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

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 $\partial_b^4 F\vert_{b=1}$ and $\partial_b^5 F\vert_{b=1}$ 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.


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 N = (2, 2) [1], 3d N = 2 [2,3], 4d N = 2 [4,5], and 5d 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 often 1 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 N ≥ 6 ABJ(M) CFTs [10,11] with gauge group U (N ) k × U (N + M ) −k and Chern-Simons level k. Like any 3d N = 6 SCFT, ABJ(M) has an SO(6) R symmetry and a U (1) global symmetry [12], so that from the N = 2 perspective the theory has an SO(4) × 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 ( √ 3; 0) (or F (1/ √ 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 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 (1.2) all evaluated at m I = 0 and b = 1, where m ± ≡ m 2 ±m 1 (or m ± ≡ m 3 ±m 1 ). 2 The quantities in red have an odd number of m ± derivatives and are pure imaginary, the quantity ∂ 2 m 2 ∂ 2 m 3 F in green is generically complex, while the remaining quantities in black are always real. For parity preserving ABJ(M) theories, which includes all N = 8 theories, ∂ 3 m ± ∂ m ∓ F in fact vanishes, while ∂ m + ∂ m − F vanishes for ABJM theory. 3 The ABJ(M) theories have N = 8 supersymmetry when k = 1, 2, in which case the SO(6) R ×U (1) global symmetry is promoted to SO(8) R . As a result ∂ 2 m 2 ∂ 2 m 3 F is related to the other real non-vanishing quantities, which are all written as derivatives of m ± , 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 N = 8 ABJ(M) theories. We can similarly relate certain higher order derivatives such as ∂ 5 b F in terms of m ± 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 × 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 × S 7 /Z k , while for large N, k and fixed M, λ ≡ N/k and then large λ is dual to weakly coupled Type IIA string theory on AdS 4 × CP 3 . 4 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 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 = 0 or b = 1 breaks the amount of supersymmetry to 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 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 ∂ 2 m ± 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]: We will match this gravity prediction for the sub-leading N 1 2 terms to our all orders in 1/N expression for ∂ 4 b F and ∂ 5 b F in 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 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 [22,23] and for 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 [23,26], 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 N = 8 ABJ(M), while for N = 6 ABJ(M) a large N expansion is still needed for ∂ 2 m 2 ∂ 2 m 3 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 ×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 ∂ 5 b 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 ∂ 4 b F and ∂ 5 b 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.  In N = 2 language, ABJM theory consists of vector multiplets for each U (N ) k × 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 × SU (2) R × U (1) flavor symmetry as shown in Table   1. Seiberg duality relates different ABJ(M) theories as which implies that M ≤ |k|. Parity then sends k → −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.
where Q = b + 1 b , the µ a , ν i correspond to the Cartans of the two gauge fields with a = 1, . . . , N + M and i = 1, . . . , 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) × SU (2), respectively.
The functions s b (x) are reviewed in Appendix A. The phase factor was computed for generic 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 ± = m 2 ± m 1 (or m ± = m 3 ± m 1 ) as As shown in [31,32], the partition function can furthermore be simplified using the Cauchy determinant formula to take the form 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 ± . One can then use standard methods from statistical mechanics to compute Z(m + , m − ) to all orders in 1/N as where the constant map function A is given by [33] A and in the second line we wrote 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: 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 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 ± derivatives are nonzero in this limit, and that these quantities are always real.

