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

We study the Grassmannian Gr_xⁿ consisting of equivalence classes of rank n algebraic vector bundles over a Riemann surface X with an holomorphic trivialization at a fixed point p. Commutative subalgebras of gl(n, H_λ), H_λ being the ring of functions holomorphic on a punctured disc about p, 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 to gl(n, H_λ). One can obtain, by the Krichever map, points of Gr_xⁿ (and solutions of mcKP) from coverings f: Y→X and other geometric data. Conversely for every point of Gr_xⁿ and for every choice of Heisenberg algebra we construct, using the cotangent bundle of Gr_xⁿ, an algebraic curve covering X 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.