Moment map

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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.

Contents

1 Formal definition
2 Hamiltonian group actions
3 Examples
4 Symplectic quotients
5 See also
6 Notes
7 References

Formal definition

Let M be a manifold with symplectic form $\omega$ . Suppose that a Lie group G acts on M via symplectomorphisms (that is, the action of each g in G preserves $\omega$ ). Let $\mathfrak{g}$ be the Lie algebra of G, $\mathfrak{g}^*$ its dual, and

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

the pairing between the two. Any $\xi$ in $\mathfrak{g}$ induces a vector field $\rho(\xi)$ on M describing the infinitesimal action of $\xi$ . 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 $\omega$ . Because G acts by symplectomorphisms, it follows that $\iota_{\rho(\xi)} \omega \,$ is closed for all $\xi$ in $\mathfrak{g}$ .

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

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

for all $\xi$ in $\mathfrak{g}$ . Here $\langle \mu, \xi \rangle$ is the function from M to $\mathbb{R}$ 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).

Hamiltonian group actions

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

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

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 $(M, \omega)$ 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

In the case of a Hamiltonian action of the circle $G = U(1)$ , 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.

Symplectic quotients

Suppose that the action of a compact Lie group G on the symplectic manifold $(M, \omega)$ 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 $\mu$ 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 pullback of $\omega$ 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.

Notes

^ The vector field $\rho(\xi)$ is called sometimes the Killing vector field relative to the action of the one-parameter subgroup generated by $\xi$ . 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

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