Fedosov Construction Research Articles

Remarks on generalized Fedosov algebras

The variant of Fedosov construction based on fairly general fiberwise product in the Weyl bundle is studied. We analyze generalized star products of functions, of sections of endomorphisms bundle, and those generating deformed bimodule structure as introduced previously by Waldmann. Isomorphisms of generalized Fedosov algebras are considered and their relevance for deriving Seiberg–Witten map is described. The existence of the trace functional is established. Explicit expressions for star products and the trace functional are provided up to the second power of deformation parameter. The example of symmetric part of noncommutativity tensor is discussed as a case with possible field-theoretic application.

Index Theory and Adiabatic Limit in QFT

The paper has the form of a proposal concerned with the relationship between the three mathematically rigorous approaches to quantum field theory: (1) local algebraic formulation of Haag, (2) Wightman formulation and (3) the perturbative formulation based on the microlocal renormalization method. In this project we investigate the relationship between (1) and (3) and utilize the known relationships between (1) and (2). The main goal of the proposal lies in obtaining obstructions for the existence of the adiabatic limit (confinement problem in the phenomenological standard model approach). We extend the method of deformation of Dutsch and Fredenhagen (in the Bordeman-Waldmann sense) and apply Fedosov construction of the formal index—an analog of the index for deformed symplectic manifolds, generalizing the Atiyah-Singer index. We present some first steps in realization of the proposal.

The eigenvalue equation for a 1-D Hamilton function in deformation quantization

The eigenvalue equation has been found for a Hamilton function in a form independent of the choice of a potential. This Letter proposes a modified Fedosov construction on a flat symplectic manifold. Necessary and sufficient conditions for solutions of an eigenvalue equation to be Wigner functions of pure states are presented. The 1-D harmonic oscillator eigenvalue equation in the coordinates time and energy is solved. A perturbation theory based on the variables time and energy is elaborated.

SEIBERG–WITTEN EQUATIONS FROM FEDOSOV DEFORMATION QUANTIZATION OF ENDOMORPHISM BUNDLE

It is shown how Seiberg–Witten equations can be obtained by means of Fedosov deformation quantization of endomorphism bundle and the corresponding theory of equivalences of star products. In such setting, Seiberg–Witten map can be iteratively computed for arbitrary gauge group up to any given degree with recursive methods of Fedosov construction. Presented approach can be also considered as a generalization of Seiberg–Witten equations to Fedosov type of noncommutativity.

Deformation Quantization of Poisson Structures Associated to Lie Algebroids

In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E , where E ! M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concreteness of the construction allows to obtain explicit formulas even for a wide class of derivations and self- equivalences of the products. Moreover, we can show that some of our products are in direct relation to the universal enveloping algebra associated to the Lie algebroid. Finally, we show that for a certain class of star products on E the integration with respect to a density with vanishing modular vector field defines a trace functional.

Deformation quantization in singular spaces

We present a method of quantizing analytic spaces X immersed in an arbitrary smooth ambient manifold M. Remarkably our approach can be applied to singular spaces. We begin by quantizing the cotangent bundle of the manifold M. Using a supermanifold framework we modify the Fedosov construction in a way such that the ⋆-product of the functions lifted from the base manifold turns out to be the usual commutative product of smooth functions on M. This condition allows us to lift the ideals associated to the analytic spaces on the base manifold to form left (or right) ideals on (OΩ1M[[ℏ]],⋆ℏ) in a way independent of the choice of generators and leading to a finite set of PDEs defining the functions in the quantum algebra associated with X. Some examples are included.

