-5q+2\right)+16p^{4}\left(23q^{3}-80q^{2}-45q+14\right)\\\
&-16p^{3}\left(91q^{4}-256q^{3}-173q^{2}+34q-18\right)\\\
&+8p^{2}\left(269q^{5}-801q^{4}+79q^{3}+307q^{2}+18\right)\\\
&-2pq^{2}\left(201q^{4}-1731q^{3}+3841q^{2}+1323q+414\right)\\\
&+q^{2}\left(-945q^{5}+1050q^{4}+3842q^{3}+852q^{2}+783q-270\right)\,,\\\
d^{(2)}_{p,q}={}&64p^{6}\left(q^{2}-5q+2\right)-128p^{5}\left(6q^{3}-21q^{2}-12q+4\right)\\\
&+16p^{4}\left(205q^{4}-570q^{3}-467q^{2}+40q-24\right)\\\
&-64p^{3}q\left(90q^{4}-244q^{3}-81q^{2}+53q+6\right)\\\
&+4p^{2}q\left(739q^{5}-2917q^{4}+2693q^{3}+2361q^{2}+840q-36\right)\\\
&+8pq^{3}\left(186q^{4}+163q^{3}-1678q^{2}-1251q-540\right)\\\
&+3q^{3}\left(-315q^{5}+140q^{4}+1500q^{3}+1162q^{2}+495q+90\right)\,.\end{split}$
(4.16)

In order to obtain this result it was useful to recognize that for different
fixed values of $m$, the coefficients can be written as

$c^{orb}_{mn}=\sum\limits_{j=0}^{m}\frac{\mu_{m,j}}{m+n-1-j}\,.$ (4.17)

We were then able to determine a general formula for $\mu_{m,j}$. Unlike
$c^{SU(N)}_{mn}$, the sum over the $d_{mn}$ is convergent, so we can
regularize the orbifold case similar to $SU(N)$ as

$\begin{split}b^{orb}_{mn}=\frac{1}{2}b^{SU(N)}_{mn}\,,\qquad
C^{orb}=\frac{1}{2}C^{SU(N)}\,,\end{split}$ (4.18)

where the value of $C^{orb}$ is again a choice, that we will find in the next
section also leads to CFT data analytic in spin down to spin zero.

For $M^{R|R^{4}}(s,t)$ and $M^{R^{4}|R^{4}}(s,t)$ we similarly convert the
position space DD to Mellin space to find the simpler structure

$\begin{split}M^{A|R^{4}}(s,t)&=\sum_{m=2}^{\infty}\left(\frac{c_{m}}{(s-2m)}+\frac{c_{m}}{(t-2m)}+\frac{c_{m}}{(u-2m)}\right)+\text{polynomial}\,,\end{split}$
(4.19)

where $A$ will be $R$ or $R^{4}$, and the polynomial (which we omit in the
results below) is any crossing symmetric polynomial of degree four for
$M^{R|R^{4}}(s,t)$ and degree seven for $M^{R^{4}|R^{4}}(s,t)$, since these
are the degrees determined by the powers of $c$ and the flat space limit. The
results for $SU(N)$ theories were given in [28] in the ’t Hooft limit888Up to
a few typos, that were corrected in [74]., while in the finite $\tau$ limit
they just differ by an overall factor:

$\begin{split}c^{R|R^{4},SU(N)}_{m}={}&-\frac{E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{2\sqrt{2}\pi^{\frac{3}{2}}}\left(63m^{4}-322m^{3}+693m^{2}-722m+300\right)\,,\\\
c^{R^{4}|R^{4},SU(N)}_{m}={}&-\frac{135E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})^{2}}{896\pi^{3}}\big{(}924m^{7}-11627m^{6}+67137m^{5}-227045m^{4}\\\
&+480151m^{3}-629468m^{2}+470408m-153720\big{)}\,.\end{split}$ (4.20)

For the orbifold case, we found that the coefficients are related to $SU(N)$
similar to the $R|R$ case as

$\begin{split}c^{R|R^{4},orb}_{m}=\frac{1}{2^{\frac{3}{4}}2}c^{R|R^{4},SU(N)}_{m}+d^{R|R^{4}}_{m}\,,\qquad
c^{R^{4}|R^{4},orb}_{m}=\frac{1}{2^{\frac{3}{2}}2}c^{R^{4}|R^{4},SU(N)}_{m}+d^{R^{4}|R^{4}}_{m}\,,\end{split}$
(4.21)

where the new terms are defined as

$\displaystyle d^{R|R^{4}}_{m}={}$
$\displaystyle\frac{E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{64\cdot
2^{\frac{1}{4}}\pi^{\frac{3}{2}}}\left(\frac{3^{3}5^{3}\left(-17488m^{3}+10854m^{2}+6031m-1506\right)}{2^{11}\left(m-\frac{15}{2}\right)_{9}}-\frac{15}{4}(3m-2)\right)\,,$
$\displaystyle d^{R^{4}|R^{4}}_{m}={}$
$\displaystyle\frac{E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})^{2}}{\sqrt{2}\pi^{3}}\frac{3^{6}5^{4}}{2^{22}\left(m-\frac{21}{2}\right)_{13}}\Big{(}1062890m^{7}-1606305m^{6}-1152985m^{5}$
$\displaystyle+3799305m^{4}-2802325m^{3}+186276\Big{)}\,,$ (4.22)

and note the orbifold factor $2^{\frac{3}{4}}$ due to the $R^{4}$ vertex.

The sum over $m$ in (4.19) for these coefficients is divergent, but can be
regulated and resummed as described in [28] to get for $SU(N)$

$\displaystyle M^{R|R^{4}}_{SU(N)}=$
$\displaystyle-\frac{E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{64\sqrt{2}\pi^{\frac{3}{2}}}\left(63s^{4}-644s^{3}+2772s^{2}-5776s+4800\right)\psi\left(2-\frac{s}{2}\right)+\text{crossed}\,,$
$\displaystyle M^{R^{4}|R^{4}}_{SU(N)}=$
$\displaystyle-\frac{135E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})^{2}}{2^{14}\cdot
7\pi^{3}}\Big{(}462s^{7}-11627s^{6}+134274s^{5}-908180s^{4}+3841208s^{3}$
$\displaystyle-10071488s^{2}+15053056s-9838080\Big{)}\psi\left(2-\frac{s}{2}\right)+\text{crossed}\,,$
(4.23)

and for the orbifold case

$\displaystyle
M^{R|R^{4}}_{orb}=\frac{1}{2^{\frac{3}{4}}2}M^{R|R^{4}}_{SU(N)}+\frac{E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{64\cdot
2^{\frac{1}{4}}\pi^{\frac{3}{2}}}\bigg{(}\frac{15}{16}(4-3s)\psi\left(2-\frac{s}{2}\right)$
$\displaystyle+\frac{3^{2}5^{2}}{2^{17}}\bigg{(}\frac{9n_{-1}(s)}{s+1}-\frac{388n_{1}(s)}{s-1}-\frac{18040n_{3}(s)}{s-3}+\frac{255788n_{5}(s)}{s-5}-\frac{995390n_{7}(s)}{s-7}+\frac{1797556n_{9}(s)}{s-9}$
$\displaystyle-\frac{1699712n_{11}(s)}{s-11}+\frac{820260n_{13}(s)}{s-13}-\frac{160083n_{15}(s)}{s-15}\bigg{)}\bigg{)}+\,\text{crossed}\,,$
$\displaystyle
M^{R^{4}|R^{4}}_{orb}=\frac{1}{2^{\frac{3}{2}}2}M^{R^{4}|R^{4}}_{SU(N)}+\frac{E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})^{2}}{4096\sqrt{2}\pi^{3}}\frac{3^{5}5^{2}}{2^{25}}\bigg{(}-\frac{3n_{-3}(s)}{s+3}+\frac{80n_{-1}(s)}{s+1}+\frac{4790n_{1}(s)}{s-1}$
$\displaystyle-\frac{2492440n_{3}(s)}{s-3}+\frac{638579095n_{5}(s)}{s-5}-\frac{15176687744n_{7}(s)}{s-7}+\frac{121176047620n_{9}(s)}{s-9}$
(4.24)
$\displaystyle-\frac{465561453840n_{11}(s)}{s-11}+\frac{996963105915n_{13}(s)}{s-13}-\frac{1260126418320n_{15}(s)}{s-15}$
$\displaystyle+\frac{937494179622n_{17}(s)}{s-17}-\frac{380559979800n_{19}(s)}{s-19}+\frac{65155115025n_{21}(s)}{s-21}\bigg{)}+\,\text{crossed}\,,$

where the functions

$\begin{split}n_{k}(s)\equiv\psi\left(2-\frac{s}{2}\right)-\psi\left(2-\frac{k}{2}\right)\,,\end{split}$
(4.25)

have a zero at $s=k$ which cancels all the apparent poles at odd integers.

###  4.3 Flat space limit

We will now compare the 1-loop Mellin amplitudes to the corresponding string
theory amplitudes in 10d using the flat-space limit formula (2.23). To apply
this formula to the 1-loop amplitudes, we should look at the regime where
$m,n,s,t,u$ all scale equally large. One can check that the terms $d_{mn}$ and
$d_{m}$ in (4.14) and (4.21) are subleading in this limit. Hence the leading
terms of each 1-loop Mellin amplitude in (4.14) and (4.21) differ for $SU(N)$
and the orbifold theories by the same powers of 2 that appear in the
respective AdS/CFT dictionaries (2.22) and it is only necessary to check
$SU(N)$. The $R|R$ term was already checked in [28], so it suffices to check
$R|R^{4}$ and $R^{4}|R^{4}$. From (4.23) we find that

$\begin{split}f_{SU(N)}^{R|R^{4}}(s,t)&=-\frac{\pi^{2}E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{2^{12}45}g_{s}^{\frac{7}{2}}\ell_{s}^{14}s^{4}\log(s)+\,\text{crossed}\,,\\\
f_{SU(N)}^{R^{4}|R^{4}}(s,t)&=-\frac{\pi^{2}E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})^{2}}{2^{20}105}g_{s}^{5}\ell_{s}^{20}s^{7}\log(s)+\,\text{crossed}\,.\end{split}$
(4.26)

We can compare this to the discontinuity of the respective 10d amplitude, as
computed from the unitarity cut formula [75]999We fix various typos in [75]
and [41], by comparing our formula to the 10d box diagram in e.g. [25].

$\begin{split}{\text{Disc}_{s}f^{A|B}(s,t)}=&(\delta_{A,B}-2)stu\frac{\pi^{2}ig_{s}^{2}l_{s}^{8}s^{7}}{15\,2^{9}}\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\sin^{7}\theta|\sin^{6}\phi|\frac{f^{A}(s,t^{\prime})}{st^{\prime}u^{\prime}}\frac{f^{B}(s,t^{\prime\prime})}{st^{\prime\prime}u^{\prime\prime}}\,,\end{split}$
(4.27)

with

$t^{\prime}=-\frac{s}{2}(1-\cos\theta)\,,\quad
t^{\prime\prime}=-\frac{s}{2}(1+\cos\theta\cos\rho+\sin\theta\cos\phi\sin\rho)\,,\quad
u^{\prime}=-s-t^{\prime}\,,\quad u^{\prime\prime}=-s-t^{\prime\prime}\,,$
(4.28)

and $\cos\rho=\frac{t-u}{s}$. Using the vertices in (2.20), we find a precise
match with (4.26).

