Exact results on the Bethe Ansatz evaluation of the SCI

Antonio Amariti; Pietro Glorioso

REVIEW 2 major objections 4 minor 51 references

Exact results on the Bethe Ansatz evaluation of the SCI

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

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Pith's one-line read For the conifold theory with SU(2)×SU(2) gauge group, the Bethe Ansatz sum over six discrete solutions, each with multiplicity eight, reproduces the direct matrix-integral evaluation of the superconformal index exactly through order t^8.

desk verdict Useful, honest computation showing discrete Bethe roots match the SCI for the conifold to O(t^8) and partially for SPP, but the 'exact' claim runs ahead of the proven completeness of the solution set. read the letter →

arxiv 2506.15296 v1 pith:RSSXN3RS submitted 2025-06-18 hep-th

Antonio Amariti , Pietro Glorioso This is my paper · ORCID

classification hep-th

keywords superconformalindexBetheAnsatzequationstoricquivergaugetheoriesconifoldsuspendedpinchpointdiscreterootscontinuoussolutionsellipticthetafunctions

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 asks when the Bethe Ansatz evaluation of the 4d N=1 superconformal index needs only discrete solutions of the Bethe Ansatz equations, without the continuous families required in earlier SU(N) N=4 SYM cases. For the conifold theory with two SU(2) gauge nodes, it shows that summing over six inequivalent discrete Bethe roots, each with multiplicity eight, exactly reproduces the direct matrix-integral index: 8 Σ_{μ=1}^{6} I_μ(t) = 1 + $10t^{2}$ + $50t^{4}$ + $200t^{6}$ + O($t^{8}$). For the suspended pinch point (SPP) theory, the same discrete-only recipe matches the direct evaluation only in a restricted region of flavor fugacities; outside that region, divergent terms fail to cancel, signaling missing discrete or continuous solutions. A sympathetic reader should care because this identifies a concrete class of toric quiver theories where the Bethe Ansatz expansion closes on discrete data, and it pinpoints where continuous solutions begin to matter.

What carries the argument

The load-bearing object is the set of Bethe Ansatz equations recast in terms of Jacobi theta functions after moving to the sum and difference variables x = u_1 + u_2 and y = u_1 - u_2. For the conifold the product over the four bifundamental chemical potentials splits into two identical equations, and a five-term Riemann identity reduces each to the product θ_1(2x) h(x;Δ) = 0, separating a continuous family h(x;Δ) = 0 from the discrete family θ_1(2x) = 0. The discrete family together with the x = 0 sector produces the six inequivalent solutions on the torus, and Weyl-group analysis assigns each a multiplicity of eight. For the SPP theory the same principle — requiring holonomy combinations to take the values 0, 1/2, ω/2, or (1+ω)/2 on the torus — generates nine candidate solutions with multiplicity sixteen and an extra factor of two for the last three.

What would settle it

Compute the O($t^{10}$) term of both the direct matrix integral (3.7) and the Bethe Ansatz sum (2.17) over the six solutions (3.28) for the conifold: any disagreement would show that continuous or missed discrete solutions contribute at that order. For SPP, search for a new solution of (4.8) outside the set (4.10) with rational or irrational holonomies that contributes in the generic-fugacity region; finding one would explain the missing terms that currently prevent the cancellation of tachyonic contributions.

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Extended reading notes

Core claim

The central claim is that, for the conifold ($T^{{1,1}}$) quiver with gauge group SU(2)×SU(2), the Bethe Ansatz formula (2.17) evaluated on the six discrete solutions (3.28) equals the matrix-integral superconformal index to the computed order: 8 Σ_{μ=1}^{6} I_μ(t) = I_{$T^{{1,1}}$}(t) + O($t^{8}$), with I_{$T^{{1,1}}$}(t) = 1 + $10t^{2}$ + $50t^{4}$ + $200t^{6}$ + O($t^{8}$). The six solutions comprise the three Hong-Liu solutions and three additional quarter-period combinations; each individual contribution contains divergent ('tachyonic') pieces that cancel only after the sum is taken with the correct multiplicity eight. For the SPP theory, the paper constructs a nine-element discrete set (4.10) guided by the conifold pattern and shows that the Bethe Ansatz sum, 16[Σ_{μ=1}^{6} I_μ + 2 Σ_{μ=7}^{9} I_μ], matches the direct evaluation I_SPP(t) = 1 + $5t^{4}$ + $4t^{6}$ + $20t^{8}$ + $6t^{9}$ + $18t^{10}$ + O($t^{11}$) only when the flavor fugacities satisfy c=d and a=$b^{{-2}}$$d^{{-2}}$ or b=$a^{{-2}}$$d^{{-2}}$. For generic fugacities the divergent terms do not cancel, indicating that some discrete or continuous solutions are missing.

