Ulf Gran

The Spinorial geometry of supersymmetric backgrounds

We propose a new method to solve the Killing spinor equations of eleven-dimensional supergravity based on a description of spinors in terms of forms and on the Spin(1,10) gauge symmetry of the supercovariant derivative. We give the canonical form of Killing spinors for N=2 backgrounds provided that one of the spinors represents the orbit of Spin(1,10) with stability subgroup SU(5). We directly solve the Killing spinor equations of N=1 and some N=2, N=3 and N=4 backgrounds. In the N=2 case, we investigate backgrounds with SU(5) and SU(4) invariant Killing spinors and compute the associated spacetime forms. We find that N=2 backgrounds with SU(5) invariant Killing spinors admit a timelike Killing vector and that the space transverse to the orbits of this vector field is a Hermitian manifold with an SU(5)-structure. Furthermore, N=2 backgrounds with SU(4) invariant Killing spinors admit two Killing vectors, one timelike and one spacelike. The space transverse to the orbits of the former is an almost Hermitian manifold with an SU(4)-structure and the latter leaves the almost complex structure invariant. We explore the canonical form of Killing spinors for backgrounds with extended, N>2, supersymmetry. We investigate a class of N=3 and N=4 backgrounds with SU(4) invariant spinors. We find that in both cases the space transverse to a timelike vector field is a Hermitian manifold equipped with an SU(4)-structure and admits two holomorphic Killing vector fields. We also present an application to M-theory Calabi-Yau compactifications with fluxes to one-dimension.

On relating multiple M2 and D2-branes

Due to the difficulties of finding superconformal Lagrangian theories for multiple M2-branes, we will in this paper instead focus on the field equations. By relaxing the requirement of a Lagrangian formulation we can explore the possibility of having structure constants fABCD satisfying the fundamental identity but which are not totally antisymmetric. We exemplify this discussion by making use of an explicit choice of a non-antisymmetric fABCD constructed from the Lie algebra structure constants fabc of an arbitrary gauge group. Although this choice of fABCD does not admit an obvious Lagrangian description, it does reproduce the correct SYM theory for a stack of N D2-branes to leading order in gYM−1 upon reduction and, moreover, it sheds new light on the centre of mass coordinates for multiple M2-branes.

The spinorial geometry of supersymmetric IIB backgrounds

We solve the Killing spinor equations of supersymmetric IIB backgrounds which admit one supersymmetry and the Killing spinor has stability subgroup G2 in Spin(9, 1) × U(1). We find that such backgrounds admit a timelike Killing vector field and the geometric structure of the spacetime reduces from Spin(9, 1) × U(1) to G2. We determine the type of G2 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 -invariant spinors, and four supersymmetries with - and with G2-invariant spinors. We show that the Killing spinor equations factorize in two sets, one involving the geometry and the 5-form flux, and the other the 3-form flux and the scalars. In the and 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 G2 case, all the fluxes vanish and the spacetime is the product of a three-dimensional Minkowski space with a holonomy G2 manifold.

The spinorial geometry of supersymmetric heterotic string backgrounds

We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection

Geometry of all supersymmetric type I backgrounds

\hat\nabla

Manifestly supersymmetric M-theory

with torsion

Type IIB 7-brane solutions from nine-dimensional domain walls

H

Systematics of IIB spinorial geometry

, the NS

N=31 is not IIB

\otimes

Systematics of M-theory spinorial geometry

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