Abstract
We investigate the Bethe-Ansatz approach to the superconformal index of \( \mathcal{N} \) = 4 supersymmetric Yang-Mills with SU(N) gauge group in the context of finite rank, N. We explicitly explore the role of the various types of solutions to the Bethe-Ansatz Equations in recovering the exact index for N = 2, 3. We classify the Bethe-Ansatz Equations solutions as standard (corresponding to a freely acting orbifold T2/ℤm × ℤn) and non-standard. For N = 2, we find that the index is fully recovered by standard solutions and displays an interesting pattern of cancellations. However, for N ≥ 3, the standard solutions alone do not suffice to reconstruct the index. We present quantitative arguments in various regimes of fugacities that highlight the challenging role played by the continuous families of non-standard solutions.
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 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].
F. Benini and P. Milan, Black Holes in 4D \( \mathcal{N} \) = 4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [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].
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].
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].
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, arXiv:1904.10009 [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. 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].
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].
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].
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].
S. Murthy, The growth of the \( \frac{1}{16} \)-BPS index in 4d \( \mathcal{N} \) = 4 SYM, arXiv:2005.10843 [INSPIRE].
P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong, AdS black holes and finite N indices, Phys. Rev. D 103 (2021) 126006 [arXiv:2005.11240] [INSPIRE].
C. Copetti, A. Grassi, Z. Komargodski and L. Tizzano, Delayed Deconfinement and the Hawking-Page Transition, arXiv:2008.04950 [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. 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 superconformal index of \( \mathcal{N} \) = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral, arXiv:2012.15208 [INSPIRE].
O. Aharony, A gravity interpretation for the Bethe ansatz expansion of the N = 4 SYM index, talk at SCGP workshop (2020).
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].
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].
V. P. Spiridonov and G. S. Vartanov, Superconformal indices of \( \mathcal{N} \) = 4 SYM field theories, Lett. Math. Phys. 100 (2012) 97 [arXiv:1005.4196] [INSPIRE].
F. Benini and P. Milan, A Bethe Ansatz type formula for the superconformal index, Commun. Math. Phys. 376 (2020) 1413 [arXiv:1811.04107] [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].
C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP 03 (2017) 074 [arXiv:1701.03171] [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].
A. Gadde, Modularity of supersymmetric partition functions, arXiv:2004.13490 [INSPIRE].
J. T. Liu, L. A. Pando Zayas, V. Rathee and W. Zhao, Toward Microstate Counting Beyond Large N in Localization and the Dual One-loop Quantum Supergravity, JHEP 01 (2018) 026 [arXiv:1707.04197] [INSPIRE].
J. T. Liu, L. A. Pando Zayas, V. Rathee and W. Zhao, One-Loop Test of Quantum Black Holes in anti-de Sitter Space, Phys. Rev. Lett. 120 (2018) 221602 [arXiv:1711.01076] [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].
F. Benini, D. Gang and L. A. Pando Zayas, Rotating Black Hole Entropy from M5 Branes, JHEP 03 (2020) 057 [arXiv:1909.11612] [INSPIRE].
N. Bobev, A. M. Charles, K. Hristov and V. Reys, The Unreasonable Effectiveness of Higher-Derivative Supergravity in AdS4 Holography, Phys. Rev. Lett. 125 (2020) 131601 [arXiv:2006.09390] [INSPIRE].
L. A. Pando Zayas and Y. Xin, Universal logarithmic behavior in microstate counting and the dual one-loop entropy of AdS4 black holes, Phys. Rev. D 103 (2021) 026003 [arXiv:2008.03239] [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: 2101.12233
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
Lezcano, A.G., Hong, J., Liu, J.T. et al. The Bethe-Ansatz approach to the \( \mathcal{N} \) = 4 superconformal index at finite rank. J. High Energ. Phys. 2021, 126 (2021). https://doi.org/10.1007/JHEP06(2021)126
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2021)126