Chern–Simons field theory on the general affine group, 3d-gravity and the extension of Cartan connections

Abstract

The purpose of this article is to study the correspondence between 3d-gravity and the Chern–Simons field theory from the perspective of geometric mechanics, specifically in the case where the structure group is the general affine group. To accomplish this, the paper discusses a variational problem of the Chern–Simons type on a principal fiber bundle with this group as its structure group. The connection to the usual Chern–Simons theory is established by utilizing a generalization, in the context of Cartan connections, of the notion of extension and reduction of connections.