Universality of Fedosov's construction for star products of Wick type on pseudo-Kähler manifolds

In this paper we construct star products on a pseudo-Kähler manifold ( M, ω, I) using a modification of the Fedosov method based on a different fibrewise product similar to the Wick product on C n . Having fixed the used connection to be the pseudo-Kähler connection these star products shall depend on certain data given by a formal series of closed two-forms on M and a certain formal series of symmetric contravariant tensor fields on M. In a first step we show that this construction is rich enough to obtain star products of every equivalence class by computing Deligne's characteristic class of these products. Among these products we uniquely characterize the ones which have the additional property to be of Wick type which means that the bidifferential operators describing the star products only differentiate with respect to holomorphic directions in the first argument and with respect to anti-holomorphic directions in the second argument. These star products are in fact strongly related relvtar products with separation of variables introduced and studied by Karabegov. This characterization gives rise to special conditions on the data that enter the Fedosov procedure. Moreover, we compare our results that are based on an obviously coordinate independent construction to those of Karabegov that were obtained by local considerations and give an independent proof of the fact that star products of Wick type are in bijection to formal series of closed two-forms of type (1, 1) on M. Using this result we finally succeed in showing that the given Fedosov construction is universal in the sense that it yields all star products of Wick type on a pseudo-Kähler manifold. Due to this result we can make some interesting observations concerning these star products; we can show that all these star products are of Vey type and in addition we can uniquely characterize the ones that have the complex conjugation incorporated as an anti-automorphism.

Quantization on a two-dimensional phase space with a constant curvature tensor

Some properties of the star product of the Weyl type (i.e., associated with the Weyl ordering) are proved. Fedosov construction of the *-product on a two-dimensional phase space with a constant curvature tensor is presented. Eigenvalue equations for momentum p and position q on a two-dimensional phase space with constant curvature tensors are solved.

Morita equivalence bimodules for Wick type star products

In this paper, the notion of star products with separation of variables on a Kähler manifold is extended to bimodule deformations of (anti-) holomorphic vector bundles over a Kähler manifold. Here the Fedosov construction is appropriately adapted using the geometric data of a connection in the vector bundle. Moreover, the relation between the star products of Wick and anti-Wick type is clarified by constructing a canonical Morita equivalence bimodule as bimodule deformation of the canonical line bundle over the Kähler manifold.

Noncommutative supergeometry, duality and deformations

Following Lett. Math. Phys. 50 (1999) 309, we introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying { Q, Q}=0). We develop the theory of connections on modules over Q-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case SO(d,d, Z ) -duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that Q-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar Q-algebras can be constructed also in the case when the deformation parameter is not formal.

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 $\mathsf E_\alpha = \nu \partial_\nu + \Lie{\xi_\alpha} + \mathsf D_\alpha$ , where the ξα are local, conformally symplectic vector fields and the $\mathsf D_\alpha$ 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 $\frac{1}{\nu}[\omega]+\frac{1}{\nu} [\Omega]$ , where $\Omega \in \nu Z^2_{{\rm dR}}(M)[[\nu]]$ 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 Moshe Flato

Morita Equivalence of Fedosov Star Products and Deformed Hermitian Vector Bundles

Based on the usual Fedosov construction of star products for a symplectic manifold M, we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over M starting with a symplectic connection on M and a connection for E. In the case of a line bundle, this gives a Morita equivalence bimodule, and the relation between the characteristic classes of the Morita equivalent star products can be found very easily within this framework. Moreover, we also discuss the case of a Hermitian vector bundle and give a Fedosov construction of the deformation of the Hermitian fiber metric.

Noncommutative gauge theories from deformation quantization

We construct noncommutative gauge theories based on the notion of the Weyl bundle, which appears in Fedosov's construction of deformation quantization on an arbitrary symplectic manifold. These correspond to D-brane worldvolume theories in non-constant B-field and curved backgrounds in string theory. All such theories are embedded into a “universal” gauge theory of the Weyl bundle. This shows that the combination of a background field and a noncommutative field strength has universal meaning as a field strength of the Weyl bundle. We also show that the gauge equivalence relation is a part of such a “universal” gauge symmetry.

Deformation Quantization of Symplectic Fibrations

A symplectic fibration is a fibre bundle in the symplectic category (a bundle of symplectic fibres over a symplectic base with a symplectic structure group). We find the relation between the deformation quantization of the base and the fibre, and that of the total space. We consider Fedosov's construction of deformation quantization. We generalize the Fedosov construction to the quantization with values in a bundle of algebras. We find that the characteristic class of deformation of a symplectic fibration is the weak coupling form of Guillemin, Lerman, and Sternberg. We also prove that the classical moment map could be quantized if there exists an equivariant connection.

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 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.