Sergey Shadchin
"Topological Quantum Field Theories and division algebras".


We consider the dimensional reduction of the minimal
supersymmetric Yang-Mills models, which exist in 4, 6 and 10
dimensions. The  result is topological field theories of Witten type
(cohomological field theories) in 2, 4 and 8 dimensions respectively.
Mathematically the supersymmetry algbra of reduced theory can be
thought of  as a realization of an equivariant differential. This
observation together with a certain deformation of the theory allows
to compute the non-perturbative corrections as equivariant Euler
characteristics. Technically this procedure appears as an equivariant
and infinedimensional version of the Gauss-Bonnet-Hopf theorem. In
four dimensions it leads to the Nekrasov approach to N=2 super
Yang-Mills theory. The goal of the talk is to show in some details how
does it work in other cases (2 and 8 dimensional) as well as to show
the connection between the structure of the non-perturbative expansion
of 2, 4 and 8 dimensional theories and complex number, quaternions and
octonions. Also some applications are dicussed: in two dimensions this
approach may shed some light to the vortex dynamics and in eight
dimensins it opens a possibility to construct some invariants of
Spin(7)-holonomy manifolds.