# Pure anti-de Sitter supergravity and the conformal bootstrap

Luis F. Alday  Mathematical Institute, University of Oxford, Woodstock Road,
Oxford, OX2 6GG, UK  Shai M. Chester  Department of Particle Physics and
Astrophysics, Weizmann Institute of Science, Rehovot, Israel

###### Abstract

We consider graviton scattering in maximal supergravity on Anti-de Sitter
space (AdS) in $d+1$ dimensions for $d=3,4,\text{and $6$}$ with no extra
compact spacetime factor. Holography suggests that this theory is dual to an
exotic maximally supersymmetric conformal field theory (CFT) in $d$ dimensions
whose only light single trace operator is the stress tensor. This contrasts
with more standard cases like Type IIB string theory on $AdS_{5}\times S^{5}$
dual to $\mathcal{N}=4$ Super-Yang-Mills, where the CFT has light single trace
operators for each Kaluza-Klein mode on $S^{5}$. We compute the 1-loop
correction to the pure AdSd+1 theory in a small Planck length expansion, which
is dual to the large central charge expansion in the CFT. We find that this
correction saturates the most general non-perturbative conformal bootstrap
bounds on this correlator in the large central charge regime for $d=3,4,6$,
while the 1-loop correction to CFTs with string/M-theory duals all lie inside
the allowed region.

##  I Introduction

The AdS/CFT duality relates quantum gravity on Anti-de Sitter (AdS) space in
$d+1$ dimensions times a compact spacetime factor, to certain supersymmetric
CFTs in $d$ dimensions Maldacena:1997re . In the simplest examples, the
compact space is simply a sphere with a similar radius as AdS, and the CFT is
maximally supersymmetric. Compactifying the graviton on the sphere generates
an infinite tower of Kaluza-Klein (KK) modes in AdS, which are dual to light
single trace operators in the CFT. It is an open question if holographic duals
exist where the radius of the sphere is parametrically smaller than that of
AdS, so that these extra dimensions would be small (See Alday:2019qrf ;
Gopakumar:2022kof  for a recent discussion). In the most extreme case, there
would simply be no compact factor at all, and the only single trace operators
in the dual CFT would be the stress tensor multiplet. No such pure AdS theory
has been constructed, despite much effort Witten:2007kt ; Maloney:2007ud ;
Hellerman:2009bu ; Keller:2014xba ; Collier:2016cls ; Afkhami-Jeddi:2019zci ;
Hartman:2019pcd ; Maxfield:2020ale ; Afkhami-Jeddi:2020ezh ; Maloney:2020nni .

We will address this question by studying the stress tensor four-point
function, which is dual to scattering of gravitons in the bulk, in maximally
supersymmetric CFTs in $d=3,4,6$ dimensions. Consider the large central charge
$c$ expansion of this correlator, where $c$ is defined as the coefficient of
the stress-tensor two-point function, and is related to the bulk as

$\begin{split}c\sim(L_{\text{AdS}}/\ell_{\text{Planck}})^{D-2}\,,\end{split}$
(1)

where $L_{\text{AdS}}$ is the radius of the AdSd+1 factor, and
$\ell_{\text{Planck}}$ is the Planck length of the full $D$-dimensional bulk
spacetime, including a possible compact factor. We can define the correlator
$\mathcal{G}$ in any such theory to any order in $1/c$ as

$\begin{split}{\mathcal{G}}&={\mathcal{G}}^{(0)}+c^{-1}{\mathcal{G}}^{R}+c^{-2}({\mathcal{G}}^{R|R}+\kappa{\mathcal{G}}^{R^{4}})+\dots\\\
&\dots+c^{-\frac{D+4}{D-2}}{\mathcal{G}}^{R^{4}}+c^{-\frac{D+8}{D-2}}{\mathcal{G}}^{D^{4}R^{4}}+\dots\,,\end{split}$
(2)

where in the first line we wrote the tree level supergravity term
$\mathcal{G}^{R}$ and the 1-loop term ${\mathcal{G}}^{R|R}$ with supergravity
vertices $R$, while in the second line we wrote tree level higher derivative
corrections that are allowed by supersymmetry 111In CFTs dual to M-theory the
lowest correction $R^{4}$ scales as $c^{-5/3}$, and was computed in
Chester:2018aca ; Chester:2018dga . In CFTs dual to string theory, this
coefficient scales like $c^{-7/4}$ at finite string coupling, and was computed
for Type IIA in Binder:2019mpb , and Type IIB in Chester:2019jas . The
$D^{4}R^{4}$ term has also been computed for M-theory in Binder:2018yvd , and
for Type IIB in Chester:2020vyz .. The expansion also includes 1-loop terms
with such higher derivative vertices, as well as higher loop terms 222The
distinction between tree and loop is ambiguous, since
$c\sim(L_{\text{AdS}}/\ell_{\text{Planck}})^{D-2}$ is the only expansion
parameter, but at low orders for some $D$ they can be distinguished by the
powers of $1/c$. The $\mathcal{G}^{R|R}$ term has an $\mathcal{G}^{R^{4}}$
type contact term with coefficient $\kappa$ as long as the scaling of the
$R^{4}$ tree level term is smaller than $R|R$, which is the case for string
and M-theory with $D=10,11$ 333This contact term has been fixed for M-theory
on $AdS_{4}\times S^{7}/\mathbb{Z}_{k}$ Alday:2021ymb ; Alday:2022rly  and
Type IIB on $AdS_{5}\times S^{5}/\mathbb{Z}_{k}$ Chester:2019pvm ;
Alday:2021vfb  for $k=1,2$., respectively, but is not for the pure AdSd+1
theory where $D=d+1$ and $d=3,4,6$. All tree and loop supergravity terms
$\mathcal{G}^{R|R|\dots}$ can be computed iteratively using the analytic
bootstrap Rastelli:2017udc ; Aharony:2016dwx , but to fix the higher
derivative corrections as well as loop contact terms such as
$\kappa{\mathcal{G}}^{R^{4}}$, we need a UV completion like string/M-theory.
These terms only affect CFT data with finite spin Heemskerk:2009pn , so at any
given order in $1/c$ we can unambiguously determine an infinite set of CFT
data for AdSd+1 duals with any (or no) compact factor. Whether or not a pure
AdSd+1 theory is also defined non-perturbatively in $c$ is a separate question
that we will address in the conclusion.

The tree level supergravity correction $\mathcal{G}^{R}$ at order $1/c$ is
unaffected by a compact spacetime factor Rastelli:2017udc ; Rastelli:2017ymc ;
Zhou:2017zaw ; Alday:2020dtb , but higher loop terms starting with
$\mathcal{G}^{R|R}$ at order $1/c^{2}$ are sensitive to the number of KK modes
Aharony:2016dwx . We will compute this 1-loop term for pure AdSd+1 theories in
$d=3,4,6$ using the analytic bootstrap, which allows us to extract all CFT
data to $O(c^{-2})$. We then can compare this $O(1/c^{2})$ data to non-
perturbative numerical bootstrap bounds Beem:2013qxa ; Beem:2015aoa ;
Chester:2014fya ; Chester:2014mea , which apply to any maximally
supersymmetric CFT, and can be computed for any $c$. We find that for all
$d=3,4,6$, the pure AdSd+1 1-loop correction precisely saturates the bootstrap
bounds in the large $c$ regime.

