Gauge theories, Simple Groups and Integrable Systems (original) (raw)

In this review we discuss interrelations between classical Hitchin integrable systems, monodromy preserving equations and topological field theories coming from N=4 supersymmetric Yang-Mills theories developed by Gukov, Kapustin and Witten. In particular, we define the systems related to bundles with nontrivial characteristic classes and discuss relations of the characteristic classes with monopole configurations in the Yang-Mills theory. 2 Classical Integrable Systems 2.1 Integrability We consider here the integrable systems of classical mechanics [17]. In this case the notion of complete integrability can be formulated correctly, while in a field theory there are subtleties in its definition. Consider a smooth symplectic manifold R of dim(R) = 2n. It means that there exists a closed non-degenerate two-form ω, and the inverse bivector π (ω a,b π bc = δ c a), such that the space of smooth functions C ∞ (R) becomes a Poisson algebra with respect to the Poisson brackets {F, G} = dF |π|dG = ∂ a F π ab ∂ b G ≡ ∂ a F π ab ∂ b G. (2.1) In terms of the bi-vector π ab the Jacobi identity for the brackets assumes the form π ab ∂ b π cd + π cb ∂ b π da + π db ∂ b π ca = 0 .