Local ν-Euler Derivations and Deligne's¶Characteristic Class of Fedosov Star Products¶and Star Products of Special Type

In this paper we explicitly construct local ν-Euler derivations , where the ξ_α are local, conformally symplectic vector fields and the are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold (M,ω) by means of which we are able to compute Deligne's characteristic class of these star products. We show that this class is given by , where is a formal series of closed two-forms on M the cohomology class of which coincides with the one introduced by Fedosov to classify his star products. Moreover, we consider star products that have additional algebraic structures and compute the effect of these structures on the corresponding characteristic classes of these star products. Specifying the constituents of Fedosov's construction we obtain star products with these special properties. Finally, we investigate equivalence transformations between such special star products and prove existence of equivalence transformations being compatible with the considered algebraic structures. Dedicated to the memory of Moshé Flato Received: 28 June 1999 / Accepted: 11 April 2002?Published online: 11 September 2002     

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?^* _ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?^* _ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables. Received: 28 April 2000 / Accepted: 6 December 2000     

On the Inverse Mapping of the Formal Symplectic Groupoid of a Deformation Quantization

To each natural star product on a Poisson manifold M we associate an antisymplectic involutive automorphism of the formal neighborhood of the zero section of the cotangent bundle of M. If M is symplectic, this mapping is shown to be the inverse mapping of the formal symplectic groupoid of the star product. The construction of the inverse mapping involves modular automorphisms of the star product.     

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 Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.     

The Characteristic Classes of Morita Equivalent Star Products on Symplectic Manifolds

In this paper we give a complete characterization of Morita equivalent star products on symplectic manifolds in terms of their characteristic classes: two star products ⋆ and ⋆' on (M,ω) are Morita equivalent if and only if there exists a symplectomorphism ψ\colon M M such that the relative class t(⋆, ψ^⋆(⋆')) is 2 π i-integral. For star products on cotangent bundles, we show that this integrality condition is related to Dirac's quantization condition for magnetic charges. Received: 19 July 2001 / Accepted: 23 January 2002     

Some remarks on -invariant Fedosov star products and quantum momentum mappings

In these notes we consider a slightly generalized Fedosov star product * on a symplectic manifold (M,ω), emanating from the fibrewise Weyl product and the triple (,Ω,s) consisting of a symplectic torsion free connection on M, a formal series ΩνZ²_dR(M)[[ν]] of closed two-forms on M, and a certain formal series s of symmetric contravariant tensor fields on M. We prove necessary and sufficient conditions for certain classical symmetries to become symmetries of the star product, only sufficient conditions having been published in special cases when this letter was written (note, however, the different proofs in [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1]). For a given symplectic vector field X on M, it is well known that (= is a sufficient condition for the Lie derivative to be a derivation of *. We prove that these conditions are in fact necessary ones, also providing a very simple proof for their being sufficient. Moreover, we prove a criterion that has first been presented by Gutt [S. Gutt, Star products and group actions, Contribution to the Bayrischzell Workshop, April 26–29, 2002] (see also [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1] for a different proof) and which specifies a necessary and sufficient condition for to be a quasi-inner derivation. The statement that this condition is a sufficient one dates back to Kravchenko [O. Kravchenko, Compos. Math. 123 (2000) 131]. Applying our results, we find necessary and sufficient criteria for a Fedosov star product to be -invariant and to admit a quantum Hamiltonian. Finally, supposing the existence of a quantum Hamiltonian, we present a cohomological condition on Ω that is equivalent to the existence of a quantum momentum mapping. In particular, our results show that the existence of a classical momentum mapping in general does not imply the existence of a quantum momentum mapping and thus give a negative answer to Xu’s question posed in [P. Xu, Commun. Math. Phys. 197 (1998) 167].     

Getzler symbol calculus and deformation quantization

In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler’s pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.     

A Formal Model of Berezin-Toeplitz Quantization

We give a new construction of symbols of the differential operators on the sections of a quantum line bundle L over a Kähler manifold M using the natural contravariant connection on L. These symbols are the functions on the tangent bundle TM polynomial on fibres. For high tensor powers of L, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on TM corresponding to some pseudo-Kähler structure on TM. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over M. We extend the star product on TM to generalized functions supported on the zero section of TM. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M.     

Fedosov’s formal symplectic groupoids and contravariant connections

Using Fedosov’s approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kähler–Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.     

Closed star products and cyclic cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C (M) and ( (E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C ^*-algebras. We also discuss the semi-classical geometry arising from these deformations.     

Hermitian Star Products are Completely Positive Deformations

Let M be a Poisson manifold equipped with a Hermitian star product. We show that any positive linear functional on C^∞(M) can be deformed into a positive linear functional with respect to the star product.     

Fedosov Star Products and One-Differentiable Deformations

We show that every star product on a symplectic manifold defines uniquely a 1-differentiable deformation of the Poisson bracket. Explicit formulas are given. As a corollary we can identify the characteristic class of any star product as a part of its explicit (Fedosov) expression.     

Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization

In this paper, we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the ^*-representation theory of such ^*-algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C^*-algebras. Throughout this paper, the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C^*-algebras, we show that the GNS construction of ^*-representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two ^*-algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate ^*-representations of the two ^*-algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over ^*-algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally, we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules.     

Formal Connections for Families of Star Products

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection, which was introduced by the first author. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled on the formal symplectic vector fields. Moreover, we showthat if the parameter space has trivial first cohomology group, any two flat formal connections are related by an automorphism of the family of star products.     

Conformal Orbifold Theories and Braided Crossed G-Categories

The Berezin-Toeplitz deformation quantization of an abelian variety is explicitly computed by the use of Theta-functions. An SL(2n,)-equivariant complex structure dependent equivalence E between the constant Moyal-Weyl product and this family of deformations is given. This equivalence is seen to be convergent on the dense subspace spanned by the pure phase functions. The Toeplitz operators associated to the equivalence E applied to a pure phase function produces a covariant constant section of the endomorphism bundle of the vector bundle of Theta-functions (for each level) over the moduli space of abelian varieties.Applying this to any holonomy function on the symplectic torus one obtains as the moduli space of U(1)-connections on a surface, we provide an explicit geometric construction of the abelian TQFT-operator associated to a simple closed curve on the surface. Using these TQFT-operators we prove an analog of asymptotic faithfulness (see [A1]) in this abelian case. Namely that the intersection of the kernels for the quantum representations is the Toreilli subgroup in this abelian case.Furthermore, we relate this construction to the deformation quantization of the moduli spaces of flat connections constructed in [AMR1] and [AMR2]. In particular we prove that this topologically defined *-product in this abelian case is the Moyal-Weyl product. Finally we combine all of this to give a geometric construction of the abelian TQFT operator associated to any link in the cylinder over the surface and we show the glueing axiom for these operators.This research was conducted in part for the Clay Mathematics Institute at University of California, Berkeley.This work was supported by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation