Commuting families in skew fields and quantization of Beauville's fibration

TL;DR

This paper constructs commuting families in algebraic structures and their quantizations, linking classical Poisson systems with quantum differential operators, especially in the context of K3 surfaces and algebraic curves.

Contribution

It introduces a novel method to construct and quantize commuting families in algebraic and geometric settings, connecting classical and quantum integrable systems.

Findings

Construction of commuting families in symmetric powers of algebras

Classical limits yield Poisson commuting families related to linear systems

Quantization produces commuting differential operators on symmetric powers of curves

Abstract

We construct commuting families in fraction fields of symmetric powers of algebras. The classical limit of this construction gives Poisson commuting families associated with linear systems. In the case of a K3 surface S, they correspond to lagrangian fibrations introduced by Beauville. When S is the canonical cone of an algebraic curve C, we construct commuting families of differential operators on symmetric powers of C, quantizing the Beauville systems.