International Journal of Geometric Methods in Modern Physics

In this paper, a theory of characteristic classes for super vector bundles over is studied. Some properties of even and odd Chern classes, constructed in this theory, are established. At last, the relevance of the theory to supergeometry is discussed.

We study quantum Hall effect within the framework of a newly proposed approach, which captures the principal results of some proposals. This can be established by considering a system of particles living on the noncommutative plane in the presence of an electromagnetic field and quantum statistical mechanically investigate its basic features. Solving the eigenvalue equation, we analytically derive the energy levels and the corresponding wavefunctions. These will be used, at low temperature and weak electric field, to determine the thermodynamical potential Ω^nc and related physical quantities. Varying Ω^nc with respect to the noncommutativity parameter θ, we define a new function that can be interpreted as a Ω^nc density. Evaluating the particle number, we show that the Hall conductivity of the system is θ-dependent. This allows us to make contact with the quantum Hall effect by offering different interpretations. We study the high temperature regime and discuss the magnetism of the system. We finally show that at , the system is sharing some common features with the Laughlin theory.

An optimal control problem for a drift-free controllable system on the Lie group SO(4) is discussed and some of its dynamical and geometrical properties are pointed out.

This paper studies the cosmological equations for a scalar field φ in the framework of a quantum gravity modified Einstein–Hilbert Lagrangian where G and Λ are dynamical variables. It is possible to show that there exists a Noether symmetry for the point Lagrangian describing this scheme in a FRW universe. Our main result is that the Noether Symmetry Approach fixes both Λ = Λ(G) and the potential V = V(φ) of the scalar field. The method does not lead, however, to easily solvable equations, by virtue of the higher dimensionality of the reduced configuration space involved, the additional variable being the running Newton coupling.

In this paper the explicit flow representation to the Bose–Hubbard Hamiltonian is given in the general case. This representation may be useful in creating cat states for the system of atoms trapped in the optical ring.

It is shown that an exact chiral symmetry can be described for Dirac–Kähler fermions using the two complexes of the geometric discretization. This principle is extended to describe exact flavor projection and it is shown that this necessitates the introduction of a new operator and two new structures of complex. To describe simultaneous chiral and flavor projection, eight complexes are needed in all and it is shown that projection leaves a single flavor of chiral field on each.

It was shown in [1], only for scalar conformal fields, that the Moyal–Weyl star product can introduce the quantum effect as the phase factor to the ordinary product.

In this paper we show that, even on the same complex structure, the Moyal–Weyl star product of two j-differentials (conformal fields of weights (j, 0)) does not vanish but it generates the quantum effect at the first order of its perturbative series.

More generally, we get the explicit expression of the Moyal–Weyl star product of j-differentials defined on any complex structure of a bi-dimensional Riemann surface Σ. We show that the star product of two j-differentials is not a j-differential and does not preserve the conformal covariance character.

This can shed some light on the Moyal–Weyl deformation quantization procedure connection's with the deformation of complex structures on a Riemann surface. Hence, the situation might relate the star products to the Moduli and Teichmüller spaces of Riemann surfaces.

From the perspective of topological field theory we explore the physics beyond instantons. We propose the fluctons as nonperturbative topological fluctuations of vacuum, from which the self-dual domain of instantons is attained as a particular case. Invoking the Atiyah–Singer index theorem, we determine the dimension of the corresponding flucton moduli space, which gives the number of degrees of freedom of the fluctons. An important consequence of these results is that the topological phases of vacuum in non-Abelian gauge theories are not necessarily associated with self-dual fields, but only with smooth fields. Fluctons in different scenarios are considered, the basic aspects of the quantum mechanical amplitude for fluctons are discussed. A possible application of fluctons in the N = 4 Topologically Twisted Supersymmetric Yang–Mills Theory is explored and the case of gravity is discussed briefly.

The global variational functional, defined by the Hilbert–Yang–Mills Lagrangian over a smooth manifold, is investigated within the framework of prolongation theory of principal fiber bundles, and global variational theory on fibered manifolds. The principal Lepage equivalent of this Lagrangian is constructed, and the corresponding infinitesimal first variation formula is obtained. It is shown, in particular, that the Noether currents, associated with isomorphisms of the underlying geometric structures, split naturally into several terms, one of which is the exterior derivative of the Komar–Yang–Mills superpotential. Consequences of invariance of the Hilbert–Yang–Mills Lagrangian under isomorphisms of underlying geometric structures such as Noether's conservation laws for global currents are then established. As an example, a general formula for the Komar–Yang–Mills superpotential of the Reissner–Nordström solution of the Einstein equations is found.

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.

The notions of uniformity and homogeneity of elastic materials are reviewed in terms of Lie groupoids and frame bundles. This framework is also extended to consider the case of Functionally Graded Media, which allows us to obtain some homogeneity conditions.

We review some basic theorems on integrability of Hamiltonian systems, namely the Liouville–Arnold theorem on complete integrability, the Nekhoroshev theorem on partial integrability and the Mishchenko–Fomenko theorem on noncommutative integrability, and for each of them we give a version suitable for the noncompact case. We give a possible global version of the previous local results, under certain topological hypotheses on the base space. It turns out that locally affine structures arise naturally in this setting.