Large $N$ Partition Functions of 3d Holographic SCFTs

We study the $S^1\times\Sigma_{\mathfrak g}$ topologically twisted index and the squashed sphere partition function of various 3d $\mathcal N\geq2$ holographic superconformal field theories arising from M2-branes. Employing numerical techniques in combination with well-motivated conjectures we provide compact closed-form expressions valid to all orders in the perturbative $1/N$ expansion for these observables. We also discuss the holographic implications of our results for the topologically twisted index for the dual M-theory Euclidean path integral around asymptotically AdS$_4$ solutions of 11d supergravity. In Lorentzian signature this leads to a prediction for the corrections to the Bekenstein-Hawking entropy of a class of static asymptotically AdS$_4$ BPS black holes.


Introduction
Supersymmetric localization is an invaluable tool to explore precision holography and has been applied with great success to many AdS/CFT setups. The key technical advantage is that supersymmetric localization often reduces the path integral of supersymmetric QFTs to finite dimensional matrix models. In favorable circumstances, and for specific background sources in the path integral, the resulting matrix models can be evaluated explicitly. The utility of this approach to the path integral of supersymmetric QFTs is clear for holographic SCFTs since it allows for the large N and strong coupling calculation of various physics observables. These results can in turn be used in holography to test the AdS/CFT correspondence and to provide concrete calculational tools to go beyond the leading supergravity approximation. Numerous examples of this approach to top-down holography have been explored recently, see [1,2] for reviews. Our goal in this work is to continue this quest in the context of AdS 4 /CFT 3 and derive new results for the partition functions of holographic 3d N = 2 SCFTs realized on the worldvolume of M2-branes.
To set the context of our discussion we begin with a summary of some supersymmetric localization results for the 3d N = 6 U(N ) k ×U(N ) −k Chern-Simons-matter theory constructed by Aharony-Bergman-Jafferis-Maldacena (ABJM) [3], which is the prime example of a 3d holographic SCFT. The supersymmetric localization matrix model for the ABJM theory on S 3 [4] can be evaluated to all orders in the perturbative 1/N expansion and the result is compactly written in terms of an Airy function which receives additional exponentially suppressed non-perturbative corrections. This result has been deduced by employing the relation of this matrix model to topological strings on local P 1 × P 1 [5], and also by using an ideal Fermi-gas formulation [6]. According to the AdS/CFT correspondence, the large N S 3 ABJM partition function should be dual to the M-theory path integral around the Euclidean AdS 4 × S 7 /Z k solution of 11d supergravity. This duality has been confirmed at the leading N 3 2 order in [7], see also [8], but also at the first N 1 2 sub-leading order [9,10] and at the universal log N order [11]. Beyond the log N correction, the S 3 ABJM partition function written in terms of an Airy function provides a remarkable prediction for the Euclidean M-theory path integral to all orders in the perturbative 1/N expansion, which seems highly non-trivial to confirm directly in M-theory. 1 These impressive results motivated several generalizations to include deformations of the ABJM theory on S 3 as well as the study of the S 3 partition function of other holographic SCFTs with similar methods. For instance, the S 3 partition function of the ABJM theory deformed by two real masses [14] and the partition function on the squashed sphere, S 3 b , with the specific value of the squashing parameter b = √ 3 [15] can also be expressed in terms of an Airy function that captures the all-order 1/N expansion of these observables in the large N limit. Similar results for the S 3 partition function in terms of an Airy function have been derived also for large N holographic SCFTs arising from D-branes and orientifold planes in type IIA string theory [16]. Recently, it was conjectured that the all-order S 3 b partition function for the large N ABJM theory deformed by three real masses and general squashing parameter is also given in terms of an Airy function, see [13,17,18].
Given these results one may wonder if similar compact expressions can be obtained for the partition functions of the ABJM theory on other compact Euclidean manifolds to all orders in the 1/N expansion. Indeed, it was recently shown in [17][18][19], see also [20] for earlier work, that this is possible for the topologically twisted index (TTI) and the superconformal index (SCI) of the theory. The TTI is the partition function of the theory on S 1 × Σ g with a partial topological twist on the Riemann surface Σ g of genus g, see [21][22][23][24]. The large N expansion of this index for the ABJM theory admits a simple compact resummation in terms of elementary functions that captures the full answer up to exponentially 1 See [12,13] for attempts, based on supersymmetric localization of 4d supergravity on Euclidean AdS4, at a holographic derivation of the full Airy function capturing the all-order 1/N expansion of the ABJM S 3 partition function.
suppressed non-perturbative contributions, see [17,18]. 2 Similar results can be derived for the all-order large N SCI of the ABJM theory in the so-called Cardy-like limit of small angular momentum fugacity [19]. These results can be successfully compared with calculations in the dual supergravity theory. The leading N 3 2 term in the ABJM TTI accounts for the Bekenstein-Hawking entropy of a dual magnetically charged static BPS AdS 4 black hole upon an appropriate change of ensembles and application of I-extremization [22]. The first subleading N 1 2 term in the TTI can be matched with the correction from the Wald entropy formula to the Bekenstein-Hawking entropy of the black hole [9,10], while the universal log N contribution can be compared with a 1-loop supergravity calculation [25]. The all-order ABJM TTI obtained in [17,18] is consistent with these supergravity results and furthermore should encode the all-order perturbative quantum correction to the dual magnetically charged AdS 4 black hole entropy. Put differently, these large N results for the ABJM TTI provide a precise prediction for the M-theory path integral around asymptotically Euclidean AdS 4 "black saddles" with an S 7 /Z k internal manifold to all orders in the perturbative 1/N expansion [17,18,26]. This is analogous to the role played by the S 3 ABJM partition function for the M-theory path integral around Euclidean AdS 4 × S 7 /Z k . Similar results are available for the ABJM SCI dual to rotating asymptotically AdS black hole solutions [19].
It is natural to ask whether the results summarized above for the deformed S 3 partition function and the TTI can be extended to other 3d holographic SCFTs. In this work we show that this is indeed possible. Using the numerical techniques of [17,18] we perform extensive numerical studies of the TTI for four classes of 3d SCFTs with known gravitational duals in M-theory. The examples we consider are the 3d N = 4 ADHM theory, as well as the Chern-Simons-matter theories dual to AdS 4 × N 0,1,0 , AdS 4 × V 5,2 , and AdS 4 × Q 1,1,1 , and appropriate orbifolds thereof. In all these examples the very precise numerical data allows us to conjecture closed form analytic expressions for the all-order large N TTI which are valid up to exponentially suppressed terms in the large N limit. Our results amount to a significant improvement of the calculations in [27][28][29] where the leading N 3 2 term and the universal log N term in the TTI for these theories were studied. Moreover, building on the recent work [17,18,30,31], we conjecture an Airy function type formula for the large N squashed S 3 partition function of the 3d N = 4 ADHM model deformed by a single real mass parameter. Finally, we consider the mABJM theory which is a 3d N = 2 SCFT obtained from the ABJM model by a superpotential mass deformation, see [32][33][34], and has a holographic description in terms of the CPW AdS 4 vacuum of M-theory [35]. Using the results for the large N ABJM TTI and squashed and mass-deformed S 3 partition function of the ABJM theory in [17,18] we deduce the corresponding expressions for the mABJM model.
Our field theory results have important implications for holography which we discuss in some detail. In particular, we obtain predictions for the corrections to the Bekenstein-Hawking entropy of various static BPS AdS 4 black hole solutions in M-theory, or alter-2 See the discussion around (2.17) below as well as Section 6 of [18] for a more precise definition of what we mean when we discuss all order TTI results in the large N limit.
natively predictions for the path integral of M-theory on Euclidean asymptotically AdS 4 gravitational backgrounds. We also use our results in conjunction with the higher-derivative supergravity analysis in [9,10] to calculate the coefficients of four-derivative terms in 4d N = 2 minimal gauged supergravity obtained as a consistent truncation on the Sasaki-Einstein manifolds we study. As a by-product of our analysis we present predictions for the subleading N 1 2 in the partition function of the four holographic SCFTs placed on the so-called spindle geometries recently discussed in the literature.
The rest of this paper is organized as follows. In Section 2 we introduce the four different 3d N ≥ 2 holographic SCFTs of interest and derive closed-form expressions for their TTI to all orders in the perturbative 1/N expansion based on the numerical techniques used in [17,18] to study the ABJM TTI. In Section 3 we consider the large N S 3 b partition function of 3d holographic SCFTs with real mass deformations and discuss the challenges for calculating it for theories with N = 2 supersymmetry. The S 3 partition function and the TTI for the 3d N = 2 mABJM theory are discussed in Section 4. We continue in Section 5 with a discussion on the holographic implications of our results for the Euclidean M-theory path integral and the all-order quantum corrections to the entropy of supersymmetric AdS 4 black holes. We summarize our main results and discuss some open questions in Section 6. Various technical details on the evaluation of the TTI and the dual AdS 4 description for four of the holographic SCFTs of interest in this work are provided in the Appendices.

