# Squashing, Mass, and Holography for 3d Sphere Free Energy

###### Abstract

We consider the sphere free energy $F(b;m_{I})$ in $\mathcal{N}=6$ ABJ(M)
theory deformed by both three real masses $m_{I}$ and the squashing parameter
$b$, which has been computed in terms of an $N$ dimensional matrix model
integral using supersymmetric localization. We show that setting
$m_{3}=i\frac{b-b^{-1}}{2}$ relates $F(b;m_{I})$ to the round sphere free
energy, which implies infinite relations between $m_{I}$ and $b$ derivatives
of $F(b;m_{I})$ evaluated at $m_{I}=0$ and $b=1$. For $\mathcal{N}=8$ ABJ(M)
theory, these relations fix all fourth order and some fifth order derivatives
in terms of derivatives of $m_{1},m_{2}$, which were previously computed to
all orders in $1/N$ using the Fermi gas method. This allows us to compute
$\partial_{b}^{4}F|_{b=1}$ and $\partial_{b}^{5}F|_{b=1}$ to all orders in
$1/N$, which we precisely match to a recent prediction to sub-leading order in
$1/N$ from the holographically dual $AdS_{4}$ bulk theory.

###### Contents

  1. 1 Introduction
  2. 2 The ABJ(M) matrix model
  3. 3 Exact relation between squashing and mass
  4. 4 Large $N$ and holography
  5. 5 Conclusion
  6. A Details of squashed sphere calculation
  7. B The large $k$ expansion

##  1 Introduction

The free energy $F(b,m)$ for a quantum field theory placed on the $d$
dimensional squashed sphere $S^{d}_{b}$ and deformed by a mass $m$ is one of
the few quantities that can be computed exactly in interacting theories. For a
rank $N$ supersymmetric gauge theory, supersymmetric localization has been
used to compute $F(b,m)$ in terms of an $N$ dimensional matrix model integral
for 2d $\mathcal{N}=(2,2)$ [1], 3d $\mathcal{N}=2$ [2, 3], 4d $\mathcal{N}=2$
[4, 5], and 5d $\mathcal{N}=1$ [6, 7] theories. The massless theory on the
round sphere, i.e. $m=0,b=1$, typically flows in the IR to a conformal field
theory. At large $N$ the CFT is often111This is generically the case for
supersymmetric gauge theories with matrix degrees of freedom. For theories
with vector degrees of freedom, the large $N$ limit is holographically dual to
weakly coupled higher spin theory on $AdS_{d+1}$, see [8] for a review. dual
to weakly coupled supergravity on $AdS_{d+1}$ [9], while turning on mass and
squashing in the CFT corresponds to suitably deforming the bulk away from
$AdS_{d+1}$. One can then study the weakly coupled gravity theory using the
large $N$ CFT either by directly comparing the deformed theories at finite
$m,b$, or by using the small $m,b$ expansion in the CFT to constrain
correlation functions on flat space, which are then holographically dual to
scattering in undeformed $AdS_{d+1}$. For either method, it is crucial to know
the explicit large $N$ expansion of $F(b,m)$, not just the matrix model
integral given by localization.

This work will focus on $F(b;m_{I})$ for the 3d $\mathcal{N}\geq 6$ ABJ(M)
CFTs [10, 11] with gauge group $U(N)_{k}\times U(N+M)_{-k}$ and Chern-Simons
level $k$. Like any 3d $\mathcal{N}=6$ SCFT, ABJ(M) has an $SO(6)_{R}$
symmetry and a $U(1)$ global symmetry [12], so that from the $\mathcal{N}=2$
perspective the theory has an $SO(4)\times U(1)$ flavor symmetry. The theory
can then be deformed by three real masses $m_{I}$, where $m_{2},m_{3}$
correspond to Cartans of the $SO(4)$ and $m_{1}$ to the Cartan of the $U(1)$.
The free energy $F(b;m_{I})$ on the squashed sphere in the presence of these
masses was computed using localization in terms of an $N$ dimensional matrix
model integral in [3]. For the massless round sphere $F(1;0)$, [13] showed
that this matrix model could be understood as a free Fermi gas with a
nontrivial potential, which allowed $F(1;0)$ to be explicitly computed to all
orders in $1/N$. This Fermi gas method was then extended to
$F(1;m_{1},m_{2},0)$ (or $F(1;m_{1},0,m_{3})$) [14], and to $F(\sqrt{3};0)$
(or $F(1/\sqrt{3};0)$) [15]. For more general $b,m_{I}$, however, the matrix
model takes a more complicated form that is not amenable to this technique. In
this work, we will use methods inspired from the Fermi gas approach to derive
the exact relation

$\begin{split}F(b;m_{1},m_{2},i\frac{b-b^{-1}}{2})=F(1;\frac{b^{-1}(m_{1}+m_{2})+b(m_{1}-m_{2})}{2},\frac{b^{-1}(m_{1}+m_{2})-b(m_{1}-m_{2})}{2},0)\,,\end{split}$
(1.1)

where the RHS is now related to the round sphere expression
$F(1;m_{1},m_{2},0)$ that was computed to all orders in $1/N$. We can then
expand both sides around the massless round sphere to derive infinite
constraints between $m_{I},b$ derivatives at each order. For instance, we find
that all combinations of 4 derivatives of $m_{I},b$ can be written in terms of
the quantities

$\begin{split}\partial_{m_{\pm}}^{4}F\,,\qquad\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F\,,\qquad\partial_{m_{\pm}}^{2}F\,,\qquad{\color[rgb]{0,0.7,0}\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F}\,,\qquad{\color[rgb]{0.8,0,0}\partial_{m_{\pm}}^{3}\partial_{m_{\mp}}F}\,,\qquad{\color[rgb]{0.8,0,0}\partial_{m_{+}}\partial_{m_{-}}F}\,,\end{split}$
(1.2)

all evaluated at $m_{I}=0$ and $b=1$, where $m_{\pm}\equiv m_{2}\pm m_{1}$ (or
$m_{\pm}\equiv m_{3}\pm m_{1}$).222All $m_{I},b$ derivatives of $F$ that are
considered in this work are assumed to be evaluated at $m_{I}=0$ and $b=1$.
The quantities in red have an odd number of $m_{\pm}$ derivatives and are pure
imaginary, the quantity $\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ in green
is generically complex, while the remaining quantities in black are always
real. For parity preserving ABJ(M) theories, which includes all
$\mathcal{N}=8$ theories, $\partial_{m_{\pm}}^{3}\partial_{m_{\mp}}F$ in fact
vanishes, while $\partial_{m_{+}}\partial_{m_{-}}F$ vanishes for ABJM
theory.333For ABJ theory with unequal rank, generically
$\partial_{m_{+}}\partial_{m_{-}}F$ is nonzero, and is related to a choice of
background Chern-Simons level [16, 17]. The ABJ(M) theories have
$\mathcal{N}=8$ supersymmetry when $k=1,2$, in which case the $SO(6)_{R}\times
U(1)$ global symmetry is promoted to $SO(8)_{R}$. As a result
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ is related to the other real non-
vanishing quantities, which are all written as derivatives of $m_{\pm}$, or
equivalently as derivatives of $F(1;m_{1},m_{2},0)$. Since
$F(1;m_{1},m_{2},0)$ was computed to all orders in $1/N$ in [14], we thus have
all orders in $1/N$ expressions for all combinations of 4 derivatives of
$m_{I},b$ in $\mathcal{N}=8$ ABJ(M) theories. We can similarly relate certain
higher order derivatives such as $\partial_{b}^{5}F$ in terms of $m_{\pm}$
derivatives, so that they too can be computed to all orders in $1/N$.

We can then compare these all orders in $1/N$ results for $F(b,m_{I})$ to the
holographic dual of $U(N)_{k}\times U(N+M)_{-k}$ ABJ(M) theory, which for
large $N$ and fixed $M,k$ is dual to weakly coupled M-theory on $AdS_{4}\times
S^{7}/\mathbb{Z}_{k}$, while for large $N,k$ and fixed $M,\lambda\equiv N/k$
and then large $\lambda$ is dual to weakly coupled Type IIA string theory on
$AdS_{4}\times\mathbb{CP}^{3}$.444Note that the finite value of $M$ does not
appear to any perturbative order in these holographic descriptions. The first
way we do this is to directly compare $F(b,m_{I})$ to the renormalized on-
shell action in the $AdS_{4}$ theory dual to these deformations. The leading
large $N$ term corresponds to the action of $\mathcal{N}=8$ gauged
supergravity on $AdS_{4}$ [18] evaluated on solutions to the equations of
motion that preserve the suitable symmetry group of the CFT deformation, which
for $m\neq 0$ or $b\neq 1$ breaks the amount of supersymmetry to
$\mathcal{N}=2$ while preserving certain abelian flavor groups. These
solutions were matched to $F(b,m_{I})$ at leading order in large $N$ for
nonzero $m_{I}$ in [19] and nonzero $b$ in [20]. The sub-leading $1/N$
corrections to $F(b,m_{I})$ correspond to higher derivative corrections to
supergravity evaluated on the corresponding solution. The first higher
derivative corrections, i.e. the four derivative terms, were recently derived
in [21] for any minimal $\mathcal{N}=2$ gauged supergravity on $AdS_{4}$ in
terms of two theory dependent coefficients. These coefficients were then fixed
for the ABJ(M) M-theory dual at finite $k$ using the large $N$ results for
$F(1,0)$ and the coefficient $c_{T}$ of the stress tensor two-point function,
which was computed using $\partial_{m_{\pm}}^{2}F$ in [22]. The free energy
could then be computed on any asymptotically $AdS_{4}$ solution to sub-leading
order in $1/N$, which for the case of squashing gave [21]:

$\begin{split}F(b;0)=\frac{\pi\sqrt{2k}}{12}\left[\left(b+\frac{1}{b}\right)^{2}\left(N^{\frac{3}{2}}+(\frac{1}{k}-\frac{k}{16})N^{\frac{1}{2}}\right)-\frac{6}{k}N^{\frac{1}{2}}\right]+O(N^{0})\,.\end{split}$
(1.3)

We will match this gravity prediction for the sub-leading $N^{\frac{1}{2}}$
terms to our all orders in $1/N$ expression for $\partial_{b}^{4}F$ and
$\partial_{b}^{5}F$ in $\mathcal{N}=8$ ABJ(M) theory, i.e. for $k=1,2$. The
further sub-leading powers of $1/N$ in our result will allow the coefficients
of future higher derivative corrections to supergravity to be similarly fixed.
Note that once these higher derivative terms are known, they can be used to
compute gravity quantities on any asymptotically $AdS_{4}$ solution, not just
that corresponding to squashing, and can even be used to compute thermodynamic
quantities like higher derivative corrections to the the black hole entropy
[21], which are much more difficult to compute directly from CFT using
holography.

