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REVIEW 2 major objections 4 minor 109 references

Constants in Sequences of M2-brane Partition Functions

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read M2-brane partition constants take exact closed form

desk verdict Highly plausible closed-form constants in ABJM/ADHM 1/N expansions, reconstructed from high-precision numerics rather than derived; the paper is honest about that, and it deserves peer review. read the letter →

arxiv 2608.04204 v1 pith:LMRSR7OE submitted 2026-08-04 hep-th

Junho Hong This is my paper · ORCID
classification hep-th
keywords M2-branesABJMtheoryADHMtopologicallytwistedindexBethepotentialconstantmapfunction1/NexpansionAiry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the previously numerical, N-independent constant terms in the all-order 1/N expansions of the ABJM Bethe potential and topologically twisted index, and of the ADHM Bethe potential, are finite linear combinations of one special function, A(k), the constant map function already known from the round three-sphere partition function. The constants are reconstructed from high-precision Bethe-Ansatz numerics: after subtracting known N-dependent terms, the residual tails are fitted, shown to match a Bernoulli-number pattern, and resummed back into integral representations and A-combinations. If the identification is exact, the missing constant sector of these holographic partition functions is no longer numerical, and through factorization relations the squashed-sphere Airy constant is fixed in closed form through the first two leading orders of large squashing. The reader should care because this removes the last undetermined piece in a web of all-order localization results for M2-brane theories and their gravity duals.

What carries the argument

The central object is the constant map function $A(k)$, defined by the integral in Eq. (12), which resums the all-genus constant-map contributions of the dual topological string. Its large-$k$ expansion has coefficients built from products of Bernoulli numbers, $|B_{2n}B_{2n+2}|$, and this signature is what lets the author recognize the same structure in the numerical tails of the Bethe potential and the index. Running that expansion backward resums the guessed general-order series into integral representations, and elementary integral identities convert those integrals into finite linear combinations of A-functions. The factorization relations of [32] are the second load-bearing mechanism, carrying the Bethe and index constants into the squashed-sphere Airy constant.

What would settle it

Compute the ABJM twisted index constant at an untested integer level, for example k=6, from Bethe-Ansatz data at fixed 't Hooft coupling with N up to about 500; the closed form predicts specific exact rational tail coefficients including f6 and f7. If subtracting the closed form leaves a stable power-law residual above the estimated non-perturbative scale of roughly $10^{-21}$, the identification is only numerically accurate, not exact.

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Extended reading notes

Core claim

At the superconformal point, the ABJM Bethe potential constant is given by $\hat g_0(k,\Delta_{\rm sc}) = A(k)-A(k/2)+\tfrac{k}{2}A(4/k)-\tfrac{k}{4}A(8/k)-\frac{\zeta(3)}{8\pi^2 k^2}$, and the ABJM topologically twisted index constant is another finite A-combination, linearly related to $\hat g_0$ with an explicit $\log k$ term. The ADHM Bethe potential constant satisfies the mirror-symmetry-preserving relation $\hat g_0^{\rm ADHM}(N_f)=\tfrac12\hat g_0(N_f)+\tfrac14\hat g_0(2N_f)+\cdots$, which at $N_f=1$ equals the ABJM value at $k=1$; its leading coefficient requires the non-elementary value $A(1/2)$. The paper reaches these formulas by fitting residual data at fixed 't Hooft coupling, recognizing exact rational tail coefficients, resumming the large-$k$ series into integrals, and identifying the pure constant within the transcendental basis $\{\zeta'(-1),\log 2,\log 4\pi\}$. Checks at $k=1,2,4$ reproduce known 20-digit values, and the remaining residuals sit at the expected non-perturbative scale.

Load-bearing premise

The argument stands on the premise that the finite fitting basis and the guessed general-order Bernoulli pattern recover the exact analytic function rather than an extremely accurate asymptotic approximation; the paper explicitly leaves a first-principles derivation for future work.

Editorial extensions

If this is right

  • The closed forms for the ABJM Bethe potential and twisted index constants replace previous numerical fits and give exact values at integer levels such as $k=1,2,4$.
  • Through the Cardy-like relation (7), the new constants transfer directly to the first two orders of the Cardy expansion of the superconformal index.
  • Through factorization, the Airy constant of the squashed three-sphere is fixed in closed form through the two leading orders in large squashing, reproducing the planar $k^2$ coefficient and the universal $\log k$ coefficient $-1/6$.
  • The ADHM and ABJM constants obey the 3d mirror symmetry constraint $\hat g_0^{\rm ADHM}(1)=\hat g_0(1)$, and the ADHM result correctly involves $A(1/2)$ rather than only integer arguments.
  • The web of exact anchors yields the functional identity $A(x)+A(2-x)=-(A(4/x)+A(2-4/x))$, verified numerically to 40 digits and derived on an infinite discrete set of squashing values.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same A-combination pattern is a natural template for the other M2-brane SCFTs listed in the paper, and repeating the high-precision fit for those theories is a direct, concrete test of whether the structure is universal.
  • If the closed forms are exact, gravity-side derivations that currently reproduce only the Airy data now have a precise target: they must generate A-combinations with the direct and inverted arguments $\{K,K/2,2K;2/K,4/K,8/K\}$, giving a sharper constraint on quantum M-theory localization.
  • The functional identity $\Phi(x)=-\Phi(4/x)$ appears to be a self-contained property of $A$; a direct proof from the integral representation would close the gap the paper leaves open and may explain why those particular arguments appear.
  • The partial result at generic flavor chemical potentials suggests, but does not prove, that the full flavor-dependent constant is again a finite A-combination with arguments shifted by the $\Delta_a$; computing the next order in $\Delta$ would settle the conjecture.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper determines closed-form expressions for the N-independent constant terms in the all-order 1/N expansions of the ABJM Bethe potential, the ABJM topologically twisted index at the superconformal point, and the ADHM Bethe potential. The central results, e.g., Eqs. (29), (32), (38), (42), and (53), express these constants as finite linear combinations of the constant map function A(k). The method is numerical: high-precision Bethe-Ansatz solutions at fixed 't Hooft coupling, subtraction of known N-dependent terms, a LinearModelFit of the residual, pattern recognition of the tail coefficients (with AI assistance), and backward resummation into integral representations. The paper also uses factorization relations to determine the first two leading orders of the large-squashing expansion of the Airy constant, and it conjectures a new functional identity for A(k). Independent checks include 20-digit matches to previously published values, mirror-symmetry consistency at N_f=1, and predicted higher-order coefficients.

