Abstract
The flavored superconformal Schur index of \( \mathcal{N} \) = 4 U(N) SYM has finite N corrections encoded in its giant graviton expansion in terms of D3 branes wrapped in AdS5 × S5. The key element of this decomposition is the non-trivial index of the theory living on the wrapped brane system. A remarkable feature of the Schur limit is that the brane index is an analytic continuation of the flavored index of \( \mathcal{N} \) = 4 U(n) SYM, where n is the total brane wrapping number. We exploit recent exact results about the Schur index of \( \mathcal{N} \) = 4 U(N) SYM to evaluate the closed form of the brane indices appearing in the giant graviton expansion. Away from the unflavored limit, they are characterized by quasimodular forms providing exact information at all orders in the index universal fugacity. As an application of these results, we present novel exact expressions for the giant graviton expansion of the unflavored Schur index in a class of four dimensional \( \mathcal{N} \) = 2 theories with equal central charges a = c, i.e. the non-Lagrangian theories \( \hat{\Gamma}\left(\textrm{SU}(N)\right) \) with Γ = E6, E7, E8.
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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].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
C. Romelsberger, Calculating the superconformal index and Seiberg duality, arXiv:0707.3702 [INSPIRE].
E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
P.C. Argyres, J.J. Heckman, K. Intriligator and M. Martone, Snowmass white paper on SCFTs, arXiv:2202.07683 [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
S. Choi, C. Hwang and S. Kim, Quantum vortices, M2-branes and black holes, arXiv:1908.02470 [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].
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 N = 4 super-Yang-Mills field theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [INSPIRE].
A. Zaffaroni, AdS black holes, holography and localization, Living Rev. Rel. 23 (2020) 2 [arXiv:1902.07176] [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.H. Lee, Trace relations and open string vacua, JHEP 02 (2024) 224 [arXiv:2312.00242] [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, JHEP 04 (2024) 146 [arXiv:2308.05191] [INSPIRE].
S. Kim and E. Lee, Holographic tests for giant graviton expansion, arXiv:2402.12924 [INSPIRE].
C.-M. Chang and Y.-H. Lin, Holographic covering and the fortuity of black holes, arXiv:2402.10129 [INSPIRE].
E. Deddo, J.T. Liu, L.A. Pando Zayas and R.J. Saskowski, The giant graviton expansion from bubbling geometry, arXiv:2402.19452 [INSPIRE].
Y. Imamura, Finite-N superconformal index via the AdS/CFT correspondence, PTEP 2021 (2021) 123B05 [arXiv:2108.12090] [INSPIRE].
D. Gaiotto and J.H. Lee, The giant graviton expansion, arXiv:2109.02545 [INSPIRE].
J.H. Lee, Exact stringy microstates from gauge theories, JHEP 11 (2022) 137 [arXiv:2204.09286] [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].
S. Murthy, Unitary matrix models, free fermions, and the giant graviton expansion, Pure Appl. Math. Quart. 19 (2023) 299 [arXiv:2202.06897] [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, 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].
D.S. Eniceicu, Comments on the giant-graviton expansion of the superconformal index, arXiv:2302.04887 [INSPIRE].
M. Beccaria and A. Cabo-Bizet, On the brane expansion of the Schur index, JHEP 08 (2023) 073 [arXiv:2305.17730] [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, Large N Schur index of N = 4 SYM from semiclassical D3 brane, JHEP 04 (2024) 110 [arXiv:2402.12172] [INSPIRE].
F.F. Gautason, V.G.M. Puletti and J. van Muiden, Quantized strings and instantons in holography, JHEP 08 (2023) 218 [arXiv:2304.12340] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d superconformal index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [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].
R. Arai, S. Fujiwara, Y. Imamura and T. Mori, Schur index of the N = 4 U(N) supersymmetric Yang-Mills theory via the AdS/CFT correspondence, Phys. Rev. D 101 (2020) 086017 [arXiv:2001.11667] [INSPIRE].
F.F. Gautason and J. van Muiden, One-loop quantization of Euclidean D3-branes in holographic backgrounds, arXiv:2402.16779 [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].
G. Eleftheriou, S. Murthy and M. Rosselló, The giant graviton expansion in AdS5 × S5, arXiv:2312.14921 [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 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 N = (1, 0) theories from AdS/CFT correspondence, JHEP 05 (2021) 221 [arXiv:2103.16094] [INSPIRE].