Load-bearing premise

The result rests on the unproven assertion that the continuous family h(x;Δ)=0 and the second sector of the conifold BAEs contribute nothing to the Bethe Ansatz sum; the only evidence is the O($t^{8}$) matching, and for SPP the analogous assertion is that the guessed nine elements are all the contributing discrete solutions.

Editorial extensions

If this is right

For the conifold, the Bethe Ansatz evaluation closes on the six discrete roots: the continuous family and the second sector, whether or not they exist, do not affect the superconformal index through order t^8.
The tachyonic divergences of individual roots cancel exactly when each solution is weighted with multiplicity eight, a multiplicity fixed by Weyl equivalence and torus identifications.
For SPP, the discrete-only Bethe Ansatz evaluation is exact only in the region c=d and a=b^{-2}d^{-2} or b=a^{-2}d^{-2}; outside this region the result signals missing discrete or continuous solutions.
The conifold is a genuine counterpoint to SU(3) N=4 SYM, where discrete roots alone fail and continuous solutions are necessary.
The methods extend the Wong-Liu solution family to multi-node toric quivers, providing new explicit solutions of the Bethe Ansatz equations beyond the previously known ones.

Reading between the lines

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

If the O(t^8) matching is the truncation of an exact formal identity, then higher orders should also match; computing the O(t^10) term of both the direct integral and the Bethe Ansatz sum for the conifold would be a quick and decisive check.
The factorization of the conifold BAEs into independent x and y equations may be the structural reason discrete roots saturate the index; other toric quivers whose BAEs factor similarly might show the same discrete-only exactness.
The SPP failure suggests that the completeness of the discrete-root set depends on flavor fugacities; a systematic scan of rational holonomy ansätze beyond the nine elements of (4.10) could reveal whether missing discrete solutions or genuine continuous families account for the discrepancy.
The region (4.13) where the SPP match holds is a codimension-one condition on the fugacities; it may correspond to an enhanced symmetry point, and checking that interpretation against the superconformal index could illuminate why the discrete set is complete only there.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The conifold matching is a real, checkable result, and the new discrete Bethe roots beyond the Hong–Liu family are worth knowing. But the word “exact” is doing work the paper hasn’t fully earned: the completeness of the discrete set is asserted on the strength of an O(t^8) agreement, with the continuous family and the second sector of the BAEs left unexamined.

What’s actually new: solving one sector of the conifold BAEs explicitly, finding six inequivalent discrete solutions (three HL plus three new ones), and showing with multiplicities that their BA sum reproduces the direct matrix integral to O(t^8). The SPP analysis is more exploratory but still useful: a nine-element guessed set, checked to solve the BAEs, matches to O(t^11) in a specific fugacity region. The appendix expansions are explicit and reproducible.

Where the soft spots are: exactly where the authors admit them. After (3.28) they say “there is no need to pursue this analysis further” because the discrete sum matches to O(t^8). That is a finite-order check, not a proof that the continuous family h(x;Δ)=0 or the second sector of (3.15) has zero contribution. A missed sector could start at any higher order. For SPP, the set (4.10) is guessed, and the paper itself says there’s no guarantee it contains all discrete solutions; the matching is restricted to region R, which is selected after seeing where the divergences cancel. That’s a post hoc scope restriction. The authors are transparent about all this, which earns them credit, but it means the title’s “Exact results” oversells what’s demonstrated.