Topologically twisted index
In this section we investigate the topologically twisted index (TTI) of various 3d N ≥ 2 holographic SCFTs on S 1 × Σ g , with Σ g a Riemann surface of genus g. Using supersymmetric localization, the path integral representation of the TTI can be recast as a matrix model. For generic N = 2 SCFTs this was first studied in [27], based on the earlier work of [21,22], and explicit examples have then been studied in [28,29]. We will first present the matrix model for the TTI of general N = 2 SCFTs and then move on to specific examples in the following subsections.
Consider an N = 2 Chern-Simons-matter quiver gauge theory with p nodes, each of which represents a gauge group U(N ) ka with Chern-Simons (CS) levels k a (a = 1, · · · , p). The theory may also include N = 2 chiral multiplets Ψ s (s = 1, · · · , I) in the R s representation of the gauge group ⊗ p a=1 U(N ) ka . In this work, we focus on the case where R s is either a pair of bi-fundamental and anti-bi-fundamental, an adjoint, or a pair of fundamental and anti-fundamental representations. To be specific, the N = 2 SCFT of interest may include • Pairs of chiral multiplets Ψ (a,b) and Ψ (b,a) transforming in the (N , N ) and (N , N ) representations of U(N ) ka ×U(N ) k b , with associated magnetic fluxes n Ψ (a,b) and n Ψ (b,a) and fugacities y Ψ (a,b) and y Ψ (b,a) , respectively, • Chiral multiplets Ψ (a,a) transforming in the adjoint representation of U(N ) ka , with associated magnetic flux n Ψ (a,a) and fugacity y Ψ (a,a) , • Pairs of chiral multiplets Ψ a and Ψ a transforming in the N and N representations of U(N ) ka , with associated magnetic fluxes n Ψa and n Ψa and fugacities y Ψa and y Ψa , respectively.
Here the magnetic fluxes on Σ g and the fugacities of the chiral multiplets are associated with the Cartan subgroup of the global symmetry group of the SCFT. The latter consists of the U(1) R superconformal R-symmetry together with various U(1) flavor symmetries. For simplicity, we will refer to them as flavor magnetic fluxes and flavor fugacities from here on.
The S 1 × S 2 TTI of the SCFT with the above matter content reads [27][28][29] x a,i where we have also included contributions from the U(1) a topological symmetry written in terms of the associated fluxes t a and fugacities ξ a . Before considering specific SCFTs, let us collect some general remarks on the above matrix model for the TTI: • The contour C in (2.1) captures the contribution from so-called Jeffrey-Kirwan residues, which can be written in terms of the Bethe Ansatz (BA) formula that we will briefly review for various examples in the following subsections. We refer to [21,22] for a more detailed discussion related to the choice of this contour.
• The generalization of the S 1 × S 2 TTI (2.1) to the S 1 × Σ g TTI with a Riemann surface of genus g can be obtained following [23]. We will provide the resulting explicit relation between Z S 1 ×S 2 and Z S 1 ×Σg for various examples in the following subsections.
• We introduce flavor chemical potentials ∆ in terms of the flavor fugacities as y = e iπ∆ . We then define the "superconformal ∆-configuration" as the set of flavor chemical potentials that extremizes the large N limit of the Bethe potential, which will be introduced explicitly below for various SCFTs. Then, since the Bethe potential as a function of chemical potentials becomes proportional to the S 3 free energy as a function of trial R-charges in the large N limit (provided chemical potentials are identified with trial R-charges [36]), and since the latter is extremized at the exact superconformal R-charges [32,37], the superconformal ∆-configuration so defined precisely matches the exact superconformal R-charges. It is worth mentioning, however, that the flavor chemical potentials for the S 1 × Σ g TTI and the trial R-charges for the S 3 partition function are only formally related to each other by the map between the Bethe potential and the free energy [36].
• We define the "universal n-configuration", also called the universal twist, as the set of flavor magnetic fluxes that ensures a non-zero magnetic flux only for the U(1) R superconformal R-symmetry and vanishing magnetic fluxes for all other U(1) flavor symmetries. This universal twist is achieved simply by setting the n-configuration to be proportional to the superconformal ∆-configuration just defined [36,38].

N = 4 ADHM theory dual to AdS
In this subsection we consider the TTI of the N = 4 SCFT with gauge group U(N ), one adjoint hypermultiplet, N f fundamental hypermultiplets, and no Chern-Simons term for the gauge field. Decomposing the N = 4 multiplets into N = 2 ones as in [39], the theory consists of three adjoint chiral multiplets Ψ I (I = 1, 2, 3) and N f pairs of fundamental and anti-fundamental chiral multiplets ψ q and ψ q (q = 1, . . . , N f ) in an N = 2 language. The superpotential is given as This SCFT can be realized as the low-energy effective worldvolume theory of N M2-branes probing a C 2 ×(C 2 /Z N f ) singularity and enjoys SU (2)×SU(N f ) flavor symmetry in addition to the SO(4) R-symmetry [16,[40][41][42]. For N f = 1, it is therefore equivalent to the ABJM theory with CS level k = 1 realized as the low-energy effective worldvolume theory of N M2-branes probing C 4 /Z k [3]. The N = 4 SCFT is often called the ADHM theory after the authors of [43], or simply the N f model as in [31,42].
To evaluate the ADHM TTI, we first briefly introduce its BA formulation in Section 2.1.1. Then in Section 2.1.2 we review the analytic calculation of the ADHM TTI in the large N limit based on the BA formula. Finally in Sections 2.1.3 and 2.1.4 we study the BA formulation numerically and deduce an analytic expression for the ADHM TTI to all orders in the perturbative 1/N expansion.

Bethe Ansatz formulation
Using the general formula (2.1), the S 1 × S 2 TTI of the ADHM theory reduces to the following matrix model: where we have assigned the same fugacities y q and yq and magnetic fluxes n q and nq for the N f pairs of fundamental and anti-fundamental multiplets for simplicity. From the marginality of the ADHM superpotential (2.2) under global symmetries, the chemical potentials and magnetic fluxes are constrained as Here we have simplified the notation compared to (2.1) by denoting 3 (2.5) The superconformal ∆-configuration and the universal n-configuration correspond to which will be confirmed later by extremizing the large N Bethe potential with respect to the chemical potentials under the constraint (2.4). For convenience, we use the notation ∆ and n to collectively represent all the flavor chemical potentials and magnetic fluxes as To obtain the BA formula for the S 1 × S 2 ADHM TTI, we first rewrite the matrix 3 The fugacity ∆m and its corresponding background flux t are defined below (2.1). The different prescription for ∆m for an even/odd N allows for the proper integer choice (2.26) below, see footnote 12 of [27] for a related discussion.

model (2.3) as
, (2.8) in terms of a large integer cut-off M (m i ≤ M − 1) and using the BA operators Using Cauchy's theorem to evaluate the contour integrals in (2.8) and using the constraints (2.4), we obtain the following BA formulation of the S 1 × S 2 ADHM TTI: where the sum is taken over solutions to the Bethe Ansatz Equations (BAE) e iB i = 1. Introducing x i = e iu i , the latter can be written as One can check that the BAE (2.11) are obtained by differentiating the following Bethe potential with respect to u i , To do so, one uses the inversion formula of the polylogarithm (2.13) given in terms of the Bernoulli polynomials B n (x) and under the assumptions (2.14) In the BA formula (2.10) we have also introduced the Jacobian matrix B as , The components of this matrix are given explicitly as To evaluate the ADHM TTI using the BA formula, one must first find all solutions to the BAE (2.11) and then substitute the BAE solution back into (2.10). Before evaluating the ADHM TTI in this way, we summarize some of the key properties of the BA formulation: • We focus on a particular solution, B 1 , to the BAE (2.11), which will be reviewed in Section 2.1.2, and on its contribution to the ADHM TTI through the BA formula (2.10). The solution B 1 provides the dominant N 3/2 contribution to log Z S 1 ×S 2 in the large N limit and our results below should be viewed as extending this solution beyond the leading N 3/2 order. The full TTI is in general a sum over all BAE solutions, B i , i.e. Z S 1 ×S 2 = e B 1 + e B 2 + . . .. Since B 1 is the dominant solution to leading order at large N we can write where we have While we do not have rigorous mathematical proof that this is the large N structure of the TTI, the known holographic results for the N 3/2 and N 1/2 leading and subleading contributions to log Z S 1 ×S 2 (summarized in Section 5.2 below) strongly suggest that this structure is correct. We therefore conclude that the contribution to log Z S 1 ×S 2 coming from any other BAE solution apart from B 1 is exponentially suppressed in the large N limit. From now on we proceed our analysis with this conclusion in mind and study the all order result for the large N limit of B 1 . We will employ the same logic for all other TTI calculations discussed in this paper.
• The general S 1 × Σ g TTI can be obtained from the S 1 × S 2 TTI as [23] log Z ADHM provided we focus on the contribution from a particular BAE solution to the ADHM TTI through (2.10). We also note that the g = 1 case must be treated separately due to the presence of additional fermionic zero-modes on the torus [23]. For this reason, we focus on g ̸ = 1 in this paper.
• From the structure of the BA formula (2.10), one can conclude that the logarithm of the ADHM TTI is linear in the flavor magnetic fluxes n provided we focus on the contribution from a particular BAE solution.
• Let us say {x ⋆ i } is a solution to the BAE (2.11) for a given ∆ = (∆ I , ∆ q , ∆q, ∆ m ) configuration. Then it is straightforward to show that {x ⋆ i e iπδ } is a solution to the BAE for a deformed ∆ = (∆ I , ∆ q − δ, ∆q + δ, ∆ m ) configuration. Substituting this new solution {x ⋆ i e iπδ } into the BA formula (2.10), one can show that the ADHM TTI is independent of δ up to the overall phase factor e iN tπδ . This implies that the modulus of the ADHM TTI is independent of the chemical potentials (∆ q , ∆q) provided they satisfy the constraint (2.4) for a given ∆ 3 ; hence we can simply set ∆ q = ∆q = 1 − ∆ 3 /2 without affecting the modulus of the ADHM TTI. Then it also becomes clear from (2.10) that the magnetic fluxes (n q , nq) can be chosen as n q = nq = 1 − n 3 /2 to satisfy the constraint (2.4) for a given n 3 without affecting the modulus of the ADHM TTI.