###  4.4 Fixing 1-loop contact term from localization

We can also use the explicit $R|R$ Mellin amplitudes and the localization
constraint in (3.1) to fix the contact term ambiguity $\bar{B}_{0}^{R|R}$. In
[15], this was done for the $SU(N)$ case by numerically computing the integral
of the explicit position space amplitude in [24]. Here, we will fix both
$SU(N)$ and the orbifold theories by computing the Mellin space integral in
(3.2) analytically. The calculation is described in Appendix C, and the result
is

$\begin{split}I_{2}[M_{SU(N)}^{R|R}]=\frac{5}{32}\,,\qquad
I_{2}[M_{orb}^{R|R}]=\frac{1}{128}\,.\end{split}$ (4.29)

We can then input the localization inputs (3.16) and (3.17) to the constraint
(3.1) to fix $\bar{B}_{0}^{R|R}$ to

$\begin{split}\bar{B}_{0,SU(N)}^{R|R}=\frac{15}{4}\,,\qquad\bar{B}_{0,orb}^{R|R}=\frac{15}{8}\,.\end{split}$
(4.30)

Curiously, these terms differ by the same factor of two as the leading $s,t$
term in $M^{R|R}$ that contributes to the flat space limit, even though these
contact terms are subleading in the AdS radius and so do not contribute to the
flat space limit.

###  4.5 Extracting CFT data

Lastly, we can extract all low-lying CFT data from the $R|R$, i.e. $c^{-2}$,
correlator using two methods. Firstly, in Appendix D we derive an inversion
integral formula for each DD in position space following [25], which allows us
to efficiently extract all CFT data above spin $\ell=0$, as expected from the
Lorentzian inversion formula [71]. Secondly, we use the projection method of
[72] to expand each correlator as written in Mellin space in conformal blocks
and extract the CFT data for all spins including $\ell=0$, using that the
contact term ambiguity $\bar{B}_{0}^{R|R}$ was fixed in the previous section.
Not only does each method agree for $\ell>0$, but we even find that the
analytic continuation of the inversion method to spin zero matches the
contribution from $M^{R|R}$, i.e. neglecting $\bar{B}_{0}^{R|R}$, for both
$SU(N)$ and the orbifold theories. We can also use the projection method to
compute the contribution from the tree level terms that we fixed in Section
3.3. We do not extract CFT data from the $R|R^{4}$ or $R^{4}|R^{4}$
correlators, since for low spins these terms are affected by contact terms for
which we have insufficient localization constraints to fix them.

Combining these various contributions, for the lowest dimension operators for
each spin $\ell$ that appear in the large $c$ and finite $\tau$ limit (which
are all double trace operators),101010These are the operators introduced in
Section 4.1 with $p=2,n=0$. we find for the lowest few $\ell$:

$\displaystyle\Delta_{0,SU(N)}^{R|R}={}$ $\displaystyle
4-\frac{4}{c}-\frac{135E({\scriptstyle\frac{3}{2}},\tau,\bar{\tau})}{7\sqrt{2}\pi^{3/2}c^{\frac{7}{4}}}+\frac{1199}{42c^{2}}-\frac{3825E({\scriptstyle\frac{5}{2}},\tau,\bar{\tau})}{32\sqrt{2}\pi^{5/2}c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,$
$\displaystyle\Delta_{2,SU(N)}^{R|R}={}$ $\displaystyle
6-\frac{1}{c}-\frac{41}{16c^{2}}-\frac{1575E({\scriptstyle\frac{5}{2}},\tau,\bar{\tau})}{22\sqrt{2}\pi^{5/2}c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,$
$\displaystyle\Delta_{4,SU(N)}^{R|R}={}$ $\displaystyle
8-\frac{12}{25c}-\frac{423}{3125c^{2}}+O(c^{-\frac{5}{2}})\,,$
$\displaystyle\Delta_{0,orb}^{R|R}={}$ $\displaystyle
4-\frac{4}{c}-\frac{135E({\scriptstyle\frac{3}{2}},\tau_{s},\bar{\tau}_{s})}{14\sqrt[4]{2}\pi^{3/2}c^{\frac{7}{4}}}+\frac{1}{c^{2}}\left(\tfrac{29129625\zeta(3)+22143194}{7680}-\tfrac{135}{14}-10688\log(2)\right)$
$\displaystyle-\frac{3825E({\scriptstyle\frac{5}{2}},\tau_{s},\bar{\tau}_{s})}{64\
2^{3/4}\pi^{5/2}c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,$ (4.31)
$\displaystyle\Delta_{2,orb}^{R|R}={}$ $\displaystyle
6-\frac{1}{c}+\frac{1}{c^{2}}\left(\tfrac{469744545375\zeta(3)+353499406166}{1720320}-769988\log(2)\right)$
$\displaystyle-\frac{1575E({\scriptstyle\frac{5}{2}},\tau_{s},\bar{\tau}_{s})}{44\
2^{3/4}\pi^{5/2}c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,$
$\displaystyle\Delta_{4,orb}^{R|R}={}$ $\displaystyle
8-\frac{12}{25c}-\frac{9}{c^{2}}\left(-\tfrac{217777351113\zeta(3)}{409600}-\tfrac{71700177905311}{179200000}+\tfrac{937053696\log(2)}{625}\right)+O(c^{-\frac{5}{2}})\,.$

One could similarly extract the OPE coefficients for these lowest twist
operators, but those are harder to bound using the numerical
bootstrap,111111This is because in order to bound OPE coefficients of
unprotected operators, one must first know their scaling dimensions, for which
there is extra uncertainty from their own bounds. so we leave it for future
work. For higher twist operators, due to the degeneracy discussed above, one
would need to look at more correlators to perform their unmixing at 1-loop
order.

##  5 Comparison to numerical bootstrap

We conclude by comparing our large $c$ results to finite $c$ numerical
bootstrap bounds. In [42, 43], bounds were computed on CFT data that appears
in the stress tensor correlator as a function of $c$. We restrict the
discussion in this section to the $SU(N)$ or $SO(2N)$ groups, since at large
$c$ and finite $\tau$ all the other orbifold gauge theories can be easily
related to $SO(2N)$. Since the value of $\tau$ could not be specified [42,
43], it was conjectured that the bounds might be saturated by either the
$SU(N)$ or $SO(2N)$ theories at one of the self-dual points under S-duality,
which are either $\tau=i$ with $\mathbb{Z}_{2}$ enhancement or
$\tau=e^{i\pi/3}$ with $\mathbb{Z}_{3}$ enhancement. We can compute the
anomalous dimensions (4.31) at the self-dual points using the formulae [76]:

$\begin{split}E(s,e^{\frac{i\pi}{3}})&=2^{1-s}3^{1-\frac{s}{2}}\zeta(s)\left(\zeta\left(s,\frac{1}{3}\right)-\zeta\left(s,\frac{2}{3}\right)\right)\,,\\\
E(s,i)&=4^{1-s}\zeta(s)\left(\zeta\left(s,\frac{1}{4}\right)-\zeta\left(s,\frac{3}{4}\right)\right)\,,\end{split}$
(5.1)

which shows that $\tau=e^{i\pi/3}$ has the largest value, which numerically is

$\begin{split}\tau=e^{i\pi/3}:\qquad\Delta_{0,SU(N)}^{R|R}&=4-\frac{4}{c}-\frac{21.7787}{c^{\frac{7}{4}}}+\frac{28.5476}{c^{2}}-\frac{22.8027}{c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,\\\
\Delta_{2,SU(N)}^{R|R}&=6-\frac{1}{c}-\frac{2.5625}{c^{2}}-\frac{13.6573}{c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,\\\
\Delta_{4,SU(N)}^{R|R}&=8-\frac{0.48}{c}-\frac{0.13536}{c^{2}}+O(c^{-\frac{5}{2}})\,,\\\
\Delta_{0,SO(2N)}^{R|R}&=4-\frac{4}{c}-\frac{12.9497}{c^{\frac{7}{4}}}+\frac{24.534}{c^{2}}-\frac{9.58737}{c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,\\\
\Delta_{2,SO(2N)}^{R|R}&=6-\frac{1}{c}-\frac{0.888511}{c^{2}}-\frac{5.74217}{c^{\frac{9}{4}}}+O(c^{-\frac{5}{2}})\,,\\\
\Delta_{4,SO(2N)}^{R|R}&=8-\frac{0.48}{c}-\frac{0.00174726}{c^{2}}+O(c^{-\frac{5}{2}})\,.\\\
\end{split}$ (5.2)

The $SO(2N)$ value is also larger than the $SU(N)$ value for large $c$, so if
any theory saturates the single correlator bounds of [42, 43], it should be
the $SO(2N)$ theory at $\tau=e^{i\pi/3}$. A numerical bootstrap study was also
done in [44] using mixed correlators between the stress tensor multiplet and
the next lowest half-BPS multiplet with $\Delta=3$. Since this multiplet is
absent for $SO(2N)$, only the $SU(N)$ theory could appear in these bounds. In
Figure 1 we show a comparison between the single and mixed correlator bounds
which we recomputed at very high bootstrap precision for low spins, and the
analytic estimates at large $c$.121212We include the explicit data points in
the attached Mathematica file. For completeness we give a full description of
our mixed correlator bootstrap setup (independent of [44]) in Appendix E. None
of the analytic estimates for $\Delta_{0}$ and $\Delta_{2}$ are close to
saturating the bounds,131313Truncating the analytic expansion to a smaller
order in large $c$ also does not help. and the single correlator and mixed
correlator bounds are in fact almost indistinguishable in this regime, which
suggests that no physical theory is saturating these bounds.

Figure 1: Upper bounds (black) on the lowest dimension scaling dimension for
spin $0,2,4$ in terms of $c$, computed using $n_{\text{max}}=62$ for the
single correlator setup, or $n_{\text{max}}=30$ for the mixed correlator setup
(the results cannot be distinguished by eye in this regime of large $c$). The
red, purple lines corresponds to the $SU(N)$, $SO(2N)$ analytic estimates,
respectively, at large $c$ and at the self dual point $\tau=e^{i\pi/3}$, as
given in (5.2).

