The spinorial geometry of supersymmetric IIB backgrounds

TL;DR

This paper classifies supersymmetric IIB supergravity backgrounds with one Killing spinor, analyzing the geometric and flux conditions for different spinor orbits, and explores specific $SU(4) times R^8$ backgrounds with pure spinors.

Contribution

It provides a detailed classification of IIB backgrounds based on spinor orbits and explicitly solves Killing spinor equations for certain cases, linking geometry and fluxes.

Findings

Existence of three spinor orbit types with specific stability subgroups.

Explicit solutions for Killing spinor equations in two orbit cases.

Identification of null, self-parallel Killing vectors in supersymmetric backgrounds.

Abstract

We investigate the Killing spinor equations of IIB supergravity for one Killing spinor. We show that there are three types of orbits of Spin(9,1) in the space of Weyl spinors which give rise to Killing spinors with stability subgroups $Spin(7)\ltimes \bR^8$, $SU(4)\ltimes \bR^8$ and $G_2$. We solve the Killing spinor equations for the $Spin(7)\ltimes \bR^8$ and $SU(4)\ltimes \bR^8$ invariant spinors, give the fluxes in terms of the geometry and determine the conditions on the spacetime geometry imposed by supersymmetry. In both cases, the spacetime admits a null, self-parallel, Killing vector field. We also apply our formalism to examine a class of $SU(4)\ltimes \bR^8$ backgrounds which admit one and two pure spinors as Killing spinors and investigate the geometry of the spacetimes.