Lectures on Gauged Supergravity and Flux Compactifications
Henning Samtleben
TL;DR
The notes present a unified, duality-covariant framework for gauged supergravities arising from flux compactifications. The central tool is the embedding tensor $\Theta_M{}^{\alpha}$, which encodes how a gauge group $\mathrm{G}_0$ embeds into the global symmetry $\mathrm{G}$ of the ungauged theory, with consistency enforced by quadratic and linear constraints. This formalism naturally produces the hierarchy of p-forms and their topological couplings, and yields a universal expression for the scalar potential in terms of $\Theta$ and the scalar matrix $\mathcal{M}$. By analyzing higher-dimensional origins (M-theory and type II) and their fluxes, the notes demonstrate how specific fluxes map to components of $\Theta$ and how dualities act on these flux parameters, including geometric and non-geometric cases. The covariant framework thus provides a powerful, compact method to study flux-induced gaugings across dimensions and relate them to flux compactifications.
Abstract
The low-energy effective theories describing string compactifications in the presence of fluxes are so-called gauged supergravities: deformations of the standard abelian supergravity theories. The deformation parameters can be identified with the various possible (geometric and non-geometric) flux components. In these lecture notes we review the construction of gauged supergravities in a manifestly duality covariant way and illustrate the construction in several examples.