Analytic approach for the large N limit
In this subsection we briefly review the analytic evaluation of the ADHM TTI in the large N limit using the BA formulation (2.10), following [27,28].
The first step is to find the solution of the BAE (2.11) in the large N limit. For that purpose, we introduce the large N ansatz This discrete ansatz is then made continuous by introducing an eigenvalue distribution where we have also introduced a continuous function t : that maps discrete indices 1, 2, · · · , N to the real interval [t ≪ , t ≫ ]. We then introduce the eigenvalue density ρ : [t ≪ , t ≫ ] → R as which satisfies the following normalization condition by construction: In the continuum limit, the large N BAE solution can be obtained by determining the eigenvalue distribution u(t) and the eigenvalue density ρ(t) along the compact support [t ≪ , t ≫ ]. These functions can be determined by extremizing the large N limit of the Bethe potential (2.12) written in terms of the continuous variables, namely 4 (2.23) We refer the reader to [27,28] for the derivation of (2.23). The resulting large N continuous BAE solution reads where µ is determined by the normalization (2.22) to be To obtain the solution (2.24), the integer n i in the BAE (2.11) has been chosen as and we have also assumed which is indeed extremized at the superconformal ∆-configuration (2.6) under the constraint (2.4). Note that the large N Bethe potential (2.28) is independent of (∆ q , ∆q) due to (2.4), and therefore their superconformal values are not determined by extremization. This was to be expected from the comment regarding the independence of the (modulus of the) TTI on (∆ q , ∆q) at the end of the previous subsection.
One can now obtain the large N limit of the ADHM TTI by substituting the BAE solution (2.24) into the BA formula (2.10), see [28] for details. The result reads .
where µ is given in (2.25) and we have defined∆ a andñ a as (2.30)

Numerical approach for the all-order 1/N expansion: results
In this subsection we improve on the known analytic result for the large N ADHM TTI (2.29) by providing an analytic expression for its all-order perturbative 1/N expansion that we infer from a detailed numerical study. Our final result reads where we have defined and also introduced the rational functions c a (∆) and d a (∆) as (2.33b) We will present numerical evidence for the all-order 1/N expansion (2.31) in the following subsection. Before doing so, we collect some general remarks on the result: • As mentioned in the beginning of Section 2.1 the ADHM TTI with N f = 1 matches the ABJM TTI with k = 1 obtained in [17,18], provided we map (∆ a ,ñ a ) defined in (2.30) to the flavor chemical potentials and flavor magnetic fluxes of the ABJM theory (∆ a , n a ). For example, it is straightforward to check that ∆ coefficient of the ABJM TTI up to an overall prefactor, see [17,18].
• We were unable to obtain the closed-form expression for the N -independent constant f 0 (N f , ∆, n), although its numerical value can be obtained with great precision for various N f and (∆, n)-configurations, see Appendix A. We leave a complete analytic determination off 0 (N f , ∆, n) for future research.
• We numerically confirmed that the non-perturbative correctionf np (N, N f , ∆, n) is indeed exponentially suppressed when focusing on the superconformal configuration (2.6), see the following subsection for details.
• The real part of the all-order 1/N expansion (2.31) is independent of (∆ q , ∆q, n q , nq) provided they satisfy the constraints (2.4) for a given ∆ 3 and n 3 . This is consistent with our remarks at the end of subsection 2.1.1. The imaginary part of (2.31) has been confirmed exactly for various configurations of flavor chemical potentials and magnetic fluxes listed in Appendix A.

Numerical approach for the all-order 1/N expansion: derivation
In this subsection we explain how we obtained the all-order 1/N expansion of the ADHM TTI (2.31) from a numerical analysis. The method parallels the analysis for the ABJM TTI studied in [18], but for completeness we repeat the procedure here. We collect the relevant numerical data in Appendix A.
To begin with, we construct a numerical solution of the BAE (2.11) with the left-hand side given by (2.26) for a given N , N f and ∆-configuration. We do so using FindRoot in Mathematica at WorkingPrecision = 200 and using the leading order solution (2.24) as initial conditions 5 . Substituting the numerical solution into the BA formula (2.10) with a given n-configuration, we obtain the numerical value of the ADHM TTI for the chosen set of N , N f , and (∆, n). After repeating this process for N = 101 ∼ 301 (step = 10), we fit the resulting data with respect to N using LinearModelFit in Mathematica. As a result, we obtain the following numerical 1/N expansion of the ADHM TTI, for a given N f and (∆, n)-configuration. The superscript "(lmf)" in the expansion coefficients emphasizes that they are numerical coefficients obtained by LinearModelFit. The upperbound L for fitting is chosen as L = 16 to minimize standard errors in estimating the coefficients f Next, repeating the LinearModelFit (2.35) with five different n-configurations satisfying the constraint (2.4), namely n = (n 1 , n 2 , n 3 , n q , nq, t) , we obtain the numerical 1/N expansion of the ADHM TTI that is linear in the flavor magnetic fluxes n. The result reads (2.37) 5 Since v(t) is not determined in the large N BAE solution (2.24), we set the initial condition for vi based on the ranges (2.27) as [29] vi initial condition = π(∆q − ∆q) 2 .
Note that the five n-configurations in (2.36) are enough to determine the numerical coefficients in (2.37) since the logarithm of the ADHM TTI is expected to be linear in the four magnetic fluxes (n I , t); recall that the ADHM TTI is independent of the other two magnetic fluxes (n q , nq) as we discussed in Section 2.1.1. In (2.37), we did not expand the numerical coefficients of the part linear in the magnetic fluxes beyond N 1 2 as it will not be needed in what follows.
The N 3 2 leading order coefficient derived in [28] and summarized in (2.29) is reproduced accurately by our numerical analysis where we find 38) for various N f and (∆, n)-configurations. Going to subleading orders, we found that the log N term has a universal coefficient which is independent of N f and (∆, n), just as in the ABJM TTI [20]. Then, by inspection of the negative integer powers of N in the numerical 1/N expansion (2.37) for various N f and (∆, n)-configurations, we observe that they can be resummed together with the universal log N contribution as producing a simple function of the shifted N parameter introduced in (2.32). The resummation (2.40) strongly motivates implementing the LinearModelFit step of the analysis with respect toN N f ,∆ rather than N . Quite remarkably, we found that such a fit then terminates at order O(1). In other words, we found that the following LinearModelFit, with only three fitting functions -(N N f ,∆ ) Finally, investigating the numerical coefficients of the improved 1/N expansion for various N f and ∆-configurations, we found with c a and d a given in (2.33). We refer the reader to Appendix A for the numerical values off (N f , ∆, n) with the corresponding analytic symbolf 0 (N f , ∆, n) and restored the non-perturbative correctionsf np (N, N f , ∆, n). The latter corrections were ignored in the above analysis since they are exponentially suppressed, as we now show.

Non-perturbative corrections to the ADHM TTI
Let us study the non-perturbative corrections in (2.31) numerically. We will focus on the superconformal and universal configuration (2.6), in which case the quantitiesN N f ,∆ , c a and d a take a simple form. To begin with, consider the following combination of O(1) contributions and non-perturbative corrections, obtained from the all-order expression (2.31). Above we have introduced obtained from (2.32) at the superconformal point. Then, using the N -independence of f 0 (N f ), one can extractf np (N, N f ) from (2.43). To be specific, we evaluate (2.43) for N = 101 ∼ 301 (step = 10) with fixed N f and then subtract the results with adjacent N values. As a result, we gather numerical data for for N = 101 ∼ 291 (step = 10). The approximation in (2.45) will be justified a posteriori. Then we use LinearModelFit to fit the resulting data with respect to N , where the upperbound L is chosen as L = 16 to minimize standard errors in estimating the numerical coefficientsf Substituting the estimate (2.47) back into the expansion (2.46), we obtain the non-perturbative behaviorf Note that the non-perturbative behavior (2.48) justifies the approximation in (2.45). The following table provides error-ratios for the leading order estimate (2.47), namely Most importantly, the exponentially suppressed non-perturbative behavior (2.48) confirms that the 1/N expansion of the ADHM TTI (2.31) indeed captures all the perturbative contributions to the TTI for the superconformal/universal configuration (2.6). We leave the analysis of non-perturbative corrections to the ADHM TTI with generic (∆, n)configurations for future research. It would be most interesting to understand the physical origin of these non-perturbative corrections as coming from appropriate instanton contributions, similar to the case of the S 3 partition function in the ABJM theory [7,44,45].

N = 3 SCFT dual to AdS
We now consider the TTI of the N = 3 SCFT dual to AdS 4 × N 0,1,0 /Z k [46][47][48][49]. This theory can be constructed as follows: we start from the ABJM quiver which consists of the gauge group U(N ) k ×U(N ) −k with the corresponding vector multiplets and 2 pairs of bi-fundamental and anti-bi-fundamental chiral multiplets A 1,2 and B 1,2 We then add r 1,2 pairs of fundamental and anti-fundamental chiral multiplets ψ (s) 1,2 and ψ (s) 1,2 (s = 1, . . . , r 1,2 ) to each gauge node with opposite CS levels ±k with r 1 + r 2 = k [49]. We find it illustrative to keep r = r 1 + r 2 independent from k in the following formulas 6 , but the constraint r = k should be kept in mind throughout. The superpotential for this model reads For r = 1 theory has SU(3) flavor symmetry in addition to the SO(3) R-symmetry. We refer to the topologically twisted index of this theory as the N 0,1,0 TTI.
To evaluate the N 0,1,0 TTI, we first briefly introduce its BA formulation in Section 2.2.1. Then, in Section 2.2.2, we review the known analytic calculation of the N 0,1,0 TTI in the large N limit. Finally, in Section 2.2.3, we provide an analytic expression for the N 0,1,0 TTI to all orders in the perturbative 1/N expansion.

