β-supergravity: a ten-dimensional theory with non-geometric fluxes, and its geometric framework

TL;DR

This work presents beta-supergravity, a ten-dimensional NSNS framework incorporating non-geometric $Q$- and $R$-fluxes and aimed at uplifting certain four-dimensional gauged supergravities. By performing a field redefinition to a beta-frame $(\tilde{g},\beta,\tilde{\phi})$ and invoking Generalized Geometry/DFT, the authors derive a manifestly covariant Lagrangian $\tilde{\mathcal{L}}_{\beta}$ that includes a new $Q$-flux connection and a second curvature term $\mathcal{R}_Q$. They develop the associated differential calculus, define the covariant derivatives $\widecheck{\nabla}$ and $\omega_Q$, and compute the equations of motion for $(\tilde{g},\tilde{\phi},\beta)$, along with a four-dimensional reduction yielding flux-driven potentials $V_Q$ and $V_R$. A toroidal example illustrates how non-geometry can be reinterpreted in the beta-frame, and the paper introduces the generalized cotangent bundle $E_{T^*}$ and a $\beta$-gauge symmetry that controls global patching. Together, these results provide a concrete ten-dimensional geometric framework for non-geometric fluxes and a route to connect them to 4D gauged supergravities, with potential pure NSNS backgrounds and future RR/gauge extensions.

Abstract

We present a ten-dimensional theory, named β-supergravity, that contains non-geometric fluxes and could uplift some four-dimensional gauged supergravities. Building on earlier work, we study here its NSNS sector, where Q- and R-fluxes are precisely identified. Interestingly, the Q-flux is captured in an analogue of the Levi-Civita spin connection, giving rise to a second curvature scalar. We reproduce the ten-dimensional Lagrangian using the Generalized Geometry formalism; this provides us with enlightening interpretations of the new structures. Then, we derive the equations of motion of our theory, and finally discuss further aspects: the dimensional reduction to four dimensions and comparison to gauged supergravities, the obtention of ten-dimensional purely NSNS solutions, the extensions to other sectors and new objects, the supergravity limit, and eventually the symmetries, in particular the βgauge transformation. We also introduce the related notion of a generalized cotangent bundle.