## Acknowledgments

We thank Ofer Aharony, Silvu Pufu, Yifan Wang, and Congkao Wen for useful
conversations. The work of LFA and TH is supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 787185). LFA is also supported in part by the
STFC grant ST/T000864/1. SMC is supported by the Zuckerman STEM Leadership
Fellowship. The authors would like to acknowledge the use of the University of
Oxford Advanced Research Computing (ARC) facility in carrying out this
work.(http://dx.doi.org/10.5281/zenodo.22558)

##  Appendix A Matrix model calculations

In this appendix we give details of the matrix model computation in Section
3.1. As discussed in [15], the mass derivatives that we consider in (3.1) are
identical for $SU(N)$ and $U(N)$, so for simplicity we will consider the
latter gauge group. For the various gauge groups we consider, we will use the
standard basis of roots given in Appendix A of [67] to write (3.3) explicitly
as

$\displaystyle
U(N):\;\;Z(m)=\int\frac{d^{N}a}{N!}\frac{e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}}{H(m)^{N}}\prod_{i<j}\frac{a_{ij}^{2}H^{2}(a_{ij})}{H(a_{ij}-m)H(a_{ij}+m)}\,,$
$\displaystyle
SO(2N):\;\;Z(m)=\int\frac{d^{N}a}{2^{N}N!}\frac{e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}}{H(m)^{N}}\prod_{i<j}\frac{a_{ij}^{2}(a_{ij}^{+})^{2}H^{2}(a_{ij})H^{2}(a_{ij}^{+})}{H(a_{ij}-m)H(a_{ij}+m)H(a_{ij}^{+}-m)H(a_{ij}^{+}+m)}\,,$
$\displaystyle
SO(2N+1):\;\;Z(m)=\int\frac{d^{N}a}{2^{N}N!}\frac{e^{-\frac{8\pi^{2}}{(\delta_{N,1}+1)g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}}{H(m)^{N}}\prod_{i}\frac{a_{i}^{2}H^{2}(a_{i})}{H(a_{i}+m)H(a_{i}-m)}$
(A.1)
$\displaystyle\qquad\qquad\qquad\quad\times\prod_{i<j}\frac{a_{ij}^{2}(a_{ij}^{+})^{2}H^{2}(a_{ij})H^{2}(a_{ij}^{+})}{H(a_{ij}-m)H(a_{ij}+m)H(a_{ij}^{+}-m)H(a_{ij}^{+}+m)}\,,$
$\displaystyle
USp(2N):\;\;Z(m)=\int\frac{d^{N}a}{2^{N}N!}\frac{e^{-\frac{16\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}}{H(m)^{N}}\prod_{i}\frac{(2a_{i})^{2}H^{2}(2a_{i})}{H(2a_{i}+m)H(2a_{i}-m)}$
$\displaystyle\qquad\qquad\qquad\quad\times\prod_{i<j}\frac{a_{ij}^{2}(a_{ij}^{+})^{2}H^{2}(a_{ij})H^{2}(a_{ij}^{+})}{H(a_{ij}-m)H(a_{ij}+m)H(a_{ij}^{+}-m)H(a_{ij}^{+}+m)}\,,$

where we define $a_{ij}=a_{i}-a_{j}$ and $a_{ij}^{+}=a_{i}+a_{j}$, and note
that the Killing form for $USp(2N)$ in our basis has extra factors of $2$
relative to the other cases, while $SO(3)$ has an extra factor of
$\frac{1}{2}$. We will compute the small $m$ expansion of $Z(m)$ using the
method of orthogonal polynomials.

Let us introduce a family of polynomials $p_{n}(a)$:

$\begin{split}p_{n}(a)\equiv\left(\frac{g_{\text{YM}}^{2}}{32\pi^{2}}\right)^{\frac{n}{2}}H_{n}\left(\frac{4\pi
a}{\sqrt{2}g_{\text{YM}}}\right)\,,\end{split}$ (A.2)

which are orthogonal with respect to the Gaussian matrix model measure:

$\begin{split}\int
dap_{m}(a)p_{n}(a)e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}a^{2}}=n!\left(\frac{g_{\text{YM}}^{2}}{16\pi^{2}}\right)^{n}\sqrt{\frac{g_{\text{YM}}^{2}}{8\pi}}\delta_{mn}\equiv
h_{n}\delta_{mn}\,.\end{split}$ (A.3)

These orthogonal polynomials are useful because they diagonalize the
Vandermonde determinant for the various Gaussian matrix models we consider:

$\begin{split}U(N):\;\;Z(0)&=\frac{1}{N!}\int
d^{N}a\prod_{i<j}{(a_{i}-a_{j})^{2}}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}\\\
&=\frac{1}{N!}\int
d^{N}a\prod_{i<j}|p_{i-1}(a_{j})|^{2}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}\\\
&=\prod_{k=0}^{N-1}h_{k}\,,\\\
&=2^{\frac{1}{2}(1-4N)N}\pi^{\frac{1}{2}(1-2N)N}G(N+1)(g_{\text{YM}}^{2})^{\frac{N^{2}}{2}}\,,\end{split}$
(A.4)

and

$\begin{split}SO(2N):\;\;Z(0)&=\frac{1}{2^{N}N!}\int
d^{N}a\prod_{i<j}{(a^{2}_{i}-a^{2}_{j})^{2}}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}\\\
&=\frac{1}{2^{N}N!}\int
d^{N}a\prod_{i<j}|p_{2(i-1)}(a_{j})|^{2}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}\\\
&=\frac{1}{2^{N}}\prod_{k=0}^{N-1}h_{2k}\,,\\\
&=\frac{A^{3/2}2^{-3N^{2}+\frac{N}{2}-\frac{1}{24}}\pi^{-2N^{2}+N+\frac{1}{4}}G\left(N+\frac{1}{2}\right)G(N+1)(g_{\text{YM}}^{2})^{N^{2}-\frac{N}{2}}}{\sqrt[8]{e}}\,,\\\
\end{split}$ (A.5)

and

$\displaystyle SO(2N+1):\;\;Z(0)$ $\displaystyle=\frac{1}{2^{N}N!}\int
d^{N}a\prod_{i<j}{(a^{2}_{i}-a^{2}_{j})^{2}}\prod_{i}a_{i}^{2}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}$
$\displaystyle=\frac{1}{2^{N}N!}\int
d^{N}a\prod_{i<j}|p_{2i-1}(a_{j})|^{2}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}$
$\displaystyle=\frac{1}{2^{N}}\prod_{k=0}^{N-1}h_{2k}\,,$ (A.6)
$\displaystyle=\frac{A^{3/2}2^{-3N^{2}-\frac{5N}{2}-\frac{1}{24}}\pi^{-2N^{2}-N-\frac{1}{4}}G(N+1)G\left(N+\frac{3}{2}\right)(g_{\text{YM}}^{2})^{N^{2}+\frac{N}{2}}}{\sqrt[8]{e}}\,,$

where $A$ is the Glaisher constant, $G(z)$ is the Barnes G function, and for
$SO(2N+1)$ we assume $N>1$ so we have the standard Gaussian measure. Here, we
replaced the Vandermonde determinant by a determinant of orthogonal
polynomials that we can write as

$\begin{split}\prod_{i<j}|p_{i-1}(a_{j})|^{2}=\sum_{\sigma_{1}\in
S_{N}}(-1)^{|\sigma_{1}|}\prod_{k_{1}=1}^{N}p_{\sigma_{1}(k_{1})-1}(a_{k_{1}})\sum_{\sigma_{2}\in
S_{N}}(-1)^{|\sigma_{2}|}\prod_{k_{2}=1}^{N}p_{\sigma_{2}(k_{2})-1}(a_{k_{2}})\,,\end{split}$
(A.7)

and then we computed each integral using (A.3). We can similarly compute
$USp(2N)$, except we now set $g^{2}_{\text{YM}}\to g^{2}_{\text{YM}}/2$ in the
polynomials $p_{n}(a)$ to account for the different Gaussian factor, which
gives

$\displaystyle USp(2N):\;\;Z(0)$ $\displaystyle=\frac{1}{2^{N}N!}\int
d^{N}a\prod_{i<j}{(a^{2}_{i}-a^{2}_{j})^{2}}\prod_{i}(2a_{i})^{2}e^{-\frac{16\pi^{2}}{g_{\text{YM}}^{2}}\sum_{i}a_{i}^{2}}$
(A.8)
$\displaystyle=2^{N}\frac{A^{3/2}2^{-3N^{2}-\frac{5N}{2}-\frac{1}{24}}\pi^{-2N^{2}-N-\frac{1}{4}}G(N+1)G\left(N+\frac{3}{2}\right)(g_{\text{YM}}^{2}/2)^{N^{2}+\frac{N}{2}}}{\sqrt[8]{e}}\,.$

We can then take the logarithm to get the results in (3.4), where note that
$c=N^{2}/4$ for $U(N)$. For $SU(N)$, we would get $c=(N^{2}-1)/4$ because
there is one less eigenvalue, as discussed in [15].

Next, we will discuss how to compute the expectation values in (3.6). Consider
an $n$-body operator ${\mathcal{O}}_{n}(a)$ that WLOG only depends on the
$a_{i}$ for $i=1,\dots n$. Due to the orthogonality of the polynomials in
$\langle{\mathcal{O}}_{n}(a)\rangle$ the only permutations
$\sigma_{1},\sigma_{2}$ in (A.7) that survive integration are those for which
$\sigma_{2}(m)=\sigma_{1}(m)$ for $m>n$. This means that in order to
contribute to the full matrix model integral,
$\\{\sigma_{2}(1),\dots,\sigma_{2}(n)\\}$ must be a permutation of
$\\{\sigma_{1}(1),\dots,\sigma_{1}(n)\\}$, which we denote by $\mu$. The
expectation value for each gauge group is

$\begin{split}&U(N):\;\;\langle{\mathcal{O}}_{n}(a)\rangle=\frac{1}{N!}\sum_{\begin{subarray}{c}\sigma\in
S_{N}\\\ \mu\in
S_{n}\end{subarray}}(-1)^{|\mu|}\int\left(\prod_{i=1}^{n}da_{i}\frac{p_{\sigma(i)-1}(a_{i})p_{\mu(\sigma(i))-1}(a_{i})}{h_{\sigma(i)-1}}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}a_{i}^{2}}\right){\mathcal{O}}_{n}(a)\,,\\\
&SO(2N):\;\;\langle{\mathcal{O}}_{n}(a)\rangle=\frac{1}{N!}\sum_{\begin{subarray}{c}\sigma\in
S_{N}\\\ \mu\in
S_{n}\end{subarray}}(-1)^{|\mu|}\int\left(\prod_{i=1}^{n}da_{i}\frac{p_{2(\sigma(i)-1)}(a_{i})p_{2(\mu(\sigma(i))-1)}(a_{i})}{h_{2(\sigma(i)-1)}}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}a_{i}^{2}}\right){\mathcal{O}}_{n}(a)\,,\\\
&SO(2N+1):\;\;\langle{\mathcal{O}}_{n}(a)\rangle=\frac{1}{N!}\sum_{\begin{subarray}{c}\sigma\in
S_{N}\\\ \mu\in
S_{n}\end{subarray}}(-1)^{|\mu|}\int\left(\prod_{i=1}^{n}da_{i}\frac{p_{2\sigma(i)-1}(a_{i})p_{2\mu(\sigma(i))-1}(a_{i})}{h_{2\sigma(i)-1}}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}a_{i}^{2}}\right){\mathcal{O}}_{n}(a)\,,\\\
&USp(2N):\;\;\langle{\mathcal{O}}_{n}(a)\rangle=\frac{1}{N!}\sum_{\begin{subarray}{c}\sigma\in
S_{N}\\\ \mu\in
S_{n}\end{subarray}}(-1)^{|\mu|}\int\left(\prod_{i=1}^{n}da_{i}\frac{p_{2\sigma(i)-1}(a_{i})p_{2\mu(\sigma(i))-1}(a_{i})}{h_{2\sigma(i)-1}}e^{-\frac{16\pi^{2}}{g_{\text{YM}}^{2}}a_{i}^{2}}\right){\mathcal{O}}_{n}(a)\,,\\\
\end{split}$ (A.9)

where note that the originally $N$-dimensional integral has reduced to an
$n$-dimensional integral, and the $p_{n}(a)$ in the $USp(2N)$ case are defined
with $g^{2}_{\text{YM}}\to g^{2}_{\text{YM}}/2$ as usual for this case. We can
now use this and the identity

$\begin{split}\int_{-\infty}^{\infty}e^{-x^{2}+yx}H_{m}(x)H_{n}(x)=e^{\frac{y^{2}}{4}}2^{m}\sqrt{\pi}m!y^{n-m}L_{m}^{n-m}(-y^{2}/2)\,,\end{split}$
(A.10)

where $L_{m}^{n}(z)$ is a generalized Laguerre polynomial, to compute the
2-body operator expectation value $\sum_{i\neq
j}\langle\cos(2\omega(a_{i}-a_{j}))\rangle$ in the $U(N)$ case:

$\displaystyle\frac{N(N-1)}{N!}\sum_{\sigma\in S_{N}}\sum_{\mu\in
S_{2}}(-1)^{|\mu|}\int\left(\prod_{i=1}^{2}da_{i}\frac{p_{\sigma(i)-1}(a_{i})p_{\mu(\sigma(i))-1}(a_{i})}{h_{\sigma(i)-1}}e^{-\frac{8\pi^{2}}{g_{\text{YM}}^{2}}a_{i}^{2}}\right)\frac{e^{2i\omega(a_{1}-a_{2})}+e^{2i\omega(a_{2}-a_{1})}}{2}$
$\displaystyle=e^{\frac{-\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\sum_{i,j=1}^{N}\left[L_{i-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{j-1}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)-(-1)^{i-j}L_{i-1}^{j-i}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)L_{j-1}^{i-j}\left({\scriptstyle\frac{\omega^{2}g_{\text{YM}}^{2}}{4\pi^{2}}}\right)\right]\,.$
(A.11)

We can then plug this into (3.7) and (3.6) to get the final result for $SU(N)$
in (3.8), which recall is the same as $U(N)$. The other cases can be computed
similarly, where $\sum_{i\neq j}\langle\cos(2\omega(a_{i}-a_{j}))\rangle$ and
$\sum_{i\neq j}\langle\cos(2\omega(a_{i}+a_{j}))\rangle$ both give the same
answer, for $SO(2N+1)$ and $USp(2N)$ we have an extra contribution from the
1-body operator in (3.6), and the $USp(2N)$ case has $g^{2}_{\text{YM}}\to
g^{2}_{\text{YM}}/2$ relative to the $SO(2N+1)$ case.

