moment map

moment map
Name	Moment map
Field	Symplectic geometry
Introduced	1970s
Key concepts	Symplectic form; Hamiltonian action; Coadjoint orbit; Marsden–Weinstein reduction; Momentum map
Notable people	Jerrold Marsden; Alan Weinstein; Jean-Marie Souriau; Victor Guillemin; Shlomo Sternberg

moment map

The moment map is a construction in symplectic geometry that associates to a Hamiltonian action of a Lie group on a symplectic manifold a conserved quantity taking values in the dual of the Lie algebra. It encodes infinitesimal generators of symmetries as functions, organizes conserved quantities under group actions, and links geometric structures on manifolds to representation-theoretic objects such as coadjoint orbits and moment polytopes. Developed in the work of Jean-Marie Souriau, Jerrold Marsden, and Alan Weinstein, the moment map plays a central role in reduction techniques, geometric quantization, and interactions between Kähler manifold theory and algebraic geometry.

Definition and basic properties

Given a symplectic manifold (M, ω) and a Lie group G acting smoothly by symplectomorphisms, a moment map μ : M → g* (where g* is the dual of the Lie algebra g of G) satisfies, for each ξ in g, the identity d⟨μ, ξ⟩ = ι_{X_ξ} ω, where X_ξ is the vector field generating the one-parameter subgroup exp(tξ). Existence and uniqueness of μ involve choices: when H^1(M; ℝ) vanishes or for connected, simply connected groups like SU(2), one often obtains a canonical μ; otherwise μ is defined up to addition of a central element of g*. The moment map intertwines the Poisson algebra of smooth functions on M with the Lie algebra structure on g via the equivariance or infinitesimal equivariance condition, relating to concepts introduced by Victor Guillemin and Shlomo Sternberg.

Examples and computations

Classical examples include the standard S^1-action on ℂ^n with symplectic form ω = (i/2)∑ dz_j ∧ d\bar z_j, where the moment map for the diagonal U(1) action is μ(z) = 1/2 ∑ |z_j|^2. The cotangent bundle T^*G for a Lie group G carries a canonical symplectic form; the left and right translations produce moment maps identifying T^*G with g* × G and recovering the canonical pairing with coadjoint coordinates, examples studied by Jean-Pierre Serre in representation contexts. For the action of SO(3) on S^2 with the area form, the inclusion S^2 ↪ ℝ^3 ≅ so(3)* is the moment map, recovering classical angular momentum. Toric varieties provide combinatorial computations: the moment map image for an effective Hamiltonian action of an n-torus T^n on a compact 2n-dimensional symplectic manifold is a convex polytope, computed via weight data appearing in works related to Atiyah and Vladimir Guillemin.

Symplectic reduction and Marsden–Weinstein theorem

The Marsden–Weinstein theorem constructs reduced phase spaces μ^{-1}(α)/G_α for a value α ∈ g*, where G_α is the stabilizer of α under the coadjoint action. Under regularity and properness hypotheses, μ^{-1}(α) is a submanifold and the quotient inherits a symplectic form ω_red uniquely characterized by pullback. This reduction procedure, foundational for constrained mechanical systems like the n-body problem analyzed by researchers at Caltech and Princeton University, permits passage from a high-dimensional phase space to a lower-dimensional moduli space while preserving symplectic structure, and it underlies constructions in moduli of flat connections and gauge theory influenced by work at Institute for Advanced Study.

Equivariance, coadjoint orbits, and Hamiltonian group actions

Equivariance of μ with respect to the given G-action and the coadjoint action on g* is a strong condition: μ(g·x) = Ad^*(g) μ(x). When equivariance fails, one can often correct μ by a cocycle valued in the center of g*, linking to central extensions studied by Souriau and the theory of projective representations of groups such as Heisenberg group and Virasoro algebra. Coadjoint orbits themselves are canonical symplectic manifolds via the Kirillov–Kostant–Souriau form; they appear as images of moment maps and classify irreducible unitary representations in geometric quantization contexts pioneered by Bertram Kostant and Kirillov.

Moment map in Kähler and algebraic geometry

On a Kähler manifold (M, ω, J) with a holomorphic action of a complex reductive group G^ℂ and a maximal compact subgroup K, moment maps relate GIT stability notions to symplectic quotients via the Kempf–Ness theorem. For projective varieties embedded in ℙ^N, the Kempf–Ness correspondence identifies GIT quotients with symplectic reductions at suitable values of μ, connecting Hilbert–Mumford stability criteria studied by David Mumford to the image of the moment map and convexity properties explored by Michael Atiyah and Hans Duistermaat.

Applications in mathematical physics and geometric quantization

Moment maps provide conserved quantities in classical mechanics—angular momentum, linear momentum, and center-of-mass integrals—and facilitate reduction of Hamiltonian systems in celestial mechanics and continuum mechanics treated in literature at Courant Institute and Harvard University. In geometric quantization, moment maps produce quantum operators representing Lie algebra actions on Hilbert spaces obtained from prequantum line bundles; the "quantization commutes with reduction" principle conjectured in part by Guillemin and Sternberg links indices of elliptic operators on reduced spaces to invariant components of quantizations before reduction. In gauge theory and moduli problems, moment maps appear in the Yang–Mills functional and the Hitchin–Kobayashi correspondence studied at Oxford University and Cambridge University, providing a bridge between infinite-dimensional symplectic reduction and algebraic stability conditions.

Category:Symplectic geometry