The second way to constrain the dual $AdS_{4}$ theory using $F(b,m)$ is using
the relation between the small $m,b$ expansion of $F(b,m)$ and integrated
correlators of the stress tensor multiplet correlator, which is dual to
scattering of gravitons on $AdS_{4}$. In particular, since both $m_{I}$ and
$b$ couple to operators in the stress tensor multiplet for 3d $\mathcal{N}=6$
SCFTs, it should be possible to relate $n$ derivatives of $F(b;m_{I})$
evaluated at $m=0,b=1$ to correlators of $n$ stress tensor multiplet operators
integrated on $S^{3}$ [17]. These integrated constraints were derived for
$\partial_{m_{\pm}}^{4}F$, $\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F$, and
$\partial_{m_{\pm}}^{2}F$ for $\mathcal{N}=8$ SCFTs in [22, 23] and for
$\mathcal{N}=6$ SCFTs in [24]. The stress tensor multiplet four point function
in ABJ(M) can then be constrained in the large $N$ limit using analyticity,
crossing symmetry, and the superconformal ward identities in terms of just a
few terms at each order [25, 26], whose coefficients can then be fixed using
the integrated constraints and the large $N$ expressions for derivatives of
$F(b;m_{I})$. One can then take the flat space limit of this holographic
correlator as in [27] and compare to the dual quantum gravity S-matrix in flat
space, where $1/N$ corrections correspond to higher derivative corrections to
supergravity. This program was carried out to sub-leading order in $1/N$ for
the M-theory limit in [26, 23], and the Type IIA limit in [24]. To go to
further orders, one needs to both derive the integrated constraints for the
remaining mass and squashing derivatives, as well as the large $N$ expansions
of the localization expressions. This paper completes the latter task for all
such fourth order derivatives for $\mathcal{N}=8$ ABJ(M), while for
$\mathcal{N}=6$ ABJ(M) a large $N$ expansion is still needed for
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$.

The rest of this paper is organized as follows. In Section 2, we review the
matrix model expression of $F(b,m)$ for $U(N)_{k}\times U(N+M)_{-k}$ ABJ(M)
theory, as well as previous all orders in $1/N$ results from the Fermi Gas
method. In Section 3, we derive the exact relation (1.1) between mass and
squashing, and use it to show that all derivatives up to fourth order as well
as $\partial_{b}^{5}F$ can be written in terms of the invariants shown in
(1.2). In Section 4 we use these relations as well as the previously derived
all orders in $1/N$ expressions for $F(1;m_{1},m_{2},0)$ to derive all orders
in $1/N$ expressions for $\partial_{b}^{4}F$ and $\partial_{b}^{5}F$, which we
will match to the gravity prediction in (1.3). We end with a discussion of our
results and future directions in Section 5. Details of our calculations are
given in various Appendices, and an attached Mathematica notebook includes our
result for $c_{T}$ in the large $k$ weak coupling expansion to $O(k^{-14})$.

##  2 The ABJ(M) matrix model

We begin by reviewing ABJ(M) theory and localization results for $F(b;m_{I})$,
including the all orders in $1/N$ results from the Fermi Gas method for the
round sphere with only $m_{\pm}$ turned on.

In $\mathcal{N}=2$ language, ABJM theory consists of vector multiplets for
each $U(N)_{k}\times U(N+M)_{-k}$ gauge group, as well as four chiral
multiplets $Z^{A},W_{A}$ for $A=1,2$ which transform under the gauge groups
and the $SU(2)_{R}\times SU(2)_{R}\times U(1)$ flavor symmetry as shown in
Table 1. Seiberg duality relates different ABJ(M) theories as

$\begin{split}U(N)_{k}\times U(N+M)_{-k}\ \ \ &\longleftrightarrow\ \ \
U(N)_{-k}\times U(N+\left\lvert k\right\rvert-M)_{k}\,,\\\ \end{split}$ (2.1)

which implies that $M\leq|k|$. Parity then sends $k\to-k$, so the $M=0$
theories can be seen to be parity invariant from the Lagrangian, while Seiberg
duality implies that the $k=M,2M$ theories must be parity invariant on the
quantum level.

field | $U(N)\times U(N+M)$ | $SU(2)\times SU(2)$ | $U(1)$
---|---|---|---
$Z^{A}$ | $(\overline{\bf N},{\bf N})$ | $({\bf 2},{\bf 1})$ | $1$
$W_{A}$ | $({\bf N},\overline{\bf N})$ | $({\bf 1},\overline{\bf 2})$ | $-1$
Table 1: Matter content of ABJM theory and their transformations under gauge
and flavor symmetries in $\mathcal{N}=2$ language.

The partition function $Z(b;m_{I})=e^{-F(b;m_{I})}$ on a squashed sphere with
squashing parameter $b$ and deformed by masses $m_{I}$ for the chiral fields
can then be computed by assembling the standard $\mathcal{N}=2$ ingredients,
as reviewed in [28], to get up to an overall $m_{I},b$ independent constant:

$\begin{split}Z(b;m_{I})&=e^{\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}\int\frac{d^{N+M}\mu\,d^{N}\nu}{N!(N+M)!}e^{i\pi
k\left[\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right]}\prod_{a>b}4\sinh\left[\pi
b(\mu_{a}-\mu_{b})\right]\sinh\left[\pi b^{-1}(\mu_{a}-\mu_{b})\right]\\\
&{}\times\prod_{i>j}4\sinh\left[\pi b(\nu_{i}-\nu_{j})\right]\sinh\left[\pi
b^{-1}(\nu_{i}-\nu_{j})\right]\prod_{i,a}\biggl{[}s_{b}\left(\frac{iQ}{4}-(\mu_{a}-\nu_{i}+\frac{m_{1}+m_{2}+m_{3}}{2})\right)\\\
&{}\times
s_{b}\left(\frac{iQ}{4}-(\mu_{a}-\nu_{i}+\frac{m_{1}-m_{2}-m_{3}}{2})\right)s_{b}\left(\frac{iQ}{4}-(-\mu_{a}+\nu_{i}+\frac{-m_{1}-m_{2}+m_{3}}{2})\right)\\\
&{}\times
s_{b}\left(\frac{iQ}{4}-(-\mu_{a}+\nu_{i}+\frac{-m_{1}+m_{2}-m_{3}}{2})\right)\biggr{]}\,,\end{split}$
(2.2)

where $Q=b+\frac{1}{b}$, the $\mu_{a},\nu_{i}$ correspond to the Cartans of
the two gauge fields with $a=1,\dots,N+M$ and $i=1,\dots,N$, and each chiral
field contributes a factor with masses $m_{I}$ determined by the charge
assignments in Table 1, so that $m_{1}$ corresponds to $U(1)$ and
$m_{2}+m_{3},m_{2}-m_{3}$ correspond to the Cartans of each factor in
$SU(2)\times SU(2)$, respectively. The functions $s_{b}(x)$ are reviewed in
Appendix A. The phase factor was computed for generic $\mathcal{N}=2$
supersymmetric gauge theories in [29] in terms of the topological anomaly,
which was given for ABJ(M) theory in [16, 30]. If we restrict to the round
sphere with $b=1$, and set $m_{3}$ (or $m_{2}$) zero, then the partition
function can be simplified using identities in A and written in terms of
$m_{\pm}=m_{2}\pm m_{1}$ (or $m_{\pm}=m_{3}\pm m_{1}$) as

$\begin{split}&Z({m_{+},m_{-}})=\int\frac{d^{N+M}\mu
d^{N}\nu}{N!(N+M)!}e^{i\pi
k\left[\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right]}\\\
&\times\frac{\prod_{a<b}\left(4\sinh^{2}\left[\pi(\mu_{a}-\mu_{b})\right]\right)\prod_{i<j}\left(4\sinh^{2}\left[\pi(\nu_{i}-\nu_{j})\right]\right)}{\prod_{i,a}\left(4\cosh\left[\pi(\mu_{a}-\nu_{i}+m_{+}/2)\right]\cosh\left[\pi(\nu_{i}-\mu_{a}+m_{-}/2)\right]\right)}\,.\end{split}$
(2.3)

As shown in [31, 32], the partition function can furthermore be simplified
using the Cauchy determinant formula to take the form

$\begin{split}Z(m_{+},m_{-})=\frac{e^{-\frac{\pi}{2}MNm_{-}}Z_{0}}{\cosh^{N}\frac{\pi
m_{+}}{2}}&\int
d^{N}y\prod_{a<b}\frac{\sinh^{2}\frac{\pi(y_{a}-y_{b})}{k}}{\cosh\left[\frac{\pi(y_{a}-y_{b})}{k}+\frac{\pi
m_{+}}{2}\right]\cosh\left[\frac{\pi(y_{a}-y_{b})}{k}-\frac{\pi
m_{+}}{2}\right]}\\\ &\times\prod_{a=1}^{N}\left(\frac{e^{i\pi
y_{a}m_{-}}}{2\cosh\left(\pi
y_{a}\right)}\prod_{l=0}^{M-1}\frac{\sinh\left[\frac{\pi\big{(}y_{a}+i(l+1/2)\big{)}}{k}\right]}{\cosh\left[\frac{\pi\big{(}y_{a}+i(l+1/2)\big{)}}{k}-\frac{\pi
m_{+}}{2}\right]}\right)\,,\end{split}$ (2.4)

where for simplicity all of the overall numerical coefficients are included in
the factor $Z_{0}$. Lastly, as shown in [13, 14, 32], one can use the Cauchy
determinant formula to further write this partition function as a free Fermi
gas with a single body Hamiltonian that depends on $m_{\pm}$. One can then use
standard methods from statistical mechanics to compute $Z(m_{+},m_{-})$ to all
orders in $1/N$ as