The 1-loop correction has also been computed for maximally supersymmetric CFTs
with string/M-theory duals. In 3d, these CFTs are $U(N)_{k}\times U(N)_{-k}$
ABJM theory with $k=1,2$, which is dual to M-theory on $AdS_{4}\times
S^{7}/\mathbb{Z}_{k}$ with $c\sim N^{3/2}$ 444The $U(N)_{2}\times U(N+1)_{-2}$
theory also has maximal supersymmetry, but this shift of the gauge factor does
not matter in the large $N$ limit. When $k>2$, the theory has $\mathcal{N}=6$
supersymmetry. Aharony:2008ug . In 4d, they are $\mathcal{N}=4$ super-Yang-
Mills (SYM) with gauge group $SU(N)$ or $SO(N)$ 555The $USp(2N)$ gauge group
is also allowed, but is similar to $SO(N)$ in the large $N$ limit., which is
dual to Type IIB string theory on $AdS_{5}\times S^{5}$ or $AdS_{5}\times
S^{5}/\mathbb{Z}_{2}$ with $c\sim N^{2}$ Maldacena:1997re , respectively. In
6d, they are $A_{N-1}$ or $D_{N}$ $(2,0)$ theories Witten:1995zh  666There are
also $(2,0)$ theories constructed from exceptional groups, but these do not
have a large $N$ limit., which are dual to $AdS_{7}\times S^{4}$ or
$AdS_{7}\times S^{4}/\mathbb{Z}_{2}$ with $c\sim N^{3}$ Witten:1998xy ;
Aharony:1998rm , respectively. The 1-loop corrections were computed in these
various cases in Alday:2017xua ; Aprile:2017bgs ; Alday:2020tgi ;
Alday:2021ymb ; Alday:2021vfb ; Alday:2022rly . In all cases, we find that
these corrections lie inside the allowed region of the bootstrap bounds for
the same regime of large $c$ where the pure AdSd+1 theory saturates the bound.

The rest of this paper is organized as follows. In Section II, we review the
constraints of maximal superconformal symmetry on the stress tensor four-point
function for $d=3,4,6$. In Section III we consider the large $c$ expansion of
this correlator and compute the 1-loop correction to pure AdSd+1 supergravity.
In Section IV we compare this correction, and the previously computed 1-loop
corrections for string/M-theory duals, to non-perturbative numerical conformal
bootstrap bounds in the large $c$ regime. We end with a discussion of our
results in Section V.

##  II Stress tensor correlator

We begin by reviewing the constraints of maximal supersymmetry in $d=3,4,6$ on
the stress tensor correlator. We consider the superconformal primary $S(x)$,
which is a scalar with $\Delta=d-2$ that transforms in the symmetric traceless
representation of the R-symmetry group $SO(8)_{R}$, $SO(6)_{R}$, and
$SO(5)_{R}$ for 3d, 4d, and 6d, respectively. Conformal and R-symmetry fixes
the four-point function to take the form

$\begin{split}&\langle
S(x_{1},Y_{1})S(x_{2},Y_{2})S(x_{3},Y_{3})S(x_{4},Y_{4})\rangle=\\\
&\qquad\qquad\qquad\frac{(Y_{1}\cdot Y_{2})^{2}(Y_{3}\cdot
Y_{4})^{2}}{|x_{12}|^{2(d-2)}|x_{34}|^{2(d-2)}}\mathcal{G}(U,V;\sigma,\tau)\,,\end{split}$
(3)

where we define the cross ratios

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

with $x_{ij}\equiv x_{i}-x_{j}$, and $Y_{i}$ are null polarization vectors
that encode the R-symmetry indices. The constraints from supersymmetry are
given by the superconformal Ward identities Dolan:2004mu , which can be
satisfied by expanding $\mathcal{G}$ in superconformal blocks as 777In 4d and
6d, we can also satisfy these Ward identities by writing
$\mathcal{G}(U,V;\sigma,\tau)$ in terms of a differential operator
$\Upsilon(U,V,\partial_{U},\partial_{V},\sigma,\tau)$ acting on a reduced
correlator $\mathcal{H}(U,V)$, which is then an R-symmetry singlet.

$\begin{split}\mathcal{G}(U,V;\sigma,\tau)=\sum_{\mathcal{M}}\lambda^{2}_{\mathcal{M}}\mathfrak{G}_{\mathcal{M}}(U,V;\sigma,\tau)\,,\end{split}$
(5)

where ${\cal M}$ runs over all the supermultiplets appearing in the $S\times
S$ OPE, the $\lambda^{2}_{\mathcal{M}}$ are the squared OPE coefficients for
each such supermultiplet $\mathcal{M}$, and the explicit form of the
superblocks can be found for each $d$ in Dolan:2004mu ; Beem:2016wfs ;
Beem:2015aoa ; Chester:2014fya . In Appendix A, for each $d$ we summarize the
multiplets $\mathcal{M}$ that appear, which we label by the scaling dimension
$\Delta$, the spin $\ell$, and the R-symmetry representation of the
superprimary. We exclude free theory multiplets, which for $d=4,6$ restricts
us to interacting theories 888In 3d, the free theory multiplet is identical to
the unitarity bound of the long multiplet, so cannot be excluded
kinematically. The $S\times S$ OPE includes long multiplets in the singlet of
the R-symmetry group with even spin $\ell$ and scaling dimension
$\Delta>d-2+\ell$, as well as protected multiplets such as the stress tensor
with fixed $\Delta$. The stress tensor $\lambda^{2}$ is fixed by the conformal
Ward identity Osborn:1993cr  to be inversely proportional to the central
charge coefficient $c$ of the stress tensor two-point function:

$\begin{split}\lambda^{2}_{\text{stress}}\propto 1/c\,,\end{split}$ (6)

where the proportionality constant is fixed in 4d so that $c$ is the conformal
anomaly Beem:2016wfs , in 6d so that a free tensor multiplet has $c=1$
Beem:2015aoa , and in 3d so that the free theory has $c=16$ Chester:2014fya .
In 4d and 6d, the existence of a protected 2d chiral algebra Beem:2013sza
fixes $\lambda^{2}_{\mathcal{M}}\propto 1/c$ for certain protected multiplets,
while the remaining protected multiplets $\mathcal{M}_{\text{prot}}$ have
$\lambda^{2}$ that remain unconstrained.

An important non-perturbative constraint on the four-point function can be
derived by swapping $1\leftrightarrow 3$ in (3), which yields the crossing
equations

$\begin{split}{\mathcal{G}}(U,V;\sigma,\tau)=\frac{U^{d-2}}{V^{d-2}}\tau^{2}{\mathcal{G}}(V,U;\sigma/\tau,1/\tau)\,,\end{split}$
(7)

which we will now use to constrain the correlator.

