A 3d-3d appetizer

TL;DR

This work tests the 3d-3d correspondence for theories labeled by Lens spaces by matching the 3d $\mathcal{N}=2$ index of $T[L(p,1)]$ with the partition function of complex Chern-Simons theory on $L(p,1)$, including the $p=1$ case where the index reproduces the $S^3$ CS result via a dual free-chiral description. It shows that for $p=0$ (i.e., $k=0$) the index aligns with the complex CS partition function, and, in the large-$p$ limit, the index becomes a constant depending only on the gauge group, $({2N-1})!!$. Through explicit computations on the squashed sphere $S^3_b$, the authors demonstrate that complex Chern-Simons theory can be viewed as two copies of $G$ Chern-Simons with a careful treatment of relative phases, but this simple product picture fails for $k\neq 0$ on general Lens spaces, necessitating nontrivial phase data $C_\alpha$ and the full $SL(2,\mathbb{Z})$ gluing framework. Collectively, these results refine the 3d-3d dictionary for Lens spaces and reveal intricate structure in the relationship between flat connections, vacua, and lens-space partition functions.

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 $G$.