$\begin{split}&Z(m_{-},m_{+})=e^{A}C^{-\frac{1}{3}}\text{Ai}\left[C^{-\frac{1}{3}}(N-B)\right]+(\text{non-
perturbative in $N$})\,,\\\
C&=\frac{2}{\pi^{2}k(1+m_{+}^{2})(1+m_{-}^{2})}\,,\qquad
B=\frac{\pi^{2}C}{3}-\frac{1}{6k}\left[\frac{1}{1+m_{+}^{2}}+\frac{1}{1+m_{-}^{2}}\right]-\frac{k}{12}+\frac{k}{2}\left(\frac{1}{2}-\frac{M}{k}\right)^{2}\,,\\\
A&=\frac{{\cal A}[k(1+im_{+})]+{\cal A}[k(1-im_{+})]+{\cal
A}[k(1+im_{-})]+{\cal A}[k(1-im_{-})]}{4}\,,\end{split}$ (2.5)

where the constant map function ${\cal A}$ is given by [33]

$\begin{split}{\cal
A}(k)&=\frac{2\zeta(3)}{\pi^{2}k}\left(1-\frac{k^{3}}{16}\right)+\frac{k^{2}}{\pi^{2}}\int_{0}^{\infty}dx\frac{x}{e^{kx}-1}\log\left(1-e^{-2x}\right)\\\
&=-\frac{\zeta(3)}{8\pi^{2}}k^{2}+2\zeta^{\prime}(-1)+\frac{\log\left[\frac{4\pi}{k}\right]}{6}+\sum_{g=0}^{\infty}\left(\frac{2\pi
i}{k}\right)^{2g-2}\frac{4^{g}B_{2g}B_{2g-2}}{(4g)(2g-2)(2g-2)!}\,,\end{split}$
(2.6)

and in the second line we wrote ${\mathcal{A}}$ in the large $k$ expansion
[33]. Note that the all orders in $1/N$ formula only depends on $M$ via the
parameter $B$.

A useful parameterization of ABJ(M) is given by the coefficient $c_{T}$ of the
two-point function of canonically normalized stress-tensors:

$\begin{split}\langle
T_{\mu\nu}(\vec{x})T_{\rho\sigma}(0)\rangle=\frac{c_{T}}{64}\left(P_{\mu\rho}P_{\nu\sigma}+P_{\nu\rho}P_{\mu\sigma}-P_{\mu\nu}P_{\rho\sigma}\right)\frac{1}{16\pi^{2}\vec{x}^{2}}\,,\qquad
P_{\mu\nu}\equiv\eta_{\mu\nu}\nabla^{2}-\partial_{\mu}\partial_{\nu}\,.\end{split}$
(2.7)

This quantity is related to the $AdS_{4}$ Planck length, and so is a more
natural expansion parameter in the holographic large $N$ limit than $N$
itself. We can compute $c_{T}$ in terms of $Z(m_{-},m_{+})$ as [34]

$\begin{split}c_{T}=&\frac{64}{\pi^{2}}\partial^{2}_{m_{\pm}}F\,,\\\
\end{split}$ (2.8)

so that it can be written to all orders in $1/N$ using (2.5). It turns out
that any other quantity computed by taking $m_{I},b$ derivatives of
$F(b;m_{I})$, when expanded at large $c_{T}$ in either the M-theory or Type
IIA limits, becomes independent of $M$. In this sense these limits are blind
to parity, which as discussed depends on the value of $M$. One can also check
that only even numbers of $m_{\pm}$ derivatives are nonzero in this limit, and
that these quantities are always real.

##  3 Exact relation between squashing and mass

We will now derive the relation between mass and squashing shown in (1.1). We
will then use this result to show that all quartic order $m_{I},b$ derivatives
of $F(b;m_{I})$ can be written in terms of the invariants (1.2), and that for
$\mathcal{N}=8$ ABJ(M) all of these invariants are known to all orders in
$1/N$ from previous Fermi gas results. For $\mathcal{N}=6$ ABJ(M), no such all
orders result is known yet for $\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$. We
will verify all these relations at both small finite $M,N$ and in the large
$k$ weak coupling expansion. Note that in general all $m_{I},b$ derivatives
are evaluated at $m_{I}=0$ and $b=1$.

We start by setting $m_{3}=i\frac{b-b^{-1}}{2}$ in (2.2) and using properties
of $s_{b}(x)$ given in Appendix A to write the partition function purely in
terms of trigonometric functions:

$\begin{split}Z(b;m_{1},&m_{2},i\frac{b-b^{-1}}{2})=e^{\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}\int\frac{d^{N+M}\mu\,d^{N}\nu}{N!(N+M)!}e^{i\pi
k\left[\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right]}\\\
&\times\prod_{a>b}4\sinh\left[\pi b(\mu_{a}-\mu_{b})\right]\sinh\left[\pi
b^{-1}(\mu_{a}-\mu_{b})\right]\prod_{i>j}4\sinh\left[\pi
b(\nu_{i}-\nu_{j})\right]\sinh\left[\pi b^{-1}(\nu_{i}-\nu_{j})\right]\\\
&\times\prod_{i,a}\frac{1}{2\cosh(\pi
b^{-1}(\mu_{a}-\nu_{i}+\frac{m_{+}}{2}))}\frac{1}{2\cosh(\pi
b(\nu_{i}-\mu_{a}+\frac{m_{-}}{2}))}\,,\end{split}$ (3.1)

where recall that $m_{\pm}=m_{2}\pm m_{1}$. In Appendix A, we then perform the
standard Fermi gas steps of writing the products of trigonometric functions as
Cauchy determinants, introducing auxiliary variables so that the $\mu,\nu$
factorize into gaussian integrals, and finally performing these integrals and
rewriting the Cauchy determinant back into the standard form. The result is

$\begin{split}Z(b;m_{1},m_{2},i\frac{b-b^{-1}}{2})=Z(b^{-1}m_{+},bm_{-})\,,\end{split}$
(3.2)

where on the RHS we wrote the simplified round sphere partition function
defined in (2.4), and note that the $b$ dependent phase that appeared in (2.2)
is precisely cancelled, so that the RHS of (3.2) depends on $b$ only through a
rescaling of the masses. We can then simply rewrite $m_{\pm}$ in terms of
$m_{1},m_{2}$ to get (1.1). This entire calculation can also be performed with
the roles of $m_{2}$ and $m_{3}$ switched with the same result on the RHS of
(3.2), which is expected since these masses both correspond to the Cartans of
the $SO(4)$ part of the flavor symmetry.

We can now expand both sides of (3.2) around $m_{I}=0$ and $b=1$ to derive
relations between derivatives of $F(b;m_{I})$. The first nonzero relation
appears at quadratic order and relates

$\begin{split}\partial_{b}^{2}F=2\partial_{m_{\pm}}^{2}F+2\partial_{m_{+}}\partial_{m_{-}}F\,,\end{split}$
(3.3)

where we used the fact that various single derivatives of $b$ and $m_{I}$
identically vanish. From the explicit single variable partition function for
$Z(m_{+},m_{-})$ as given in (2.4), we see that

$\begin{split}\bar{Z}(m_{+},m_{-})=(-1)^{MN}Z(-m_{+},m_{-})\,.\end{split}$
(3.4)

This implies that any odd number of derivatives of $m_{+}$ is pure imaginary,
such as $\partial_{m_{+}}\partial_{m_{-}}F$. We thus conclude that

$\begin{split}\text{Re}\,\partial_{b}^{2}F=2\partial_{m_{\pm}}^{2}F\,,\end{split}$
(3.5)

where recall that $\partial_{m_{\pm}}^{2}F$ is manifestly real. This relation
is expected from the general results of [17, 35], which showed that the real
part of two derivatives of any parameter that couples to the stress tensor
multiplet should be related to $c_{T}$, where the precise relation in our case
was given in (2.8).

At cubic order, we similarly find the nonzero relations

$\begin{split}i\partial_{m_{1}}\partial_{m_{2}}\partial_{m_{3}}F=-4\partial_{m_{\pm}}^{2}F\,,\qquad\partial_{b}^{3}F=-6\partial_{m_{\pm}}^{2}F-6\partial_{m_{+}}\partial_{m_{-}}F\,.\end{split}$
(3.6)

Since conformal symmetry fixes both three point and two point functions of the
stress tensor to be proportional to $c_{T}$ [36], we therefore expect that all
three derivative terms can be written as linear combinations of two derivative
terms. The factor of $i$ on the LHS of the first relation follows from the
fact that the real cubic casimir invariant for the $SO(4)\times U(1)$ flavor
symmetry (from the $\mathcal{N}=2$ perspective) is $im_{1}m_{2}m_{3}$.

At quartic order, the full list of nonzero relations is

$\begin{split}\partial_{m_{2,3}}^{4}F&=\qquad\qquad\;\;\,2\partial_{m_{\pm}}^{4}F+6\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F\qquad\qquad\quad\;\;+8\partial_{m_{\pm}}^{3}\partial_{m_{\mp}}F\,,\\\
\partial_{m_{1}}^{4}F&=\qquad\qquad\;\;\,2\partial_{m_{\pm}}^{4}F+6\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F\qquad\qquad\quad\;\;-8\partial_{m_{\pm}}^{3}\partial_{m_{\mp}}F\,,\\\
\partial_{m_{1}}^{2}\partial_{m_{2,3}}^{2}F&=\qquad\qquad\;\;\,2\partial_{m_{\pm}}^{4}F-2\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F\,,\\\
\partial_{b}^{2}\partial_{m_{2,3}}^{2}F&=\;\;8\partial_{m_{\pm}}^{2}F\qquad\qquad\qquad\qquad\qquad\,+\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F\,,\\\
\partial_{b}^{2}\partial_{m_{1}}^{2}F&=\;\;8\partial_{m_{\pm}}^{2}F+2\partial_{m_{\pm}}^{4}F-2\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F\,,\\\
\partial_{b}^{4}F&=78\partial_{m_{\pm}}^{2}F-2\partial_{m_{\pm}}^{4}F-6\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F+6\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F-8\partial_{m_{\pm}}^{3}\partial_{m_{\mp}}F-30\partial_{m_{+}}\partial_{m_{-}}F\,,\\\
\end{split}$ (3.7)

which are all written in terms of the six invariants in (1.2). As discussed
above, the two invariants $\partial_{m_{+}}\partial_{m_{-}}F$ and
$\partial^{3}_{m_{\pm}}\partial_{m_{\mp}}F$ are both pure imaginary, since
they involve an odd number of derivatives of $m_{+}$, and they vanish for ABJM
theory with equal rank. The one invariant that cannot generically be written
as derivatives of $m_{\pm}$ is $\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$.
For $\mathcal{N}=8$ ABJ(M), however, the flavor group $SO(4)\times U(1)$ is
enhanced to $SO(6)$, which implies that
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ must be related to the other mass
derivatives as [23]

$\begin{split}\mathcal{N}=8:\qquad\qquad\partial_{m_{I}}^{2}\partial_{m_{J}}^{2}F=2\partial_{m_{+}}^{4}F-2\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F\qquad\text{for}\qquad
I\neq J\,.\end{split}$ (3.8)

In this case, we find that
$\partial_{b}^{2}\partial_{m_{2,3}}^{2}F=\partial_{b}^{2}\partial_{m_{1}}^{2}F$
in (3.7) as expected.

