Self-duality of the SL_2 Hitchin integrable system at genus two

Abstract

We revisit the Hitchin integrable system whose phase space is the bundle cotangent to the moduli space $N$ of holomorphic $SL_2$-bundles over a smooth complex curve of genus two. $N$ may be identified with the 3-dimensional projective space of theta functions of the second order, We prove that the Hitchin system on $T^*N$ possesses a remarkable symmetry: it is invariant under the interchange of positions and momenta. This property allows to complete the work of van Geemen-Previato which, basing on the classical results on geometry of the Kummer quartic surfaces, specified the explicit form of the Hamiltonians of the Hitchin system. The resulting integrable system resembles the classic Neumann systems which are also self-dual. Its quantization produces a commuting family of differential operators of the second order acting on homogeneous polynomials in four complex variables. As recently shown by van Geemen-de Jong, these operators realize the Knizhnik-Zamolodchikov-Bernard-Hitchin connection for group SU(2) and genus 2 curves.