# Two-Valued Groups, Kummer Varieties, and Integrable Billiards

## 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.

## Keywords

2-valued groups Kummer varieties Hyperelliptic Jacobians Integrable billiards Semi-stable bundles## Mathematics Subject Classification

20N20 14H40 14H70## Notes

### Acknowledgements

The authors would like to thank the reviewers for their helpful remarks. The research of one of the authors (V. D.) was partially supported by the Serbian Ministry of Education, Science, and Technological Development, Project 174020 *Geometry and Topology of Manifolds, Classical Mechanics, and Integrable Dynamical Systems*.

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