Hitchin Systems at Low Genera

The paper gives a quick account of the simplest cases of the Hitchin integrable systems and of the Knizhnik-Zamolodchikov-Bernard connection at genus 0, 1 and 2. In particular, we construct the action-angle variables of the genus 2 Hitchin system with group SL2 by exploiting its relation to the classical Neumann integrable systems. 1 Hitchin systems As was realized by Hitchin in [16], a large family of integrable systems may be obtained by a symplectic reduction of a chiral 2-dimensional gauge theory. Let Σ denote a closed Riemann surface of genus g and let G be a complex Lie group which we shall assume simple, connected and simply connected. We shall denote by A the space of Lie(G)-valued 0,1-gauge fields1 A = Az dz on Σ. Hitchin’s construction [16] associates to Σ and G an integrable system obtained by a symplectic reduction of the infinite-dimensional complex symplectic manifold T ∗A of pairs (A,Φ) where Φ = Φz dz is a Lie(G)-valued 1,0-Higgs field. The holomorphic symplectic form on T ∗A is ∫