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

M. R. Adams, M. J. Bergvelt

Research output: Contribution to journalArticlepeer-review


We study the Grassmannian Grxn 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, Hλ), Hλ 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, Hλ). One can obtain, by the Krichever map, points of Grxn (and solutions of mcKP) from coverings f: Y→X and other geometric data. Conversely for every point of Grxn and for every choice of Heisenberg algebra we construct, using the cotangent bundle of Grxn, 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 languageEnglish (US)
Pages (from-to)265-305
Number of pages41
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Jun 1993
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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