At quintic and higher order of $b,m_{I}$ derivatives, the relation (3.2) is
not sufficient to write all derivatives in terms of just $m_{\pm}$ derivatives
even for $\mathcal{N}=8$. For instance, at quintic order
$\partial_{m_{1}}\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ cannot be further
simplified. Nevertheless, at quintic order for $\mathcal{N}=8$ we can use
(3.2) and (3.8) to write $\partial_{b}^{5}F$ as

$\begin{split}\partial_{b}^{5}F=-660\partial_{m_{\pm}}^{2}F-100\partial_{m_{\pm}}^{4}F+180\partial^{2}_{m_{+}}\partial_{m_{-}}^{2}F\,.\end{split}$
(3.9)

At higher order $a>5$, we can no longer write $\partial_{b}^{a}F$ in terms of
just $m_{\pm}$ derivatives.

We checked all the relations discussed in this section for $U(N+M)_{k}\times
U(N)_{-k}$ at finite $M,N,k$ as well as in the large $k$ weak coupling
expansion. For instance, in Table 2 we show
$\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F$ and
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ for the $U(1)_{k}\times U(2)_{-k}$
theory for $k=1,\dots 10$, as computed from the explicit partition function in
(2.2). As expected, for $k=1,2$ when the theory is $\mathcal{N}=8$, these
quantities are identical and real. For all higher $k$, when the theory is only
$\mathcal{N}=6$, these quantities are distinct and
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ is complex. Finally, in Appendix B
we show that $\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F$ and
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ differ explicitly in the large $k$
weak coupling expansion, which is automatically $\mathcal{N}=6$. We also
computed $c_{T}$ to $O(k^{-14})$ using an efficient algorithm for the weak
coupling expansion, which improves the $O(k^{-2})$ result of [37].

$k$ | $\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F$ | $\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$
---|---|---
1 | -24.3523 | -24.3523
2 | -29.2656 | -29.2656
3 | -33.3494 | $-32.7324\,-1.05679i$
4 | -36.5672 | $-35.4997\,-2.0899i$
5 | -39.0269 | $-37.748\,-2.83569i$
6 | -40.8938 | $-39.5664\,-3.29976i$
7 | -42.317 | $-41.0312\,-3.5507i$
8 | -43.4129 | $-42.2112\,-3.65555i$
9 | -44.2669 | $-43.1652\,-3.66535i$
10 | -44.9409 | $-43.941\,-3.61564i$
Table 2: Explicit free energy derivatives for the $U(1)_{k}\times U(2)_{-k}$
ABJ theory for $k=1,\dots 10$, where all derivatives as usual are evaluated at
$m_{I}=0$ and $b=1$.

##  4 Large $N$ and holography

In the previous section, we showed that for $\mathcal{N}=8$ ABJ(M) theory, we
can relate $\partial_{b}^{4}F$ and $\partial_{b}^{5}F$ to derivatives of
$F(m_{-},m_{+})$ using (3.7) and (3.8). As reviewed in Section 2, this
quantity was computed to all orders in $1/N$ using Fermi gas methods, which
implies that we can also compute $\partial_{b}^{4}F$ and $\partial_{b}^{5}F$
to all orders in $1/N$. For the $U(N)_{k}\times U(N)_{-k}$ theory with finite
$k=1,2$, in which case we have $\mathcal{N}=8$ supersymmetry, we find

$\begin{split}\partial_{b}^{4}F=&10\sqrt{2k}\pi
N^{3/2}-\frac{5\pi\left(k^{2}-16\right)}{4\sqrt{2}\sqrt{k}}\sqrt{N}+39k^{2}\mathcal{A}^{\prime\prime}(k)-5k^{4}\mathcal{A}^{{}^{\prime\prime\prime\prime}}(k)-9\\\
&+\frac{\pi\left(5\left(k^{2}-32\right)k^{2}+704\right)}{384\sqrt{2}k^{3/2}}\frac{1}{\sqrt{N}}+O(N^{-1})\,,\\\
\partial_{b}^{5}F=&-60\sqrt{2k}\pi
N^{3/2}+\frac{15\pi\left(k^{2}-16\right)}{2\sqrt{2}\sqrt{k}}\sqrt{N}-330k^{2}\mathcal{A}^{\prime\prime}(k)+50k^{4}\mathcal{A}^{{}^{\prime\prime\prime\prime}}(k)+30\\\
&-\frac{5\pi\left(k^{4}-32k^{2}+448\right)}{64\sqrt{2}k^{3/2}}\frac{1}{\sqrt{N}}+O(N^{-1})\,,\end{split}$
(4.1)

while it is straightforward to compute higher orders in $1/N$. The constant
map $\mathcal{A}$ was defined in (2.6), and its derivatives can be computed
exactly for any integer value of $k$. In particular, for the $k=1,2$ values
that are relevant here, we find that

$\begin{split}{\cal
A}^{\prime\prime}(1)&=\frac{1}{6}+\frac{\pi^{2}}{32}\,,\qquad\qquad\,\,\,\,{\cal
A}^{\prime\prime}(2)=\frac{1}{24}\,,\\\ {\cal
A}^{\prime\prime\prime\prime}(1)&=1+\frac{4\pi^{2}}{5}-\frac{\pi^{4}}{32}\,,\qquad{\cal
A}^{\prime\prime\prime\prime}(2)=\frac{1}{16}+\frac{\pi^{2}}{80}\,.\end{split}$
(4.2)

The leading and sub-leading terms in (4.1) exactly match $\partial_{b}^{4}F$
and $\partial_{b}^{5}F$ as computed from the bulk prediction (1.3). The bulk
prediction is in fact for any $k$, not just the $k=1,2$ with $\mathcal{N}=8$
supersymmetry that we could compute here. This implies that (4.1) must hold
for any value of $k$ up to $O(N^{\frac{1}{2}})$. As discussed above, in the
large $N$ limit it is more natural to expand quantities in terms of $c_{T}$
than $N$. Using the large $N$ expansion for $c_{T}$ given by (2.8) and (2.5),
we find that $\partial_{b}^{4}F$ and $\partial_{b}^{5}F$ can be expanded to
all orders in $1/c_{T}$ as

$\begin{split}\frac{1}{c_{T}^{2}}\partial_{b}^{4}F&=\frac{15\pi^{2}}{32c_{T}}+\frac{24k^{2}\mathcal{A}^{\prime\prime}(k)-5k^{4}\mathcal{A}^{{}^{\prime\prime\prime\prime}}(k)+6}{c_{T}^{2}}+4\left(\frac{6\pi}{k^{2}}\right)^{2/3}\frac{1}{c_{T}^{\frac{7}{3}}}+O(c_{T}^{\frac{8}{3}})\,,\\\
\frac{1}{c_{T}^{2}}\partial_{b}^{5}F&=-\frac{45\pi^{2}}{16c_{T}}+\frac{-240k^{2}\mathcal{A}^{\prime\prime}(k)+50k^{4}\mathcal{A}^{{}^{\prime\prime\prime\prime}}(k)-60}{c_{T}^{2}}-40\left(\frac{6\pi}{k^{2}}\right)^{2/3}\frac{1}{c_{T}^{\frac{7}{3}}}+O(c_{T}^{\frac{8}{3}})\,.\\\
\end{split}$ (4.3)

Here, the $1/c_{T}$ corresponds to the tree level supergravity correction, the
$1/c_{T}^{2}$ corresponds to the 1-loop supergravity correction, and the
$1/c_{T}^{\frac{7}{3}}$ term corresponds to the tree level $D^{6}R^{4}$
correction. Curiously, the $1/c_{T}^{\frac{5}{3}}$ correction, which would
correspond to the $R^{4}$ correction, vanishes.

Finally, we can also consider the limit of large $N,k$ at fixed $\lambda\equiv
N/k$ and then large $\lambda$, which is dual to weakly coupled Type IIA string
theory on $AdS_{4}\times\mathbb{CP}^{3}$. Using the large $k$ expansion of
$\mathcal{A}(k)$ given on the second line of (2.6), we find that

$\begin{split}\frac{1}{c_{T}^{2}}\partial_{b}^{4}F=&\frac{1}{c_{T}}\left[\frac{15\pi^{2}}{32}-\frac{9\zeta(3)}{32\sqrt{2}\pi\lambda^{\frac{3}{2}}}+O(\lambda^{-3})\right]\\\
&+\frac{1}{c_{T}^{2}}\left[5+\frac{9\zeta(3)}{\pi^{2}\lambda}-\frac{15\zeta(3)}{2\sqrt{2}\pi^{3}\lambda^{\frac{3}{2}}}+O(\lambda^{-3})\right]+O(c_{T}^{-3})\,,\\\
\frac{1}{c_{T}^{2}}\partial_{b}^{5}F=&\frac{1}{c_{T}}\left[-\frac{45\pi^{2}}{16}+\frac{45\zeta(3)}{16\sqrt{2}\pi\lambda^{\frac{3}{2}}}+O(\lambda^{-3})\right]\\\
&+\frac{1}{c_{T}^{2}}\left[-50-\frac{90\zeta(3)}{\pi^{2}\lambda}+\frac{75\zeta(3)}{\sqrt{2}\pi^{3}\lambda^{\frac{3}{2}}}+O(\lambda^{-3})\right]+O(c_{T}^{-3})\,.\\\
\end{split}$ (4.4)

Here, the $c_{T}^{-1}$ term corresponds to the tree level supergravity
correction, the $c_{T}^{-1}\lambda^{-\frac{3}{2}}$ term corresponds to tree
level $R^{4}$, while the various $c_{T}^{-2}$ terms correspond to 1-loop
corrections. Unlike the M-theory expansion in (4.3), we find that the $R^{4}$
correction no longer vanishes. This result can also be compared to future bulk
calculations in this background.

