4d Index to 3d Index and 2d TQFT

TL;DR

This paper computes the 4d superconformal index for certain gauge theories on lens spaces, revealing reductions to 3d indices and proposing a 2d TQFT correspondence for 4d N=2 theories.

Contribution

It introduces a method to compute 4d superconformal indices on lens spaces and connects these indices to 3d indices and a conjectured 2d TQFT structure.

Findings

4d N=1,2 index reduces to 3d N=2,4 index in large p limit

Index on S^1 x L(p,1) relates to 3d partition functions on squashed lens spaces

Proposal of a 2d TQFT on Riemann surface matching 4d N=2 index

Abstract

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 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, and conjecture the existence of a 2d Topological Quantum Field Theory on the Riemann surface whose correlation function coincides with the 4d N=2 index on S^1 x L(p,1).