Abstract
We analyze a set of contributions to the superconformal index of 4d 𝒩 = 4 SU(N) super Yang-Mills using the Bethe Ansatz approach. These contributions dominate at the large N limit, where their leading order in N reproduces various supersymmetric Euclidean black hole saddles in the dual theory, and they also dominate for finite N in high temperature Cardy-like limits. We compute the O(N0) terms, including those exponentially suppressed in the Cardy limit, and show that there are no 1/N corrections beyond them. Under certain assumptions, it implies that the gravitational perturbative series around these black hole saddles is 1-loop exact.
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F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
F. Benini and E. Milan, Black Holes in 4D 𝒩 = 4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [INSPIRE].
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 [arXiv:1810.11442] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
M. Honda, Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys. Rev. D 100 (2019) 026008 [arXiv:1901.08091] [INSPIRE].
A. Arabi Ardehali, Cardy-like asymptotics of the 4d 𝒩 = 4 index and AdS5 blackholes, JHEP 06 (2019) 134 [arXiv:1902.06619] [INSPIRE].
J. Kim, S. Kim and J. Song, A 4d 𝒩 = 1 Cardy Formula, JHEP 01 (2021) 025 [arXiv:1904.03455] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The asymptotic growth of states of the 4d 𝒩 = 1 superconformal index, JHEP 08 (2019) 120 [arXiv:1904.05865] [INSPIRE].
A. Amariti, I. Garozzo and G. Lo Monaco, Entropy function from toric geometry, Nucl. Phys. B 973 (2021) 115571 [arXiv:1904.10009] [INSPIRE].
A. González Lezcano and L.A. Pando Zayas, Microstate counting via Bethe Ansätze in the 4d 𝒩 = 1 superconformal index, JHEP 03 (2020) 088 [arXiv:1907.12841] [INSPIRE].
A. Lanir, A. Nedelin and O. Sela, Black hole entropy function for toric theories via Bethe Ansatz, JHEP 04 (2020) 091 [arXiv:1908.01737] [INSPIRE].
A. Cabo-Bizet and S. Murthy, Supersymmetric phases of 4d 𝒩 = 4 SYM at large N , JHEP 09 (2020) 184 [arXiv:1909.09597] [INSPIRE].
A. Arabi Ardehali, J. Hong and J.T. Liu, Asymptotic growth of the 4d 𝒩 = 4 index and partially deconfined phases, JHEP 07 (2020) 073 [arXiv:1912.04169] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The large-N limit of the 4d 𝒩 = 1 superconformal index, JHEP 11 (2020) 150 [arXiv:2005.10654] [INSPIRE].
S. Murthy, The growth of the \( \frac{1}{16} \)-BPS index in 4d 𝒩 = 4 SYM, arXiv:2005.10843 [INSPIRE].
P. Agarwal et al., AdS black holes and finite N indices, Phys. Rev. D 103 (2021) 126006 [arXiv:2005.11240] [INSPIRE].
F. Benini et al., Superconformal indices at large N and the entropy of AdS5 × SE5 black holes, Class. Quant. Grav. 37 (2020) 215021 [arXiv:2005.12308] [INSPIRE].
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 [arXiv:2007.12604] [INSPIRE].
C. Copetti, A. Grassi, Z. Komargodski and L. Tizzano, Delayed deconfinement and the Hawking-Page transition, JHEP 04 (2022) 132 [arXiv:2008.04950] [INSPIRE].
A. Cabo-Bizet, From multi-gravitons to Black holes: The role of complex saddles, arXiv:2012.04815 [INSPIRE].
A. Amariti, M. Fazzi and A. Segati, The SCI of 𝒩 = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral, JHEP 06 (2021) 132 [arXiv:2012.15208] [INSPIRE].
S.M. Hosseini and A. Zaffaroni, Large N matrix models for 3d 𝒩 = 2 theories: twisted index, free energy and black holes, JHEP 08 (2016) 064 [arXiv:1604.03122] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS4, Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].
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 [arXiv:1611.09374] [INSPIRE].
A. Cabo-Bizet, V.I. Giraldo-Rivera and L.A. Pando Zayas, Microstate counting of AdS4 hyperbolic black hole entropy via the topologically twisted index, JHEP 08 (2017) 023 [arXiv:1701.07893] [INSPIRE].
F. Azzurli et al., A universal counting of black hole microstates in AdS4, JHEP 02 (2018) 054 [arXiv:1707.04257] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Passias, Holographic microstate counting for AdS4 black holes in massive IIA supergravity, JHEP 10 (2017) 190 [arXiv:1707.06884] [INSPIRE].