##  Appendix B Closed form for $R|R$ Mellin amplitude in $SU(N)$

In this appendix we compute explicitly the double sum

$\begin{split}&\Phi(s,t)=\sum_{m,n=2}^{\infty}\frac{c_{mn}}{(s-2m)(t-2n)}\,,\end{split}$
(B.1)

where the coefficients $c_{mn}=c_{nm}$ are those relevant for $SU(N)$, given
in (4.12). The first remark is that this sum needs to be regularized.
Different regularisations differ from each other by ambiguities of the form
$\alpha+\beta(8-3s-3t)$. Below we will choose a specific regularisation and
give $\alpha$ in that case. $\beta$ will not be needed for our purposes. More
precisely, consider the convergent double sum

$\begin{split}&\Phi_{reg}(s,t)=\sum_{m,n=2}^{\infty}\left(\frac{c_{mn}}{(s-2m)(t-2n)}-\delta_{mn}\right)\,,\end{split}$
(B.2)

where

$\delta_{mn}=\frac{3mn}{2(m+n)^{3}}+\frac{3mt-4m+3ns-4n}{4(m+n)^{3}}\,.$ (B.3)

Note that the subleading/second term in $\delta_{mn}$ vanishes in the
symmetric combination that enters the Mellin amplitude (4.11)

$M_{SU(N)}^{R|R}(s,t)=\Phi_{reg}(s,t)+\Phi_{reg}(t,u)+\Phi_{reg}(u,s)+C^{SU(N)}\,,$
(B.4)

but it is necessary to regularize a single double sum, very much as what
happens with the box function in flat space. In order to compute
$\Phi_{reg}(s,t)$ we will follow the same strategy as in [32]. The answer is
expected to have the form

$\displaystyle\Phi_{reg}(s,t)={}$ $\displaystyle
R_{0}(s,t)\left(\psi^{(1)}\left(2-\frac{s}{2}\right)+\psi^{(1)}\left(2-\frac{t}{2}\right)-\left(\psi^{(0)}\left(2-\frac{s}{2}\right)-\psi^{(0)}\left(2-\frac{t}{2}\right)\right)^{2}\right)$
$\displaystyle+R_{1}(s,t)\psi^{(0)}\left(2-\frac{s}{2}\right)+R_{1}(t,s)\psi^{(0)}\left(2-\frac{t}{2}\right)+R_{2}(s,t)\,,$
(B.5)

where the rational functions $R_{0}(s,t)$, $R_{1}(s,t)$, $R_{2}(s,t)$ are free
of poles at $s,t=4,6,\cdots$. $R_{0}(s,t)$ is fixed by the requirement that
the residues at simultaneous poles are reproduced, which simply implies that
$R_{0}(s,t)$ is given by $c_{mn}$ upon setting $m\to s/2$ and $n\to t/2$:

$\begin{split}R_{0}(s,t)=\frac{P_{0}(s,t)}{40(s+t-10)(s+t-8)(s+t-6)(s+t-4)(s+t-2)}\,,\end{split}$
(B.6)

with $P_{0}(s,t)$ a symmetric polynomial of total degree six

$\displaystyle P_{0}(s,t)={}$ $\displaystyle
15s^{4}t^{2}+30s^{3}t^{3}-360s^{3}t^{2}+15s^{2}t^{4}-360s^{2}t^{3}+2304s^{2}t^{2}-70s^{4}t+1096s^{3}t$
$\displaystyle-5048s^{2}t+88s^{4}-1024s^{3}+3552s^{2}-70st^{4}+1096st^{3}-5048st^{2}+8640st$
$\displaystyle-4736s+88t^{4}-1024t^{3}+3552t^{2}-4736t+2048\,.$ (B.7)

Next, let us focus on $R_{1}(s,t)$. This is fixed by the requirement that at
each pole $s=2m$, we get the precise $t$-dependence, namely an infinite sum
over poles with the correct residues. This implies

$\begin{split}R_{1}(s,t)=\frac{P_{1}(s,t)}{40(s+t-10)(s+t-8)(s+t-4)(s+t-2)}\,,\end{split}$
(B.8)

with $P_{1}(s,t)$ a polynomial of total degree 5.

$\displaystyle P_{1}(s,t)={}$
$\displaystyle-105s^{3}t^{2}-45s^{2}t^{3}+1200s^{2}t^{2}-75s^{4}t+1350s^{3}t-7720s^{2}t-15s^{5}+400s^{4}$
$\displaystyle-3532s^{3}+13008s^{2}+250st^{3}-3980st^{2}+17200st-21248s-368t^{3}$
$\displaystyle+4192t^{2}-13312t+12800\,.$ (B.9)

Finally, $R_{2}(s,t)$ can be constrained as follows. Note that because of the
form of $R_{0}(s,t)$ and $R_{1}(s,t)$, spurious poles are introduced at
$s+t=2,4,6,8,10$ and $R_{2}(s,t)$ must cancel these poles. Its polar structure
together with the flat space limit (which constraints the large $s,t$
behaviour) allow to write $R_{2}(s,t)$ in the following form

$\begin{split}R_{2}(s,t)=\frac{P_{2}(s,t)}{240(s+t-10)(s+t-8)(s+t-6)(s+t-4)(s+t-2)}+\alpha+\beta(8-3s-3t)\,,\end{split}$
(B.10)

where $P_{2}(s,t)$ is a symmetric polynomial of total degree 6. It turns out
that $P_{2}(s,t)$ is completely fixed by requiring cancellation at the
spurious poles, up to the ambiguity of the form $\alpha+\beta(8-3s-3t)$ (times
the denominator), exactly as expected. For the regularisation above we obtain

$\displaystyle
P_{2}(s,t)=-45s^{4}t^{2}-630s^{3}t^{2}-45s^{2}t^{4}-630s^{2}t^{3}+19404s^{2}t^{2}-72s^{5}t+585s^{4}t$
$\displaystyle+8640s^{3}t-125604s^{2}t-27s^{6}+477s^{5}-1062s^{4}-28908s^{3}+239688s^{2}-72st^{5}+585st^{4}$
$\displaystyle+8640st^{3}-125604st^{2}+520848st-683424s-27t^{6}+477t^{5}-1062t^{4}-28908t^{3}$
$\displaystyle+239688t^{2}-683424t+642816+\pi^{2}\left(45s^{4}t^{2}+520s^{3}t^{2}+45s^{2}t^{4}+520s^{2}t^{3}-14544s^{2}t^{2}\right.$
$\displaystyle+54s^{5}t-400s^{4}t-7056s^{3}t+89808s^{2}t+9s^{6}-164s^{5}-648s^{4}+25984s^{3}-172656s^{2}$
$\displaystyle+54st^{5}-400st^{4}-7056st^{3}+89808st^{2}-354528st+458560s+9t^{6}-164t^{5}-648t^{4}$
$\displaystyle\left.+25984t^{3}-172656t^{2}+458560t-419328\right)\,,$ (B.11)

and for our specific normalisation $\alpha=-3\zeta(3)$. Note that $\beta$ will
disappear once we consider the combination of all three channels. The Mellin
amplitude presented here is also included in the attached Mathematica file.

##  Appendix C 1-loop localization integral in Mellin space

In this Appendix we will compute the integral (3.2) for the Mellin amplitudes
$M^{R|R}_{SU(N)}$ and $M^{R|R}_{orb}$ as described in Section 4.2. Our
strategy is to first compute it for a general simultaneous pole in $s$ and $t$

$\begin{split}I_{2}\left(\frac{1}{(s-2m)(t-2n)}\right)\,.\end{split}$ (C.1)

To this end, we first use the integral representation of the Gamma function

$\begin{split}\Gamma(x)=\int\limits_{0}^{\infty}dt\,t^{x-1}e^{-t}\,,\end{split}$
(C.2)

for $\Gamma\left(\tfrac{u}{2}\right)$ and $\Gamma\left(2-\tfrac{u}{2}\right)$
in order to factorize the $s$ and $t$ integrals, and then do these integrals
by summing residues

$\begin{split}I_{2}\left(\tfrac{1}{(s-2m)(t-2n)}\right)=-\int\limits_{0}^{\infty}dt_{1}dt_{2}\frac{t_{1}\Gamma(m)\Gamma(n)\,_{2}F_{1}\left(2,m;m+1;-\frac{t_{1}}{t_{2}}\right)\,_{2}F_{1}\left(2,n;n+1;-\frac{t_{1}}{t_{2}}\right)}{8e^{t_{1}+t_{2}}t_{2}\Gamma(m+1)\Gamma(n+1)}\,.\end{split}$
(C.3)

By doing the integral for many values of $m$ and $n$, we found the general
formula

$\displaystyle I_{2}\left(\tfrac{1}{(s-2m)(t-2n)}\right)={}$
$\displaystyle\frac{1}{4}(m-1)(n-1)(m+n-1)\left(\frac{\pi^{2}}{12}\left(2H_{m+n-2}-H_{m-2}-H_{n-2}\right)-\zeta(3)\right)$
$\displaystyle-\frac{\pi^{2}}{96}(m+n)\left(m^{2}+mn+n^{2}-2m-2n+1\right)+r_{m,n}\,,$
(C.4)

where $H_{n}$ is the $n$-th harmonic number and $r_{m,n}$ is symmetric under
exchange of $m$ and $n$ and determined by the recursion relation

$\displaystyle
r_{m,n}=-\frac{(n-1)n}{(n-3)(n-2)}r_{m,n-2}+\frac{2(n-1)(m(n-1)+(n-3)n+1)}{(n-2)^{2}(m+n-2)}r_{m,n-1}$
(C.5)
$\displaystyle+\frac{m\left(6m^{2}+3m-2\right)(2n-3)+4n-9-6m(m+1)(m-1)^{2}(2n-3)\left(\psi^{(1)}(m-1)-\tfrac{\pi^{2}}{6}\right)}{48(n-3)(n-2)^{2}(m+n-2)}\,,$

together with the initial values

$\begin{split}r_{2,2}=\frac{53}{48}\,,\qquad r_{2,3}=\frac{101}{32}\,,\qquad
r_{3,3}=\frac{223}{32}\,.\end{split}$ (C.6)

Next we have to deal with the double sum over $m$ and $n$. In order to improve
the convergence of the sum for the $SU(N)$ case we subtract the first three
terms in the large $m\sim n$ expansion of the summand (4.12). These terms are
polynomial Mellin amplitudes that can be easily summed and integrated
separately. We then cut off the sum of the subtracted terms as follows (using
crossing symmetry of the integrand of (3.2))

$\displaystyle I_{2}[M^{R|R}]\approx{}$
$\displaystyle\sum\limits_{m=2}^{M}\sum\limits_{n=2}^{m}\left(3(2-{{\delta}}_{mn})c^{SU(N)}_{mn}I_{2}\left(\tfrac{1}{(s-2m)(t-2n)}\right)-b^{SU(N)}_{mn}I_{2}(1)-I_{2}(\text{subtractions}_{mn})\right)$
$\displaystyle+\sum\limits_{m,n=2}^{\infty}I_{2}(\text{subtractions}_{mn})+C^{SU(N)}I_{2}(1)\,.$
(C.7)

For the cutoff $M=5000$ we find agreement with the result of [15] to 4 digits,
as shown in (4.30).

