[Paper Review] A symplectic slice theorem

Juan‐Pablo Ortega, Tudor S. Raţiu|arXiv (Cornell University)|Oct 8, 2001

Homotopy and Cohomology in Algebraic Topology9 references4 citations

TL;DR

This paper establishes a symplectic slice theorem for canonical proper Lie group actions on symplectic manifolds without requiring a global momentum map. Instead, it uses the Chu map—a canonical, always-existing 2-cocycle derived from the symplectic form—to construct a tubular model around orbits, enabling a normal form for the symplectic structure and Hamiltonian vector fields. The key contribution is a sufficient condition for local Hamiltonian behavior (tubewise Hamiltonian actions), characterized by the exactness of a g*-valued one-form derived from the Chu map.

ABSTRACT

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model it can be used the so called Chu map instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.

Motivation & Objective

• To generalize the symplectic slice theorem beyond Hamiltonian actions by replacing the momentum map with the Chu map.
• To provide a normal form for the symplectic structure near orbits of canonical proper group actions.
• To derive reconstruction equations for Hamiltonian vector fields in tubular coordinates.
• To identify sufficient conditions under which a canonical action is tubewise Hamiltonian, i.e., admits a local momentum map in G-invariant neighborhoods.

Proposed method

• Construct the symplectic normal space Vm as the quotient of the annihilator of the orbit tangent space.
• Define the Chu map Ψ: M → Z²(g) via Ψ(m)(ξ,η) = ω(m)(ξM(m), ηM(m)), which is always defined for canonical actions.
• Introduce the subalgebra k ⊂ g consisting of elements η such that ηM(m) ∈ (g·m)ω, ensuring compatibility with the symplectic structure.
• Build a local model Yr = G ×H (m*r × (Vm)r) using the isotropy group H, the symplectic normal space Vm, and the restricted momentum map.
• Derive the reconstruction equations for G-invariant Hamiltonian vector fields in terms of the tubular coordinates.
• Establish exactness of a g*-valued one-form γ as a sufficient condition for the existence of a standard momentum map in the local model.

Experimental results

Research questions

• RQ1 Can a symplectic slice theorem be formulated without requiring a global momentum map for canonical proper group actions?
• RQ2 How can the Chu map replace the momentum map in constructing a tubular model for symplectic group actions?
• RQ3 Under what conditions does a canonical action admit a local momentum map in a G-invariant neighborhood (tubewise Hamiltonian)?
• RQ4 What is the role of the g*-valued one-form γ in determining the existence of a standard momentum map in the local model?
• RQ5 How do the reconstruction equations for Hamiltonian vector fields simplify in the tubular coordinates defined by the symplectic slice theorem?

Key findings

• The Chu map provides a canonical, always-existing 2-cocycle that replaces the momentum map in the construction of the symplectic slice theorem for canonical proper actions.
• The symplectic normal space Vm is a symplectic vector space with the induced form ωVm, and carries a canonical H-action with an associated momentum map JVm.
• The subalgebra k ⊂ g defined by k = {η ∈g | ηM(m) ∈(g·m)ω} is closed under the Lie bracket, ensuring structural consistency.
• The reconstruction equations for Hamiltonian vector fields in tubular coordinates take a particularly simple form, generalizing known bundle equations.
• A G-action is tubewise Hamiltonian at m if and only if the g*-valued one-form γ defined by ⟨γ(g)·TeLg·η,ξ⟩= −ω(m)(Adg−1ξM(m), ηM(m)) is exact.
• If H¹(G) = 0 or the orbit G·m is isotropic, then the G-action is tubewise Hamiltonian at m, as the exactness condition is trivially satisfied.