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

TL;DR

This paper analyzes the large-N behavior of the topologically twisted index of N=4 SU(N) Super-Yang-Mills theory on T^2×S^2, linking it to black hole microstates via the Farey tail and holography.

Contribution

It provides a holographic interpretation of the twisted index using the black hole Farey tail and constructs extremal solutions in a dual supergravity model, extending microstate counting.

Findings

Large-N asymptotics of the twisted index computed using Bethe-Ansatz

Construction of 5d extremal solutions from the black hole Farey tail

Comparison of gravitational partition function with the twisted index in the large-N limit

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.