The citation pattern and the math look fine. The derivations are standard theta-function manipulations, the Riemann identity use is clean, and the expansions are consistent. I don’t see fitted constants or circular benchmarks; the direct integral is an independent check.

Who this is for: people working on the Bethe Ansatz expansion of superconformal indices at small rank, especially SU(2) toric quivers. It’s a useful data point for that community.

Recommendation: send to peer review. A good referee should ask for either a completeness argument for the discrete solutions or a rephrasing that limits the claim to the checked order. The SPP part also needs a more systematic solution search or a clear statement that it’s a conjecture. With those, the paper becomes a solid contribution.

Referee Report

2 major / 4 minor

Summary. This paper studies the Bethe Ansatz (BA) evaluation of the 4d N=1 superconformal index for two toric quiver gauge theories with SU(2) gauge factors: the conifold theory with gauge group SU(2)×SU(2) and the suspended pinch point (SPP) theory with gauge group SU(2)^3. For the conifold, the authors reduce one sector of the BAEs to the equation θ1(2x)h(x;Δ)=0, solve the discrete branch to obtain the six-element set (3.28) with multiplicity eight, and show that the BA sum reproduces the direct matrix-integral result I_{T^1,1}(t)=1+10t^2+50t^4+200t^6+O(t^8), as displayed in eqs. (3.31)–(3.32). For SPP, they propose a nine-element solution set (4.10), verify that each element solves the BAEs, and find that the BA sum with the stated multiplicities matches the direct index inside a restricted fugacity region R defined by (4.13), up to O(t^{11}) at equal fugacities. The paper concludes that for these two models the discrete Bethe roots suffice, in contrast to SU(N≥3) N=4 SYM where continuous solutions are required.

Significance. If the completeness assumptions were established, the conifold result would be a valuable data point in the small-rank Bethe Ansatz program: a multi-node toric quiver whose superconformal index is reproduced by discrete Bethe roots alone, with an explicit solution set and no fitted parameters. The comparison against the independent direct matrix-integral evaluation is the right benchmark, and the paper is transparent about the conditional nature of the SPP result. The verification of the Hong-Liu solutions for arbitrary N in Appendix B and the explicit low-order expansions in Appendix C are useful and reproducible. The main significance is therefore real, but it is tied to a completeness claim that the manuscript does not actually prove.

major comments (2)

[Section 3, eqs. (3.15)–(3.32)] The central claim that the conifold SCI is exactly reproduced by the discrete roots in (3.28) is not established, because the derivation solves only the first sector of (3.15). The continuous family h(x;Δ)=0 from (3.21) and the second sector of (3.15) are left unanalyzed; the text after (3.28) asserts 'there is no need to pursue this analysis further' solely on the basis of the O(t^8) agreement in (3.32). A finite-order match cannot exclude a missed isolated solution or a contribution from the continuous locus, for which the ordinary BA formula (2.17) is not even directly applicable because H=0 there. Please either provide a proof (or a reference-level argument) that these sectors contribute zero, or explicitly state the conifold result as a matching to the computed order rather than as an exact identity.
[Section 4, eqs. (4.10)–(4.15)] For the SPP theory the solution set is a guess, as the paper itself acknowledges after (4.10): 'This prescription does not guarantee that all the discrete solutions to (4.8) have been obtained.' Consequently the matching (4.15) tests only the nine-element set (4.10), not the full BA formula; a further discrete solution could change the sum inside the region R at order t^{11} or beyond. The region R in (4.13) is also selected after observing where the divergent terms cancel, which is a post hoc restriction. The SPP conclusion should either be accompanied by a completeness proof for (4.10) within R, or be presented explicitly as a numerical match for a selected subset of solutions in a chosen kinematic region.

minor comments (4)

