Flux formulation of DFT on group manifolds and generalized Scherk-Schwarz compactifications

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 ∈ 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 .

Exceptional field theory: SL(5)

“…However, there is still discussion in the literature, whether one should use other approaches to describe non-toroidal backgrounds. One of them is the so called WZW Double Field Theory, that attempts to construct a DFT on a group manifold [59][60][61][62]. hence, in this context it would be interesting to expand the ideas of exceptional field theory to DFT W ZW and look for possible uplifts.…”

Enhanced gauge symmetry and winding modes in double field theory

“…18 Here we do not need to know the way in which the algebra is realized. It could be in the way presented in the previous section, or in whatever other way one may come up with, like for example by a compactification on an S 3 with H flux [37,38], or a compactification on S 3 L × S 3 R , under an ordinary 6-dimensional Lie bracket [39].…”

Double Field Theory on Group Manifolds in a Nutshell

“…In order to make contact with the traditional formulation of DFT, we need to impose an extra constraint [27,28]. Generally, there exist two possible ways to do this.…”

“…All of these observations suggest that DFT WZW can be considered as a generalization of DFT. It can be reformulated in form of a generalized metric formulation [27], thus extrapolating the theory to all orders in fields, and in form of a flux formulation [28]. Through the consequent use of covariant derivatives in this theory, it possesses an additional 2D-diffeomorphism invariance as a result of the rigorous splitting between background fields and fluctuations in DFT WZW [27].…”

“…In order to resolve these issues, an alternative theory with doubled coordinate space called Double Field Theory on Group Manifolds (DFT WZW ) has been proposed [26,27,28,33]. Starting from Closed String Field Theory (CSFT) calculations at tree level up to cubic order in fields and leading order of α , an action and the according gauge transformations have been derived from a Wess-Zumino-Witten (WZW) model and its corresponding Kač-Moody algebra [26].…”