Simone Gutt

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

Abstract We give a geometric interpretation of Berezins symbolic calculus on Kahler manifolds in the framework of geometric quantization. Berezins covariant symbols are defined in terms of coherent states and we study a function ϴ on the manifold which is the central object of the theory. When this function is constant Berezins quantization rule coincides with the prescription of geometric quantization for the quantizable functions. It is defined on a larger class of functions. We show in the compact homogeneous case how to extend Berezins procedure to a dense subspace of the algebra of smooth functions.

An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a *-product on the cotangent bundle T*G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a *-product on IR2n.

Quantization of Kähler manifolds II

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.

Equivalence of star products

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 Fedosovs 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.

Quantization of Kähler manifolds. III

We use Berezins dequantization procedure to define a formal *-product on the algebra of smooth functions on the unit disk in ℂ. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.

Local cohomology of the algebra of C∞ functions on a connected manifold

A multilinear version of Peetres theorem on local operators is the key to prove the equality between the local and differentiable Hochschild cohomology on the one hand, and on the other hand the equality between the second and third local Chevalley cohomology groups and their differentiable counterpart.

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenkos sufficient condition for a moment map for a Fedosov star product is also necessary.

Quantization of Khler manifolds. IV

We use Berezins dequantization procedure to define a formal *-product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.

Regular * representations of lie algebras

We prove the existence of a * product on the cotangent bundle of a parallelizable manifold M. When M is a Lie group the properties of this * product allow us to define a linear representation of the Lie algebra of this group on L2(G), which is, in fact, the one corresponding to the usual regular representation of G.

On tangential star products for the coadjoint Poisson structure

We derive necessary conditions on a Lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the Lie algebra for the coadjoint Poisson structure which is both differential and tangential to all the coadjoint orbits. In particular we show that when the Lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its Lie algebra.