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Mathematical Research Letters
Volume 26 (2019)
Number 1
Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices
Pages: 9 – 33
DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n1.a2
Authors
Abstract
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