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On the classification of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids

Maarten Mol

Abstract

In this paper we study a natural generalization of symplectic toric manifolds in the context of regular Poisson manifolds of compact types. To be more precise, we consider a class of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids that we call faithful. Given such a groupoid, we classify its faithful multiplicity-free Hamiltonian actions in terms of what we call Delzant subspaces of its orbit space -- certain `suborbifolds with corners' satisfying the Delzant condition relative to the integral affine orbifold structure of the orbit space. This encompasses both the classification of symplectic toric manifolds (due to Delzant) in terms of Delzant polytopes and the classification of proper Lagrangian fibrations over an integral affine base manifold (due to Duistermaat) in terms of a sheaf cohomology group. Each Delzant subspace comes with an orbifold version of this cohomology, the degree one part of which classifies faithful multiplicity-free Hamiltonian actions with momentum map image equal to the Delzant subspace, provided there exists such an action. The obstruction to existence is encoded by a degree two class in this cohomology: the Lagrangian Dixmier-Douady class. In addition to the above, we introduce another invariant, which leads to a variation of our classification result involving only classical sheaf cohomology and the group cohomology of certain modules for the isotropy groups of the groupoid.

On the classification of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids

Abstract

In this paper we study a natural generalization of symplectic toric manifolds in the context of regular Poisson manifolds of compact types. To be more precise, we consider a class of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids that we call faithful. Given such a groupoid, we classify its faithful multiplicity-free Hamiltonian actions in terms of what we call Delzant subspaces of its orbit space -- certain `suborbifolds with corners' satisfying the Delzant condition relative to the integral affine orbifold structure of the orbit space. This encompasses both the classification of symplectic toric manifolds (due to Delzant) in terms of Delzant polytopes and the classification of proper Lagrangian fibrations over an integral affine base manifold (due to Duistermaat) in terms of a sheaf cohomology group. Each Delzant subspace comes with an orbifold version of this cohomology, the degree one part of which classifies faithful multiplicity-free Hamiltonian actions with momentum map image equal to the Delzant subspace, provided there exists such an action. The obstruction to existence is encoded by a degree two class in this cohomology: the Lagrangian Dixmier-Douady class. In addition to the above, we introduce another invariant, which leads to a variation of our classification result involving only classical sheaf cohomology and the group cohomology of certain modules for the isotropy groups of the groupoid.
Paper Structure (49 sections, 72 theorems, 350 equations, 2 figures)

This paper contains 49 sections, 72 theorems, 350 equations, 2 figures.

Table of Contents

  1. Delzant subspaces and the momentum map image
  2. Classification of maximally toric representations
  3. Statement of the classification theorem
  4. Isomorphism versus equivalence of symplectic representations
  5. Maximally toric representations of tori
  6. Splittings of abelian group extensions
  7. The classification of maximally toric representations
  8. Delzant subspaces of integral affine orbifolds
  9. Background on orbifolds
  10. Background on integral affine structures on orbifolds
  11. Background on the integral affine orbifold associated to a regular proper symplectic groupoid
  12. Delzant subspaces
  13. On the image of the momentum map
  14. The symplectic normal representations and the momentum image
  15. Proofs of Theorem \ref{['momimtoricthm']} and Proposition \ref{['conedescrpsympnormrepprop']}
  16. ...and 34 more sections

Key Result

Theorem 1

Let $(\mathcal{T},\Omega)$ be a symplectic torus bundle over $M$ with induced integral affine structure $\Lambda$ on $M$. So, there is a canonical bijection:

Figures (2)

  • Figure 1: A sketch of the boundary $\partial\Delta$ (coloured in green) of the Delzant subspace in Example \ref{['example:exoticdelzsubsp']}.
  • Figure 2: A fundamental domain for the $\Gamma$-action on $\Delta$, partitioned by collecting points with the same isotropy group. For connected components of members with non-trivial isotropy groups, the corresponding stalks of $\mathcal{I}^1$ are depicted.

Theorems & Definitions (212)

  • Theorem 1: Classification of faithful toric spaces
  • Example 1
  • Example 2
  • Definition
  • Theorem 2
  • Theorem 3: First structure theorem
  • Theorem 4: Splitting theorem
  • Remark 1
  • Example 3
  • Example 4
  • ...and 202 more