For the orbifold case we can reuse the $SU(N)$ result in (4.30) times $1/2$
and only need to add the terms involving $d_{mn}$. We expect the main
contribution to the sum to come from the region where either $m$ or $n$ is
small. In order to improve convergence in this region we expand the summand at
large $n$

$\begin{split}\frac{d_{mn}}{(s-2m)(t-2n)}=\text{sub}_{n}^{(4)}(d_{mn})+O(n^{-5})\,.\end{split}$
(C.8)

The subtractions defined in this way have terms of the form

$\begin{split}\frac{t^{k}}{s-2m}\,,\quad k=0,1,2\,,\end{split}$ (C.9)

for which we compute the integral (3.2):

$\begin{split}I_{2}\left(\frac{1}{s-2m}\right)={}&\frac{1}{144}\left(-6m^{3}-3m^{2}+2m-2\right)+\frac{1}{24}(m-1)^{2}m(m+1)\psi^{(1)}(m-1)\,,\\\
I_{2}\left(\frac{t}{s-2m}\right)={}&\frac{30m^{5}-75m^{4}+20m^{3}+80m^{2}-41m+22}{720(m-2)}\\\
&-\frac{1}{24}(m-2)(m-1)^{2}m(m+1)\psi^{(1)}(m-2)\,,\\\
I_{2}\left(\frac{t^{2}}{s-2m}\right)={}&\frac{-90m^{6}+345m^{5}-360m^{4}-160m^{3}+443m^{2}-194m+16}{1800(m-2)}\\\
&+\frac{1}{60}(m-2)(m-1)^{2}m(m+1)(3m-4)\psi^{(1)}(m-2)\,.\end{split}$ (C.10)

The integral for $M_{orb}^{R|R}$ is then approximated by

$\displaystyle I_{2}(M_{orb}^{R|R}(s,t))\approx{}$
$\displaystyle\frac{5}{64}+\sum\limits_{m=2}^{100}\sum\limits_{n=2}^{300}3(2-{{\delta}}_{mn})\left(d_{mn}I_{2}\left(\tfrac{1}{(s-2m)(t-2n)}\right)-I_{2}(\text{sub}_{n}^{(4)}(d_{mn}))\right)$
$\displaystyle={}$
$\displaystyle\frac{5}{64}-0.023437499999\approx\frac{5}{64}-\frac{3}{128}\,,$
(C.11)

as shown in (4.30).

##  Appendix D Lorentzian inversion

In this Appendix we explain how to extract the CFT-data to one-loop from the
double discontinuity for the $SU(N)$ and orbifold 1-loop terms. We will follow
a streamlined version of the method of [25]. The function that describes the
CFT-data is given by

$c(h,\bar{h})=\int_{0}^{1}\frac{dz}{z^{2}}k_{1-h}(z)\int_{0}^{1}\frac{d\bar{z}}{\bar{z}^{2}}\frac{r_{\bar{h}}^{2}}{\bar{h}-1/2}k_{\bar{h}}(\bar{z})\frac{dDisc[(z\bar{z})(\bar{z}-z)\mathcal{G}(U,V)]}{4\pi^{2}}\,,$
(D.1)

where $h=\frac{\Delta-\ell+2}{2}$, $\bar{h}=\frac{\Delta+\ell+4}{2}$ and
$k_{h}(z)=z^{h}\,_{2}F_{1}(h,h,2h,z)$ and we have also introduced

$r_{h}=\frac{\Gamma(h)^{2}}{\Gamma(2h-1)}\,.$ (D.2)

A given double trace operator with quantum numbers $\Delta=4+2n+\ell+\gamma$
and $\ell$ produces a pole at $h=3+n+\gamma/2$, $\bar{h}=4+n+\ell+\gamma/2$.
For small $\gamma$ in a perturbative expansion we have

$c(h,h+\ell+1)=\left\langle\frac{a}{2(3+n+\gamma/2-h)}\right\rangle_{n,\ell}\,,$
(D.3)

where the average is among all the nearly degenerate intermediate operators
(degenerate at zeroth order). The inversion integral then works as follows.
The OPE data is encoded in poles in the $h-$plane. These arise from the $z\to
0$ limit of the integration, and different powers of $z$ in that limit yield
different twists. The integral over $\bar{z}$, on the other hand, is dual to
$\bar{h}$ and provides the spin dependence for each twist.

We will consider the inversion integral in a $1/c$ expansion and are
interested in leading twist operators (of approximate twist 4). To order
$1/c^{2}$ their contribution to the correlator will be of the form

$dDisc[\mathcal{G}(U,V)]=4\pi^{2}(z\bar{z})^{2}\left(h_{0}(\bar{z})\log^{2}z+h_{1}(\bar{z})\log
z+h_{2}(\bar{z})\right)+\cdots$ (D.4)

we can then perform the integral over $z$ keeping the relevant poles in a
small $z$ expansion. We obtain

$c(h,\bar{h})=-\frac{R_{0}(\bar{h})}{(h-1)^{3}}-\frac{R_{1}(\bar{h})}{(h-1)^{2}}-\frac{R_{2}(\bar{h})}{h-1}+\cdots$
(D.5)

where we disregard all poles in $\bar{h}$ except the ones at $\bar{h}=1$ and

$\displaystyle R_{0}(\bar{h})$
$\displaystyle=\int_{0}^{1}\frac{d\bar{z}}{\bar{z}^{2}}\frac{r_{\bar{h}}^{2}}{\bar{h}-1/2}k_{\bar{h}}(\bar{z})(2\bar{z}^{4}h_{0}(\bar{z}))\,,$
(D.6) $\displaystyle R_{1}(\bar{h})$
$\displaystyle=\int_{0}^{1}\frac{d\bar{z}}{\bar{z}^{2}}\frac{r_{\bar{h}}^{2}}{\bar{h}-1/2}k_{\bar{h}}(\bar{z})(\bar{z}^{4}h_{1}(\bar{z}))\,,$
$\displaystyle R_{2}(\bar{h})$
$\displaystyle=\int_{0}^{1}\frac{d\bar{z}}{\bar{z}^{2}}\frac{r_{\bar{h}}^{2}}{\bar{h}-1/2}k_{\bar{h}}(\bar{z})(\bar{z}^{4}h_{2}(\bar{z}))\,.$

These residues are related to the anomalous dimensions as follows

$\displaystyle\langle a^{(0)}\left(\gamma^{(1)}\right)^{2}\rangle_{\ell}$
$\displaystyle=8R_{0}(\bar{h})\,,$ (D.7) $\displaystyle\langle
a^{(0)}\gamma^{(2)}+a^{(1)}\gamma^{(1)}\rangle_{\ell}$
$\displaystyle=4R_{1}(\bar{h})+4\partial_{\bar{h}}R_{0}(\bar{h})\,.$

Since we are only interested in the anomalous dimension, we will only compute
$R_{0}(\bar{h})$ and $R_{1}(\bar{h})$ for the $SU(N)$ and orbifold theories.
For both theories we obtain

$\displaystyle h_{0}(\bar{z})=6\frac{6-6\bar{z}+\bar{z}^{2}}{\bar{z}^{6}}\,.$
(D.8)

$R_{0}(\bar{h})$ can then be easily computed and gives

$\displaystyle R_{0}(\bar{h})=\frac{24r_{\bar{h}}}{(\bar{h}+2)(\bar{h}-3)}\,.$
(D.9)

From this we can compute

$\langle
a^{(0)}\left(\gamma^{(1)}\right)^{2}\rangle_{\ell}=a^{(0)}_{\ell}\frac{576}{(\bar{h}+2)^{2}(\bar{h}-3)^{2}}\,.$
(D.10)

Next, we want to extract the one-loop anomalous dimension. From Section 4.1 we
find

$h_{1}^{SU(N)}(\bar{z})=\frac{18\left({\bar{z}}^{4}+29{\bar{z}}^{3}-204{\bar{z}}^{2}+350{\bar{z}}-175\right)}{{\bar{z}}^{8}}\,.$
(D.11)

Plugging this into the formula above we find

$\displaystyle
R_{1}(\bar{h})=\frac{36\left({\bar{h}}^{2}-{\bar{h}}-55\right)}{({\bar{h}}-5)({\bar{h}}-3)({\bar{h}}+2)({\bar{h}}+4)}r_{\bar{h}}\,.$
(D.12)

Plugging this into the formula for $\langle
a^{(0)}\gamma^{(2)}+a^{(1)}\gamma^{(1)}\rangle_{\ell}$, subtracting the
corresponding tree level pieces and dividing by the GFFT coefficients, we get
the $c^{-2}$ values shown in (4.31), except for spin zero where the analytic
continuation only gives the contribution $\frac{287}{6}$ that neglects the
contact term ambiguity.

We can now repeat the same computation for orbifold case, where we find

$\displaystyle
h_{1}^{orb}(\bar{z})=\frac{1}{262144(1-\bar{z})\bar{z}^{8}}\Big{(}36018675\bar{z}^{11}-81959850\bar{z}^{10}+59928960\bar{z}^{9}-14292880\bar{z}^{8}$
$\displaystyle+347968\bar{z}^{7}-43008\bar{z}^{6}-10932224\bar{z}^{5}-8949760\bar{z}^{4}+455073792\bar{z}^{3}-1260945408\bar{z}^{2}$
$\displaystyle+1238630400\bar{z}-412876800\Big{)}+\frac{\text{arctanh}\left(\sqrt{1-\bar{z}}\right)}{262144(1-\bar{z})^{3/2}\bar{z}^{6}}\Big{(}36018675\bar{z}^{10}-105972300\bar{z}^{9}$
$\displaystyle+111367200\bar{z}^{8}-48484800\bar{z}^{7}+7083648\bar{z}^{6}-61440\bar{z}^{5}+442368\bar{z}^{4}-10616832\bar{z}^{3}$
$\displaystyle+38535168\bar{z}^{2}-47185920\bar{z}+18874368\Big{)}\,.$ (D.13)

The only difference is that now the integrals cannot be done for all $\ell$,
but case by case they can be done, and one can also analytically continue to
spin zero. The results are shown in (4.31), except for spin zero we again only
get the contribution $\frac{29129625\zeta(3)+22143194}{7680}-10688\log(2)$
that neglects the contact term ambiguity.