Bethe Ansatz formulation
Following the general formula (2.1), the S 1 × S 2 N 0,1,0 TTI is given by the matrix model where we have assigned the same fugacities y q 1,2 and yq 1,2 and magnetic fluxes n q 1,2 and nq 1,2 for the r 1,2 pairs of fundamental and anti-fundamental multiplets for simplicity, as in the ADHM case above. Note that we have also introduced the fugacities (−1) N +1 associated with the U(1) topological symmetries for later convenience. From the marginality of the N 0,1,0 superpotential (2.50) and its invariance under global symmetries, the chemical potentials and magnetic fluxes are constrained according to 1 = n 1 + n 4 = n 2 + n 4 , 1 = n q 1 + nq 1 = n q 2 + nq 2 , where we have relabeled the fugacities in (2.1) as The superconformal ∆-configuration and the universal n-configuration correspond to as can be derived from matching the S 3 free energy with the above trial R-charges and the holographically dual regularized on-shell action [50]. For convenience, we write ∆ and n to collectively represent all the chemical potentials and the magnetic fluxes as To present the BA formulation of the S 1 × S 2 N 0,1,0 TTI, we first rewrite the matrix model (2.51) as . (2.57) The sign ambiguities σ i ,σ j can be fixed to as explained in Appendix D. Evaluating the integrals in (2.56) by residues and using the constraints (2.52), we obtain the following BA formula for the S 1 × S 2 N 0,1,0 TTI, where the BAE are e iB i = e iB j = 1. Introducing x i = e iu i andx j = e iũ j , the latter can be written as In the BA formula (2.59) we have also introduced the Jacobian matrix B as in analogy with the ADHM case. In view of the BA formula (2.59), it is clear that the first three remarks we have spelled out for the ADHM TTI below (2.16) apply to the N 0,1,0 TTI as well.

Analytic approach for the large N limit
The large N limit of the N 0,1,0 TTI was studied analytically in [27,28], following the same method we reviewed in the ADHM case. We skip the derivation and simply quote the final result, where we have focused on the superconformal point for simplicity. Even at large N , the full dependence on (∆, n) is complicated. 7 Because of this difficulty, we will only focus on the superconformal point when investigating the all-order 1/N expansion of the N 0,1,0 TTI. Recall also that the large N TTI of the SCFT holographically dual to AdS 4 × N 0,1,0 /Z k is given by (2.62) with k = r [49]. For r = 0, (2.62) reduces to the large N ABJM TTI [22] as expected from the quiver description.

Numerical approach for the all-order 1/N expansion
Now we improve on the leading order analytic result (2.62) by providing an analytic expression for the all-order perturbative 1/N expansion of the N 0,1,0 TTI. Our result reads (2.63) 7 An expression for the case k = r = 1, ∆3 = ∆4, and n3 = n4 can be found in [28] where we have chosen r 1 = r 2 = r/2 and defined The numerical analysis leading to (2.63) proceeds just as in the ADHM case and we refer to Appendix B for the relevant numerical data. We close this section with a few remarks: • We have been unable to obtain the closed-form expression for the N -independent constantsf 0 (k, r) in (2.63), although their numerical values can be obtained with high precision for various k and r, see Appendix B.
• The imaginary part of the N 0,1,0 TTI can be determined analytically as in (2.63) from the BA formula (2.59) and the property of a numerical BAE solution which we have also numerically confirmed.
• We have been able to numerically deduce the leading non-perturbative behavior of the N 0,1,0 TTI following the procedure described for the ADHM TTI in Section 2.1.4 to findf although the N 0,1,0 TTI involves larger numerical errors compared to the ADHM TTI and the outlying case (k, r) = (2, 4) comes with especially large numerical errors. We leave a complete numerical confirmation of the exponentially suppressed behavior (2.65) and the corresponding physical interpretation for future research.

N = 2 SCFT dual to AdS
We now consider the TTI of the N = 2 SCFT dual to AdS 4 × V 5,2 /Z N f , which has been constructed in two different ways [51,52] dubbed as Model I and Model II respectively in [28]. In Model I, the theory has a U(N ) N f ×U(N ) −N f gauge group and consists of two pairs of bi-fundamental A 1,2 and anti-bi-fundamental B 1,2 chiral multiplets together with two adjoint chiral multiplets Ψ 1,2 for each gauge node. The superpotential reads In Model II, the theory has gauge group U(N ) with a vanishing CS level and consists of three adjoint chiral multiplets Ψ I (I = 1, 2, 3) and N f pairs of fundamental and antifundamental chiral multiplets ψ q and ψ q (q = 1, . . . , N f ). The superpotential reads Note that Model II has exactly the same matter content as the ADHM theory studied in Section 2.1 but the different superpotential breaks the supersymmetry to N = 2. We will use Model II to study the V 5,2 TTI. Note that for N f = 1 the theory has an SO(5) flavor symmetry in addition to the U(1) R-symmetry.

Bethe Ansatz formulation
Since Model II has the same field content as the ADHM theory, we can simply use the BA formula for the ADHM TTI spelled out in Section 2.1.1 to study the V 5,2 TTI. The constraints on the chemical potentials and magnetic fluxes are different in the two SCFTs, however, since the two theories have different superpotentials. For the SCFT dual to AdS 4 × V 5,2 /Z N f , we have the following constraints from the marginality of the V 5,2 superpotential (2.67). The ∆-configuration and the universal n-configuration are also different from the ADHM case and read These values can be obtained by extremizing the large N Bethe potential (2.28) with respect to the chemical potentials under the constraint (2.68).

Analytic approach for the large N limit
Since the BA formula for the V 5,2 TTI is the same as the one for the ADHM TTI, the large N limit of the V 5,2 TTI is readily obtained from (2.29). The result is [28] log where µ is given in (2.25) and we have defined∆ a andñ a in (2.30). The only difference with the ADHM case is that chemical potentials and magnetic fluxes satisfy the set of constraints (2.68).

Numerical approach for the all-order 1/N expansion
Just as in the previous cases, a numerical analysis allows us to propose an analytic expression for the all-order perturbative 1/N expansion of the V 5,2 TTI: where we have introduced the rational functions a I (∆), b I (∆) (with I = 1, 2) as , (2.72b) and the shifted N parameterN N f ,∆ is defined in (2.32). We only focused on the real part of log Z V 5,2 S 1 ×S 2 since the imaginary part is the same as (2.31). The numerical analysis leading to this result follows the same logic as that in the ADHM case. The only difference is that the set of n-configurations used for the LinearModelFit is chosen to satisfy (2.68), n = (n 1 , n 2 , n 3 , n q , nq, t) , , (2.73) in contrast with (2.36). As before, we were unable to obtain the analytic form of the Nindependent termf 0 (N f ,∆,ñ), although we could compute it to high precision for various values of N f and (∆,ñ). We refer the reader to Appendix C for the relevant numerical data. The first three remarks summarized in Section 2.1.3 also apply to the V 5,2 TTI.

Non-perturbative corrections to the V 5,2 TTI
Here we numerically explore the non-perturbative corrections in (2.71), focusing on the superconformal and universal configuration (2.69). Just as in the ADHM case, the combination of the N -independent contribution and the non-perturbative corrections is given by where we have omitted (∆, n) in the argument to lighten the notation and introduced the quantityN N f = N + N f 6 + 1 4N f based on (2.32) and (2.69). We then follow the same procedure as described in Section 2.1.4, using LinearModelFit to fit the non-perturbative correctionf np (N, N f ) with respect to N and with L = 16: The following table provides error-ratios for the leading order estimate (2.76), namely 78) for N f ∈ {1, 2, 3, 4, 5}: Most importantly, the exponentially suppressed non-perturbative behavior (2.77) confirms that the 1/N expansion (2.71) indeed captures all the perturbative contributions to the TTI at the superconformal point (2.69). We leave a more general analysis of nonperturbative corrections to the V 5,2 TTI with generic (∆, n)-configurations, as well as their physical interpretation, for future research.

N = 2 SCFT dual to AdS
As our last example, we study the TTI of the N = 2 SCFT dual to AdS 4 × Q 1,1,1 /Z N f [40,[53][54][55]. In the conventions of [40,55], the theory has a U(N )×U(N ) gauge group with vanishing CS levels together with two pairs of bi-fundamental A 1,2 and anti-bi-fundamental a are fundamental and anti-fundamental with respect to the second and first U(N ) factors, so that they yield a single trace operator together with A a (a = 1, 2). The superpotential is given by (2.79) For N f = 1 the theory has SU(2) 3 flavor symmetry in addition to the U(1) R-symmetry.

