Flux Formulation of DFT on Group Manifolds and Generalized Scherk-Schwarz Compactifications

TL;DR

This paper develops a flux formulation of Double Field Theory on group manifolds, enabling generalized Scherk-Schwarz compactifications that connect to gauged supergravities, by utilizing the Maurer-Cartan form for Lie groups.

Contribution

It introduces a flux formulation of DFT on group manifolds with a novel background flux split, and constructs the twist using the Maurer-Cartan form, solving the twist construction problem.

Findings

Derived flux formulation for DFT on group manifolds.

Connected background fluxes to Lie group structures.

Explicitly calculated the vielbein for compact embeddings in O(3,3).

Abstract

A flux formulation of Double Field Theory on group manifold is derived and applied to study generalized Scherk-Schwarz compactifications, which give rise to a bosonic subsector of half-maximal, electrically gauged supergravities. In contrast to the flux formulation of original DFT, the covariant fluxes split into a fluctuation and a background part. The latter is connected to a $2D$-dimensional, pseudo Riemannian manifold, which is isomorphic to a Lie group embedded into O($D,D$). All fields and parameters of generalized diffeomorphisms are supported on this manifold, whose metric is spanned by the background vielbein $E_A{}^I \in$ GL($2D$). This vielbein takes the role of the twist in conventional generalized Scherk-Schwarz compactifications. By doing so, it solves the long standing problem of constructing an appropriate twist for each solution of the embedding tensor. Using the geometric structure, absent in original DFT, $E_A{}^I$ is identified with the left invariant Maurer-Cartan form on the group manifold, in the same way as it is done in geometric Scherk-Schwarz reductions. We show in detail how the Maurer-Cartan form for semisimple and solvable Lie groups is constructed starting from the Lie algebra. For all compact embeddings in O($3,3$), we calculate $E_A{}^I$.