International Journal of Geometric Methods in Modern Physics

In this paper, we analyze from the Poisson dynamics and geometry point of view the Ishii dynamical system.

The main purpose of this paper is to describe the stochastic fractional differential equations associated to Euler top system. The numerical integration for the stochastic fractional model associated to Euler top system is presented.

We introduce a new covariant differential ^A∇ adapted to an almost paracontact structure. We prove its relation with a Weyl structure and give some necessary and sufficient conditions for its harmonicity. We also characterize the P-Sasakian condition in terms of ^A∇.

We prove the generalized Hyers–Ulam stability of mapping on normed spaces for the following 2-dimensional quadratic functional equation:

It is shown that if the contribution of flat connections on the dimension of the moduli spaces of Yang–Mills instantons and anti-instantons is appropriately taken into the account, then the inadmissible cases of negative dimensions may be reduced to zero-dimensional moduli spaces, corresponding to a collection of points, and whose counting will correspond to the Donaldson invariant of the base manifold. These results will lead to a possible description of that invariant in terms of flat connections with diverse applications, for example for testing the conjecture on its equivalence to the Seiberg–Witten invariant, and for the study of the qualitative and quantitative aspects of the gauge/gravity duality.

A Kenmotsu 3-manifold M admitting a Ricci soliton (g, w) with a transversal potential vector field w (orthogonal to the Reeb vector field) is of constant sectional curvature -1. A cosymplectic 3-manifold admitting a Ricci soliton with the Reeb potential vector field or a transversal vector field is of constant sectional curvature 0.

It is shown that, given a system of 2n first-order (or of n second-order) ODEs, there exists an infinite number of symplectic structures and Hamiltonian functions such that the corresponding Hamilton equations are locally equivalent to the given system of equations, without restrictions analogous to the Helmholtz conditions.

We consider systems of local variational problems defining nonvanishing cohomology classes. Symmetry properties of the Euler–Lagrange expressions play a fundamental role since they introduce a cohomology class which adds up to Noether currents; they are related with invariance properties of the first variation, thus with the vanishing of a second variational derivative. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order that such a current be global.

We find the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi domain as the sum of the Kähler two-form on ℂ and the one on the Siegel ball . The classical motion and quantum evolution on determined by a linear Hamiltonian in the generators of the Jacobi group is described by a Riccati equation on and a linear first-order differential equation in z ∈ ℂ, where H₁ denotes the three-dimensional Heisenberg group. When the transformation FC is applied, the first-order differential equation for the variable z ∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel–Jacobi space , where denotes the Siegel upper half-plane.

Conformal Killing equations and their integrability conditions for expanding hyperheavenly spaces with Λ in spinorial formalism are studied. It is shown that any conformal Killing vector reduces to homothetic or isometric Killing vector. Reduction of respective Killing equation to one master equation is presented. Classification of homothetic and isometric Killing vectors is given. Type [D] ⊗ [any] is analyzed in detail and some expanding complex metrics of types [III, N] ⊗ [III, N] with Λ admitting isometric Killing vectors are found.

Chern–Weil theory provides for each invariant polynomial on a Lie algebra 𝔤 a map from 𝔤-connections to differential cocycles whose volume holonomy is the corresponding Chern–Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and to dg-bundles and that the Chern–Simons action functional associated this way to an n-symplectic manifold is the action functional of the AKSZ σ-model whose target space is the given n-symplectic manifold (examples of this are the Poisson σ-model or the Courant σ-model, including ordinary Chern–Simons theory, or higher-dimensional Abelian Chern–Simons theory). Here we show how, within the framework of the higher Chern–Weil theory in smooth ∞-groupoids, this result can be naturally recovered and enhanced to a morphism of higher stacks, the same way as ordinary Chern–Simons theory is enhanced to a morphism from the stack of principal G-bundles with connections to the 3-stack of line 3-bundles with connections.

In this paper we present a model where the modified Landau-like levels of charged particles in a magnetic field are determined due to the modified smoothness of ℝ⁴ as underlying structure of the Minkowski spacetime. The standard smoothness of ℝ⁴ is shifted to the exotic , k = 2p, p = 1, 2, …. This is achieved by superstring theory using gravitational backreaction induced from a strong, almost constant magnetic field on standard ℝ⁴. The exact string background containing flat ℝ⁴ is replaced consistently by the curved geometry of SU(2)_k × ℝ as part of the modified exact backgrounds. This corresponds to the change of smoothness on ℝ⁴ from the standard ℝ⁴ to some exotic . The calculations of the spectra are using the CFT marginal deformations and Wess–Zumino–Witten (WZW) models. The marginal deformations capture the effects of the magnetic field as well as its gravitational backreactions. The spectra depend on even level k of WZW on SU(2). At the same time the WZ term as element of H³(SU(2), ℝ) determines also the exotic smooth . As the consequence we obtain that a nonzero mass-gap induced by the exotic emerges in the spectrum.

In this paper we investigate a variant of the classical mixmaster universe model of anisotropic cosmology, where the spatial sections are noncommutative 3-tori. We consider ways in which the discrete dynamical system describing the mixmaster dynamics can be extended to act on the noncommutative torus moduli, and how the resulting dynamics differs from the classical one, for example, in the appearance of exotic smooth structures. We discuss properties of the spectral action, focussing on how the slow-roll inflation potential determined by the spectral action affects the mixmaster dynamics. We relate the model to other recent results on spectral action computation and we identify other physical contexts in which this model may be relevant.