##  Appendix E Mixed Correlator Bootstrap Setup

In this appendix we describe the setup for the numerical bootstrap with mixed
correlators between the stress tensor multiplet and the next lowest half-BPS
multiplet. The definitions collected from various places in the literature
should be general enough to make this a useful reference for more general
setups as well. We write four-point functions of $\frac{1}{2}$-BPS operators
of dimension $p_{i}$ as

$\displaystyle\langle S_{p_{1}}(x_{1},y_{1})\cdots
S_{p_{4}}(x_{4},y_{4})\rangle$ $\displaystyle\equiv
T_{p_{1}p_{2}p_{3}p_{4}}\mathcal{G}_{\\{p_{i}\\}}(z,\bar{z},{\alpha},\bar{\alpha})\,,$
(E.1) $\displaystyle T_{p_{1}p_{2}p_{3}p_{4}}$
$\displaystyle=g_{12}^{\frac{p_{1}+p_{2}}{2}}g_{34}^{\frac{p_{3}+p_{4}}{2}}\left(\frac{g_{24}}{g_{14}}\right)^{\frac{p_{2}-p_{1}}{2}}\left(\frac{g_{13}}{g_{14}}\right)^{\frac{p_{3}-p_{4}}{2}}\,,$

where

$g_{ij}=\frac{Y_{i}\cdot
Y_{j}}{x_{ij}^{2}}\,,\qquad\alpha\bar{\alpha}=\frac{1}{\sigma}\,,\qquad(1-\alpha)(1-\bar{\alpha})=\frac{\tau}{\sigma}\,.$
(E.2)

The superconformal Ward identities imply that the correlator can generally be
written in the form [77, 5]

$\displaystyle\mathcal{G}_{\\{p_{i}\\}}(z,\bar{z},{\alpha},\bar{\alpha})$
$\displaystyle=k\chi(z,{\alpha})\chi(\bar{z},\bar{\alpha})+\frac{(z-{\alpha})(z-\bar{\alpha})(\bar{z}-{\alpha})(\bar{z}-\bar{\alpha})}{({\alpha}-\bar{\alpha})(z-\bar{z})}$
$\displaystyle\times\left(-\frac{\chi(\bar{z},\bar{\alpha})f(z,{\alpha})}{{\alpha}z(\bar{z}-\bar{\alpha})}+\frac{\chi(\bar{z},{\alpha})f(z,\bar{\alpha})}{\bar{\alpha}z(\bar{z}-{\alpha})}+\frac{\chi(z,\bar{\alpha})f(\bar{z},{\alpha})}{{\alpha}\bar{z}(z-\bar{\alpha})}-\frac{\chi(z,{\alpha})f(\bar{z},\bar{\alpha})}{\bar{\alpha}\bar{z}(z-{\alpha})}\right)$
$\displaystyle+\frac{(z-{\alpha})(z-\bar{\alpha})(\bar{z}-{\alpha})(\bar{z}-\bar{\alpha})}{({\alpha}\bar{\alpha})^{2}}H_{\\{p_{i}\\}}(z,\bar{z},{\alpha},\bar{\alpha}),$
(E.3)

where $k_{\\{p_{i}\\}}$ is called the unit contribution, $f_{\\{p_{i}\\}}$ is
the chiral correlator and $H_{\\{p_{i}\\}}$ is the reduced correlator. These
different parts can be extracted from $\mathcal{G}_{\\{p_{i}\\}}$ as follows

$\displaystyle k_{\\{p_{i}\\}}$
$\displaystyle={\mathcal{G}}_{\\{p_{i}\\}}(z,\bar{z},z,\bar{z}),$ (E.4)
$\displaystyle f_{\\{p_{i}\\}}(\bar{z},\bar{\alpha})$
$\displaystyle=\frac{\bar{\alpha}\bar{z}}{\bar{z}-\bar{\alpha}}\left({\mathcal{G}}_{\\{p_{i}\\}}(z,\bar{z},z,\bar{\alpha})-k_{\\{p_{i}\\}}\chi_{\\{p_{i}\\}}(\bar{z},\bar{\alpha})\right),$

and $H_{\\{p_{i}\\}}$ is obtained by subtracting the other contributions from
(E.3). The function $\chi_{\\{p_{i}\\}}$ is given by

$\chi_{\\{p_{i}\\}}(z,{\alpha})=\left(\frac{z}{\alpha}\right)^{\max(|p_{21}|,|p_{34}|)/2}\left(\frac{1-{\alpha}}{1-z}\right)^{\max(p_{21}+p_{34},0)/2}.$
(E.5)

We would like to bootstrap the contributions of long superconformal multiplets
which only contribute to $H_{\\{p_{i}\\}}$. The contributions of short
multiplets are completely fixed by the free theory. Combining the
$k_{\\{p_{i}\\}}$ and $f_{\\{p_{i}\\}}$ from free theory with the
$1\leftrightarrow 3$ crossing equation for the correlator, we derive the
following crossing equations for the reduced correlators

$\displaystyle V^{2}H_{\\{2222\\}}(U,V)-U^{2}H_{\\{2222\\}}(V,U)$
$\displaystyle=\frac{(V-U)(c(U+V)+1)}{c}\,,$ (E.6) $\displaystyle
V^{5/2}H_{\\{2323\\}}(U,V)-U^{5/2}H_{\\{2323\\}}(V,U)$
$\displaystyle=\frac{(V-U)\sqrt{UV}(2c(U+V)+3)}{2c}\,,$ $\displaystyle
V^{5/2}H_{\\{2233\\}}(U,V)-U^{2}H_{\\{3223\\}}(V,U)$
$\displaystyle=\frac{\sqrt{V}(6V-3U-2c)}{2c}\,,$ $\displaystyle
V^{2}H_{\\{3223\\}}(U,V)-U^{5/2}H_{\\{2233\\}}(V,U)$
$\displaystyle=\frac{\sqrt{U}(3V-6U+2c)}{2c}\,,$

where we used the notation

$\displaystyle H_{\\{2222\\}}(z,\bar{z},{\alpha},\bar{\alpha})$
$\displaystyle=H_{\\{2222\\}}(U,V)\,,$ $\displaystyle\qquad
H_{\\{2323\\}}(z,\bar{z},{\alpha},\bar{\alpha})$
$\displaystyle=\frac{1}{\sqrt{{\alpha}\bar{\alpha}}}H_{\\{2323\\}}(U,V)\,,$
(E.7) $\displaystyle H_{\\{2233\\}}(z,\bar{z},{\alpha},\bar{\alpha})$
$\displaystyle=H_{\\{2233\\}}(U,V)\,,$ $\displaystyle\qquad
H_{\\{3223\\}}(z,\bar{z},{\alpha},\bar{\alpha})$
$\displaystyle=\frac{1}{\sqrt{{\alpha}\bar{\alpha}}}H_{\\{3223\\}}(U,V)\,.$

The reduced correlator for $\langle 3333\rangle$ is the only one with a non-
trivial dependence on the R-symmetry cross-ratios. Its crossing equation is

$\displaystyle V^{3}H_{\\{3333\\}}(U,V,\sigma,\tau)-\tau
U^{3}H_{\\{3333\\}}\left(V,U,\frac{\sigma}{\tau},\frac{1}{\tau}\right)=\frac{9(4c-7)\left(V^{2}-\tau
U^{2}\right)}{c(4c-3)}+U^{4}(\sigma+\tau-1)$
$\displaystyle+U^{3}(-\sigma-2\tau+V(-2\sigma-\tau+1)+1)+UV^{3}(2\sigma-\tau+1)+V^{4}(-\sigma+\tau-1)$
$\displaystyle+V^{3}(\sigma-\tau+2)+\frac{9}{4c}\Big{(}-U^{3}+\tau\left(-U\left(U^{2}+1\right)+(U-1)V+V^{3}\right)$
(E.8)
$\displaystyle+\sigma(U-V)\left(U^{2}+V^{2}-1\right)-UV+U+V^{3}+V\Big{)}.$

In order to make use of the full superconformal symmetry when bootstrapping
the long multiplets, we have to subtract the contribution of the short
multiplets to the reduced correlator

$H_{\\{p_{i}\\}}^{\text{long}}=H_{\\{p_{i}\\}}-H_{\\{p_{i}\\}}^{\text{short}}\,.$
(E.9)

The next subsections describe the derivation of
$H_{\\{p_{i}\\}}^{\text{short}}$ for the mixed correlator bootstrap with
external operators $S_{2}$ and $S_{3}$. This is done in several steps. First
one expands the free theory correlators in terms of superconformal blocks and
then takes into account multiplet recombination in order to determine which of
the short multiplets remain present when transitioning from the free to the
interacting theory. These steps were done for the correlators $\langle
2222\rangle$, $\langle 2233\rangle$ and $\langle 3333\rangle$ in [78] and we
only have to supplement these results with the corresponding ones for $\langle
2323\rangle$ and $\langle 3223\rangle$. Once the contributions of the short
multiplets are determined, we need to find their contributions to the reduced
correlator. The decomposition (E.3) and in particular the function (E.5) were
chosen in [5] in such a way that the superconformal Casimir equation commutes
with the decomposition, which ensures that the superconformal blocks
themselves have a nice decomposition. Using this representation of the
superconformal blocks it is easy to read of their contribution to the reduced
correlator.

###  E.1 Superconformal blocks

The superconformal multiplets that can be exchanged in the four-point
functions we will consider can be labeled by the conformal dimension
${{\Delta}}$, spin $\ell$ and SU(4) representation $[m,n-m,m]$ of the
superconformal primary of the multiplet. For expanding free theory correlators
or using the recombination identity (E.30) it is also useful to label
superconformal multiplets in a different way which handles the different types
of multiplets on the same footing. Following [78] we can label them by their
twist $\gamma$ and a representation of $GL(2|2)$ which is labeled by a Young
diagram ${\underline{\smash{\lambda}}}$ of “fat hook” shape, meaning that with
$\lambda_{i}$ the length of the $i$th row and $\lambda^{T}_{j}$ the height of
the $j$th column of the Young diagram,

$\lambda_{i}\leq 2\,,\text{for }i\geq 3\,,\qquad\lambda^{T}_{j}\leq
2\,,\text{for }j\geq 3\,.$ (E.10)

One can translate between the two ways of labeling using the table below.

$\begin{array}[]{|c||c|c|c|c|}\hline\cr GL(2|2)\text{ rep
}{\underline{\smash{\lambda}}}&\text{dim }{{\Delta}}&\text{spin
}\ell&SU(4)\text{ rep }[m,n-m,m]&\text{multiplet}\\\
\hline\cr[0]&\gamma&0&[0,\gamma,0]&\frac{1}{2}\text{-BPS}\\\
\hline\cr[\lambda,1^{\mu}]\ (\lambda\geq
2)&\gamma{+}\lambda{-}2&\lambda{-}2&[\mu,\gamma{-}2\mu{-}2,\mu]&\text{semi-
short}\\\
\left[1^{\mu}\right]&\gamma&0&[\mu,\gamma{-}2\mu,\mu]&\frac{1}{4}\text{-BPS}\\\
\hline\cr{[\lambda_{1},\lambda_{2},2^{\mu_{2}},1^{\mu_{1}}]\ (\lambda_{2}\geq
2)}&\gamma{+}\lambda_{1}{+}\lambda_{2}{-}4&\lambda_{1}{-}\lambda_{2}&[\mu_{1}{-}\mu_{2},\gamma{-}2\mu_{1}-4,\mu_{1}{-}\mu_{2}]&\text{long}\\\
\hline\cr\end{array}$ Table 1: Translation between ${\mathcal{N}}=4$
superconformal reps and superfields
${\mathcal{O}}^{\gamma{\underline{\smash{\lambda}}}}$.

Note that for long multiplets there is a degeneracy in the description in
terms of $\gamma,{\underline{\smash{\lambda}}}$ since there are certain shift
symmetries that leave ${{\Delta}},\ell,m,n$ unchanged.

