Double Field Theory and Membrane Sigma-Models

TL;DR

This paper explores the geometric structure of double field theory (DFT) through a doubled membrane sigma-model, linking it with Courant algebroids and generalized geometry, and introduces a new DFT algebroid.

Contribution

It introduces a new DFT algebroid, constructs a membrane sigma-model for DFT, and clarifies the geometric origin of noncommutative and nonassociative deformations in string theory.

Findings

Unified description of geometric and non-geometric flux backgrounds

Clarification of noncommutative and nonassociative geometry in string theory

Construction of a gauge-invariant doubled membrane sigma-model

Abstract

We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the corresponding operations in DFT. This describes precisely how the geometric structure of DFT lies in between two Courant algebroids and is reconciled with generalized geometry. We construct the membrane sigma-model that corresponds to DFT, and demonstrate how the standard T-duality orbit of geometric and non-geometric flux backgrounds is captured by its action functional in a unified way. This also clarifies the appearence of noncommutative and nonassociative deformations of geometry in non-geometric closed string theory. Gauge invariance of the DFT membrane sigma-model is compatible with the flux formulation of DFT and its strong constraint, whose geometric origin is explained. Our approach leads to a new generalization of a Courant algebroid, that we call a DFT algebroid and relate to other known generalizations, such as pre-Courant algebroids and symplectic nearly Lie 2-algebroids. We also describe the construction of a gauge-invariant doubled membrane sigma-model that does not require imposing the strong constraint.