3d-3d correspondence

$M_3\mathrm{\&}T[M_3]$
Compactifying 6d (2,0) theory on $M_3$ gives a 3d $\mathcal{N}=2$ theory $T[M_3]$ in Type IIB string theory, which isn't on $M_3$ but only depends on the topology of $M_3$.

1. ${\mathcal{M}}_{SUSYvacua}(T[M_3;G])\simeq {\mathcal{M}}_{flat}(M_3;G_{\mathbb{C}})$, with the former on ${\mathbb{R}}^2\times S^1$.

2. $Z_{T[M_3;G]}[L(k,1{)}_b]=Z_{CS}^{(k,\sigma )}[M_3;G_{\mathbb{C}}]$. (WRT invariant)
   eg. $L(k=0,1)=S^1\times S^2$ gives the superconformal index of $T[M_3]$; when $M=S^3$, $T[M_3]$ is dual to free chiral multiplets (in the sense of brane construction, $Z$-function (partition function on $S^3$) and superconformal index); when $M=L(p,1)$, it's dual to moduli space of $N$ vortex in a $U(p)$ gauge theory ${\mathcal{V}}_{N,p}$

3. GPPV invariant and ${\mathcal{H}}_a[M_3]$ (categorification)

ref: [1503.04809] A 3d-3d appetizer
[1701.06567] BPS spectra and 3-manifold invariants