Genus-2 Holographic Correlator on $AdS_5 \times S^5$ from Localization

We consider the four-point function of the stress tensor multiplet superprimary in $\mathcal{N}=4$ super-Yang-Mills (SYM) with gauge group $SU(N)$ in the large $N$ and large 't Hooft coupling $\lambda\equiv g_\text{YM}^2N$ limit, which is holographically dual to the genus expansion of IIB string theory on $AdS_5\times S^5$. In \cite{Binder:2019jwn} it was shown that the integral of this correlator is related to derivatives of the mass deformed $\mathcal{N}=2^*$ sphere free energy, which was computed using supersymmetric localization to leading order in $1/N^2$ for finite $\lambda$. We generalize this computation to any order in $1/N^2$ for finite $\lambda$ using topological recursion, and use this any order constraint to fix the $R^4$ correction to the holographic correlator to any order in the genus expansion. We also use it to complete the derivation of the 1-loop supergravity correction, and show that analyticity in spin fails at zero spin in the large $N$ expansion as predicted from the Lorentzian inversion formula. In the flat space limit, the $R^4$ term in the holographic correlator matches that of the IIB S-matrix in 10d, which is a precise check of AdS$_5$/CFT$_4$ for local operators at genus-one. Using the flat space limit and localization we then fix $D^4R^4$ in the holographic correlator to any order in the genus expansion, which is nontrivial at genus-two, i.e. $1/N^6$. This is the first result at two orders beyond the planar limit at strong coupling for a holographic correlator.


