Geometric algebra techniques in flux compactifications (II)

Abstract

We study constrained generalized Killing spinors over the metric cone and cylinder of a (pseudo-)Riemannian manifold, developing a toolkit which can be used to investigate certain problems arising in supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions for the metric and fluxes of the unit section of such cylinders and cones into differential and algebraic constraints on collections of differential forms defined on the cylinder or cone. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As a non-trivial application, we consider the most general N=2 compactification of eleven-dimensional supergravity on eight-manifolds.Comment: 56 pages, 2 figures, some commutative diagram

Singular foliations for M-theory compactification

“…Section 2 gives a brief review of the class of compactifications we consider, in order to fix notations and conventions. Section 3 discusses a geometric characterization of Majorana spinors ξ on M which is inspired by the rigorous approach developed in [32][33][34] for the method of bilinears [35], in the case when the spinor ξ is allowed to be chiral at some loci. It also gives the Kähler-Atiyah parameterizations of this spinor which correspond to the approach of [8] and to that of [1] and describes the relevant G-structures using both spinors and idempotents in the Kähler-Atiyah algebra of M .…”

“…All fiber bundles we consider are smooth. 2 We use freely the results and notations of [8,[32][33][34], with the same conventions as there. To simplify notation, we write the geometric product ⋄ of [32][33][34] simply as juxtaposition while indicating the wedge product of differential forms through ∧.…”

“…We can in fact take B to be a scalar product on S and denote the corresponding norm by || || (see [32,33] for details). Requiring that the background preserves exactly N = 1 supersymmetry amounts to asking that dim K(D, Q) = 1.…”

The landscape of G-structures in eight-manifold compactifications of M-theory

“…Hp supersymmetry equations (which were also derived in [25]) using the nine-dimensional formalism. Reference [26] makes intensive use of an assumption (equation (3.9) of loc.…”

“…The rest of the paper is devoted to the detailed study of the latter case. Section 2 discusses the scalar and one-form bilinears which can be constructed using a basis of K when dim K = 2 and introduces two cosmooth generalized distributions D and D 0 (where D 0 ⊂ D) which 1 Such N = 2 backgrounds were considered in [25] using a nine-dimensional formalism and were also discussed in [26] with similar methods, but without carefully studying the corresponding geometry of the eight-manifold. Certain N = 1 compactifications down to three-dimensional Minkowski space but with torsion-full SU(4) structure were studied in [27, section 3].…”