Bethe Ansatz formulation
According to the general formula (2.1), the Q 1,1,1 TTI takes the form of a matrix model, where we have again assigned the same fugacities y q 1,2 and yq 1,2 and magnetic fluxes n q 1,2 and nq 1,2 for the N f pairs of fundamental and anti-fundamental multiplets for simplicity. Note that we have also included the fugacities (−1) N +N f −2⌊N f /2⌋ associated with the U(1) topological symmetries for later convenience. From the marginality of the Q 1,1,1 superpotential (2.79) and its invariance under global symmetries, the chemical potentials and magnetic fluxes are constrained to satisfy 2 = 4 a=1 n a , 2 = n 1 + n q 1 + nq 1 = n 2 + n q 2 + nq 2 , where we have used the notation y Ψ (a,b) → y a = e iπ∆a , y Ψa → y qn = e iπ∆q n , y Ψa → yq n = e iπ∆q n . (2.82) The superconformal ∆-configuration and the universal n-configuration correspond to This can be obtained in principle by extremizing the large N Bethe potential with respect to chemical potentials as in the ADHM case, but the dependence of the Bethe potential on (∆, n) is much more involved for the Q 1,1,1 TTI. An alternative way to derive the result is to match the S 3 free energy with trial R-charges chosen as in (2.83) and the holographically dual regularized on-shell action [32]. For notational convenience, we use ∆ and n to collectively represent all the chemical potentials and the magnetic fluxes as ∆ = (∆ a , ∆ q 1,2 , ∆q 1,2 ) , n = (n a , n q 1,2 , nq 1,2 ) . (2.84) Following the procedure detailed in Section 2.1.1, the BA formula for the S 1 ×S 2 Q 1,1,1 TTI reads in terms of the Jacobian matrix B given in (2.61). The explicit BAE can be written as follows in terms of the variables u i andũ j introduced via Once again, the remarks we have spelled out in the ADHM case below (2.16) apply mutatis mutandis to the BA formula for the Q 1,1,1 TTI.

Analytic approach for the large N limit
The large N limit of the Q 1,1,1 TTI follows the same logic as in the other cases. In the derivation, we do however correct a couple of subtle points in the intermediate steps of the Bethe potential analysis conducted in [27,28]. To streamline the presentation, we include some details in Appendix D. These corrections do not affect the final large N result. For the superconformal ∆-configuration in (2.83) and arbitrary n, it reads [28] log Observe that, because of (2.81) and (2.83), the left hand side can only depend on the chemical potential ∆ 1 at the superconformal point. The upshot of (2.87) is that ∆ 1 and n are in fact flat directions for the large N TTI. We will see below that this remains true in the all-order 1/N expansion. See [32] for a similar phenomenon in the S 3 partition function of the Q 1,1,1 theory.
Similarly to the N 0,1,0 case discussed in Section 2.2, the full dependence on ∆ is unwieldy even at large N . A partial result obtained by setting some of the chemical potentials to be equal can be found in [28]. Because of this complication, we will only study the all-order 1/N expansion of the Q 1,1,1 TTI at the superconformal point, but we will keep the magnetic fluxes n arbitrary modulo the constraint (2.81).

Numerical approach for the all-order 1/N expansion
To go beyond the analytic result for the large N Q 1,1,1 TTI (2.87), we use the same numerical method as in the other cases. This leads us to an analytic expression for the all-order perturbative 1/N expansion of the TTI. At the superconformal point where the only independent chemical potential is ∆ 1 , the result reads where we have definedN Once again, we have focused on the real part of the TTI, see the comment below (2.64).
Our numerical investigations confirm that the Q 1,1,1 TTI is in fact independent of (∆ 1 , n). The configurations of magnetic fluxes used in the numerical fits have been chosen to satisfy the constraints (2.81), (n 1 , n 2 , n 3 , n 4 , n q 1 , nq 1 , n q 2 , nq 2 ) We refer the reader to Appendix D for the relevant numerical data used in the derivation of (2.88). As in the other cases, we have not yet found the closed-form expression for the Nindependent constantf 0 (N f ). The numerical values can be obtained with a high precision, see Appendix D where we show they are independent of (∆ 1 , n). The numerical analysis of the non-perturbative corrections parallels the ADHM case. We found the following behavior at the superconformal point, for N f ∈ {1, 2, 3, 5}. Numerical errors in the non-perturbative corrections for N f = 4 are however too large to confirm (2.91) in that case. We leave a physical interpretation of the above non-perturbative behavior and further numerical study for the future.

Sphere partition functions
In this section we study another observable for the 3d N ≥ 2 SCFTs introduced in Section 2, namely their S 3 b partition functions with real mass deformations. In Section 3.1 we focus on the perturbative part of the S 3 b partition function of the ADHM theory with an N = 4 symmetry breaking mass parameter µ. Then in Section 3.2 we briefly comment on the S 3 b partition function of other N ≥ 2 holographic SCFTs with real mass deformations, highlighting some obstacles that remain to obtain the all-order expression for the partition functions in an 1/N expansion.

N = 4 ADHM theory
The sphere partition function of the ADHM theory can be computed using supersymmetric localization, which yields a matrix model [16,56]. It is possible to introduce geometric and mass deformations to this result by squashing the sphere with a parameter b (where b = 1 corresponds to the round sphere) and turning on an N = 4 symmetry breaking mass parameter µ [31]. In this case, the matrix model is given as [31,57,58] where the double sine function s b (x) is defined in e.g. App. A of [15]. For µ = 0 and b = 1, the matrix model (3.1) reduces to the one for the S 3 partition function of the undeformed N = 4 ADHM theory, which has been evaluated to all orders in the perturbative 1/N expansion in terms of an Airy function using the ideal Fermi-gas formulation [16,56], The function A above is the constant map contribution to the partition function of ABJM theory on the round S 3 [6]. Its explicit expression will not be needed in what follows. The matrix model (3.1) has an interesting property for the special value of the N = 4 symmetry breaking mass parameter µ = 3) 8 We mainly follow the conventions of [31] but for the precise normalization of Z ADHM we refer to the other two references.

Combining the relation (3.3) with the Airy formula (3.2) shows that the perturbative part of the
is also given in terms of an Airy function: ADHM partition function with generic mass parameter µ is given by an Airy function as where the functions C b and B b read 2 ) terms of the µ = 0 squashed sphere partition function that were obtained in [10] in the dual bulk supergravity theory.
• The Airy formula (3.5) with N f = 1 becomes equivalent to the Airy formula for the S 3 b partition function of real mass deformed ABJM theory conjectured in [17,18] where the CS level and the real masses are set to k = 1 and (m 1 , m 2 , m 3 ) = (0, 0, µ), respectively.
• For µ = 0 and b 2 = 3 it was shown in [15] that the supersymmetric localization matrix model admits a Fermi gas description and its perturbative part in the large N expansion can be resummed into an Airy function. It is easy to check that our conjecture in (3.5) and (3.6) reduces to the results in [15] for µ = 0 and b 2 = 3.
In addition, one can use our Airy proposal to compute the coefficient c T of the stress tensor 2-point function in the undeformed ADHM theory following [59,60], (3.7) Upon taking the derivatives, we obtain which matches the result of [57] derived using the ideal Fermi-gas approach.
On top of the above consistency checks, the conjecture (3.5) displays an interesting property. In the b → 0 limit (or equivalently b → ∞ since there is a b ↔ b −1 symmetry apparent in the matrix model (3.1)) with fixed µ b+b −1 , the argument of the Airy function in (3.5) becomes The RHS is now identical to the shifted N parameter for the S 1 × Σ g ADHM TTI (2.32) provided we identify the mass µ and the chemical potentials∆ as (3.10) Note that this identification is consistent with the remark around (2.34): the flavor chemical potentials∆ in the ADHM theory are identified with the chemical potentials ∆ in the ABJM theory based on the duality between the two theories at k = N f = 1, and the ABJM ∆ are given precisely by (3.10) for (m 1 , m 2 , m 3 ) = (0, 0, µ) as described in [17,18]. The correspondence between the shifted N for the S 3 b partition function and the shifted N for the S 1 × Σ g TTI upon taking the b → 0 limit is not specific to the ADHM model and can also be deduced for the ABJM theory. Using the results in [17,18] we find the following analogous expressions for the ABJM theory 11) where N − B b (k, ∆) is the shift for the S 3 partition function andN ∆ is the shift for S 1 × Σ g TTI. It would be interesting to understand the physics behind this relation between different observables in these holographic SCFTs in more detail.
Considering that the S 3 b partition function of the ABJM theory with generic real mass deformations also admits an Airy form [17,18], and that the ADHM theory with N f = 1 is dual to the ABJM theory with k = 1, we expect that the S 3 b partition function of the ADHM theory deformed by the real masses of the adjoint/fundamental hypermultiplets and by the FI parameter in addition to µ will also admit an Airy form. The latter should then be valid to all orders in the perturbative 1/N expansion, although we have been unable to formulate a precise conjecture at this stage.

