# Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models

## TL;DR

This paper introduces new integrable systems derived from multiplicative quiver varieties, generalizing the Ruijsenaars-Schneider model, with explicit Poisson-commuting functions and connections to gauge theory and algebraic structures.

## Contribution

It constructs and explicitly describes new integrable systems from multiplicative quiver varieties for cyclic quivers, extending the classical Ruijsenaars-Schneider model.

## Key findings

- Explicit Poisson-commuting functions in Darboux coordinates
- Generalization of Ruijsenaars-Schneider system for m>1
- Connections to supersymmetric gauge theory and algebraic structures

## Abstract

We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for $m>1$ we obtain new integrable systems that generalise the Ruijsenaars-Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs, as well as in the context of the Macdonald theory.