##  5 Conclusion

The main result of this work is the exact relation between the mass and
squashing deformed sphere free energy $F(b;m_{I})$ given in (1.1) for all
$\mathcal{N}=6$ ABJ(M) theories. This relation implies infinite relations
between derivatives of $F(b;m_{I})$ evaluated at $m=0,b=1$, such as the fact
that all four derivatives can be written in terms of the six quantities listed
in (1.2). For the $\mathcal{N}=8$ ABJ(M) theories, these relations allowed us
to compute $\partial_{b}^{4}F(b;m_{I})$ and $\partial_{b}^{5}F(b;m_{I})$ to
all orders in $1/N$, which at sub-leading order match the prediction given in
(1.3) from M-theory compactified on $AdS_{4}\times S^{7}/\mathbb{Z}_{k}$ and
expanded to leading order beyond the supergravity limit [21]. Our results
provide constraints at further orders in $1/N$ that will allow more higher
derivative corrections to supergravity to be derived following the program
outlined in [21].

It is instructive to compare the results of this work for $F(b;m_{I})$ in
ABJ(M) to similar results in 4d $\mathcal{N}=4$ SYM. The free energy
$F(b;m;\tau)$ in this theory was computed using localization in [4, 5] in
terms of an $N$ dimensional matrix model integral that depends on the
complexified gauge coupling $\tau$, a single mass $m$, and the squashing $b$,
all of which couple to operators in the $\mathcal{N}=4$ stress tensor
multiplet. In [38], it was found that all four derivatives of these three
parameters can be written in terms of the three invariants

$\begin{split}\partial_{m}^{4}F(b;m;\tau)\,,\qquad\partial_{m}^{2}\partial_{b}^{2}F(b;m;\tau)\,,\qquad
c\,,\end{split}$ (5.1)

where $c$ is the conformal anomaly and the coefficient of the canonically
normalized stress tensor two-point function. Recall that for $\mathcal{N}=8$
ABJ(M) theory, we also found that all four derivatives of $F(b;m_{I})$ could
be written in terms of the three quantities shown in black in (1.2), where the
similarity to 4d becomes even tighter once we use Table 3.7 to exchange
$\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F$ for
$\partial_{m_{\pm}}^{2}\partial_{b}^{2}F$, and we note that the 3d analog of
$c$ is $c_{T}$, which is proportional to the third invariant
$\partial_{m_{\pm}}^{2}F$. The fact that there are just three independent
quartic derivatives for maximally supersymmetric theories in both 3d and 4d is
in some sense expected, as in both cases the unprotected $D^{8}R^{4}$ term in
the large $N$ expansion of the stress tensor correlator can be fixed in terms
of four coefficients [39, 26], so if there were four independent quartic
derivatives then one could have derived an unprotected quantity from protected
localization constraints. Another similarity between 3d and 4d is that in [40]
it was shown that for a special value of the mass $F(b;m;\tau)$ obeys

$\begin{split}F(b;i\frac{b-b^{-1}}{2};\tau)=F(1;0;\tau)\,,\end{split}$ (5.2)

i.e. it becomes independent of the squashing and mass, just as in (1.1) we
showed that the exact same relation between $m_{3}$ and $b$ made $F(b;m_{I})$
equivalent to the round sphere free energy with $m_{3}=0$, although in 3d the
dependence on $m_{3}$ and $b$ is now captured by $m_{1}$ and $m_{2}$. It would
be interesting to find a deeper geometric explanation for this simplification
in all theories where $m,b$ both couple to the stress tensor multiplet.

One application of our results that we did not explore in this work is the
relation between $n$ derivatives of $m_{I},b$ of $F(b;m_{I})$ and correlators
of $n$ stress tensor multiplets. For $\mathcal{N}=8$ ABJM theory,
$\partial_{m_{\pm}}^{4}F$ and $\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F$ were
related in [23] to integrated constraints on the stress tensor four point
function, which were used to derive the large $N$ expansion up to order
$D^{4}R^{4}$ in the bulk language. One further constraint is needed to fix the
$D^{6}R^{4}$ term,555This term could equivalently be fixed using the flat
space limit relation to the M-theory S-matrix, which has been computed to
order $D^{6}R^{4}$. which is the highest order protected term, and it is
possible that the integrated constraint from
$\partial_{b}^{2}\partial_{m}^{2}F$ will be sufficient to fix this term.

For $\mathcal{N}=6$ ABJ(M), from the list of independent quartic derivatives
in (1.2), we expect that there will now be six total independent constraints.
Recall that $\partial_{m_{+}}\partial_{m_{-}}F$ and
$\partial_{m_{\pm}}^{3}\partial_{m_{\mp}}F$ are known to vanish in the Fermi
gas expression that describes both the M-theory and Type IIA string theory
limits, so we expect just four constraints in these cases. The integrated
constraints for $\partial_{m_{\pm}}^{4}F$ and
$\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}F$ were derived in [24] and used to
fix the stress tensor correlator in both the M-theory and Type IIA limit to
order $R^{4}$. To fix the correlator to order $D^{4}R^{4}$ just from CFT
results, one would need six constraints, which is probably more than are even
in principle independent. On the other hand, if one uses the known Type IIA
amplitude in the flat space limit to fix two of these constraints, then just
four more constraints are required, which matches the four invariants we found
in this work. To complete this program, one would need to derive the large $N$
expansion of $\partial^{2}_{m_{2}}\partial_{m_{3}}^{2}F$, which remains
unknown for $\mathcal{N}=6$ ABJ(M).666Recall that for $\mathcal{N}=8$ ABJ(M)
this quantity was related by symmetry to derivatives of $m_{+}$ and $m_{-}$
which are known to all orders in $1/N$. It would be nice if the Fermi gas
method for $n$-body operators as initiated in [41] could be used to compute
this quantity. At strong coupling, one could also try to compute them at large
$N$ and finite $\lambda\equiv N/k$ using topological recursion as was done for
Wilson loops and the free energy in [42]. Topological recursion for the ABJM
matrix model is quite complicated, however, especially for the multi-body
operators we consider, so one could instead try to guess the large $N$ and
finite $\lambda$ result from the small $\lambda$, i.e. large $k$, weak
coupling expansion, which can be computed to very large order using the
algorithm introduced in Appendix B of this work. A first step would be
guessing the finite $\lambda$ resummation for $c_{T}$, which we computed to
$O(k^{-14})$ in this work.

One final interesting limit of $\mathcal{N}=6$ ABJ(M) that we have not yet
considered is the large $M,k$ limit at fixed $\tilde{\lambda}=M/k$ and $N$,
which is holographically dual to weakly coupled $\mathcal{N}=6$ higher spin
theory [43]. Unlike the M-theory and string theory limits, this limit is
sensitive to the value of $M$, and thus to parity. The parity violating
quartic invariants shown in red in (1.2) can also be computed in this limit
following [44], and could potentially be used to constrain the correlator.
This limit was recently considered in the context of the 3d $\mathcal{N}=6$
numerical bootstrap in [32], and will be further discussed in upcoming work.

## Acknowledgments

We thank Ofer Aharony, Itamar Yaakov, Zohar Komargodski, Silviu Pufu, Damon
Binder, Marcos Marino, Yifan Wang, Alba Grassi, Masazumi Honda, Erez Urbach
and Ohad Mamroud for useful conversation, and Ofer Aharony for reading through
the manuscript. We also thank the organizers of “Bootstrap 2019” and Perimeter
Institute for Theoretical Physics for its hospitality during the course of
this work. SMC is supported by the Zuckerman STEM Leadership Fellowship. This
work was supported in part by an Israel Science Foundation center for
excellence grant (grant number 1989/14).

##  Appendix A Details of squashed sphere calculation

Let us start by reviewing the properties of the double sine function
$s_{b}(x)$ (for reviews see for instance [45, 15]). This function is defined
as

$s_{b}(x)=\exp\left[-\frac{i\pi}{2}x^{2}-\frac{i\pi}{24}\left(b^{2}+b^{-2}\right)+\int_{\mathbb{R}+i0}\frac{dt}{4t}\frac{e^{-2itx}}{\sinh\left(bt\right)\sinh\left(t/b\right)}\right]\,,$
(A.1)

where the integration contour evades the the pole at $t=0$ by going into the
upper half–plane. This function obeys several identities:

  1. 1.

$s_{b}^{-1}(x)=s_{b}(-x)$.

  2. 2.

$s_{b^{-1}}(x)=s_{b}(x)$.

  3. 3.

$s_{b}\left(\frac{ib}{2}-\sigma\right)s_{b}\left(\frac{ib}{2}+\sigma\right)=\frac{1}{2\cosh(\pi
b\sigma)}$.

The last identity is what we used to get (2.3) and (3.1).

