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

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My question is motivated by the 3d-3d correspondence. I want to consider M5 branes living on time ×M3×D2 (6d in total) and where the ambient space is time × cotangent bundle of M3 × Taub-Nut space. Then to preserve susy, among other things, I require M3 to be (special) Lagrangian. 

Ok, now I can reduce the M5-brane 6d (2,0) theory on M3 to get a 3d supersymmetric gauge theory called T[M3] (in analogy with class S-theories where we compactify on a Riemann surface). Now, this theory has a moduli space of supersymmetric vacua MT[M3] which apparently (see Gukov in this video) is isomorphic to the space of flat SL2(C) connection on M3 which is isomorphic to the Hitchin moduli space of pairs (E,ϕ) where E is a vector bundle and ϕ 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 M3)? 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 M3 with the flat connections on it and the Hitchin moduli space on it would be extremely appreciated :)

asked Feb 3, 2017 in Theoretical Physics by conformal_gk (3,625 points) [ revision history ]
edited Feb 3, 2017 by conformal_gk

1 Answer

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Let G be a compact Lie group of ADE type. Compactifying the N=(2,0) 6d SCFT of type G on a 3-manifold M3 gives a N=2 3d QFT TG[M3]. More precisely, to have SUSY in 3d, one needs to make a topological twist along M3, which is the one obtained from an embedding M3TM3 in a M-theory realization.

The claim is that the moduli space of SUSY vacua of TG[M3] on R2×S1 is Mflat(M3,GC), the moduli space of flat connections on M3 for the complexified gauge group GC. 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 S1. One then gets at low energy maximally, i.e. 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 M3, 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 M3, which can be naturally combined with the gauge field on M3 to obtained a complexified gauge field on M3. Supersymmetry requires this compexified gauge field on M3 to be flat.

answered Feb 12, 2017 by 40227 (5,140 points) [ no revision ]

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