Real Forms of Holomorphic Hamiltonian Systems

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Authors: Arathoon, Philip, Fontaine, Marine
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212778
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Real Forms of Holomorphic Hamiltonian Systems. Philip Arathoon and Marine Fontaine. SIGMA 20 (2024), 114, 24 pages

By complexifying a Hamiltonian system, one obtains dynamics on a holomorphic symplectic manifold. To invert this construction, we present a theory of real forms that not only recovers the original system but also yields different real Hamiltonian systems that 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.