Significance. If the closed forms are correct, they fill a long-standing gap in the exact large-N description of M2-brane partition functions: the N-independent constants were previously available only numerically. The results are concrete, falsifiable, and supply exact special values and predicted coefficients that can be tested further. The paper is commendably transparent about its numerical provenance, and the use of independent anchors (published values, mirror symmetry, exact A-values at integer arguments) strengthens confidence. However, as the paper itself states, no first-principles derivation is provided, so the exactness of the closed forms is not established.

major comments (2)
  1. [III.B, IV.B, Appendix A] The all-order exactness of the closed forms (29), (32), (38), (42), (53), and (56) is not established. The general-order Bernoulli-number formulas are inferred from a finite set of fitted rationals, and the backward resummation in Appendix A reproduces the guessed series by construction, so it provides no independent evidence for the all-order claim. Numerical agreement at the 10^-17 to 10^-20 level cannot exclude an exponentially small remainder such as O(e^{-c k}) with large c, nor a term that vanishes beyond the quoted precision but is analytically nonzero; both would alter the claimed exact expressions. The paper explicitly leaves a first-principles derivation for future work (Section III.B.1 and the analogous paragraph in Section IV.B). I therefore request that the abstract and the text moderate the claim from 'determine in closed form' to 'conjecture with strong numerical evidence', or that a derivation be supplied.
  2. [III.B.2, Eq. (39)] The constant f0 = -8ζ'(-1) - (5/2) log 2 - (2/3) log 4π is presented as part of the closed-form result (38), but it is not derived. The text explains that the fitted value is 'identified' within the basis {ζ'(-1), log 2, log 4π}, motivated by analogy with the -log 2/6 identification in (29). The claimed independent confirmation via Eq. (43) uses the A-function representation (42), whose derivation already incorporates the identification of f0; while the match with published values is a nontrivial consistency check, it does not prove that the fitted constant equals this combination. A closed-form claim for f0 should follow from the integral representation rather than from a numerical fit and a guess of the transcendental basis.
minor comments (4)
  1. [Eq. (12)] The equality signs connecting the integral representation, the large-k asymptotic expansion, and the small-k expansion are misleading: the second and third expressions are asymptotic series valid in different regimes and are not equal as convergent series. Using '~' or '=' with an explicit qualifier would be clearer.
  2. [Section III.B.1] The pattern-recognition step is described as being performed with the help of an AI assistant, but no details of the prompts, the candidate families considered, or the selection criteria are provided. Since this step is the basis for the general-order coefficient formulas, some additional documentation would improve reproducibility.
  3. [Section V, Eq. (75)] The functional identity Φ(x) = -Φ(4/x) is a new mathematical statement, but it is only verified numerically (to 40 digits) and is not proved. The text should label it as a conjecture rather than a 'functional identity', unless a proof is included.
  4. [Abstract] The phrase 'verify them down to the level of non-perturbative corrections' is slightly overstated: the residuals are at the level of 10^-19 to 10^-20, while the leading non-perturbative correction is estimated as ~10^-21 at the chosen 't Hooft coupling; the verification is close to but not strictly at that level.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the constants are transparently reconstructed from numerics and checked against independent values; the guessed all-order formula is explicitly left unproven, not smuggled in as a derivation.

full rationale

The derivation chain in this paper is a numerical reconstruction rather than a first-principles derivation, and the paper says so explicitly ("We leave a first-principles derivation of the closed-form (29) for future work"). The closed forms (29), (38), and (53) are obtained by subtracting known N-dependent terms from finite-N Bethe-Ansatz data, fitting the residual with LinearModelFit over {k^2, log k, 1, k^{-2n}}, guessing a Bernoulli-number general-order formula from the first few fitted rationals, and then resumming the guessed series backward into integral representations. No equation is set equal to its target by construction: the guessed all-order formula could have failed the higher-order fits and the pointwise comparisons, and the "predicted" higher-order coefficients are in-sample consistency checks rather than out-of-sample predictions. The A-function representations (32), (42), and (56) are derived from the integral representations by explicit integral identities in Appendix B; the constant map function A(k) is defined independently in (12), so these are not self-definitional. There are genuine independent anchors: the 20-digit values at k=1,2,4 from Appendix C.2 of [30] and Appendix C of [13], the mirror-symmetry constraint at N_f=1, the exact special values (43), and consistency with the planar and universal log k results. The self-citations to [13,14,15,30,32] carry substantive prior results; the factorization application in Section V does rely on [32], but the primary constants of Sections III and IV do not depend on that citation. The residual epistemic weakness is that exactness of the all-order coefficient formulas is conjectural, not circular. Score 2 reflects the minor in-sample "prediction" language and load-bearing self-citation in the secondary Airy-constant application, neither of which amounts to a circular derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new free parameters are introduced: the constants are fixed by data or by the prior function A. The load-bearing inputs are the prior all-order expansion formulas, the numerical vacuum selection, the pattern-recognition assumption, the factorization relations of [32], and the numerically verified functional identity (75).

