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SIGMA, 2024 Volume 20, 114, 24 pp. (Mi sigma2116)

Real Forms of Holomorphic Hamiltonian Systems

Philip Arathoona, Marine Fontaineb

a Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
b Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

Abstract: By complexifying a Hamiltonian system, one obtains dynamics on a holomorphic symplectic manifold. To invert this construction, we present a theory of real forms which not only recovers the original system but also yields different real Hamiltonian systems which share the same complexification. This provides a notion of real forms for holomorphic Hamiltonian systems analogous to that of real forms for complex Lie algebras. Our main result is that the complexification of any analytic mechanical system on a Grassmannian admits a real form on a compact symplectic manifold. This produces a ‘unitary trick’ for Hamiltonian systems which curiously requires an essential use of hyperkähler geometry. We demonstrate this result by finding compact real forms for the simple pendulum, the spherical pendulum, and the rigid body.

Keywords: Hamiltonian dynamics, integrable systems, hyperkähler geometry.

MSC: 53D20, 14J42

Received: June 12, 2024; in final form December 10, 2024; Published online December 21, 2024

Language: English

DOI: 10.3842/SIGMA.2024.114


ArXiv: 2009.10417


© Steklov Math. Inst. of RAS, 2025