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

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 $4(1+\text{dim}G)$ bosonic and $4(1+$\text{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,, $\text{dim}$, G)}{SO(3)\times SO($\text{dim}$, G)}$,andthereisa$SU(2)\times G$ gauge group. We then construct $N=4$ Chern-Simons $(SO(3)\ltimes $\mathbf{R}$^3)\times (G\ltimes $\mathbf{R}$^{$\text{dim}$G})$ three dimensional gauged supergravity with scalar manifold $\frac{SO(4,1+\text{dim}G)}{SO(4)\times SO(1+\text{dim}G)}$ 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 ${\mathbf{R}}^3\times {\mathbf{R}}^{\text{dim}G}$.