
The minimal supersymmetric gauge theories can exist only in 3,4, 6 and 10 dimensions. By compactifying 2 dimensions we get topological field theories of Witten type (Cohomological Field Theories, CohFT). The dimensions of their spacetime is 1,2,4 and 8, which is just dimensions of the only existing division algebras: R, C, Q and O. In this talk I show that this is not a coincidence and there is a certain relation between division algebras and corresponding field theory. The Qcase corresponds to N=2, d=4 super YangMills, and is well studied both by physicists (SeibergWitten prepotential theory) and mathematicians (Donaldson invariant theory). The same objects are expected in 8 and 2 dimensions. Namely, in 8 dimensions (Ocase) it is possible to propose some invariants of Spin(7) holonomy 8dimensional manifolds (8d Joyce manifolds), and in 2 dimensions (Ccase) it is possible to compute nonperturbative corrections to the twisted superpotential. The interest of 2d case is also based on the observation that the corresponding TQFT describes the dynamics of vortices, which are beleived to be used in description of rational quantum Hall effect. 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 nonperturbative 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.
