Abstract
The existence of genuinely non-geometric backgrounds, i.e. ones without geometric dual, is an important question in string theory. In this paper we examine this question from a sigma model perspective. First we construct a particular class of Courant algebroids as protobialgebroids with all types of geometric and non-geometric fluxes. For such structures we apply the mathematical result that any Courant algebroid gives rise to a 3D topological sigma model of the AKSZ type and we discuss the corresponding 2D field theories. It is found that these models are always geometric, even when both 2-form and 2-vector fields are neither vanishing nor inverse of one another. Taking a further step, we suggest an extended class of 3D sigma models, whose world volume is embedded in phase space, which allow for genuinely non-geometric backgrounds. Adopting the doubled formalism such models can be related to double field theory, albeit from a world sheet perspective.
Highlights
For such structures we apply the mathematical result that any Courant algebroid gives rise to a 3D topological sigma model of the AKSZ type and we discuss the corresponding 2D field theories
The extended nature of the fundamental degrees of freedom in string theory leads to duality symmetries, whose consequences are unconventional from a traditional field theory viewpoint
This approach is inspired by previous work along these lines in the string theory literature [4, 6], which we extended and generalized
Summary
We define and construct Courant algebroids that accommodate 3-index twists of any type, which will be identified with geometric and non-geometric fluxes that appear in string theory. We call such structures “Courant-Roytenberg” algebroids, since they were introduced in refs. Our approach in the presentation is to provide some basic definitions first, apply them to construct a class of cases interesting for string theory, and to present in detail an explicit example
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