Gravity from Lie algebroid morphisms

TL;DR

This paper explores how Lie algebroid structures can define metrics on spacetime manifolds, leading to new gravity models that generalize and embed known lower-dimensional theories, inspired by the Poisson Sigma Model.

Contribution

It introduces a framework for constructing gravity models from Lie algebroid morphisms, generalizing 2d and 3d gravity models through compatibility conditions on E-tensors.

Findings

Identifies conditions for bilinear forms to serve as metrics on Sigma.

Connects Lie algebroid structures with Riemannian foliations and sub-Riemannian structures.

Provides a broad class of new gravity models in various dimensions.

Abstract

Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from TSigma to any Lie algebroid E, where Sigma is regarded as d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X) A_i A_j, so as to permit an interpretation as a metric on Sigma. This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the additional structure is identified with a Riemannian foliation of M, in the Poisson case E=T^*M with a sub-Riemannian structure which is Poisson invariant with respect to its annihilator bundle. (For integrable image of S, this means that the induced Riemannian leaves should be invariant with respect to all Hamiltonian vector fields of functions which are locally constant on this foliation). This provides a huge class of new gravity models in d dimensions, embedding known 2d and 3d models as particular examples.