Wrapped NS5-Branes, Consistent Truncations and In\"on\"u-Wigner Contractions

Abstract

We construct consistent Kaluza-Klein truncations of type IIA supergravity on (i) $\Sigma_2\times S^3$ and (ii) $\Sigma_3\times S^3$, where $\Sigma_2 = S^2/\Gamma$, $\mathbb{R}^2/\Gamma$, or $\mathbb{H}^2/\Gamma$, and $\Sigma_3 = S^3/\Gamma$, $\mathbb{R}^3/\Gamma$, or $\mathbb{H}^3/\Gamma$, with $\Gamma$ a discrete group of symmetries, corresponding to NS5-branes wrapped on $\Sigma_2$ and $\Sigma_3$. The resulting theories are a $D=5$, $\mathcal{N}=4$ gauged supergravity coupled to three vector multiplets with scalar manifold $SO(1,1)\times SO(5,3)/(SO(5)\times SO(3))$ and gauge group $SO(2)\times\left(SO(2)\ltimes_{\Sigma_2}\mathbb{R}^4\right)$ which depends on the curvature of $\Sigma_2$, and a $D=4$, $\mathcal{N}=2$ gauged supergravity coupled to one vector multiplet and two hypermultiplets with scalar manifold $SU(1,1)/U(1)\times G_{2(2)}/SO(4)$ and gauge group $\mathbb{R}^+\times\mathbb{R}^+$ for truncations (i) and (ii) respectively. Instead of carrying out the truncations at the 10-dimensional level, we show that they can be obtained directly by performing In\"on\"u-Wigner contractions on the 5 and 4-dimensional gauged supergravity theories that come from consistent truncations of 11-dimensional supergravity associated with M5-branes wrapping $\Sigma_2$ and $\Sigma_3$. This suggests the existence of a broader class of lower-dimensional gauged supergravity theories related by group contractions that have a 10 or 11-dimensional origin.