Next, we will show how (3.1) is related to (2.3) as in (3.2), by adapting the
usual Fermi gas steps for ABJ as discussed in [46, 30]. We start from a
slightly modified version of (3.1):

$\displaystyle\mathcal{Z}\equiv Z\left(b;\frac{b\
m_{+}-b^{-1}m_{-}}{2},\frac{b\
m_{+}+b^{-1}m_{-}}{2},\frac{{i}\left(b-b^{-1}\right)}{2}\right)=\mathcal{N}_{1}\int
d^{M+N}\mu\ d^{N}\nu\ e^{\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}$
$\displaystyle
e^{-\frac{{i}k}{4\pi}\left(\sum_{j}\mu_{j}^{2}-\sum_{l}\nu_{l}^{2}\right)}\times\left(\prod_{j<l}2\
\text{sinh}\left(b\left(\frac{\mu_{j}-\mu_{l}}{2}\right)\right)2\
\text{sinh}\left(b^{-1}\left(\frac{\mu_{j}-\mu_{l}}{2}\right)\right)\right)\left(\prod_{h<g}2\
\text{sinh}\left(b\left(\frac{\nu_{h}-\nu_{g}}{2}\right)\right)\right.$
$\displaystyle\left.2\
\text{sinh}\left(b^{-1}\left(\frac{\nu_{h}-\nu_{g}}{2}\right)\right)\right)\times\prod_{r,s}\left[\frac{1}{2\
\text{cosh}\left(b\left(\frac{\mu_{r}-\nu_{s}}{2}\right)-\frac{\pi
m_{-}}{2}\right)}\frac{1}{2\
\text{cosh}\left(b^{-1}\left(\frac{\mu_{r}-\nu_{s}}{2}\right)+\frac{\pi
m_{+}}{2}\right)}\right]\,,$ (A.2)

where for convenience we changed variables
$\left(\mu,\nu\right)\rightarrow\left(\mu,\nu\right)/\left(2\pi\right)$
relative to (3.1), and defined the numerical constant

$\displaystyle\mathcal{N}_{1}$
$\displaystyle=\frac{1}{\left(2\pi\right)^{M+2N}\left(M+N\right)!\ N!}\,.$
(A.3)

Our goal is to show that $\mathcal{Z}$ is independent of $b$. Once this is
done, plugging in the right values of $m_{\pm}$ will give (3.2). Our first
step will be to use the Cauchy determinant formula [14, 46] to turn the
integrand into a product of two determinants. From here, a clever change of
integration variables will let us replace one of the determinants with the
product of the diagonal elements of the corresponding matrix. We will then
express the integrand as a Fourier transform; as is routinely done in Fermi
gas derivations. This allows us do the $\mu,\nu$ integrals (these are simple
Gaussian integrals at this stage), which gives a simple $b$-dependent phase
factor that exactly cancels a similar factor (A.2), thus making $\mathcal{Z}$
independent of $b$. Since the object of our derivation is to exhibit the
$b$-independence of (A.2), we do not need to keep track of any $b$-independent
factors that are produced along the way. Hence, we will employ a series of
normalization factors $\mathcal{N}_{i}$ that soak up all such $b$-independent
factors.

We now begin the calculation by using the Cauchy determinant formula as given
in [14, 46] to turn (A.2) into

$\displaystyle\mathcal{Z}$ $\displaystyle=\mathcal{N}_{1}\int d^{M+N}\mu\
d^{N}\nu\
e^{-\frac{{i}k}{4\pi}\left(\sum_{j}\mu^{2}_{j}-\sum_{l}\nu^{2}_{l}\right)-\frac{M}{2}\sum_{r}\left(Q\mu_{r}+\pi
m_{+}-\pi
m_{-}\right)+\frac{M}{2}\sum_{s}Q\nu_{s}+\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}$
$\displaystyle\qquad\left[\text{det}\left(\Theta_{N,l}\frac{1}{2\
\text{cosh}\frac{b^{-1}\left(\mu_{j}-\nu_{l}\right)+\pi
m_{+}}{2}}+\Theta_{l,N+1}e^{\left(M+N-l+\frac{1}{2}\right)\left(b^{-1}\mu_{j}+\pi
m_{+}\right)}\right)\right]$
$\displaystyle\qquad\qquad\left[\text{det}\left(\Theta_{N,s}\frac{1}{2\
\text{cosh}\frac{b\left(\mu_{r}-\nu_{s}\right)-\pi
m_{-}}{2}}+\Theta_{s,N+1}e^{\left(M+N-s+\frac{1}{2}\right)\left(b\mu_{r}-\pi
m_{-}\right)}\right)\right]\,,$ (A.4)

where $\Theta_{r,s}=\Theta\left(r-s\right)$ is the step function, and the
indices $\left(j,l,r,s\right)$ run from $1$ to $N+M$. Now we use the following
identity (similar to the one in Appendix A of [47]):

$\displaystyle\int d^{M+N}\mu\ d^{N}\nu\
\text{det}\left(f\left((\mu_{j},\nu_{l}\right)\right)\times\text{det}\left(f\left((\mu_{j},\nu_{l}\right)\right)$
$\displaystyle\qquad\qquad\qquad\qquad=\left(M+N\right)!\int d^{M+N}\mu\
d^{N}\nu\
\left(\prod_{j=1}^{M+N}f\left((\mu_{j},\nu_{j}\right)\right)\times\text{det}\left(f\left((\mu_{r},\nu_{s}\right)\right)$
(A.5)

to get:

$\displaystyle\mathcal{Z}$ $\displaystyle=\mathcal{N}_{2}\sum_{\sigma\in
S_{M+N}}\left(-1\right)^{\sigma}\int d^{M+N}\mu\ d^{N}\nu\
e^{-\frac{{i}k}{4\pi}\left(\sum_{j}\mu^{2}_{j}-\sum_{l}\nu^{2}_{l}\right)-\frac{M}{2}\sum_{r}\left(Q\mu_{r}+\pi
m_{+}-\pi
m_{-}\right)+\frac{M}{2}\sum_{s}Q\nu_{s}+\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}$
$\displaystyle\qquad\left(\prod_{j=1}^{N}\frac{1}{2\
\text{cosh}\frac{b^{-1}\left(\mu_{j}-\nu_{j}\right)+\pi
m_{+}}{2}}\right)\left(\prod_{l=N+1}^{M+N}e^{\left(M+N-l+\frac{1}{2}\right)\left(b^{-1}\mu_{l}+\pi
m_{+}\right)}\right)$
$\displaystyle\qquad\qquad\left(\prod_{r=1}^{N}\frac{1}{2\
\text{cosh}\frac{b\left(\mu_{\sigma(r)}-\nu_{r}\right)-\pi
m_{-}}{2}}\right)\left(\prod_{s=N+1}^{M+N}e^{\left(M+N-s+\frac{1}{2}\right)\left(b\mu_{\sigma(s)}-\pi
m_{-}\right)}\right)\,,$ (A.6)

where we’ve also expanded the remaining determinant term. We now rewrite the
integrals as Fourier transforms:

$\displaystyle\mathcal{Z}$ $\displaystyle=\mathcal{N}_{3}\int d^{M+N}\mu\
d^{N}\nu\ d^{N}p\ d^{M+N}q\
e^{-\frac{{i}k}{4\pi}\left(\sum_{j}\mu^{2}_{j}-\sum_{l}\nu^{2}_{l}\right)-\frac{M}{2}\sum_{r}\left(Q\mu_{r}+\pi
m_{+}-\pi
m_{-}\right)+\frac{M}{2}\sum_{s}Q\nu_{s}+\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}$
$\displaystyle\qquad\left(\prod_{j=1}^{N}\frac{e^{{i}\frac{p_{j}\left(b^{-1}\left(\mu_{j}-\nu_{j}\right)+\pi
m_{+}\right)}{2\pi}}}{2\ \text{cosh}\
\frac{p_{j}}{2}}\right)\left(\prod_{l=N+1}^{M+N}e^{\left(M+N-l+\frac{1}{2}\right)\left(b^{-1}\mu_{l}+\pi
m_{+}\right)}\right)$
$\displaystyle\qquad\qquad\left(\prod_{r=N+1}^{M+N}\delta\left(2\pi{i}\left(M+N-r+\frac{1}{2}\right)+q_{r}\right)e^{-{i}\frac{q_{r}m_{-}\pi}{2\pi}}\right)\left(\prod_{s=1}^{N}\frac{e^{-{i}\frac{q_{s}\left(\pi
m_{-}+b\nu_{s}\right)}{2\pi}}}{2\ \text{cosh}\ \frac{q_{s}}{2}}\right)$
$\displaystyle\qquad\qquad\qquad\left(\sum_{\sigma\in
S_{M+N}}\left(-1\right)^{\sigma}\prod_{t=1}^{M+N}e^{{i}\frac{q_{t}\left(b\mu_{\sigma(t)}\right)}{2\pi}}\right)\,.$
(A.7)

The $\mu,\nu$ integrals are now easy to do and generate some mass-dependent
phases, which are absorbed into the definition of $\mathcal{N}_{4}$. The
resulting expression may be massaged into the following form:

$\displaystyle\mathcal{Z}$
$\displaystyle=\left(\prod_{j=N+1}^{M+N}e^{\frac{{i}}{4\pi
k}\left(-2\pi{i}\left(M+N-j+1/2\right)b^{-1}+{i}\pi
MQ\right)^{2}}\right)e^{\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}\times\mathcal{N}_{4}\sum_{\sigma\in
S_{M+N}}\left(-1\right)^{\sigma}\int d^{M+N}q\ d^{N}p$
$\displaystyle\qquad\left(\prod_{l=N+1}^{M+N}e^{\frac{{i}}{4\pi
k}\left(-4{i}\pi q_{\sigma(l)}\left(M+N-l+1/2\right)+q_{l}^{2}b^{2}+2{i}\pi
MQbq_{l}\right)}\right)\left(\prod_{r=1}^{N}\frac{e^{{i}\frac{p_{r}}{2\pi}\left(\frac{q_{\sigma(r)}-q_{r}}{k}+\pi
m_{+}\right)}}{2\ \text{cosh}\ \frac{p_{r}}{2}}\right)$
$\displaystyle\qquad\qquad\left(\prod_{s=1}^{N}\frac{e^{-{i}\frac{q_{s}m_{-}}{2}}}{2\
\text{cosh}\
\frac{q_{s}}{2}}\right)\left(\prod_{t=N+1}^{M+N}\delta\left(2\pi{i}\left(M+N-t+\frac{1}{2}\right)+q_{t}\right)\right)\,.$

We now observe that the $b$-dependent terms under the integral sign are
functions of only of those $q_{j}$’s with $j\in\\{N+1,...,M+N\\}$. These may
be pulled out of the integral with the help of the delta functions – leaving
behind an integral that is completely independent of $b$ (and hence absorbed
into the definition of $\mathcal{N}_{5}$):

