ABSTRACT
We present a holographic derivation of the entropy of supersymmetric asymptotically AdS5 black holes. We define a BPS limit of black hole thermodynamics by first focussing on a supersymmetric family of complexified solutions and then reaching extremality. We show that in this limit the black hole entropy is the Legendre transform of the on-shell gravitational action with respect to three chemical potentials subject toa constraint. This constraint follows from supersymmetry and regularity in the Euclidean bulk geometry. Further, we calculate, using localization, the exact partition function of the dual \( \mathcal{N} \) = 1 SCFT on a twisted S1 × S3 with complexified chemical potentials obeying this constraint. This defines a generalization of the supersymmetric Casimir energy, whose Legendre transform at large N exactly reproduces the Bekenstein-Hawking entropy of the black hole.
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References
A. Sen, Extremal black holes and elementary string states, Mod. Phys. Lett.A 10 (1995) 2081 [hep-th/9504147] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett.B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4from supersymmetric localization, JHEP05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
J.B. Gutowski and H.S. Reall, Supersymmetric AdS5black holes, JHEP02 (2004) 006 [hep-th/0401042] [INSPIRE].
J.B. Gutowski and H.S. Reall, General supersymmetric AdS5black holes, JHEP04 (2004) 048 [hep-th/0401129] [INSPIRE].
Z.W. Chong, M. Cvetic, H. Lu and C.N. Pope, General non-extremal rotating black holes in minimal five-dimensional gauged supergravity, Phys. Rev. Lett.95 (2005) 161301 [hep-th/0506029] [INSPIRE].
Z.W. Chong, M. Cvetic, H. Lu and C.N. Pope, Five-dimensional gauged supergravity black holes with independent rotation parameters, Phys. Rev.D 72 (2005) 041901 [hep-th/0505112] [INSPIRE].
H.K. Kunduri, J. Lucietti and H.S. Reall, Supersymmetric multi-charge AdS5black holes, JHEP04 (2006) 036 [hep-th/0601156] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Ph ys.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].
R.A. Janik and M. Trzetrzelewski, Supergravitons from one loop perturbative N = 4 SYM, Phys. Rev.D 77 (2008) 085024 [arXiv:0712.2714] [INSPIRE].
L. Grant , P.A. Grassi, S. Kim and S. Minwalla, Comments on 1/16 BPS Quantum States and Classical Configurations, JHEP 05 (2008) 049 [arXiv: 0803. 4183] [INSPIRE].
C.-M. Chang and X. Yin, 1/16 BPS states in N = 4 super-Yang-Mills theory, Phys. Rev.D 88 (2013) 106005 [arXiv:1305 .6314] [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys.117 (1988) 353 [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th /0206161] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys.313 (2012) 71 [arXiv: 0712.2824] [INSPIRE].
B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP08 (2014) 123 [arXiv: 1405.5144] INSPIRE].
B. Assel, D. Cassani, L. DiPietro, Z. Komargodski, J. Lorenzen and D. Martelli, The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP07 (2015) 043 [arXiv: 1503.05537] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
A. Sen, Entropy Function and AdS2/CFT1Correspondence, JHEP11 (2008) 075 [arXiv: 0805.0095] [INSPIRE].
P.J. Silva, Thermodynamics at the BPS bound for Black Holes in AdS, JHEP10 (2006) 022 [hep-th/0607056] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Zaffaroni, An extremization principle for the entropy of rotating BPS black holes in AdS5, JHEP07 (2017) 106 [arXiv:1705. 05383] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS4, Phys. Lett.B 771 (2017) 462 [arXiv: 1608. 07294] [INSPIRE].
F. Azzurli, N. Bobev, P.M. Crichigno, V.S. Min and A. Zaffaroni, A universal counting of black hole microstates in AdS4, JHEP02 (2018) 054 [arXiv: 1707.04257] [ INSPIRE].
N. Halmagyi and S. Lal, On the on-shell: the action of AdS4black holes, JHEP03 (2018) 146 [arXiv:1710.09580] [INSPIRE].
A. Cabo-Bizet, U. Kol, L.A. Pando Zayas, I. Papadimitriou and V. Rathee, Entropy functional and the holographic attractor mechanism, JHEP 05 (2018) 155 [arXiv: 1712.01849] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv: 1810.12067 [INSPIRE].
M. Honda, Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys. Rev.D 100 (2019) 026008 [arXiv:1901.08091] [IINSPIRE].
A. Arabi Ardehali, Cardy-like asymptotics of the 4d \( \mathcal{N} \) = 4 index and AdS5blackholes, JHEP06 (2019) 134 [arXiv:1902.06619] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Comments on deconfinement in AdS/CFT, arXiv: 1811.08646 [INSPIRE].
