9 research outputs found
An algebraic index theorem for Poisson manifolds
The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map
Formality theorems for Hochschild complexes and their applications
We give a popular introduction to formality theorems for Hochschild complexes
and their applications. We review some of the recent results and prove that the
truncated Hochschild cochain complex of a polynomial algebra is non-formal.Comment: Submitted to proceedings of Poisson 200
Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds
In this paper we investigate equivariant Morita theory for algebras with
momentum maps and compute the equivariant Picard groupoid in terms of the
Picard groupoid explicitly. We consider three types of Morita theory:
ring-theoretic equivalence, *-equivalence and strong equivalence. Then we apply
these general considerations to star product algebras over symplectic manifolds
with a Lie algebra symmetry. We obtain the full classification up to
equivariant Morita equivalence.Comment: 28 pages. Minor update, fixed typos
Wick type deformation quantization of Fedosov manifolds
A coordinate-free definition for Wick-type symbols is given for symplectic
manifolds by means of the Fedosov procedure. The main ingredient of this
approach is a bilinear symmetric form defined on the complexified tangent
bundle of the symplectic manifold and subject to some set of algebraic and
differential conditions. It is precisely the structure which describes a
deviation of the Wick-type star-product from the Weyl one in the first order in
the deformation parameter. The geometry of the symplectic manifolds equipped by
such a bilinear form is explored and a certain analogue of the
Newlander-Nirenberg theorem is presented. The 2-form is explicitly identified
which cohomological class coincides with the Fedosov class of the Wick-type
star-product. For the particular case of K\"ahler manifold this class is shown
to be proportional to the Chern class of a complex manifold. We also show that
the symbol construction admits canonical superextension, which can be thought
of as the Wick-type deformation of the exterior algebra of differential forms
on the base (even) manifold. Possible applications of the deformed superalgebra
to the noncommutative field theory and strings are discussed.Comment: 20 pages, no figure