Introduction
The graviton S-matrix is one of the simplest observables in flat space quantum gravity. In IIB string theory in ten flat dimensions, the S-matrix can be computed for small string coupling g s and finite string length s using a genus expansion of the worldsheet. When IIB string theory is compactified on AdS 5 × S 5 , the scattering of gravitons is holographically dual to four point functions of single-trace half-BPS operators in the boundary CFT, which is maximally supersymmetric N = 4 super-Yang-Mills (SYM) with gauge group SU (N ) [2]. In the 't Hooft limit where N → ∞ for fixed 't Hooft coupling λ = g 2 YM N and then λ → ∞, the CFT correlators can be computed using classical IIB supergravity on AdS 5 × S 5 , where 1/N 2 corrections correspond to higher genus corrections in string theory and 1/λ corrections correspond to higher derivative corrections to supergravity. Unlike the flat space S-matrix, there is no systematic way to compute all these terms. At leading order in 1/λ and 1/N , i.e. tree level supergravity, the correlators can be computed using Witten diagrams [3][4][5][6][7][8][9][10], but this becomes difficult at loop level or for higher derivative corrections to supergravity since the contact terms are not fully known (see however [11][12][13][14] for partial results). Recently, some of these terms have been computed using supersymmetric localization [1,15], the flat space limit [16][17][18][19][20][21][22], and unitarity methods [23][24][25][26][27][28][29][30][31], which do not require an explicit bulk action as in the original analytic bootstrap 1 paper [49] as first applied to N = 4 SYM in [50]. In this work, we will use an extension of the localization method of [1] along with the flat space limit to fix more of these terms, including the genus-one R 4 and the genus-two D 4 R 4 contact terms, as well as the complete 1-loop supergravity term.
Before we discuss the expansion of the holographic correlator in AdS 5 × S 5 , it is simpler to consider the IIB S-matrix that is related to this correlator in the flat space limit. The IIB S-matrix has been computed in a small g 2 s expansion to genus-two for finite s [51,52], and to genus-three [53] to the lowest few orders in s . We will consider the following terms in the small g s and s expansion of this amplitude: where we normalized the amplitude by the genus-zero supergravity term A SG , and s, t, u = −s − t are Mandelstam variables. Higher orders in s can come from contact terms of higher derivative correction to supergravity, which are analytic in s, t, u and have an expansion in g s , as well as loops, which are non-analytic in s, t, u. The first couple higher derivative terms are R 4 and D 4 R 4 . These terms are protected, and so only receive corrections at genus-zero as well as genus-one and two for R 4 and D 4 R 4 , respectively, which take the form [54,55] f 0 stu(s 2 + t 2 + u 2 ) . (1. 2) The only loop term shown in (1.1) is the 1-loop term with two supergravity vertices, which can be computed in terms of genus-zero supergravity using unitarity [56].
the bottom component of the stress tensor multiplet, and is a scalar with dimension 2 that transforms in the 20 of the SU (4) R-symmetry. Superconformal Ward identities fix this correlator in terms of a single function of the conformal cross ratios [57], whose Mellin transform [20,58] we denote by M(s, t). In the strong coupling 't Hooft limit, M has an expansion in 1/N and 1/λ, whose scaling we can determine from (1.1) using the AdS/CFT dictionary L 4 4 s = λ = g 2 YM N , .
Crossing symmetry and the analytic structure of Witten diagrams in Mellin space [19,20,58,59] fix the allowed terms in this expansion to be 2  derivative corrections, as well as the 1-loop supergravity Mellin amplitude M SG|SG that is non-analytic in s, t, u. The coefficient of the supergravity term is fixed by a Ward identity from the conservation of the stress tensor, and the non-analytic 1-loop term is fixed in terms of tree level supergravity using unitarity [24,25,29] up to a constant ambiguity M 0 .
The remaining coefficients in (1.4) have been addressed using two methods. Firstly, in the flat space limit (1.4) can be related to the IIB S-matrix (1.1), which was originally used in [22] to fix the genus-zero R 4 term. In [1], this term was also fixed by relating the integral of SSSS to derivatives ∂ 2 m ∂ λ −1 F | m=0 of the mass m deformed N = 2 * free energy F on S 4 , which can be computed from a matrix model using supersymmetric localization [15]. This quantity was computed to leading order in the 't Hooft limit in [60], and used along with the flat space limit in [1] to fix the coefficients of both R 4 and D 4 R 4 at genus-zero to get To fix coefficients beyond genus-zero, ∂ 2 m ∂ λ −1 F | m=0 must be computed to higher orders. In 3d, the holographic correlator in ABJM theory [61] with gauge group U (N ) k × U (N ) −k and Chern-Simons level k, which is dual to IIA string theory on AdS 4 × CP 3 in the large N and large λ = N/k expansion, was also related to derivatives of the mass deformed S 3 free energy [42]. This quantity was computed using localization [62] in terms of a matrix model that is quite complicated, but can still be expanded to all order in 1/N using the Fermi gas method [63,64]. This all orders constraint could then be used to fix the R 4 correction to all orders in the genus expansion [43]. 3 In our 4d case, no such Fermi gas method exists for the matrix model of the mass deformed N = 2 * sphere free energy F , which also takes a complicated form. Instead, in this work we take advantage of the fact that the m = 0 free energy for N = 4 SYM is just a free Gaussian matrix model, so ∂ 2 m ∂ λ −1 F | m=0 can be written as an expectation value in this free theory. This expectation value can then be computed to any order in 1/N at finite or perturbative λ using topological recursion [65,66], and also at finite N and λ (if we ignore non-perturbative instantons in the Nekrasov partition function) using orthogonal polynomials [67].
We then use this any order in 1/N method to fix the coefficient of M 0 to any genus. For R 4 , we compute the coefficient B 0 0 of the genus-one correction, which scales as √ λc −2 , and is the only nonzero correction to R 4 beyond genus-zero. We can also fix the coefficient B SG|SG 0 of the constant ambiguity in the 1-loop supergravity term, which completes the derivation of [24,25,29]. This constant term only contributes to scalar CFT data. In [25], it was conjectured that B SG|SG 0 = 0 4 so that the anomalous dimension of the lowest twist doubletrace operator would be analytic in spin down to zero spin. We find that B SG|SG 0 = 0, so analyticity in spin fails at zero spin, as expected from the Lorentzian inversion formula in the large N limit of SYM [68].
We also use the flat space limit and the known IIB S-matrix to fix the leading s, t contribution to each Mellin amplitude. For R 4 , we get the same answer for the genus-one term, which is the first genus-one check of AdS 5 / CFT 4 for local operators that could not be determined from genus-zero. 5 We are also able to fix D 4 R 4 at genus-two, which is the 3 The R 4 correction has also been computed in the finite k ABJM theory [40] and the 6d (2, 0) theory [41], which are both dual to M-theory. 4 This coefficient was called α 16 in that paper, since they expand in 4c = N 2 − 1. 5 In [28,29,31] various non-analytic genus-one terms in the holographic correlator were matched to the only nonzero correction to D 4 R 4 beyond genus-zero, by combining the two constraints from the flat space limit and localization. This is the first correction to a holographic correlator computed at genus-two, which is two orders beyond the planar limit.
The rest of this paper is organized as follows. In Section 2, we review properties of the stress tensor multiplet four-point function in the strong coupling limit, including the flat space limit and the relation to the N = 2 * sphere free energy. In Section 3, we show how topological recursion can be used to efficiently compute this quantity to any order in 1/N 2 , which we explicitly carry out to the lowest five orders, and check for finite N using the method of orthogonal polynomials. In Section 4, we use this all orders constraint to fix the higher genus coefficients in the holographic correlator. We conclude with a discussion in Section 5.
We include many explicit results from the localization calculation in the appendices.

