${\rm SDiff}(S^2)$ and the orbit method

TL;DR

This paper explores the representation theory of the infinite-dimensional Lie group ${\rm SDiff}(S^2)$, relevant in hydrodynamics and gravity, using coadjoint orbits and fixed point formulas, revealing divergence issues and their resolutions.

Contribution

It provides a detailed analysis of the coadjoint orbit method for ${\rm SDiff}(S^2)$, including Casimir functions, the Cartan algebra, and trace evaluations, connecting mathematical structures to physical applications.

Findings

Identification of Casimir functions and Cartan algebra for ${\rm SDiff}(S^2)$

Evaluation of traces using Atiyah-Bott fixed point formula with divergence analysis

Connection of mathematical results to black hole physics and symmetry groups

Abstract

The group of area preserving diffeomorphisms of the two sphere, ${\rm SDiff}(S^2)$, is one of the simplest examples of an infinite dimensional Lie group. It plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related, newly defined symmetry groups. We investigate its representation theory using the method of coadjoint orbits. We describe the Casimir functions and the Cartan algebra. Then we evaluate the trace of a simple ${\rm SDiff}(S^2)$ operator using the Atiyah-Bott fixed point formula. The trace is divergent but we show that it has well-defined truncations related to the structure of ${\rm SDiff}(S^2)$. Finally, we relate our results back to the recent appearances of ${\rm SDiff}(S^2)$ in black hole physics.