Spectral curves, opers and integrable systems.

摘要

We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural isomorphism between a moduli space of spectral data and a moduli space of differential data, each equipped with an infinite collection of commuting flows. The spectral data are principal G-bundles on an algebraic curve, equipped with an abelian reduction near one point. The flows come from the action of a Cartan subgroup of the loop group. The differential data are flat connections known as opers, and the flows on them form a generalized Drinfeld-Sokolov hierarchy. Thus, we obtain a geometric description of the entire phase space of the hierarchy. The above isomorphism extends the Krichever construction of special algebro-geometric solutions of the nth KdV hierarchy corresponding to G = SL_n.; An interesting feature is the appearance of formal spectral curves, replacing the projective spectral curves of the classical approach. To each such curve corresponds a homogeneous space of the loop group and a soliton system. Moreover the flows of the system have interpretations in terms of Jacobians of formal curves. The geometry of these (usually singular) curves reflects the fine structure of loop groups, in particular the detailed classification of their Cartan subgroups, which we establish up to conjugacy by the positive half of the loop group. We also apply this classification to the study of fixed point sets on affine Grassmannians.