Riemannian geometry of ${\rm Diff}(S^1)/S^1$ and representations of the Virasoro algebra

TL;DR

This paper computes the Ricci curvature of the homogeneous space ${ m Diff}(S^1)/S^1$ using classical Riemannian geometry methods, providing new insights into its geometric structure without relying on Kähler symmetries.

Contribution

It offers a novel computation of Ricci curvature for ${ m Diff}(S^1)/S^1$ by applying classical finite-dimensional techniques, diverging from previous Kähler-based approaches.

Findings

Ricci curvature of ${ m Diff}(S^1)/S^1$ computed explicitly

Utilizes classical homogeneous space geometry methods

Provides new geometric insights into the Virasoro algebra representations

Abstract

The main result of the paper is a computation of the Ricci curvature of $\DS/S^1$. Unlike earlier results on the subject, we do not use the K\"{a}hler structure symmetries to compute the Ricci curvature, but rather rely on classical finite-dimensional results of Nomizu et al on Riemannian geometry of homogeneous spaces.