##  III One-loop from tree level

We will now restrict to the pure AdSd+1 theory, and consider the large $c$
expansion of the correlator ${\mathcal{G}}$ shown in (2), where we expand long
multiplet CFT data as

$\begin{split}\Delta_{n,\ell}&=2(d-2)+2n+\ell+\gamma^{R}_{n,\ell}/c+\gamma^{R|R}_{n,\ell}c^{2}+\dots\,,\\\
\lambda_{n,\ell}^{2}&=(\lambda^{(0)}_{n,\ell})^{2}+(\lambda^{R}_{n,\ell})^{2}/c+(\lambda^{R|R}_{n,\ell})^{2}/c^{2}+\dots\,.\\\
\end{split}$ (8)

A similar expansion exists for the OPE coefficients of the protected
operators, although of course their scaling dimensions are fixed. The long
multiplets that appear in (8) are all double trace operators $[SS]_{n,\ell}$
of the schematic form

$\begin{split}[SS]_{n,\ell}=S\Box^{n}\partial_{\mu_{1}}\dots\partial_{\mu_{\ell}}S\,,\end{split}$
(9)

with $\Delta^{(0)}_{n,\ell}=2(d-2)+2n+\ell$ in the $c\to\infty$ generalized
free field theory (GFFT). Note that if the bulk theory had a compact factor,
e.g. $AdS_{5}\times S^{5}$, then we could use the higher KK modes to construct
more such long operators, which would be degenerate in the GFFT and thus mix
in the $1/c$ expansion. The GFFT and tree correlators, which are insensitive
to the bulk factor, were computed in each $d$ in Dolan:2001tt ; Zhou:2017zaw ;
Heslop:2004du ; Arutyunov:2002ff  and used to extract tree level data, which
we summarize in Table 1. For theories with higher KK modes, we can only
extract the average long multiplet anomalous dimensions
$\langle\lambda^{2}_{n,\ell}\gamma_{n,\ell}^{R}\rangle$, due to the degeneracy
at GFFT. For protected multiplets, we can obtain the unique CFT data for all
such large $c$ theories.

At 1-loop level, we can expand the superblock expansion (5) to get

$\begin{split}&\mathcal{G}^{R|R}=\sum_{n=0}^{\infty}\sum_{\ell\in\text{Even}}\Big{[}\frac{1}{8}{(\lambda^{(0)}_{n,\ell})^{2}(\gamma^{R}_{n,\ell})^{2}}\log^{2}U\\\
&+{(\lambda^{(0)}_{n,\ell})^{2}\gamma^{R|R}_{n,\ell}}\frac{\log
U}{2}+\dots\Big{]}\mathfrak{G}_{n,\ell}+\sum_{{\mathcal{M}}_{\text{prot}}}(\lambda^{R|R}_{\mathcal{M}})^{2}\mathfrak{G}_{\mathcal{M}}\,,\end{split}$
(10)

where the ellipses refers to other other combinations of tree and loop data,
and recall that ${\mathcal{M}}_{\text{prot}}$ denotes protected multiplets
whose OPE coefficients are not $1/c$ exact. The significance of the
$\log^{2}U$ term is that it is the only term at this order that has a double
discontinuity (DD) as $U\to 0$ 999This is true for every known maximally
supersymmetric CFT in $d=3,4,6$ except the $U(N)_{1}\times U(N)_{-1}$ ABJM
theory, for which additional contributions come from odd twist long multiplet
OPE coefficients, which can also be computed from tree level data. See
Alday:2022rly  for more details.. The Lorentzian inversion formula Caron-
Huot:2017vep  shows that all CFT data with sufficiently large $\ell$ can be
extracted from the DD as $V\to 0$, so we can obtain this DD from the
$\log^{2}U$ terms after applying crossing (7). For instance, we can compute
the 1-loop correction to the OPE coefficient of 3d protected multiplets
${(A,+)_{\ell}}$ as

$\begin{split}&(\lambda^{R|R}_{(A,+)_{\ell}})^{2}=\frac{12(2\ell+5)\Gamma(\ell+3)^{4}}{\Gamma\left(\ell+\frac{5}{2}\right)^{2}\Gamma\left(\ell+\frac{7}{2}\right)^{2}}\\\
&\times\int_{0}^{1}\frac{d\bar{z}}{\bar{z}}g_{\ell+4,\ell+2}(\bar{z})\text{dDisc}[{\cal
G}^{[0040]}(z\bar{z},1-\bar{z},)|_{z}]\,,\end{split}$ (11)

where ${\cal G}^{[0040]}|_{z}$ is the leading twist term in the highest weight
representation of $SO(8)_{R}$, we define the lightcone blocks
$g_{\Delta,\ell}(z)$ in Appendix C, and we introduce the variables
$U=z\bar{z}$ and $V=(1-z)(1-\bar{z})$. We compute dDisc acting on
$\log^{2}V\sim\log^{2}(1-\bar{z})$ as

$\begin{split}{\rm
dDisc}\,[f(z,\bar{z})\log^{2}{1-\bar{z}}]=4\pi^{2}f(z,\bar{z})\,,\end{split}$
(12)

where we assume $f(z,\bar{z})$ is analytic as $\bar{z}\to 1$ (i.e. $V\to 0$ in
a small $U$ expansion). We give the inversion formulae for the other CFT data
in Appendix C. Note that in the string/M-theory cases, the inversion formula
does not converge for low spins, which corresponds to the existence of the
contact terms $\kappa\mathcal{G}^{R^{4}}$ in (2). In the pure AdSd+1 case we
do not have such contact terms as discussed above, so we can in fact extract
all CFT data at 1-loop order.

To apply these inversion formulae, we need to compute the $\log^{2}U$ terms in
(10) for finite $U$, expand to leading in $U$ in the crossed channel (7), and
perform the integral of the resulting resummed $V\sim 1-\bar{z}$ expression.
We compute the $\log^{2}U$ terms in a small $U$ expansion using the ansatz

$\begin{split}&\frac{1}{8}\sum_{n=0}^{\infty}\sum_{\ell\in\text{Even}}{(\lambda^{(0)}_{n,\ell})^{2}(\gamma^{R}_{n,\ell})^{2}}\mathfrak{G}_{n,\ell}=\sum_{n=0}^{\infty}U^{d-2+n}\Big{[}p_{1}\\\
&+p_{2}\log
V+p_{3}\log^{2}V+p_{4}\text{Li}_{2}(1-V)\Big{]}+\dots\,,\end{split}$ (13)

where here we showed the singlet channel, while the dots denote the other
R-symmetry channels $R$ that will start at higher powers of $U$ and have
nontrivial $\sigma,\tau$ dependence given by the structures
$Y_{R}(\sigma,\tau)$, as given in Eq. B.14 of Nirschl:2004pa . The
coefficients $p_{i}$ are polynomials in $V$ divided by monomials in $V$. We
then perform crossing, expand to leading order $U$, and resum the expansion in
$V\sim 1-\bar{z}$ to get the relevant DDs. The final expressions are inverse
trigonometric functions of $\bar{z}$ times high degree polynomials in
$\bar{z}$, whose explicit form we give in the attached Mathematica file. We
then plug these into the inversion formula to obtain the 1-loop correction to
CFT data. For the lowest spin in each multiplet we find

