A 3d-3d appetizer

TL;DR

This paper investigates the 3d-3d correspondence for theories labeled by Lens spaces, demonstrating a match between 3d ${ m N}=2$ theory indices and complex Chern-Simons partition functions, revealing new insights into their relationship.

Contribution

It provides a detailed analysis of the 3d-3d correspondence for Lens space theories, including explicit calculations and the behavior of indices and partition functions for various p values.

Findings

Index of $T[L(p,1)]$ matches complex Chern-Simons partition function on $L(p,1)$

For p=1, the index reproduces the $S^3$ Chern-Simons partition function

At large p, the index becomes p-independent

Abstract

We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d ${\cal N}=2$ "Lens space theory" $T[L(p,1)]$ and the partition function of complex Chern-Simons theory on $L(p,1)$. In particular, for $p=1$, we show how the familiar $S^3$ partition function of Chern-Simons theory arises from the index of a free theory. For large $p$, we find that the index of $T[L(p,1)]$ becomes a constant independent of $p$. In addition, we study $T[L(p,1)]$ on the squashed three-sphere $S^3_b$. This enables us to see clearly, at the level of partition function, to what extent $G_\mathbb{C}$ complex Chern-Simons theory can be thought of as two copies of Chern-Simons theory with compact gauge group $G$.