Exact relation between squashing and mass
We will now derive the relation between mass and squashing shown in (1.1). We will then We start by setting m 3 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: where recall that m ± = m 2 ± 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 µ, ν factorize into gaussian integrals, and finally performing these integrals and rewriting the Cauchy determinant back into the standard form. The result is 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 ± 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 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 This implies that any odd number of derivatives of m + is pure imaginary, such as ∂ m + ∂ m − F .
We thus conclude that where recall that ∂ 2 m ± 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 At quartic order, the full list of nonzero relations is which are all written in terms of the six invariants in (1.2). As discussed above, the two invariants ∂ m + ∂ m − F and ∂ 3 m ± ∂ m ∓ 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 ± is ∂ 2 m 2 ∂ 2 m 3 F . For N = 8 ABJ(M), however, the flavor group SO(4) × U (1) is enhanced to SO(6), which implies that In this case, we find that 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 ± derivatives even for N = 8. For instance, at quintic order ∂ m 1 ∂ 2 m 2 ∂ 2 m 3 F cannot be further simplified. Nevertheless, at quintic order for N = 8 we can use (3.2) and (3.8) to write ∂ 5 b F as At higher order a > 5, we can no longer write ∂ a b F in terms of just m ± derivatives. We checked all the relations discussed in this section for U (N + M ) k × 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 . . 10, as computed from the explicit partition function in (2.2). As expected, for k = 1, 2 when the theory is N = 8, these quantities are identical and real. For all higher k, when the theory is only N = 6, these quantities are distinct and ∂ 2 m 2 ∂ 2 m 3 F is complex. Finally, in Appendix B we show that ∂ 2 m 1 ∂ 2 m 2 F and ∂ 2 m 2 ∂ 2 m 3 F differ explicitly in the large k weak coupling expansion, which is automatically 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].

Large N and holography
In the previous section, we showed that for N = 8 ABJ(M) theory, we can relate ∂ 4 b F and ∂ 5 b 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 ∂ 4 b F and ∂ 5 b F to all orders in 1/N . For the U (N ) k × U (N ) −k theory with finite k = 1, 2, in which case we have N = 8 supersymmetry, we find while it is straightforward to compute higher orders in 1/N . The constant map 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 (4. 2) The leading and sub-leading terms in (4.1) exactly match ∂ 4 b F and ∂ 5 b 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 N = 8 supersymmetry that we could compute here. This implies that (4.1) must hold for any value of k up to O(N 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 ∂ 4 b F and ∂ 5 b F can be expanded to all orders in 1/c T as  Here, the 1/c T corresponds to the tree level supergravity correction, the 1/c 2 T corresponds to the 1-loop supergravity correction, and the 1/c 7 3 T term corresponds to the tree level D 6 R 4 correction. Curiously, the 1/c 5 3 T correction, which would correspond to the R 4 correction, vanishes.
Finally, we can also consider the limit of large N, k at fixed λ ≡ N/k and then large λ, which is dual to weakly coupled Type IIA string theory on AdS 4 × CP 3 . Using the large k expansion of A(k) given on the second line of (2.6), we find that

(4.4)
Here, the c −1 T term corresponds to the tree level supergravity correction, the c −1 T λ − 3 2 term corresponds to tree level R 4 , while the various c −2 T 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.

Conclusion
The main result of this work is the exact relation between the mass and squashing deformed 3) from M-theory compactified on AdS 4 × S 7 /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 N = 4 SYM. The free energy F (b; m; τ ) 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 τ , a single mass m, and the squashing b, all of which couple to operators in the 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 where c is the conformal anomaly and the coefficient of the canonically normalized stress tensor two-point function. Recall that for 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 F , and we note that the 3d analog of c is c T , which is proportional to the third invariant ∂ 2 m ± 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 [26,39], 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; τ ) obeys 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 N = 8 ABJM theory, ∂ 4 m ± F and ∂ 2 m + ∂ 2 m − 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, 5 which is the highest order protected term, and it is possible that the integrated constraint from ∂ 2 b ∂ 2 m F will be sufficient to fix this term. For 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 ∂ m + ∂ m − F and ∂ 3 m ± ∂ m ∓ 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 ∂ 4 m ± F and ∂ 2 m + ∂ 2 m − 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 ∂ 2 m 2 ∂ 2 m 3 F , which remains unknown for N = 6 ABJ(M). 6 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 λ ≡ 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 λ result from the small λ, 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 λ resummation for c T , which we computed to O(k −14 ) in this work.
One final interesting limit of N = 6 ABJ(M) that we have not yet considered is the large M, k limit at fixedλ = M/k and N , which is holographically dual to weakly coupled 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 N = 6 numerical bootstrap in [32], and will be further discussed in upcoming work. roud 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).

