Systematics of M-theory spinorial geometry

Abstract

We reduce the classification of all supersymmetric backgrounds in eleven dimensions to the evaluation of the supercovariant derivative and of an integrability condition, which contains the field equations, on six types of spinors. We determine the expression of the supercovariant derivative on all six types of spinors and give in each case the field equations that do not arise as the integrability conditions of Killing spinor equations. The Killing spinor equations of a background become a linear system for the fluxes, geometry and spacetime derivatives of the functions that determine the spinors. The solution of the linear system expresses the fluxes in terms of the geometry and specifies the restrictions on the geometry of spacetime for all supersymmetric backgrounds. We also show that the minimum number of field equations that is needed for a supersymmetric configuration to be a solution of eleven-dimensional supergravity can be found by solving a linear system. The linear systems of the Killing spinor equations and their integrability conditions are given in both a timelike and a null spinor basis. We illustrate the construction with examples.

The spinorial geometry of supersymmetric heterotic string backgrounds

“…This is similar to the calculation of IIB-supergravity and M-theory in [31]. We use the following conventions for the indices: A, B, C ∈ {+, −,1, ..,4, 1, .., 4}, α, β ∈ {1, .., 4} and k, l, m, n ∈ {1, .., 4} with the restriction k, l, m, n = α, β.…”

Supersymmetric wrapped membranes,AdS2spaces, and bubbling geometries

“…The derivation of the supersymmetry conditions is very similar to that for threefolds in the generic case, and technical details are omitted. The metric is given by 20) withŵ = e 3 . The torsion conditions we derive are…”

Aspects of Spinorial Geometry

“…Using the expressions (2.14), (2.15) and (2.16) for the components of D A e i 1 ···i I in the basis (2.13) which appear in square brackets in (2.17), one can easily compute the components of D A e i 1 ···i I in the canonical basis (2.7). The explicit expressions for the different basis elements are given in [19]. Substituting (2.17) into (2.10) and setting each component in the basis (2.7) equal to zero, one derives a linear system with variables the spin connection Ω of the geometry and the fluxes that appear in the supercovariant derivative, in this case the four-form F .…”