Momentum maps and morita equivalence

We introduce quasi-symplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including ordinary Hamiltonian G-spaces, Lu’s momentum maps of Poisson group actions, and the group-valued momentum maps of Alekseev-Malkin-Meinrenken. More precisely, we carry out the following program: (1) We define and study properties of quasi-symplectic groupoids. (2) We study the momentum map theory defined by a quasisymplectic groupoid γ ⇒ P. In particular, we study the reduction theory and prove that J⁻¹(𝒪)/γ is a symplectic manifold for any Hamiltonian γ-space (X →^J P, ω_X) (even though ω_X ∈ ω²(X) may be degenerate), where 𝒪 ⊂ P is a groupoid orbit. More generally, we prove that the intertwiner space (X₁×P X^¯ ₂)/γ between two Hamiltonian γ-spaces X₁ and X₂ is a symplectic manifold (whenever it is a smooth manifold). (3)We study Morita equivalence of quasi-symplectic groupoids. In particular, we prove that Morita equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories. Moreover the intertwiner space (X₁ ×P X₂)/γ depends only on the Morita equivalence class. As a result, we recover various wellknown results concerning equivalence of momentum maps including the Alekseev-Ginzburg-Weinstein linearization theorem and the Alekseev-Malkin-Meinrenken equivalence theorem between quasi-Hamiltonian spaces and Hamiltonian loop group spaces.