Prym varieties and integrable systems

Abstract

A new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of GLn defined on certain infinite-dimensional Grassmannians is studied. It is proved that every finite-dimensional orbit of the action of traceless elements of these commutative Lie algebras is isomorphic to the Prym variety associated with a morphism of algebraic curves. Conversely, it is shown that every Prym variety can be realized as a finite-dimensional orbit of the action of traceless diagonal elements of the formal loop algebra, which defines the multicomponent KP system.

Equations of Hurwitz schemes in the infinite Grassmannian

“…Similar tasks have been carried out in the literature (see [2,13,14,17]) where the set of points of Gr(V ) defined by certain geometric data has been characterized.…”

“…Our approach to the Hurwitz functor is closely related to those given in [2,6,13]. Let us review their approaches in a very concise way.…”

“…This has been studied in works authored by M. R. Adams and M. J. Bergvelt ([2]), M. J. Bergvelt and A. P. E. ten Kroode ( [6]), and Y. Li and M. Mulase ( [13,14]). The soliton hierarchies appearing naturally in this problem are given by the flows defined by the Heisenberg algebras in sl(n, C) (the affine Kac-Moody algebra associated with the loop group of Sl(n, C)).…”

Quantum Curves

“…It consists of subspaces V of H s = H ⊗ C s (of direct sum of s copies of H) such that the natural projection of V to H s + = H + ⊗ C s is an isomorphism. It is easy to generalize Theorem 2.1 and Theorem 2.3 to this case; see [14], Th.6.2, [11], Prop 2.1.…”

“…However, here m ∈ Z, all operators are pseudodifference operators, D is replaced by Λ. Notice, that formula (14) can be considered also as an action of γ on the space of monic difference operators. To solve the equation KL = qLK where L is a monic difference operator we consider the flag V n = SH n where S is defined by the formula (12).…”