The spinorial geometry of supersymmetric IIB backgrounds

Abstract

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