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

We examine the topologically twisted index of $\mathcal{N}=4$ super-Yang-Mills with gauge group $SU(N)$ on $T^2\times S^2$, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds $T^2\mathrm{/}{\mathbb{Z}}_m\times {\mathbb{Z}}_n$ where $N=mn$. After summing over these sectors, the index can be expressed as the elliptic genus of a two-dimensional $\mathcal{N}=(0,2)$ 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 $\mathcal{N}=(0,2)$ theory.