assumptions (5)
  • domain assumption The all-order 1/N expansions (19), (21), (50), and (52) hold, with N-independent constants g0 and f0 well-defined as stated in [13,14,30].
    These formulas define the quantities whose constants are determined; they are taken from related work by the same group and are not re-derived here. Section III.A and IV.A.
  • domain assumption The numerical BAE solution obtained by Newton iteration from the leading large-N solution is the exact solution of the relevant Bethe vacuum, and other vacua do not contribute to the extracted constants.
    The extraction assumes the standard large-N vacuum used in [3,13,30]; the paper does not prove vacuum selection. Section III.B.
  • ad hoc to paper The pattern-recognition step, based on Bernoulli-number structure and resummation of fitted series, recovers the exact analytic function rather than only an asymptotic approximation.
    The closed forms are obtained by fitting coefficients and running the large-k expansion backward; the exactness of this reconstruction is not proven. Sections III.B and Appendix A.
  • domain assumption The factorization relations of [32] correctly encode the N-independent constants of the index and TTI into the squashed sphere Airy constant.
    Used in Section V to translate the new constants into the Airy constant; [32] is cited, not re-derived.
  • ad hoc to paper The new functional identity Phi(x) = -Phi(4/x) for Phi(x) = A(x) + A(2-x) holds; it is confirmed numerically to 40 digits but not proved from the integral representation.
    Derived from the consistency of the anchor web (74)-(75) and numerically verified, but not proven. Section V.B.

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Cite this review

Pith. "Pith review of Constants in Sequences of M2-brane Partition Functions." pith.science (2026). https://pith.science/paper/LMRSR7OE

@misc{pith2026260804204,
  author       = {Pith},
  title        = {Pith review of: Constants in Sequences of M2-brane Partition Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMRSR7OE}},
  note         = {Machine review of arXiv:2608.04204}
}
abstract

We determine in closed form the $N$-independent constant terms in the all-order $1/N$ expansions of the topologically twisted index and of the associated Bethe potential for the ABJM theory, as well as the Bethe potential constant of the ADHM theory. We reconstruct these constants analytically from high-precision Bethe-Ansatz numerics, verify them down to the level of non-perturbative corrections, and find them to be closely related to the constant map function $A$ governing the round three-sphere partition function. The resulting expressions for the ADHM and ABJM constants pass the non-trivial test dictated by 3d mirror symmetry. Via recently established factorization relations, they also determine in closed form the $N$-independent constant contribution to the squashed three-sphere partition function, through the first two leading orders in its large-squashing expansion. These constants supply precisely the piece left undetermined in the recent exact results for 3d supersymmetric partition functions to all orders in the $1/N$ expansion, and thereby mark an important step toward completing them, both in field theory and in the dual quantum gravity description.

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Works this paper leans on

109 extracted references · 1 canonical work pages

  1. [13]

    Benini and E

    F. Benini and E. Milan, Black Holes in 4DN=4 Super- Yang-Mills Field Theory, Phys. Rev. X10, 021037 (2020), arXiv:1812.09613 [hep-th]

    arXiv 2020

  2. [30]

    Closset, H

    C. Closset, H. Kim, and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP03(2017), 074, arXiv:1701.03171 [hep-th]

    arXiv 2017

  3. [56]

    Hatsuda, ABJM on ellipsoid and topological strings, JHEP07(2016), 026, arXiv:1601.02728 [hep-th]

    Y. Hatsuda, ABJM on ellipsoid and topological strings, JHEP07(2016), 026, arXiv:1601.02728 [hep-th]

    arXiv 2016

  4. [80]

    F. F. Gautason, V. G. M. Puletti, and J. van Muiden, Quantized strings and instantons in holography, JHEP 08(2023), 218, arXiv:2304.12340 [hep-th]

    arXiv 2023

  5. [1]

    The expression (29) was obtained in the following steps

    The Bethe potential constant Superconformal point.Our main result for the Bethe potential constant at the superconformal point is the closed-form expression ˆg0(k,∆ sc) =− ζ(3) 32π2k2− log 2 6 (29) + k π2 Z ∞ 0 dxlog(1−e −kx) log coshx =− ζ(3) 32π2k2− log 2 6 + ∞X n=1 2π k 2n (−1)n4n(4n−1)|B 2nB2n+2| n(2n+ 2)! , where the second equality is the asymptotic...

