Momentum map

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In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map [1] ) 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 momentum 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.

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Formal definition

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

the pairing between the two. Any in induces a vector field on describing the infinitesimal action of . To be precise, at a point in the vector is

where is the exponential map and denotes the -action on . [2] Let denote the contraction of this vector field with . Because acts by symplectomorphisms, it follows that is closed (for all in ).

Suppose that is not just closed but also exact, so that for some function . If this holds, then one may choose the to make the map linear. A momentum map for the -action on is a map such that

for all in . Here is the function from to defined by . The momentum map is uniquely defined up to an additive constant of integration (on each connected component).

An -action on a symplectic manifold is called Hamiltonian if it is symplectic and if there exists a momentum map.

A momentum map is often also required to be -equivariant, where acts on via the coadjoint action, and sometimes this requirement is included in the definition of a Hamiltonian group action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the momentum 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 , as first described by Souriau (1970).

Examples of momentum maps

In the case of a Hamiltonian action of the circle , the Lie algebra dual is naturally identified with , and the momentum map is simply the Hamiltonian function that generates the circle action.

Another classical case occurs when is the cotangent bundle of and is the Euclidean group generated by rotations and translations. That is, is a six-dimensional group, the semidirect product of and . The six components of the momentum map are then the three angular momenta and the three linear momenta.

Let be a smooth manifold and let be its cotangent bundle, with projection map . Let denote the tautological 1-form on . Suppose acts on . The induced action of on the symplectic manifold , given by for is Hamiltonian with momentum map for all . Here denotes the contraction of the vector field , the infinitesimal action of , with the 1-form .

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

Some facts about momentum maps

Let be Lie groups with Lie algebras , respectively.

  1. Let be a coadjoint orbit. Then there exists a unique symplectic structure on such that inclusion map is a momentum map.
  2. Let act on a symplectic manifold with a momentum map for the action, and be a Lie group homomorphism, inducing an action of on . Then the action of on is also Hamiltonian, with momentum map given by , where is the dual map to ( denotes the identity element of ). A case of special interest is when is a Lie subgroup of and is the inclusion map.
  3. Let be a Hamiltonian -manifold and a Hamiltonian -manifold. Then the natural action of on is Hamiltonian, with momentum map the direct sum of the two momentum maps and . Here , where denotes the projection map.
  4. Let be a Hamiltonian -manifold, and a submanifold of invariant under such that the restriction of the symplectic form on to is non-degenerate. This imparts a symplectic structure to in a natural way. Then the action of on is also Hamiltonian, with momentum map the composition of the inclusion map with 's momentum map.

Symplectic quotients

Suppose that the action of a Lie group on the symplectic manifold is Hamiltonian, as defined above, with equivariant momentum map . From the Hamiltonian condition, it follows that is invariant under .

Assume now that acts freely and properly on . It follows that is a regular value of , so and its quotient are both smooth manifolds. The quotient inherits a symplectic form from ; that is, there is a unique symplectic form on the quotient whose pullback to equals the restriction of to . Thus, the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, after ( Marsden & Weinstein 1974 ), symplectic quotient, or symplectic reduction of by and is denoted . Its dimension equals the dimension of minus twice the dimension of .

More generally, if G does not act freely (but still properly), then ( Sjamaar & Lerman 1991 ) showed that is a stratified symplectic space, i.e. a stratified space with compatible symplectic structures on the strata.

Flat connections on a surface

The space of connections on the trivial bundle on a surface carries an infinite dimensional symplectic form

The gauge group acts on connections by conjugation . Identify via the integration pairing. Then the map

that sends a connection to its curvature is a moment map for the action of the gauge group on connections. In particular the moduli space of flat connections modulo gauge equivalence is given by symplectic reduction.

See also

Notes

  1. Moment map is a misnomer and physically incorrect. It is an erroneous translation of the French notion application moment. See this mathoverflow question for the history of the name.
  2. 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 )

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