Abstract
We explore the supersymmetric partition functions of 6d SCFTs on ℝ4 × T2 with non-vanishing charges for compatible global symmetries. We utilize the elliptic genera for self-dual strings and compute the free energy of 6d SCFTs in the Cardy limit. For a 6d (2,0) theory on N M5-brane, we obtain the free energy proportional to N3. We find that the origin of N3 comes from the condensation of the self-dual strings, whose total number is proportional to \( \frac{N^3-N}{6} \). We further extend our analysis to the general E-string theory and obtain its Cardy free energy.
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C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].
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].
E. Witten, Some comments on string dynamics, in Strings ′95: future perspectives in string theory, (1995) [hep-th/9507121] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].
S. Bolognesi and K. Lee, 1/4 BPS string junctions and N3 problem in 6-dim (2, 0) superconformal theories, Phys. Rev. D 84 (2011) 126018 [arXiv:1105.5073] [INSPIRE].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
J. Kallen, J.A. Minahan, A. Nedelin and M. Zabzine, N3-behavior from 5D Yang-Mills theory, JHEP 10 (2012) 184 [arXiv:1207.3763] [INSPIRE].
J.A. Minahan, A. Nedelin and M. Zabzine, 5D super Yang-Mills theory and the correspondence to AdS7/CFT6, J. Phys. A 46 (2013) 355401 [arXiv:1304.1016] [INSPIRE].
H.-C. Kim, S. Kim, S.-S. Kim and K. Lee, The general M5-brane superconformal index, arXiv:1307.7660 [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir energy and the anomaly polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
D. Freed, J.A. Harvey, R. Minasian and G.W. Moore, Gravitational anomaly cancellation for M-theory five-branes, Adv. Theor. Math. Phys. 2 (1998) 601 [hep-th/9803205] [INSPIRE].
T. Maxfield and S. Sethi, The conformal anomaly of M5-branes, JHEP 06 (2012) 075 [arXiv:1204.2002] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
K. Intriligator, 6d, N = (1, 0) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [INSPIRE].
S.M. Hosseini, I. Yaakov and A. Zaffaroni, Topologically twisted indices in five dimensions and holography, JHEP 11 (2018) 119 [arXiv:1808.06626] [INSPIRE].
H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [arXiv:1110.2175] [INSPIRE].
S. Kim and J. Nahmgoong, Asymptotic M5-brane entropy from S-duality, JHEP 12 (2017) 120 [arXiv:1702.04058] [INSPIRE].
J. Nahmgoong, 6d superconformal Cardy formulas, JHEP 02 (2021) 092 [arXiv:1907.12582] [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].
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].
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [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].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP 12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Comments on deconfinement in AdS/CFT, arXiv:1811.08646 [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
S. Choi and S. Kim, Large AdS6 black holes from CFT5, arXiv:1904.01164 [INSPIRE].
G. Kántor, C. Papageorgakis and P. Richmond, AdS7 black-hole entropy and 5D N = 2 Yang-Mills, JHEP 01 (2020) 017 [arXiv:1907.02923] [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].
P.S. Howe, N.D. Lambert and P.C. West, The selfdual string soliton, Nucl. Phys. B 515 (1998) 203 [hep-th/9709014] [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
M. Del Zotto, J. Gu, M.-X. Huang, A.-K. Kashani-Poor, A. Klemm and G. Lockhart, Topological strings on singular elliptic Calabi-Yau 3-folds and minimal 6d SCFTs, JHEP 03 (2018) 156 [arXiv:1712.07017] [INSPIRE].
J. Kim, K. Lee and J. Park, On elliptic genera of 6d string theories, JHEP 10 (2018) 100 [arXiv:1801.01631] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].
O.J. Ganor and A. Hanany, Small E8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
J. Kim, S. Kim, K. Lee, J. Park and C. Vafa, Elliptic genus of E-strings, JHEP 09 (2017) 098 [arXiv:1411.2324] [INSPIRE].
W. Cai, M.-X. Huang and K. Sun, On the elliptic genus of three E-strings and heterotic strings, JHEP 01 (2015) 079 [arXiv:1411.2801] [INSPIRE].
