The topologically twisted index of $\mathcal{N}=4$ SU($N$) Super-Yang-Mills theory and a black hole Farey tail

Abstract

We investigate the large-$N$ asymptotics of the topologically twisted index of $\mathcal{N}=4$ SU($N$) Super-Yang-Mills (SYM) theory on $T^2\times S^2$ and provide its holographic interpretation based on the black hole Farey tail. In the field theory side, we use the Bethe-Ansatz (BA) formula, which gives the twisted index of $\mathcal{N}=4$ SYM theory as a discrete sum over Bethe vacua, to compute the large-$N$ asymptotics of the twisted index. In a dual $\mathcal{N}=2$ gauged STU model, we construct a family of 5d extremal solutions uplifted from the 3d black hole Farey tail, and compute the regularized on-shell actions. The gravitational partition function given in terms of these regularized on-shell actions is then compared with a canonical partition function derived from the twisted index by the inverse Laplace transform, in the large-$N$ limit. This extends the previous microstate counting of an AdS${}_5$ black string by the twisted index and thereby improves holographic understanding of the twisted index.

v1: 50 pages; v2: references added

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