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 [15,45]). This function is defined as where the integration contour evades the the pole at t = 0 by going into the upper half-plane.
This function obeys several identities: 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 [30,46]. We start from a slightly modified version of (3.1): where for convenience we changed variables (µ, ν) → (µ, ν) /(2π) relative to (3.1), and defined the numerical constant Our goal is to show that Z is independent of b. Once this is done, plugging in the right values of m ± 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 µ, ν 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 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 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 where Θ r,s = Θ (r − s) is the step function, and the indices (j, l, r, s) run from 1 to N + M . Now we use the following identity (similar to the one in Appendix A of [47]): to get: where we've also expanded the remaining determinant term. We now rewrite the integrals as Fourier transforms: The µ, ν integrals are now easy to do and generate some mass-dependent phases, which are absorbed into the definition of N 4 . The resulting expression may be massaged into the following form: where note in the second line that the b-dependent phase in (A.2) has cancelled. We have is independent of b, so we are now free to plug in b = 1 to get:

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 , and O(ν i , µ 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 which brings our expressions to the form where dX, dY are the Haar measures 7 for U (N ) and . In order to perform computations, we must start by explicitly expanding f and O in 1/k.
For example: f (x, y) = 1 + 1 k And similarly for 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 i x k i and a y l a for k, l ∈ 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 7 These include the standard Vandermonde determinant factors.
compute expectation values of the form in a free Gaussian matrix model, for some integers ν 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 multitrace 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 where ν i ∈ N. The expectation value we are computing is explicitly (B.10) for κ = iπ 2 . We begin by defining 2n = j∈N jν j (note that for odd j∈N jν j this correlator vanishes).
We can think ofν as a partition of 2n,ν = [1 ν 1 ...(2n) ν 2n ]. Since partitions of 2n are in one-toone correspondence with classes of the permutation group S 2n , we will use the same notation for both. Thus, for example, the correlator (trX) m corresponds to the identity element 1 ∈ S m , while the correlator trX m corresponds to the longest cycle (12...m) ∈ S m .
Denoting by χ Y the irreducible characters of S 2n with Y denoting a class of S 2n , 8 the authors of [48] found The sum here is over all classes Y of S 2n , and ch Y (1) is the dimension of the su(n) representation associated with Y . 9 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 O by using equation (2.8): where primes denote derivatives by m + = m 2 + m 1 . The derivatives are given by (B.14) Note that Z vanishes since it is odd under ν, µ → −ν, µ, and so it is enough to compute Z .
The corresponding operator O can be read off from (B.14): 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 ∂ 2 m 2 ∂ 2 m 3 F from equation (1.2) in perturbation theory. Specifically, we would like to compare it to another independent quantity, ∂ 2 m 1 ∂ 2 m 2 F , and to show that they are not the same in perturbation theory in large k (where we have N = 6 SUSY). Explicitly, ∂ 2 m 2 ∂ 2 m 3 F is given by Using the relation (1.1), we find that we can write the first term as the ratio of two numbers. To find the numerator, we start by inserting n into the box in the top-left corner, We thus have to compute the derivatives ∂ 2 m − ∂ 2 b Z, ∂ 4 m + Z, ∂ 2 m + ∂ 2 m − Z, ∂ 2 m + Z, ∂ m + ∂ m − Z. First, we write the operators O(x i , y a ) corresponding to each term: a,i 1 8 π 2 sech 2 (g s r ai ) 20g s r ai tanh(g s r ai ) − 2 4g 2 s r 2 ai + π 2 + 4 + 3 4g 2 s r 2 ai + π 2 sech 2 (g s r ai ) + π 2 4 b,j tanh (g s r bj ) a,i sech 2 (g s r ai ) 6g s r ai − 4g 2 s r 2 ai − cosh (2g s r ai ) + π 2 − 1 tanh (g s r ai ) a,i 4g 2 s r 2 ai + 2g s r ai sinh(2g s r ai ) + π 2 sech 2 (g s r ai ) .

(B.26)
We would like to compare this result to ∂ 2 m 1 ∂ 2 m 2 F . Using the definitions of m ± we find Using the results above we can compute this as well: In general, we find that ∂ 2 m 2 ∂ 2 m 3 F = ∂ 2 m 1 ∂ 2 m 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