F. Benini and P. Milan, Black holes in 4d \( \mathcal{N} \) = 4 Super-Yang-Mills, arXiv:1812.09613 [INSPIRE].
A. Buchel and J.T. Liu, Gauged supergravity from type JIB string theory on Y**p,q manifolds, Nucl. Phys.B 771 (2007) 93 [hep-th/0608002], [INSPIRE].
J.P. Gauntlett and O. Varela, Consistent Kaluza- Klein reductions for general supersymmetric AdS solutions, Phys. Rev.D 76 (2007) 126007 [arXiv:0707.2315] [INSPIRE].
W. Chen, H. Lu and C.N. Pope, Mass of rotating black holes in gauged supergravities, Phys. Rev.D 73 (2006) 104036 [hep-th/0510081] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev.D 15 (1977) 2752 [INSPIRE].
H.K. Kunduri and J. Lucietti, Notes on non-extremal, charged, rotating black holes in minimal D = 5 gauged supergravity, Nucl. Phys.B 724 (2005) 343 [hep-th/0504158] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
J. Choi, S. Lee and S. Lee, Near Horizon Analysis of Extremal AdS5 Bla ck Holes, JHEP 05 (2008) 002 [arXiv:0802.3330] [INSPIRE].
S. Kim and K.-M. Lee, 1/ 16-BPS Black Holes and Giant Gravitons in the AdS5 x S5Space, JHEP 12 (2006) 077 [hep-th/0607085] [INSPIRE].
M. Cvetic, H. Lu and C.N. Pope, Charged Kerr-de Sitter black holes in five dimensions, Phys. Lett.B 598 (2004) 273 [hep-th/0406196] [INSPIRE].
S.W. Hawking, C.J. Hunter and M. Taylor, Rotation and the AdS/CFT correspondence, Phys. Rev.D 59 (1999) 064005 [hep-th/9811056] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on Curved Spaces and Holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].
D. Cassani, C. Klare, D. Martelli, A. Tomasiello and A. Zaffaroni, Supersymmetry in Lorent zian Curved Spaces and Holography, Commun. Math. Phys.327 (2014) 577 [arXiv: 1207.2181] [INSPIRE].
M.F. Sohnius and P.C. West, An Alternative Minimal Off-Shell Version of N = 1 Supergravity, Phys. Lett.105B (1981) 353 [INSPIRE].
M. Sohnius and P.C. West, The Tensor Calculus and Matter Coupli ng of the Alt ernative Minimal Auxiliary Field Formulation of N = 1 Supergravity, Nucl. Phys.B 198 (1982) 493 [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring Curved Superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].
I. Papadimitriou, Supercurrent anomalies in 4d SCFTs, JHEP07 (2017) 038 [arXiv: 1703.04299] [INSPIRE].
O.S. An, Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization, JHEP 12 (2017) 107 [arXiv:1703.09607] [INSPIRE].
P. Benetti Genolini, D. Cassani, D. Martelli and J. Sparks, The holographic supersymmetric Casimir energy, Phys. Rev.D 95 (2017) 021902 [arXiv:1606.02724] [INSPIRE].
P. Benetti Genolini, D. Cassani, D. Martelli and J. Sparks, Holographic renormalization and supersymmetry, JHEP02 (2017) 132 [arXiv: 1612.06761] [INSPIRE].
S. Nawata, Localization of N = 4 Superconformal Field Theory on S1 × S3and Index, JHEP11 (2011) 144 [arXiv:1104.4470] [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev.D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
C. Closset and I. Shamir, The \( \mathcal{N} \)= 1 Chiral Multiplet on T2 × S2and Supersymmetric Localization, JHEP03 (2014) 040 [arXiv: 1311.2430] [INSPIRE].
S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys.B 821 (2009) 241 [Erratum ibid.B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
K. Hosomichi, A Review on SUSY Gauge Theories onS3 , in New Dualities of Supersymmetric Gauge Theories, J. Teschner eds., Springer, Berlin Germany (2016), pg. 307 [arXiv: 1412. 7128] [INSPIRE].
G. Felder and A. Varchenko, The elliptic gamma function and SL(3, Z) ⋉ Z3, math/9907061.
A. Arabi Ardehali, J.T. Liu and P. Szepietowski, High-Temperature Expansion of Supersymmetric Partition Functions, JHEP 07 (2015) 113 [arXiv:1502.07737] [INSPIRE].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
D. Martelli and J. Sparks, The character of the supersymmetric Casimir energy, JHEP08 (2016) 117 [arXiv:1512.02521] [INSPIRE].
