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

We compute the 4d superconformal index for N=1, 2 gauge theories on S¹×L(p,1), where $L(p,1)$ is a lens space. We find that the 4d N=1, 2 index on S¹×L(p,1) reduces to a 3d N=2, 4 index on S¹×S² in the large p limit, and to a 3d partition function on a squashed $L(p,1)$ when the size of the temporal S¹ shrinks to zero. As an application of our index, we study 4d N=2 superconformal field theories arising from the 6d N=(2,0) A₁ theory on a punctured Riemann surface Σ, and conjecture the existence of a 2d topological quantum field theory on Σ whose correlation function coincides with the 4d N=2 index on S¹×L(p,1).