Moment map - Knowlpedia

Let $G$ be a Lie group with a symplectic action on $(M,\omega)$ , and let $\mathfrak g$ be its Lie algebra. Write $\mathfrak g^*=\operatorname{Hom}_{\mathbb R}(\mathfrak g,\mathbb R)$ for its linear dual.

For $\xi\in\mathfrak g$ , use the fundamental vector field $\xi_M(x)=\left.\frac{d}{dt}\right|_{t=0}\exp(t\xi)\cdot x$ , where $\exp$ is the exponential map.

A moment map for the action is a smooth map

\mu\colon M \to \mathfrak{g}^*

such that for every $\xi\in\mathfrak{g}$ ,

d\langle \mu,\xi\rangle \;=\; \iota_{\xi_M}\omega,

where $d$ is the exterior derivative and $\iota_{\xi_M}$ denotes contraction by the vector field $\xi_M$ .

Conventions and uniqueness

This definition does not impose equivariance. An equivariant moment map also satisfies $\mu(gx)=\operatorname{Ad}_g^*\mu(x)$ for the coadjoint action. The Hamiltonian-action entry uses this stronger convention; some authors call an action Hamiltonian as soon as it has a moment map in the present sense. A different sign convention may replace the displayed identity by its negative.

If $M$ is connected, any two moment maps satisfying the displayed identity differ by a constant in $\mathfrak{g}^*$ . If both are equivariant, that constant is fixed by the coadjoint action.

For an equivariant moment map, a useful reformulation is that in equivariant cohomology (Cartan model), the pair $(\omega,\mu)$ combines into an equivariantly closed degree-2 element.

Examples

Rotation of the plane. For the standard counterclockwise $S^1$ -action on $\mathbb{R}^2\cong \mathbb{C}$ , the unsigned left-action generator, and $\omega=dx\wedge dy$ , a moment map is $\mu(z)=-\tfrac12|z|^2$ (identifying $(\mathfrak{u}(1))^*\cong \mathbb{R}$ ).
Height on the 2-sphere. For the standard rotation action of $S^1$ on $S^2$ around the vertical axis with the area form, a moment map is the height function (up to an additive constant).
Cotangent lift. If $G$ acts on a manifold $Q$ , the induced action on $T^*Q$ with its canonical symplectic form is Hamiltonian, with moment map $\mu(q,p)(\xi)=p(\xi_Q(q))$ .