Research Statement
In the past few years, increased attention has been paid to gauge field equations in
spacetime of dimension greater than four, with a view to obtaining physically interesting
theories via dimensional reduction. Such equations and connected with one the
Bogomol'nyiPrasadSommerfield states appear in the supergravity, in the lowenergy string theory and Mtheory.
Using solutions of the YangMills equations makes possible to obtain soliton solutions in these theories. Therefore
the search and investigate new solutions of the higherdimensional selfdual
YangMills equations having applications in particle physics is one of my
interests.
My current research concerns also various aspects of classical solutions in 11D supergravity.
There has recently been a great deal of interest in finding de Sitter and antide Sitter vacua of supergravity and string theory.
This is motivated in part by the desire to construct possible models for latetime cosmology.
Interestingly, the gravitymatter system may be reinterpreted as a pure gravity theory with a torsion
that are given locally by the Cayley structure constants
of a geodesic loop in the affinely connected space. Thus, we may study the Bose sector of
11D supergravity using not only geometric but also algebraic methods.
Methods of the Cayley algebra also may be applied to research of supersymmetric string
solitons. It is well known that in addition to the fivebrane
solutions onebrane and twobrane solutions of heterotic string theory are existed. The
construction of these solutions involves crucially the properties of octonions. In particular,
a connection between strings and octonions that is provided by the existence of super
YangMills theories and GreenSchwarz superstring action in D=3,4,6,10 with the number of
transverse dimensions (1,2,4,8) coinciding with the dimensions of the four Hurwitz algebras,
cannot be accidental. Therefore the role of octonions and other exotic algebraic structures in
superstring theory is also interesting for me.
From the viewpoint of mathematical physics, some above works has made most conspicuous the
possibly central role played by octonions and mathematical structures connected with
one. This topic goes in and out of fashion and from time to time somebody takes it up and
pushes it a little further. The octonions have a sort of magical attraction and the feeling is
that one day their very characteristic special features will find a natural application. It is
certainly true that some of the special features involve exceptional Lie algebras,
which play a role in the particle physics because of its relation with GUT, supergravity and superstring
theory.