S. Kim and K. Lee, Indices for 6 dimensional superconformal field theories, J. Phys. A 50 (2017) 443017 [arXiv:1608.02969] [INSPIRE].
M. Bullimore, H.-C. Kim and P. Koroteev, Defects and quantum Seiberg-Witten geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].
M. Bullimore and H.-C. Kim, The superconformal index of the (2, 0) theory with defects, JHEP 05 (2015) 048 [arXiv:1412.3872] [INSPIRE].
B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].
J. Kim, S. Kim and K. Lee, Little strings and T-duality, JHEP 02 (2016) 170 [arXiv:1503.07277] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
M.V. Berry, Stokes’s phenomenon for superfactorial asymptotic series, Proc. Roy. Soc. Lond. A 435 (1991) 437.
M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Modular anomaly equation, heat kernel and S-duality in N = 2 theories, JHEP 11 (2013) 123 [arXiv:1307.6648] [INSPIRE].
N.M. Temme, Asymptotic methods for integrals, World Scientific, Singapore (2014).
J.S. Lee, C. Kwon and H. Park, Modified saddle-point integral near a singularity for the large deviation function, J. Stat. Mech. 2013 (2013) P11002.
K.E. Oughstun and G.C. Sherman, Analysis of the phase function and its saddle points, in Electromagnetic pulse propagation in causal dielectrics, Springer, Berlin, Heidelberg, Germany (1994), pg. 192.
P. Hořava and E. Witten, Heterotic and type-I string dynamics from eleven-dimensions, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209] [INSPIRE].
A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, Nucl. Phys. B Proc. Suppl. 58 (1997) 177 [hep-th/9607139] [INSPIRE].
J.A. Minahan, D. Nemeschansky, C. Vafa and N.P. Warner, E-strings and N = 4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581 [hep-th/9802168] [INSPIRE].
A. Gadde, B. Haghighat, J. Kim, S. Kim, G. Lockhart and C. Vafa, 6d string chains, JHEP 02 (2018) 143 [arXiv:1504.04614] [INSPIRE].
J. Kim, S. Kim and K. Lee, Higgsing towards E-strings, JHEP 01 (2021) 110 [arXiv:1510.03128] [INSPIRE].
H.-C. Kim, S. Kim and J. Park, 6d strings from new chiral gauge theories, arXiv:1608.03919 [INSPIRE].
C.-M. Chang, M. Fluder, Y.-H. Lin and Y. Wang, Proving the 6d Cardy formula and matching global gravitational anomalies, arXiv:1910.10151 [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, JHEP 05 (2014) 110 [arXiv:1311.2935] [INSPIRE].
J. Kim, S. Kim and J. Song, A 4d N = 1 Cardy formula, JHEP 01 (2021) 025 [arXiv:1904.03455] [INSPIRE].
O. Aharony, M. Berkooz, S. Kachru, N. Seiberg and E. Silverstein, Matrix description of interacting theories in six-dimensions, Adv. Theor. Math. Phys. 1 (1998) 148 [hep-th/9707079] [INSPIRE].
O. Aharony, M. Berkooz and N. Seiberg, Light cone description of (2, 0) superconformal theories in six-dimensions, Adv. Theor. Math. Phys. 2 (1998) 119 [hep-th/9712117] [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].
F. Benini and P. Milan, Black holes in 4D N = 4 super-Yang-Mills field theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for superconformal field theories in 3, 5 and 6 dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Zaffaroni, A note on the entropy of rotating BPS AdS7 × S4 black holes, JHEP 05 (2018) 121 [arXiv:1803.07568] [INSPIRE].
R. Gopakumar and C. Vafa, M-theory and topological strings. 2, hep-th/9812127 [INSPIRE].
J. Gu, M.-X. Huang, A.-K. Kashani-Poor and A. Klemm, Refined BPS invariants of 6d SCFTs from anomalies and modularity, JHEP 05 (2017) 130 [arXiv:1701.00764] [INSPIRE].
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Lee, K., Nahmgoong, J. Cardy limits of 6d superconformal theories. J. High Energ. Phys. 2021, 118 (2021). https://doi.org/10.1007/JHEP05(2021)118
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DOI: https://doi.org/10.1007/JHEP05(2021)118