Abstract
We consider the refined Schur superconformal index of 4d \( \mathcal{N} \) = 4 U(N) SYM and the first term of its giant-graviton expansion, first predicted in arXiv:2001.11667 using indirect superconformal algebra considerations and analytic continuation of fugacities. This correction is the leading non-perturbative correction to the index at large N and we reproduce it from the semiclassical partition function of quantum D3 brane wrapped on S1 × S3 in a twisted modification of the AdS5 × S5 string background, depending on the index R-symmetry fugacity. Our calculation does not exploit directly supersymmetry. It is based on the determination of the partition function of the various bosonic and fermionic fluctuations on the wrapped brane whose action is conformal with specific constant holonomies along thermal cycle. We show how those partition functions may be obtained by adapting the operator counting method of Cardy to the twisted background.
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J.M. Maldacena and A. Strominger, AdS3 Black Holes and a Stringy Exclusion Principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [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].
D. Gaiotto and J.H. Lee, The Giant Graviton Expansion, arXiv:2109.02545 [INSPIRE].
R. Arai and Y. Imamura, Finite N Corrections to the Superconformal Index of S-fold Theories, PTEP 2019 (2019) 083B04 [arXiv:1904.09776] [INSPIRE].
R. Arai, S. Fujiwara, Y. Imamura and T. Mori, Finite N corrections to the superconformal index of orbifold quiver gauge theories, JHEP 10 (2019) 243 [arXiv:1907.05660] [INSPIRE].
R. Arai, S. Fujiwara, Y. Imamura and T. Mori, Finite N corrections to the superconformal index of toric quiver gauge theories, PTEP 2020 (2020) 043B09 [arXiv:1911.10794] [INSPIRE].
R. Arai, S. Fujiwara, Y. Imamura and T. Mori, Schur index of the \( \mathcal{N} \) = 4U(N) supersymmetric Yang-Mills theory via the AdS/CFT correspondence, Phys. Rev. D 101 (2020) 086017 [arXiv:2001.11667] [INSPIRE].
R. Arai et al., Finite-N corrections to the M-brane indices, JHEP 11 (2020) 093 [arXiv:2007.05213] [INSPIRE].
S. Fujiwara, Y. Imamura and T. Mori, Flavor symmetries of six-dimensional \( \mathcal{N} \) = (1, 0) theories from AdS/CFT correspondence, JHEP 05 (2021) 221 [arXiv:2103.16094] [INSPIRE].
Y. Imamura, Finite-N superconformal index via the AdS/CFT correspondence, PTEP 2021 (2021) 123B05 [arXiv:2108.12090] [INSPIRE].
Y. Imamura and S. Murayama, Holographic index calculation for Argyres-Douglas and Minahan-Nemeschansky theories, PTEP 2022 (2022) 113B01 [arXiv:2110.14897] [INSPIRE].
Y. Imamura, Analytic continuation for giant gravitons, PTEP 2022 (2022) 103B02 [arXiv:2205.14615] [INSPIRE].
S. Fujiwara et al., Simple-Sum Giant Graviton Expansions for Orbifolds and Orientifolds, PTEP 2024 (2024) 023B02 [arXiv:2310.03332] [INSPIRE].
N. Bobev, F.F. Gautason and J. van Muiden, The conformal manifold of S-folds in string theory, JHEP 03 (2024) 167 [arXiv:2312.13370] [INSPIRE].
S. Murthy, Unitary matrix models, free fermions, and the giant graviton expansion, Pure Appl. Math. Quart. 19 (2023) 299 [arXiv:2202.06897] [INSPIRE].
D.S. Eniceicu, R. Mahajan and C. Murdia, Complex eigenvalue instantons and the Fredholm determinant expansion in the Gross-Witten-Wadia model, JHEP 01 (2024) 129 [arXiv:2308.06320] [INSPIRE].
J.T. Liu and N.J. Rajappa, Finite N indices and the giant graviton expansion, JHEP 04 (2023) 078 [arXiv:2212.05408] [INSPIRE].
D.S. Eniceicu, Comments on the Giant-Graviton Expansion of the Superconformal Index, arXiv:2302.04887 [INSPIRE].
J.H. Lee, Exact stringy microstates from gauge theories, JHEP 11 (2022) 137 [arXiv:2204.09286] [INSPIRE].
J.H. Lee, Trace relations and open string vacua, JHEP 02 (2024) 224 [arXiv:2312.00242] [INSPIRE].
G. Eleftheriou, S. Murthy and M. Rosselló, The giant graviton expansion in AdS5 × S5, arXiv:2312.14921 [INSPIRE].
S. Choi, S. Kim, E. Lee and J. Lee, From giant gravitons to black holes, JHEP 11 (2023) 086 [arXiv:2207.05172] [INSPIRE].
M. Beccaria and A. Cabo-Bizet, Large black hole entropy from the giant brane expansion, arXiv:2308.05191 [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].
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].
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. Cabo-Bizet and S. Murthy, Supersymmetric phases of 4d \( \mathcal{N} \) = 4 SYM at large N, JHEP 09 (2020) 184 [arXiv:1909.09597] [INSPIRE].
V. Balasubramanian, M.-X. Huang, T.S. Levi and A. Naqvi, Open strings from N = 4 superYang-Mills, JHEP 08 (2002) 037 [hep-th/0204196] [INSPIRE].
V. Balasubramanian, D. Berenstein, B. Feng and M.-X. Huang, D-branes in Yang-Mills theory and emergent gauge symmetry, JHEP 03 (2005) 006 [hep-th/0411205] [INSPIRE].