N = 2, 3 holographic SCFTs
In this subsection we briefly present some of the difficulties one faces when trying to derive or conjecture an Airy formula for the S 3 b partition function of N = 2, 3 holographic SCFTs with real mass deformations. We focus on the theories introduced in Sections 2.2, 2.3, and 2.4.
For the N = 3 SCFT dual to AdS 4 × N 0,1,0 /Z k , the Airy formula for the undeformed S 3 partition function with r 1 = r 2 = r/2 was derived in [6] and is given by where r is kept independent from k to keep the discussion general, but one should ultimately identify r = k. Generalizing the Airy formula (3.12) to the S 3 b partition function with real mass deformations is non-trivial, mainly due to the lack of a relation of the form (3.3) in the N = 3 SCFT of interest. Without such a constraint, it is difficult to conjecture an Airy formula for the S 3 b partition function of the N 0,1,0 theory with real mass deformations. Furthermore, the duality between the ADHM theory with N f = 1 and the ABJM theory with k = 1 was crucial in formulating our conjecture (3.5). In contrast, the N = 3 SCFT dual to AdS 4 ×N 0,1,0 /Z k does not exhibit such a duality as far as we are aware.
The situation is even worse for the N = 2 SCFTs dual to AdS 4 × V 5,2 /Z N f and AdS 4 × V 5,2 /Z k since there are no known Airy formula even for their undeformed S 3 partition functions. The matrix models for their S 3 partition functions can be written explicitly based on [4], but the ideal Fermi-gas formulation of [6] has not yet been successfully applied to obtain the all-order perturbative 1/N expansion of the S 3 partition functions from the corresponding matrix models.

Intermezzo: The mABJM theory
We now discuss another example of a holographic 3d N = 2 SCFT for which the TTI and the S 3 partition function can be computed in the large N limit via supersymmetric localization. The theory in question is a close cousin of the ABJM model at k = 1, 2 and can be obtained from it by adding a superpotential mass term for one of the four bi-fundamental chiral superfields of the ABJM theory. 9 In the IR the RG flow ends at a strongly interacting 3d N = 2 SCFT which we will refer to as the mABJM theory, see [32][33][34]61] for more details. In addition to the U(1) R-symmetry the theory also enjoys SU(3) flavor symmetry. Its holographic dual description is provided by an AdS 4 vacuum of 11d supergravity, see [35], which can be found by uplifting one of the non-trivial vacua of the 4d N = 8 SO(8) gauged supergravity found in [62].
We conjecture that the S 3 partition function and the TTI of the mABJM theory to all orders in the 1/N expansion can be obtained from the corresponding results in the ABJM model after a particular specialization of the real mass parameters, the electric fugacities, and the background magnetic fluxes.
We start with the S 3 partition function. For the ABJM theory in the presence of the three real masses, m i , and the supersymmetric preserving squashing deformation it was conjectured in [13,17,18] that the S 3 partition function takes the form where Ai is the Airy function and the expression above captures all terms perturbative in 1/N and receives non-trivial instanton corrections. The parameters in (4.1) are defined as follows: where b is the squashing parameter and we have introduced the constrained quantities such that a ∆ a = 2, and The function A b (k) in (4.1) for general (m i , b) is currently not known but its form has been obtained in various limits. See [6,14,15,30,45] for more details on the calculation of the ABJM S 3 partition function for different values of the parameters (m i , b). 9 To form a gauge invariant superpotential one needs to multiply the bi-fundamental chiral superfield with a monopole operator. To have a relevant superpotential deformation the monopole operator has to have a sufficiently low conformal dimension. This is possible for k = 1 and k = 2, see [61] for a detailed discussion. Thus, in this Section, all formulas apply only for the values k = 1, 2.
Once we introduce the superpotential mass term and flow to the IR, the SO(8) global symmetry of the ABJM theory is broken to SU(3)×U(1) which in turn restricts the allowed values of the real mass parameters in the following way Fixing ∆ 1 = 1 is a simple manifestation of the fact that this real mass parameter is no longer present once we integrate out one of the chiral superfield. The other two mass parameters are associated with the SU(3) flavor symmetry of the mABJM theory in the IR. We conjecture that the all-order 1/N expansion of the mABJM S 3 partition function can be obtained from (4.1)-(4.4) by restricting the parameters ∆ a as in (4.5). Note that the mABJM theory preserves the full N = 2 superconformal invariance when the real masses vanish, i.e. when ∆ 2 = ∆ 3 = ∆ 4 = 1/3. These values of ∆ a can be obtained either by a direct inspection of the superpotential or by using F -maximization. We now proceed with the TTI. For the ABJM TTI the all-order 1/N expression was found in [17,18] and reads where c a are given by andf 0 is a function that does not depend on N and was computed numerically for some value of the parameters in [17,18]. Note that we have introduced the quantitŷ (4.8) in terms of which the TTI admits a nice compact form. This expression for the TTI was obtained numerically in [17,18] where it was extensively checked with high accuracy and was shown to be valid up to O(e − √ N ) corrections. The TTI for the mABJM theory can be obtained from the expression in (4.6) by taking k = 1, 2 and restricting the fugacities ∆ a to obey (4.5) and in addition constraining the magnetic fluxes to obey Note that for the ABJM theory there is a single constraint on the magnetic fluxes determining the TTI in (4.6), namely a n a = 2(1 − g), which is now modified since the flavor symmetry is partially broken in the mABJM model which in turn fixes n 1 = (1 − g). The superconformal configuration for the mABJM TTI is obtained by setting (4.10) It is possible to obtain these expression for the mABJM TTI by simply taking limits of the ABJM TTI in (4.6) because the supersymmetric localization formula for the TTI depends only on the matter content, R-charges, and flavor fluxes of the theory and is thus to a large extent similar for two SCFTs related by a supersymmetric RG flow. We end our discussion of the mABJM theory by pointing out that based on the recent results in [19] we can also obtain some explicit results for the SCI of this model. It was shown in [19] that the ABJM SCI in the Cardy-like limit of small angular fugacity ω takes the following explicit form 10 k,∆ +ĝ 0 (k, ∆) (4.11) Based on the discussion above we the SCI for the mABJM theory can be obtained from this expression by fixing k = 1, 2 and restricting ∆ a and n a as in (4.5) and (4.9), respectively. The expressions for the partition functions of the mABJM theory presented above can be compared with available results in the literature. The leading N 3 2 behavior of both the TTI and the S 3 partition function for general values of ∆ a and n a agree with the holographic calculations in [32][33][34]. For the SCI, the holographic result is only available for the so-called universal limit corresponding to the superconformal configuration, see [63], and it agrees with the corresponding limit of (4.11). From the perspective of the matrix model for the TTI and SCI it is clear from the analysis of [17][18][19] that the results for the mABJM theory can be obtained as a limit of the ABJM theory results. Finally, as we discuss in Section 5.2 below, the N 1 2 terms in the expansion of all three partition functions of the mABJM theory for the superconformal configuration are consistent with the 4d minimal gauged supergravity higher-derivative holographic analysis of [9,10].

Holography
In this section we discuss some holographic implications of our all-order perturbative 1/N expansions for the TTI of the various holographic SCFTs studied above. In Section 5.1 we discuss the holographic dual description of the all-order TTI in terms of the Euclidean Mtheory path integral and the BPS magnetic Reissner-Nordström AdS 4 black hole entropy. In Section 5.2 we explain how our TTI results can be combined with the recent analysis of the 4-derivative corrections to N = 2 minimal gauged supergravity [9,10] to yield results for the subleading corrections to other types of SCFT partition functions in the large N limit.

