Abstract
The superconformal index of the N=4 SU(N) supersymmetric Yang-Mills theory counts the 1/16-BPS (Bogomol’nyi-Prasad-Sommerfield) states in this theory, and has been used via the AdS/CFT correspondence to count black hole microstates of 1/16-BPS black holes. On one hand, this index may be related to the Euclidean partition function of the theory on S3×S1 with complex chemical potentials, which maps by the AdS/CFT correspondence to a sum over Euclidean gravity solutions. On the other hand, the index may be expressed as a sum over solutions to Bethe Ansatz (BA) equations. We show that the solutions to the BA equations that are known to have a good large N limit, for the case of equal chemical potentials for the two angular momenta, have a one-to-one mapping to (complex) Euclidean black hole solutions on the gravity side. This mapping captures both the leading contribution from the classical gravity action (of order N2), as well as nonperturbative corrections in 1/N, which on the gravity side are related to wrapped D3-branes. Some of the BA solutions map to orbifolds of the standard Euclidean black hole solutions (which obey exactly the same boundary conditions as the other solutions). there are many more gravitational solutions than Bethe Ansatz solutions, but we show that, by considering the nonperturbative effects, the extra solutions are ruled out, leading to a precise match between the solutions on both sides. Published by the American Physical Society 2021
Highlights
AND SUMMARYThe AdS=CFT correspondence [1,2,3,4] maps black holes in asymptotically anti–de Sitter (AdS) spacetimes to coarse-grained descriptions of states in conformal field theories (CFTs), such that the Bekenstein-Hawking entropy of the black holes may be given a statistical mechanics interpretation as a counting of CFT microstates
In the last few years, considerable progress has been made [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] in performing this counting for 1=16-BPS (Bogomol’nyi-PrasadSommerfield) black holes in type IIB string theory on AdS5 × S5, which map to 1=16-BPS states in the N 1⁄4 4 SUðNÞ supersymmetric Yang-Mills (SYM) theory on S3
The counting of 1=16-BPS states is based on computing the superconformal index [49,50], which is a sum over these states with chemical potentials for four of the five charges carried by the black holes [the black holes carry three global symmetry charges R1;2;3 in the Cartan algebra of the SUð4ÞR symmetry, and two angular momenta J1;2 in AdS5]
Summary
The AdS=CFT correspondence [1,2,3,4] maps black holes in asymptotically anti–de Sitter (AdS) spacetimes to coarse-grained descriptions of states in conformal field theories (CFTs), such that the Bekenstein-Hawking entropy of the black holes may be given a statistical mechanics interpretation as a counting of CFT microstates. Some of the known solutions to these equations give contributions which in the large N limit resemble our expectations from the gravity side; there is a leading term (in the logarithm of the contribution to the partition function) of order N2, power-law corrections in 1=N, and nonperturbative corrections of order e−N This raises a natural question—can we identify the sum over Bethe Ansatz solutions in the CFT computation of the index, with the sum over gravitational solutions? There are other Euclidean wrapped D3-branes that preserve supersymmetry, which wrap an S1 inside the S5 and an S3 in the AdS5 coordinates These additional D3branes are not related to nonperturbative terms in the Bethe Ansatz contribution of the Hong-Liu solutions, but they still give corrections to the gravitational partition function. Appendix A contains reference material, Appendices B–D contain details on field theory and gravity computations, while in Appendix E we present the new Lorentzian D3-brane giant graviton configurations
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