The topologically twisted index of $N=4$ super-Yang-Mills on $T^2\times S^2$ and the elliptic genus

We examine the topologically twisted index of $N=4$ super-Yang-Mills with gauge group $SU\left(N\right)$ on $T^2\times S^2$, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds $T^2/Z_m\times Z_n$ where $N=mn$. After summing over these sectors, the index can be expressed as the elliptic genus of a two-dimensional $N=(0,2)$ theory resulting from Kaluza-Klein reduction on $S^2$. This provides an alternate path to the 'high-temperature' limit of the index, and confirms the connection to the right-moving central charge of the $N=(0,2)$ theory.