International Journal of Modern Physics A

This paper gives an overview of recently developed model for the QCD analytic invariant charge. Its underlying idea is to bring the analyticity condition, which follows from the general principles of local Quantum Field Theory, in perturbative approach to renormalization group (RG) method. The concrete realization of the latter consists in explicit imposition of analyticity requirement on the perturbative expansion of β function for the strong running coupling, with subsequent solution of the corresponding RG equation. In turn, this allows one to avoid the known difficulties originated in perturbative approximation of the RG functions. Ultimately, the proposed approach results in qualitatively new properties of the QCD invariant charge. The latter enables one to describe a wide range of the strong interaction processes both of perturbative and intrinsically nonperturbative nature.

The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor exp[-βV^{eff cl}(x₀)], where V^{eff cl}(x₀) is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average . We show how to generalize this concept to paths q^μ(τ) in curved space with metric g_μν(q), and calculate perturbatively the high-temperature expansion of V^{eff cl}(q₀). The requirement of independence under coordinate transformations q^μ(τ)→ q′^μ(τ) introduces subtleties in the definition and treatment of the path average , and covariance is achieved only with the help of a suitable Faddeev–Popov procedure.

We present a universal normal algebra suitable for constructing and classifying Calabi–Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a "dual" construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi–Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi–Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan–Lie algebras. This Universal Calabi–Yau algebra is a powerful tool for deciphering the Calabi–Yau genome in all dimensions.

We propose a method in the context of BFFT approach that leads to truncation of the infinite series regarded as constraints in the extended phase space, as well as other physical quantities (such as Hamiltonian). This has been done for cases where the matrix of Poisson brackets among the constraints is symplectic or constant. The method is applied to the Proca model, single self dual chiral bosons and chiral Schwinger models as examples.

We study quark confinement in a system of two parallel domain walls interpolating different color dielectric media. We use the phenomenological approach in which the confinement of quarks appears considering the QCD vacuum as a color dielectric medium. We explore this phenomenon in QCD₂, where the confinement of the color flux between the domain walls manifests, in a scenario where two 0-branes (representing external quark and antiquark) are connected by a QCD string. We obtain solutions of the equations of motion via first-order differential equations. We find a new color confining potential that increases monotonically with the distance between the domain walls.

We discuss the possibility to absorb all anomalies in the supersymmetry algebra of the N=(1,1) Wess–Zumino model in d=1+1 by a local counterterm. This counterterm corresponds to the change of the vacuum parameter in the model and the transition to an unconventional but admissible renormalization scheme. It does not modify the physical consequences such as BPS saturation, and thus the situation is rather different from gauge theory where local counterterms are required to absorb spurious gauge anomalies.