Supersymmetric Backgrounds, the Killing Superalgebra, and Generalised Special Holonomy

TL;DR

This paper establishes a correspondence between supersymmetric flux backgrounds in M theory and type II string theory and integrable generalised G structures, extending the concept of special holonomy to generalised geometry.

Contribution

It introduces the Kosmann-Dorfman bracket and demonstrates that supersymmetric backgrounds correspond to integrable generalised G structures, generalising special holonomy in flux backgrounds.

Findings

Supersymmetric flux backgrounds correspond to integrable generalised G structures.

The internal Killing superalgebra simplifies and its closure is proven.

Eleven-dimensional Killing superalgebra is the supertranslational part of super-Poincaré algebra.

Abstract

We prove that, for M theory or type II, generic Minkowski flux backgrounds preserving $\mathcal{N}$ supersymmetries in dimensions $D\geq4$ correspond precisely to integrable generalised $G_{\mathcal{N}}$ structures, where $G_{\mathcal{N}}$ is the generalised structure group defined by the Killing spinors. In other words, they are the analogues of special holonomy manifolds in $E_{d(d)} \times\mathbb{R}^+$ generalised geometry. In establishing this result, we introduce the Kosmann-Dorfman bracket, a generalisation of Kosmann's Lie derivative of spinors. This allows us to write down the internal sector of the Killing superalgebra, which takes a rather simple form and whose closure is the key step in proving the main result. In addition, we find that the eleven-dimensional Killing superalgebra of these backgrounds is necessarily the supertranslational part of the $\mathcal{N}$-extended super-Poincar\'e algebra.