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Singular Poisson-K\"ahler geometry of stratified K\"ahler spaces and quantization

By Johannes Huebschmann and U Lille

Abstract

In the presence of classical phase space singularities the standard methods are insufficient to attack the problem of quantization.In certain situations the difficulties can be overcome by means of K\"ahler quantization on stratified K\"ahler spaces. Such a space is a stratified symplectic space together with a complex analytic structure which is compatible with the stratified symplectic structure; in particular each stratum is a K\"ahler manifold in an obvious fashion. Holomorphic quantization on a stratified K\"ahler space then yields a costratified Hilbert space, a quantum object having the classical singularities as its shadow. Given a K\"ahler manifold with a hamiltonian action of a compact Lie group that also preserves the complex structure, reduction after quantization coincides with quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond but the invariant unreduced and reduced quantum observables as wellComment: Lecture Notes, International School on Geometry and Quantization, University of Luxembourg, August 31 - September 5, 2009, 37 pages, 16 figure

Topics: Mathematics - Symplectic Geometry, Mathematical Physics, 53D50 32Q15 53D17 53D20 81S10 81T25
Year: 2011
OAI identifier: oai:arXiv.org:1103.1584
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