Séance Séminaire

Séminaire Gaston Darboux

Localization in different dimensions 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, also known as Joyce manifolds.