Summary
This is the first exposition of the quantization theory of singular symplectic (i.e., Marsden-Weinstein) quotients and their applications to physics in book form. A preface by J. Marsden and A. Weinstein precedes individual refereed contributions by M.T. Benameur and V. Nistor, M. Braverman, A. Cattaneo and G. Felder, B. Fedosov, J. Huebschmann, N.P. Landsman, R. Lauter and V. Nistor, M. Pflaum, M. Schlichenmaier, V. Schomerus, B. Schroers, and A. Sengupta. This book is intended for mathematicians and mathematical physicists working in quantization theory, algebraic, symplectic, and Poisson geometry, the analysis and geometry of stratified spaces, pseudodifferential operators, low-dimensional topology, operator algebras, noncommutative geometry, or Lie groupoids, and for theoretical physicists interested in quantum gravity and topological quantum field theory. The subject matter provides a remarkable area of interaction between all these fields, highlighted in the example of the moduli space of flat connections, which is discussed in detail. The reader will acquire an introduction to the various techniques used in this area, as well as an overview of the latest research approaches. These involve classical differential and algebraic geometry, as well as operator algebras and noncommutative geometry. Thus one will be amply prepared to follow future developments in this fascinating and expanding field, or enter it oneself. It is to be expected that the quantization of singular spaces will become a key theme in 21st century (concommutative) geometry.
Table of Contents
Preface |
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vii | |
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Some comments on the history, theory, and applications of symplectic reduction |
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1 | (20) |
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Homology of complete symbols and non-commutative geometry |
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21 | (26) |
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Cohomology of the Mumford quotient |
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47 | (14) |
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Poisson sigma models and symplectic groupoids |
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61 | (34) |
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Pseudo-differential operators and deformation quantization |
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95 | (24) |
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Singularities and Poisson geometry of certain representation spaces |
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119 | (18) |
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Quantized reduction as a tensor product |
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137 | (44) |
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Analysis of geometric operator on open manifolds: a groupoid approach |
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181 | (50) |
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Smooth structures on stratified spaces |
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231 | (28) |
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Singular projective varities and quantization |
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259 | (24) |
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Poisson structure and quantization of Chern-Simons theory |
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283 | (24) |
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Combinatorial quantization of Euclidean gravity in three dimensions |
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307 | (22) |
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The Yang-Mills measure and symplectic structure over spaces of connections |
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329 | |
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