Mathematical Research Letters

Volume 26 (2019)

Number 1

Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices

Pages: 9 – 33



Peter Crooks (Department of Mathematics, Northeastern University, Boston, Massachusetts, U.S.A.)

Steven Rayan (Department of Mathematics & Statistics, University of Saskatchewan, Saskatoon, SK, Canada)


We study holomorphic integrable systems on the hyperkähler manifold $G \times S_{\mathrm{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\mathrm{reg}}$ is the Slodowy slice determined by a regular $\mathfrak{sl}_2 (\mathbb{C})$-triple. Our main result is that this manifold carries a canonical abstract integrable system, a foliation-theoretic notion recently introduced by Fernandes, Laurent–Gengoux, and Vanhaecke. We also construct traditional integrable systems on $G \times S_{\mathrm{reg}}$, some of which are completely integrable and fundamentally based on Mishchenko and Fomenko’s argument shift approach.

Received 16 June 2017

Accepted 5 November 2017

Published 7 June 2019