  6. [2]

    The derivation logic parallels the Bethe potential case

    The topologically twisted index constant Superconformal point.Our main result for the TTI constant at the superconformal point is the closed-form expression ˆf0(k,∆ sc,n sc) =− 3ζ(3) 8π2 k2 + 1 6 logk+f 0 + k π2 Z ∞ 0 dxlog 1−e −kx 12xcoth 2x−2xcothx−4 + 4 log sinhx x =− 3ζ(3) 8π2 k2 + 1 6 logk+f 0 + ∞X n=1 2π k 2n (−1)n4n+1 3n·4 n−n+ 1 |B2nB2n+2| n(2n+ 2...

  7. [3]

    The Bethe potential constant Our main result for the ADHM Bethe potential con- stant is the closed-form expression ˆgADHM 0 (Nf,e∆sc) =a 2N2 f − log 2 8 + Nf π2 Z ∞ 0 dxlog 1−e −Nfx 1 2 log coshx+ 1 4 log coshx 2 =a 2N2 f − log 2 8 + ∞X n=1 2π Nf 2n (−1)n(4n−1)(2·4 n + 1)|B2nB2n+2| 4n(2n+ 2)! (53) with the leading order coefficient a2 = 3ζ(3) 32π2 + 7 32 ...

  8. [4]

    The topologically twisted index constant Applying the same analysis to the ADHM TTI data and extracting theN-independent constant through (52), we find that ˆfADHM 0 at the superconformal configuration (49) organizes as ˆfADHM 0 (Nf,e∆sc,ensc) =k 2N2 f + 1 3 logN f +k 0 + ∞X n=1 2π Nf 2n tn,(57) with (quoted from theλ= 30 data) k2 =−0.159517636940496, k0 ...

Show all 109 references
  1. [5]

    Cabo-Bizet, D

    A. Cabo-Bizet, D. Cassani, D. Martelli, and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes, JHEP10(2019), 062, arXiv:1810.11442 [hep-th]

    2019 arXiv

  2. [6]

    ABJM constants The large-kexpansion in (29) follows from the chain k π2 Z ∞ 0 dxL(x) log coshx(A3) =− k π2 ∞X m=1 1 m Z ∞ 0 dxe−mkx log coshx =− 1 π2 ∞X n=1 4n(4n−1)B 2n 2n ζ(2n+ 2)k −2n = ∞X n=1 2π k 2n (−1)n4n(4n−1)|B 2nB2n+2| n(2n+ 2)! , where we have usedB 2n = (−1)n+1|B2n...

  3. [7]

    ADHM constant By (A2), the two-level ADHM kernel of (53) carries the Taylor coefficient (4n−1)(2·4n+1)B2n 8n(2n)! at orderx 2n, so the chain (A3) applied verbatim gives Nf π2 Z ∞ 0 dxlog 1−e −Nfx (A5) × 1 2 log coshx+ 1 4 log coshx 2 = ∞X n=1 2π Nf 2n (−1)n(4n−1)(2·4 n + 1)|B ...

  4. [8]

    Pestun, Localization of gauge theory on a four- sphere and supersymmetric Wilson loops, Commun

    V. Pestun, Localization of gauge theory on a four- sphere and supersymmetric Wilson loops, Commun. Math. Phys.313, 71 (2012), arXiv:0712.2824 [hep-th]

    2012 arXiv

  5. [9]

    Pestunet al., Localization techniques in quan- tum field theories, J

    V. Pestunet al., Localization techniques in quan- tum field theories, J. Phys. A50, 440301 (2017), arXiv:1608.02952 [hep-th]

    2017 arXiv

  6. [10]

    Benini, K

    F. Benini, K. Hristov, and A. Zaffaroni, Black hole microstates in AdS 4 from supersymmetric localization, JHEP05(2016), 054, arXiv:1511.04085 [hep-th]

    2016 arXiv

  7. [11]

    S. Choi, J. Kim, S. Kim, and J. Nahmgoong, Large AdS black holes from QFT (2018), arXiv:1810.12067 [hep- th]

    2018 arXiv

  8. [12]

    Bobev, J

    N. Bobev, J. Hong, and V. Reys, Large N Partition Functions, Holography, and Black Holes, Phys. Rev. Lett.129, 041602 (2022), arXiv:2203.14981 [hep-th]

    2022 arXiv

  9. [14]

    Kapustin, B

    A. Kapustin, B. Willett, and I. Yaakov, Exact Re- sults for Wilson Loops in Superconformal Chern- Simons Theories with Matter, JHEP03(2010), 089, arXiv:0909.4559 [hep-th]

    2010 arXiv

  10. [15]

    — gives A= 1 4 h A 2 b2 +A 2− 2 b2 +A(3−b 2) +A(b 2−1) i = 1 4 h A(2b2−2)−A(2b 2) +A(3−b 2) +A(b 2−1) i ,(74) whose equality, upon using the odd parityA(−k) = −A(k), is precisely the functional identity Φ(x) =−Φ 4 x ,Φ(x)≡A(x) +A(2−x),(75) evaluated atx= 2 b2 . Since (2.33) of...

