A 3d-3d appetizer

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 $\mathcal{N}=2$ “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 $S_b^3$. 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 $G$.