$\begin{split}(\lambda^{R|R}_{(B,+)})^{2}&=793.76\,,\qquad(\lambda^{R|R}_{(A,+)_{0}})^{2}=97.766\,,\\\
(\lambda^{R|R}_{(B,2)})^{2}&=3968.8\,,\qquad\;(\lambda^{R|R}_{(A,2)_{1}})^{2}=570.50\,,\\\
\gamma^{R|R}_{0,0}&=21555\,,\qquad\qquad\,\Delta^{R|R}_{3d,2}=2713.6\,,\\\
\end{split}$ (14)

where here we show 5 digits of precision, but we can compute arbitrary
precision. In 4d, the only nontrivial data is the anomalous dimensions, which
were already computed for pure AdS5 in Alday:2017xua  for $\ell\geq 0$:

$\begin{split}\gamma^{R|R}_{0,\ell}=\frac{24\left(7\ell^{5}+116\ell^{4}+725\ell^{3}+2044\ell^{2}+2292\ell+288\right)}{(\ell+1)^{2}(\ell+2)(\ell+3)(\ell+4)(\ell+5)(\ell+6)^{3}}\,.\end{split}$
(15)

In 6d, we compute the lowest few spins for the multiplets with non-trivial
$1/c$ expansions to get

$\begin{split}(\lambda^{R|R}_{\mathcal{B}[0,2]_{1}})^{2}&=-4.2372\,,\qquad(\lambda^{R|R}_{\mathcal{B}[0,2]_{3}})^{2}=-0.1531\,,\\\
\gamma^{R|R}_{0,0}&=-54695\,,\qquad\qquad\;\Delta^{R|R}_{6d,2}=-644.25\,,\\\
\gamma^{R|R}_{0,4}&=-18.918\,,\qquad\;\,(\lambda^{R|R}_{\mathcal{D}[0,4]})^{2}=-822.70\,.\end{split}$
(16)

##  IV Numerical conformal bootstrap

We will now compare these 1-loop corrections to the numerical bootstrap bounds
on CFT data in the stress tensor correlator for $d=3,4,6$, which were computed
for $d=3,4$ in Alday:2021ymb ; Alday:2021vfb , and which we compute now for 6d
following Beem:2015aoa . These bounds come from optimizing the infinite set of
constraints imposed by the crossing equations (7) on the superblock expansion
in (5), for more details in each case see the original works Beem:2013qxa ;
Beem:2015aoa ; Chester:2014fya , and Poland:2018epd ; Chester:2019wfx ;
Simmons-Duffin:2016gjk ; Poland:2022qrs  for recent reviews. The convergence
of these bounds is monotonic and given by the parameter $\Lambda$ originally
defined in Chester:2014fya , which counts how many derivatives are used in the
expansion of conformal blocks around the crossing symmetric point 101010For
comparison, the most precise Ising model bounds were computed with
$\Lambda=43$ in Landry:2019qug , while all the bounds shown here use at least
twice that precision.. These bounds apply to any theory with maximal
supersymmetry in the given $d$ and are computed as a function of $c$, which is
related to the stress tensor OPE coefficient as in (6). Since these bounds are
non-perturbative in $c$, we will look at the large $c$ regime where we expect
the $1/c$ expansion of the previous section to be good. The large $c$
expansion of CFT data is asymptotic, which means that after a few orders the
expansion will actually get worse, unless we look at very large values of $c$.
We observe that the $1/c^{2}$ corrections get smaller relative to $1/c$ tree
corrections as the spin increases, which implies that the asymptotic expansion
is getting more accurate at this order. We do not want to look at very high
spin data, however, because then the difference between each order will be
hard to observe. As a compromise, we will focus on the lowest spin CFT data
for which the Lorentzian inversion converges for the string/M-theory CFTs. We
summarize the comparison of the analytic $1/c$ expansion to fits in the large
$c$ regime of the bootstrap bounds in Table 1 111111A rough diagnostic for the
error of these fits is given by how close the $1/c$ tree level correction
matches the known answer. The range of $c$ used for the fits was motivated to
give such a tree level match, such that the 1-loop term is then a prediction..

3d: |  $\Delta_{0,2}$: Exact | $4-49.931/c+2713.6/c^{2}$
---|---|---
|  Fit | $3.99996-49.82/c+2619.4/c^{2}$
|  $\lambda^{2}_{(A,2)_{1}}$: Exact | $9.7523-98.764/c+570.43/c^{2}$
|  Fit | $9.7523-98.772/c+580.443/c^{2}$
|  $\lambda^{2}_{(A,+)_{0}}$: Exact | $7.1111+48.448/c+97.768/c^{2}$
|  Fit | $7.1111+48.445/c+103.35/c^{2}$
4d: |  $\Delta_{0,2}$: Exact | $6-1/c+0.12976/c^{2}$
|  Fit | $6.0000-0.99929/c+0.14718/c^{2}$
6d: |  $\Delta_{0,2}$: Exact | $10-10.909/c-258.79/c^{2}$
|  Fit | $10.000-11.209/c+270.96/c^{2}$
|  $\Delta_{0,4}$: Exact | $12-3.1648/c-17.157/c^{2}$
|  Fit | $12.000-3.1956/c-17.832/c^{2}$
|  $\lambda^{2}_{\mathcal{B}[02]_{1}}$: Exact | $0.75757-0.98484/c-4.2372/c^{2}$
|  Fit | $0.75757-0.98009/c-3.9446/c^{2}$
|  $\lambda^{2}_{\mathcal{B}[02]_{3}}$: Exact | $0.43076-0.15440/c-0.15313/c^{2}$
|  Fit | $0.43076-0.15432/c-0.17448/c^{2}$
Table 1: Fits of the numerical bootstrap bounds at large $c$, compared to
exact $O(1/c^{2})$ values for the pure AdSd+1 theory for $d=3,4,6$.

Figure 1: Upper and lower numerical bootstrap bounds (in black) on the
$\lambda_{(A,+)_{0}}^{2}$ and $\lambda_{(A,2)_{1}}^{2}$ OPE coefficients, as
well as upper bounds on the scaling dimension $\Delta_{0,2}$ of the lowest
dimension spin 2 long multiplet, made with precision $\Lambda=83$. These
bounds apply to any 3d $\mathcal{N}=8$ CFT, and are plotted in terms of the
stress-tensor coefficient $c$ in the large $c$ regime, where $c=16$ for the
free theory. The gray dotted line denotes the large $c$ expansion to order
tree level supergravity $O(c^{-1})$, which does not depend on the compact
factor in the bulk. The purple, blue, and orange dashed lines also include the
1-loop supergravity correction $O(c^{-2})$ on $AdS_{4}\times S^{7}$,
$AdS_{4}\times S^{7}/\mathbb{Z}_{2}$, and AdS4, respectively.