N = 4 stress-tensor four-point function
We begin by reviewing what is already known about the stress tensor multiplet four-point function. First we discuss general constraints from the N = 4 superconformal group. Then we discuss Mellin space, the large N strong coupling expansion for SYM, and the flat space limit. Lastly, we review the relation derived in [1] between the integrated stress tensor correlator and the N = 2 * free energy on S 4 .

Setup
Let us denote the bottom component of the stress tensor multiplet as S, which is a dimension 2 scalar in the 20 of the SU (4) R-symmetry. We can express this operator as a traceless symmetric tensor S IJ of SO(6) ∼ = SU (4) fundamental indices I, J = 1, . . . , 6. To avoid carrying around indices, it is convenient to contract them with auxiliary polarization vectors Y I that are constrained to be null, i.e. Y · Y = 0, so that we define where x denotes the position dependence. We normalize S so that its two-point function is corresponding non-analytic term in the IIB S-matrix, but in both flat space and AdS 5 × S 5 these quantities are completely fixed by genus-zero data.
We are interested in studying the four-point function SSSS , which is fixed by conformal and SU (4) symmetry to take the form where the conformally invariant cross ratios U, V and the SO(6) invariants σ, τ are 3) is a degree 2 polynomial in each Y i , the quantity S(U, V ; σ, τ ) is a degree 2 polynomial in σ, τ . The superconformal Ward identity further requires that S(U, V ; σ, τ ) be written in terms of the SU (4)-independent quantity T (U, V ) as where S free (U, V ; σ, τ ) is the "free theory" correlator 6 All the non-trivial interacting information in the correlator is contained in T (U, V ), which will be our main focus of study in this paper.