$\displaystyle\mathcal{Z}$
$\displaystyle=\mathcal{N}_{5}\left(\prod_{j=N+1}^{M+N}e^{\frac{{i}}{4\pi
k}\left(\left(-2\pi{i}\left(M+N-j+1/2\right)b^{-1}+{i}\pi
MQ\right)^{2}-4\pi^{2}\left(M+N-j+1/2\right)^{2}b^{2}+4\pi^{2}MQb\left(M+N-j+1/2\right)\right)}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\times
e^{\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}$ $\displaystyle=\mathcal{N}_{6}\
e^{-\frac{{i}\pi}{k}\left(b^{2}+\frac{1}{b^{2}}-2\right)\frac{M\left(M^{2}-1\right)}{12}}\times
e^{\frac{i\pi}{12k}(b-b^{-1})^{2}M(M^{2}-1)}$
$\displaystyle=\mathcal{N}_{6}\,,$ (A.9)

where note in the second line that the $b$-dependent phase in (A.2) has
cancelled. We have thus shown that $\mathcal{Z}=Z\left(b;\frac{b\
m_{+}-b^{-1}m_{-}}{2},\frac{b\
m_{+}+b^{-1}m_{-}}{2},\frac{{i}\left(b-b^{-1}\right)}{2}\right)$ is
independent of $b$, so we are now free to plug in $b=1$ to get:

$\displaystyle Z\left(b;\frac{b\ m_{+}-b^{-1}m_{-}}{2},\frac{b\
m_{+}+b^{-1}m_{-}}{2},\frac{{i}\left(b-b^{-1}\right)}{2}\right)$
$\displaystyle=Z\left(1;\frac{m_{+}-m_{-}}{2},\frac{m_{+}+m_{-}}{2},0\right)$
$\displaystyle=Z\left(1;m_{1},m_{2},0\right)\,.$ (A.10)

This result is valid for any value of $m_{+}$ and $m_{-}$, so we can set
$m_{+}\rightarrow b^{-1}m_{+}$ and $m_{-}\rightarrow bm_{-}$ to get (3.2).

##  Appendix B The large $k$ expansion

In this appendix we compute some observables in ABJ(M) in perturbation theory
in the CS level $k$. The computation follows standard procedure, for a recent
example see [37].

Our starting point is the partition function deformed by real masses on the
squashed sphere (2.2). Taking derivatives with respect to masses and the
squashing parameter, and setting them to zero, we can define observables in
the matrix model. In general, this procedure should lead to some expectation
value in the matrix model of the form

$\langle\mathcal{O}\rangle=\int\frac{d^{N}\nu d^{N+M}\mu}{N!(N+M)!}e^{i\pi
k\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)}Z_{1-\operatorname{loop}}(\nu_{i},\mu_{a})\mathcal{O}(\nu_{i},\mu_{a})\;.$
(B.1)

where

$Z_{1-\operatorname{loop}}\left(\nu_{i},\mu_{b}\right)=\frac{\prod_{i<j}4\sinh^{2}\left[\pi\left(\nu_{i}-\nu_{j}\right)\right]\prod_{a<b}4\sinh^{2}\left[\pi\left(\mu_{a}-\mu_{b}\right)\right]}{\prod_{i,a}4\cosh\left(\pi\left(\nu_{i}-\mu_{a}\right)\right)\cosh\left(\pi\left(\nu_{i}-\mu_{a}\right)\right)}\;,$
(B.2)

and $\mathcal{O}(\nu_{i},\mu_{a})$ is some operator.

We are interested in computing such objects in perturbation theory in $1/k$.
To simplify notation we first perform the change of variables

$x_{i}=\pi\sqrt{k}\nu_{i},\quad y_{a}=\pi\sqrt{k}\mu_{a}\;,$ (B.3)

which brings our expressions to the form

$\langle\mathcal{O}\rangle=\int
dXdYe^{\frac{i}{\pi}\left(\sum_{i}x_{i}^{2}-\sum_{a}y_{a}^{2}\right)}f(x,y)\mathcal{O}(x,y)\;,$
(B.4)

where $dX,dY$ are the Haar measures777These include the standard Vandermonde
determinant factors. for $U(N)$ and

$f(x,y)=\prod_{i<j}\frac{k\sinh^{2}\left(k^{-\frac{1}{2}}\left(x_{i}-x_{j}\right)\right)}{\left(x_{i}-x_{j}\right)^{2}}\prod_{a<b}\frac{k\sinh^{2}\left(k^{-\frac{1}{2}}\left(y_{a}-y_{b}\right)\right)}{\left(y_{a}-y_{b}\right)^{2}}\frac{1}{\prod_{i,a}\cosh^{2}\left(k^{-\frac{1}{2}}\left(x_{i}-y_{a}\right)\right)}\;.$
(B.5)

We will normalize all expectation values as
$\langle\mathcal{O}\rangle\to\langle\mathcal{O}\rangle/Z_{0}$ where

$Z_{0}=\int
dXdYe^{\frac{i}{\pi}\left(\sum_{i}x_{i}^{2}-\sum_{a}y_{a}^{2}\right)}\;.$
(B.6)

Note that the expression (B.4) has the form of an expectation value of
$f(x,y)\mathcal{O}(x,y)$ in a matrix model which is the product of two
decoupled free Gaussian matrix models.

In order to perform computations, we must start by explicitly expanding $f$
and $\mathcal{O}$ in $1/k$. For example:

$\displaystyle f(x,y)$
$\displaystyle=1+\frac{1}{k}\left(\frac{1}{3}\left(\sum_{i<j}x_{ij}^{2}+\sum_{a<b}y_{ij}^{2}\right)-\sum_{i,a}(x_{i}-y_{a})^{2}\right)+O(k^{-2})\;,$
(B.7)

And similarly for $\mathcal{O}(x,y)$. Plugging these expansions into (B.4), it
is clear that we will end up with the expectation value of a sum of products
of the form $\sum_{i}x_{i}^{k}$ and $\sum_{a}y_{a}^{l}$ for $k,l\in\mathbb{N}$
in a product of Gaussian matrix models. Since the $X$ and $Y$ Gaussian matrix
models are decoupled, these expectation values decouple. So it is enough to be
able to compute expectation values of the form

$\langle\left(\sum_{i_{1}}x_{i_{1}}\right)^{\nu_{1}}\left(\sum_{i_{2}}x_{i_{2}}^{2}\right)^{\nu_{2}}\left(\sum_{i_{3}}x_{i_{3}}^{3}\right)^{\nu_{3}}...\rangle=\langle(\text{tr}X)^{\nu_{1}}(\text{tr}X^{2})^{\nu_{2}}(\text{tr}X^{3})^{\nu_{3}}...\rangle$
(B.8)

in a free Gaussian matrix model, for some integers $\nu_{i}$, $i=1,2,...$ (and
similarly for $y_{a}$). Here we have used the fact that the $x_{i}$’s are the
eigenvalues of $X$.

We have thus reduced the problem to that of computing expectation values of
multi-trace operators in a free Gaussian matrix model. These expectation
values have closed form expressions [48], which we now review. Focus on the
$X$ matrix model, and consider the computation of the expectation value of the
operator

$t_{\bar{\nu}}=(\text{tr}X)^{\nu_{1}}(\text{tr}X^{2})^{\nu_{2}}(\text{tr}X^{3})^{\nu_{3}}...$
(B.9)

where $\nu_{i}\in\mathbb{N}$. The expectation value we are computing is
explicitly

$\langle t_{\bar{\nu}}\rangle=\frac{\int
dXt_{\bar{\nu}}(X)e^{-\frac{1}{2\kappa}\text{tr}X^{2}}}{\int
dXe^{-\frac{1}{2\kappa}\text{tr}X^{2}}}\;.$ (B.10)

for $\kappa=\frac{i\pi}{2}$.

We begin by defining $2n=\sum_{j\in\mathbb{N}}j\nu_{j}$ (note that for odd
$\sum_{j\in\mathbb{N}}j\nu_{j}$ this correlator vanishes). We can think of
$\bar{\nu}$ as a partition of $2n$,
$\bar{\nu}=[1^{\nu_{1}}...(2n)^{\nu_{2n}}]$. Since partitions of $2n$ are in
one-to-one correspondence with classes of the permutation group $S_{2n}$, we
will use the same notation for both. Thus, for example, the correlator
$(\text{tr}X)^{m}$ corresponds to the identity element $1\in S_{m}$, while the
correlator $\text{tr}X^{m}$ corresponds to the longest cycle $(12...m)\in
S_{m}$.

Denoting by $\chi^{Y}$ the irreducible characters of $S_{2n}$ with $Y$
denoting a class of $S_{2n}$,888The character tables can be generated in
Mathematica for small $n$. For example, the character table for $S_{4}$ is
obtained using the command FiniteGroupData[“S4”, “CharacterTable”]. the
authors of [48] found

$\langle
t_{\bar{\nu}}\rangle=(2n-1)!!\kappa^{n}\sum_{Y}\frac{\chi^{Y}(\bar{\nu})\chi^{Y}([2^{n}])}{\chi^{Y}([1^{2n}])}\text{ch}_{Y}(1)\;.$
(B.11)

The sum here is over all classes $Y$ of $S_{2n}$, and $\text{ch}_{Y}(1)$ is
the dimension of the $su(n)$ representation associated with $Y$.999This can be
computed by mapping the classes $Y$ to Young tableaux. The dimension is then
given by the ratio of two numbers. To find the numerator, we start by
inserting $n$ into the box in the top-left corner, and then filling in the
rest of the boxes such that a step to the right increases the number by $1$,
while a step down decreases it by $1$. The numerator is then the product of
these integers. The denominator is just the usual hook length of the diagram.
Equation (B.11) represents a fast and efficient way of computing many-body
correlators in the free Gaussian matrix model, and thus allows us to compute
the observables discussed above to high orders.

