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arXiv · 0706.1471

Singular unitarity in "quantization commutes with reduction"

Hui Li
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Abstract

Let $M$ be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group $G$. Let $M//G=ϕ^{-1}(0)/G=M_0$ be the symplectic quotient at value 0 of the moment map $ϕ$. 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.

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Hui Li. 2008-01-09. Singular unitarity in "quantization commutes with reduction". https://doi.org/10.1016/j.geomphys.2008.01.006

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