Strong coupling expansion and the flat space limit
We now specify to SYM, and discuss the strong coupling 't Hooft limit, where we take N → ∞ (or c → ∞) with λ ≡ g 2 YM N fixed and then λ → ∞. Recall from the introduction that the double expansion in c −1 and λ − 1 2 is dual to the IIB expansion in g 2 s 8 s (counting supergraviton loops) and 2 s (counting higher derivatives) according to the AdS/CFT dictionary (1.3). Higher powers in c −1 can thus correspond to either higher genus loop Witten diagrams or corrections to contact Witten diagrams, as we will see below.
It is convenient to express the strong coupling expansion in Mellin space. The Mellin transforms M(s, t; σ, τ ) and M(s, t) of the connected full and reduced correlators S(U, V ; σ, τ ) conn and T (U, V ), respectively, are defined by [50]: is then related to M(s, t; σ, τ ) by the Mellin space version of Θ(U, V ; σ, τ ) in (2.5), which takes the form of a difference operator given in [50] whose explicit form we will not use. The degree of a given term in M(s, t) is four less than that of M(s, t; σ, τ ) in the large s, t limit due to this difference operator.
The main utility of the Mellin amplitude M(s, t) for us is that it provides an easy way to relate the holographic correlator SSSS on AdS 5 × S 5 to the IIB S-matrix A according to the flat space limit formula [1,19,20,40,42]: where f (s, t) was defined in (1.1) as A(s, t) = A SG (s, t)f (s, t) so that the leading supergravity term is normalized to one, and the momenta of the flat space S-matrix is here restricted to (2.9) Secondly, large N counting [23] requires that M(s, t) receives contributions from exchange Witten diagrams of only single and double-trace operators at tree level, and at most (n + 1)trace operators at n-loop level, so only poles corresponding to the twists of these operators may appear at each order. These conditions severely restrict the allowed Mellin amplitudes at each order, and lead to the strong coupling expansion shown in (1.4), which can then be transformed to position space using (2.7) to get where the position space expression can be found by taking the inverse Mellin transform (2.7) and is written in terms of the functionsD r 1 ,r 2 ,r 3 ,r 3 (U, V ) defined in [3]. The coefficient of M SG is fixed by requiring that the unprotected R-symmetry singlet of dimension two that appears in the conformal block decomposition of the free part S free (U, V ; σ, τ ) is not present in the full correlator [50]. In our conventions [1], this amounts to setting the coefficient of At higher order in 1/λ contact Witten diagrams contribute whose vertices are higher derivative corrections to tree level supergravity of the form D 2n R 4 , which scale as c −1 λ − n+3 2 . In Mellin space these terms must be crossing symmetric degree m polynomials M m in s, t, u subject to s + t + u = 4, where the flat space limit requires that m ≤ n. The first couple of terms are [38] both M 0 and M 2 can contribute with coefficients B 2 0 and B 2 2 , respectively. These coefficients were fixed using localization and the relation to the known IIB S-matrix in the flat space limit [1], and we gave the results in (1.5). Note that at O(c −1 ) only the genus-zero coefficients of these contact Witten diagrams appear, and there could be higher genus terms at higher order in 1/c, which would still be tree level in the bulk correlator.
At 1-loop, both single and double-trace operators can be exchanged. The single-trace poles were already fixed by the conformal Ward identity to only appear in M SG at order c −1 , so they do not appear at any other order. The double-trace pole contribution in position space comes from a 1-loop Witten diagram that can be computed by "squaring" the contribution of tree level double-trace anomalous dimensions according to the unitarity method of [23], and so their coefficients are entirely fixed by tree level data. For instance, the leading order in 1/λ double-trace pole contribution T SG|SG comes from 1-loop Witten diagrams with pairs of supergravity vertices, and so scales as c −2 . The log 2 U and log 2 V terms 7 in T SG|SG were fixed from summing supergravity double-trace anomalous dimensions in [24], and in [25] these terms were completed to the full T SG|SG using an ansatz that was verified in [31]. The explicit form of T SG|SG is extremely complicated, so we refer the reader to [25] for the explicit definition. The Mellin space expression M SG|SG , which was derived in [29] from the previous position space expressions, takes the much simpler form These can be understood as the terms that contribute to the double discontinuity [68] at O(c −2 ).
which has poles at all the expected double-trace twists, and where c mn is c mn =30m 2 n 2 (m + n) 2 − 10mn(7m 3 + 36m 2 n + 36mn 2 + 7n 3 ) − 296(m + n) + 64 + (44m 4 + 548m 3 n + 1152m 2 n 2 + 548mn 3 + 44n 4 ) − 2(128m 3 + 631m 2 n + 631mn 2 + 128n 3 ) + 12(37m 2 + 90mn + 37n 2 ) . (2.14) In [29], this expression was given without d mn or C, and had a divergence that was independent of s, t. We can choose d mn so as to cancel this divergence, and then fix C so that M SG|SG is actually the Mellin transform of T SG|SG as defined in [25]. In Appendix A we show that one (non-unique) choice is It was shown in [29] that M SG|SG is asymptotically linear in s, t, as expected for a c −2 term from the flat space limit, so the full c −2 contribution includes a constant M 0 whose coefficient , this coefficient parameterizes the finite counterterm from regulating the divergence 8 of 1-loop supergravity on AdS 5 ×S 5 . The next lowest 1-loop term is M SG|R 4 genus-0 , which scales as c −2 λ − 3 2 and was computed using similar methods in [28,29], but we will not consider 1-loop terms at this order.  .10)). To do this we will use the relation to the known IIB S-matrix using the flat space limit formula (2.8), as well as the relation to the N = 2 * free energy on S 4 that was shown in [1], which we will review next.

