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Published **2001**
by Birkhäuser Basel, Imprint, Birkhäuser in Basel .

Written in English

- Mathematics

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.

**Edition Notes**

Statement | edited by N.P. Landsman, M. Pflaum, M. Schlichenmaier |

Series | Progress in Mathematics -- 198, Progress in Mathematics -- 198. |

Contributions | Pflaum, M., Schlichenmaier, M. |

Classifications | |
---|---|

LC Classifications | QA1-939 |

The Physical Object | |

Format | [electronic resource] / |

Pagination | 1 online resource (XII, 355 pages). |

Number of Pages | 355 |

ID Numbers | |

Open Library | OL27084025M |

ISBN 10 | 3034883641 |

ISBN 10 | 9783034883641 |

OCLC/WorldCa | 840290513 |

Search within book. Front Matter. Pages i-xii. PDF. Comments on the history, theory, and applications of symplectic reduction Combinatorial quantization of Euclidean gravity in three dimensions. B. J. Schroers. Pages The Yang-Mills measure and symplectic structure over spaces of connections. Ambar N. Sengupta. Pages About. This is the first exposition of the quantization theory of singular symplectic (Marsden-Weinstein) quotients and their applications to physics. These involve classical differential and algebraic geometry, as well as operator algebras and noncommutative geometry. Abstract: Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is eduevazquez.com: Eckhard Meinrenken, Reyer Sjamaar. SINGULAR REDUCTION AND QUANTIZATION 5 The symplectic quotients of Mhave a natural stratiﬁcation by symplectic orbifolds determined by the inﬁnitesimal orbit types of M. (See Section ) Deﬁnition A point µ∈g∗ is a quasi-regular value of Φ if the G-orbits in Φ−1(Gµ) all have the same eduevazquez.com: Eckhard Meinrenken, Reyer Sjamaar.

Quantization of the symplectic groupoid. We present the geometric quantization of the standard Podlè s sphere by using a multiplicative real polarization of the symplectic groupoid. We introduce the concept of multiplicative integrability of the modular function as one key point of the construction. The symplectic structure on moduli space (published in The Floer Memorial Volume, ) The modular automorphism group of a Poisson manifold (appeared in Journal of Geometry and Physics) Tangential deformation quantization and polarized symplectic groupoids (appeared in Deformation Theory and Symplectic Geometry, S. Gutt, J. Rawnsley, and D. Sternheimer, eds., Kluwer, . Jul 01, · Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is eduevazquez.com by: QUANTIZATION OF SYMPLECTIC REDUCTION MICHAEL ANSHELEVICH Abstract. Symplectic reduction, also known as Marsden-Weinstein reduction, is an important construction in Poisson geometry. Following N.P. Landsman [22], we propose a quantization of this procedure by means of M. Rieﬀel’s theory of induced representations. Here to an equivariant.

singular symplectic quotients M 0 with coeƒcients in certain complex line bundles. These bundles include the trivial line bundle, the Riemann—Roch number of which we call the arithmetic genus of M 0, and the prequantum line ÔÔbundleÕÕ, the Riemann—Roch number of which is the dimension of the quantization of M 0. On the other hand, quantization of singular spaces, see e.g. [5], [11], [12], [13], [10], [18] is an upcoming topic in Mathematical Physics, in particular in view of the interest of reduction. We then de ne the quantization of the singular quotient T G==AdGas the kernel of the (twisted) Dolbeault{Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space L2(T)W(G;T). Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The “quantization commutes with reduction” theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann–Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient Cited by:

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