Consistent truncations to 3-dimensional supergravity

Abstract

We show how to construct consistent truncations of 10-/11-dimensional supergravity to 3-dimensional gauged supergravity, preserving various amounts of supersymmetry. We show, that as in higher dimensions, consistent truncations can be defined in terms of generalised $G$-structures in Exceptional Field Theory, with $G \subset E_{8(8)}$ for the 3-dimensional case. Differently from higher dimensions, the generalised Lie derivative of $E_{8(8)}$ Exceptional Field Theory requires a set of "covariantly constrained" fields to be well-defined, and we show how these can be constructed from the $G$-structure itself. We prove several general features of consistent truncations, allowing us to rule out a higher-dimensional origin of many 3-dimensional gauged supergravities. In particular, we show that the compact part of the gauge group can be at most $\mathrm{SO}(9)$ and that there are no consistent truncations on a 7-or 8-dimensional product of spheres such that the full isometry group of the spheres is gauged. Moreover, we classify which matter-coupled ${\cal N} \geq 4$ gauged supergravities can arise from consistent truncations. Finally, we give several examples of consistent truncations to three dimensions. These include the truncations of IIA and IIB supergravity on $S^7$ leading to two different ${\cal N}=16$ gauged supergravites, as well as more general IIA/IIB truncations on $H^{p,7-p}$. We also show how to construct consistent truncations on compactifications of IIB supergravity on $S^5$ fibred over a Riemann surface. These result in 3-dimensional ${\cal N}=4$ gauged supergravities with scalar manifold ${\cal M} = \frac{\mathrm{SO}(6,4)}{\mathrm{SO}(6) \times \mathrm{SO}(4)} \times \frac{\mathrm{SU}(2,1)}{\mathrm{S}(\mathrm{U}(2)\times\mathrm{U}(1))}$ with a $\mathrm{ISO}(3)\times\mathrm{U}(1)^4$ gauging and for hyperboloidal Riemann surfaces contain ${\cal N}=(2,2)$ AdS$_3$ vacua.