Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity

Abstract

We introduce the notion of εη-Einstein ε-contact metric three-manifold, which includes as particular cases η-Einstein Riemannian and Lorentzian (para) contact metric three-manifolds, but which in addition allows for the Reeb vector field to be null. We prove that the product of an εη-Einstein Lorentzian ε-contact metric three-manifold with an εη-Einstein Riemannian contact metric three-manifold carries a bi-parametric family of Ricci-flat Lorentzian metric-compatible connections with isotropic, totally skew-symmetric, closed and co-closed torsion, which in turn yields a bi-parametric family of solutions of six-dimensional minimal supergravity coupled to a tensor multiplet. This result allows for the systematic construction of families of Lorentzian solutions of six-dimensional supergravity from pairs of εη-Einstein contact metric three-manifolds. We classify all left-invariant εη-Einstein structures on simply connected Lie groups, paying special attention to the case in which the Reeb vector field is null. In particular, we show that the Sasaki and K-contact notions extend to ε-contact structures with null Reeb vector field but are however not equivalent conditions, in contrast to the situation occurring when the Reeb vector field is not light-like. Furthermore, we pose the Cauchy initial-value problem of an ε-contact εη-Einstein structure, briefly studying the associated constraint equations in a particularly simple decoupling limit. Altogether, we use these results to obtain novel families of six-dimensional supergravity solutions, some of which can be interpreted as continuous deformations of the maximally supersymmetric solution on Sl˜(2,R)×S3.