[Paper Review] Stacky Hamiltonian actions and symplectic reduction

Benjamin Hoffman, Reyer Sjamaar+1 more|arXiv (Cornell University)|Aug 2, 2018

Homotopy and Cohomology in Algebraic Topology29 references3 citations

TL;DR

This paper extends classical symplectic geometry theorems—Kirwan convexity, Meyer-Marsden-Weinstein reduction, and Duistermaat-Heckman—to the setting of Hamiltonian actions by étale Lie group stacks on étale symplectic stacks. It introduces stacky moment maps and proves that under a cleanness condition, the moment map image is a convex polytope whose normal fan need not be rational, generalizing the Atiyah-Guillemin-Sternberg theorem to non-lattice cocharacter groups. The symplectic reduction theorem holds without compactness or properness assumptions, and the Duistermaat-Heckman theorem is extended to the linear variation of reduced forms.

ABSTRACT

We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of the Kirwan convexity theorem, the Meyer-Marsden-Weinstein symplectic reduction theorem, and the Duistermaat-Heckman theorem in this context.

Motivation & Objective

• To generalize Kirwan's convexity theorem to Hamiltonian actions of étale Lie group stacks on étale symplectic stacks.
• To establish a symplectic reduction theorem for such actions without requiring compactness or properness of the group stack.
• To extend the Duistermaat-Heckman theorem to stacky torus actions, focusing on the linear variation of reduced symplectic forms.
• To develop foundational tools for Hamiltonian actions on differentiable stacks, including stacky Lie groups, moment maps, and basic differential forms.
• To show that every simple stacky polytope arises as the moment polytope of a toric symplectic stack, generalizing toric algebraic stacks to the C∞-setting.

Proposed method

• Introduces Hamiltonian actions of étale Lie group stacks on étale symplectic stacks using 0-symplectic groupoids and moment maps in a 2-categorical framework.
• Defines stacky tori and maximal stacky tori via crossed modules and Lie 2-groups, allowing non-lattice cocharacter groups.
• Applies the Bott connection and basic differential forms on foliation groupoids to characterize transverse symplectic structures.
• Uses weak fibred products of Lie group stacks (via C. Zhu’s appendix) to construct and analyze reductions.
• Applies strictification theorems (Theorem 6.8.1) to reduce general stacky actions to strict actions, simplifying analysis.
• Employs the theory of 2-categories and 2-group actions to formalize moment maps and equivariance in higher geometry.

Experimental results

Research questions

• RQ1 Can the Kirwan convexity theorem be generalized to Hamiltonian actions of étale Lie group stacks on étale symplectic stacks, even when the cocharacter group is not a lattice?
• RQ2 Under what conditions is the symplectic reduction of a symplectic stack by a Hamiltonian action of an étale Lie group stack again a symplectic stack, without assuming compactness or properness?
• RQ3 How does the Duistermaat-Heckman theorem extend to stacky torus actions, particularly regarding the variation of the reduced symplectic form?
• RQ4 What is the structure of the moment map image (the stacky moment body) in the case of a stacky torus action, and how does it differ from the classical rational polytope?
• RQ5 Can every simple stacky polytope be realized as the moment polytope of a toric symplectic stack, and how does this construction relate to Prato’s quasifolds?

Key findings

• The image of the moment map under a Hamiltonian action of a stacky torus is a convex polytope, even when the normal fan is not rational, due to the cocharacter group being a quasi-lattice rather than a lattice.
• The symplectic reduction theorem holds for Hamiltonian actions of étale Lie group stacks on étale symplectic stacks under a regularity hypothesis on the moment map, without requiring compactness or properness of the group or action.
• The linear variation of the reduced symplectic form in the Duistermaat-Heckman theorem is generalized to stacky torus actions, establishing a key part of the classical theorem in the stacky setting.
• The Lie 2-algebra of vector fields on an étale stack is equivalent to a Lie algebra, a structural result that simplifies the analysis of symmetries.
• A strictification theorem (Theorem 6.8.1) shows that every stacky action is equivalent to a strict action, enabling the use of classical techniques in a higher-categorical context.
• Every simple stacky polytope arises as the moment polytope of a toric symplectic stack, generalizing Prato’s quasifolds and providing a C∞-counterpart to algebraic toric stacks.