[Abstract and Section 3] The word 'exact' in the abstract and Conclusion is stronger than the displayed check: eq. (3.32) contains an O(t^8) remainder, so the conifold matching is verified through order t^6. Please state the achieved order explicitly wherever 'exact matching' is used.
[Section 2] There are several typos, e.g. 'techinique' and 'evalutation' in Section 2 and 'beacuse' in Section 3; these should be corrected.
[Section 3 and Section 4] The expansion variables are q=t^2 for the conifold and q=t^5 for SPP; it would help the reader if this distinction were repeated in the captions or in the text of Appendix C.
[Appendix C] The series in Appendix C are very long; consider stating that they were generated and verified symbolically, or moving the full expressions to ancillary files, while keeping the cancellations visible in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Bethe-Ansatz sums are checked against independent direct matrix-integral evaluations, and the admitted completeness gaps are not circular reductions.

full rationale

The load-bearing comparison in the conifold case is Eq. (3.32), where the Bethe-Ansatz sum over the six discrete solutions (3.28), each with multiplicity 8, is equated to the independently evaluated matrix integral (3.31). The discrete solution set is obtained by solving the first sector of the square-rooted BAE (3.13), not by matching the index, and no parameter is fitted to the direct integral. The SPP check (4.15) similarly compares the explicitly verified nine-element set (4.10) with the direct integral, with the region R defined in (4.13) chosen post hoc from the cancellation of tachyonic terms; restricting the fugacity domain is a scope restriction, not a construction of the claimed result. The paper's own admissions, that the continuous family h(x;Delta)=0 from (3.21) and the second sector of (3.15) are left unanalysed and that the SPP set is not proven complete, are genuine rigor gaps in the claimed exactness proof, but they are completeness risks rather than circular reductions: the target index is not used to define the Bethe solutions or the evaluated sum. The only self-citation, ref. [36], appears in the introduction and plays no role in the derivation. Therefore no circular step is exhibited and the paper is self-contained against its external direct-integral benchmarks.

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

No free parameters are fitted to data; the fugacity rescalings a,b,c,d in (3.30) and (4.11) are bookkeeping choices for the series expansion, not fitted constants. No new entities are introduced. The main external inputs are the Bethe Ansatz formula and standard special-function identities, plus two ad hoc completeness assumptions that are explicitly acknowledged by the authors.

assumptions (5)

domain assumption The Bethe Ansatz formula (2.17) of Benini-Milan [6], including the reduction of the index integral to a sum over contributing solutions MBAE with the stated Weyl-group exclusions.
The paper assumes the residue/pole structure of (2.11) reduces to summing over MBAE. This is a theorem in [6] under the discrete-solutions assumption, and part of what is being tested here.
standard math The Riemann quintuple identity (A.12) and the theta-function properties (A.7)-(A.9).
Used to factor the BAEs in Section 3; standard results from [51].
ad hoc to paper For the conifold, the second sector of (3.15) and the continuous family h(x;Δ)=0 in (3.21) contribute nothing to the BA sum.
After (3.28), the paper asserts the discrete set suffices without analyzing these sectors; the O(t^8) matching (3.32) is used as evidence, not a derivation.
ad hoc to paper For SPP, the guessed set (4.10) contains all discrete solutions contributing to the index in the region R.
Section 4 states the prescription does not guarantee all discrete solutions are obtained; the matching (4.15) in region R is conditional on this set being complete.
domain assumption The constraints Σ Δ = 2ω for the superpotential-invariant flavor and R-charge assignments.
Standard for N=1 SCFTs; used in (2.9), (3.6), (4.2), (B.15), (B.17).

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Pith. "Pith review of Exact results on the Bethe Ansatz evaluation of the SCI." pith.science (2026). https://pith.science/paper/RSSXN3RS

@misc{pith2026250615296,
  author       = {Pith},
  title        = {Pith review of: Exact results on the Bethe Ansatz evaluation of the SCI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSSXN3RS}},
  note         = {Machine review of arXiv:2506.15296}
}

abstract

We evaluate the superconformal index using the Bethe Ansatz (BA) approach for 4d $\mathcal{N}=1$ toric quiver gauge theories with a small amount of gauge groups. We restrict to $\mathrm{SU}(2)$ gauge factors and compare the results with the ones obtained by a direct evaluation of the index. The answer obtained from the BA approach using only discrete solutions for the BA equations does not always reproduce the direct evaluation result. A similar problem affects the case of $\mathrm{SU}(N)$ $\mathcal{N}=4$ SYM for $N \geq 3$, and it requires to introduce continuous solutions to the BA equations. However, we find that for $T^{1,1}$ and partially for the suspended pinch point the matching is exact in absence of the contributions from continuous solutions.

