DBundles and Integrable Hierarchies
Abstract
We study the geometry of Dbundleslocally projective Dmoduleson algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent KadomtsevPetviashvili (KP) and spin CalogeroMoser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of Dbundles; in particular, we prove that the local structure of Dbundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP are therefore captured by Dbundles on cubic curves E, that is, irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1. We develop a FourierMukai transform describing Dmodules on cubic curves E in terms of (complexes of) sheaves on a twisted cotangent bundle over E. We then apply this transform to classify Dbundles on cubic curves, identifying their moduli spaces with phase spaces of general CM particle systems (realized through the geometry of spectral curves in our twisted cotangent bundle). Moreover, it is immediate from the geometric construction that the flows of the KP and CM hierarchies are thereby identified and that the poles of the KP solutions are identified with the positions of the CM particles. This provides a geometric explanation of a muchexplored, puzzling phenomenon of the theory of integrable systems: the poles of meromorphic solutions to KP soliton equations move according to CM particle systems.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2006
 DOI:
 10.48550/arXiv.math/0603720
 arXiv:
 arXiv:math/0603720
 Bibcode:
 2006math......3720B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Quantum Algebra;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems