4d index to 3d index and 2d topological quantum field theory

We compute the 4d superconformal index for N = 1, 2 gauge theories on S-1 x L(p, 1), where L(p, 1) is a lens space. We find that the 4d N = 1, 2 index on S-1 x L(p, 1) reduces to a 3d N = 2, 4 index on S-1 x S-2 in the large p limit, and to a 3d partition function on a squashed L(p, 1) when the size of the temporal S-1 shrinks to zero. As an application of our index, we study 4d N = 2 superconformal field theories arising from the 6d N = (2, 0) A(1) theory on a punctured Riemann surface Sigma, and conjecture the existence of a 2d topological quantum field theory on Sigma whose correlation function coincides with the 4d N = 2 index on S-1 x L(p, 1).