The Torsion of Spinor Connections and Related Structures

Abstract

Abstract. In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Furthermore, we discuss applications of this notion of torsion.

SUSY structures on deformed supermanifolds

“…The step of induction is done by showing that ı(•), Y (• k−1 ) = Y (• k ) with Y (•) as in (38) or (39). The fact that all summands appear follows with (10).…”

“…This is the price for the fact that the class of connections we will consider does not preserve the charge conjugation in general, see definition 3.7, remark 3.8, and proposition 3.10. For the proofs of the statements and an extended discussion of the objects which we recall below we cordially refer the reader to [10] where the torsion of spinor connections has been introduced.…”

“…In particular, we recall that the choice of charge conjugation on the spinor bundle S over the reduced manifold M red is in such a way that ∆ 1 = 1. Some of the relations we will use in the following have also been used in [10] when we discussed second order commutators and their relations to the Bianchi identities cf. proposition 3.6.…”

“…This is achieved by imposing vanishing conditions on some of the higher order terms. When we discussed brane metrics in [10] the vanishing of D(T , •, •) restricted the parallel spinors to those contained in the kernel of the basic vector field of the metric. In view of future work one might construct actions of the infinitesimal automorphism S of M q on sets of fields which allow defining generalized invariant Lagrangians.…”

Connections on Cahen-Wallach Spaces

“…For this we have to consider enlargements of the Lie algebra of infinitesimal isometries of a given space to a super Lie algebra. The odd part of it is characterized by spinors that are parallel with respect to a given connection, see [1,15,16,17], for example, in addition to the references from the introduction. Starting from the text at hand, the next natural step is the classification of super algebras that extend the isometries of CW-spaces.…”