Three dimensional construction of the Virasoro-Bott group

TL;DR

This paper constructs the Virasoro-Bott group as a 3D geometric object, linking it to diffeomorphisms, boundary conditions, and topological terms, and explores its relation to Chern-Simons theory and gravity.

Contribution

It provides a novel 3D geometric construction of the Virasoro-Bott group using boundary conditions and topological terms, connecting it to Chern-Simons theory and gravity.

Findings

Realization of the Virasoro-Bott group as a quotient of diffeomorphisms with boundary conditions

Identification of the Lie algebra with the Virasoro algebra

Extension to a semidirect product with the Heisenberg algebra under generalized boundary conditions

Abstract

We present a three-dimensional geometric construction of the Virasoro-Bott group, which is a central extension of the group of diffeomorphisms of the circle. Our approach is analogous to the well-known construction of a central extension of the loop group by means of the Wess-Zumino topological term. In particular, the Virasoro-Bott group is realized as a quotient group of diffeomorphisms of the disc with special boundary conditions. We identify the Lie algebra corresponding to our group with the Virasoro algebra. We also show that for generalized boundary conditions the Virasoro algebra is extended to a semidirect product with the Heisenberg algebra. We discuss the relation between our construction, the Chern-Simons theory, and the three-dimensional gravity.