A Note on Coherent States for Virasoro Orbits

TL;DR

This paper constructs and analyzes coherent states for specific Virasoro orbits, providing explicit formulas and exploring their properties, with potential applications to (2+1)-dimensional gravity.

Contribution

It introduces a method to construct coherent states for Virasoro orbits using geometric quantization and provides explicit formulas for the K"ahler potential.

Findings

Explicit formula for the K"ahler potential.

Construction of coherent states via geometric quantization.

Discussion on applications to (2+1)-dimensional gravity.

Abstract

There are two sets of orbits of the Virasoro group which admit a K\"ahler structure. We consider the construction of coherent states for the orbit $\widehat{{\rm diff}\,S^1}$/$SL(2, {\mathbb R})$ which furnishes unitary representations of the group. The procedure is analogous to geometric quantization using a holomorphic polarization. We also give an explicit formula for the K\"ahler potential for this orbit and comment on normalization of the coherent states. We further explore some of the properties of these states, including the definition of symbols corresponding to operators and their star products. Some comments which touch upon the possibility of applying this to gravity in (2+1) dimensions are also given.