Schur index of the $\mathcal{N}=4$ $U(N)$ SYM via the AdS/CFT correspondence

We calculate the Schur index of the $\mathcal{N}=4$ $U(N)$ SYM with finite $N$ via the AdS/CFT correspondence as the contribution of D3-branes wrapped on contractible cycles in ${\mathit{S}}^5$ on some assumptions motivated by preliminary analyses. As far as we have checked numerically it agrees with the index calculated on the gauge theory side. In a certain limit it reproduces the analytic result given by Bourdier, Drukker, and Felix.