Abstract
There is a correspondence between the protected local operators in the 3d SCFTs describing the geometry ℂ2 probed by a stack of N M2-branes and plane partitions of trace N. We give combinatorial expressions of the indices which count the local operators parametrizing ℂ2/ℤk probed by N M2-branes in the canonical and grand canonical ensembles in terms of generating functions for plane partitions. We derive the asymptotic behaviors of the grand potential in the high-temperature limit and the scaling dimension in the large N limit.
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O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M2-branes at toric CY4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [INSPIRE].
D. Bashkirov and A. Kapustin, Supersymmetry enhancement by monopole operators, JHEP 05 (2011) 015 [arXiv:1007.4861] [INSPIRE].
M. Mezei and S. S. Pufu, Three-sphere free energy for classical gauge groups, JHEP 02 (2014) 037 [arXiv:1312.0920] [INSPIRE].
S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
A. Kapustin and B. Willett, Generalized superconformal index for three dimensional field theories, arXiv:1106.2484 [INSPIRE].
S. S. Razamat and B. Willett, Down the rabbit hole with theories of class \( \mathcal{S} \), JHEP 10 (2014) 099 [arXiv:1403.6107] [INSPIRE].
H. Hayashi, T. Nosaka and T. Okazaki, Dualities and flavored indices of M2-brane SCFTs, arXiv:2206.05362 [INSPIRE].
S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].
J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, Z) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].
Y. Imamura and S. Yokoyama, N = 4 Chern-Simons theories and wrapped M-branes in their gravity duals, Prog. Theor. Phys. 121 (2009) 915 [arXiv:0812.1331] [INSPIRE].
Y. Imamura and K. Kimura, N = 4 Chern-Simons theories with auxiliary vector multiplets, JHEP 10 (2008) 040 [arXiv:0807.2144] [INSPIRE].
R. P. Stanley, Theory and application of plane partitions. Part 1, Stud. Appl. Math. 50 (1971) 167.
R. P. Stanley, Theory and application of plane partitions. Part 2, Stud. Appl. Math. 50 (1971) 259.
G. E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, U.K. (1998).
D. E. Littlewood, The theory of group characters and matrix representations of groups, American Mathematical Society, Providence, RI, U.S.A. (2006).
R. P. Stanley, The conjugate trace and trace of a plane partition, J. Combinat. Theor. A 14 (1973) 53.
D. Gaiotto and T. Okazaki, Sphere correlation functions and Verma modules, JHEP 02 (2020) 133 [arXiv:1911.11126] [INSPIRE].
R. Kodera and H. Nakajima, Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras, Proc. Symp. Pure Math. 98 (2018) 49 [arXiv:1608.00875] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, sixth edition, Oxford University Press, Oxford, U.K. (2008).
P. A. MacMahon, IX. Memoir on the theory of the partitions of numbers — part VI. Partitions in two-dimensional space, to which is added an adumbration of the theory of the partitions in three-dimensional space, Phil. Trans. Roy. Soc. Lond. A 211 (1912) 345.
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].
B. Henning, X. Lu, T. Melia and H. Murayama, Operator bases, S-matrices, and their partition functions, JHEP 10 (2017) 199 [arXiv:1706.08520] [INSPIRE].
T. Melia and S. Pal, EFT asymptotics: the growth of operator degeneracy, SciPost Phys. 10 (2021) 104 [arXiv:2010.08560] [INSPIRE].
G. Meinardus, Asymptotische Aussagen über Partitionen (in German), Math. Z. 59 (1953) 388.
B. L. Granovsky and D. Stark, A Meinardus theorem with multiple singularities, Commun. Math. Phys. 314 (2012) 329.
E. M. Wright, Asymptotic partition formulae: (II) weighted partitions, Proc. Lond. Math. Soc. s2-36 (1934) 117.
R. Arai, S. Fujiwara, Y. Imamura, T. Mori and D. Yokoyama, Finite-N corrections to the M-brane indices, JHEP 11 (2020) 093 [arXiv:2007.05213] [INSPIRE].