F. Benini, H. Khachatryan and E. Milan, Black hole entropy in massive Type IIA, Class. Quant. Grav. 35 (2018) 035004 [arXiv:1707.06886] [INSPIRE].
S.M. Hosseini, I. Yaakov and A. Zaffaroni, Topologically twisted indices in five dimensions and holography, JHEP 11 (2018) 119 [arXiv:1808.06626] [INSPIRE].
P.M. Crichigno, D. Jain and B. Willett, 5d Partition Functions with A Twist, JHEP 11 (2018) 058 [arXiv:1808.06744] [INSPIRE].
S.M. Hosseini, K. Hristov, A. Passias and A. Zaffaroni, 6D attractors and black hole microstates, JHEP 12 (2018) 001 [arXiv:1809.10685] [INSPIRE].
M. Suh, Supersymmetric AdS6 black holes from matter coupled F (4) gauged supergravity, JHEP 02 (2019) 108 [arXiv:1810.00675] [INSPIRE].
M. Fluder, S.M. Hosseini and C.F. Uhlemann, Black hole microstate counting in Type IIB from 5d SCFTs, JHEP 05 (2019) 134 [arXiv:1902.05074] [INSPIRE].
D. Gang, N. Kim and L.A. Pando Zayas, Precision Microstate Counting for the Entropy of Wrapped M5-branes, JHEP 03 (2020) 164 [arXiv:1905.01559] [INSPIRE].
G. Kántor, C. Papageorgakis and P. Richmond, AdS7 black-hole entropy and 5D 𝒩 = 2 Yang-Mills, JHEP 01 (2020) 017 [arXiv:1907.02923] [INSPIRE].
S. Choi, C. Hwang and S. Kim, Quantum vortices, M2-branes and black holes, arXiv:1908.02470 [INSPIRE].
N. Bobev and P.M. Crichigno, Universal spinning black holes and theories of class ℛ, JHEP 12 (2019) 054 [arXiv:1909.05873] [INSPIRE].
J. Nian and L.A. Pando Zayas, Microscopic entropy of rotating electrically charged AdS4 black holes from field theory localization, JHEP 03 (2020) 081 [arXiv:1909.07943] [INSPIRE].
F. Benini, D. Gang and L.A. Pando Zayas, Rotating Black Hole Entropy from M5 Branes, JHEP 03 (2020) 057 [arXiv:1909.11612] [INSPIRE].
M. David, A. Lezcano González, J. Nian and L.A. Pando Zayas, Logarithmic corrections to the entropy of rotating black holes and black strings in AdS5, JHEP 04 (2022) 160 [arXiv:2106.09730] [INSPIRE].
M. David, J. Nian and L.A. Pando Zayas, Gravitational Cardy Limit and AdS Black Hole Entropy, JHEP 11 (2020) 041 [arXiv:2005.10251] [INSPIRE].
A. Amariti and A. Segati, Kerr-Newman black holes from 𝒩 = 1∗, JHEP 06 (2023) 216 [arXiv:2210.03015] [INSPIRE].
S.M. Hosseini, I. Yaakov and A. Zaffaroni, The joy of factorization at large N: five-dimensional indices and AdS black holes, JHEP 02 (2022) 097 [arXiv:2111.03069] [INSPIRE].
J. Hong, The topologically twisted index of 𝒩 = 4 SU(N) Super-Yang-Mills theory and a black hole Farey tail, JHEP 10 (2021) 145 [arXiv:2108.02355] [INSPIRE].
K. Goldstein et al., Residues, modularity, and the Cardy limit of the 4d 𝒩 = 4 superconformal index, JHEP 04 (2021) 216 [arXiv:2011.06605] [INSPIRE].
V. Jejjala, Y. Lei, S. van Leuven and W. Li, SL(3, ℤ) Modularity and New Cardy limits of the 𝒩 = 4 superconformal index, JHEP 11 (2021) 047 [arXiv:2104.07030] [INSPIRE].
V. Jejjala, Y. Lei, S. van Leuven and W. Li, Modular factorization of superconformal indices, JHEP 10 (2023) 105 [arXiv:2210.17551] [INSPIRE].
S. Choi, S. Jeong, S. Kim and E. Lee, Exact QFT duals of AdS black holes, JHEP 09 (2023) 138 [arXiv:2111.10720] [INSPIRE].
F. Benini and E. Milan, A Bethe Ansatz type formula for the superconformal index, Commun. Math. Phys. 376 (2020) 1413 [arXiv:1811.04107] [INSPIRE].