Relation to
The action of any 4d N = 4 SCFT can be deformed by the complex marginal operator Φ that couples to the complex conformal manifold parameter τ . We can also deform by the dimension two scalar S and the dimension three scalar P in the stress tensor multiplet, which both couple to a real mass m and break the supersymmetry to N = 2. Since correlators of all the operators in the stress tensor multiplet are related by supersymmetry [57,71], the parameters m, τ , andτ are sources for all stress tensor multiplet correlators. In [1], it was shown that the correlator SSSS integrated over S 4 was related to the S 4 free energy F (m, τ,τ ) as where T is the interacting part of SSSS as defined in (2.5), and I[G] is the S 4 integral As shown by Pestun [15], the S 4 partition function Z = exp(−F ) of mass deformed N = 4 SYM, i.e. the N = 2 * theory, with gauge group SU (N ) can be computed using supersymmetric localization through a matrix model that takes the form where a ij ≡ a ij , the delta function enforces that the SU (N ) eigenvalues a i have zero trace, is a product of two Barnes G-functions. The quantity |Z inst | 2 represents the contribution to the localized partition function coming from instantons located at the North and South poles of S 4 [72][73][74][75], and can be ignored in the 't Hooft limit because it is exponentially small when g YM → 0. At m = 0, the partition function describes a Gaussian matrix model, whose free energy takes the form [15] The RHS of the integrated constraint (2.16) can then be simplified for SU (N ) SYM in the 't Hooft limit to where F pert denotes the perturbative free energy that ignores instantons. This quantity was computed to leading order in 1/N 2 in the 't Hooft limit in [60] using a large N saddle point approximation to get It was then expanded to any order in 1/λ in [1] to get where we converted from 1/N 2 to 1/c using c = N 2 −1 4 , and divided by c 2 to take into account that factor on the LHS of (2.16). This O(c −1 ) expression along with the integrated constraint (2.16) and the flat space limit was used in [1] to fix the O(c −1 ) coefficients B 0 0 , B 2 0 , and B 2 2 in (1.4). In the next section, we will generalize (2.21) to all orders in 1/N 2 and 1/λ, which can then be used to fix the remaining coefficients shown in (1.4).
3 N = 2 * free energy on S 4 The goal of this section is to compute the quantity F in (2.20) to all orders in 1/N 2 and 1/λ using the N = 2 * SU (N ) free energy F (m, λ) on S 4 , where we can ignore the contribution from instantons in the 't Hooft limit. From the localized partition function (2.18), we see that can be expressed as a matrix model expectation value of a 2-body operator where K(z) ≡ − H (z) H(z) , and K (z) can be simply expressed using its Fourier transform The expectation value should be taken with respect to the matrix model of the m = 0 theory, i.e. N = 4 SU (N ) SYM. Since the operator K (a ij ) only depends on the difference between eigenvalues, its expectation value is in fact the same for SU (N ) or U (N ) SYM, so for simplicity we will consider the U (N ) partition function We can also ignore the −1 term in (3.2), since we take derivatives of λ in (3.1). Our goal is then to compute the expectation value where we used the fact that the sum is symmetric in i, j. This expectation value is very similar to that of N = 4 Wilson loops, which have been computed in an 1/N 2 expansion for finite λ using topological recursion [76][77][78], and also for finite N and λ using orthogonal polynomials [79,80]. We will now apply the same methods to (3.4), and then take the integral over the auxiliary variable ω in (3.2) and take the λ derivatives in (3.1) to recover F.

1/N 2 expansion from topological recursion
The strategy of this calculation is to express the 2-body expectation value (3.4) in terms of a quantity called the resolvent, which has a known expansion to all orders in 1/N . We do this by expressing (3.4) in terms its inverse Laplace transform with respect to each argument: where the inverse Laplace transform is defined as with γ i chosen so that the contour lies to the right of all singularities in the integrand, and we have included the i = j term in (3.5), which is independent of λ and so does not affect F. The resolvent W (y 1 , . . . , y n ) is defined as the connected expectation value which has the large N expansion W n (y 1 , . . . , y n ) ≡ ∞ m=0 1 N 2m W n m (y 1 , . . . , y n ) . (3.8) The coefficients W n m are generating functions of "genus" m discrete surfaces with n boundaries, so this expansion is called the "genus" expansion [81]. 12 These W n m can be computed for any n, m in a Gaussian matrix model using the topological recursion method of [65,66].
We start with the "genus" zero 1-body resolvent 12 This "genus" expansion is just the 1/N 2 expansion, which differs from the 1 c = 4 N 2 −1 expansion that counts higher genus corrections in the holographic correlator. To avoid confusion between the two uses of genus, we will refer to "genus" in the resolvent expansion using quotes.
The other "genus" zero resolvents can be computed from the recursion formula where R n = {2, . . . , n} and R n l are subsets of R n of size l. The higher "genus" resolvents can then be computed from the recursion formulae for m ≥ 1: and for m ≥ 1 and n ≥ 2: ∂ y l W n−1 m (y 2 , . . . , y l , . . . , y n ) − W n−1 m (y 2 , . . . , y 1 , . . . , y n ) y l − y 1 .

