Singular unitarity in "quantization commutes with reduction"

TL;DR

This paper investigates the relationship between quantum Hilbert spaces associated with a Kähler manifold and its symplectic quotient, establishing asymptotic unitarity without regularity assumptions using a metaplectic correction.

Contribution

It demonstrates asymptotic unitarity of a modified isomorphism between quantum spaces of singular quotients with a metaplectic correction, extending previous regularity-dependent results.

Findings

Asymptotic unitarity established to leading order in Planck's constant.

The relation holds without regularity assumptions on the quotient.

Metaplectic correction is crucial for the unitarity result.

Abstract

Let $M$ be a connected compact quantizable K\"ahler manifold equipped with a Hamiltonian action of a connected compact Lie group $G$. Let $M//G=\phi^{-1}(0)/G=M_0$ be the symplectic quotient at value 0 of the moment map $\phi$. The space $M_0$ may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over $M_0$ and the $G$-invariant subspace of the quantum Hilbert space over $M$. In this paper, without any regularity assumption on the quotient $M_0$, we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck's constant of a modified map of the above isomorphism under a ``metaplectic correction'' of the two quantum Hilbert spaces.