Geometric algebra techniques in flux compactifications (II)

TL;DR

This paper develops geometric algebra methods to analyze supersymmetric flux compactifications in supergravity, focusing on Killing spinors and Fierz identities, with applications to eleven-dimensional supergravity on eight-manifolds.

Contribution

It introduces a new toolkit using geometric algebra for translating supersymmetry conditions into differential form constraints, enhancing computational and conceptual understanding.

Findings

Provides a synthetic description of Fierz identities.

Develops effective methods for supersymmetry condition translation.

Applies techniques to N=2 supergravity compactifications.

Abstract

We study constrained generalized Killing spinors over the metric cone and cylinder of a (pseudo-)Riemannian manifold, developing a toolkit which can be used to investigate certain problems arising in supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions for the metric and fluxes of the unit section of such cylinders and cones into differential and algebraic constraints on collections of differential forms defined on the cylinder or cone. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As a non-trivial application, we consider the most general N=2 compactification of eleven-dimensional supergravity on eight-manifolds.