D. Gaiotto and J. H. Lee, The giant graviton expansion, arXiv:2109.02545 [INSPIRE].
N. Nekrasov, Magnificent four, Adv. Theor. Math. Phys. 24 (2020) 1171 [arXiv:1712.08128] [INSPIRE].
G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, U.K. (2004).
E. R. Gansner, The Hillman-grassl correspondence and the enumeration of reverse plane partitions, J. Combinat. Theor. A 30 (1981) 71.
E. R. Gansner, Matrix correspondences of plane partitions, Pacific J. Math. 92 (1981) 295.
E. A. Bender and D. E. Knuth, Enumeration of plane partitions, J. Combinat. Theor. A 13 (1972) 40.
P. A. MacMahon, Combinatory analysis, volume I and II bound in one volume, Dover Publications Inc., Mineola, NY, U.S.A. (2004).
R. P. Stanley, Symmetries of plane partitions, J. Combinat. Theor. A 43 (1986) 103.
I. R. Klebanov and A. A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].
H. Fuji, S. Hirano and S. Moriyama, Summing up all genus free energy of ABJM matrix model, JHEP 08 (2011) 001 [arXiv:1106.4631] [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].
C. P. Herzog, I. R. Klebanov, S. S. Pufu and T. Tesileanu, Multi-matrix models and tri-Sasaki Einstein spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].
R. C. Santamaria, M. Mariño and P. Putrov, Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories, JHEP 10 (2011) 139 [arXiv:1011.6281] [INSPIRE].
D. Martelli and J. Sparks, The large N limit of quiver matrix models and Sasaki-Einstein manifolds, Phys. Rev. D 84 (2011) 046008 [arXiv:1102.5289] [INSPIRE].
S. Cheon, H. Kim and N. Kim, Calculating the partition function of N = 2 gauge theories on S3 and AdS/CFT correspondence, JHEP 05 (2011) 134 [arXiv:1102.5565] [INSPIRE].
D. L. Jafferis, I. R. Klebanov, S. S. Pufu and B. R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
M. Gabella, D. Martelli, A. Passias and J. Sparks, The free energy of N = 2 supersymmetric AdS4 solutions of M-theory, JHEP 10 (2011) 039 [arXiv:1107.5035] [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. Azzurli, N. Bobev, P. M. Crichigno, V. S. Min and A. Zaffaroni, A universal counting of black hole microstates in AdS4, JHEP 02 (2018) 054 [arXiv:1707.04257] [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].
S. Choi, C. Hwang and S. Kim, Quantum vortices, M2-branes and black holes, arXiv:1908.02470 [INSPIRE].
K. Bringmann, W. Craig, J. Males and K. Ono, Distributions on partitions arising from Hilbert schemes and hook lengths, arXiv:2109.10394.
G. Cesana, W. Craig and J. Males, Asymptotic equidistribution for partition statistics and topological invariants, arXiv:2111.13766.
D. Wood, The computation of polylogarithms, Tech. Rep. 15-92*, Computing Laboratory, University of Kent, Canterbury, U.K., June 1992.
A. Grassi and M. Mariño, M-theoretic matrix models, JHEP 02 (2015) 115 [arXiv:1403.4276] [INSPIRE].
Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].
L. Mutafchiev, Asymptotic analysis of expectations of plane partition statistics, Abh. Math. Semin. Univ. Hambg. 88 (2018) 255.
W. K. Hayman, A generalisation of Stirling’s formula., J. Reine Angew. Math. 1956 (1956) 67.
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Okazaki, T. M2-branes and plane partitions. J. High Energ. Phys. 2022, 28 (2022). https://doi.org/10.1007/JHEP07(2022)028
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DOI: https://doi.org/10.1007/JHEP07(2022)028