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Works this paper leans on

51 extracted references · 4 canonical work pages

[1]

Kinney, J.M

J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju,An Index for 4 dimensional super conformal theories, Commun. Math. Phys.275 (2007) 209 [hep-th/0510251]

arXiv 2007
[2]

Romelsberger,Counting chiral primaries in N = 1, d=4 superconformal field theories, Nucl

C. Romelsberger,Counting chiral primaries in N = 1, d=4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060]

arXiv 2006
[3]

Benini and E

F. Benini and E. Milan,Black Holes in 4DN=4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [1812.09613]

arXiv 2020
[4]

S. Choi, J. Kim, S. Kim and J. Nahmgoong,Large AdS black holes from QFT, 1810.12067

arXiv
[5]

Closset, H

C. Closset, H. Kim and B. Willett,N = 1 supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [1707.05774]. – 26 –

arXiv 2017
[6]

Benini and E

F. Benini and E. Milan,A Bethe Ansatz type formula for the superconformal index, Commun. Math. Phys.376 (2020) 1413 [1811.04107]

arXiv 2020
[7]

Honda,Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys

M. Honda,Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys. Rev. D100 (2019) 026008 [1901.08091]

arXiv 2019
[8]

Arabi Ardehali, J

A. Arabi Ardehali, J. Hong and J.T. Liu,Asymptotic growth of the 4dN = 4 index and partially deconfined phases, JHEP 07 (2020) 073 [1912.04169]

arXiv 2020

Show all 51 references

[9]

Arabi Ardehali,Cardy-like asymptotics of the 4dN = 4 index and AdS5 blackholes, JHEP 06 (2019) 134 [1902.06619]

A. Arabi Ardehali,Cardy-like asymptotics of the 4dN = 4 index and AdS5 blackholes, JHEP 06 (2019) 134 [1902.06619]

2019 arXiv
[10]

Cabo-Bizet, D

A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy,The asymptotic growth of states of the 4dN = 1 superconformal index, JHEP 08 (2019) 120 [1904.05865]

2019 arXiv
[11]

Amariti, I

A. Amariti, I. Garozzo and G. Lo Monaco,Entropy function from toric geometry, Nucl. Phys. B 973 (2021) 115571 [1904.10009]

2021 arXiv
[12]

J. Kim, S. Kim and J. Song,A 4d N=1 Cardy Formula, 1904.03455

1904 arXiv
[13]

Amariti, M

A. Amariti, M. Fazzi and A. Segati,The SCI ofN = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral, JHEP 06 (2021) 132 [2012.15208]

2021 arXiv
[14]

Benini, E

F. Benini, E. Colombo, S. Soltani, A. Zaffaroni and Z. Zhang,Superconformal indices at largeN and the entropy of AdS5× SE5 black holes, Class. Quant. Grav.37 (2020) 215021 [2005.12308]

2020 arXiv
[15]

S. Choi, S. Kim and J. Song,LargeN Universality of 4dN = 1 Superconformal Index and AdS Black Holes, 2309.07614

arXiv
[16]

Cabo-Bizet, D

A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy,The large-N limit of the 4dN = 1 superconformal index, JHEP 11 (2020) 150 [2005.10654]

2020 arXiv
[17]

Cabo-Bizet, D

A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy,Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes, JHEP 10 (2019) 062 [1810.11442]

2019 arXiv
[18]

Amariti and A

A. Amariti and A. Zanetti,S-duality in the Cardy-like limit of the superconformal index, JHEP 02 (2024) 218 [2307.08391]

2024 arXiv
[19]

Jejjala, Y

V. Jejjala, Y. Lei, S. van Leuven and W. Li,SL(3, Z) Modularity and New Cardy limits of theN = 4 superconformal index, JHEP 11 (2021) 047 [2104.07030]

