KP hierarchy for the cyclic quiver

TL;DR

This paper generalizes the KP hierarchy using cyclic quivers and Cherednik algebras, linking solutions to quiver varieties and classical Calogero-Moser systems, including multi-component cases.

Contribution

It introduces a new hierarchy related to cyclic quivers, extending KP theory and connecting it to quiver varieties and integrable systems.

Findings

Hierarchy depends on m parameters, generalizing KP hierarchy.

Solutions are parameterized by quiver varieties.

Pole dynamics follow Calogero-Moser systems and their spin versions.

Abstract

We introduce a generalisation of the KP hierarchy, closely related to the cyclic quiver and the Cherednik algebra $H_k(\mathbb Z_m)$. This hierarchy depends on $m$ parameters (one of which can be eliminated), with the usual KP hierarchy corresponding to the $m=1$ case. Generalising the result of G. Wilson, we show that our hierarchy admits solutions parameterised by suitable quiver varieties. The pole dynamics for these solutions is shown to be governed by the classical Calogero-Moser system for the wreath-product $\mathbb Z_m\wr S_n$ and its new spin version. These results are further extended to the case of the multi-component hierarchy.