Anomalies, black strings and the charged Cardy formula
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Received: July 14, 2020 Accepted: September 1, 2020 Published: September 25, 2020
Seyed Morteza Hosseini,a Kiril Hristov,b Yuji Tachikawaa and Alberto Zaffaronic,d a
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan b Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria c Dipartimento di Fisica, Universit` a di Milano-Bicocca, I-20126 Milano, Italy d INFN, sezione di Milano-Bicocca, I-20126 Milano, Italy
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We derive the general anomaly polynomial for a class of two-dimensional CFTs arising as twisted compactifications of a higher-dimensional theory on compact manifolds Md , including the contribution of the isometries of Md . We then use the result to perform a counting of microstates for electrically charged and rotating supersymmetric black strings in AdS5 × S 5 and AdS7 × S 4 with horizon topology BTZnS 2 and BTZnS 2 × Σg , respectively, where Σg is a Riemann surface. We explicitly construct the latter class of solutions by uplifting a class of four-dimensional rotating black holes. We provide a microscopic explanation of the entropy of such black holes by using a charged version of the Cardy formula. Keywords: AdS-CFT Correspondence, Anomalies in Field and String Theories, Black Holes in String Theory, Field Theories in Higher Dimensions ArXiv ePrint: 2006.08629
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)167
JHEP09(2020)167
Anomalies, black strings and the charged Cardy formula
Contents 1 Introduction
2
2 Charged Cardy formula 2.1 Non-supersymmetric case 2.2 Supersymmetric case and its relation to the anomaly polynomial
4 4 6 7 7 9 9 11 11 12 13 14 14 15 17
4 Rotating black strings in AdS5 and AdS7 19 4.1 The 4d supergravity model 20 4.2 Near horizon parameters and the BPS Higgsing process 22 4.2.1 Supersymmetry conditions coming from the vector multiplets 22 4.2.2 Supersymmetry conditions coming from the universal hypermultiplet 23 4.3 Explicit solution 25 5 4.4 Recovering rotating black strings in AdS5 × S 26 5 Black strings microstates and the charged Cardy formula 5.1 Black strings in AdS5 × S 5 5.2 Black strings in AdS7 × S 4
27 28 30
6 Discussion and outlook
33
A Entropy function from gravitational blocks
34
–1–
JHEP09(2020)167
3 Anomaly polynomial and the dimensional reduction 3.1 Generalities 3.1.1 Bott-Cattaneo formula for S 2k 3.1.2 Equivariant integration 3.2 Going from 4d to 2d 3.2.1 Free chiral fermions 3.2.2 General N = 1 theories and the charged Cardy formula 3.2.3 N = 4 super Yang-Mills in the large N limit 3.3 Going from 6d to 2d 3.3.1 (2, 0) theory on S 2 × Σg in the large N limit 3.3.2 (2,0) theory on toric K¨ahler surfaces 3.3.3 Examples of compactifications (2,0) theory on toric K¨ahler surfaces
1
Introduction
The charged Cardy formula, s c c J2 log ρ(n, J) ≈ 2π n− − , 6 24 2k
(1.1)
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