A Model of Topological Affine Gravity in Two Dimensions

TL;DR

This paper introduces a topological two-dimensional gravity model based solely on a linear connection, which, through equations of motion, relates to metrics and vector fields satisfying constant curvature conditions.

Contribution

It presents a novel topological gravity model in two dimensions that depends only on a linear connection and explores its solutions and equivalent formulations.

Findings

Classical solutions described by Weyl group orbits.

Model admits an equivalent formulation with independent metric and connection.

Solutions involve metrics and vector fields satisfying constant curvature.

Abstract

A model of two--dimensional gravity with an action depending only on a linear connection is considered. This model is a topological one, in the sense that the classical action does not contain a metric or zweibein at all. A metric and an additional vector field are instead introduced in the process of solving equations of motion for the connection. They satisfy the constant curvature equation. It is shown that the general solution of these equations of motion can be described by using the space of orbits under the action of the Weyl group in the functional space containing all pairs formed by a metric and a vectorfield. It is shown also that this model admits an equivalent description by using a family of actions depending on the metric and the connection as independent variables.