"Physics" Seminar, February 22, 1999
David Ben Zvi (Harvard University),
Spectral Curves, Opers and Integrable Systems.
We describe some relations between
loop groups, soliton equations, and algebraic
geometry. Soliton equations (such as the
Korteweg-deVries hierarchy) are naturally
expressed as flows on spaces of flat connections (differential data).
On the other hand, integrable systems in algebraic
geometry are often expressed, using the theory
of spectral curves, as linear flows on Jacobians (spectral data).
We present a simple and general equivalence between
moduli of differential data and moduli of spectral data, which
provides a comprehensive geometric picture for a wide range
of systems. An interesting feature is the role played by
formal versions of spectral curves, which are also responsible
for some interesting phenomena in loop groups.