Weyl Bundle Research Articles

О локальном представлении синектических связностей на расслоениях Вейля

Synectic extensions of complete lifts of linear connections in tangent bundles were introduced by A. P. Shirokov in the seventies of the last century [1; 2]. He established that these connections are linear and are real realizations of linear connections on first-order tangent bundles en­do­wed with a smooth structure over the algebra of dual numbers. He also pro­ved the existence of a smooth structure on tangent bundles of arbitrary or­der on a smooth manifold M over the algebra of plu­ral numbers. Studying holomorphic linear connections on over an algebra , A. P. Shirokov obtained real realizations of these con­nec­tions, which he called Synectic extensions of a linear connection defi­ned on M. A natural generalization of the algebra of plural numbers is the A. Weyl algebra, and a generalization of the tangent bundle is the A. Weyl bundle. It was shown in [3] that a synectic extension of linear connections defined on M a smooth manifold can also be constructed on A. Weyl bundles , where is the A. Weyl algebra. The geometry of these bundles has been studied by many authors — A. Morimoto, V. V. Shu­rygin and others. A detailed analysis of these works can be found in [3]. In this paper, we study synectic lifts of linear connections defined on A. Weyl bundles.

Bargmann–Fock Sheaves on Kähler Manifolds

Fedosov used flat sections of the Weyl bundle on a symplectic manifold to construct a star product $\star$ which gives rise to a deformation quantization. By extending Fedosov's method, we give an explicit, analytic construction of a sheaf of Bargmann-Fock modules over the Weyl bundle of a K\"ahler manifold $X$ equipped with a compatible Fedosov abelian connection, and show that the sheaf of flat sections forms a module sheaf over the sheaf of deformation quantization algebras defined $(C^\infty_X[[\hbar]], \star)$. This sheaf can be viewed as the $\hbar$-expansion of $L^{\otimes k}$ as $k \to \infty$, where $L$ is a prequantum line bundle on $X$ and $\hbar = 1/k$.

The Weyl map and bundle gerbes

We introduce the notion of a general cup product bundle gerbe and use it to define the Weyl bundle gerbe on T×SU(n)∕T. The Weyl map from T×SU(n)∕T to SU(n) is then used to show that the pullback of the basic bundle gerbe on SU(n) defined by the second two authors is stably isomorphic to the Weyl bundle gerbe as SU(n)-equivariant bundle gerbes. Both bundle gerbes come equipped with connections and curvings and by considering the holonomy of these we show that these bundle gerbes are not D-stably isomorphic.

Batalin–Vilkovisky quantization and the algebraic index

Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the Batalin–Vilkovisky (BV) quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.

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.

The Fedosov ∗-product in Mathematica

The computer program ‘Fecom.nb’ implementing the Fedosov ∗-product in Darboux coordinates is presented. It has been written in Mathematica 6.0 but it can be easily modified to be run in some earlier version of Mathematica. To optimize computations elements of the Weyl algebra are treated as polynomials. Several procedures which order the terms are included. New version program summary Program title: Fecom.nb Catalogue identifier: AEBU_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBU_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8107 No. of bytes in distributed program, including test data, etc.: 42 175 Distribution format: tar.gz Programming language: Mathematica v.6.0 Computer: All Operating system: All RAM: Sufficient for installation of Mathematica Classification: 17.16 Catalogue identifier of previous version: AEBU_v1_0 Journal reference of previous version: Comput. Phys. Comm. 179 (2008) 924 Does the new version supersede the previous version?: Yes Nature of problem: Computing the ∗-product of the Weyl type in the Fedosov formalism in a Darboux chart. Solution method: Inputting the dimension of the phase space, coefficients of the symplectic connection, the range of approximation and the functions to be multiplied; computing the Abelian connection and the flat sections of the Weyl bundle representing the multiplied functions; calculating the projection of the ○-product of these flat sections on the phase space. Reasons for new version: Optimization and including the trivial cases – deformations of the 0th and the 1st order. Summary of revisions: In case we calculate the ∗-product up to the odd power h k , it is necessary to know the Abelian connection and the flat sections up to the degree 2 k − 1 . But for the even power h k , it is sufficient to know the Abelian connection and the flat sections up to the degree 2 k − 2 . In the new version of the program we use this observation. Now the running time of the program for even powers h k is shorter than in the previous version. The 0th and the 1st order of the deformation are trivial – the pointwise product of functions and the Poisson bracket of them respectively. But to make the program complete we added the possibility of calculating also these trivial situations. The new version of the program works then also for the maximal power of h equal 0 and 1. Running time: The test run, provided with the distribution, took approximately 2 minutes to run using Mathematica 7.0 on a Windows XP machine.

The Fedosov ∗-product in Mathematica

The computer program ‘Fecom.nb’ implementing the Fedosov ∗-product in Darboux coordinates is presented. It has been written in Mathematica 6.0 but it can be easily modified to be run in some earlier version of Mathematica. To optimize computations elements of the Weyl algebra are treated as polynomials. Several procedures which order the terms are included. Program summary Program title: Fecom.nb Catalogue identifier: AEBU_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBU_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4758 No. of bytes in distributed program, including test data, etc.: 40 585 Distribution format: tar.gz Programming language: Mathematica v.6.0 Computer: Any computer able to run Mathematica Operating system: Any operating system capable of running Mathematica RAM: Sufficient for the installation of Mathematica Classification: 17.16 Nature of problem: Computing the ∗-product of the Weyl type in the Fedosov formalism in a Darboux chart. Solution method: Inputting the dimension of the phase space, coefficients of the symplectic connection, the range of approximation and the functions to be multiplied; computing the Abelian connection and the flat sections of the Weyl bundle representing the multiplied functions; calculating the projection of the o- product of these flat sections on the phase space. Running time: Details of the time taken for various example runs are given in Section 4 of the manuscript. The test run provided with the distribution took approximately 2 minutes using Mathematica 5.1 on a Windows XP machine.

The Weyl bundle as a differentiable manifold

Construction of an infinite dimensional differentiable manifold ${\mathbb R}^{\infty}$ not modelled on any Banach space is proposed. Definition, metric and differential structures of a Weyl algebra and a Weyl algebra bundle are presented. Continuity of the $\circ$-product in the Tichonov topology is proved. Construction of the $*$-product of the Fedosov type in terms of theory of connection in a fibre bundle is explained.

Geometrical origin of the [formula omitted]-product in the Fedosov formalism

The construction of the ∗ -product proposed by Fedosov is implemented in terms of the theory of fibre bundles. The geometrical origin of the Weyl algebra and the Weyl bundle is shown. Several properties of the product in the Weyl algebra are proved. Symplectic and abelian connections in the Weyl algebra bundle are introduced. Relations between them and the symplectic connection on a phase space M are established. Elements of differential symplectic geometry are included. Examples of the Fedosov formalism in quantum mechanics are given.

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.

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.