(3.12)
These last two recursion formulae compute all W n m in terms of W n m with n + m < n + m. The recursion formulae can be used to efficiently compute resolvents to any order. For instance, the "genus" zero 2-body resolvent is [82] In Appendix B we give the other resolvents we need to compute (3.5) to O(N −6 ), i.e. W 1 m and W 2 m−1 for m = 1, . . . , 4. In general, all W n m factor in terms of their arguments y i , except for W 2 0 . For the former resolvents, it is straightforward to take the inverse Laplace transforms for each y i separately, which yield a Bessel function for each y i . For instance, the inverse Laplace transform of W 1 0 in (3.9) with argument 2iω is and we give the results for the other factorizable resolvents in Appendix B. For W 2 0 (y 1 , y 2 ), we must perform the two-dimensional integral in (3.6) with the specific arguments b 1 = −b 2 = 2ωi to get 13 .
where we included the overall N 2 in (3.5) and

(3.17)
For F 0 we ignore the first term in (3.17) and use the inverse Laplace transform of W 1 0 in (3.14) to get

(3.20)
Note that the λ − 1 2 term vanishes for all F m , the λ − n 2 terms for positive odd n > 1 have coefficient ζ(n), and a constant term only shows up in F 0 .
The "genus" expansion for the expectation value (3.4) is naturally an expansion in 1/N 2 , since this quantity is the same for U (N ) or SU (N ) SYM, but for SU (N ) SYM we are interested in the 1 c = 4 N 2 −1 expansion: (3.21) We can convert the 1/N 2 expansion coefficients F m in (3.16) into the 1/c expansion coefficients F m as

Finite N from orthogonal polynomials
Instead of expanding F in a 't Hooft expansion for large N (or large c), one can compute it for finite N in terms of a single finite sum using the method of orthogonal polynomials [67].
While this finite N answer is not the full answer for the mass deformed free energy, since we neglected instantons in the definition (2.20) of F, it can still serve as a nontrivial check on the 't Hooft expansion of the previous section.
We begin with the 2-body expectation value (3.4), which we can write as since the expectation value is the same for each pair of eigenvalues a i = a j . We now introduce a family of polynomials p n (a) using the Hermite polynomials H n (x): , (3.24) in https://mathoverflow.net/questions/315264/asymptotic-expansion-of-bessel-function-integral.
which are orthogonal with respect to the Gaussian measure da p m (a)p n (a)e − 8π 2 N λ a 2 = n! λ 16π 2 N n λ 8πN δ mn ≡ h n δ mn .
where we expanded the determinant in terms of permutations of its matrix elements. We can now perform each a i integral in (3.3) using the orthogonality relation (3.25) to get Let us now consider an n-body operator O n (a) that without loss of generality only depends on a i for i = 1, . . . n. We can write this expectation value using (3.26) as  to get the expectation value λ. The 1/N 2 expansion is most accurate for small g 2 YM = λ/N and large N , but appears to be very precise for all range of parameters. Note that each subsequent 1/N 2 correction in (3.19) improves the match to the finite N result.

Constraining the holographic correlator
We will now use the single constraint from the mass deformed free energy computed to all orders in 1/c and 1/λ in the previous section, as well as the single constraint from the relation where recall that u = 4 − s − t and M SG|SG has a more complicated s, t, u dependence that we gave in (2.13).

