Moment maps of infinitesimal actions #

Let σ be a two-form on a manifold M and let a : Z ↦ Z_M be an infinitesimal action of a real Lie algebra 𝔤 on M (TauCeti.InfinitesimalAction). A moment map of a is a map μ : M → 𝔤* whose components μ_Z = ⟨μ, Z⟩ are smooth functions with Hamiltonian vector fields Z_M, in the convention ι_{X_f} σ = df:

ι_{Z_M} σ = d⟨μ, Z⟩ for every Z ∈ 𝔤

(TauCeti.SmoothTwoForm.IsMomentMap). No equivariance is required, so this is the infinitesimal notion; for the fundamental vector fields of a group action, coadjoint equivariance is a separate condition.

This file proves the basic laws of moment maps:

a moment map stays a moment map after adding a constant ν ∈ 𝔤*, after restricting the action along a Lie algebra homomorphism 𝔥 → 𝔤 (composing with the dual map), and, negated, for the opposite form -σ;
a moment map pulls back to a moment map along a smooth map relating two actions, so equivariant symplectomorphisms transport moment maps (TauCeti.SmoothTwoForm.IsMomentMap.mpullback), and the moment maps μ and ν of actions of one Lie algebra on two manifolds add up to the moment map (x, y) ↦ μ x + ν y of the diagonal action on the product (TauCeti.SmoothTwoForm.IsMomentMap.prod);
on a preconnected manifold any two moment maps of the same action differ by a constant, so the moment maps of an action are exactly the constant shifts of any one of them (TauCeti.SmoothTwoForm.IsMomentMap.isMomentMap_iff_exists_eq_add);
for a nondegenerate form on a finite-dimensional manifold, Z_M is the Hamiltonian vector field of μ_Z, the Poisson bracket with μ_Z is the derivative along Z_M, so {μ_Z, μ_Z'} is σ(Z_M, Z'_M), and Noether's theorem holds: the derivative of a function along Z_M vanishes exactly when it Poisson-commutes with μ_Z;
for a closed nondegenerate form, {μ_Z, μ_Z'} has Hamiltonian vector field (⁅Z, Z'⁆)_M (TauCeti.SmoothTwoForm.IsMomentMap.hamiltonianVectorField_poissonBracket).

The differential dμ_x : T_x M → 𝔤* of a moment map (TauCeti.SmoothTwoForm.IsMomentMap.differential) satisfies ⟨dμ_x(v), Z⟩ = σ(Z_M x, v). Hence its kernel is the σ-orthogonal (𝔤 · x)^σ of the orbit tangent space (TauCeti.SmoothTwoForm.IsMomentMap.ker_differential) and, where σ is nondegenerate on a finite-dimensional manifold, its image is the annihilator 𝔤_x^0 of the isotropy algebra (TauCeti.SmoothTwoForm.IsMomentMap.range_differential). So dμ_x is surjective exactly when 𝔤_x = 0, that is, when the action is locally free at x, and the rank of dμ_x is the dimension of the orbit tangent space. These identities describe the tangent spaces of the levels of a moment map, the starting point of symplectic reduction. Along a Hamiltonian vector field X of a function f the differential is ⟨dμ_x(X x), Z⟩ = -df_x(Z_M x) (TauCeti.SmoothTwoForm.IsMomentMap.differential_apply_of_isHamiltonianVectorField), so X is tangent to the level at x exactly when df_x vanishes on every Z_M x (TauCeti.SmoothTwoForm.IsMomentMap.mem_ker_differential_iff_of_isHamiltonianVectorField).

The failure of a moment map to be infinitesimally equivariant is measured by its cocycle, the Lie algebra 2-cochain c(Z, Z') = σ(Z_M, Z'_M) - μ_{⁅Z, Z'⁆} on 𝔤 at each point, with values in the trivial module ℝ (TauCeti.SmoothTwoForm.momentCocycle). For a nondegenerate form it is {μ_Z, μ_Z'} - μ_{⁅Z, Z'⁆}. A constant shift μ + ν changes it by the coboundary d ν, and it is compatible with restriction of the action. For a closed form, d(σ(Z_M, Z'_M)) = dμ_{⁅Z, Z'⁆} (TauCeti.SmoothTwoForm.IsMomentMap.mvfderiv_eval₂), so the cocycle is constant on a preconnected manifold (TauCeti.SmoothTwoForm.IsMomentMap.momentCocycle_eq), and it is a Lie algebra 2-cocycle (TauCeti.SmoothTwoForm.IsMomentMap.momentCocycle_mem_twoCocycle). The cocycle vanishes at x exactly when σ(Z_M x, Z'_M x) = μ(x)(⁅Z, Z'⁆) for all Z, Z' (TauCeti.SmoothTwoForm.momentCocycle_eq_zero_iff), and then dμ_x(Z_M x) = ⁅Z, μ(x)⁆, so the isotropy algebra 𝔤_x lies in the coadjoint stabilizer of μ(x) (TauCeti.SmoothTwoForm.IsMomentMap.isotropy_le_lieAnnihilator). For a nondegenerate form, if moreover μ(x) is coadjoint-fixed, the components of μ pairwise Poisson-commute at x (TauCeti.SmoothTwoForm.IsMomentMap.poissonBracket_eq_zero).

The first example is the linear action of the symplectic Lie algebra sp(V, ω), the skew-adjoint endomorphisms of a finite-dimensional symplectic vector space (V, ω), by the vector fields x ↦ A x. Its moment map is μ_A(x) = ½ ω(A x, x) (TauCeti.SymplecticForm.isMomentMap_linearMomentMap), and it is infinitesimally equivariant: {μ_A, μ_B} = μ_{⁅A, B⁆} (TauCeti.SymplecticForm.poissonBracket_linearMomentMap).

The second example is the action of an abelian Lie algebra 𝔤 on (V, ω) by the translations generated by a linear map τ : 𝔤 → V. Its moment map is μ_Z(x) = ω(τ Z, x) (TauCeti.SymplecticForm.isMomentMap_translationMomentMap), and every moment map of this action has the Heisenberg cocycle c(Z, Z') = ω(τ Z, τ Z') (TauCeti.SymplecticForm.momentCocycle_translation), so none is infinitesimally equivariant unless ω vanishes on the image of τ.

Main definitions #

TauCeti.SmoothTwoForm.IsMomentMap σ a μ: μ is a moment map of the infinitesimal action a for σ.
TauCeti.SmoothTwoForm.IsMomentMap.differential: the differential dμ_x : T_x M → 𝔤*.
TauCeti.SmoothTwoForm.momentCocycle σ a μ x: the cocycle σ(Z_M, Z'_M) - μ_{⁅Z, Z'⁆} at x, a 2-cochain of 𝔤 with trivial coefficients.
TauCeti.SymplecticForm.linearMomentMap ω: the moment map μ_A(x) = ½ ω(A x, x) of the linear action of sp(V, ω).
TauCeti.SymplecticForm.translationMomentMap ω τ: the moment map μ_Z(x) = ω(τ Z, x) of a translation action.

References #

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984, Section 26.
A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001, Lecture 22, and Lecture 23 (the kernel and image of the differential of a moment map, in the proof of the Marsden–Weinstein–Meyer theorem).
D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Section 5.2 (moment maps, and the cocycle measuring their failure to be equivariant).

Moment maps of infinitesimal actions #

Main definitions #

References #

The differential of a moment map #

The cocycle of a moment map #