A model of affine gravity in two dimensions and plurality of topology

A new model of two-dimensional gravity with an action depending only on a linear connection is suggested. 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 generated in the process of solving the equations of motion for the connection. The general solution of these equations of motion is given by an arbitrary Weyl connection which can be described by using the space of orbits under the action of the conformal group in the functional space containing all pairs formed by a metric and a vector field. By choosing a gauge one obtains a constant curvature equation. It is shown that this model admits an equivalent description by using a family of Lagrangians depending on the metric and the connection as independent variables. We show that nonlinear Lagrangians in the first order formalism lead to plurality of topology for the manifolds under consideration and give a simple general mechanism of governing topology change.