Matrix Model and Stationary Problem in Toda Chain

We analyze the stationary problem for the Toda chain, and show that arising geometric data exactly correspond to the multi-support solutions of one-matrix model with a polynomial potential. For the first nontrivial examples the Hamiltonians and symplectic forms are calculated explicitly, and the consistency checks are performed. The corresponding quantum problem is formulated and some its properties and perspectives are discussed.Comment: 11 pages, LaTeX; Based on talks at "Classical and quantum integrable systems", Dubna, January 2005 and "Selected topics of modern mathematical physics", St.Petersburg, June 2005, and a lecture for the minicourse: "Toda lattices: basics and perspectives", Fields Institute, Toronto, April 200

Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed.Comment: 41 pages, 5 figures, LaTeX; some revision of exposition, misprints corrected, the version to appear in Commun. Math. Phy

Period Integrals, Quantum Numbers and Confinement in SUSY QCD

We present a direct computation of the period integrals on degenerate Seiberg-Witten curves for supersymmetric QCD, and show how these periods determine the changes in the quantum numbers of the states, when passing from the weak to the strong-coupling domains in the mass moduli space of the theory. The confinement of monopoles at strong coupling is discussed, and we demonstrate that the ambiguities in choosing the way in the moduli space do not influence to the physical conclusions on confinement of monopoles in the phase with the condensed light dyons.Comment: 16 pages, contribution to special volume on Integrable Systems in Quantum Theor

On Microscopic Origin of Integrability in Seiberg-Witten Theory

We discuss microscopic origin of integrability in Seiberg-Witten theory, following mostly the results of hep-th/0612019, as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbation in the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformization of the spectral curve.Comment: 24 pages, 1 figure, LaTeX, based on the talks at 'Geometry and Integrability in Mathematical Physics', Moscow, May 2006; 'Quarks-2006', Repino, May 2006; Twente conference on Lie groups, December 2006 and 'Classical and Quantum Integrable Models', Dubna, January 200

Matrix Models, Complex Geometry and Integrable Systems. II

We consider certain examples of applications of the general methods, based on geometry and integrability of matrix models, described in hep-th/0601212. In particular, the nonlinear differential equations, satisfied by quasiclassical tau-functions are investigated. We also discuss a similar quasiclassical geometric picture, arising in the context of multidimensional supersymmetric gauge theories and the AdS/CFT correspondence.Comment: 44 pages, 10 figures, based on several lecture courses and the talks at "Complex geometry and string theory" and the Polivanov memorial seminar; misprints corrected, references adde

Extended Seiberg-Witten Theory and Integrable Hierarchy

The prepotential of the effective N=2 super-Yang-Mills theory perturbed in the ultraviolet by the descendents of the single-trace chiral operators is shown to be a particular tau-function of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory with 2N-2 fundamental hypermultiplets at the appropriate locus of the moduli space of vacua) or a theory on a single fractional D3 brane at the ADE singularity the hierarchy is the dispersionless Toda chain. We present its explicit solutions. Our results generalize the limit shape analysis of Logan-Schepp and Vershik-Kerov, support the prior work hep-th/0302191 which established the equivalence of these N=2 theories with the topological A string on CP^1 and clarify the origin of the Eguchi-Yang matrix integral. In the higher rank case we find an appropriate variant of the quasiclassical tau-function, show how the Seiberg-Witten curve is deformed by Toda flows, and fix the contact term ambiguity.Comment: 49 page

On some algebraic examples of Frobenius manifolds

We construct some explicit quasihomogeneous algebraic solutions to the associativity (WDVV) equations by using analytical methods of the finite gap integration theory. These solutions are expanded in the uniform way to non-semisimple Frobenius manifolds.Comment: 14 page

Estimating the First-year Corrosion Losses of Structural Metals for Continental Regions of the World

The knowledge of the first-year corrosion losses of metals (K1) in various regions of the world is of great importance in engineering applications. The K1 values are used to determine the categories of atmospheric corrosivity, and K1 is also the main parameter in models for the prediction of long-term corrosion losses of metals. In the absence of experimental values of K1, their values can be predicted on the basis of meteorological and aerochemical parameters of the atmosphere using the dose-response functions (DRF). Currently, the DRFs presented in ISO 9223:2012(E) /1/ standard are used for predicting K1 in any region of the world, along with the unified DRFs /2/ and the new DRFs /3/. The predicted values of corrosion losses (K1pr) of carbon steel, zinc, copper and aluminum obtained by various DRFs for various continental regions of the world are presented. In this work we used the atmosphere corrosivity parameters and experimental data on the corrosion losses of metals for the first year of exposure (K1exp) for the locations of the tests performed under the international UN/ECE program, the MICAT project, and the Russian program. For the first time, a comparative assessment of the reliability of various DRFs is given by comparing the values of K1pr and K1ex using graphical and statistical methods. The statistical indicators of reliability of predicting the corrosion losses of metals are calculated for various categories of atmosphere corrosivity. It is shown that the new dose-response functions offer the highest reliability for all categories of atmosphere corrosivity

Matrix Models, Complex Geometry and Integrable Systems. I

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of the quasiclassical integrable systems, solved by construction of tau-functions or prepotentials. The complex curves and tau-functions of one- and two- matrix models are discussed in detail.Comment: 52 pages, 19 figures, based on several lecture courses and the talks at "Complex geometry and string theory" and the Polivanov memorial seminar; misprints corrected, references adde

Pseudoclassical description of scalar particle in non-Abelian background and path-integral representations

Path-integral representations for a scalar particle propagator in non-Abelian external backgrounds are derived. To this aim, we generalize the procedure proposed by Gitman and Schvartsman 1993 of path-integral construction to any representation of SU(N) given in terms of antisymmetric generators. And for arbitrary representations of SU(N), we present an alternative construction by means of fermionic coherent states. From the path-integral representations we derive pseudoclassical actions for a scalar particle placed in non-Abelian backgrounds. These actions are classically analyzed and then quantized to prove their consistency