Real Forms of Holomorphic Hamiltonian Systems
Philip Arathoon, Marine Fontaine
TL;DR
Real Forms of Holomorphic Hamiltonian Systems develops a framework for real forms of holomorphic Hamiltonian dynamics by combining holomorphic symplectic and Poisson geometry with hyperkähler brane techniques. It proves that the complexification of analytic mechanical systems on Grassmannians admits real forms on compact symplectic manifolds, enabling a unitary-like rotation of dynamics while preserving complex structure, and demonstrates this with compact real forms for the simple pendulum, spherical pendulum, and rigid body. The approach relies on Poisson reduction, hyperkähler reduction, and a curious diffeomorphism between coadjoint orbits and Grassmannian cotangent bundles, linking physically relevant systems to compact geometries such as $S^2$, $S^2\times S^2$, and $\mathbb{C}P^3$. It further shows how holomorphic integrability transfers to real integrability on real forms and provides a framework to transfer dynamics across real forms via complexification, suggesting broad generalizations to other coadjoint orbits and related integrable models.
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.