Two-valued groups, Kummer varieties and integrable billiards

TL;DR

This paper explores algebraic two-valued group structures on Kummer varieties of genus 2 hyperelliptic Jacobians, linking them to integrable billiard systems and introducing new algebraic and geometric frameworks.

Contribution

It introduces a novel approach to two-valued groups on Kummer varieties using sigma-functions, geometric addition laws, and moduli of semi-stable bundles, expanding understanding of integrable systems.

Findings

Established connection between two-valued groups and integrable billiards.

Developed a new notion of n-groupoid as a multivalued algebraic structure.

Realized two-valued structures within the moduli space of semi-stable bundles.

Abstract

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of $\sigma$-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of $n$-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were "virtual", while in the present case the resulting bundles are effectively realizable.