[Paper Review] Dirac geometry, quasi-Poisson actions and D/G-valued moment maps

Henrique Bursztyn, Marius Crainic|ArXiv.org|Oct 2, 2007

Homotopy and Cohomology in Algebraic Topology25 references4 citations

TL;DR

This paper establishes a unified framework for $D/G$-valued moment maps by showing that Dirac geometry and quasi-Poisson geometry—previously seen as distinct approaches—yield isomorphic categories of Hamiltonian spaces. It proves that both formalisms, rooted in Manin pairs $(\mathfrak{d},\mathfrak{g})$ and isotropic connections on $D$, lead to equivalent moment map theories via a canonical equivalence of Hamiltonian categories.

ABSTRACT

We study Dirac structures associated with Manin pairs (\d,\g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of "double" in each context.

Motivation & Objective

• To unify two distinct geometric frameworks—Dirac geometry and quasi-Poisson geometry—for $D/G$-valued moment maps.
• To explain the geometric origin of $G$-valued moment maps and their equivalence between twisted 2-form and quasi-Poisson formulations.
• To establish a canonical equivalence between the categories of Hamiltonian spaces in the Dirac and quasi-Poisson settings.
• To clarify the role of isotropic connections and equivariant cohomology in constructing $D/G$-valued moment maps.
• To demonstrate that presymplectic groupoids and doubles arise naturally in both frameworks, linking to the notion of 'double' in generalized geometry.

Proposed method

• Construct a $G$-manifold $S = D/G$ from a Manin pair $(\mathfrak{d}, \mathfrak{g})$ integrated by a group pair $(D, G)$, using the dressing action.
• Define a $\phi_S$-twisted Dirac structure $L_S \subset TS \oplus T^*S$ via an isotropic connection $\theta \in \Omega^1(D, \mathfrak{g})$ on the principal $G$-bundle $D \to D/G$.
• Use the identification of the trivial $\mathfrak{d}$-bundle over $S$ as an exact Courant algebroid to relate $\theta$ to a splitting of $TS \oplus T^*S$.
• Construct a closed equivariant 3-form $\phi_S$ on $S$ from the curvature of $\theta$, which defines the twist in the Dirac structure.
• Define Hamiltonian spaces via morphisms $J: M \to S$ from Dirac manifolds $M$ to $S$, with moment maps valued in $S$.
• Prove equivalence between the Hamiltonian categories of Dirac and quasi-Poisson geometries by constructing a functorial isomorphism using compatible splittings and the linear algebra of Manin pairs.

Experimental results

Research questions

• RQ1 How are the Dirac geometric and quasi-Poisson approaches to $D/G$-valued moment maps related at the level of Hamiltonian spaces?
• RQ2 Can the equivalence between twisted 2-form and quasi-Poisson formulations of $G$-valued moment maps be generalized to $D/G$-valued moment maps?
• RQ3 What is the role of isotropic connections in constructing $\phi_S$-twisted Dirac structures on $D/G$?
• RQ4 How do presymplectic groupoids and doubles emerge from both the Dirac and quasi-Poisson frameworks?
• RQ5 Is there a canonical equivalence between the categories of Hamiltonian spaces in the two formalisms?

Key findings

• The categories of Hamiltonian spaces with $D/G$-valued moment maps in the Dirac and quasi-Poisson frameworks are isomorphic.
• An isotropic connection $\theta$ on $D \to D/G$ induces a closed equivariant 3-form $\phi_S$ and a $\phi_S$-twisted Dirac structure $L_S$ on $S = D/G$.
• The Dirac structure $L_S$ corresponds to the subbundle $\mathfrak{g} \subset \mathfrak{d}_S = \mathfrak{d} \times S$ under a splitting induced by $\theta$.
• The equivalence between the two formalisms is established via a canonical functor that preserves moment map morphisms and compatibility conditions.
• The theory unifies $G$-valued moment maps and symmetric-space valued moment maps as special cases of $D/G$-valued moment maps.
• The construction shows that the notion of 'double' in Dirac geometry (via $TS \oplus T^*S$) and in quasi-Poisson geometry (via the Lie algebroid of the action) are geometrically equivalent.