As an example, we compute $c_{T}$ to order $k^{-14}$. We can find the operator
$\mathcal{O}$ by using equation (2.8):

$c_{T}=\frac{64}{\pi^{2}}\partial^{2}_{m_{+}}F=-\frac{64}{\pi^{2}}\left(\frac{Z^{\prime\prime}}{Z}-\left(\frac{Z^{\prime}}{Z}\right)^{2}\right)\;,$
(B.12)

where primes denote derivatives by $m_{+}=m_{2}+m_{1}$. The derivatives are
given by

$\displaystyle Z^{\prime}=$ $\displaystyle-\int\frac{d^{N}\nu
d^{N+M}\mu}{N!(N+M)!}e^{i\pi
k\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)}Z_{1-\operatorname{loop}}\left(\nu_{i},\mu_{a}\right)\frac{\pi}{2}\sum_{a,i}\tanh\pi\left(\nu_{i}-\mu_{a}\right)\;,$
(B.13) $\displaystyle Z^{\prime\prime}=$ $\displaystyle\int\frac{d^{N}\nu
d^{N+M}\mu}{N!(N+M)!}e^{i\pi
k\left(\sum_{i}\nu_{i}^{2}-\sum_{a}\mu_{a}^{2}\right)}Z_{1-\operatorname{loop}}\left(\nu_{i},\mu_{a}\right)$
$\displaystyle\times\frac{\pi^{2}}{4}\left(\left(\sum_{a,i}\tanh\left(\pi\left(\nu_{i}-\mu_{a}\right)\right)\right)^{2}-\sum_{a,i}\frac{1}{\cosh^{2}\left(\pi\left(\nu_{i}-\mu_{a}\right)\right)}\right)\;.$
(B.14)

Note that $Z^{\prime}$ vanishes since it is odd under $\nu,\mu\to-\nu,\mu$,
and so it is enough to compute $Z^{\prime\prime}$. The corresponding operator
$\mathcal{O}$ can be read off from (B.14):

$\mathcal{O}(\nu,\mu)=\frac{\pi^{2}}{4}\left(\left(\sum_{a,i}\tanh\left(\pi\left(\nu_{i}-\mu_{a}\right)\right)\right)^{2}-\sum_{a,i}\frac{1}{\cosh^{2}\left(\pi\left(\nu_{i}-\mu_{a}\right)\right)}\right)\,.$
(B.15)

Following the algorithm above, we computed $c_{T}$ to $O(k^{-14})$. Due to the
length of the expression, we give it in an attached Mathematica file.

We will next be interested in computing
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$ from equation (1.2) in
perturbation theory. Specifically, we would like to compare it to another
independent quantity, $\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F$, and to show
that they are not the same in perturbation theory in large $k$ (where we have
$\mathcal{N}=6$ SUSY). Explicitly, $\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F$
is given by

$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F=-\frac{\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}Z}{Z}+\frac{\partial_{m_{2}}^{2}Z}{Z}\frac{\partial_{m_{3}}^{2}Z}{Z}\;.$
(B.16)

Using the relation (1.1), we find that we can write the first term as

$\frac{\partial_{m_{2}}^{2}\partial_{3}^{2}Z}{Z}=4\frac{\partial_{m_{-}}^{2}\partial_{b}^{2}Z}{Z}-\frac{\left(2\partial_{m_{+}}^{4}-2\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}+16\partial_{m_{+}}^{2}\right)Z}{Z}\;,$
(B.17)

while we find

$\frac{\partial_{m_{2}}^{2}Z}{Z}=\frac{\partial_{m_{3}}^{2}Z}{Z}=2\frac{\partial_{m_{+}}^{2}}{Z}+2\frac{\partial_{m_{+}}\partial_{m_{-}}Z}{Z}\;.$
(B.18)

We thus have to compute the derivatives
$\partial_{m_{-}}^{2}\partial_{b}^{2}Z$, $\partial_{m_{+}}^{4}Z$,
$\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}Z$, $\partial_{m_{+}}^{2}Z$,
$\partial_{m_{+}}\partial_{m_{-}}Z$. First, we write the operators
$\mathcal{O}(x_{i},y_{a})$ corresponding to each term:

$\displaystyle\partial_{m_{-}}^{2}\partial_{b}^{2}Z\to$
$\displaystyle\sum_{a,i}\frac{1}{8}\pi^{2}\text{sech}^{2}(g_{s}r_{ai})\left(20g_{s}r_{ai}\tanh(g_{s}r_{ai})-2\left(4g_{s}^{2}r_{ai}^{2}+\pi^{2}+4\right)+3\left(4g_{s}^{2}r_{ai}^{2}+\pi^{2}\right)\text{sech}^{2}(g_{s}r_{ai})\right)$
$\displaystyle+\frac{\pi^{2}}{4}\sum_{b,j}\tanh\left(g_{s}r_{bj}\right)\sum_{a,i}\text{sech}^{2}\left(g_{s}r_{ai}\right)\left(6g_{s}r_{ai}-\left(4g_{s}^{2}r_{ai}^{2}-\cosh\left(2g_{s}r_{ai}\right)+\pi^{2}-1\right)\tanh\left(g_{s}r_{ai}\right)\right)$
$\displaystyle+O_{b}O_{+}(0)$ (B.19) $\displaystyle\partial_{m_{+}}^{4}Z\to$
$\displaystyle
O_{+}^{2}(0)-2\partial_{m}O_{+}(m)\frac{\pi}{2}\sum_{a,i}\tanh\left(\pi
r_{ai}\right)+\partial_{m}^{2}O_{+}(m)$ (B.20)
$\displaystyle\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}Z\to$ $\displaystyle
O_{+}^{2}(0)$ (B.21) $\displaystyle\partial_{m_{+}}^{2}Z\to$ $\displaystyle
O_{+}(0)$ (B.22) $\displaystyle\partial_{m_{+}}\partial_{m_{-}}Z\to$
$\displaystyle-\frac{\pi^{2}}{4}\left(\sum_{a,i}\tanh\left(g_{s}r_{ai}\right)\right)^{2}$
(B.23)

where $r_{ai}\equiv x_{a}-y_{i}$. Here

$\displaystyle O_{+}(m)=$
$\displaystyle\frac{\pi^{2}}{4}\left(\left(\sum_{a,i}\tanh\left(g_{s}r_{ai}+\frac{\pi
m}{2}\right)\right)^{2}-\sum_{a,i}\frac{1}{\cosh^{2}\left(g_{s}r_{ai}+\frac{\pi
m}{2}\right)}\right)\;,$ (B.24) $\displaystyle O_{b}=$
$\displaystyle\sum_{a<b}\left(2g_{s}x_{ab}\left(\mathop{\rm
coth}\nolimits\left(g_{s}x_{ab}\right)-g_{s}x_{ab}\text{csch}^{2}\left(g_{s}x_{ab}\right)\right)\right)+\left(x_{ab}\leftrightarrow
y_{ij}\right)$
$\displaystyle-\frac{1}{2}\sum_{a,i}\left(4g_{s}^{2}r_{ai}^{2}+2g_{s}r_{ai}\sinh(2g_{s}r_{ai})+\pi^{2}\right)\text{sech}^{2}(g_{s}r_{ai})\;.$
(B.25)

Now we can compute the expectation values. Plugging these into equation
(B.16), we find

$\partial_{m_{2}}^{2}\partial_{3}^{2}F=-\frac{1}{4}N(M+N)\pi^{4}-\frac{1}{6}iN(M+N)\pi^{3}\left(-28M+4M^{3}+3M\pi^{2}\right)\frac{1}{k}+\frac{1}{24}N(M+N)\pi^{3}\left(-96M^{2}\pi-5\pi^{3}+13M^{2}\pi^{3}+5MN\pi^{3}+5N^{2}\pi^{3}\right)\frac{1}{k^{2}}+\frac{1}{36}iN(M+N)\pi^{3}\left(-232M\pi^{2}-132M^{3}\pi^{2}+4M^{5}\pi^{2}-56M^{2}N\pi^{2}+8M^{4}N\pi^{2}-56MN^{2}\pi^{2}+8M^{3}N^{2}\pi^{2}+21M\pi^{4}+15M^{3}\pi^{4}+15M^{2}N\pi^{4}+15MN^{2}\pi^{4}\right)\frac{1}{k^{3}}+\frac{N(M+N)\pi^{3}}{1440}\left(15360\pi^{3}+26880M^{2}\pi^{3}+2880M^{4}\pi^{3}-19200MN\pi^{3}+1920M^{3}N\pi^{3}-19200N^{2}\pi^{3}+5760M^{2}N^{2}\pi^{3}+7680MN^{3}\pi^{3}+3840N^{4}\pi^{3}-1644\pi^{5}-2545M^{2}\pi^{5}-363M^{4}\pi^{5}+2680MN\pi^{5}-717M^{3}N\pi^{5}+2680N^{2}\pi^{5}-1753M^{2}N^{2}\pi^{5}-2072MN^{3}\pi^{5}-1036N^{4}\pi^{5}\right)\frac{1}{k^{4}}\;.$
(B.26)

We would like to compare this result to
$\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F$. Using the definitions of
$m_{\pm}$ we find

$\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}Z=2\partial_{m_{+}}^{4}Z-2\partial_{m_{+}}^{2}\partial_{m_{-}}^{2}Z\;.$
(B.27)

Using the results above we can compute this as well:

$\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F=-\frac{1}{4}N(M+N)\pi^{4}+\frac{1}{24}N(M+N)\pi^{4}\left(-5\pi^{2}+M^{2}\pi^{2}+5MN\pi^{2}+5N^{2}\pi^{2}\right)\frac{1}{k^{2}}+\frac{N(M+N)\pi^{4}}{2073600}\left(-293760\pi^{4}+396000M^{2}\pi^{4}-4320M^{4}\pi^{4}+1267200MN\pi^{4}-168480M^{3}N\pi^{4}+1267200N^{2}\pi^{4}-1141920M^{2}N^{2}\pi^{4}-1946880MN^{3}\pi^{4}-973440N^{4}\pi^{4}\right)\frac{1}{k^{4}}\;.$
(B.28)

In general, we find that
$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F\neq\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F$.
For example, for ABJM (where $M=0$), they agree up to order $1/k^{3}$, with
the first difference appearing at order $1/k^{4}$. Specifically, for ABJM we
find

$\partial_{m_{2}}^{2}\partial_{m_{3}}^{2}F-\partial_{m_{1}}^{2}\partial_{m_{2}}^{2}F=-\frac{\pi^{6}N^{2}}{12}\left(3\pi^{2}-32\right)\left(N^{4}-5N^{2}+4\right)\frac{1}{k^{4}}+O\left(\frac{1}{k^{6}}\right)\;.$
(B.29)

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