Moment map

Formal definition

Let M be a manifold with symplectic form ω. Suppose that a Lie group G acts on M via symplectomorphisms (that is, the action of each g in G preserves ω). Let g be the Lie algebra of G, g ∗ its dual, and

the pairing between the two. Any ξ in g induces a vector field ρ(ξ) on M describing the infinitesimal action of ξ. To be precise, at a point x in M the vector ρ ( ξ ) x is

where exp : g → G is the exponential map and ⋅ denotes the G-action on M. Let ι ρ ( ξ ) ω denote the contraction of this vector field with ω. Because G acts by symplectomorphisms, it follows that ι ρ ( ξ ) ω is closed for all ξ in g .

A moment map for the G-action on (M, ω) is a map μ : M → g ∗ such that

for all ξ in g . Here ⟨ μ , ξ ⟩ is the function from M to R defined by ⟨ μ , ξ ⟩ ( x ) = ⟨ μ ( x ) , ξ ⟩ . The moment map is uniquely defined up to an additive constant of integration.

A moment map is often also required to be G-equivariant, where G acts on g ∗ via the coadjoint action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the moment map coadjoint equivariant; however in general the coadjoint action must be modified to make the map equivariant (this is the case for example for the Euclidean group). The modification is by a 1-cocycle on the group with values in g ∗ , as first described by Souriau (1970).

Hamiltonian group actions

The definition of the moment map requires ι ρ ( ξ ) ω to be closed. In practice it is useful to make an even stronger assumption. The G-action is said to be Hamiltonian if and only if the following conditions hold. First, for every ξ in g the one-form ι ρ ( ξ ) ω is exact, meaning that it equals d H ξ for some smooth function

If this holds, then one may choose the H ξ to make the map ξ ↦ H ξ linear. The second requirement for the G-action to be Hamiltonian is that the map ξ ↦ H ξ be a Lie algebra homomorphism from g to the algebra of smooth functions on M under the Poisson bracket.

If the action of G on (M, ω) is Hamiltonian in this sense, then a moment map is a map μ : M → g ∗ such that writing H ξ = ⟨ μ , ξ ⟩ defines a Lie algebra homomorphism ξ ↦ H ξ satisfying ρ ( ξ ) = X H ξ . Here X H ξ is the vector field of the Hamiltonian H ξ , defined by

Examples of moment maps

In the case of a Hamiltonian action of the circle G = U ( 1 ) , the Lie algebra dual g ∗ is naturally identified with R , and the moment map is simply the Hamiltonian function that generates the circle action.

Another classical case occurs when M is the cotangent bundle of R 3 and G is the Euclidean group generated by rotations and translations. That is, G is a six-dimensional group, the semidirect product of S O ( 3 ) and R 3 . The six components of the moment map are then the three angular momenta and the three linear momenta.

Let N be a smooth manifold and let T ∗ N be its cotangent bundle, with projection map π : T ∗ N → N . Let τ denote the tautological 1-form on T ∗ N . Suppose G acts on N . The induced action of G on the symplectic manifold ( T ∗ N , d τ ) , given by g ⋅ η := ( T π ( η ) g − 1 ) ∗ η for g ∈ G , η ∈ T ∗ N is Hamiltonian with moment map − ι ρ ( ξ ) τ for all ξ ∈ g . Here ι ρ ( ξ ) τ denotes the contraction of the vector field ρ ( ξ ) , the infinitesimal action of ξ , with the 1-form τ .

The facts mentioned below may be used to generate more examples of moment maps.

Some facts about moment maps

Let G , H be Lie groups with Lie algebras g , h , respectively.

1. Let O ( F ) , F ∈ g ∗ be a coadjoint orbit. Then there exists a unique symplectic structure on O ( F ) such that inclusion map O ( F ) ↪ g ∗ is a moment map.

2. Let G act on a symplectic manifold ( M , ω ) with Φ G : M → g ∗ a moment map for the action, and ψ : H → G be a Lie group homomorphism, inducing an action of H on M . Then the action of H on M is also Hamiltonian, with moment map given by ( d ψ ) e ∗ ∘ Φ G , where ( d ψ ) e ∗ : g ∗ → h ∗ is the dual map to ( d ψ ) e : h → g ( e denotes the identity element of H ). A case of special interest is when H is a Lie subgroup of G and ψ is the inclusion map.

3. Let ( M 1 , ω 1 ) be a Hamiltonian G -manifold and ( M 2 , ω 2 ) a Hamiltonian H -manifold. Then the natural action of G × H on ( M 1 × M 2 , ω 1 × ω 2 ) is Hamiltonian, with moment map the direct sum of the two moment maps Φ G and Φ H . Here ω 1 × ω 2 := π 1 ∗ ω 1 + π 2 ∗ ω 2 , where π i : M 1 × M 2 → M i denotes the projection map.

4. Let M be a Hamiltonian G -manifold, and N a submanifold of M invariant under G such that the restriction of the symplectic form on M to N is non-degenerate. This imparts a symplectic structure to N in a natural way. Then the action of G on N is also Hamiltonian, with moment map the composition of the inclusion map with M 's moment map.

Symplectic quotients

Suppose that the action of a compact Lie group G on the symplectic manifold (M, ω) is Hamiltonian, as defined above, with moment map μ : M → g ∗ . From the Hamiltonian condition it follows that μ − 1 ( 0 ) is invariant under G.

Assume now that 0 is a regular value of μ and that G acts freely and properly on μ − 1 ( 0 ) . Thus μ − 1 ( 0 ) and its quotient μ − 1 ( 0 ) / G are both manifolds. The quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to μ − 1 ( 0 ) equals the restriction of ω to μ − 1 ( 0 ) . Thus the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, symplectic quotient or symplectic reduction of M by G and is denoted M / / G . Its dimension equals the dimension of M minus twice the dimension of G.

See also: GIT quotient, Quantization commutes with reduction.