3D gauged supergravity from SU(2) reduction of $N=1$ 6D supergravity

TL;DR

This paper derives three-dimensional gauged supergravity theories with SU(2) and G gauge groups from six-dimensional supergravity via group manifold reduction, and demonstrates their on-shell equivalence after certain integrations.

Contribution

It presents a consistent reduction from 6D to 3D supergravity with explicit construction of the scalar manifold and gauge groups, including the derivation of an equivalent Chern-Simons supergravity theory.

Findings

Derived 3D supergravity with SU(2) and G gauge groups from 6D supergravity.

Constructed an equivalent Chern-Simons supergravity theory with scalar manifold.

Showed on-shell equivalence after integrating out certain scalars and gauge fields.

Abstract

We obtain Yang-Mills $SU(2)\times G$ gauged supergravity in three dimensions from $SU(2)$ group manifold reduction of (1,0) six dimensional supergravity coupled to an anti-symmetric tensor multiplet and gauge vector multiplets in the adjoint of $G$. The reduced theory is consistently truncated to $N=4$ 3D supergravity coupled to $4(1+\textrm{dim}\, G)$ bosonic and $4(1+\textrm{dim}\, G)$ fermionic propagating degrees of freedom. This is in contrast to the reduction in which there are also massive vector fields. The scalar manifold is $\mathbf{R}\times \frac{SO(3,\, \textrm{dim}\, G)}{SO(3)\times SO(\textrm{dim}\, G)}$, and there is a $SU(2)\times G$ gauge group. We then construct $N=4$ Chern-Simons $(SO(3)\ltimes \mathbf{R}^3)\times (G\ltimes \mathbf{R}^{\textrm{dim}G})$ three dimensional gauged supergravity with scalar manifold $\frac{SO(4,\,1+\textrm{dim}G)}{SO(4)\times SO(1+\textrm{dim}G)}$ and explicitly show that this theory is on-shell equivalent to the Yang-Mills $SO(3)\times G$ gauged supergravity theory obtained from the $SU(2)$ reduction, after integrating out the scalars and gauge fields corresponding to the translational symmetries $\mathbf{R}^3\times \mathbf{R}^{\textrm{dim}\, G}$.