Higher Asymptotics of Unitarity in "Quantization Commutes with Reduction"

TL;DR

This paper provides detailed asymptotic expansions for functions measuring the unitarity of the Guillemin--Sternberg map and its metaplectically corrected version in geometric quantization, as h-bar approaches zero.

Contribution

It derives explicit formulas and asymptotic expansions for the unitarity functions in the context of geometric quantization with symmetry reduction.

Findings

Asymptotic expansions of unitarity functions as h-bar approaches zero.

Explicit expressions for the unitarity functions in the Guillemin--Sternberg setting.

Confirmation that metaplectic correction improves unitarity in the semiclassical limit.

Abstract

Let M be a compact Kaehler manifold equipped with a Hamiltonian action of a compact Lie group G. In [Invent. Math. 67 (1982), no.~3, 515--538], Guillemin and Sternberg showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M//G. This map, though, is not in general unitary, even to leading order in h-bar. In [Comm. Math. Phys. 275 (2007), no.~2, 401--422], Hall and the author showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed h-bar, becomes unitary in the semiclassical limit that h-bar goes to zero. (cf. the work of Ma and Zhang in [C. R. Math. Acad. Sci. Paris 341 (2005), no.~5, 297--302], and [Ast\'erisque No. 318 (2008), viii+154 pp.]). The unitarity of the classical Guillemin--Sternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M//G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as h-bar goes to zero.