The correlator $\mathcal{G}_{\\{p_{i}\\}}(z,\bar{z},{\alpha},\bar{\alpha})$
can be expanded in superconformal blocks

$\bar{F}^{\gamma{\underline{\smash{\lambda}}}}=\left(\frac{z\bar{z}}{{\alpha}\bar{\alpha}}\right)^{\frac{\gamma}{2}}F^{{\alpha}{\beta}\gamma{\underline{\smash{\lambda}}}}\,,\qquad{\alpha}=\frac{\gamma+r}{2}\,,\
{\beta}=\frac{\gamma+s}{2}\,,$ (E.11)

where $r=p_{21}\equiv p_{2}-p_{1}$, $s=p_{34}$ and the definition of
$F^{{\alpha}{\beta}\gamma{\underline{\smash{\lambda}}}}$ is given in [78]. The
same superconformal blocks have a representation directly in terms of the
components of the decomposition (E.3) due to [5]. These have separate formulae
for the different types of multiplets. We can read off from table 1141414We
checked for that the definition of the superblocks (E.11) matches with the one
below for all the cases that we are using.

$\displaystyle\tfrac{1}{2}\text{-BPS}:$
$\displaystyle\quad\bar{F}^{\gamma[0]}$
$\displaystyle=\mathcal{B}_{0,\Delta}^{r,s}\,,$ (E.12)
$\displaystyle\text{semi-short}:$
$\displaystyle\quad\bar{F}^{\gamma[\lambda,1^{\mu}]}$
$\displaystyle=\mathcal{C}_{\lambda-2,\mu,\gamma-\mu-2}^{r,s}\,,\quad$
$\displaystyle\lambda\geq 2\,,$ $\displaystyle\tfrac{1}{4}\text{-BPS}:$
$\displaystyle\quad\bar{F}^{\gamma[1^{\mu}]}$
$\displaystyle=\mathcal{B}_{\mu,\gamma-\mu}^{r,s}=\mathcal{C}_{-1,\mu-1,\gamma-\mu-1}^{r,s}\,,$
$\displaystyle\text{long}:$
$\displaystyle\quad\bar{F}^{\gamma[\lambda_{1},\lambda_{2},2^{\mu_{2}},1^{\mu_{1}}]}$
$\displaystyle=\mathcal{A}_{\gamma+\lambda_{1}+\lambda_{2}-4,\lambda_{1}-\lambda_{2},\mu_{1}-\mu_{2},\gamma-\mu_{1}-\mu_{2}-4}^{r,s}\,,\quad$
$\displaystyle\lambda_{2}\geq 2\,.$

The block for half BPS multiplets is given by151515Here we fixed several
typos.

$\displaystyle\mathcal{B}_{0,\Delta}^{r,s}=\left\\{\begin{array}[]{l}\displaystyle
k=1,\qquad
f(z,{\alpha})=\displaystyle\sum_{\begin{subarray}{c}i=\max(|r|,|s|)\\\
\text{even or odd}\end{subarray}}^{\Delta-2}k_{0,i}^{r,s}(z,{\alpha}),\\\
\displaystyle
H^{\mathcal{B}}_{{{\Delta}}}(z,\bar{z},\alpha,\bar{\alpha})=(z\bar{z})^{-2}\sum_{\begin{subarray}{c}i=\max(|r|,|s|)\\\
\text{even or
odd}\end{subarray}}^{\Delta-4}\sum_{j=0}^{(\Delta-4-i)/2}(-1)^{j}G^{r,s}_{i+j+4,j}(z,\bar{z})Z_{j,i+j}^{r,s}({\alpha},\bar{\alpha}).\end{array}\right.$
(E.13)

The superblocks for semi-short and quarter BPS multiplets are given by

$\displaystyle\mathcal{C}_{\ell,m,n}^{r,s}=\left\\{\begin{array}[]{l}\displaystyle
k=0,\qquad f(z,{\alpha})=(-1)^{m}\ k_{\ell+m+2,n-m}^{r,s}(z,{\alpha}),\\\
\displaystyle
H^{\mathcal{C}}_{\ell,m,n}(z,\bar{z},{\alpha},\bar{\alpha})=(z\bar{z})^{-2}\sum_{i=1}^{m}(-1)^{i-1}\
G^{r,s}_{6+\ell+m+n-i,\ell+i}(z,\bar{z})Z_{m-i,n-i}^{r,s}({\alpha},\bar{\alpha})\\\
\hskip 71.13188pt+\displaystyle(z\bar{z})^{-2}\sum_{i=0}^{i_{\rm
max}}(-1)^{m+i}\
G_{4+\ell+n-i,2+\ell+m+i}^{r,s}(z,\bar{z})Z_{i,n-m-i-2}^{r,s}({\alpha},\bar{\alpha})\end{array}\right.$
(E.14)

with $i_{\rm max}=(n-m-2-\max(|r|,|s|))/2$. Finally we have for long
multiplets

$\displaystyle\mathcal{A}_{{{\Delta}},\ell,m,n}^{r,s}=\left\\{\begin{array}[]{l}\displaystyle
k=0,\qquad f(z,{\alpha})=0,\\\ \displaystyle
H^{\mathcal{A}}_{{{\Delta}},\ell,m,n}(z,\bar{z},{\alpha},\bar{\alpha})=(z\bar{z})^{-2}G^{r,s}_{{{\Delta}}+4,\ell}(z,\bar{z})Z_{m,n}^{r,s}({\alpha},\bar{\alpha})\end{array}\right..$
(E.15)

These definitions are in terms of the usual $SO(4,2)$ conformal blocks

$\displaystyle G_{\Delta,\ell}^{r,s}(z,\bar{z})$
$\displaystyle=\frac{z\bar{z}}{\bar{z}-z}\left[k_{\frac{\Delta-\ell-2}{2}}^{r,s}(z)k_{\frac{\Delta+\ell}{2}}^{r,s}(\bar{z})-k_{\frac{\Delta+\ell}{2}}^{r,s}(z)k_{\frac{\Delta-\ell-2}{2}}^{r,s}(\bar{z})\right],$
(E.16) $\displaystyle k_{h}^{r,s}(z)$
$\displaystyle=z^{h}\;_{2}F_{1}\left(h+\frac{r}{2},h+\frac{s}{2};2h,z\right),$
(E.17)

and the $S_{5}$ spherical harmonics, which are given in terms of the same
functions

$Z_{m,n}^{r,s}({\alpha},\bar{\alpha})=(-1)^{m}G_{-n,m}^{-r,-s}({\alpha},\bar{\alpha})\,.$
(E.18)

We also used the notation

$\displaystyle k_{j,m}^{r,s}(z,{\alpha})$
$\displaystyle=k_{1+\frac{m}{2}+j}^{r,s}(z)k_{-\frac{m}{2}}^{-r,-s}({\alpha})\,.$
(E.19)

###  E.2 Free theory

We begin with the correlators in the free theory

$\displaystyle\left\langle 2222\right\rangle={}$ $\displaystyle
a_{1}(g_{12}^{2}g_{34}^{2}+g_{13}^{2}g_{24}^{2}+g_{14}^{2}g_{23}^{2})+a_{2}(g_{12}g_{23}g_{34}g_{41}+g_{13}g_{32}g_{21}g_{14}+g_{13}g_{34}g_{42}g_{21})\,,$
$\displaystyle\left\langle 2233\right\rangle={}$ $\displaystyle
b_{1}g_{12}^{2}g_{34}^{3}+b_{2}\left(g_{12}g_{14}g_{23}g_{34}^{2}+g_{12}g_{13}g_{24}g_{34}^{2}\right)+b_{3}g_{13}g_{14}g_{23}g_{24}g_{34}\,,$
$\displaystyle\left\langle 3333\right\rangle={}$ $\displaystyle
c_{1}\left(g_{14}^{3}g_{23}^{3}+g_{13}^{3}g_{24}^{3}+g_{12}^{3}g_{34}^{3}\right)+c_{2}(g_{13}g_{14}^{2}g_{24}g_{23}^{2}+g_{12}g_{14}^{2}g_{34}g_{23}^{2}+g_{13}^{2}g_{14}g_{24}^{2}g_{23}$
$\displaystyle+g_{12}^{2}g_{14}g_{34}^{2}g_{23}+g_{12}^{2}g_{13}g_{24}g_{34}^{2}+g_{12}g_{13}^{2}g_{24}^{2}g_{34})+c_{3}g_{12}g_{13}g_{14}g_{23}g_{24}g_{34}\,,$
(E.20)

with

$\displaystyle a_{1}$ $\displaystyle=1,\,a_{2}=\frac{4}{N^{2}-1}\,,$ (E.21)
$\displaystyle b_{1}$
$\displaystyle=1,\,b_{2}=\frac{6}{N^{2}-1},\,b_{3}=\frac{12}{N^{2}-1}\,,$
$\displaystyle c_{1}$
$\displaystyle=1,\,c_{2}=\frac{9}{N^{2}-1},\,c_{3}=\frac{18(N^{2}-12)}{(N^{2}-1)(N^{2}-4)}\,.$

The SCPW expansions of these three correlators are given by [78]

$\displaystyle\left\langle 2222\right\rangle={}$ $\displaystyle
T_{2222}\left(1+\sum\limits_{\lambda\geq
0}^{\infty}A^{a_{2}}_{2[\lambda]}\bar{F}^{2[\lambda]}+\sum\limits_{\lambda_{1}\geq\lambda_{2}\geq
0}^{\infty}A^{a_{1},a_{2}}_{4[\lambda_{1},\lambda_{2}]}\bar{F}^{4[\lambda_{1},\lambda_{2}]}\right)\,,$
$\displaystyle\left\langle 2233\right\rangle={}$ $\displaystyle
T_{2233}\left(1+\sum\limits_{\lambda\geq
0}^{\infty}A^{b_{2}}_{2[\lambda]}\bar{F}^{2[\lambda]}+\sum\limits_{\lambda_{1}\geq\lambda_{2}\geq
0}^{\infty}A^{0,b_{3}}_{4[\lambda_{1},\lambda_{2}]}\bar{F}^{4[\lambda_{1},\lambda_{2}]}\right)\,,$
(E.22) $\displaystyle\left\langle 3333\right\rangle={}$ $\displaystyle
T_{3333}\left(1+\sum\limits_{\lambda\geq
0}^{\infty}A^{c_{2}}_{2[\lambda]}\bar{F}^{2[\lambda]}+\sum\limits_{\lambda_{1}\geq\lambda_{2}\geq
0}^{\infty}A^{c_{2},c_{3}}_{4[\lambda_{1},\lambda_{2}]}\bar{F}^{4[\lambda_{1},\lambda_{2}]}+\sum\limits_{\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq
0}^{\infty}A_{6[\lambda_{1},\lambda_{2},\lambda_{3}]}\bar{F}^{6[\lambda_{1},\lambda_{2},\lambda_{3}]}\right),$

where the OPE coefficients for the twist 2 and 4 multiplets only differ by
their dependence on the color factors

$\displaystyle
A^{a}_{2[\lambda]}=\frac{1+(-1)^{\lambda}}{2}\,\frac{2a(\lambda!)^{2}}{(2\lambda)!}\,,$
(E.23) $\displaystyle
A^{a,b}_{4[\lambda_{1},\lambda_{2}]}=\frac{1+(-1)^{\lambda_{1}-\lambda_{2}}}{2}\,\frac{\lambda_{1}!\left(\lambda_{1}+1\right)!\left(\lambda_{2}!\right){}^{2}\left(a\left(\lambda_{1}-\lambda_{2}+1\right)\left(\lambda_{1}+\lambda_{2}+2\right)+b(-1)^{\lambda_{2}}\right)}{\left(2\lambda_{2}\right)!\left(2\lambda_{1}+1\right)!}\,,$

and the coefficients for the twist 6 operators in the $\langle 3333\rangle$
correlator are given by

