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 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.
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 formula_1 be the Lie algebra of "G", formula_2 its dual, and
the pairing between the two. Any ξ in formula_1 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 formula_5 is
where formula_7 is the exponential map and formula_8 denotes the "G"-action on "M". Let formula_9 denote the contraction of this vector field with ω. Because "G" acts by symplectomorphisms, it follows that formula_9 is closed (for all ξ in formula_1).
Suppose that formula_9 is not just closed but also exact, so that formula_13 for some function formula_14. Suppose also that the map formula_15 sending formula_16 is a Lie algebra homomorphism. Then a momentum map for the "G"-action on ("M", ω) is a map formula_17 such that
for all ξ in formula_1. Here formula_20 is the function from "M" to R defined by formula_21. The momentum map is uniquely defined up to an additive constant of integration.
A momentum map is often also required to be "G"-equivariant, where "G" acts on formula_2 via the coadjoint 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 formula_2, as first described by Souriau (1970).
Hamiltonian group actions.
The definition of the momentum map requires formula_24 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 formula_1 the one-form formula_24 is exact, meaning that it equals formula_27 for some smooth function
If this holds, then one may choose the formula_14 to make the map formula_16 linear. The second requirement for the "G"-action to be Hamiltonian is that the map formula_16 be a Lie algebra homomorphism from formula_1 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 momentum map is a map formula_33 such that writing formula_34 defines a Lie algebra homomorphism formula_16 satisfying formula_36. Here formula_37 is the vector field of the Hamiltonian formula_14, defined by
Examples of momentum maps.
In the case of a Hamiltonian action of the circle formula_40, the Lie algebra dual formula_2 is naturally identified with formula_42, and the momentum map is simply the Hamiltonian function that generates the circle action.
Another classical case occurs when formula_43 is the cotangent bundle of formula_44 and formula_45 is the Euclidean group generated by rotations and translations. That is, formula_45 is a six-dimensional group, the semidirect product of formula_47 and formula_44. The six components of the momentum map are then the three angular momenta and the three linear momenta.
Let formula_49 be a smooth manifold and let formula_50 be its cotangent bundle, with projection map formula_51. Let formula_52 denote the tautological 1-form on formula_50. Suppose formula_45 acts on formula_49. The induced action of formula_45 on the symplectic manifold formula_57, given by formula_58 for formula_59 is Hamiltonian with momentum map formula_60 for all formula_61. Here formula_62 denotes the contraction of the vector field formula_63, the infinitesimal action of formula_64, with the 1-form formula_52.
The facts mentioned below may be used to generate more examples of momentum maps.
Some facts about momentum maps.
Let formula_66 be Lie groups with Lie algebras formula_67, respectively.
1. Let formula_68 be a coadjoint orbit. Then there exists a unique symplectic structure on formula_69 such that inclusion map formula_70 is a momentum map.
2. Let formula_45 act on a symplectic manifold formula_72 with formula_73 a momentum map for the action, and formula_74 be a Lie group homomorphism, inducing an action of formula_75 on formula_43. Then the action of formula_75 on formula_43 is also Hamiltonian, with momentum map given by formula_79, where formula_80 is the dual map to formula_81 (formula_82 denotes the identity element of formula_75). A case of special interest is when formula_75 is a Lie subgroup of formula_45 and formula_86 is the inclusion map.
3. Let formula_87 be a Hamiltonian formula_45-manifold and formula_89 a Hamiltonian formula_75-manifold. Then the natural action of formula_91 on formula_92 is Hamiltonian, with momentum map the direct sum of the two momentum maps formula_93 and formula_94. Here formula_95, where formula_96 denotes the projection map.
4. Let formula_43 be a Hamiltonian formula_45-manifold, and formula_49 a submanifold of formula_43 invariant under formula_45 such that the restriction of the symplectic form on formula_43 to formula_49 is non-degenerate. This imparts a symplectic structure to formula_49 in a natural way. Then the action of formula_45 on formula_49 is also Hamiltonian, with momentum map the composition of the inclusion map with formula_43's momentum 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 momentum map formula_33. From the Hamiltonian condition it follows that formula_109 is invariant under "G".
Assume now that 0 is a regular value of μ and that "G" acts freely and properly on formula_109. Thus formula_109 and its quotient formula_112 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 formula_109 equals the restriction of ω to formula_109. Thus the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, symplectic quotient or symplectic reduction of "M" by "G" and is denoted formula_115. Its dimension equals the dimension of "M" minus twice the dimension of "G".
Flat Connections on a Surface.
The space formula_116 of connections on the trivial bundle formula_117 on a surface carries an infinite dimensional symplectic form
The gauge group formula_119 acts on connections by conjugation formula_120. Identify formula_121 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 formula_123 is given by symplectic reduction.