2021 arXiv
[20]

Ardehali and J

A.A. Ardehali and J. Hong,Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions, JHEP 01 (2022) 062 [2110.01538]

2022 arXiv
[21]

Goldstein, V

K. Goldstein, V. Jejjala, Y. Lei, S. van Leuven and W. Li,Residues, modularity, and the Cardy limit of the 4dN = 4 superconformal index, 2011.06605

2011 arXiv
[22]

Cabo-Bizet and S

A. Cabo-Bizet and S. Murthy,Supersymmetric phases of 4dN = 4 SYM at largeN, JHEP 09 (2020) 184 [1909.09597]

2020 arXiv
[23]

S. Choi, S. Kim and J. Song,Supersymmetric Spectral Form Factor and Euclidean Black Holes, Phys. Rev. Lett.131 (2023) 151602 [2206.15357]. – 27 –

2023 arXiv
[24]

Arabi Ardehali and S

A. Arabi Ardehali and S. Murthy,The 4d superconformal index near roots of unity and 3d Chern-Simons theory, JHEP 10 (2021) 207 [2104.02051]

2021 arXiv
[25]

Cassani and Z

D. Cassani and Z. Komargodski,EFT and the SUSY Index on the 2nd Sheet, SciPost Phys. 11 (2021) 004 [2104.01464]

2021 arXiv
[26]

Amariti, M

A. Amariti, M. Fazzi and A. Segati,Expanding on the Cardy-like limit of the SCI of 4dN = 1 ABCD SCFTs, JHEP 07 (2021) 141 [2103.15853]

2021 arXiv
[27]

Aharony, F

O. Aharony, F. Benini, O. Mamroud and E. Milan,A gravity interpretation for the Bethe Ansatz expansion of theN = 4 SYM index, Phys. Rev. D104 (2021) 086026 [2104.13932]

2021 arXiv
[28]

Mamroud,The SUSY index beyond the Cardy limit, JHEP 01 (2024) 111 [2212.11925]

O. Mamroud,The SUSY index beyond the Cardy limit, JHEP 01 (2024) 111 [2212.11925]

2024 arXiv
[29]

Aharony, O

O. Aharony, O. Mamroud, S. Nowik and M. Weissman,The Bethe Ansatz for the superconformal index with unequal angular momenta, 2402.03977

arXiv
[30]

González Lezcano and L.A

A. González Lezcano and L.A. Pando Zayas,Microstate counting via Bethe Ansätze in the 4dN = 1 superconformal index, JHEP 03 (2020) 088 [1907.12841]

2020 arXiv
[31]

Lanir, A

A. Lanir, A. Nedelin and O. Sela,Black hole entropy function for toric theories via Bethe Ansatz, JHEP 04 (2020) 091 [1908.01737]

2020 arXiv
[32]

González Lezcano, J

A. González Lezcano, J. Hong, J.T. Liu and L.A. Pando Zayas,Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions, JHEP 01 (2021) 001 [2007.12604]

2021 arXiv
[33]

Copetti, A

C. Copetti, A. Grassi, Z. Komargodski and L. Tizzano,Delayed Deconfinement and the Hawking-Page Transition, 2008.04950

2008 arXiv
[34]

Y. Chen, M. Heydeman, Y. Wang and M. Zhang,Probing supersymmetric black holes with surface defects, JHEP 10 (2023) 136 [2306.05463]

2023 arXiv
[35]

Cabo-Bizet, M

A. Cabo-Bizet, M. David and A. González Lezcano,Thermodynamics of black holes with probe D-branes, 2312.12533

arXiv
[36]

Amariti, P

A. Amariti, P. Glorioso, D. Morgante and A. Zanetti,Cardy matches Bethe on the surface: A tale of a brane and a black hole, Nucl. Phys. B 1010 (2025) 116773 [2403.17190]

2025 arXiv
[37]

Agarwal, S

P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong,AdS black holes and finite N indices, Phys. Rev. D103 (2021) 126006 [2005.11240]

2021 arXiv
[38]

Murthy,Growth of the 1 16-BPS index in 4dN = 4 supersymmetric Yang-Mills theory, Phys

