Moment map

In mathematics, specifically in symplectic geometry, the momentum map (or moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

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 ${\mathfrak {g}}$ be the Lie algebra of G, $\mathfrak{g}^*$ its dual, and

$\langle, \rangle : \mathfrak{g}^* \times \mathfrak{g} \to \mathbf{R}$

the pairing between the two. Any ξ in ${\mathfrak {g}}$ 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 $\rho(\xi)_x$ is

$\left.\frac{d}{dt}\right|_{t = 0} \exp(t \xi) \cdot x,$

where $\exp : \mathfrak{g} \to G$ is the exponential map and $\cdot$ denotes the G-action on M.^[1] Let $\iota_{\rho(\xi)} \omega \,$ denote the contraction of this vector field with ω. Because G acts by symplectomorphisms, it follows that $\iota_{\rho(\xi)} \omega \,$ is closed for all ξ in ${\mathfrak {g}}$.

A moment map for the G-action on (M, ω) is a map $\mu : M \to \mathfrak{g}^*$ such that

$d(\langle \mu, \xi \rangle) = \iota_{\rho(\xi)} \omega$

for all ξ in ${\mathfrak {g}}$. Here $\langle \mu, \xi \rangle$ is the function from M to R defined by $\langle \mu, \xi \rangle(x) = \langle \mu(x), \xi \rangle$. 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 $\mathfrak{g}^*$ 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 $\mathfrak{g}^*$, as first described by Souriau (1970).

Hamiltonian group actions

The definition of the moment map requires $\iota_{\rho(\xi)} \omega$ 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 ${\mathfrak {g}}$ the one-form $\iota_{\rho(\xi)} \omega$ is exact, meaning that it equals $dH_\xi$ for some smooth function

$H_\xi : M \to \mathbf{R}.$

If this holds, then one may choose the $H_\xi$ to make the map $\xi \mapsto H_\xi$ linear. The second requirement for the G-action to be Hamiltonian is that the map $\xi \mapsto H_\xi$ be a Lie algebra homomorphism from ${\mathfrak {g}}$ 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 $\mu : M\to \mathfrak{g}^*$ such that writing $H_\xi = \langle \mu, \xi \rangle$ defines a Lie algebra homomorphism $\xi \mapsto H_\xi$ satisfying $\rho(\xi) = X_{H_\xi}$. Here $X_{H_\xi}$ is the vector field of the Hamiltonian $H_\xi$, defined by

$\iota_{X_{H_\xi}} \omega = d H_\xi.$

Examples of moment maps

In the case of a Hamiltonian action of the circle $G = \mathcal{U}(1)$, the Lie algebra dual $\mathfrak{g}^*$ is naturally identified with $\mathbb {R}$, 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 $\mathbb {R} ^{3}$ and $G$ is the Euclidean group generated by rotations and translations. That is, $G$ is a six-dimensional group, the semidirect product of $SO(3)$ and $\mathbb {R} ^{3}$. 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 $\pi : T^*N \rightarrow N$. Let $\tau$ denote the tautological 1-form on $T^*N$. Suppose $G$ acts on $N$. The induced action of $G$ on the symplectic manifold $(T^*N, \mathrm{d}\tau)$, given by $g \cdot \eta := (T_{\pi(\eta)}g^{-1})^* \eta$ for $g \in G, \eta \in T^*N$ is Hamiltonian with moment map $-\iota_{\rho(\xi)} \tau$ for all $\xi \in \mathfrak{g}$. Here $\iota_{\rho(\xi)}\tau$ denotes the contraction of the vector field $\rho(\xi)$, the infinitesimal action of $\xi$, with the 1-form $\tau$.

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 $\mathfrak{g}, \mathfrak{h}$, respectively.

1. Let $\mathcal{O}(F), F \in \mathfrak{g}^*$ be a coadjoint orbit. Then there exists a unique symplectic structure on ${\mathcal {O}}(F)$ such that inclusion map $\mathcal{O}(F) \hookrightarrow \mathfrak{g}^*$ is a moment map.

2. Let $G$ act on a symplectic manifold $(M, \omega)$ with $\Phi_G : M \rightarrow \mathfrak{g}^*$ a moment map for the action, and $\psi : H \rightarrow G$ 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 $(\mathrm{d}\psi)_{e}^* \circ \Phi_G$, where $(\mathrm{d}\psi)_{e}^* : \mathfrak{g}^* \rightarrow \mathfrak{h}^*$ is the dual map to $(\mathrm{d}\psi)_{e} : \mathfrak{h} \rightarrow \mathfrak{g}$ ($e$ denotes the identity element of $H$). A case of special interest is when $H$ is a Lie subgroup of $G$ and $\psi$ is the inclusion map.

3. Let $(M_1, \omega_1)$ be a Hamiltonian $G$-manifold and $(M_2, \omega_2)$ a Hamiltonian $H$-manifold. Then the natural action of $G\times H$ on $(M_1 \times M_2, \omega_1 \times \omega_2)$ is Hamiltonian, with moment map the direct sum of the two moment maps $\Phi_G$ and $\Phi_H$. Here $\omega_1 \times \omega_2 := \pi_1^*\omega_1 + \pi_2^*\omega_2$, where $\pi_i : M_1 \times M_2 \rightarrow M_i$ 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 $\mu : M\to \mathfrak{g}^*$. From the Hamiltonian condition it follows that $\mu^{-1}(0)$ is invariant under G.

Assume now that 0 is a regular value of μ and that G acts freely and properly on $\mu^{-1}(0)$. Thus $\mu^{-1}(0)$ and its quotient $\mu^{-1}(0) / G$ 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 $\mu^{-1}(0)$ equals the restriction of ω to $\mu^{-1}(0)$. 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.

See also

• Poisson-Lie group
• Toric manifold
• Geometric Mechanics
• Kirwan map

Notes

1. The vector field ρ(ξ) is called sometimes the Killing vector field relative to the action of the one-parameter subgroup generated by ξ. See, for instance, (Choquet-Bruhat & DeWitt-Morette 1977)

References

• J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970. ISSN 0750-2435.
• S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Science Publications, 1990. ISBN 0-19-850269-9.
• Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, Oxford Science Publications, 1998. ISBN 0-19-850451-9.
• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4
• Ortega, Juan-Pablo; Ratiu, Tudor S. (2004). Momentum maps and Hamiltonian reduction. Progress in Mathematics 222. Birkhauser Boston. ISBN 0-8176-4307-9.