Supersymmetric Backgrounds and Generalised Special Holonomy

TL;DR

This paper introduces a new concept of generalised special holonomy in the context of supersymmetric flux compactifications in M theory and Type II string theory, linking it to preserved supersymmetry and providing a unified geometric framework.

Contribution

It defines intrinsic torsion in generalised geometry and establishes a correspondence between minimal supersymmetry and generalised special holonomy groups in flux compactifications.

Findings

Preservation of supersymmetry corresponds to specific generalised holonomy groups.

For D≥4, minimal supersymmetry implies the manifold has generalised special holonomy.

Example: N=1 in D=4 corresponds to SU(7) holonomy, extending G2 structures.

Abstract

We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form $\mathbb{R}^{D-1,1}\times M$. Using the language of $E_{d(d)}\times\mathbb{R}^+$ generalised geometry, we show that, for $D\geq 4$, preserving minimal supersymmetry is equivalent to the manifold $M$ having generalised special holonomy and list the relevant holonomy groups. We conjecture that this result extends to backgrounds preserving any number of supersymmetries. As a prime example, we consider $\mathcal{N}=1$ in $D=4$. The corresponding generalised special holonomy group is $SU(7)$, giving the natural M theory extension to the notion of a $G_2$ manifold, and, for Type II backgrounds, reformulating the pure spinor $SU(3)\times SU(3)$ conditions as an integrable structure.