## Abstract

We study the Grassmannian Gr_{x}^{n} 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_{x}^{n} (and solutions of mcKP) from coverings f: Y→X and other geometric data. Conversely for every point of Gr_{x}^{n} and for every choice of Heisenberg algebra we construct, using the cotangent bundle of Gr_{x}^{n}, 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.

Original language | English (US) |
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Pages (from-to) | 265-305 |

Number of pages | 41 |

Journal | Communications in Mathematical Physics |

Volume | 154 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1993 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics