Gauged Supergravities from Twisted Doubled Tori and Non-Geometric String Backgrounds

TL;DR

This work builds a unified, duality-covariant framework for gauged supergravities via twisted doubled tori and megatorus constructions, linking all known fluxes to the embedding tensor. By embedding Scherk–Schwarz twists into ${ m O}(n,n)$ (and ${ m E}_{7(7)}$ for M-theory), it demonstrates how NS-NS, geometric, and non-geometric fluxes arise as generalized geometric fluxes ${ m T}_{MN}{}^P$ and reveals when doubled twists correspond to ordinary geometry or genuinely non-geometric backgrounds. The study provides explicit realizations for string theory on ${ m T}^3$ with $h, au,Q,R$ fluxes, connects these to ${ m N}=4$ gaugings through the embedding tensor, and extends the construction to M-theory with a $56$-dimensional megatorus and ${ m E}_{7(7)}$ covariant gaugings. Together, these results clarify the higher-dimensional origins of gauged supergravities and supply a practical prescription to extract the embedded geometry from duality-twisted compactifications.

Abstract

We propose a universal geometric formulation of gauged supergravity in terms of a twisted doubled torus. We focus on string theory (M-theory) reductions with generalized Scherk-Schwarz twists residing in the O(n,n) (E_{7(7)}) duality group. The set of doubled geometric fluxes, associated with the duality twists and identified naturally with the embedding tensor of gauged supergravity, captures all known fluxes, i.e. physical form fluxes, ordinary geometric fluxes, as well as their non-geometric counterparts. Furthermore, we propose a prescription for obtaining the effective geometry embedded in the string theory twisted doubled torus or in the M-theory megatorus and apply it for several models of geometric and non-geometric flux compactifications.

Figures (1)

• Figure 1: The tangent space on a torus $T({\mathbb T}^2) = \{\lambda \frac{\partial}{\partial y} + \mu \frac{\partial}{\partial \widetilde{y}}, \lambda,\mu \in {\mathbb R}\}$ splits into the tangent space on its base circle $T({\mathbb S}^1) = \{\lambda \frac{\partial}{\partial y}, \lambda \in {\mathbb R}\}$ plus its cotangent space $T^*({\mathbb S}^1) = \{\lambda dy \simeq \lambda \frac{\partial}{\partial \widetilde{y}}, \lambda \in {\mathbb R}\}$ when there is a natural O(1,1) inner product pairing $\frac{\partial}{\partial y}$ and $\frac{\partial}{\partial \widetilde{y}}$.