01.01.06 (Mathematical logic, algebra, and number theory)
gauge theories; symmetry breaking; analytic loops; alternative and Maltcev algebras.
The pure Yang–Mills theory defined in the four-dimensional Euclidean space has a rich and interesting structure even at the classical level. The discovery of regular solutions to the Yang–Mills field equations, which correspond to absolute minimum of the action, has led to an intensive study of such a classical theory. One hopes that a deep understanding of the classical theory will be invaluable when one tries to quantize such a theory. In the past few years, increased attention has been paid to gauge field equations in space-time of dimension greater than four, with a view to obtaining physically interesting theories via dimensional reduction. A telling illustration of this is the geometrical Higgs mechanism. At the same time, an analog of (anti-)self-dual Yang–Mills equations in 8D has been obtained. Later there were found its several solutions which were then used to construct string and membrane solitons. In recent works, the 8D equations have been applied to construct a topological Yang–Mills theory on Joyce manifolds as an 8D counterpart of the Donaldson-Witten theory. It is also recently discussed that self-dual Yang–Mills gauge fields depending only upon time play a role in the context of M-theory. From the viewpoint of mathematical physics, the above works has made most conspicuous the possibly central role played by octonions and their attending Lie groups. The algebra of octonions (Cayley numbers) is the most known example of nonassociative alternative algebras. The alternative algebras are closely associated with the Malcev algebras and analytic Moufang loops. These algebraic structures and their application in physics of gauge fields are a subject of our researches.
Graduated from Faculty of Physics of Ivanovo State University in 1985 (department of theoretic physics). Ph.D. thesis on a speciality "Mathematical logic, algebra and number theory" was defended in 1994.
Loginov E. K. Analytic loops and gauge fields // Nucl. Phys., 2001, B606, 636–646 (hep-th/0109209).
Loginov E. K. On linear representations of Moufang loops // Commun. Algebra, 1993, 21, 2527–2536.