Compatible Poisson brackets for Lax equations and classical ζ -matrices

We construct a family of compatible linear Poisson brackets for Lax equations and equations of zero curvature with a spectral parameter.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 176–188, 1987.     

Finite-mode regularization in supersymmetric theories and anomalies

Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 80, No. 2, pp. 313–320, August, 1989.     

Self-dual Yang-Mills equations in split signature

We study the self-dual Yang-Mills equations in split signature. We give a special solution, called the basic split instanton, and describe the ADHM construction in the split signature. Moreover a split version of t'Hooft ansatz is described.     

On integrable systems and supersymmetric gauge theories

We discuss the properties ofN=2 supersymmetric gauge theories underlying the Seiberg-Witten hypothesis. We consider the main points of the theory that describes the finite-gap solutions to integrable equations in terms of complex curves and generating differentials. We clarify the invariant meaning of these definitions. This formalism is applied to the exact nonperturbative solutions found recently in theN=2 supersymmetric non-Abelian gauge theories. In the known cases, we compare this formalism with the results that can be obtained using standard quantum field-theory methods. The paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 3–46, July, 1997.     

Higher covariant derivative regularization for calculations in supersymmetric theories

A variant of the higher covariant derivative regularization is used for calculation of a two-loop β-function for the general renormalizable N = 1 supersymmetric theory. It is shown that the β-function is given by integrals of total derivatives. Partially this can be explained by substituting solutions of Slavnov-Taylor identities into the Schwinger-Dyson equations.     

Gauge-invariant discrete models of Yang-Mills equations

Two discrete models of Yang-Mill equations are constructed in the space ℝⁿ for some matrix-valued Lie group. A gauge-invariant discrete model is examined. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 742–754, May, 1997. Translated by O. V. Sipacheva     

Solutions of Yang-Mills and Yang-Mills-Higgs equations

V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 89, No. 3, pp. 402–412, December, 1991.     

Global solutions of Yang-Mills field equations

Survey of recent results on the global solutions of hyperbolic Yang-Mills equations.     

Low-dimensional Yang-Mills theories: Matrix models and emergent geometry

In a simple example of a bosonic three-matrix model, we show how a background geometry can condense as the temperature or coupling constant passes through a critical value. We show that this example belongs to a new universality class of phase transitions where the background geometry is itself emergent.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.