    2024

  11. [16]

    D. L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP05(2012), 159, arXiv:1012.3210 [hep-th]

    2012 arXiv

  12. [17]

    N. Hama, K. Hosomichi, and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP03(2011), 127, arXiv:1012.3512 [hep-th]

    2011 arXiv

  13. [18]

    N. Hama, K. Hosomichi, and S. Lee, SUSY Gauge The- ories on Squashed Three-Spheres, JHEP05(2011), 014, arXiv:1102.4716 [hep-th]

    2011 arXiv

  14. [19]

    Imamura and D

    Y. Imamura and D. Yokoyama, N=2 supersymmetric theories on squashed three-sphere, Phys. Rev. D85, 025015 (2012), arXiv:1109.4734 [hep-th]

    2012 arXiv

  15. [20]

    Bobev, J

    N. Bobev, J. Hong, and V. Reys, Large N partition functions of the ABJM theory, JHEP02(2023), 020, arXiv:2210.09318 [hep-th]

    2023 arXiv

  16. [21]

    Bobev, J

    N. Bobev, J. Hong, and V. Reys, Large N partition func- tions of 3d holographic SCFTs, JHEP08(2023), 119, arXiv:2304.01734 [hep-th]

    2023 arXiv

  17. [22]

    Bobev, P.-J

    N. Bobev, P.-J. De Smet, J. Hong, V. Reys, and X. Zhang, An Airy tale at large N, JHEP07(2025), 123, arXiv:2502.04606 [hep-th]

    2025 arXiv

  18. [23]

    Hristov, 4dN= 2 supergravity observables from Nekrasov-like partition functions, JHEP02(2022), 079, arXiv:2111.06903 [hep-th]

    K. Hristov, 4dN= 2 supergravity observables from Nekrasov-like partition functions, JHEP02(2022), 079, arXiv:2111.06903 [hep-th]

    2022 arXiv

  19. [24]

    Hristov, ABJM at finite N via 4d supergravity, JHEP 10(2022), 190, arXiv:2204.02992 [hep-th]

    K. Hristov, ABJM at finite N via 4d supergravity, JHEP 10(2022), 190, arXiv:2204.02992 [hep-th]

    2022 arXiv

  20. [25]

    H. Fuji, S. Hirano, and S. Moriyama, Summing Up All Genus Free Energy of ABJM Matrix Model, JHEP08 (2011), 001, arXiv:1106.4631 [hep-th]

    2011 arXiv

  21. [26]

    Marino and P

    M. Marino and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech.1203, P03001 (2012), arXiv:1110.4066 [hep-th]

    2012 arXiv

  22. [27]

    Benini and A

    F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP07(2015), 127, arXiv:1504.03698 [hep-th]

    2015 arXiv

  23. [28]

    Benini and A

    F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96, 13 (2017), arXiv:1605.06120 [hep-th]

    2017 arXiv

  24. [29]

    Closset and H

    C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP08(2016), 059, arXiv:1605.06531 [hep-th]

    2016 arXiv

  25. [31]

    Closset, H

    C. Closset, H. Kim, and B. Willett, Seifert fibering op- erators in 3dN= 2 theories, JHEP11(2018), 004, arXiv:1807.02328 [hep-th]

    2018 arXiv

  26. [32]

    Closset and H

    C. Closset and H. Kim, Three-dimensionalN= 2 su- persymmetric gauge theories and partition functions on Seifert manifolds: A review, Int. J. Mod. Phys. A34, 1930011 (2019), arXiv:1908.08875 [hep-th]

    2019 arXiv

  27. [33]

    Hong, Perturbatively exact supersymmetric parti- tion functions of ABJM theory on Seifert manifolds and holography, JHEP01(2025), 194, arXiv:2411.09006 [hep-th]

    J. Hong, Perturbatively exact supersymmetric parti- tion functions of ABJM theory on Seifert manifolds and holography, JHEP01(2025), 194, arXiv:2411.09006 [hep-th]

    2025 arXiv

  28. [34]

    Bhattacharya, S

    J. Bhattacharya, S. Bhattacharyya, S. Minwalla, and S. Raju, Indices for Superconformal Field Theories in 3,5 and 6 Dimensions, JHEP02(2008), 064, arXiv:0801.1435 [hep-th]

    2008 arXiv

  29. [35]

    Kim, The Complete superconformal index for N=6 Chern-Simons theory, Nucl

    S. Kim, The Complete superconformal index for N=6 Chern-Simons theory, Nucl. Phys. B821, 241 (2009), [Erratum: Nucl.Phys.B 864, 884 (2012)], arXiv:0903.4172 [hep-th]

    2009 arXiv

  30. [36]

    Imamura and S

    Y. Imamura and S. Yokoyama, Index for three di- mensional superconformal field theories with gen- eral R-charge assignments, JHEP04(2011), 007, arXiv:1101.0557 [hep-th]

    2011 arXiv

  31. [37]

    Bobev, S

    N. Bobev, S. Choi, J. Hong, and V. Reys, Large N su- perconformal indices for 3d holographic SCFTs, JHEP 02(2023), 027, arXiv:2210.15326 [hep-th]

    2023 arXiv

  32. [38]

    Bobev, S

    N. Bobev, S. Choi, J. Hong, and V. Reys, Superconfor- mal indices of 3dN= 2 SCFTs and holography, JHEP 10(2024), 121, arXiv:2407.13177 [hep-th]

    2024 arXiv

  33. [39]

    Bobev, S

    N. Bobev, S. Choi, J. Hong, and V. Reys, Towards OSV in AdS (2026), arXiv:2606.23893 [hep-th]

    2026 arXiv

  34. [40]

    Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP04(2012), 120, arXiv:1111.6905 [hep-th]

    S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP04(2012), 120, arXiv:1111.6905 [hep-th]

    2012 arXiv

  35. [41]

    C. Beem, T. Dimofte, and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP12(2014), 177, arXiv:1211.1986 [hep-th]

    2014 arXiv

  36. [42]

