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.

Formal definition

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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

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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

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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

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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

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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

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Notes

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  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)

References

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  • 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. Vol. 222. Birkhauser Boston. ISBN 0-8176-4307-9.
  • Audin, Michèle (2004), Torus actions on symplectic manifolds, Progress in Mathematics, vol. 93 (Second revised ed.), Birkhäuser, ISBN 3-7643-2176-8
  • Guillemin, Victor; Sternberg, Shlomo (1990), Symplectic techniques in physics (Second ed.), Cambridge University Press, ISBN 0-521-38990-9
  • Woodward, Chris (2010), Moment maps and geometric invariant theory, Les cours du CIRM, vol. 1, EUDML, pp. 55–98, arXiv:0912.1132, Bibcode:2009arXiv0912.1132W
  • Bruguières, Alain (1987), "Propriétés de convexité de l'application moment" (PDF), Astérisque, Séminaire Bourbaki, 145–146: 63–87
  • Marsden, Jerrold; Weinstein, Alan (1974), "Reduction of symplectic manifolds with symmetry", Reports on Mathematical Physics, 5 (1): 121–130, Bibcode:1974RpMP....5..121M, doi:10.1016/0034-4877(74)90021-4
  • Sjamaar, Reyer; Lerman, Eugene (1991), "Stratified symplectic spaces and reduction", Annals of Mathematics, 134 (2): 375–422, doi:10.2307/2944350, JSTOR 2944350