Traces for star products on symplectic manifolds (original) (raw)
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Progress of Theoretical Physics Supplement, 2002
In this paper, I describe about a direct elementary proof of the existence of an essentially unique trace for an arbitrary star product on a symplectic manifold and about the construction of traces for some star products on the dual of a Lie algebra. The first topic is a joint work with J. Rawnsley and will appear in [J. Geom. Phys.] the second topic is part of a joint project with P. Bieliavsky, M. Bordemann and S. Waldmann [math. QA/0202126]. §1. Traces for star products on symplectic manifolds Let * be a star product (which we always assume here to be defined by bidifferential operators) on a Poisson manifold (M, P). In the algebra of smooth functions on M , consider the ideal C ∞ 0 (M) of compactly supported functions. A trace is a C[[ν]]-linear map τ : C ∞ 0 (M)[[ν]] → C[ν −1 , ν]] satisfying τ (u * v) = τ (v * u). The question of existence and uniqueness of such traces has been solved in the symplectic framework by the following result.
Traces for star products on the dual of a Lie algebra
2003
In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.
A Riemann-Roch-Hirzebruch formula for traces of differential operators
Annales scientifiques de l'École normale supérieure, 2008
Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology HH 2n (Dn, D * n) of the algebra of differential operators on a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann-Roch-Hirzebruch formula. Résumé. Soit D un opérateur différentiel holomorphe opérant sur les sections d'un fibré vectoriel holomorphe sur une variété complexe de dimension n. Nous démontrons une formule, conjecturée par Feigin et Shoikhet, donnant le nombre de Lefschetz de D comme intégrale d'une forme différentielle sur la variété. La classe de cette forme différentielle est obtenue, via la géométrie différentielle formelle, du générateur canonique de la cohomologie de Hochschild HH 2n (Dn, D * n) de l'algèbre des opérateurs différentiels sur un entourage formel d'un point. Si D est l'identité, la formule se réduità la formule de Riemann-Roch-Hirzebruch. Contents 1. Introduction 1 2. Hochschild homology of the algebra of differential operators 3 3. The third trace 10 4. The first trace is proportional to the third.. . 11 5.. .. and so is the second 14 6. Asymptotic topological quantum mechanics 15 Appendix A. Triangulations and signs 20 Appendix B. Heat kernel estimates and asymptotic expansion 20 References 30
Some naturally defined star products for Kähler manifolds by
2012
We give for the Kähler manifold case an overview of the constructions of some naturally defined star products. In particular, the Berezin-Toeplitz, Berezin, Geometric Quantization, Bordemann-Waldmann, and Karbegov standard star product are introduced. With the exception of the Geometric Quantization case they are of separation of variables type. The classifying Karabegov forms and the Deligne-Fedosov classes are given. Besides the Bordemann-Waldmann star product they are all equivalent.
On representations of star product algebras over cotangent spaces on Hermitian line bundles
Journal of Functional Analysis, 2003
For every formal power series B = B 0 + λB 1 + O(λ 2 ) of closed two-forms on a manifold Q and every value of an ordering parameter κ ∈ [0, 1] we construct a concrete star product ⋆ B κ on the cotangent bundle π : T * Q → Q. The star product ⋆ B κ is associated to the formal symplectic form on T * Q given by the sum of the canonical symplectic form ω and the pull-back of B to T * Q. Deligne's characteristic class of ⋆ B κ is calculated and shown to coincide with the formal de Rham cohomology class of π * B divided by iλ. Therefore, every star product on T * Q corresponding to the Poisson bracket induced by the symplectic form ω + π * B 0 is equivalent to some ⋆ B κ . It turns out that every ⋆ B κ is strongly closed. In this paper we also construct and classify explicitly formal representations of the deformed algebra as well as operator representations given by a certain global symbol calculus for pseudodifferential operators on Q. Moreover, we show that the latter operator representations induce the formal representations by a certain Taylor expansion. We thereby obtain a compact formula for the WKB expansion. *
A trace formula for symmetric spaces
Duke Mathematical Journal, 1993
Now there is a semisimple group G' and a maximal torus A' with Weyl group W' and root system R', and an isomorphism of A' onto A which takes W' to W and R' to R,. The usual Chevalley restriction theorem implies that the set of closed G'-conjugacy classes in G' can be identified with the spectrum of F[A'] w' that is, with the orbits of W' in A'. This gives the required bijection. We note that the isomorphic algebras F[A] w" and F[A'] w' are polynomial algebras, but the group G' need not be simply connected. This is the general principle of comparison that guides the particular example to be considered below. Of course, over a global or a local field, it must be modified suitably. Assuming for now its validity over a number field F, it is natural to postulate the existence of a trace formula identity of the form K(h, h2) dh dh2 ft K'(O', 9') do'. '(F) \G'(F,) Here K and K' are the cuspidal kernels, or rather the "discrete parts" of the respective trace formulas. They are associated to functions f and f' on G and G', respectively; the above equality is supposed to be true if f and f' have "matching orbital integrals". The only representations which contribute to the left-hand side are those which are distinguished with respect to H, that is, contain a vector such that the integral is nonzero. The equality then should characterize distinguished representations as functorial images from representations of G'. The purpose of this paper is to explore this idea in a simple case. However, one discovers quickly that such a simple scheme does not work. A more correct formula might take the form or K(h, h2)O(hx) dh dh2 ; K'(9', 9') do' K(h, h2) dhx dh2 f K'(0', O')0'(0') do', where 0 and 0' are suitable automorphic forms on H and G' respectively, which serve as "weights" in the formula.
Classical and Quantum Gravity, 1997
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.
Classification of traces and hypertraces on spaces of classical pseudodifferential operators
Journal of Noncommutative Geometry, 2013
Let M be a closed manifold and let CL • (M) be the algebra of classical pseudodifferential operators. The aim of this note is to classify trace functionals on the subspaces CL a (M) ⊂ CL • (M) of operators of order a. CL a (M) is a CL 0 (M)-module for any real a; it is an algebra only if a is a non-positive integer. Therefore, it turns out to be useful to introduce the notions of pretrace and hypertrace. Our main result gives a complete classification of pre-and hypertraces on CL a (M) for any a ∈ R, as well as the traces on CL a (M) for a ∈ Z, a ≤ 0. We also extend these results to classical pseudodifferential operators acting on sections of a vector bundle. As a byproduct we give a new proof of the well-known uniqueness results for the Guillemin-Wodzicki residue trace and for the Kontsevich-Vishik canonical trace. The novelty of our approach lies in the calculation of the cohomology groups of homogeneous and log-polyhomogeneous differential forms on a symplectic cone. This allows to give an extremely simple proof of a generalization of a Theorem of Guillemin about the representation of homogeneous functions as sums of Poisson brackets. Contents