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We investigate the large-$N$ asymptotics of the topologically twisted indexof $\mathcal N=4$ SU($N$) Super-Yang-Mills (SYM) theory on $T^2\times S^2$ andprovide its holographic interpretation based on the black hole Farey tail. Inthe field theory side, we use the Bethe-Ansatz (BA) formula, which gives thetwisted 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 $\mathcalN=2$ gauged STU model, we construct a family of 5d extremal solutions upliftedfrom the 3d black hole Farey tail, and compute the regularized on-shellactions. The gravitational partition function given in terms of theseregularized on-shell actions is then compared with a canonical partitionfunction derived from the twisted index by the inverse Laplace transform, inthe large-$N$ limit. This extends the previous microstate counting of anAdS$_5$ black string by the twisted index and thereby improves holographicunderstanding of the twisted index.

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