Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 025, 15 pages      arXiv:1410.5529      https://doi.org/10.3842/SIGMA.2015.025

Metaplectic-c Quantomorphisms

Jennifer Vaughan
Department of Mathematics, University of Toronto, Canada

Received December 13, 2014, in final form March 16, 2015; Published online March 24, 2015

Abstract
In the classical Kostant-Souriau prequantization procedure, the Poisson algebra of a symplectic manifold $(M,\omega)$ is realized as the space of infinitesimal quantomorphisms of the prequantization circle bundle. Robinson and Rawnsley developed an alternative to the Kostant-Souriau quantization process in which the prequantization circle bundle and metaplectic structure for $(M,\omega)$ are replaced by a metaplectic-c prequantization. They proved that metaplectic-c quantization can be applied to a larger class of manifolds than the classical recipe. This paper presents a definition for a metaplectic-c quantomorphism, which is a diffeomorphism of metaplectic-c prequantizations that preserves all of their structures. Since the structure of a metaplectic-c prequantization is more complicated than that of a circle bundle, we find that the definition must include an extra condition that does not have an analogue in the Kostant-Souriau case. We then define an infinitesimal quantomorphism to be a vector field whose flow consists of metaplectic-c quantomorphisms, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen for the infinitesimal quantomorphisms of a prequantization circle bundle. In particular, this space is isomorphic to the Poisson algebra $C^\infty(M)$.

Key words: geometric quantization; metaplectic-c prequantization; quantomorphism.

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