We start with the the bounds on 3d $\mathcal{N}=8$ CFTs, which were computed
with $\Lambda=83$. In Figure 1 we show upper and lower bounds on OPE
coefficients for the protected $(A,+)_{\ell}$ and $(A,2)_{\ell}$ multiplets
for the lowest spins $\ell=0$ and $\ell=1$, respectively. Both upper and lower
bounds exist for the OPE coefficients, because their protected scaling
dimensions $\Delta=\ell+2$ are separated from the continuum of long
multiplets. The lower bounds are the nontrivial bounds in this case, as the
upper bounds simply interpolate between the GFFT values at $c\to\infty$ and
the free theory values at $c=16$. We also show upper bounds on the lowest
dimension scaling dimension $\Delta_{0,\ell}$ of the long multiplet with spin
$\ell=2$. We compare these bounds to the 1-loop data for the pure AdS4 theory
as given in (14), as well as for the $AdS_{4}\times S^{7}$ and $AdS_{4}\times
S^{7}/\mathbb{Z}_{2}$ theories Alday:2021ymb ; Alday:2022rly , which we review
in Appendix B. We find that the pure AdS4 1-loop correction at $1/c^{2}$
noticeably improves the universal tree correction at $1/c$ and approximately
saturates the numerical bounds, unlike the $AdS_{4}\times S^{7}$ and
$AdS_{4}\times S^{7}/\mathbb{Z}_{2}$ 1-loop corrections, which lie inside the
allowed region.

Figure 2: Upper bounds (in black) on the scaling dimension $\Delta_{0,2}$ of
the lowest dimension spin 2 long multiplet, made with precision $\Lambda=123$.
These bounds apply to any interacting 4d $\mathcal{N}=4$ CFT, and are plotted
in terms of the stress-tensor coefficient $c$ in the large $c$ regime, where
$c=3/4$ for the minimal interacting theory $SU(2)$ SYM. The gray dotted line
denotes the large $c$ expansion to order tree level supergravity $O(c^{-1})$,
which does not depend on the compact factor in the bulk. The purple, blue, and
orange dashed lines also include the 1-loop supergravity correction
$O(c^{-2})$ on $AdS_{5}\times S^{5}$, $AdS_{5}\times S^{5}/\mathbb{Z}_{2}$,
and AdS5, respectively.

Next, we consider the bounds on 4d $\mathcal{N}=4$ CFTs, which were computed
with $\Lambda=123$. In Figure 2 we show upper bounds on the lowest dimension
scaling dimension $\Delta_{0,\ell}$ of the long multiplet with spin $\ell=2$.
We compare these bounds to the 1-loop data for the pure AdS5 theory as given
in (15), as well as for the $AdS_{5}\times S^{5}$ and $AdS_{5}\times
S^{5}/\mathbb{Z}_{2}$ theories Aprile:2017bgs ; Alday:2017xua ; Alday:2021vfb
, which we review in Appendix B. Again, we find that the pure AdS5 1-loop
correction noticeably saturates the numerical bounds relative to the tree,
$AdS_{5}\times S^{5}$, or $AdS_{5}\times S^{5}/\mathbb{Z}_{2}$ expressions.
The correction is particularly striking in this case, as the tree level
correction lies below the upper bound, and only the pure AdS5 1-loop
correction is positive.

Figure 3: Upper and lower numerical bootstrap bounds (in black) on the
$\lambda_{\mathcal{B}[02]_{\ell}}^{2}$ OPE coefficients for $\ell=1,3$, as
well as upper bounds on the scaling dimension $\Delta_{0,\ell}$ of the lowest
dimension $\ell=2,4$ long multiplets, made with precision $\Lambda=91$. These
bounds apply to any interacting 6d $(2,0)$ CFT, and are plotted in terms of
the stress-tensor coefficient $c$ for $c\geq 25$, which is the value for the
minimal interacting theory $A_{1}$. The gray dotted line denotes the large $c$
expansion to order tree level supergravity $O(c^{-1})$, which does not depend
on the compact factor in the bulk. The purple, blue, and orange dashed lines
also include the 1-loop supergravity correction $O(c^{-2})$ on $AdS_{7}\times
S^{4}$, $AdS_{7}\times S^{4}/\mathbb{Z}_{2}$, and AdS7, respectively.

Finally, we consider the bounds on 6d $(2,0)$ CFTs, which we computed with
$\Lambda=91$ 121212See Lemos:2021azv  for a recent numerical bootstrap study
of this correlator that compared the bounds to the finite $c$ inversion
formula iteratively acted on the protected CFT data.. The bootstrap is
generically less converged as $d$ increases, so in this case we show bounds on
two low values of spin for each nontrivial multiplet to show the improvement
of the match. Since $c$ is generically bigger for physical 6d CFTs, e.g. the
minimal interacting CFT is the $A_{1}$ theory with $c=25$ Beem:2015aoa , we
plot the entire allowed range of $c$. In Figure 3 we show upper bounds on the
OPE coefficients for the protected $\mathcal{B}[0,2]_{\ell}$ multiplet for the
lowest spins $\ell=1,3$. While we cannot compute lower bounds as in 3d,
because this multiplet is not separated from the continuum of long multiplets,
the upper bound in this case is now nontrivial. We also show upper bounds on
the lowest dimension scaling dimension $\Delta_{0,\ell}$ of the long multiplet
with spin $\ell=2,4$. We compare these bounds to the 1-loop data for the pure
AdS7 theory as given in (16), as well as for the $AdS_{7}\times S^{4}$ and
$AdS_{7}\times S^{4}/\mathbb{Z}_{2}$ theories Alday:2020tgi , which we review
in Appendix B. Again, we find that the pure AdS7 1-loop correction noticeably
saturates the numerical bounds relative to the tree, $AdS_{7}\times S^{4}$, or
$AdS_{7}\times S^{4}/\mathbb{Z}_{2}$ expressions. We also computed an upper
bound on $c$ (i.e. a lower bound on the stress tensor OPE coefficient), which
applies to any interacting 6d $(2,0)$ CFT, and got

$\begin{split}c\geq 21.6441\,,\end{split}$ (17)

which is weaker than the bound $c\gtrsim 25$ conjectuerd in Beem:2015aoa .
This latter bound was found by extrapolating bounds computed at lower values
of $\Lambda$ to $\Lambda\to\infty$, and was used as evidence that these
general bootstrap bounds were saturated by the physical $A_{1}$ theory with
$c=25$. We use a different definition of $\Lambda$ than Beem:2015aoa
131313Compare our definition in 6.13 of Chester:2014fya , to their definition
in 5.9 of Beem:2015aoa . We thank Balt van Rees for pointing this out., so it
is hard check their conjectured extrapolation against our bound, but since in
3d 141414For the 3d $\mathcal{N}=8$ stress tensor bootstrap, a kink was found
at $c\approx 22.2735$ even at the high value of $\Lambda=43$ Agmon:2019imm ,
which is close but different from the lowest known interacting theory of
$U(2)_{2}\times U(1)_{-1}$ ABJ with $c\approx 21.3333$. The value of $c$ at
this kink was also shown to be the lowest value allowed by a mixed correlator
bootstrap that kinematically ruled out $U(2)_{2}\times U(1)_{-1}$ ABJ, which
strongly suggests that even at infinite $\Lambda$ the kink will not correspond
to this theory. and 4d 151515For the 4d stress tensor bootstrap for $c=3/4$,
corresponding to the lowest interacting $SU(2)$ SYM theory, it was shown in
Chester:2021aun  that bounds obtained after imposing localization constraints
for this theory strictly rule out the more general bounds like those in this
paper. we know that the general bounds are not saturated by the
string/M-theory theory duals for the smallest such values of $c$, it seems
likely that this general 6d bound is also not saturated by the $A_{1}$ theory
even at $\Lambda\to\infty$.