Holographic duals of the TTI
For simplicity, we will first focus on the superconformal and universal configurations of flavor chemical potentials and magnetic fluxes to discuss holographic duals of the TTI. In this special case, the real part of the all-order TTI for all examples studied above simplify to Note that we have fixed r = k for the N 0,1,0 TTI and have used the relation (2.18) to generalize the previous results from S 1 × S 2 to S 1 × Σ g . Recall that we do not consider the case g = 1, which should be treated separately. Our goal now is to discuss a holographic interpretation for the results in (5.1).
The AdS/CFT correspondence in the context of M2-branes leads to a duality between a 3d CFT describing the low-energy physics of the M2-branes at the tip of the cone over a 7d internal manifold Y 7 on one hand, and M-theory on a dual asymptotically AdS 4 × f Y 7 background 11 obtained in the near horizon limit of the supergravity solution sourced by the M2-branes, on the other. The 3d SCFTs whose TTI has been analyzed in Section 2 correspond to the low-energy effective worldvolume theories of N M2 branes on the cone over the 7d Sasaki-Einstein manifolds S 7 /Z N f , N 0,1,0 /Z k , V 5,2 /Z N f and Q 1,1,1 /Z N f . On the M-theory side, the dual 11d Euclidean backgrounds have the following metric and 4-form flux, see for example [64]: where ds 2 6 is a locally Kähler-Einstein metric such that R (6) µν = 8g (6) µν and σ = dJ with the Kähler form J normalized as J 3 = 3! vol 6 . In (5.2) we have kept the original convention of the Hodge star in Lorentzian signature. Some known facts about the relevant 7d Sasaki-Einstein geometries are collected in Appendix E.
The holographic dual of the TTI with the superconformal configuration (5.1) is given by the Euclidean M-theory path integral around the 11d background (5.2) where the 4d external space corresponds to the Euclidean Romans background of N = 2 minimal gauged supergravity [26,65,66]. The field strength F = dA and the metric for this background take the form Here κ = 1, 0, −1 is the normalized curvature of the Riemann surface Σ g with g = 0, g = 1, and g > 1, respectively. The Euclidean solution above admits a Lorentzian black hole interpretation upon Wick rotating and taking Q → 0. This limit produces a regular Lorentzian solution only when κ = −1 [26]. In this case, one obtains a magnetically charged asymptotically AdS 4 supersymmetric black hole with fixed charge and a hyperbolic horizon. To leading order in N , the Euclidean M-theory path integral around the 11d background (5.2) can be approximated in terms of the regularized two-derivative on-shell action of the 4d background (5.3). The latter can be computed in terms of the scale L and the 4d Newton constant [36] Notably this result is independent of the electric charge Q, or equivalently of the periodicity β of the coordinate τ , and is valid both for the family of Euclidean solutions in (5.3) and the Lorentzian supersymmetric black hole solution. We can now use the standard holographic dictionary to relate the gravitational quantities to the field theory parameters. To leading two-derivative order, this yields Using the volumes computed in Appendix E, we find perfect agreement between the TTI (5.1) and the Euclidean M-theory path integral in the large N limit for the various Sasaki- Note that since the imaginary part of log Z SCFT S 1 ×Σg is defined modulo 2πZ it does not affect the leading term in the large N expansion.
At finite N , the field theory TTI receives perturbative 1/N corrections as studied in detail in Section 2. On the gravity side, the Euclidean M-theory path integral around the 11d background (5.2) receives higher-derivative and loop corrections to the saddle point approximation (5.6). In addition, the holographic dictionary receives quantum corrections. These quantum gravitational effects should combine to account for the subleading o(N 3 2 ) terms in (5.5). Therefore, our field theory results for the TTI in (5.1) can be interpreted as a prediction for the full set of perturbative 1/N corrections to the M-theory path integral. It is worth noting that there has been recent progress in computing the first perturbative correction, at order N 1 2 , directly in supergravity by including four-derivative terms. This was explained for both the ABJM and ADHM theories in [9,10]. In Section 5.2 below we discuss how the same four-derivative supergravity analysis can be extended to more general N = 2, 3 SCFTs. The universal 1 2 log N logarithmic correction to the TTI has been obtained from a one-loop bulk computation in [20,25] for the ABJM theory, i.e. for S 7 as internal manifold. It would of course be very interesting to understand how to include higher-derivative couplings and higher loop orders in the 11d supergravity path integral in order to reproduce the all-order perturbative 1/N expansions in (5.1). This appears to be a formidable task at present. Finally, we would like to point out that the universal relation between the sphere free energy and the TTI for 3d N = 2 holographic SCFTs observed in [36,38] for the leading term in the large N expansion is no longer valid when one includes also the subleading terms in (5.1).
As mentioned below (5.3), the Euclidean Romans background admits a continuation to a Lorentzian magnetic BPS AdS 4 charged black hole for a suitable choice of parameters. In this context, our results (5.1) can be used to obtain all the perturbative corrections to the Bekenstein-Hawking entropy of the black hole. Since this background is a solution of minimal gauged supergravity there is no need to change ensembles or invoke I-extremization in order to relate the Euclidean on-shell action (or equivalently the TTI) to the black hole entropy, see [22,67]. We therefore conclude that AdS/CFT dictates the following result for the black hole entropy S S = Re log Z SCFT where the perturbative part of the right sand side is given by (5.1) to all orders in the 1/N expansion. Through the holographic dictionary, this translates to an infinite set of perturbative corrections to the Bekenstein-Hawking entropy contained in the left hand side.
To go beyond the superconformal values of TTI parameters, corresponding to minimal 4d N = 2 gauged supergravity, one has to turn on suitable deformations in the SCFTs that are captured by supergravity vector and hyper multiplets. In the holographic context we are then faced with the hard problem of either finding appropriate 4d consistent truncations of 11d supergravity that include these additional multiplets or construct the corresponding supergravity backgrounds directly in 11d. For the four 11d AdS 4 vacua discussed in this work we do not know how to solve this problem in general. From the 4d perspective there are no known consistent truncations that capture all relevant deformations, see e.g. [64,68,69], while constructing 11d generalizations of the background in (5.2) and (5.3) to include these additional deformations is prohibitively hard since one has to solve coupled system of PDEs. This state of affairs should be contrasted with the situation for the ABJM theory where there is a truncation of 11d supergravity on S 7 to 4d maximal gauged supergravity which can be exploited to construct supersymmetric Lorentzian black hole solutions with non-trivial profiles for fields in vector and hyper multiplets, see for example [22,33,67], as well as more general Euclidean supersymmetric solutions that are dual to the general ABJM TTI to leading order in the large N limit, see [26]. Note also that the continuation of Euclidean supergravity backgrounds to Lorentzian black hole solutions that are free of pathologies is subtle in matter-coupled gauged supergravity theories. As shown in [26,33], in the context of 4d N = 8 gauged supergravity the space of Euclidean supergravity solutions that capture the (∆, n)-dependent TTI is larger than the space of known supersymmetric Lorentzian black holes. It is therefore not immediately clear which 11d black holes can be studied using the TTI of the 3d N ≥ 2 SCFTs discussed above away from the superconformal point.
Even if these difficulties are overcome and more general supergravity backgrounds dual to the (∆, n)-dependent TTI for the four holographic SCFTs we studied are constructed, one should consider the additional subtlety of specifying a choice of ensemble. The entropy of Lorentzian black holes is usually formulated in the microcanonical ensemble of fixed charges while the TTI is written in the ensemble with fixed flavor chemical potentials. One must therefore further implement an inverse Laplace transform on the (∆, n)-dependent TTIs with respect to ∆ in order to obtain the perturbative corrections to the Bekenstein-Hawking entropy of the dual black holes. At the leading O(N 3 2 ) order, this change of ensemble has been implemented for the ABJM TTI and dubbed "I-extremization", and the resulting expression matches the dual AdS 4 black hole entropy precisely [22,67]. 12 Going beyond this leading order implementation of the inverse Laplace transform is still an open problem. Nevertheless, we emphasize that the TTI results presented in this paper should provide a very useful guide to understand how to overcome part (or all) of the difficulties summarized above in computing the corrections to the semi-classical entropy of AdS 4 black holes in M-theory using holography.

Subleading corrections to SCFT partition functions from holography
The interplay between four-derivative couplings in 4d N = 2 minimal gauged supergravity and subleading corrections to the large N limit of SCFT partition functions has been recently used to great efficacy in the ABJM and ADHM theories [9,10]. Here we extend the same analysis to more general N = 2, 3 holographic SCFTs. We also discuss the applica-4d Euclidean geometry F χ AdS 4 w. S 3 bdry 1 1 Taub-NUT-AdS 4 w. S 3 b bdry tion of these results to partition functions on the so-called spindles, see [70] and references thereof.
The main result of the four-derivative supergravity analysis in [9,10] is that the large N partition function of a 3d holographic SCFT on any compact 3d manifold M 3 can be written in the form where F and χ are the two-derivative regularized on-shell action 13 and the Euler number of the dual 4d Euclidean background whose conformal boundary corresponds to M 3 . We list some examples of F and χ in Table 1. Note that the metric for the Kerr-Newman-AdS black hole can be found in [71], that of the AdS 4 -Taub-NUT space is presented in Section 3.2.3 of [10], and the data for the black hole with spindle horizon can be found in [72]. 14 The information about the details of the holographic SCFT enters the result in (5.8) through the (A, B, C) coefficients. For a given SCFT, these coefficients can be deduced from the large N expansion of any two distinct physical observables that can be written as in (5.8). From this, one can deduce the first subleading correction to the SCFT partition function on any 3-manifold using (5.8) and the quantities F and χ computed from the bulk supergravity. This strategy was used in [9,10] where the coefficients (A, B, C) for the ABJM and ADHM theories were fixed using the known large N expansions of the partition functions on the round and squashed 3-spheres.
Using the all-order results for the TTI we have obtained in this paper, we can now extend this analysis to other SCFTs. For the N 0,1,0 N = 3 theory, we can combine the large N expansion of (5.1) and the S 3 partition function in (3.12) to read off the (A, B, C) coefficients. For the mABJM theory discussed in Section 4 we can use the S 3 partition function and the TTI in a similar fashion to find (A, B, C). Note that these values are compatible with (5.8) and the SCI for the superconformal configuration of the mABJM 13 We normalize this action by dividing it by πL 2 /(2GN ) so that for empty Euclidean AdS4 with an S 3 boundary we have F = 1.
14 Compared to [72], we have rescaled ω there = −2πiω here and used the lower sign in their result for F. theory computed using (4.11). The results for (A, B, C) for these SCFTs are summarized in Table 2.

SCFT SUSY
As explained in Section 3.2, we currently do not have access to the subleading corrections in the S 3 partition functions of the V 5,2 and Q 1,1,1 theories, so we cannot apply the above strategy to uniquely fix their (A, B, C) coefficients. It would be most interesting to derive such corrections and combine them with our (5.1) to obtain the coefficients entering (5.8) for these N = 2 theories. In the absence of such a calculation we note that for V 5,2 and Q 1,1,1 we can use the results for the TTI in (5.1) to find the linear combination B + C by reading off the coefficient of the N 1 2 term in the large N expansion and using (5.8) with χ = 2F = 2(1 − g). This results in the following values It was recently observed that the Cardy-like expansion ω → i0 + of the SCI for the ABJM and ADHM theories is controlled by the so-called Bethe potential and the TTI [19]. It is highly likely that more general N = 2 SCIs have the same property, and we plan to study this generalization soon [73]. At this stage, it is worth pointing out that the N 1 2 term in the N 0,1,0 SCI obtained by substituting the (A, B, C) coefficients and the (F, χ) for the dual KN-AdS background into the formula (5.8) is already in perfect agreement with this expectation.
Finally, we note that our results lead to a prediction for the N 1 2 term in the partition function of the holographic SCFTs discussed here when they are placed on the recently discussed spindle geometries, see [70] and references thereof. To obtain these results one has to use the expression in (5.8) with the appropriate (F, χ) characterizing the spindle geometry, see Table 1, and the values of (A, B, C) from Table 2 for the SCFT of interest. It will be most interesting to reproduce this supergravity prediction for the N 3 2 and N 1 2 terms in the spindle free energy by an independent calculation in the dual SCFTs based on the recently proposed localization formula in [74].

