Abstract
We study the superconformal index of \( \mathcal{N} \) = 1 quiver theories at large-N for general values of electric charges and angular momenta, using both the Bethe Ansatz formulation and the more recent elliptic extension method. We are particularly interested in the case of unequal angular momenta, J1 ≠ J2, which has only been partially considered in the literature. We revisit the previous computation with the Bethe Ansatz formulation with generic angular momenta and extend it to encompass a large class of competing exponential terms. In the process, we also provide a simplified derivation of the original result. We consider the newly-developed elliptic extension method as well; we apply it to the J1 ≠ J2 case, finding a good match with the Bethe Ansatz results. We also investigate the relation between the two different approaches, finding in particular that for every saddle of the elliptic action there are corresponding terms in the Bethe Ansatz formula that match it at large-N.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.D. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737 [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
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 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].
A. Zaffaroni, AdS black holes, holography and localization, Living Rev. Rel. 23 (2020) 2 [arXiv:1902.07176] [INSPIRE].
J.B. Gutowski and H.S. Reall, Supersymmetric AdS5 black holes, JHEP 02 (2004) 006 [hep-th/0401042] [INSPIRE].
J.B. Gutowski and H.S. Reall, General supersymmetric AdS5 black holes, JHEP 04 (2004) 048 [hep-th/0401129] [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].
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].
H.K. Kunduri, J. Lucietti and H.S. Reall, Supersymmetric multi-charge AdS5 black holes, JHEP 04 (2006) 036 [hep-th/0601156] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [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].
Y. Nakayama, Index for orbifold quiver gauge theories, Phys. Lett. B 636 (2006) 132 [hep-th/0512280] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, On the Superconformal Index of N = 1 IR Fixed Points: A Holographic Check, JHEP 03 (2011) 041 [arXiv:1011.5278] [INSPIRE].
R. Eager, J. Schmude and Y. Tachikawa, Superconformal Indices, Sasaki-Einstein Manifolds, and Cyclic Homologies, Adv. Theor. Math. Phys. 18 (2014) 129 [arXiv:1207.0573] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
F. Benini and E. Milan, Black Holes in 4D \( \mathcal{N} \) = 4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Zaffaroni, An extremization principle for the entropy of rotating BPS black holes in AdS5, JHEP 07 (2017) 106 [arXiv:1705.05383] [INSPIRE].
A. Cabo-Bizet and S. Murthy, Supersymmetric phases of 4d \( \mathcal{N} \) = 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 \( \mathcal{N} \) = 4 index and partially deconfined phases, JHEP 07 (2020) 073 [arXiv:1912.04169] [INSPIRE].
F. Benini, E. Colombo, S. Soltani, A. Zaffaroni and Z. Zhang, Superconformal indices at large N and the entropy of AdS5 × SE5 black holes, Class. Quant. Grav. 37 (2020) 215021 [arXiv:2005.12308] [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].
O. Aharony, F. Benini, O. Mamroud and E. Milan, A gravity interpretation for the Bethe Ansatz expansion of the \( \mathcal{N} \) = 4 SYM index, Phys. Rev. D 104 (2021) 086026 [arXiv:2104.13932] [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 AdS5 blackholes, JHEP 06 (2019) 134 [arXiv:1902.06619] [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].
K. Goldstein, V. Jejjala, Y. Lei, S. van Leuven and W. Li, Residues, modularity, and the Cardy limit of the 4d \( \mathcal{N} \) = 4 superconformal index, JHEP 04 (2021) 216 [arXiv:2011.06605] [INSPIRE].
A. Amariti, M. Fazzi and A. Segati, The SCI of \( \mathcal{N} \) = 4USp(2Nc) and SO(Nc) SYM as a matrix integral, JHEP 06 (2021) 132 [arXiv:2012.15208] [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].
V. Jejjala, Y. Lei, S. van Leuven and W. Li, SL(3, Z) Modularity and New Cardy limits of the \( \mathcal{N} \) = 4 superconformal index, JHEP 11 (2021) 047 [arXiv:2104.07030] [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].
D. Cassani and L. Papini, The BPS limit of rotating AdS black hole thermodynamics, JHEP 09 (2019) 079 [arXiv:1906.10148] [INSPIRE].
A. González Lezcano and L.A. Pando Zayas, Microstate counting via Bethe Ansätze in the 4d \( \mathcal{N} \) = 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, D. Cassani, D. Martelli and S. Murthy, The large-N limit of the 4d \( \mathcal{N} \) = 1 superconformal index, JHEP 11 (2020) 150 [arXiv:2005.10654] [INSPIRE].
J. Kim, S. Kim and J. Song, A 4d \( \mathcal{N} \) = 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 \( \mathcal{N} \) = 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. Amariti, M. Fazzi and A. Segati, Expanding on the Cardy-like limit of the SCI of 4d \( \mathcal{N} \) = 1 ABCD SCFTs, JHEP 07 (2021) 141 [arXiv:2103.15853] [INSPIRE].
C. Closset, H. Kim and B. Willett, \( \mathcal{N} \) = 1 supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [arXiv:1707.05774] [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].
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].
A. Cabo-Bizet, From multi-gravitons to Black holes: The role of complex saddles, arXiv:2012.04815 [INSPIRE].
G. Felder and A. Varchenko, The elliptic gamma function and SL(3, Z) ⋉ Z3, Adv. Math. 156 (2000) 44 [math/9907061].
A. Weil, Elliptic functions according to eisenstein and kronecker, Ergeb. Math. Grenzgeb. A 8 (1976) [DOI].
W. Duke, On a formula of bloch, Funct. Approximatio Comment. Math. 37 (2007) 109.
V. Paşol and W. Zudilin, A study of elliptic gamma function and allies, Res. Math. Sci. 5 (2018) [arXiv:1801.00210] [INSPIRE].
J.J. Duistermaat and G.J. Heckman, On the Variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259.
J. Duistermaat and G. Heckman, Addendum to “on the variation in the cohomology of the symplectic form of the reduced phase space”, Invent. Math. 72 (1983) 153.
E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].
N. Berline and M. Vergne, Zeros d’un champ de vecteurs et classes caracteristiques equivariantes, Duke Math. J. 50 (1983) 539.
M.F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1 [INSPIRE].
J. Hong and J.T. Liu, The topologically twisted index of \( \mathcal{N} \) = 4 super-Yang-Mills on T2 × S2 and the elliptic genus, JHEP 07 (2018) 018 [arXiv:1804.04592] [INSPIRE].
A.G. Lezcano, J. Hong, J.T. Liu and L.A. Pando Zayas, The Bethe-Ansatz approach to the \( \mathcal{N} \) = 4 superconformal index at finite rank, JHEP 06 (2021) 126 [arXiv:2101.12233] [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].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2110.01911
Rights and permissions
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.
About this article
Cite this article
Colombo, E. The large-N limit of 4d superconformal indices for general BPS charges. J. High Energ. Phys. 2022, 13 (2022). https://doi.org/10.1007/JHEP12(2022)013
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2022)013