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

We investigate the large- N asymptotics of the topologically twisted index of $$ \mathcal{N} $$ N = 4 SU( N ) Super-Yang-Mills (SYM) theory on T 2 × S 2 and provide its holographic interpretation based on the black hole Farey tail [1]. In the field theory side, we use the Bethe-Ansatz (BA) formula, which gives the twisted index of $$ \mathcal{N} $$ N = 4 SYM theory as a discrete sum over Bethe vacua, to compute the large- N asymptotics of the twisted index. In a dual $$ \mathcal{N} $$ 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.