Discussion
In this work we found closed form expressions for the all-order perturbative 1/N expansion for the TTI of N ≥ 2 holographic SCFTs arising from M2-branes in M-theory. We focused on five examples corresponding to the four Sasaki-Einstein orbifolds together with the mABJM theory which arises from a superpotential mass-deformation of the ABJM model. In all these examples the 1/N expansion for the TTI resums into a simple compact expression that takes a similar form to the one we recently found for the ABJM theory in [17,18]. We also conjectured a closed form expression in terms of an Airy function for the large N S 3 partition function of the ADHM theory on the squashed sphere in the presence of arbitrary real mass deformation. Our results suggest several possible generalizations and questions for future exploration some of which we discuss below.
• While we exhibited very strong numerical evidence for the validity of the TTI expressions presented in this work it is very important to establish these results through more direct analytical methods. In addition to providing insights into how to calculate the TTI for large N holographic SCFTs, such an analytic understanding will also shed light on how to extend our results to configurations with general values for the real mass parameters some of which we set to specific values in the examples treated above. A better analytic control of our TTI calculations will also allow to understand the role of any other solutions of the BAE. In essence, we have focused on the solution to the BAE equation that dominates in the large N limit and have ignored the contributions of any other possible BAE solutions, see the discussion around (2.17) above. It will be most interesting to investigate the space of solutions of the TTI BAE for the holographic SCFTs studied here and to uncover their possible role in holography and black hole physics. Since such a study of the BAE seems quite non-trivial in general, focusing on gauge groups of low ranks could be helpful in order to delineate the structure of the BAE solution space, as in the case for the SCI of the 4d N = 4 Super-Yang-Mills theory [75,76].
• To this end it is also desirable to understand the non-perturbative contributions to the TTI. Our numerical approach was sufficiently accurate to estimate the leading nonperturbative correction to the TTI but further insights will require other methods that will hopefully lead to better analytic control over the infinite series of such corrections and its possible resummation.
• Given the conjecture for the squashed sphere partition function of the ADHM and mABJM models with real mass deformation presented in this work and the analogous results for the ABJM theory in [17,18], it is clearly desirable to develop analytic or numerical methods to compute these large N observables from first principle for 3d N = 2 holographic SCFTs. Perhaps the analytical techniques studied in [77] together with the results in [30,57] can prove useful in this regard. We are also currently exploring various numerical methods to study this problem [78].
• The results for the sphere partition function, TTI, and SCI discussed here and in our previous work suggest that there are closed form expressions for the large N partition functions of holographic SCFTs on other compact Euclidean 3-manifolds. It will be desirable to systematically derive such results and find an underlying organizing principle. Our work, together with the results in [13,[79][80][81][82], should provide useful guidance in this pursuit.
• An important feature of all results for the large N partition functions discussed above is that they are naturally written in terms of a "shifted N ". This strongly suggest that there is an underlying mechanism in M-theory that makes this "shifted N " the natural large N expansion parameter. For the ABJM round sphere partition function a partial explanation of this shift in N can be attributed to a modification of the M2-brane charge quantization prescription in the presence of the well-known topological 8-derivative term in the M-theory effective action, see [83]. It is clear that this cannot be the full answer since this modified charge quantization condition does not account for the full shift in N and, as discussed above, the shift in N can depend on continuous parameters which suggest that it has a non-topological origin. It will be very interesting to further clarify this aspect of our results.
• Our focus in this work was on 3d N = 2 holographic SCFTs in the M-theory limit. It will be interesting to explore whether similar explicit closed form expressions can be found for the partition functions of other holographic SCFTs with type IIB or IIA gravitational dual descriptions. A first step in this direction could be the careful exploration of the type IIA limit of the large N observables we discussed above.
• Our results pose several open questions in the holographic context. The most important one is to reproduce the all-order 1/N expansion of the TTI, SCI, and sphere partition functions by using the M-theory path integral. This would constitute a very stringent precision test of AdS/CFT beyond the large N limit. This appears to be a daunting task and has currently been achieved only for the N 3 2 , N 1 2 , and log N terms in some of the examples discussed above. Given our incomplete understanding of M-theory, a promising avenue to extend these results further is to use the 4d gauged supergravity localization framework put forward in [12,84,85]. We hope that our results above will provide a valuable benchmark in pursuing this open problem and will ultimately lead to a precise OSV-type conjecture for asymptotically AdS gravitational solutions. Lastly, we note that if there exist other BAE solutions that contribute to the TTI, they should also be understood holographically. Such solutions should correspond to new asymptotically AdS 4 saddle points whose contribution to the M-theory path integral are exponentially suppressed compared to the backgrounds considered in Section 5.1. Recently the authors of [86] identified a family of such saddle points for the ABJM SCI in the generalized Cardy-like limit, and it would be most interesting to understand the relation between these saddle points and more general BAE solutions for the TTI exploiting the map between the SCI and the TTI used in [19].
We hope to explore some of these questions in the near future. (A.1) For all the above configurations, we compared numerical estimateŝ obtained from the LinearModelFit (2.41) with the corresponding analytic expressions given in the RHS of (2.42), namelŷ where∆ a ,ñ a are defined in (2.30) and c a , d a are defined in (2.33). To estimate the precision of our results, we checked that the error ratios Here we provide numerical data that supports the all-order 1/N expansion of the N 0,1,0 TTI given in (2.63). The list of (k, r)-configurations for which we confirmed (2.63) with r 1 = r 2 = r/2 is given as follows: Note that the choice r 1 = r 2 = r/2 is crucial to obtain the all-order results (2.63) since the subleading contribution depends on both parameters (r 1 , r 2 ) in general unlike the N 3 2 leading order (2.62) completely determined by their sum r = r 1 + r 2 . For the above listed (k, r)-configurations, we compared numerical estimateŝ obtained from the LinearModelFit for the N 0,1,0 TTI similar to the one for the ADHM TTI (2.41), namely 3) with the corresponding analytic expressions in (2.63), We checked that the error ratios Here we also provide some numerical errors involved in the estimate for the leading non-perturbative behavior (2.65) following the definition (2.49).
where∆ a ,ñ a are defined in (2.30) and a I , b I are defined in (2.72). To estimate the precision, we checked that the error ratios are small enough for all the configurations listed in (C.1,C.2,C.3). Below we provide some examples to show how small the error ratios (C.6) are, and also present numerical estimates for the constant termf

topologically twisted index
We begin with a review of the analytic derivation of the large N result for the TTI (2.87). Along the way, we correct a few minor errors in [27,28].
To derive the BA formula for the S 1 × S 2 Q 1,1,1 TTI, we first rewrite the matrix model (2.80) as x j y qn 1 −x j y qn in terms of a large integer cut-off M (m i ≤ M −1,m j ≥ 1−M ) and using the BA operators .

(D.2)
The sign ambiguities σ i ,σ j in the BAE are defined as Anticipating the following results, we will show below that the analytic large N BAE solution imposes a range on the values of (x i ,x j ) and y a such that the ambiguities take values σ i =σ j = (−1) N , (D.4) on the BAE solution. Thus, it is consistent to use this choice in our analysis. Note that the same remark holds in the N 0,1,0 theory, see (2.58). As discussed in the ADHM case, the residues picked up by the integrals in (D.2) lead to the BA formula (2.85) for the Q 1,1,1 TTI, where the BAE are given in (2.86). One can check that these BAE can be obtained by differentiating the following Bethe potential with respect to u i andũ j : Li 2 (e i(ũ j −u i +π∆a) ) − a=1,2 Li 2 (e i(ũ j −u i −π∆a) ) (D.5) Li 2 (e i(−u i +π∆q n ) ) − Li 2 (e i(−ũ i −π∆q n ) ) + π 2 u i ∆q n + π 2ũ i (∆ qn − 2) .
We note that this Bethe potential is slightly different from the one written previously in the literature [27][28][29]. Our result is equivalent to the previous ones in the large N limit, but we stress that (D.5) is the exact Bethe potential consistent with the BA operators (D.2) at finite N . This slight modification leads to a different set of integers (D.9) chosen to find a BAE solution instead of (n i ,ñ j ) = (1 − i, j − N ). The large N limit of the solutions to the BAE (2.86) has been constructed following the same procedure as described in Section 2.2.2. This involves the construction of a large N ansatz and its continuum version in the large N limit, see [28] for details. Here we only summarize the resulting large N BAE solution for the superconformal ∆-configuration in (2.83): where δv(t) =ṽ(t) − v(t) and the end points for the eigenvalue distribution are given by The parameter µ is determined by the normalization (2.22) and reads µ = N f 3 π . (D.8) To find the solution (D.6), the integers n i ,ñ j in the BAE (2.86) have been chosen as and we have also assumed These ranges lead to (D.4). One then obtains the large N limit of the Q 1,1,1 TTI by substituting (D.6) into the large N limit of the BA formula. The result is given in (2.87).

E A compendium of 7d Sasaki-Einstein manifolds
In this appendix, we collect some known facts regarding the geometry and volumes of various Sasaki-Einstein manifolds. All 6d Kähler-Einstein metrics with line element ds 2 6 have been normalized such that R (6) µν = 8 g (6) µν , (E. 1) in what follows. See [87] for an early reference on Freund-Rubin type AdS 4 solutions in 11d supergravity with internal Sasaki-Einstein manifolds. (E.