Two-Dimensional Supersymmetric Sigma Models on Almost-Product Manifolds and Non-Geometry

We show that the superconformal symmetries of the (1,1) sigma model decompose into a set of more refined symmetries when the target space admits projectors $P_{\pm}$, and the orthogonal complements $Q_{\pm}$, covariantly constant with respect to the two natural torsionful connections $\nabla^{(\pm)}$ that arise in the sigma model. Surprisingly the new symmetries still close to form copies of the superconformal algebra, even when the projectors are not integrable, so one is able to define a superconformal theory not associated with a particular geometry, but rather with non-integrable projectors living on a larger manifold. We show that this notion of non-geometry encompasses the locally non-geometric examples that arise in the T-duality inspired doubled formulations, with the benefit that it is more generally applicable, as it does not depend on the existence of isometries, or invariant structures beyond $P_{\pm}$ and $Q_{\pm}$. We derive the conditions for (2,2) supersymmetry in the projective sense, thus extending the relation between (2,2) theories and bi-Hermitian target spaces to the non-geometric setting. In the bosonic subsector we propose a BRST type approach to defining the physical degrees of freedom in the non-geometric scenario.