[Paper Review] Courant morphisms and moment maps

Henrique Bursztyn, David Iglesias Ponte+1 more|ArXiv.org|Jan 10, 2008

Homotopy and Cohomology in Algebraic Topology14 references4 citations

TL;DR

This paper introduces an intrinsic notion of Hamiltonian spaces for Manin pairs $(E, A)$, where $E$ is a Courant algebroid and $A \subset E$ is a Dirac structure, using Courant morphisms. It establishes a unified framework that naturally recovers both quasi-Poisson and Dirac geometric approaches to moment maps, proving their equivalence through a functorial correspondence. The key contribution is a construction linking presymplectic and quasi-Poisson groupoids via moment map reduction.

ABSTRACT

We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived. As an application, we discuss the relationship between quasi-Poisson and presymplectic groupoids.

Motivation & Objective

• To provide an intrinsic, data-independent formulation of Hamiltonian spaces for Manin pairs $(E, A)$, avoiding reliance on noncanonical choices.
• To unify two distinct geometric formulations of moment maps: quasi-Poisson actions and Dirac maps into homogeneous spaces.
• To establish a functorial correspondence between Hamiltonian spaces in quasi-Poisson and Dirac geometries.
• To construct a bridge between presymplectic and quasi-Poisson groupoids via moment map reduction.
• To generalize moment map theories using Courant algebroid morphisms and Dirac structures with support.

Proposed method

• Define Hamiltonian spaces as triples $(X, J, K)$, where $J: X \to S$ is a moment map and $K$ is a Dirac structure on $(TX \oplus T^*X) \times E$ supported on $\mathrm{graph}(J)$, satisfying compatibility with the Manin pair $(E, A)$.
• Use morphisms of Manin pairs to formalize the notion of Hamiltonian spaces, generalizing both quasi-Poisson and Dirac-geometric frameworks.
• Apply the theory to the double of a Lie quasi-bialgebroid to recover Hamiltonian quasi-Poisson spaces.
• Apply the theory to Courant algebroids defined by closed 3-forms to recover Hamiltonian spaces via strong Dirac maps.
• Construct a correspondence between presymplectic groupoids and quasi-Poisson groupoids via the moment map reduction of Hamiltonian spaces.
• Use the integration of Dirac structures to a Lie groupoid $\mathcal{G}$, and show that the induced bivector field $\Pi_{\mathcal{G}}$ matches the multiplicative quasi-Poisson structure.

Experimental results

Research questions

• RQ1 Can a unified, intrinsic notion of Hamiltonian spaces for Manin pairs be defined without relying on additional geometric data?
• RQ2 How can the equivalence between quasi-Poisson and Dirac-geometric moment map theories be conceptually explained?
• RQ3 What is the precise relationship between presymplectic and quasi-Poisson groupoids in terms of moment maps and reduction?
• RQ4 How do Courant morphisms and Dirac structures on product spaces encode Hamiltonian actions?
• RQ5 Can the integration of Dirac structures lead to a canonical construction of quasi-Poisson groupoids from presymplectic groupoids?

Key findings

• The intrinsic notion of Hamiltonian spaces for Manin pairs $(E, A)$ is defined via Courant morphisms and Dirac structures on $X \times S$, providing a unified framework for moment map theories.
• The category of Hamiltonian spaces for $(E, A)$ is isomorphic to both the quasi-Poisson and Dirac-geometric categories, regardless of auxiliary choices, explaining their equivalence.
• When $E = A \oplus A^*$ is the double of a Lie quasi-bialgebroid, Hamiltonian spaces correspond precisely to Hamiltonian quasi-Poisson spaces.
• When $E = TS \oplus T^*S$ with a closed 3-form, Hamiltonian spaces correspond to Hamiltonian spaces via strong Dirac maps into $S$.
• The construction yields a functorial correspondence between presymplectic and quasi-Poisson groupoids, explicitly linking their moment map structures.
• The quasi-Poisson bivector field $\Pi_{\mathcal{G}}$ on the integrated groupoid $\mathcal{G}$ matches the multiplicative bivector field from the Lie quasi-bialgebroid, proving consistency of the construction.