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We examine the topologically twisted index of $\mathcal N=4$ super-Yang-Millswith gauge group $SU(N)$ on $T^2\times S^2$, and demonstrate that it receivescontributions from multiple sectors corresponding to the freely actingorbifolds $T^2/\mathbb Z_m\times\mathbb Z_n$ where $N=mn$. After summing overthese sectors, the index can be expressed as the elliptic genus of atwo-dimensional $\mathcal N=(0,2)$ theory resulting from Kaluza-Klein reductionon $S^2$. This provides an alternate path to the 'high-temperature' limit ofthe index, and confirms the connection to the right-moving central charge ofthe $\mathcal N=(0,2)$ theory.

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