St-Petersburg State University Research Institute of Physics Faculty of Physics High Energy and Elementary Particles Physics
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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 space-time 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 Q-case corresponds to N=2, d=4 super Yang-Mills, and
is well studied both by physicists (Seiberg-Witten prepotential
theory) and mathematicians (Donaldson invariant theory). The same
objects are expected in 8 and 2 dimensions. Namely, in 8 dimensions
(O-case) it is possible to propose some invariants of Spin(7) holonomy
8-dimensional manifolds (8d Joyce manifolds), and in 2 dimensions
(C-case) it is possible to compute non-perturbative 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 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.
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March 2009