Abstract
We study “constrained generalized Killing (s)pinors,” which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As an application, we show how our approach can be used to efficiently treatN=1compactification ofM-theory on eight manifolds and prove that we recover results previously obtained in the literature.
Highlights
A fundamental problem in the study of flux compactifications of M-theory and string theory is to give efficient geometric descriptions of supersymmetric backgrounds in the presence of fluxes
We show that the geometric analysis of supersymmetry conditions for flux backgrounds can be formulated efficiently in this language, thereby uncovering structure whose implications have remained largely unexplored
An example with a single algebraic constraint Qξ = 0 is discussed in Section 6, which the reader can consult first as an illustration motivating the formal developments taken up in the rest of the paper. We show how such supersymmetry conditions can be translated efficiently and briefly into a system of differential and algebraic constraints for a collection of inhomogeneous differential forms expressed as (s)pinor bilinears, displaying the underlying structure in a form which is conceptually clear as well as highly amenable to computation
Summary
A fundamental problem in the study of flux compactifications of M-theory and string theory is to give efficient geometric descriptions of supersymmetric backgrounds in the presence of fluxes. The supersymmetry equations of eleven-dimensional supergravity involve the supercovariant connection D, which acts on sections of the bundle S of Majorana spinors (a.k.a. real pinors) defined in eleven dimensions; this corresponds to the differential constraint Dξ = 0, without any algebraic constraint. When considering a compactification down from ten dimensions, the constraints Q1ξ = Q2ξ = 0 descend to similar constraints for the internal part of ξ, while the constraint Dξ = 0 induces both a differential and an algebraic constraint for the internal part; the compactification procedure produces a differential constraint while increasing the number of algebraic constraints, the resulting equations being again of constrained generalized Killing type, but formulated for sections of some bundle of (s)pinors defined over the internal space of the compactification. The reader who is familiar with geometric algebra may wish to concentrate on Sections 3.5, 3.6, 3.7, 3.9, and 3.10 and especially on our treatment of parallelism and orthogonality for twisted (anti-)self-dual forms, which is important for applications
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