$\displaystyle A_{6[\lambda_{1},\lambda_{2}]}={}$ $\displaystyle
m^{+}_{\lambda_{1},\lambda_{2}}\frac{1}{2}\Big{(}c_{1}\left(\lambda_{1}+2\right)\left(\lambda_{1}+3\right)\left(\lambda_{1}-\lambda_{2}+1\right)\left(\lambda_{2}+1\right)\left(\lambda_{2}+2\right)\left(\lambda_{1}+\lambda_{2}+4\right)$
$\displaystyle+4c_{2}\left(\left((-1)^{\lambda_{2}}+1\right)\lambda_{1}\left(\lambda_{1}+5\right)+8(-1)^{\lambda_{2}}+\left((-1)^{\lambda_{2}}-1\right)\lambda_{2}\left(\lambda_{2}+3\right)+4\right)\Big{)}\,,$
$\displaystyle A_{6[\lambda_{1},\lambda_{2},1]}={}$ $\displaystyle
m^{-}_{\lambda_{1},\lambda_{2}}\frac{1}{4}\Big{(}c_{1}\left(\lambda_{1}+1\right)\left(\lambda_{1}+4\right)\left(\lambda_{1}-\lambda_{2}+1\right)\lambda_{2}\left(\lambda_{2}+3\right)\left(\lambda_{1}+\lambda_{2}+4\right)$
$\displaystyle+4c_{2}\left((-1)^{\lambda_{2}}-1\right)\left(\lambda_{1}-\lambda_{2}+1\right)\left(\lambda_{1}+\lambda_{2}+4\right)\Big{)}\,,$
(E.24) $\displaystyle A_{6[\lambda_{1},\lambda_{2},2]}={}$ $\displaystyle
m^{+}_{\lambda_{1},\lambda_{2}}\frac{1}{12}\Big{(}c_{1}\lambda_{1}\left(\lambda_{1}+5\right)\left(\lambda_{1}-\lambda_{2}+1\right)\left(\lambda_{2}-1\right)\left(\lambda_{2}+4\right)\left(\lambda_{1}+\lambda_{2}+4\right)$
$\displaystyle+4c_{2}\left(\left((-1)^{\lambda_{2}}+1\right)\lambda_{1}\left(\lambda_{1}+5\right)+\left((-1)^{\lambda_{2}}-1\right)\left(\lambda_{2}-1\right)\left(\lambda_{2}+4\right)\right)\Big{)}\,,$

with

$\displaystyle
m^{\pm}_{\lambda_{1},\lambda_{2}}=\frac{1\pm(-1)^{\lambda_{1}-\lambda_{2}}}{2}\,\frac{\left(\lambda_{1}+2\right)!{}^{2}\left(\lambda_{2}+1\right)!{}^{2}}{\left(2\lambda_{1}+4\right)!\left(2\lambda_{2}+2\right)!}.$
(E.25)

The correlators $\langle 2323\rangle$ and $\langle 3223\rangle$ are related to
$\langle 2233\rangle$ in (E.20) by crossing and we obtain their SCPW expansion
by using the elegant method from [78] where one expands both the
superconformal blocks as well as the correlator in terms of super Schur
polynomials and equates the coefficients. We find the expansion

$\left\langle 2323\right\rangle=T_{2323}\left(\sum\limits_{\lambda\geq
0}^{\infty}A_{3[\lambda]}\bar{F}^{3[\lambda]}+\sum\limits_{\lambda_{1}\geq\lambda_{2}\geq
0}^{\infty}A_{5[\lambda_{1},\lambda_{2}]}\bar{F}^{5[\lambda_{1},\lambda_{2}]}\right)\,,$
(E.26)

with the coefficients

$\displaystyle A_{3[\lambda]}={}$ $\displaystyle\frac{\pi
2^{-4\lambda-3}\left(b_{2}(-1)^{\lambda}(\lambda+1)+b_{3}\right)\Gamma(2\lambda+3)}{\Gamma\left(\lambda+\frac{3}{2}\right)^{2}}\,,$
$\displaystyle A_{5[\lambda_{1},\lambda_{2}]}={}$
$\displaystyle\frac{\pi\Gamma\left(\lambda_{1}+3\right)\Gamma\left(\lambda_{2}+2\right)}{2^{2(\lambda_{1}+\lambda_{2})+5}\Gamma\left(\lambda_{1}+\frac{5}{2}\right)\Gamma\left(\lambda_{2}+\frac{3}{2}\right)}\Big{(}2b_{2}\left((-1)^{\lambda_{1}}\left(\lambda_{1}+2\right)+(-1)^{\lambda_{2}}\left(\lambda_{2}+1\right)\right)$
$\displaystyle+b_{1}(-1)^{\lambda_{1}-\lambda_{2}}\left(\lambda_{1}+2\right)\left(\lambda_{1}-\lambda_{2}+1\right)\left(\lambda_{2}+1\right)\left(\lambda_{1}+\lambda_{2}+3\right)\Big{)}\,.$
(E.27)

Similarly we find the following expansion for the correlator $\langle
3223\rangle$

$\left\langle 3223\right\rangle=T_{3223}\left(\sum\limits_{\lambda\geq
0}^{\infty}A_{3[\lambda]}\bar{F}^{3[\lambda]}+\sum\limits_{\lambda_{1}\geq\lambda_{2}\geq
0}^{\infty}A_{5[\lambda_{1},\lambda_{2}]}\bar{F}^{5[\lambda_{1},\lambda_{2}]}\right)\,.$
(E.28)

The OPE coefficients are in this case

$\displaystyle A_{3[\lambda]}={}$ $\displaystyle\frac{\pi
2^{-4\lambda-3}\left(b_{2}(\lambda+1)+b_{3}(-1)^{\lambda}\right)\Gamma(2\lambda+3)}{\Gamma\left(\lambda+\frac{3}{2}\right)^{2}}\,,$
$\displaystyle A_{5[\lambda_{1},\lambda_{2}]}={}$
$\displaystyle\frac{\pi\Gamma\left(\lambda_{1}+3\right)\Gamma\left(\lambda_{2}+2\right)}{2^{2(\lambda_{1}+\lambda_{2})+5}\Gamma\left(\lambda_{1}+\frac{5}{2}\right)\Gamma\left(\lambda_{2}+\frac{3}{2}\right)}\Big{(}2b_{2}\left((-1)^{\lambda_{1}}\left(\lambda_{2}+1\right)+(-1)^{\lambda_{2}}\left(\lambda_{1}+2\right)\right)$
$\displaystyle+b_{1}\left(\lambda_{1}+2\right)\left(\lambda_{1}-\lambda_{2}+1\right)\left(\lambda_{2}+1\right)\left(\lambda_{1}+\lambda_{2}+3\right)\Big{)}\,.$
(E.29)

###  E.3 Short sector in the interacting theory

In order to identify the short sector in the interacting theory it is
necessary to resolve an ambiguity between short multiplets and long multiplets
at the unitarity bound which is due to the recombination identity

$\displaystyle\bar{F}_{\text{long}}^{\gamma[\lambda+1,1^{\nu+1}]}\equiv\text{lim}_{\rho\rightarrow
1}\bar{F}^{\gamma[\lambda+\rho,\rho,1^{\nu}]}=\bar{F}^{\gamma-2[\lambda+2,1^{\nu}]}+\bar{F}^{\gamma[\lambda+1,1^{\nu+1}]}\,.$
(E.30)

Here the LHS is the analytic continuation of a long multiplet with
$\rho=2,3,\ldots$ to the unitarity bound where it becomes equal to the sum of
two short multiplets. This can be used to replace short multiplets by
multiplets of higher twist $\gamma$.

For the correlators $\langle 2222\rangle$, $\langle 2233\rangle$ and $\langle
3333\rangle$ the short sectors in the interacting theory were determined
already in [78] and we repeat them here for convenience, supplemented with the
results for $\langle 2323\rangle$ and $\langle 3223\rangle$. In order to
resolve this ambiguity we follow section 2.4 of [27]. There it is argued that
in a large $c$ expansion the OPE coefficients of protected multiplets with
twist $\gamma<\min(p_{1}+p_{2},p_{3}+p_{4})$ have to be $O(c^{-2})$. However,
in the free theory correlators above, all twist 2 and twist 3 semi-short
multiplets have OPE coefficients with contributions only at $O(1)$ and
$O(c^{-1})$, so we have to use (E.30) to remove them completely and replace
them by twist 4 and 5 multiplets. Another way to argue is that there are no
double trace semi-short operators of the form
$S_{q}\partial^{\ell}S_{\tilde{q}}$ (with $S_{q}$ and $S_{\tilde{q}}$ half BPS
operators) of twist $q+\tilde{q}=2$ or $3$ that could be protected. For the
correlators involving $S_{2}$ we thus have

$\displaystyle\mathcal{G}_{\\{2222\\}/\\{2233\\}}^{\text{short}}={}$
$\displaystyle 1+A_{2[0]}\bar{F}^{2[0]}+\sum_{\lambda\geq
0}^{\infty}A_{4[\lambda]}\bar{F}^{4[\lambda]}+\sum_{\lambda\geq
1}^{\infty}A^{\prime}_{4[\lambda,1]}\bar{F}^{4[\lambda,1]}\,,$ (E.31)
$\displaystyle\mathcal{G}_{\\{2323\\}/\\{3223\\}}^{\text{short}}={}$
$\displaystyle A_{3[0]}\bar{F}^{3[0]}+\sum\limits_{\lambda\geq
0}^{\infty}A_{5[\lambda]}\bar{F}^{5[\lambda]}+\sum\limits_{\lambda\geq
1}^{\infty}A^{\prime}_{5[\lambda,1]}\bar{F}^{5[\lambda,1]}\,,$

where

$\displaystyle A^{\prime}_{4[\lambda,1]}$
$\displaystyle=A_{4[\lambda,1]}-A_{2[\lambda+1]}\,,$ (E.32) $\displaystyle
A^{\prime}_{5[\lambda,1]}$
$\displaystyle=A_{5[\lambda,1]}-A_{3[\lambda+1]}\,,$

and it is understood that the different OPE coefficients for each correlator
from section E.2 are used.

For the correlator $\langle 3333\rangle$ the same argument works for the twist
2 semi-short multiplets, however the OPE coefficients of the twist 4
multiplets have terms at ${\mathcal{O}}(c^{-2})$, so they can be present in
the interacting theory. Since these operators are non-degenerate, these OPE
coefficients are however determined by OPE coefficients that appear in other
correlators [78]

$\displaystyle\tilde{A}_{4[\lambda]}=\frac{\left(A_{4[\lambda]}^{2233}\right)^{2}}{A_{4[\lambda]}^{2222}}=\frac{1+(-1)^{\lambda}}{2}\,\frac{144\lambda!(\lambda+1)!}{(N^{2}-1)(2\lambda+1)!\left(-\lambda(\lambda+3)+(\lambda+1)(\lambda+2)N^{2}+2\right)},$
$\displaystyle\tilde{A}_{4[\lambda,1]}=\frac{\left(A_{4[\lambda,1]}^{{}^{\prime}2233}\right)^{2}}{A_{4[\lambda,1]}^{{}^{\prime}2222}}=\frac{1-(-1)^{\lambda}}{2}\,\frac{576((\lambda+1)!)^{2}}{(N^{2}-1)(2\lambda+2)!\left(\lambda(\lambda+3)\left(N^{2}-1\right)-12\right)}.$