Abstract
We show that the superconformal index of \( \mathcal{N}=1 \) superconformal field theories in four dimensions has an asymptotic growth of states which is exponential in the charges. Our analysis holds in a Cardy-like limit of large charges, for which the index is dominated by small values of chemical potentials. In this limit we find the saddle points of the integral that defines the superconformal index using two different methods. One method, valid for finite N, is to first take the Cardy-like limit and then find the saddle points. The other method is to analyze the saddle points at large N and then take the Cardy-like limit. The result of both analyses is that the asymptotic growth of states of the superconformal index exactly agrees with the Bekenstein-Hawking entropy of supersymmetric black holes in the dual AdS5 theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5black holes, arXiv:1810.11442 [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Comments on deconfinement in AdS/CFT, arXiv:1811.08646 [INSPIRE].
F. Benini and P. Milan, Black holes in 4d \( \mathcal{N}=4 \)Super-Yang-Mills, arXiv:1812.09613 [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 \( \mathcal{N}=4 \)index and AdS 5blackholes, JHEP06 (2019) 134 [arXiv:1902.06619] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Zaffaroni, An extremization principle for the entropy of rotating BPS black holes in AdS 5, JHEP07 (2017) 106 [arXiv:1705.05383] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS 4from supersymmetric localization, JHEP05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
J.B. Gutowski and H.S. Reall, Supersymmetric AdS 5black holes, JHEP02 (2004) 006 [hep-th/0401042] [INSPIRE].
J.B. Gutowski and H.S. Reall, General supersymmetric AdS 5black holes, JHEP04 (2004) 048 [hep-th/0401129] [INSPIRE].
Z.W. Chong, M. Cvetič, H. Lü and C.N. Pope, General non-extremal rotating black holes in minimal five-dimensional gauged supergravity, Phys. Rev. Lett.95 (2005) 161301 [hep-th/0506029] [INSPIRE].
Z.W. Chong, M. Cvetič, H. Lü and C.N. Pope, Five-dimensional gauged supergravity black holes with independent rotation parameters, Phys. Rev.D 72 (2005) 041901 [hep-th/0505112] [INSPIRE].
H.K. Kunduri, J. Lucietti and H.S. Reall, Supersymmetric multi-charge AdS 5black holes, JHEP04 (2006) 036 [hep-th/0601156] [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].
B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP08 (2014) 123 [arXiv:1405.5144] [INSPIRE].
B. Assel, D. Cassani, L. Di Pietro, Z. Komargodski, J. Lorenzen and D. Martelli, The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP07 (2015) 043 [arXiv:1503.05537] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys.B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
A. Sen, Entropy Function and AdS 2/CF T 1Correspondence, JHEP11 (2008) 075 [arXiv:0805.0095] [INSPIRE].
S. Murthy and B. Pioline, A Farey tale for N = 4 dyons, JHEP09 (2009) 022 [arXiv:0904.4253] [INSPIRE].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
A. Arabi Ardehali, High-temperature asymptotics of supersymmetric partition functions, JHEP07 (2016) 025 [arXiv:1512.03376] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett.B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
E. Shaghoulian, Modular Invariance of Conformal Field Theory on S 1 × S 3and Circle Fibrations, Phys. Rev. Lett.119 (2017) 131601 [arXiv:1612.05257] [INSPIRE].
S. Kim and K.-M. Lee, 1/16-BPS Black Holes and Giant Gravitons in the AdS 5× S 5Space, JHEP12 (2006) 077 [hep-th/0607085] [INSPIRE].
J. Kim, S. Kim and J. Song, A 4d N = 1 Cardy Formula, arXiv:1904.03455 [INSPIRE].
L. Di Pietro and M. Honda, Cardy Formula for 4d SUSY Theories and Localization, JHEP04 (2017) 055 [arXiv:1611.00380] [INSPIRE].
A. Arabi Ardehali, J.T. Liu and P. Szepietowski, High-Temperature Expansion of Supersymmetric Partition Functions, JHEP07 (2015) 113 [arXiv:1502.07737] [INSPIRE].
S.M. Hosseini, A. Nedelin and A. Zaffaroni, The Cardy limit of the topologically twisted index and black strings in AdS 5, JHEP04 (2017) 014 [arXiv:1611.09374] [INSPIRE].
C. Hwang, S. Lee and P. Yi, Holonomy Saddles and Supersymmetry, Phys. Rev.D 97 (2018) 125013 [arXiv:1801.05460] [INSPIRE].
A. Parkes and P.C. West, Finiteness in Rigid Supersymmetric Theories, Phys. Lett.138B (1984) 99 [INSPIRE].
A.J. Parkes and P.C. West, Three Loop Results in Two Loop Finite Supersymmetric Gauge Theories, Nucl. Phys. B256 (1985) 340 [INSPIRE].
P.C. West, The Yukawa β-function in N = 1 Rigid Supersymmetric Theories, Phys. Lett.137B (1984) 371 [INSPIRE].
G. Felder and A. Varchenko, The elliptic gamma function and SL(3, Z) × Z 3, math/9907061.
V.P. Spiridonov, Elliptic beta integrals and solvable models of statistical mechanics, Contemp. Math.563 (2012) 181 [arXiv:1011.3798] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions, JHEP06 (2012) 016 [arXiv:1203.5677] [INSPIRE].
E.M. Rains, Limits of elliptic hypergeometric integrals, Ramanujan J.18 (2007) 257 [math/0607093].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
K.A. Intriligator and B. Wecht, The Exact superconformal R symmetry maximizes a, Nucl. Phys.B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys.B 536 (1998) 199 [hep-th/9807080] [INSPIRE].
S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks, An Infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals, JHEP06 (2005) 064 [hep-th/0411264] [INSPIRE].
S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, Gauge theories from toric geometry and brane tilings, JHEP01 (2006) 128 [hep-th/0505211] [INSPIRE].
A. Butti, D. Forcella and A. Zaffaroni, The Dual superconformal theory for L p,q,rmanifolds, JHEP09 (2005) 018 [hep-th/0505220] [INSPIRE].
S. Benvenuti and M. Kruczenski, From Sasaki-Einstein spaces to quivers via BPS geodesics: L p,q|r , JHEP04 (2006) 033 [hep-th/0505206] [INSPIRE].
A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions, math/0306164.
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] [INSPIRE].
F.A. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys.B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1904.05865
On leave at the Galileo Galilei Institute, Largo Enrico Fermi, 2, 50125 Firenze, Italy. (Dario Martelli)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Cabo-Bizet, A., Cassani, D., Martelli, D. et al. The asymptotic growth of states of the 4d \( \mathcal{N}=1 \) superconformal index. J. High Energ. Phys. 2019, 120 (2019). https://doi.org/10.1007/JHEP08(2019)120
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2019)120