The G_2 spinorial geometry of supersymmetric IIB backgrounds

TL;DR

This paper classifies supersymmetric IIB backgrounds with specific G_2, Spin(7), and SU(4) structures, revealing geometric reductions, flux conditions, and explicit spacetime descriptions for these highly symmetric solutions.

Contribution

It provides a detailed geometric analysis of IIB backgrounds with various supersymmetries and stabilizer groups, including explicit forms of the metric and fluxes, extending previous classifications.

Findings

G_2 backgrounds admit a time-like Killing vector and G_2 structure.

Spin(7) and SU(4) backgrounds have null vector fields and special holonomy manifolds.

G_2 backgrounds have fluxes vanishing and are products of Minkowski space with G_2 manifolds.

Abstract

We solve the Killing spinor equations of supersymmetric IIB backgrounds which admit one supersymmetry and the Killing spinor has stability subgroup G_2 in Spin(9,1) x U(1). We find that such backgrounds admit a time-like Killing vector field and the geometric structure of the spacetime reduces from Spin(9,1) x U(1) to G_2. We determine the type of G_2 structure that the spacetime admits by computing the covariant derivatives of the spacetime forms associated with the Killing spinor bilinears. We also solve the Killing spinor equations of backgrounds with two supersymmetries and Spin(7)\ltimes R^8-invariant spinors, and four supersymmetries with SU(4)\ltimes R^8- and with G_2-invariant spinors. We show that the Killing spinor equations factorize in two sets, one involving the geometry and the five-form flux, and the other the three-form flux and the scalars. In the Spin(7)\ltimes R^8 and SU(4)\ltimes R^8 cases, the spacetime admits a parallel null vector field and so the spacetime metric can be locally described in terms of Penrose coordinates adapted to the associated rotation free, null, geodesic congruence. The transverse space of the congruence is a Spin(7) and a SU(4) holonomy manifold, respectively. In the G_2 case, all the fluxes vanish and the spacetime is the product of a three-dimensional Minkowski space with a holonomy G_2 manifold.