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

[摘要] We introduce the notion of epsilon eta-Einstein epsilon-contact metric three-manifold, which includes as particular cases eta-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 epsilon eta-Einstein Lorentzian epsilon-contact metric three-manifold with an epsilon eta-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 epsilon eta-Einstein contact metric three-manifolds. We classify all left-invariant epsilon eta-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 epsilon-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 epsilon-contact epsilon eta-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) over tilde (2, R) x S-3. (C) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2020-12-01

[关键词] Lorentzian geometry with torsion;Contact metric manifolds;eta-Einstein manifolds;Supergravity