The spinorial geometry of supersymmetric heterotic string backgrounds

Abstract

We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection∇ with torsion H, the NS⊗NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing spinors of the null backgrounds have stability subgroups K ⋉ R 8 in Spin(9, 1), for K = Spin(7), SU (4), Sp(2), SU (2)×SU (2) and {1}, and the Killing spinors of the timelike backgrounds have stability subgroups G 2 , SU (3), SU (2) and {1}. The former admit a single null∇-parallel vector field while the latter admit a timelike and two, three, five and nine spacelike∇-parallel vector fields, respectively. The spacetime of the null backgrounds is a Lorentzian two-parameter family of Riemannian manifolds B with skew-symmetric torsion. If the rotation of the null vector field vanishes, the holonomy of the connection with torsion of B is contained in K. The spacetime of time-like backgrounds is a principal bundle P with fibre a Lorentzian Lie group and base space a suitable Riemannian manifold with skew-symmetric torsion. The principal bundle is equipped with a connection λ which determines the non-horizontal part of the spacetime metric and of H. The curvature of λ takes values in an appropriate Lie algebra constructed from that of K. In addition dH has only horizontal components and contains the Pontrjagin class of P . We have computed in all cases the Killing spinor bilinears, expressed the fluxes in terms of the geometry and determine the field equations that are implied by the Killing spinor equations.

Geometry and supersymmetry of heterotic warped flux AdS backgrounds

“…Such theories are the common sector of type II supergravities and the heterotic supergravity. One simplification that occurs in the common sector and heterotic supergravities is that all solutions of the KSEs are known and the geometry of the backgrounds has been determined [25]- [27]. In particular, AdS 3 configurations arise naturally in the context of classification of supersymmetric heterotic backgrounds [25]- [27] and their geometry has also been examined 1 in [28].…”

“…One simplification that occurs in the common sector and heterotic supergravities is that all solutions of the KSEs are known and the geometry of the backgrounds has been determined [25]- [27]. In particular, AdS 3 configurations arise naturally in the context of classification of supersymmetric heterotic backgrounds [25]- [27] and their geometry has also been examined 1 in [28]. The AdS backgrounds are also special cases of the heterotic horizons in [29,30] though the latter has been explored only for a closed 3-form field strength H. In addition, there is a no-go theorem for AdS 4 heterotic backgrounds that has been established in [31,32] without assuming supersymmetry but imposing smoothness on the fields and compactness of the transverse 6-dimensional space.…”

“…If the solution of the KSEs is determined by the σ ± spinors, the investigation of the geometry of M 7 can be done as a special case of that of heterotic horizons in [29,30] which utilized the classification results of [25]- [27]. To see this first note that h = − 2 ℓ dz and so the constant k which enters in the description of geometry for the heterotic horizons is…”

“…This is because it is a consequence of the gravitino and dilatino KSEs that these vector fields close to a su(2) algebra [25]- [27]. The metric and torsion are given as in (2.35) but now the formulae are valid up to order α ′2 .…”

Non-existence of supersymmetric AdS 5 black rings

“…The unique maximally supersymmetric solution preserving all supersymmetries is AdS 5 . Systematic classifications of 1/2-supersymmetric solutions were investigated in [19,20] using spinorial geometry techniques first implemented in the study of higher dimensional solutions in [21][22][23]. Using these methods, a restricted class of half-supersymmetric near-horizon geometries was analysed in [24].…”