Non-geometric Backgrounds and the First Order String Sigma Model

Abstract: We study the first order form of the NS string sigma model allowing for worldsheet couplings corresponding on the target space to a bi-vector, a two-form and an inverse metric. Lifting the topological sector of this action to three dimensions produces several Wess-Zumino like terms which encode the bi-vector generalization of the Courant bracket. This bracket may be familiar to physicists through the (H_{ijk},F_{ij}^{k},Q_i^{jk},R^{ijk}) notation for non-geometric backgrounds introduced by Shelton-Taylor-Wecht. The non-geometricity of the string theory in encoded in the global properties of the bi-vector, when the bi-vector is a section then the string theory is geometric. Another interesting situation emerges when one considers membrane actions which are not equivalent to string theories on the boundary of the membrane. Such a situation arises when one attempts to describe the so-called R-space (the third T-dual of a T^3 with H_3 flux). This model appears to be, at least classically, described by a membrane sigma model, not a string theory. Examples of geometric backgrounds with bi-vector couplings and non-vanishing Q-coefficients are provided by gauged WZW models.