(M-theory-)Killing spinors on symmetric spaces

Abstract

Abstract. We show how the theory of invariant principal bundle connections for reductive homogeneous spaces can be applied to determine the holonomy of generalised Killing spinor covariant derivatives of the form D = ∇ + Ω in a purely algebraic and algorithmic way, whereis a left-invariant homomorphism. Specialising this to the case of symmetric M −theory backgrounds (i.e. (M, g, F ) with (M, g) a symmetric space and F an invariant closed 4-form), we derive several criteria for such a background to preserve some supersymmetry and consequently find all supersymmetric symmetric M −theory backgrounds.

On integrability of strings on symmetric spaces

“…1 In the sense that the full moduli spaces for many AdS3 and AdS2 backgrounds were not determined. 2 For example in the eleven-dimensional case the list of backgrounds found in [6] was analyzed recently for supersymmetry in [7] and no new examples were found. 3 For the cases without NSNS flux (table 1) integrability of the corresponding supercoset models follows from the construction of [3].…”

“…16 Using the conservation equations for the components of the Noether current in eqs. (4.69)-(4.71), the superspace constraints for the supercoset subspace (with H abc = 0) 7) and the superisometry algebra in eqs. (2.10) and (4.49) we find for the terms in the curvature of L proportional to P a…”

All symmetric space solutions of eleven-dimensional supergravity

“…The general form of the superisometry algebra for symmetric space backgrounds is also derived.He gave a list of all the AdS geometries that can occur and determined the fluxes completely for all cases except backgrounds of the form AdS 2 × (S 2 ) n × T 9−2n for which the analysis was too involved. The supersymmetry of the backgrounds listed in [10] was later analyzed in [11].Here we will revisit this problem and determine all symmetric space solutions of elevendimensional supergravity including the AdS 2 × (S 2 ) n × T 9−2n ones. On the way we will correct some mistakes in the analysis of [10].…”

“…Solutions with È 2 factors are obtained as special cases of the above analysis. With two È 2 factors we find the solutions11 …”

Spencer Cohomology and 11-Dimensional Supergravity

“…The consensus seems to be that, at present, the classification of all supersymmetric backgrounds is inaccessible, whereas that of highly supersymmetric backgrounds seems tantalisingly in reach. In particular, backgrounds with N > 16 are now known to be locally homogeneous [11] and this brings to bear the techniques of homogeneous geometry to classify certain kinds of backgrounds; e.g., symmetric [12,13] or homogeneous under a given Lie group [14,15], at least when the group is semisimple. This paper is a first step in a Lie algebraic approach at the classification problem.…”