Systematics of IIB spinorial geometry

TL;DR

Systematics of IIB spinorial geometry develops a unified framework to classify all supersymmetric IIB backgrounds by turning the Killing spinor equations and their integrability conditions into linear systems on five fundamental spinor types $\\sigma_I$. This approach expresses fluxes and geometry in terms of spacetime data and the functions determining Killing spinors, enabling the determination of background geometry for arbitrary numbers of supersymmetries and identifying which field equations are implied by supersymmetry. The paper analyzes maximal and half-maximal $H$-invariant backgrounds, showing a clean factorization for maximal cases and a nonlinear, $z$-dependent structure for half-maximal ones, with explicit treatments of $N=2$ cases for $SU(4)\ltimes \mathbb{R}^8$ and maximally supersymmetric $Spin(7)\ltimes\mathbb{R}^8$ and $SU(4)\ltimes\mathbb{R}^8$ backgrounds. Together, these results provide a practical manual to compute geometry and flux constraints for a broad class of IIB backgrounds and to determine which equations are automatically satisfied by supersymmetry.

Abstract

We reduce the classification of all supersymmetric backgrounds of IIB supergravity to the evaluation of the Killing spinor equations and their integrability conditions, which contain the field equations, on five types of spinors. This extends the work of [hep-th/0503046] to IIB supergravity. We give the expressions of the Killing spinor equations on all five types of spinors. In this way, the Killing spinor equations become a linear system for the fluxes, geometry and spacetime derivatives of the functions that determine the Killing spinors. This system can be solved to express the fluxes in terms of the geometry and determine the conditions on the geometry of any supersymmetric background. Similarly, the integrability conditions of the Killing spinor equations are turned into a linear system. This can be used to determine the field equations that are implied by the Killing spinor equations for any supersymmetric background. We show that these linear systems simplify for generic backgrounds with maximal and half-maximal number of $H$-invariant Killing spinors, $H\subset Spin(9,1)$. In the maximal case, the Killing spinor equations factorize, whereas in the half-maximal case they do not. As an example, we solve the Killing spinor equations of backgrounds with two $SU(4)\ltimes \bR^8$-invariant Killing spinors. We also solve the linear systems associated with the integrability conditions of maximally supersymmetric $Spin(7)\ltimes\bR^8$- and $SU(4)\ltimes\bR^8$-backgrounds and determine the field equations that are not implied by the Killing spinor equations.