    Hwang, H.-C

    C. Hwang, H.-C. Kim, and J. Park, Factorization of 17 the 3d superconformal index, JHEP08(2014), 018, arXiv:1211.6023 [hep-th]

    2014 arXiv

  37. [43]

    Hristov, Equivariant localization and gluing rules in 4d N = 2 higher derivative supergravity (2024) arXiv:2406.18648 [hep-th]

    K. Hristov, Equivariant localization and gluing rules in 4d N = 2 higher derivative supergravity (2024) arXiv:2406.18648 [hep-th]

    2024 arXiv

  38. [44]

    Cassia and K

    L. Cassia and K. Hristov, Constant maps in equivariant topological strings and geometric modeling of fluxes, J. Phys. A58, 495201 (2025), arXiv:2502.20444 [hep-th]

    2025

  39. [45]

    Cassia and K

    L. Cassia and K. Hristov, M2-brane partition functions and HD supergravity from equivariant volumes, JHEP 03(2026), 100, arXiv:2508.21619 [hep-th]

    2026

  40. [46]

    Benetti Genolini, J

    P. Benetti Genolini, J. P. Gauntlett, and J. Sparks, Equivariant Localization in Supergravity, Phys. Rev. Lett.131, 121602 (2023), arXiv:2306.03868 [hep-th]

    2023 arXiv

  41. [47]

    Benetti Genolini, J

    P. Benetti Genolini, J. P. Gauntlett, Y. Jiao, A. L¨ uscher, and J. Sparks, Localization of the Free Energy in Supergravity, Phys. Rev. Lett.133, 141601 (2024), arXiv:2407.02554 [hep-th]

    2024 arXiv

  42. [48]

    Benetti Genolini, F

    P. Benetti Genolini, F. Gaar, J. P. Gauntlett, and J. Sparks, Equivariant localization for higher deriva- tive supergravity, in14th School of Physics Roberto A. Salmeron and 2nd Workshop on Quantum and Statisti- cal Physics(2026) arXiv:2604.08656 [hep-th]

    2026 arXiv

  43. [49]

    Benetti Genolini, F

    P. Benetti Genolini, F. Gaar, J. P. Gauntlett, J. Park, and J. Sparks, Airy functions from quantum M-theory (2026), arXiv:2607.07255 [hep-th]

    2026 arXiv

  44. [50]

    Marino and P

    M. Marino and P. Putrov, Exact Results in ABJM The- ory from Topological Strings, JHEP06(2010), 011, arXiv:0912.3074 [hep-th]

    2010 arXiv

  45. [51]

    Drukker, M

    N. Drukker, M. Marino, and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306, 511 (2011), arXiv:1007.3837 [hep-th]

    2011 arXiv

  46. [52]

    Hanada, M

    M. Hanada, M. Honda, Y. Honma, J. Nishimura, S. Shiba, and Y. Yoshida, Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling con- stant, JHEP05(2012), 121, arXiv:1202.5300 [hep-th]

    2012 arXiv

  47. [53]

    Hatsuda and K

    Y. Hatsuda and K. Okuyama, Probing non- perturbative effects in M-theory, JHEP10(2014), 158, arXiv:1407.3786 [hep-th]

    2014 arXiv

  48. [54]

    Grassi, Y

    A. Grassi, Y. Hatsuda, and M. Marino, Topologi- cal Strings from Quantum Mechanics, Annales Henri Poincare17, 3177 (2016), arXiv:1410.3382 [hep-th]

    2016 arXiv

  49. [55]

    Nosaka, Instanton effects in ABJM theory with general R-charge assignments, JHEP03(2016), 059, arXiv:1512.02862 [hep-th]

    T. Nosaka, Instanton effects in ABJM theory with general R-charge assignments, JHEP03(2016), 059, arXiv:1512.02862 [hep-th]

    2016 arXiv

  50. [57]

    N. Kubo, T. Nosaka, and Y. Pang, Exact large N ex- pansion of mass deformed ABJM theory on squashed sphere, JHEP02(2025), 106, arXiv:2411.07334 [hep- th]

    2025 arXiv

  51. [58]

    N. Kubo, T. Nosaka, and Y. Pang, Exact large N ex- pansion of N=4 circular quiver Chern-Simons theories, Phys. Rev. D112, 046023 (2025), arXiv:2504.04402 [hep-th]

    2025 arXiv

  52. [59]

    Bobev, F

    N. Bobev, F. F. Gautason, and J. van Muiden, Holographic Tests of theµEnsemble (2026), arXiv:2607.06493 [hep-th]

    2026 arXiv

  53. [60]

    Aganagic, A

    M. Aganagic, A. Klemm, M. Marino, and C. Vafa, Ma- trix model as a mirror of Chern-Simons theory, JHEP 02, 010, arXiv:hep-th/0211098

    arXiv

  54. [61]

    Drukker, M

    N. Drukker, M. Marino, and P. Putrov, Nonper- turbative aspects of ABJM theory, JHEP11, 141, arXiv:1103.4844 [hep-th]

    arXiv

  55. [62]

    Hatsuda, S

    Y. Hatsuda, S. Moriyama, and K. Okuyama, Instan- ton Effects in ABJM Theory from Fermi Gas Approach, JHEP01(2013), 158, arXiv:1211.1251 [hep-th]

    2013 arXiv

  56. [63]

    S. M. Hosseini, To appear (2026)

    2026

  57. [64]

    D. Z. Freedman and S. S. Pufu, The holography ofF- maximization, JHEP03(2014), 135, arXiv:1302.7310 [hep-th]

    2014 arXiv

  58. [65]

    S. M. Hosseini and A. Zaffaroni, LargeNmatrix mod- els for 3dN= 2 theories: twisted index, free energy and black holes, JHEP08(2016), 064, arXiv:1604.03122 [hep-th]

    2016 arXiv

  59. [66]

    L. A. Pando Zayas and Y. Xin, Universal logarithmic behavior in microstate counting and the dual one-loop entropy ofAdS 4 black holes, Phys. Rev. D103, 026003 (2021), arXiv:2008.03239 [hep-th]

    2021 arXiv

  60. [67]

    Bhattacharya and S

    J. Bhattacharya and S. Minwalla, Superconformal In- dices for N = 6 Chern Simons Theories, JHEP01 (2009), 014, arXiv:0806.3251 [hep-th]

    2009 arXiv

  61. [68]

    Krattenthaler, V

    C. Krattenthaler, V. P. Spiridonov, and G. S. Var- tanov, Superconformal indices of three-dimensional the- ories related by mirror symmetry, JHEP06(2011), 008, arXiv:1103.4075 [hep-th]

    2011 arXiv

  62. [69]

    Kapustin and B

    A. Kapustin and B. Willett, Generalized Superconfor- mal Index for Three Dimensional Field Theories (2011), arXiv:1106.2484 [hep-th]

    2011 arXiv

  63. [70]

    Aharony, S

    O. Aharony, S. S. Razamat, N. Seiberg, and B. Willett, 3d dualities from 4d dualities, JHEP07(2013), 149, arXiv:1305.3924 [hep-th]

    2013 arXiv

  64. [71]

    S. Choi, C. Hwang, and S. Kim, Quantum vortices, M2-branes and black holes, JHEP09(2024), 096, arXiv:1908.02470 [hep-th]

    2024 arXiv

  65. [72]

    Choi and C

    S. Choi and C. Hwang, Universal 3d Cardy Block and Black Hole Entropy, JHEP03(2020), 068, arXiv:1911.01448 [hep-th]

    2020 arXiv

  66. [73]

    Nian and L

    J. Nian and L. A. Pando Zayas, Microscopic en- tropy of rotating electrically charged AdS 4 black holes from field theory localization, JHEP03(2020), 081, arXiv:1909.07943 [hep-th]

    2020 arXiv

  67. [74]

    Kapustin, B

    A. Kapustin, B. Willett, and I. Yaakov, Nonperturba- tive Tests of Three-Dimensional Dualities, JHEP10 (2010), 013, arXiv:1003.5694 [hep-th]

    2010 arXiv

  68. [75]

    Hatsuda and T

    Y. Hatsuda and T. Okazaki, Fermi-gas correlators of ADHM theory and triality symmetry, SciPost Phys.12, 005 (2022), arXiv:2107.01924 [hep-th]

    2022 arXiv

  69. [76]

    S. M. Chester, S. S. Pufu, Y. Wang, and X. Yin, Boot- strapping M-theory orbifolds, JHEP06(2024), 001, arXiv:2312.13112 [hep-th]

    2024

  70. [77]

    Bershadsky, S

    M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys.165, 311 (1994), arXiv:hep-th/9309140

    1994 arXiv

  71. [78]

    Aharony, O

    O. Aharony, O. Bergman, D. L. Jafferis, and J. Mal- dacena, N=6 superconformal Chern-Simons-matter the- ories, M2-branes and their gravity duals, JHEP10 (2008), 091, arXiv:0806.1218 [hep-th]

    2008 arXiv

  72. [79]

    Cagnazzo, D

    A. Cagnazzo, D. Sorokin, and L. Wulff, String in- stanton in AdS(4) x CP**3, JHEP05(2010), 009, arXiv:0911.5228 [hep-th]

    2010 arXiv

  73. [81]

    Beccaria, S

    M. Beccaria, S. Giombi, and A. A. Tseytlin, Instan- ton contributions to the ABJM free energy from quan- tum M2 branes, JHEP10(2023), 029, arXiv:2307.14112 18 [hep-th]

    2023 arXiv

  74. [82]

    F. F. Gautason and J. van Muiden, Localization of the M2-Brane, Phys. Rev. Lett.135, 101601 (2025), arXiv:2503.16597 [hep-th]

    2025 arXiv

  75. [83]

    van Muiden, Quantum M2-branes and Holography (2026) arXiv:2603.14544 [hep-th]

    J. van Muiden, Quantum M2-branes and Holography (2026) arXiv:2603.14544 [hep-th]

    2026

  76. [84]

    Bobev, M

    N. Bobev, M. David, J. Hong, V. Reys, and X. Zhang, A compendium of logarithmic corrections in AdS/CFT, JHEP04(2024), 020, arXiv:2312.08909 [hep-th]

    2024 arXiv

  77. [85]

    J. T. Liu, L. A. Pando Zayas, V. Rathee, and W. Zhao, Toward Microstate Counting Beyond Large N in Local- ization and the Dual One-loop Quantum Supergravity, JHEP01(2018), 026, arXiv:1707.04197 [hep-th]

    2018 arXiv

  78. [86]