O. Aharony, F. Benini and O. Mamroud, work in progress.
J. Hong and J.T. Liu, The topologically twisted index of 𝒩 = 4 super-Yang-Mills on T 2 × S2 and the elliptic genus, JHEP 07 (2018) 018 [arXiv:1804.04592] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The Geometry of Supersymmetric Partition Functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].
O. Aharony, F. Benini, O. Mamroud and E. Milan, A gravity interpretation for the Bethe Ansatz expansion of the 𝒩 = 4 SYM index, Phys. Rev. D 104 (2021) 086026 [arXiv:2104.13932] [INSPIRE].
D. Cassani and Z. Komargodski, EFT and the SUSY Index on the 2nd Sheet, SciPost Phys. 11 (2021) 004 [arXiv:2104.01464] [INSPIRE].
A. Arabi Ardehali and S. Murthy, The 4d superconformal index near roots of unity and 3d Chern-Simons theory, JHEP 10 (2021) 207 [arXiv:2104.02051] [INSPIRE].
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] [INSPIRE].
A. Dabholkar, N. Drukker and J. Gomes, Localization in supergravity and quantum AdS4/CF T3 holography, JHEP 10 (2014) 090 [arXiv:1406.0505] [INSPIRE].
K. Hristov, I. Lodato and V. Reys, On the quantum entropy function in 4d gauged supergravity, JHEP 07 (2018) 072 [arXiv:1803.05920] [INSPIRE].
K. Hristov, I. Lodato and V. Reys, One-loop determinants for black holes in 4d gauged supergravity, JHEP 11 (2019) 105 [arXiv:1908.05696] [INSPIRE].
K. Hristov and V. Reys, Factorization of log-corrections in AdS4/CFT3 from supergravity localization, JHEP 12 (2021) 031 [arXiv:2107.12398] [INSPIRE].
L.V. Iliesiu, S. Murthy and G.J. Turiaci, Black hole microstate counting from the gravitational path integral, arXiv:2209.13602 [INSPIRE].
N. Bobev, J. Hong and V. Reys, Large N partition functions of the ABJM theory, JHEP 02 (2023) 020 [arXiv:2210.09318] [INSPIRE].
N. Bobev, J. Hong and V. Reys, Large N Partition Functions, Holography, and Black Holes, Phys. Rev. Lett. 129 (2022) 041602 [arXiv:2203.14981] [INSPIRE].
N. Bobev, S. Choi, J. Hong and V. Reys, Large N superconformal indices for 3d holographic SCFTs, JHEP 02 (2023) 027 [arXiv:2210.15326] [INSPIRE].
C. Closset, H. Kim and B. Willett, 𝒩 = 1 supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [arXiv:1707.05774] [INSPIRE].
F. Benini and G. Rizi, Superconformal index of low-rank gauge theories via the Bethe Ansatz, JHEP 05 (2021) 061 [arXiv:2102.03638] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
G. Felder and A. Varchenko, The elliptic gamma function and SL(3, ℤ) ⋉ z3, Adv. Math. 156 (2000) 44 [math/9907061].
Acknowledgments
The author thanks Erez Urbach and Tal Sheaffer for useful discussions, and is grateful to Francesco Benini and Ofer Aharony for many useful conversations and for comments on the manuscript. The work of the author was supported in part by an Israel Science Foundation (ISF) center for excellence grant (grant number 2289/18), by ISF grant no. 2159/22, by Simons Foundation grant 994296 (Simons Collaboration on Confinement and QCD Strings), by grant no. 2018068 from the United States-Israel Bi- national Science Foundation (BSF), by the Minerva foundation with funding from the Federal German Ministry for Education and Research, by the German Research Foundation through a German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”, and by a research grant from Martin Eisenstein. The author would like to thank SISSA for its hospitality, funded by ERC-COG grant NP-QFT No. 864583. The author also thanks the Simons Center for Geometry and Physics, Stony Brook University for organizing the “Supersymmetric Black Holes, Holography and Microstate Counting” workshop which helped facilitate this paper. The author is also supported by the ERC-COG grant NP-QFT No. 864583 “Non-perturbative dynamics of quantum fields: from new deconfined phases of matter to quantum black holes”, by the MUR-FARE2020 grant No. R20E8NR3HX “The Emergence of Quantum Gravity from Strong Coupling Dynamics”. OM is also partially supported by the INFN “Iniziativa Specifica GAST”.
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Mamroud, O. The SUSY index beyond the Cardy limit. J. High Energ. Phys. 2024, 111 (2024). https://doi.org/10.1007/JHEP01(2024)111
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DOI: https://doi.org/10.1007/JHEP01(2024)111