##  V Discussion

Our results show that pure AdSd+1 maximal supergravity saturates the most
general non-perturbative bootstrap bounds in the large $c$ regime, while CFTs
with string/M-theory duals lie in the allowed region. This suggests that to
study the latter theories, one needs to disallow the existence of the pure
AdSd+1 theory by either looking at mixed correlator with other single trace
operators Agmon:2019imm ; Bissi:2020jve , or imposing theory specific
constraints like supersymmetric localization Pestun:2016zxk . Indeed, in 3d
one can strengthen these general bootstrap bounds by inputting the OPE
coefficients of the $(B,2)$ and $(B,+)$ multiplets for the $U(N)_{k}\times
U(N)_{-k}$ ABJM theory for $k=1,2$, as computed to all orders in $1/N$ using
localization in Agmon:2017xes , in which case the 1-loop data for the dual
$AdS_{4}\times S^{7}/\mathbb{Z}_{k}$ theories then saturates the bounds
Alday:2021ymb ; Alday:2022rly . In 4d, one can input the two localization
inputs for $SU(N)$ SYM derived in Binder:2019jwn ; Chester:2020dja , which are
a function of the complexified coupling $\tau$, in which case the the bounds
in Chester:2021aun  match 4-loop weak coupling results Fleury:2019ydf  in the
appropriate regime, and exclude the general bootstrap bounds shown here for
all $\tau$. In 6d there is no localization, but for correlators of single
trace operators other than the stress tensor one can input nontrivial OPE
coefficients given by the protected 2d chiral algebra Beem:2014kka ;
Chester:2018dga  for the $A_{N-1}$ or $D_{N}$ theories

We can also use the general bootstrap bounds themselves to further study the
pure AdSd+1 theory, assuming it continues to saturate the bounds to higher
order in $1/c$. In particular, by applying a fit to the large $c$ regime of
the numerical bounds, one could read off higher derivative corrections to
supergravity such as the $\mathcal{G}^{R^{4}}$ term discussed in the
introduction, to help determine a putative UV completion. Since
$\mathcal{G}^{R^{4}}$ occurs at the same order as higher loop corrections in
some cases, e.g. $c^{-3}$ for pure AdS5 (2), it will be necessary to compute
these higher loops, as was recently recently done for the 2-loop correction on
$AdS_{5}\times S^{5}$ Huang:2021xws ; Drummond:2022dxw . The pure AdSd+1 case
should be much easier due to the lack of mixing, and so could even guide the
calculation in the more physical cases with compact factors. More ambitiously,
we can non-perturbatively define the pure AdSd+1 theory as whatever saturates
the bootstrap bounds at finite $c$; it would be fascinating to find
independent evidence for or against the existence of such a theory.

Finally, we can ask what theory saturates the stress tensor correlator
bootstrap bound with less than maximal supersymmetry. In 3d, the
$\mathcal{N}=6$ bootstrap bounds were found in Binder:2020ckj ; Binder:2021cif
to be saturated by $U(1)_{2N}\times U(1+N)_{-2N}$ ABJ theory Aharony:2008gk
for all $N$, which has a vector-like large $N$ limit dual to supersymmetric
higher spin gravity Chang:2012kt ; Aharony:2020omh ; Aharony:2021ovo . With no
supersymmetry, it was observed in El-Showk:2014dwa ; ElShowk:2012ht ;
Kos:2013tga  that critical $O(N)$ vector models saturate the bound on $c$
161616See Chester:2015lej ; Chester:2015qca  for similar results on
$\mathcal{N}=2$ critical $O(N)$ vector models., so it is likely that the 3d
stress tensor correlator bounds in general are saturated by interacting vector
model CFTs. In higher dimensions, however, there are no interacting unitary
vector models 171717The critical $O(N)$ vector model can be defined also in
$4<d<6$ Fei:2014yja , but it is non-unitary Giombi:2019upv . Nonetheless, it
can be non-rigorously bootstrapped with some success Chester:2014gqa ;
Li:2016wdp ; Nakayama:2014yia ., so it is possible that the most general non-
supersymmetric stress tensor bounds could be saturated by pure AdSd+1 Einstein
gravity with $d>3$. It would be fascinating to check this by generalizing the
non-supersymmetric stress tensor bootstrap in 3d Dymarsky:2017yzx  to higher
$d$. If such non-supersymmetric pure AdSd+1 theories exist for any $d$, then
they suggest that unitary interacting CFTs can be constructed for any $d$,
unlike supersymmetric CFTs which only exist for $d\leq 6$.

## Acknowledgments

We thank Anatoly Dymarsky, Balt van Rees, Joao Penedones, and Leonardo
Rastelli for useful conversations, Himanshu Raj for collaboration on related
projects, and Ofer Aharony for reviewing the manuscript. We also thank the
organizers of the 2022 Bootstrap conference in Porto, during which this
project was completed. SMC is supported by the Weizmann Senior Postdoctoral
Fellowship. The work of LFA 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. The authors would like to acknowledge the use of the
WEXAC cluster in carrying out this work.

##  Appendix A Multiplets

Type | $(\Delta,\ell)$ |  ${SO}(8)_{R}$ irrep | spin $\ell$ |  $1/c$ exact
---|---|---|---|---
$(B,+)$ | $(2,0)$ | ${\bf 294}_{c}=[0040]$ | $0$ | no
$(B,2)$ | $(2,0)$ | ${\bf 300}=[0200]$ | $0$ | no
$(B,+)$ | $(1,0)$ | ${\bf 35}_{c}=[0020]$ | $0$ | yes
$(A,+)$ | $(\ell+2,\ell)$ | ${\bf 35}_{c}=[0020]$ | even | no
$(A,2)$ | $(\ell+2,\ell)$ | ${\bf 28}=[0100]$ | odd | no
Long | $\Delta>\ell+1$ | ${\bf 1}=[0000]$ | even | no
Id | $(0,0)$ | ${\bf 1}=[0000]$ | even | N/A
Table 2: The possible superconformal multiplets in the $S\times S$ OPE for 3d
$\mathcal{N}=8$ CFTs. The quantum numbers are those of the superconformal
primary in each multiplet.

