Geometry of type II common sector N=2 backgrounds

TL;DR

This paper classifies the geometries of all type II N=2 backgrounds with two supersymmetries, detailing their spacetime structures, stability subgroups, and how T-duality relates different geometries, with implications for string world-sheet conformal symmetry.

Contribution

It provides a comprehensive geometric classification of type II N=2 backgrounds with two supersymmetries, including the effects of T-duality and the structure of the underlying manifolds.

Findings

Backgrounds with $K\ltimes \bR^8$ stability subgroup are pp-waves in $K$-structured manifolds.

Spacetimes with $K$-invariant Killing spinors are fiber bundles over 8D $K$-structured bases.

T-duality exchanges backgrounds with $K$- and $K\ltimes \bR^8$-invariant Killing spinors.

Abstract

We describe the geometry of all type II common sector backgrounds with two supersymmetries. In particular, we determine the spacetime geometry of those supersymmetric backgrounds for which each copy of the Killing spinor equations admits a Killing spinor. The stability subgroups of both Killing spinors are $Spin(7)\ltimes \bR^8$, $SU(4)\ltimes \bR^8$ and $G_2$ for IIB backgrounds, and $Spin(7)$, SU(4) and $G_2\ltimes \bR^8$ for IIA backgrounds. We show that the spacetime of backgrounds with spinors that have stability subgroup $K\ltimes \bR^8$ is a pp-wave propagating in an eight-dimensional manifold with a $K$-structure. The spacetime of backgrounds with $K$-invariant Killing spinors is a fibre bundle with fibre spanned by the orbits of two commuting null Killing vector fields and base space an eight-dimensional manifold which admits a $K$-structure. Type II T-duality interchanges the backgrounds with $K$- and $K\ltimes\bR^8$-invariant Killing spinors. We show that the geometries of the base space of the fibre bundle and the corresponding space in which the pp-wave propagates are the same. The conformal symmetry of the world-sheet action of type II strings propagating in these N=2 backgrounds can always be fixed in the light-cone gauge.