A new approach to N = 2 supersymmetry based on the concept of harmonic superspace is proposed and is used to give an unconstrained superfield geometric description of N = 2 super Yang-Mills and supergravity theories as well as of matter N = 2 hypermultiplets. The harmonic N = 2 superspace has as independent coordinates, in addition to the usual ones, the isospinor harmonics u : on the sphere SU(2)/U(l). The role of u : is to relate the SU(2) group realised on the component fields to a U ( l ) group acting on the relevant superfields. Their introduction makes it possible to SU(2)-covariantise the notion of Grassmann analyticity. Crucial for our construction is the existence of an analytic subspace of the general harmonic N = 2 superspace. The hypermultiplet superfields and the true prepotentials (pre-prepotentials) of N = 2 super Yang-Mills and supergravity are unconstrained superfunctions over this analytic subspace. The pre-prepotentials have a clear geometric interpretation as gauge connections with respect to the internal SU(2)/U( 1) directions. A radically new feature arises: the number of gauge and auxiliary degrees of freedom becomes infinite while the number of physical degrees of freedom remains finite. Other new results are the massive N = 2 Yang-Mills theory and various off-shell selfinteractions of hypermultiplets. The propagators for matter and Yang-Mills superfields are given.

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In the paper we study the Yang–Mills effective action in the four-dimensional spacetime by using background field formalism. We give an explicit way of cutoff regularization procedure, then do a two-loop renormalization and calculate a second β-function coefficient. We also show that the two-loop singularity contains only logarithmic part in the first degree. At the same time additional properties of a Green function regular part are obtained.

517.9HSlder estimates for weak solutions of doubly nonlinear parabolic equations of the type of fast diffusion with coefficients satisfying only natural growth conditions and the monotonicity requirement are obtained. Bibliography: 17 titles.