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

Abstract

We study the GrassmannianGr x n 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, Hλ),H λ 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, Hλ). One can obtain, by the Krichever map, points ofGr x n (and solutions of mcKP) from coveringsf: Y→X and other geometric data. Conversely for every point ofGr x n and for every choice of Heisenberg algebra we construct, using the cotangent bundle ofGr x n , 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.