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