The 5d superconformal index at large N and black holes
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Springer
Received: June 22, 2020 Accepted: August 17, 2020 Published: September 18, 2020
P. Marcos Crichignoa and Dharmesh Jainb a
Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1090 GL, Amsterdam, The Netherlands b Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake City, Kolkata 700106, India
E-mail: [email protected], [email protected] Abstract: We study the large N limit of the superconformal index of a large class of 5d N = 1 superconformal field theories and show it is given by the square of the partition function on the squashed five-sphere. We show this simple relation implies a Cardy formula in 5d, which is valid in an “extended” regime in which fugacities are finite and N is large. For theories with weakly coupled gravity duals we conjecture this large N Cardy formula universally accounts for the microscopic entropy of spinning black holes in AdS 6 . We check this explicitly for known black hole solutions in massive type IIA and type IIB string theory, carrying two angular momenta and one electric charge, and predict the entropy of black holes carrying multiple electric charges, yet to be constructed. We also discuss large N properties of the Sb3 × Σg partition function, extending previous results to theories with type IIB duals. Keywords: AdS-CFT Correspondence, Black Holes in String Theory, Conformal Field Models in String Theory, D-branes ArXiv ePrint: 2005.00550
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)124
JHEP09(2020)124
The 5d superconformal index at large N and black holes
Contents 1 Introduction and summary
1
2 Review of the 5d Nekrasov partition function and S 5 2.1 Mass deformed squashed S 5
3 5 8 11 12
4 Black hole entropy 4.1 The universal spinning black hole 4.2 A mesonic spinning black hole in massive type IIA
14 14 16
5 The Sb3 × Σg partition function
17
A Superconformal algebra
20
B Special functions and asymptotics B.1 Asymptotics of the elliptic gamma function
21 23
1
Introduction and summary
The large N expansion is one of the most powerful nonperturbative tools in quantum field theory [1]. In the N → ∞ limit, some theories become exactly solvable or can be recast in terms of a dual description with a standard perturbative expansion. The large N expansion is, however, more than a technical tool; it has also led to enormous insights into the dynamics of quantum field theories when the number of degrees of freedom at each point is large. A prime example is the gauge/gravity duality in which a large number of quantum degrees of freedom are reorganized into semiclassical gravitational degrees of freedom. In this paper, we study universal aspects of 5d N = 1 supersymmetric gauge theories that emerge at large N . Our calculations are purely field theoretic and do not assume holography. However, they have interesting holographic applications, in particular for the counting of microstates of black holes i
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