I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

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Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

Deformation quantization of Poisson manifolds, I

I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII. Further Applications.- VIII. Product Preserving Functors.- IX. Bundle Functors on Manifolds.- X. Prolongation of Vector Fields and Connections.- XI. General Theory of Lie Derivatives.- XII. Gauge Natural Bundles and Operators.- References.- List of symbols.- Author index.

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Natural operations in differential geometry

All purely bosonic supersymmetric solutions of minimal supergravity in five dimensions are classified. The solutions preserve either one half or all of the supersymmetry. Explicit examples of new solutions are given, including a large family of plane-fronted waves and a maximally supersymmetric analogue of the Godel universe which lifts to a solution of 11-dimensional supergravity that preserves 20 supersymmetries.

All supersymmetric solutions of minimal supergravity in five- dimensions

Introduction to lie algebras and representation theory

Quantization of Kähler manifolds I: geometric interpretation of Berezin's quantization

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Lorentzian symmetric spaces

We give an elementary proof of the fact that equivalence classes of smooth or differentiable star products on a symplectic manifold M are parametrized by sequences of elements in the second de Rham cohomology space of the manifold. The parametrization is given explicitly in terms of Fedosov's construction which yields a star product when one chooses a symplectic connection and a sequence of closed 2-forms on M. We also show how derivations of a given star product, modulo inner derivations, are parametrized by sequences of elements in the first de Rham cohomology space of M.

https://iopscience.iop.org/article/10.1088/0264-9381/14/1A/008/pdf

Equivalence of star products

We use Berezin’s dequantization procedure to define a formal *- product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous Kahler manifold M. We prove that this formal *-product isconvergent when M isa hermitian symmetric space. © 1993 American Mathematical Society.

https://www.ams.org/tran/1993-337-01/S0002-9947-1993-1179394-9/S0002-9947-1993-1179394-9.pdf

Michel Cahen

Papers

Quantization of Kähler manifolds I: geometric interpretation of Berezin's quantization

Lorentzian symmetric spaces

Equivalence of star products

Quantization of Kähler manifolds II

Lorentz manifolds modelled on a Lorentz symmetric space