In this appendix we review the supermultiplets that appear in the OPE $S\times
S$ for $d=3,4,6$ interacting theories. In 3d, $S$ is a $(B,+)$ type multiplet
that transforms in the $[0020]$ of $SO(8)_{R}$, and we show the possible
multiplets in Table 2. In this case, none of the protected multiplets are
$1/c$ exact except trivially the stress tensor multiplet itself.

Type | $(\Delta,\ell)$ |  ${SU}(4)_{R}$ irrep | spin $\ell$ |  $1/c$ exact
---|---|---|---|---
$\mathcal{B}$ | $(2,0)$ | ${\bf 20^{\prime}}=[020]$ | $0$ | yes
$\mathcal{B}$ | $(4,0)$ | ${\bf 105}=[040]$ | $0$ | yes
$\mathcal{B}$ | $(4,0)$ | ${\bf 84}=[202]$ | $0$ | yes
$\mathcal{C}$ | $(\ell+4,\ell)$ | ${\bf 20^{\prime}}=[020]$ | even | yes
$\mathcal{C}$ | $(\ell+4,\ell)$ | ${\bf 15}=[101]$ | odd | yes
Long | $\Delta>\ell+2$ | ${\bf 1}=[000]$ | even | no
Id | $(0,0)$ | ${\bf 1}=[000]$ | even | N/A
Table 3: The possible superconformal multiplets in the $S\times S$ OPE for 4d
$\mathcal{N}=4$ CFTs. The quantum numbers are those of the superconformal
primary in each multiplet, and for familiarity we use $SU(4)$ conventions for
the Dynkin labels.

In 4d, $S$ is a $\mathcal{B}$ type multiplet in the $[020]$ of $SU(4)_{R}$,
and we show the possible multiplets in Table 3. Here, there are no non-trivial
protected multiplets.

Type | $(\Delta,\ell)$ |  ${SO}(5)_{R}$ irrep | spin $\ell$ |  $1/c$ exact
---|---|---|---|---
$\mathcal{D}$ | $(4,0)$ | ${\bf 14}=[20]$ | $0$ | yes
$\mathcal{D}$ | $(8,0)$ | ${\bf 35^{\prime}}=[04]$ | $0$ | no
$\mathcal{D}$ | $(8,0)$ | ${\bf 55}=[40]$ | $0$ | yes
$\mathcal{B}$ | $(\ell+8,\ell)$ | ${\bf 14}=[20]$ | even | yes
$\mathcal{B}$ | $(\ell+8,\ell)$ | ${\bf 10}=[02]$ | odd | no
Long | $\Delta>\ell+6$ | ${\bf 1}=[00]$ | even | no
Id | $(0,0)$ | ${\bf 1}=[00]$ | even | N/A
Table 4: The possible superconformal multiplets in the $S\times S$ OPE for 6d
$(2,0)$ CFTs. The quantum numbers are those of the superconformal primary in
each multiplet.

In 6d, $S$ is a $\mathcal{D}$ type multiplet in the $[20]$ of $SO(5)_{R}$, and
we show the possible multiplets in Table 4. Here, the non-trivial protected
multiplets are $\mathcal{D}[04]$ and $\mathcal{B}[02]_{\ell}$ with odd $\ell$,
which are identical to the long multiplets at their unitarity value $\ell=6$.

##  Appendix B CFT data

In this appendix, we collect previous results for 1-loop CFT data in $d=3,4,6$
for string/M-theory duals, which we will use in the main text. In 3d, the
1-loop corrections were computed for $U(N)_{k}\times U(N)_{-k}$ ABJM dual to
$AdS_{4}\times S^{7}/\mathbb{Z}_{k}$ for $k=1,2$ in Alday:2021ymb ;
Alday:2022rly  to get for the $k=1$ theory

$\begin{split}&AdS_{4}\times
S^{7}:\qquad\qquad\qquad\quad\;\;\gamma^{R|R}_{0,2}=-39254.4\,,\\\
&(\lambda^{R|R}_{(A,+)_{0}})^{2}=513.49\,,\qquad(\lambda^{R|R}_{(A,2)_{1}})^{2}=5221.3\,,\\\
\end{split}$ (18)

and for the $k=2$ theory

$\begin{split}&AdS_{4}\times
S^{7}/\mathbb{Z}_{2}:\qquad\qquad\quad\;\;\;\,\gamma^{R|R}_{0,2}=-16740.9\,,\\\
&(\lambda^{R|R}_{(A,+)_{0}})^{2}=285.32\,,\qquad(\lambda^{R|R}_{(A,2)_{1}})^{2}=2239.9\,.\\\
\end{split}$ (19)

In 4d, the 1-loop corrections were computed for $\mathcal{N}=4$ SYM with
$SU(N)$ Alday:2017xua ; Aprile:2017bgs  and $SO(N)$ Alday:2021vfb  gauge group
dual to $AdS_{5}\times S^{5}$ and $AdS_{5}\times S^{5}/\mathbb{Z}_{2}$,
respectively, to get

$\begin{split}&AdS_{5}\times
S^{5}:\qquad\quad\;\;\gamma^{R|R}_{0,2}=-2.5625\,,\\\ &AdS_{5}\times
S^{5}/\mathbb{Z}_{2}:\qquad\gamma^{R|R}_{0,2}=-0.88851\,.\\\ \end{split}$ (20)

In 6d, the 1-loop corrections were computed for $A_{N-1}$ and $D_{N}$ CFTs
dual to $AdS_{7}\times S^{4}$ and $AdS_{7}\times S^{4}/\mathbb{Z}_{2}$,
respectively, to get for the former theory

$\begin{split}&AdS_{7}\times S^{4}:\\\
&\quad\;\;\,\gamma^{R|R}_{0,2}=-1171.1\,,\qquad\quad\;\;\;\;\gamma^{R|R}_{0,4}=-25.414\,,\\\
&(\lambda^{R|R}_{\mathcal{B}[02]_{1}})^{2}=-12.388\,,\qquad(\lambda^{R|R}_{\mathcal{B}[02]_{3}})^{2}=-0.18697\,,\end{split}$
(21)

and for the latter theory

$\begin{split}&AdS_{7}\times S^{4}/\mathbb{Z}_{2}:\\\
&\quad\;\;\;\,\gamma^{R|R}_{0,2}=-644.25\,,\qquad\quad\;\;\,\,\gamma^{R|R}_{0,4}=-18.918\,,\\\
&(\lambda^{R|R}_{\mathcal{B}[02]_{1}})^{2}=-7.6294\,,\qquad(\lambda^{R|R}_{\mathcal{B}[02]_{3}})^{2}=-0.15983\,.\end{split}$
(22)