F.A. Dolan and H. Osborn, Applications of the Superconformal Inde x for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys.B 818 (2009) 137 [arXiv: 0801.4947] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn-deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys.8 (2004) 603 [hep-th/0310285] [INSPIRE].
B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys.B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP11 (2007) 050 [hep-th/0608050] [INSPIRE].
D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys.280 (2008) 611 [hep-th/0603021] [INSPIRE].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP05 (2013) 144 [arXiv: 1206.6339] [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir Energy and the Anomaly Polynomial, JHEP09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
A. Sen, Black Hol e Entropy Function, Attractors and Precision Counting of Microstates, Gen. Rel. Grav.40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP06 (2011) 019 [arXiv:1012.0265] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Locali zation & Exact Holography, JHEP04 (2013) 062 [arXiv:1111.1161] [INSPIRE].
A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv: 1208.4074 [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometric integrals and 't Hooft anomaly matching conditions, JHEP06 (2012) 016 [arXiv:1203.5677] [INSPIRE].
E. Shaghoulian, Modular Invariance of Conformal Field Theory on S1x S3and Circle Fibrations, Phys. Rev. Lett.119 (2017) 131601 [arXiv: 1612.05257] [INSPIRE].
A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP09 (2005) 038 [hep-th/0506177] [INSPIRE].
J.F. Morales and H. Samtleben, Entropy function and attractors for AdS black holes, JHEP10 (2006) 074 [hep-th/0608044] [INSPIRE].
O.J.C. Dias and P.J. Silva, Euclidean analysis of the entropy functional formalism, Phys. Rev.D 77 (2008) 084011 [arXiv:0704.1405] [INSPIRE].
N.V. Suryanarayana and M.C. Wapler, Charges from Attractors, Class. Quant. Grav.24 (2007) 5047 [arXiv:0704.0955] [INSPIRE].
S.M. Hosseini and A. Zaffaroni, Large N matrix models for 3d \( \mathcal{N} \) = 2 theories: twisted index, free energy and black holes, JHEP08 (2016) 064 [arXiv: 1604.03122] [INSPIRE].
J. Lorenzen and D. Martelli, Comments on the Casimir energy in supersymmetric field theories, JHEP07 (2015) 001 [arXiv:1412.7463] [INSPIRE].
J.L. Blazquez-Salcedo, J. Kunz, F. Navarro-Lerida and E. Radu, Squashed, magnetized black holes in D = 5 minimal gauged supergravity, JHEP02 (2018) 061 [arXiv:1711.10483] [INSPIRE].
D. Cassani and L. Papini, Squashing the boundary of supersymmetric AdS5black holes, JHEP12 (2018) 037 [arXiv:1809.02149] [INSPIRE].
J. Markeviciute and J.E. Santos, Evidence for the existence of a novel class of supersymmetric black holes with AdS5 x S5asymptotics, Class. Quant. Grav.36 (2019) 02LT01 [arXiv:1806.01849] [INSPIRE].
J. Markeviciute, Rotating Hairy Black Holes in AdS5 x S5, JHEP03 (2019) 110 [arXiv: 1809.04084] INSPIRE].
S.M. Hosseini, K. Hristov and A. Zaffaroni, A note on the entropy of rotating BPS AdS7 X S4black holes, JHEP 05 (2018) 121 [arXiv: 1803.07568] [INSPIRE].
J.P. Gauntlett and J.B. Gutowski, All supersymmetric solutions of minimal gauged supergravity in five-dimensions, Phys. Rev.D 68 (2003) 105009 [Erratum ibid.D 70 (2004) 089901] [hep-th/0304064] [INSPIRE].
D. Cassani, J. Lorenzen and D. Martelli, Comments on supersymmetric solutions of minimal gauged supergravity in five dimensions, Class. Quant. Grav.33 (2016) 115013 [arXiv: 1510.01380] [INSPIRE].
T. Ortfn, A Note on Lie-Lorentz derivatives, Class. Quant. Grav.19 (2002) L143 [hep-th/0206159] [INSPIRE].
V.P. Spiridonov, Elliptic beta integrals and solvable models of statistical mechanics, Contemp. Math.563 (2012) 181 [arXiv: 1011.3798] [INSPIRE].
E. Friedman and S. Ruijsenaars, Shintani-barnes zeta and gamma functions, Adv. Math.187 (2004) 362.
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ARXIV EPRINT: 1810.11442
On leave at the Galileo Galilei Institute, Largo Enrico Fermi, 2, 50125 Firenze, Italy. (Dario Martelli)
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Cabo-Bizet, A., Cassani, D., Martelli, D. et al. Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes. J. High Energ. Phys. 2019, 62 (2019). https://doi.org/10.1007/JHEP10(2019)062
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DOI: https://doi.org/10.1007/JHEP10(2019)062