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.
Highlights
String backgrounds that are T-dual to each other may correspond to target spaces with different geometry and topology
A double field theory (DFT), where both coordinates conjugate to momentum modes and dual coordinates conjugate to winding modes of the closed string are implemented, was constructed in [18, 19] and more recently in [22,23,24,25]
Since the latter are the natural arena for the general AKSZ construction in three worldvolume dimensions [44], this essentially means that given the data of a Courant algebroid one can construct, uniquely up to isomorphism, a membrane sigma-model which is a three-dimensional topological field theory
Summary
We will derive the O(d, d)-invariant open membrane sigma-model associated to DFT, whose boundary dynamics will govern the motion of closed strings in backgrounds with both geometric and non-geometric fluxes in a manifestly T-duality invariant way
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