arXiv subjects
Publications and source records attributed to S. Waldmann.
This paper deals with two aspects of the theory of characteristic classes of star products: first, on an arbitrary Poisson manifold, we describe Morita equivalent star products in terms of their Kontsevich classes; second, on symplectic manifolds, we describe the relationship between Kontsevich's and Fedosov's characteristic classes of star products.
For a coisotropic (or first-class) submanifold C of a Poisson manifold X we consider star-products for which the vanishing ideal I of C becomes a left ideal in the deformed algebra thus defining a left module structure on the space of smooth functions on C. We show how this can be deduced from a formality conjecture a la Tamarkin generalized to cochains compatible with C. To this end we first prove a theorem a la Hochschild-Kostant-Rosenberg between the space of compatible multivector fields and compatible multidifferential operators. We then equip the latter with a G-infinity structure, and prove that the obstructions to the formality are controlled by certain cohomology groups which we reduce in the case of C being a subvectorspace of the vectorspace M. In codimension 1 we conjecture -encouraged by low-dimensional examples and Gloessner's representation theorem (1998)- that formality holds. For higher codimensions it is not impossible that obstructions occur which in the symplectic case are linked to the Atiyah-Molino class of a regular foliation.
In this contribution we review the formal GNS construction developped in a previous preprint (q-alg/9607019), and formulate the usual WKB-expansion in flat 2n-dimensional phase space in terms of a GNS construction with a positive linear functional with support on a projectable Lagrangean submanifold defined as a graph of an exact one form dS. The main trick is a suitable form of the star-exponential of S.
In this letter we compute some elementary properties of the Fedosov star product of Weyl type, such as symmetry and order of differentiation. Moreover, we define the notion of a star product of Wick type on every Kähler manifold by a straight forward generalization of the corresponding star product in $\mathbb C^n$: the corresponding sequence of bidifferential operators differentiates its first argument in holomorphic directions and its second argument in antiholomorphic directions. By a Fedosov type procedure we give an existence proof of such star products for any Kähler manifold.
Based on a closed formula for a star product of Wick type on $\CP^n$, which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the uniformly dense subspace of representative functions with respect to the canonical action of the unitary group) that consists of {\em converging} power series in the formal parameter, thereby giving an elementary algebraic proof of a convergence result already obtained by Cahen, Gutt, and Rawnsley. In this subalgebra the formal parameter can be substituted by a real number $α$: the resulting associative algebras are infinite-dimensional except for the case $α=1/K$, $K$ a positive integer, where they turn out to be isomorphic to the finite-dimensional algebra of linear operators in the $K$th energy eigenspace of an isotropic harmonic oscillator with $n+1$ degrees of freedom. Other examples like the $2n$-torus and the Poincaré disk are discussed.
We derive a closed formula for a star-product on complex projective space and on the domain $SU(n+1)/S(U(1)\times U(n))$ using a completely elementary construction: Starting from the standard star-product of Wick type on $C^{n+1} \setminus \{ 0 \}$ and performing a quantum analogue of Marsden-Weinstein reduction, we can give an easy algebraic description of this star-product. Moreover, going over to a modified star-product on $C^{n+1} \setminus \{ 0 \}$, obtained by an equivalence transformation, this description can be even further simplified, allowing the explicit computation of a closed formula for the star-product on $\CP^n$ which can easily transferred to the domain $SU(n+1)/S(U(1)\times U(n))$.