Geometry of all supersymmetric type I backgrounds

TL;DR

This paper classifies all supersymmetric type I backgrounds by analyzing the geometric structures dictated by Killing spinor equations, revealing how supersymmetry levels relate to spinor isotropy groups and holonomy reductions.

Contribution

It provides a comprehensive geometric classification of all supersymmetric type I backgrounds using spinorial geometry and Killing spinor equations, including detailed relations between supersymmetry, isotropy groups, and holonomy.

Findings

Classification of backgrounds based on isotropy groups in Spin(9,1)

Relation between the number of parallel spinors and preserved supersymmetries

Holonomy reductions imply further constraints on backgrounds

Abstract

We find the geometry of all supersymmetric type I backgrounds by solving the gravitino and dilatino Killing spinor equations, using the spinorial geometry technique, in all cases. The solutions of the gravitino Killing spinor equation are characterized by their isotropy group in Spin(9,1), while the solutions of the dilatino Killing spinor equation are characterized by their isotropy group in the subgroup Sigma(P) of Spin(9,1) which preserves the space of parallel spinors P. Given a solution of the gravitino Killing spinor equation with L parallel spinors, L = 1,2,3,4,5,6,8, the dilatino Killing spinor equation allows for solutions with N supersymmetries for any 0 < N =< L. Moreover for L = 16, we confirm that N = 8,10,12,14,16. We find that in most cases the Bianchi identities and the field equations of type I backgrounds imply a further reduction of the holonomy of the supercovariant connection. In addition, we show that in some cases if the holonomy group of the supercovariant connection is precisely the isotropy group of the parallel spinors, then all parallel spinors are Killing and so there are no backgrounds with N < L supersymmetries.