Connections on Cahen-Wallach Spaces

Abstract

Abstract. We systematically discuss connections on the spinor bundle of Cahen-Wallach symmetric spaces. A large class of these connections is closely connected to a quadratic relation on Clifford algebras. This relation in turn is associated to the symmetric linear map that defines the underlying space. We present various solutions of this relation. Moreover, we show that the solutions we present here provide a complete list with respect to a particular algebraic condition on the parameters that enter into the construction.

A family of non-restricted D=11 geometric supersymmetries

“…Here C (W ) denotes the Clifford algebra of W . In [16] such connections are discussed in detail, and we will state the result in Propositions 4.1 and 4.3. In Appendix A we recall our conventions for the Clifford algebra.…”

“…In [16] we showed that pairs of this type play an essential role, when we look for flat connections. 5 From the physics point of view the three-form -or more precisely the combination Γ + sc ,d (eµ) -can be interpreted as the additional ingredient in the variation of the gravitino that comes from the curvature of the three-form potential in minimal eleven dimensional supergravity theories, see, for example, [6].…”

“…In [16] we discuss in detail a large class of pairs (c, d) that solve condition (29), the so called quadratic Clifford pairs. Furthermore we give an additional condition that makes the list of solutions we present complete.…”

Eleven-dimensional symmetric supergravity backgrounds, their geometric superalgebras, and a common reduction

“…From [9] we know that any connection D on S that is compatible with the symmetric structure of M B is described by a pair of elements (c, d) of the Clifford algebra of V . It is given by…”