R. de Mello Koch, J. Smolic and M. Smolic, Giant Gravitons — with Strings Attached (I), JHEP 06 (2007) 074 [hep-th/0701066] [INSPIRE].
M. Beccaria, S. Giombi and A.A. Tseytlin, (2, 0) theory on S5 × S1 and quantum M2 branes, Nucl. Phys. B 998 (2024) 116400 [arXiv:2309.10786] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Large N expansion of superconformal index of k = 1 ABJM theory and semiclassical M5 brane partition function, Nucl. Phys. B 1001 (2024) 116507 [arXiv:2312.01917] [INSPIRE].
M. Beccaria and A. Cabo-Bizet, On the brane expansion of the Schur index, JHEP 08 (2023) 073 [arXiv:2305.17730] [INSPIRE].
J. Bourdier, N. Drukker and J. Felix, The \( \mathcal{N} \) = 2 Schur index from free fermions, JHEP 01 (2016) 167 [arXiv:1510.07041] [INSPIRE].
J. Bourdier, N. Drukker and J. Felix, The exact Schur index of \( \mathcal{N} \) = 4 SYM, JHEP 11 (2015) 210 [arXiv:1507.08659] [INSPIRE].
Y. Pan and W. Peelaers, Exact Schur index in closed form, Phys. Rev. D 106 (2022) 045017 [arXiv:2112.09705] [INSPIRE].
J.L. Cardy, Operator content and modular properties of higher dimensional conformal field theories, Nucl. Phys. B 366 (1991) 403 [INSPIRE].
D. Kutasov and F. Larsen, Partition sums and entropy bounds in weakly coupled CFT, JHEP 01 (2001) 001 [hep-th/0009244] [INSPIRE].
O. Aharony et al., The Hagedorn — deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
B. Assel et al., The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].
B. Assel, D. Cassani and D. Martelli, Supersymmetric counterterms from new minimal supergravity, JHEP 11 (2014) 135 [arXiv:1410.6487] [INSPIRE].
D. Cassani and D. Martelli, The gravity dual of supersymmetric gauge theories on a squashed S1 × S3, JHEP 08 (2014) 044 [arXiv:1402.2278] [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir Energy and the Anomaly Polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
C. Beem et al., Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
F.A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [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. Mikhailov, Giant gravitons from holomorphic surfaces, JHEP 11 (2000) 027 [hep-th/0010206] [INSPIRE].
A. Mikhailov, Nonspherical giant gravitons and matrix theory, hep-th/0208077 [INSPIRE].
M.C. Abbott, J. Murugan, A. Prinsloo and N. Rughoonauth, Meromorphic Functions and the Topology of Giant Gravitons, Phys. Lett. B 730 (2014) 215 [arXiv:1312.4900] [INSPIRE].
E. Bergshoeff and P.K. Townsend, Super D-branes, Nucl. Phys. B 490 (1997) 145 [hep-th/9611173] [INSPIRE].
G.W. Gibbons, M.J. Perry and C.N. Pope, Partition functions, the Bekenstein bound and temperature inversion in anti-de Sitter space and its conformal boundary, Phys. Rev. D 74 (2006) 084009 [hep-th/0606186] [INSPIRE].
S. Giombi, I.R. Klebanov and A.A. Tseytlin, Partition Functions and Casimir Energies in Higher Spin AdSd+1/CFTd, Phys. Rev. D 90 (2014) 024048 [arXiv:1402.5396] [INSPIRE].
D. Marolf, L. Martucci and P.J. Silva, Fermions, T duality and effective actions for D-branes in bosonic backgrounds, JHEP 04 (2003) 051 [hep-th/0303209] [INSPIRE].
D. Marolf, L. Martucci and P.J. Silva, Actions and Fermionic symmetries for D-branes in bosonic backgrounds, JHEP 07 (2003) 019 [hep-th/0306066] [INSPIRE].
D. Marolf, L. Martucci and P.J. Silva, The explicit form of the effective action for F1 and D-branes, Class. Quant. Grav. 21 (2004) S1385 [hep-th/0404197] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Higher spins in AdS5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT, JHEP 11 (2014) 114 [arXiv:1410.3273] [INSPIRE].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].
H.-C. Kim, S. Kim, S.-S. Kim and K. Lee, The general M5-brane superconformal index, arXiv:1307.7660 [INSPIRE].
R. Lehoucq, J.-P. Uzan and J. Weeks, Eigenmodes of Lens and Prism Spaces, math/0202072.
M. Lachieze-Rey, Laplacian eigenmodes for the three-sphere, J. Phys. A 37 (2004) 5625.
Y. Hatsuda and T. Okazaki, \( \mathcal{N} \) = 2* Schur indices, JHEP 01 (2023) 029 [arXiv:2208.01426] [INSPIRE].
Acknowledgments
We thank Arkady A. Tseytlin, Yosuke Imamura, and Ji Hoon Lee for useful discussions related to various aspects of this work. MB and ACB are supported by the INFN grants GSS and GAST. ACB would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Black holes: bridges between number theory and holographic quantum information, where work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1.
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Beccaria, M., Cabo-Bizet, A. Large N Schur index of \( \mathcal{N} \) = 4 SYM from semiclassical D3 brane. J. High Energ. Phys. 2024, 110 (2024). https://doi.org/10.1007/JHEP04(2024)110
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DOI: https://doi.org/10.1007/JHEP04(2024)110