Genus-one from localization
We start by using the integrated constraint ( (4. 2) The first three expressions were computed in [1], while the last was computed in this work by evaluating the integral numerically to high precision using the explicit position space expression in [25]. 15 It is remarkable that such a complicated expression has such a simple 15 I thank Hynek Paul for sending me an explicit formula for this very complicated expression.
integral, which hints at a simpler hidden structure.
The RHS of (2.20) is derivatives of the mass deformed free energy evaluated at zero mass, which was computed to O(N −6 ) in (3.20), and can be written in the 1/c expansion (3.21) using (3.22). To the order we considered in (1.4) we found where the constraints on the genus-zero tree level coefficients B 0 0 , B 2 0 , and B 2 2 were already derived in this way in [1]. The genus-one R 4 coefficient B 0 0 completes the CFT derivation of R 4 in the strong coupling 't Hooft limit, which receives no perturbative higher genus corrections. These higher genus R 4 corrections would scale as λ 2n− 7 2 c n for n > 2, which do not appear in F as verified to genus-four in (3.20). The coefficient B SG|SG 0 of the constant ambiguity at order 1/c 2 , along with the non-analytic 1-loop term T SG|SG in position space [24,25] (or M SG|SG in Mellin space [29]), completes the 1-loop supergravity term. To fix the D 4 R 4 terms we must next consider the flat space limit.

Genus-two from localization and string theory
We can fix the leading large s, t coefficient at each order by taking the flat space limit (2.8) where the constraints on the genus-zero coefficients B 0 0 and B 2 2 were already derived in this way in [22]. The genus-one R 4 coefficient B

Unprotected CFT data to order O(c −3 )
Now that SSSS has been fixed to the order shown in (4.1), we can use it to extract any CFT data to this order that we like. For instance, we find the anomalous dimensions γ j of the unique lowest twist even spin j double-trace operators [S∂ µ 1 . . . ∂ µ j S] to be where the three O(c −1 ) terms were computed in [83,84], [22], and [1], respectively. Contact terms with n-derivatives only contribute to operators up to spin n/2 − 4, as explained in [49].
The 1-loop supergravity term at order 1/c 2 was originally computed for all spins in [25], where it was conjectured that B SG|SG 0 was zero so that the 1/c 2 term would be analytic in spin down to j = 0. In fact, analyticity in spin is only expected for j > 0 in strongly coupled N = 4 SYM at 1-loop order [32], 16 and the fact that j = 0 differs from j > 0 is a striking validation of this fact. For j > 0, the contributions to the anomalous dimension from the loop amplitudes M SG|R 4 genus-0 and M SG|D 4 R 4 genus-0 have also been computed in [1,28], but the ambiguities needed to compute all spins have not yet been fixed. For higher twist there are many degenerate double-trace operators, so one would need to compute many different half-BPS correlators to determine their anomalous dimensions [24,25].

