Two-dimensional supersymmetric sigma models on almost-product manifolds and non-geometry

Abstract

In this paper we study the relation between the target space geometry of (1, 1) sigma models and the factorization of the associated superconformal theory, at the classical level. It turns out that the superconformal currents T± factorize as , with and conserved separately, provided that the target space admits projectors P(±) and Q(±) (that is, a pair of almost-product structures), compatible with the Riemannian structure of the target space, and covariantly constant with respect to the two torsionful connections ∇(±) that arise naturally from the sigma model. It is a surprising result that the integrability of the projectors is not an obstruction for the associated symmetries and to form copies of the superconformal algebra. While one expects to be able to define a superconformal theory associated with a particular Riemannian submanifold defined by an integrable projector, a consequence of the above result is that there are no obstructions to defining a superconformal theory associated with non-integrable projectors. We show that this notion of non-geometry encompasses the locally non-geometric examples that arise in the T-duality inspired doubled formulations. In addition, we derive the general conditions for (2, 2) supersymmetry to be realized in the projective sense. This extends the relation between (2, 2) sigma models and bi-Hermitian geometry to the non-geometric setting. For the bosonic subsector we propose a BRST-type approach to defining the physical degrees of freedom in the non-geometric scenario.