SDiff(S 2) and the orbit method

Robert Penna

doi:10.1063/1.5140475

SDiff(S ²) and the orbit method

Robert Penna

College of Arts & Sciences

SUNY Polytechnic Institute

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

The group of area preserving diffeomorphisms of the two sphere, SDiff(S²), 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 SDiff(S²) 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 SDiff(S²). Finally, we relate our results back to the recent appearances of SDiff(S²) in black hole physics.

Original language	English
Article number	012301
Journal	Journal of Mathematical Physics
Volume	61
Issue number	1
DOIs	https://doi.org/10.1063/1.5140475
State	Published - Jan 1 2020

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10.1063/1.5140475

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abstract = "The group of area preserving diffeomorphisms of the two sphere, SDiff(S2), 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 SDiff(S2) 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 SDiff(S2). Finally, we relate our results back to the recent appearances of SDiff(S2) in black hole physics.",

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SDiff(S ²) and the orbit method

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