##  Appendix C Inversion formulae

In this appendix we collect the inversion formulae from Alday:2021ymb ;
Alday:2020tgi  that we apply to the DDs computed for the 1-loop pure AdSd+1
correlator for $d=3,6$ to get the CFT data reported in the main text. Recall
that for 4d, the pure AdS5 results are already available from Alday:2017xua .
For 3d, the $(A,+)_{\ell}$ formula was given in (11), while the $(A,2)_{\ell}$
result can be extracted from the formula

$\begin{split}&\frac{12(\ell+1)^{2}(\ell+2)^{2}}{(2\ell+1)(2\ell+3)^{2}(2\ell+5)}\lambda^{2}_{(A,2)_{\ell-1}}\\\
&+\frac{2(\ell+2)(\ell+3)}{(2\ell+3)(2\ell+7)}\lambda^{2}_{(A,+)_{\ell}}+\frac{3}{4}\lambda^{2}_{(A,2)_{\ell+1}}=\\\
&\frac{12(2\ell+5)\Gamma(\ell+3)^{4}}{\Gamma\left(\ell+\frac{5}{2}\right)^{2}\Gamma\left(\ell+\frac{7}{2}\right)^{2}}\\\
&\times\int_{0}^{1}\frac{d\bar{z}}{\bar{z}}g_{\ell+4,\ell+2}(\bar{z})\text{dDisc}[{\cal
G}^{[0200]}(z\bar{z},1-\bar{z},)|_{z}]\,,\end{split}$ (23)

after plugging in the results for $\lambda^{2}_{(A,+)_{\ell}}$, and using the
lightcone block with normalization

$\begin{split}&g_{\Delta,\ell}(1-V)=\frac{\Gamma(\ell+1/2)}{4^{\Delta}\sqrt{\pi}\ell!}(1-V)^{\ell}\\\
&\quad\times{}_{2}F_{1}\left(\frac{\Delta+\ell}{2},\frac{\Delta+\ell}{2},\Delta+\ell,1-V\right)\,.\\\
\end{split}$ (24)

The $(B,+)$ and $(B,2)$ OPE coefficients then correspond to the values

$\begin{split}\lambda^{2}_{(B,2)}=\lambda^{2}_{(A,2)_{-1}}\,,\qquad\lambda^{2}_{(B,+)}=\lambda^{2}_{(A,+)_{-2}}\,.\end{split}$
(25)

We can extract the anomalous dimension from the formula

$\begin{split}\gamma^{R|R}_{0,\ell}=&\frac{1}{(\lambda^{(0)}_{2,\ell})^{2}}\Big{(}4R^{[0040]}_{1,R|R}(\ell)+\frac{1}{2}\partial_{\ell}\big{[}(\lambda^{(0)}_{0,\ell})^{2}(\gamma^{R}_{0,\ell})^{2}\big{]}\\\
&-(\lambda^{R}_{0,\ell})^{2}\gamma^{R}_{0,\ell}\Big{)}\,,\end{split}$ (26)

where we have the inversion integral

$\begin{split}&R^{[0040]}_{1,R|R}(\ell)=\frac{512(\ell+1)(\ell+2)(2\ell+3)\Gamma(\ell+1)^{4}}{\Gamma\left(\ell+\frac{1}{2}\right)^{2}\Gamma\left(\ell+\frac{5}{2}\right)^{2}}\\\
&\times\int_{0}^{1}{d\bar{z}}{\bar{z}}g_{\ell+6,\ell}(\bar{z})\text{dDisc}\left.{\cal
G}_{R|R}^{[0040]}(z\bar{z},1-\bar{z})\right|_{z^{3}\log z}\,,\end{split}$ (27)

and the tree and GFFT formula needed above take a more complicated form that
we give in the attached Mathematica file.

For 6d, it is convenient to solve the superconformal Ward identities by
writing $\mathcal{G}(U,V;\sigma,\tau)$ in (3) as

$\begin{split}\mathcal{G}(U,V;\sigma,\tau)=\mathcal{F}(U,V;\sigma,\tau)+\Upsilon\circ\mathcal{H}(U,V)\,,\end{split}$
(28)

where $\mathcal{F}$ is the free theory correlator, $\Upsilon$ is a complicated
differential operator defined in Dolan:2004mu , and $\mathcal{H}(U,V)$ is an
R-symmetry singlet called the reduced correlator. In terms of this reduced
correlator, we can extract the 1-loop OPE coefficient as

$\begin{split}&\lambda^{2}_{\mathcal{B}[02]_{\ell}}=-\frac{\pi(\ell+1)(\ell+4)\Gamma(\ell+5)\Gamma(\ell+7)}{2^{4\ell+19}(\ell+2)\Gamma\left(\ell+\frac{11}{2}\right)\Gamma\left(\ell+\frac{13}{2}\right)}\\\
&\times\int_{0}^{1}d\bar{z}\bar{z}^{4}g_{\ell+11,\ell+1}^{-2,0}(\bar{z})\text{dDisc}\left.\mathcal{H}(z\bar{z},1-\bar{z})\right|_{z^{0}}\,,\end{split}$
(29)

where we define the mixed lightcone block in the 6d normalization as

$\begin{split}&g_{\Delta,\ell}^{-2,0}(1-V)=(V-1)^{\ell}\\\
&\qquad\times{}_{2}F_{1}\left(\frac{\Delta+\ell+2}{2},\frac{\Delta+\ell}{2},\Delta+\ell,1-V\right)\,.\\\
\end{split}$ (30)

The $\mathcal{D}[04]$ OPE coefficient then corresponds to the limit

$\begin{split}\lambda^{2}_{\mathcal{D}[04]}=\lim_{\ell\to-1}(\ell+1)\lambda^{2}_{\mathcal{B}[02]_{\ell}}\,.\end{split}$
(31)

We can extract the anomalous dimension from the formula

$\begin{split}&\gamma^{R|R}_{0,\ell}=\hat{\gamma}^{R|R}_{0,\ell}+\gamma^{\text{extra}}_{\ell}\,,\\\
&\gamma^{\text{extra}}_{\ell}\equiv-\frac{298598400(2\ell+11)\left(\ell^{2}+11\ell+14\right)}{(\ell+1)^{3}(\ell+2)^{3}(\ell+9)^{3}(\ell+10)^{3}}\,,\end{split}$
(32)

where we compute

$\begin{split}&\hat{\gamma}^{R|R}_{0,\ell}=-\frac{\sqrt{\pi}45\
2^{-7-2\ell}\Gamma(\ell+5)}{(\ell+1)(\ell+2)(\ell+9)(\ell+10)\Gamma\left(\ell+\frac{13}{2}\right)}\\\
&\times\int_{0}^{1}d\bar{z}\bar{z}^{4}g_{\ell+11,\ell+1}^{-2,0}(\bar{z})\Big{[}\text{dDisc}\left.\mathcal{H}(z\bar{z},1-\bar{z})\right|_{\log
z}\\\
&-\frac{3456\left(397\bar{z}^{3}-2910\bar{z}^{2}+5730\bar{z}-3305\right)}{\bar{z}^{5}(1-\bar{z})^{-1}}\Big{]}\,.\end{split}$
(33)

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