Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus

TL;DR

This paper explores the 3d-3d correspondence by relating superconformal indices of certain 3d theories to SL(2,C) Chern-Simons partition functions on mapping cylinders and tori, revealing deep geometric and quantum connections.

Contribution

It demonstrates the equivalence of superconformal indices and Chern-Simons partition functions for mapping cylinders and tori, using duality domain wall theories and ideal triangulations.

Findings

Superconformal indices match SL(2,C) Chern-Simons partition functions.

Equality for the mapping torus reduces to a basis change in the Hilbert space.

The approach connects 3d field theories with quantum topology.

Abstract

We probe the 3d-3d correspondence for mapping cylinder/torus using the superconformal index. We focus on the case when the fiber is a once-punctured torus (\Sigma_{1,1}). The corresponding 3d field theories can be realized using duality domain wall theories in 4d N=2* theory. We show that the superconformal indices of the 3d theories are the SL(2,C) Chern-Simons partition function on the mapping cylinder/torus. For the mapping torus, we also consider another realization of the corresponding 3d theory associated with ideal triangulation. The equality between the indices from the two descriptions for the mapping torus theory is reduced to a basis change of the Hilbert space for the SL(2,C) Chern-Simons theory on Rx\Sigma_{1,1}.