${\cal N}=(4,4)$ supersymmetric AdS$_3$ solutions in $d=11$

Abstract

We derive necessary and sufficient conditions for AdS$_3$ solutions of $d=11$ supergravity to preserve ${\cal N}=(1,1)$ supersymmetry in terms of G-structures. Such solutions necessarily support an SU(3)-structure on the internal 8-manifold M$_8$, in terms of which we phrase the conditions for supersymmetry preservation. We use this to derive the local form of all ${\cal N}=(4,4)$ supersymmetric AdS$_3$ solutions in $d=11$, for which M$_8$ decomposes as a foliation of a 3-sphere over a 5 dimensional base. There are 3 independent classes, 2 of which preserve the small superconformal algebra and one preserving its large counterpart for which M$_5$ contains a second 3-sphere. We show that for each solution with large (4,4) supersymmetry there are two corresponding solutions with small $(4,4)$, one for which M$_5$ maintains its 3-sphere, one where this blows up to $\mathbb{R}^3$ which can be compactified to $\mathbb{T}^3$. We use our results to construct several new solutions that lie within our derived classes as well as recovering some existing solutions.