Triality and the consistent reductions on ${\rm AdS}_3\times S^3$

We study compactifications on ${\rm AdS}_3\times S^3$ and deformations thereof. We exploit the triality symmetry of the underlying duality group ${\rm SO}(4,4)$ of three-dimensional supergravity in order to construct and relate new consistent truncations. For non-chiral $D=6$, ${\cal N}_{\rm 6d}=(1,1)$ supergravity, we find two different consistent truncations to three-dimensional supergravity, retaining different subsets of Kaluza-Klein modes, thereby offering access to different subsectors of the full nonlinear dynamics. As an application, we construct a two-parameter family of ${\rm AdS}_3\times M^3$ backgrounds on squashed spheres preserving ${\rm U}(1)^2$ isometries. For generic value of the parameters, these solutions break all supersymmetries, yet they remain perturbatively stable within a non-vanishing region in parameter space. They also contain a one-parameter family of ${\cal N}=(0,4)$ supersymmetric ${\rm AdS}_3\times M^3$ backgrounds on squashed spheres with ${\rm U}(2)$ isometries. Using techniques from exceptional field theory, we determine the full Kaluza-Klein spectrum around these backgrounds.