Y. Imamura and S. Murayama, Holographic index calculation for Argyres-Douglas and Minahan-Nemeschansky theories, PTEP 2022 (2022) 113B01 [arXiv:2110.14897] [INSPIRE].
S. Fujiwara et al., Simple-sum giant graviton expansions for orbifolds and orientifolds, PTEP 2024 (2024) 023B02 [arXiv:2310.03332] [INSPIRE].
Y. Imamura, Analytic continuation for giant gravitons, PTEP 2022 (2022) 103B02 [arXiv:2205.14615] [INSPIRE].
J. Bourdier, N. Drukker and J. Felix, The exact Schur index of 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].
Y. Hatsuda and T. Okazaki, N = 2∗ Schur indices, JHEP 01 (2023) 029 [arXiv:2208.01426] [INSPIRE].
B.-N. Du, M.-X. Huang and X. Wang, Schur indices for N = 4 super-Yang-Mills with more general gauge groups, JHEP 03 (2024) 009 [arXiv:2311.08714] [INSPIRE].
C. Beem, S.S. Razamat and P. Singh, Schur indices of class S and quasimodular forms, Phys. Rev. D 105 (2022) 085009 [arXiv:2112.10715] [INSPIRE].
S.S. Razamat, On a modular property of N = 2 superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, JHEP 08 (2018) 114 [arXiv:1707.07679] [INSPIRE].
M.-X. Huang, Modular anomaly equation for Schur index of N = 4 super-Yang-Mills, JHEP 08 (2022) 049 [arXiv:2205.00818] [INSPIRE].
H. Zheng, Y. Pan and Y. Wang, Surface defects, flavored modular differential equations, and modularity, Phys. Rev. D 106 (2022) 105020 [arXiv:2207.10463] [INSPIRE].
M. Buican and T. Nishinaka, N = 4 SYM, Argyres-Douglas theories, and an exact graded vector space isomorphism, JHEP 04 (2022) 028 [arXiv:2012.13209] [INSPIRE].
M.J. Kang, C. Lawrie and J. Song, Infinitely many 4D N = 2 SCFTs with a = c and beyond, Phys. Rev. D 104 (2021) 105005 [arXiv:2106.12579] [INSPIRE].
S. Cecotti and M. Del Zotto, Infinitely many N = 2 SCFT with ADE flavor symmetry, JHEP 01 (2013) 191 [arXiv:1210.2886] [INSPIRE].
S. Cecotti, M. Del Zotto and S. Giacomelli, More on the N = 2 superconformal systems of type Dp(G), JHEP 04 (2013) 153 [arXiv:1303.3149] [INSPIRE].
H. Jiang, Modularity in Argyres-Douglas theories with a = c, arXiv:2403.05323 [INSPIRE].
M. Buican and T. Nishinaka, Conformal manifolds in four dimensions and chiral algebras, J. Phys. A 49 (2016) 465401 [arXiv:1603.00887] [INSPIRE].
B. Feigin, E. Feigin and I. Tipunin, Fermionic formulas for (1, p) logarithmic model characters in Φ2,1 quasiparticle realisation, arXiv:0704.2464 [INSPIRE].
B.L. Feigin and I.Y. Tipunin, Characters of coinvariants in (1, p) logarithmic models, arXiv:0805.4096 [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
J.A. Minahan and D. Nemeschansky, Superconformal fixed points with En global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
D. Gaiotto and T. Okazaki, Dualities of corner configurations and supersymmetric indices, JHEP 11 (2019) 056 [arXiv:1902.05175] [INSPIRE].
R. Stanley, Enumerative combinatorics, volume 2, Cambridge University Press, Cambridge, U.K. (2023).
D. Xie, W. Yan and S.-T. Yau, Chiral algebra of the Argyres-Douglas theory from M5 branes, Phys. Rev. D 103 (2021) 065003 [arXiv:1604.02155] [INSPIRE].
Acknowledgments
We thank Arkady Tseytlin, Matthew Buican, Yosuke Imamura, and Ji Hoon Lee for useful discussions related to various aspects of this work. Financial support from the INFN grant GAST is acknowledged. 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. Giant graviton expansion of Schur index and quasimodular forms. J. High Energ. Phys. 2024, 282 (2024). https://doi.org/10.1007/JHEP05(2024)282
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DOI: https://doi.org/10.1007/JHEP05(2024)282