Conclusion
In this work we computed the four point function SSSS of the superprimary of the stress tensor multiplet in N = 4 SYM with gauge group SU (N ) in the strong coupling 't Hooft limit at large c ∼ N 2 and large λ, which is holographically dual to scattering of supergravitons in IIB string theory on AdS 5 × S 5 expanded for small g s and s . The integral of SSSS was related in [1] to derivatives ∂ 2 m ∂ τ ∂τ F of the mass deformed N = 2 * free energy F on S 4 , which can be expressed using supersymmetric localization as an expectation value in a free Gaussian matrix model [15]. This quantity was previously only known to leading order in the 't Hooft limit [60]. Our main technical result was a derivation of ∂ 2 m ∂ τ ∂τ F to any order in 1/N at finite or perturbative λ using topological recursion [65,66], which we verified at finite N and λ using orthogonal polynomials [67].
We used this any order result to fix the R 4 correction to SSSS to all orders in the genus expansion, of which only the genus-zero and one are nonzero. We were also able to fix the constant ambiguity in the 1-loop supergravity contribution, which completes the derivation of this term as initiated in [24,25,29], and shows that analyticity in spin for the lowest twist anomalous dimension fails for zero spin, as expected from the Lorentzian inversion formula [68]. We also used the known IIB S-matrix, which is related to the flat space limit of SSSS , to constrain this correlator. This gave the same result for R 4 , which is the first genus-one check of AdS 5 / CFT 4 for local operators that could not be determined from genus-zero. By combining localization and the flat space limit we then fixed D 4 R 4 to all orders in the genus expansion, and verified that only the genus-zero and two contributions are nonzero. This genus-two term scales as λ There are many future directions to this work. One could generalize the any order in 1/N localization computation to SYM with gauge group SO(N ) or Sp(N ), whose matrix model is also known [15]. While these other gauge group lead to the same holographic correlator for tree level supergravity, they likely differ for the higher order corrections considered in this work. One could also generalize the computation to correlators S 2 S 2 S p S p of two stress tensor multiplets and two single-trace half-BPS multiplets whose superprimary has dimension p > 2, which was related in [1] to derivatives ∂ 2 m ∂ τp ∂τ p F of the sphere free energy deformed by the coupling τ p to the top components of these other half-BPS multiplets. The partition function (not counting instantons) for this deformation was computed using localization in terms of a matrix model in [85], and resembles the p = 2 matrix model considered in this work except that the Gaussian term in (2.18) is replaced by where the coefficient τ p in general differs from the coupling τ p due to operator mixing on S 4 .
While topological recursion in fact applies to any polynomial potential [65,66] such as this one, it is difficult to resolve this operator mixing beyond the leading order in 1/N 2 unmixing that was done in [1,86]. If this quantity could be computed, then it could be used to fix the constant ambiguity in the 1-loop supergravity contribution to S 2 S 2 S p S p , which was explicitly derived for p = 3 in [27]. The other linear in s, t ambiguity that appears in this term could then be fixed from the flat space limit, since it is the same order in large s, t as the non-analytic terms.
One could also consider new integrated constraints that come from four mass derivatives of F , or derivatives in terms of the squashing parameter b for the free energy F b on the squashed sphere, which was also computed in terms of a matrix model using localization in [87]. All these quantities take the form of expectation values in the undeformed free Gaussian matrix model for N = 4 SYM, and so could be computed to any order in 1/N using the methods of this paper. These additional constraints could then be used to fix SSSS up to genus-three. 17 They could also be used to fix the ambiguities in the 1-loop term M SG|R 4 genus-0 with one supergravity vertex and one genus-zero R 4 vertex [28,29], which is asymptotically degree 4 in s, t, u and so contains four polynomial ambiguities that must be fixed.
In this work we considered the strong coupling 't Hooft limit, but one could also consider the holographic limit where N → ∞ and τ = θ 2π + 4πi The coefficients for each expansion must be SL(2, Z) invariants of τ and τ s , respectively, and indeed the coefficients of the protected R 4 , D 4 R 4 , and D 6 R 4 in the IIB S-matrix involve non-holomorphic Eisenstein series [88][89][90][91]. In [1], the flat space limit was used to show that the coefficient of the R 4 term in SSSS must also be an Eisenstein series. To derive this from the mass deformed partition function one would need to consider the contribution of the instantons in the Nekrasov partition function to the matrix model expectation value. It would nice to see if this is possible to compute with our methods to any order in 1/N .
Lastly, while the application of integrated constraints and localization to holographic correlators has been perturbative in this paper and the original work [1], these relations are in fact non-perturbative, and so could be applied to the numerical bootstrap for N = 4 SYM [92,93]. For this purpose, the finite N formula for the perturbative part of the mass deformed free energy, as derived using orthogonal polynomials in this work, will be especially useful, especially if one could augment it with a similar formula for the contribution from the Nekrasov partition function. These constraints could allow one to impose the values of τ andτ in the numerical bootstrap for finite N , just as N was imposed in the original studies [92,93] using the conformal anomaly c, and thereby solve N = 4 SYM numerically for all τ ,τ and N .

A Comparing M SG|SG and T SG|SG
In this appendix we check that M SG|SG as defined in (2.13) with d mn and C fixed in (2.15) is equivalent to T SG|SG as given in [25]. In Appendix D of [29], it was checked that the U 2 log U for finite V term in M SG|SG yields the same lowest twist double-trace anomalous dimension for spin j > 0 as given in [25]. Since this check was only done for j > 0, it was not sensitive to the constant terms d mn and C that only contributes j = 0 data. Indeed, in [29] d mn and C were not specified, so it was impossible to compare the results in [29] and [25] for j = 0.
Our strategy here is to extract the leading U 2 log U log V term, which contributes to j = 0 CFT data, from M SG|SG and compare it directly to T SG|SG . From the Mellin transform (2.7), we see that to extract this term we must take the s = 4 and t = 4 poles. After taking these poles and performing the sums over m and n, which are finite due to the d mn term, we find that the U 2 log U log V term is This can be matched with the relevant term in the small U, V expansion of T SG|SG that can be extracted from the explicit expression in [25].