Relation between moduli spaces of susy vacua-Higgs bundles-flat connections

Question

My question is motivated by the 3d-3d correspondence. I want to consider M5 branes living on time $\times M_3 \times D^2$ (6d in total) and where the ambient space is time $\times$ cotangent bundle of $M_3$ $\times$ Taub-Nut space. Then to preserve susy, among other things, I require $M_3$ to be (special) Lagrangian. 

Ok, now I can reduce the M5-brane 6d (2,0) theory on $M_3$ to get a 3d supersymmetric gauge theory called $T[M_3]$ (in analogy with class $S$-theories where we compactify on a Riemann surface). Now, this theory has a moduli space of supersymmetric vacua $\mathcal{M}_{T[M3]}$ which apparently (see Gukov in this video) is isomorphic to the space of flat $SL_2(\mathbb{C})$ connection on $M_3$ which is isomorphic to the Hitchin moduli space of pairs $(E,\phi)$ where $E$ is a vector bundle and $\phi$ is a Higgs field (element of Lie algebra - well of the adjoint bundle - tensor canonical bundle). 

My question is what is the precise relation of these moduli spaces (on $M_3$)? I know how moduli Hitchin moduli space and moduli space of flat connections are related on arbitrary $X$ but then I do not get how to specify on this arbitrary 3-manifold which also is related to the 3d-3d correspondence. 

Any reference on the specific relation of  moduli space of 3d susy vacua on $M_3$ with the flat connections on it and the Hitchin moduli space on it would be extremely appreciated :)

Answer 1

Let $G$ be a compact Lie group of ADE type. Compactifying the $\mathcal{N}=(2,0)$ 6d SCFT of type $G$ on a 3-manifold $M_3$ gives a $\mathcal{N}=2$ 3d QFT $T_G[M_3]$. More precisely, to have SUSY in 3d, one needs to make a topological twist along $M_3$, which is the one obtained from an embedding $M_3 \subset T^{*}M_3$ in a M-theory realization.

The claim is that the moduli space of SUSY vacua of $T_G[M_3]$ on $\mathbb{R}^2 \times S^1$ is $M_{flat}(M_3,G_{\mathbb{C}})$, the moduli space of flat connections on $M_3$ for the complexified gauge group $G_{\mathbb{C}}$. See for example section 5 of

https://arxiv.org/abs/1108.4389

To understand this fact, it is better to start in 6d and to first compactify on $S^1$. One then gets at low energy maximally, i.e. $\mathcal{N}=2$ , 5d super-Yang-Mills theory of group $G$. The advantage is that this 5d theory has an explicit Lagrangian description, which is not the case for the 6d theory. One can then compactify 5d super-Yang-Mills on $M_3$, with appropriate topological twist. In 5d super-Yang-Mills on flat space, one has 5 real scalar fields in the adjoint representation. After the topological twist, 3 of them becomes 1-forms on $M_3$, which can be naturally combined with the gauge field on $M_3$ to obtained a complexified gauge field on $M_3$. Supersymmetry requires this compexified gauge field on $M_3$ to be flat.