The Krichever map,vector bundles over algebraic curves,and Heisenberg algebras

We study the GrassmannianGr_xⁿ consisting of equivalence classes of rankn algebraic vector bundles over a Riemann surfaceX with an holomorphic trivialization at a fixed pointp. Commutative subalgebras ofgl(n, H),H being the ring of functions holomorphic on a punctured disc aboutp, define flows on the Grassmannian, giving rise to classes of solutions to multi-component KP hierarchies. These commutative subalgebras correspond to Heisenberg algebras in the Kac-Moody algebra associated togl(n, H₎. One can obtain, by the Krichever map, points ofGr_xⁿ (and solutions of mcKP) from coveringsf: YX and other geometric data. Conversely for every point ofGr_xⁿ and for every choice of Heisenberg algebra we construct, using the cotangent bundle ofGr_xⁿ, an algebraic curve coveringX and other data, thus inverting the Krichever map. We show the explicit relation between the choice of Heisenberg algebra and the geometry of the covering space.The research was partially supported by US Army grant DAA L03-87-K-0110 and NSF grant DMS 9106938