$\mathcal D$-bundles and integrable hierarchies

Abstract

Abstract. We study the geometry of D-bundles-locally projective D-modules-on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of D-bundles; in particular, we prove that the local structure of D-bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP are therefore captured by D-bundles on cubic curves E, that is, irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1. We develop a FourierMukai transform describing D-modules on cubic curves E in terms of (complexes of) sheaves on a twisted cotangent bundle E over E. We then apply this transform to classify D-bundles on cubic curves, identifying their moduli spaces with phase spaces of general CM particle systems (realized through the geometry of spectral curves in E ). 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 much-explored, puzzling phenomenon of the theory of integrable systems: the poles of meromorphic solutions to KP soliton equations move according to CM particle systems.

$${\mathcal{W}}$$ -Symmetry of the Adèlic Grassmannian

“…The Sato Grassmannian GR [S] parametrizes subspaces of K which are complementary to subspaces commensurable with O (see [AMP,BN2] for algebraic constructions of GR. Note that this Grassmannian, which is a scheme, is quite different from the ind-scheme Gr(K), the thin Grassmannian, parametrizing subspaces commensurable with O.…”

“…The same construction applies with GR = GR(K) replaced by the (isomorphic) Grassmannian GR n = GR(K n ), line bundles replaced by vector bundles and K = gl 1 (K) replaced by the loop algebra gl n (K). The KP flows on GR are now replaced by the multicomponent KP flows on GR n , corresponding to maximal tori in gl n (K) [KvdL] (see [Pl] and [BN2] for more on the geometry of multicomponent KP).…”

“…For related studies of (and background on) the geometry of D-bundles we refer the reader to [BN1,BN2,BN3], which originated as an attempt to understand geometrically and generalize the relations between bispectrality, the adèlic Grassmannian, ideals in the Weyl algebra and Calogero-Moser spaces first uncovered and explored in work of W2,BW1,BW2]. In [BN1], the Cannings-Holland description of D-modules on curves and their cuspidal quotients is revisited from a more algebro-geometric viewpoint (and the Morita equivalence of Smith-Stafford generalized to arbitrary dimension).…”

“…In [BN1], the Cannings-Holland description of D-modules on curves and their cuspidal quotients is revisited from a more algebro-geometric viewpoint (and the Morita equivalence of Smith-Stafford generalized to arbitrary dimension). In [BN2], two constructions of KP solutions from D-bundles are explained-one (which is most relevant to this paper) related to line bundles on cusp curves and the other related to CalogeroMoser systems. (This description of moduli spaces of D-bundles or ideals in D as Calogero-Moser spaces is established for arbitrary curves in [BN3].)…”

“…(This description of moduli spaces of D-bundles or ideals in D as Calogero-Moser spaces is established for arbitrary curves in [BN3].) As explained in [BN2], in genus zero the two constructions are evidently exchanged by the Fourier transform and thus, by [BW1], by Wilson's bispectral involution [W1]. However, in higher genus the constructions are completely independent.…”