    L. A. Pando Zayas and Y. Xin, Topologically twisted index in the ’t Hooft limit and the dual AdS 4 black hole entropy, Phys. Rev. D100, 126019 (2019), arXiv:1908.01194 [hep-th]

    2019 arXiv

  79. [87]

    S. M. Hosseini, To appear; companion to [56] (2026)

    2026

  80. [88]

    Benini, C

    F. Benini, C. Closset, and S. Cremonesi, Chiral fla- vors and M2-branes at toric CY4 singularities, JHEP 02(2010), 036, arXiv:0911.4127 [hep-th]

    2010 arXiv

  81. [89]

    Bashkirov and A

    D. Bashkirov and A. Kapustin, Supersymmetry en- hancement by monopole operators, JHEP05(2011), 015, arXiv:1007.4861 [hep-th]

    2011 arXiv

  82. [90]

    Mezei and S

    M. Mezei and S. S. Pufu, Three-sphere free energy for classical gauge groups, JHEP02(2014), 037, arXiv:1312.0920 [hep-th]

    2014 arXiv

  83. [91]

    Grassi and M

    A. Grassi and M. Marino, M-theoretic matrix models, JHEP02(2015), 115, arXiv:1403.4276 [hep-th]

    2015 arXiv

  84. [92]

    M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Y. I. Manin, Construction of Instantons, Phys. Lett. A65, 185 (1978)

    1978

  85. [93]

    Minahan, U

    J. Minahan, U. Naseer, and C. Thull, Squashing and su- persymmetry enhancement in three dimensions, SciPost Phys.12, 025 (2022), arXiv:2107.07151 [hep-th]

    2022 arXiv

  86. [94]

    Hristov and V

    K. Hristov and V. Reys, Factorization of log-corrections in AdS4/CFT3 from supergravity localization, JHEP12 (2021), 031, arXiv:2107.12398 [hep-th]

    2021 arXiv

  87. [95]

    S. M. Chester, R. R. Kalloor, and A. Sharon, Squashing, Mass, and Holography for 3d Sphere Free Energy, JHEP 04(2021), 244, arXiv:2102.05643 [hep-th]

    2021 arXiv

  88. [96]

    Closset, T

    C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komar- godski, and N. Seiberg, Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories, JHEP10(2012), 053, arXiv:1205.4142 [hep- th]

    2012 arXiv

  89. [97]

    Closset, T

    C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Ko- margodski, Supersymmetric Field Theories on Three- Manifolds, JHEP05(2013), 017, arXiv:1212.3388 [hep- th]

    2013 arXiv

  90. [98]

    N. B. Agmon, S. M. Chester, and S. S. Pufu, Solving M- theory with the Conformal Bootstrap, JHEP06(2018), 159, arXiv:1711.07343 [hep-th]

    2018 arXiv

  91. [99]

    S. M. Chester, S. S. Pufu, and X. Yin, The M-Theory S- Matrix From ABJM: Beyond 11D Supergravity, JHEP 08(2018), 115, arXiv:1804.00949 [hep-th]

    2018 arXiv

  92. [100]

    D. J. Binder, S. M. Chester, and S. S. Pufu, Absence of D4R4 in M-Theory From ABJM, JHEP04(2020), 052, arXiv:1808.10554 [hep-th]

    2020 arXiv

  93. [101]

    D. J. Binder, S. M. Chester, and S. S. Pufu, AdS4/CFT3 from weak to strong string coupling, JHEP01(2020), 034, arXiv:1906.07195 [hep-th]

    2020 arXiv

  94. [102]

    N. B. Agmon, S. M. Chester, and S. S. Pufu, The M-theory Archipelago, JHEP02(2020), 010, arXiv:1907.13222 [hep-th]

    2020 arXiv

  95. [103]

    S. M. Chester, R. Dempsey, and S. S. Pufu, Higher-derivative corrections in M-theory from pre- cision numerical bootstrap, JHEP07(2025), 096, arXiv:2412.14094 [hep-th]

    2025 arXiv

  96. [104]

    Gang and N

    D. Gang and N. Kim, LargeNtwisted partition func- tions in 3d-3d correspondence and Holography, Phys. Rev. D99, 021901 (2019), arXiv:1808.02797 [hep-th]

    2019 arXiv

  97. [105]

    D. Gang, N. Kim, and L. A. Pando Zayas, Precision Microstate Counting for the Entropy of Wrapped M5- branes, JHEP03(2020), 164, arXiv:1905.01559 [hep- th]

    2020 arXiv

  98. [106]

    Bobev, A

    N. Bobev, A. M. Charles, D. Gang, K. Hristov, and V. Reys, Higher-derivative supergravity, wrapped M5- branes, and theories of classR, JHEP04(2021), 058, arXiv:2011.05971 [hep-th]

    2021 arXiv

  99. [107]

    Gaiotto and A

    D. Gaiotto and A. Tomasiello, The gauge dual of Ro- mans mass, JHEP01(2010), 015, arXiv:0901.0969 [hep- th]

    2010 arXiv

  100. [108]

    Guarino, D

    A. Guarino, D. L. Jafferis, and O. Varela, String The- ory Origin of Dyonic N=8 Supergravity and Its Chern- Simons Duals, Phys. Rev. Lett.115, 091601 (2015), arXiv:1504.08009 [hep-th]

    2015 arXiv

  101. [109]

    I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, 7th ed., edited by A. Jeffrey and D. Zwillinger (Academic Press, Amsterdam, 2007)

    2007

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