Deformation Quantization of Almost Kahler Models and Lagrange-Finsler Spaces

Finsler and Lagrange spaces can be equivalently represented as almost Kahler manifolds enabled with a metric compatible canonical distinguished connection structure generalizing the Levi Civita connection. The goal of this paper is to perform a natural Fedosov-type deformation quantization of such geometries. All constructions are canonically derived for regular Lagrangians and/or fundamental Finsler functions on tangent bundles.Comment: the latex 2e variant of the manuscript accepted for JMP, 11pt, 23 page

BRST quantization of quasi-symplectic manifolds and beyond

We consider a class of \textit{factorizable} Poisson brackets which includes almost all reasonable Poisson structures. A particular case of the factorizable brackets are those associated with symplectic Lie algebroids. The BRST theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case. This can be viewed as an extension of the Fedosov deformation quantization to a wide class of \textit{irregular} Poisson structures. In a more general case, the factorizable Poisson brackets are shown to be closely connected with the notion of $n$-algebroid. A simple description is suggested for the geometry underlying the factorizable Poisson brackets basing on construction of an odd Poisson algebra bundle equipped with an abelian connection. It is shown that the zero-curvature condition for this connection generates all the structure relations for the $n$-algebroid as well as a generalization of the Yang-Baxter equation for the symplectic structure.Comment: Journal version, references and comments added, style improve

Higher order relations in Fedosov supermanifolds

Higher order relations existing in normal coordinates between affine extensions of the curvature tensor and basic objects for any Fedosov supermanifolds are derived. Representation of these relations in general coordinates is discussed.Comment: 11 LaTex pages, no figure

Scalar Casimir Energies of Tetrahedra

New results for scalar Casimir self-energies arising from interior modes are presented for the three integrable tetrahedral cavities. Since the eigenmodes are all known, the energies can be directly evaluated by mode summation, with a point-splitting regulator, which amounts to evaluation of the cylinder kernel. The correct Weyl divergences, depending on the volume, surface area, and the corners, are obtained, which is strong evidence that the counting of modes is correct. Because there is no curvature, the finite part of the quantum energy may be unambiguously extracted. Dirichlet and Neumann boundary conditions are considered and systematic behavior of the energy in terms of geometric invariants is explored.Comment: Talk given at QFEXT 1

Symplectic geometries on supermanifolds

Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a non-degenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar symplectic structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of symplectic geometries on supermanifolds.Comment: LaTex, 1o pages, LaTex, changed conten

Direct Detection of Electroweak-Interacting Dark Matter

Assuming that the lightest neutral component in an SU(2)L gauge multiplet is the main ingredient of dark matter in the universe, we calculate the elastic scattering cross section of the dark matter with nucleon, which is an important quantity for the direct detection experiments. When the dark matter is a real scalar or a Majorana fermion which has only electroweak gauge interactions, the scattering with quarks and gluon are induced through one- and two-loop quantum processes, respectively, and both of them give rise to comparable contributions to the elastic scattering cross section. We evaluate all of the contributions at the leading order and find that there is an accidental cancellation among them. As a result, the spin-independent cross section is found to be O(10^-(46-48)) cm^2, which is far below the current experimental bounds.Comment: 19 pages, 7 figures, published versio

A zeta function approach to the relation between the numbers of symmetry planes and axes of a polytope

A derivation of the Ces\`aro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries. Several results are obtained that relate the coefficients, $b_i$, in the Shephard-Todd polynomial to the geometry of the fundamental domain. For the 3-sphere we show that $b_4$ is given by the ratio of the volume of the fundamental tetrahedron to its Schl\"afli reciprocal.Comment: Plain TeX, 26 pages (eqn. (86) corrected

∗\ast-SDYM fields and heavenly spaces: II. Reductions of the ∗\ast-SDYM system

Reductions of self-dual Yang-Mills (SDYM) system for $\ast$-bracket Lie algebra to the Husain-Park (HP) heavenly equation and to sl(N,{\boldmath{C}) SDYM equation are given. An example of a sequence of $su(N)$ chiral fields ($N\geq 2$) tending for $N\to\infty$ to a curved heavenly space is found.Comment: 18 page

Cohomologies of the Poisson superalgebra

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $R^{2n}$ ($C^{2n}) are investigated under suitable continuity restrictions on cochains. The first and second cohomology spaces in the trivial representation and the zeroth and first cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a constant nondegenerate Poisson superbracket for arbitrary n>0. The third cohomology space in the trivial representation and the second cohomology space in the adjoint representation of this superalgebra are found for arbitrary n>1.Comment: Comments: 40 pages, the text to appear in Theor. Math. Phys. supplemented by computation of the 3-rd trivial cohomolog

Weyl-Underhill-Emmrich quantization and the Stratonovich-Weyl quantizer

Weyl-Underhill-Emmrich (WUE) quantization and its generalization are considered. It is shown that an axiomatic definition of the Stratonovich-Weyl (SW) quantizer leads to severe difficulties. Quantization on the cylinder within the WUE formalism is discussed.Comment: 15+1 pages, no figure