The topologically twisted index of N=4 super-Yang-Mills on T2 x S2 and the elliptic genus

We examine the topologically twisted index of N = 4 super-Yang-Mills with gauge group SU(N) on T(2)xS(2), and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds T-2/Z(m) x 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.