S. Murthy,Growth of the 1 16-BPS index in 4dN = 4 supersymmetric Yang-Mills theory, Phys. Rev. D105 (2022) L021903 [2005.10843]

2022 arXiv
[39]

Benini and G

F. Benini and G. Rizi,Superconformal index of low-rank gauge theories via the Bethe Ansatz, JHEP 05 (2021) 061 [2102.03638]

2021 arXiv
[40]

Lezcano, J

A.G. Lezcano, J. Hong, J.T. Liu and L.A. Pando Zayas,The Bethe-Ansatz approach to theN = 4 superconformal index at finite rank, JHEP 06 (2021) 126 [2101.12233]. – 28 –

2021 arXiv
[41]

Deddo, L.A

E. Deddo, L.A. Pando Zayas and W. Zhou,The superconformal index and black hole instabilities, JHEP 05 (2025) 170 [2502.01614]

2025 arXiv
[42]

S. Choi, D. Jain, S. Kim, V. Krishna, G. Kwon, E. Lee et al.,Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index, 2501.17217
[43]

Cabo-Bizet and W

A. Cabo-Bizet and W. Li,Generalized Bethe expansions of superconformal indices, 2411.12018
[44]

Hong and J.T

J. Hong and J.T. Liu,The topologically twisted index ofN = 4 super-Yang-Mills on T 2×S2 and the elliptic genus, JHEP 07 (2018) 018 [1804.04592]

2018 arXiv
[45]

Hosseini, A

S.M. Hosseini, A. Nedelin and A. Zaffaroni,The Cardy limit of the topologically twisted index and black strings in AdS5, JHEP 04 (2017) 014 [1611.09374]

2017 arXiv
[46]

Sundborg,The Hagedorn transition, deconfinement and N=4 SYM theory, Nucl

B. Sundborg,The Hagedorn transition, deconfinement and N=4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001]

2000 arXiv
[47]

Aharony, J

O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys.8 (2004) 603 [hep-th/0310285]

2004 arXiv
[48]

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 (2012) 71 [0712.2824]

2012 arXiv
[49]

Closset, T.T

C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski,The Geometry of Supersymmetric Partition Functions, JHEP 01 (2014) 124 [1309.5876]

2014 arXiv
[50]

Assel, D

B. Assel, D. Cassani and D. Martelli,Localization on Hopf surfaces, JHEP 08 (2014) 123 [1405.5144]

2014 arXiv
[51]

Kharchev and A

S. Kharchev and A. Zabrodin,Theta vocabulary i, Journal of Geometry and Physics 94 (2015) 19–31. – 29 –

2015

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Kinney, J.M

J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju,An Index for 4 dimensional super conformal theories, Commun. Math. Phys.275 (2007) 209 [hep-th/0510251]

arXiv 2007

[2] [2]

Romelsberger,Counting chiral primaries in N = 1, d=4 superconformal field theories, Nucl

C. Romelsberger,Counting chiral primaries in N = 1, d=4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060]

arXiv 2006

[3] [3]

Benini and E

F. Benini and E. Milan,Black Holes in 4DN=4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [1812.09613]

arXiv 2020

[4] [4]

S. Choi, J. Kim, S. Kim and J. Nahmgoong,Large AdS black holes from QFT, 1810.12067

arXiv

[5] [5]

Closset, H

C. Closset, H. Kim and B. Willett,N = 1 supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [1707.05774]. – 26 –

arXiv 2017

[6] [6]

Benini and E

F. Benini and E. Milan,A Bethe Ansatz type formula for the superconformal index, Commun. Math. Phys.376 (2020) 1413 [1811.04107]

arXiv 2020

[7] [7]

Honda,Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys

M. Honda,Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys. Rev. D100 (2019) 026008 [1901.08091]

arXiv 2019

[8] [8]

Arabi Ardehali, J

A. Arabi Ardehali, J. Hong and J.T. Liu,Asymptotic growth of the 4dN = 4 index and partially deconfined phases, JHEP 07 (2020) 073 [1912.04169]

arXiv 2020