With E. Frenkel. **Spectral Curves, Opers and Integrable
Systems**.

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 on the
spectral side come from the action of a Heisenberg subgroup of the
loop group. The differential data are flat connections known as
opers. The flows on the differential side come from a generalized
Drinfeld-Sokolov hierarchy. Our isomorphism between the two sides
provides a geometric description of the entire phase space of the
Drinfeld-Sokolov hierarchy. It extends the Krichever construction of
special algebro-geometric solutions of the n-th 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. The geometry of these (usually singular)
curves reflects the fine structure of loop groups, in particular the
detailed classification of their Cartan subgroups. 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.