# Supergravity Fluxes and Generalised Geometry

## TL;DR

This paper reviews how supergravity theories' internal sectors can be described using generalised geometry, focusing on supersymmetric backgrounds and recent advances in heterotic flux vacua involving holomorphic Courant algebroids and deformation theory.

## Contribution

It introduces a novel approach to studying heterotic flux vacua via holomorphic Courant algebroids and $L_$-algebras, connecting superpotential functionals to holomorphic Chern-Simons theory.

## Key findings

- Reformulation of the Hull-Strominger system using $L_$-algebras.
- Identification of the superpotential as a holomorphic Chern-Simons functional.
- Advancement in understanding supersymmetric backgrounds through generalized geometry.

## Abstract

We briefly review the description of the internal sector of supergravity theories in the language of generalised geometry and how this gives rise to a description of supersymmetric backgrounds as integrable geometric structures. We then review recent work, featuring holomorphic Courant algebroids, on the description of $\mathcal N=1$ heterotic flux vacua. This work studied the finite deformation problem of the Hull-Strominger system, guided by consideration of the superpotential functional on the relevant space of geometries. It rewrote the system in terms of the Maurer-Cartan set of a particular $L_\infty$-algebra associated to a holomorphic Courant algebroid, with the superpotential itself becoming an analogue of a holomorphic Chern-Simons functional.