Singular unitarity in "quantization commutes with reduction"

Details Singular unitarity in "quantization commutes with reduction" Singular unitarity in "quantization commutes with reduction"

2007-06-11

arxiv.org/abs/0706.1471v3

Mathematics

Symplectic Geometry

28 pages

Scientific paper

10.1016/j.geomphys.2008.01.006

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.

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