Multidimensional integrable vacuum cosmology with two curvatures

The vacuum cosmological model on the manifold $R \times M_1 \times \ldots \times M_n$ describing the evolution of $n$ Einstein spaces of non-zero curvatures is considered. For $n = 2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_1 =$dim $M_1, N_2 =$ dim$M_2) = (6,3), (5,5), (8,2)$. The Kasner-like behaviour of the solutions near the singularity $t_s \to +0$ is considered ($t_s$ is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary $n$. For $n=2$ these solutions are attractors for other ones, when $t_s \to + \infty$. For dim $M = 10, 11$ and $3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n=2$ ones using "curvature-splitting" trick. In the case $n=2$, $(N_1, N_2)= (6,3)$ a family of non-singular solutions with the topology $R^7 \times M_2$ is found.Comment: 21 pages, LaTex. 5 figures are available upon request (hard copy). Submitted to Classical and Quantum Gravit

Multi-Dimensional Sigma-Functions

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by K.Weierstrass and F.Klein was aimed to extend to higher genera Weierstrass theory of elliptic functions based on the Weierstrass $\sigma$-functions. Our development was motivated by the recent achievements of mathematical physics and theory of integrable systems that were based of the results of classical theory of multi-dimensional theta functions. Both theta and sigma-functions are integer and quasi-periodic functions, but worth to remark the fundamental difference between them. While theta-function are defined in the terms of the Riemann period matrix, the sigma-function can be constructed by coefficients of polynomial defining the curve. Note that the relation between periods and coefficients of polynomials defining the curve is transcendental. Since the publication of our 1997-review a lot of new results in this area appeared (see below the list of Recent References), that promoted us to submit this draft to ArXiv without waiting publication a well-prepared book. We complemented the review by the list of articles that were published after 1997 year to develop the theory of $\sigma$-functions presented here. Although the main body of this review is devoted to hyperelliptic functions the method can be extended to an arbitrary algebraic curve and new material that we added in the cases when the opposite is not stated does not suppose hyperellipticity of the curve considered.Comment: 267 pages, 4 figure

QCD coupling below 1 GeV from quarkonium spectrum

In this paper we extend the work synthetically presented in Ref.[1] and give theoretical details and complete tables of numerical results. We exploit calculations within a Bethe-Salpeter (BS) formalism adjusted for QCD, in order to extract an ``experimental'' strong coupling \alpha_s^{exp}(Q^2) below 1 GeV by comparison with the meson spectrum. The BS potential follows from a proper ansatz on the Wilson loop to encode confinement and is the sum of a one-gluon-exchange and a confinement terms. Besides, the common perturbative strong coupling is replaced by the ghost-free expression \alpha_E(Q^2) according to the prescription of Analytic Perturbation Theory (APT). The agreement of \alpha_s^{exp}(Q^2) with the APT coupling \alpha_E(Q^2) turns out to be reasonable from 1 GeV down to the 200 MeV scale, thus confirming quantitatively the validity of the APT prescription. Below this scale, the experimental points could give a hint on the vanishing of \alpha_s(Q^2) as Q approaches zero. This infrared behaviour would be consistent with some lattice results and a ``massive'' generalization of the APT approach. As a main result, we claim that the combined BS-APT theoretical scheme provides us with a rather satisfactory correlated understanding of very high and rather low energy phenomena from few hundreds MeV to few hundreds GeV.Comment: Preliminary revision. Typos corrected, comments and references adde