Algebraic Index Theorem for Families

Nest, Ryszard; Tsygan, Boris

doi:10.1006/aima.1995.1037

Cited by 118 publications

(132 citation statements)

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“…The existence of a differential star product on any symplectic manifold was first proven in 1983 by De Wilde and Lecomte [8] whilst the fact that equivalence classes of differential star products are parametrized by series of elements in the second de Rham cohomology space of M appeared first in Nest and Tsygan [20], in Bertelson, Cahen and Gutt [3,4] and in Deligne [6]. In the first two cited papers, the correspondence relies on Fedosov's [10] geometrical construction of a star product; Fedosov takes a symplectic connection, extends it as a connection in the Weyl bundle whose curvature lies in the centre and builds from this a star product whose equivalence class is determined by the cohomology class of this central curvature.…”

Section: Introductionmentioning

confidence: 99%

Equivalence of star products on a symplectic manifold; an introduction to Deligne's Čech cohomology classes

Gutt

Rawnsley

1999

Journal of Geometry and Physics

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These notes grew out of the Quantisation Seminar 1997-98 on Deligne's paper [6] and the lecture of the first author in the Workshop on Quantization and Momentum Maps at the University of Warwick in December 1997.We recall the definitions of the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and show the properties of and relations between these classes by elementary methods based onČech cohomology.

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Section: Introductionmentioning

confidence: 99%

Equivalence of star products on a symplectic manifold; an introduction to Deligne's Čech cohomology classes

Gutt

Rawnsley

1999

Journal of Geometry and Physics

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“…This question was finally settled in the affirmative by Kontsevich [154] on the basis of his "formality conjecture". Yet another approach to formal deformation quantization on a symplectic manifold can be found in Karasev and Maslov [145]; star products with some additional properties (admitting a formal trace) are discussed in Connes, Flato and Sternheimer [64] and Flato and Sternheimer [92], and classification results are also available [34], [71], [183].…”

Section: Deformation Quantizationmentioning

confidence: 99%

Quantization Methods: A Guide for Physicists and Analysts

2005

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This survey is an overview of some of the better known quantization techniques (for systems with finite numbers of degrees-of-freedom) including in particular canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. We have attempted to give an account sufficiently in depth to convey the general picture, as well as to indicate the mutual relationships between various methods, their relative successes and shortcomings, mentioning also open problems in the area. Finally, even for approaches for which lack of space or expertise prevented us from treating them to the extent they would deserve, we have tried to provide ample references to the existing literature on the subject. In all cases, we have made an effort to keep the discussion accessible both to physicists and to mathematicians, including non-specialists in the field.

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“…The classification up to equivalence was first obtained for the symplectic case by Nest and Tsygan [109,110], Bertelson, Cahen and Gutt [15], Deligne [50] (see also [81,112]), and Weinstein and Xu [139]. Here the equivalence classes are shown to be in canonical bijection with formal series in the second (complex) de Rham cohomology: One has a characteristic class 4) where the origin of the above affine space is chosen by convention and two symplectic star products are equivalent if and only if their characteristic classes coincide.…”

Section: Deformation Quantizationmentioning

confidence: 99%

“…The case where the classical phase space is a general Poisson manifold turned out to be much more difficult and was finally solved by Kontsevich [98], see also [100,101] by proving his formality conjecture [99]. The classification of star products was obtained again first for the symplectic case by Nest and Tsygan [109,110], Bertelson, Cahen and Gutt [15], Deligne [50], Weinstein and Xu [139]. The classification in the Poisson case follows also from the formality theorem of Kontsevich [98].…”

Section: Introductionmentioning

confidence: 99%