Momentum map – Wikipedia

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[edit]

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

${displaystyle {mathfrak {g}}}$

be the Lie algebra of G,

${displaystyle {mathfrak {g}}^{*}}$

its dual, and

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

the pairing between the two. Any ξ in

${displaystyle {mathfrak {g}}}$

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

${displaystyle rho (xi )_{x}}$

is

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

where

${displaystyle exp :{mathfrak {g}}to G}$

is the exponential map and

${displaystyle cdot }$

denotes the G-action on M.^[2] Let

${displaystyle iota _{rho (xi )}omega ,}$

denote the contraction of this vector field with ω. Because G acts by symplectomorphisms, it follows that

${displaystyle iota _{rho (xi )}omega ,}$

is closed (for all ξ in

${displaystyle {mathfrak {g}}}$

).

Suppose that

${displaystyle iota _{rho (xi )}omega ,}$

is not just closed but also exact, so that

${displaystyle iota _{rho (xi )}omega =dH_{xi }}$

for some function

${displaystyle H_{xi }:Mto mathbb {R} }$

. If this holds, then one may choose the

${displaystyle H_{xi }}$

to make the map

${displaystyle xi mapsto H_{xi }}$

linear. A momentum map for the G-action on (M, ω) is a map

${displaystyle mu :Mto {mathfrak {g}}^{*}}$

such that

${displaystyle d(langle mu ,xi rangle )=iota _{rho (xi )}omega }$

for all ξ in

${displaystyle {mathfrak {g}}}$

. Here

${displaystyle langle mu ,xi rangle }$

is the function from M to R defined by

${displaystyle langle mu ,xi rangle (x)=langle mu (x),xi rangle }$

. The momentum map is uniquely defined up to an additive constant of integration (on each connected component).

An

${displaystyle G}$

-action on a symplectic manifold

${displaystyle (M,omega )}$

is called Hamiltonian if it is symplectic and if there exists a momentum map.

A momentum map is often also required to be

${displaystyle G}$

-equivariant, where G acts on

${displaystyle {mathfrak {g}}^{*}}$

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

${displaystyle {mathfrak {g}}^{*}}$

, as first described by Souriau (1970).

Examples of momentum maps[edit]

In the case of a Hamiltonian action of the circle

${displaystyle G=U(1)}$

, the Lie algebra dual

${displaystyle {mathfrak {g}}^{*}}$

is naturally identified with

${displaystyle mathbb {R} }$

, and the momentum map is simply the Hamiltonian function that generates the circle action.

Another classical case occurs when

${displaystyle M}$

is the cotangent bundle of

${displaystyle mathbb {R} ^{3}}$

and

${displaystyle G}$

is the Euclidean group generated by rotations and translations. That is,

${displaystyle G}$

is a six-dimensional group, the semidirect product of

${displaystyle SO(3)}$

and

${displaystyle mathbb {R} ^{3}}$

. The six components of the momentum map are then the three angular momenta and the three linear momenta.

Let

${displaystyle N}$

be a smooth manifold and let

${displaystyle T^{*}N}$

be its cotangent bundle, with projection map

${displaystyle pi :T^{*}Nrightarrow N}$

. Let

${displaystyle tau }$

denote the tautological 1-form on

${displaystyle T^{*}N}$

. Suppose

${displaystyle G}$

acts on

${displaystyle N}$

. The induced action of

${displaystyle G}$

on the symplectic manifold

${displaystyle (T^{*}N,mathrm {d} tau )}$

, given by

${displaystyle gcdot eta :=(T_{pi (eta )}g^{-1})^{*}eta }$

for

${displaystyle gin G,eta in T^{*}N}$

is Hamiltonian with momentum map

${displaystyle -iota _{rho (xi )}tau }$

for all

${displaystyle xi in {mathfrak {g}}}$

. Here

${displaystyle iota _{rho (xi )}tau }$

denotes the contraction of the vector field

${displaystyle rho (xi )}$

, the infinitesimal action of

${displaystyle xi }$

, with the 1-form

${displaystyle tau }$

.

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

Some facts about momentum maps[edit]

Let

${displaystyle G,H}$

be Lie groups with Lie algebras

${displaystyle {mathfrak {g}},{mathfrak {h}}}$

, respectively.

1. Let
   ${displaystyle {mathcal {O}}(F),Fin {mathfrak {g}}^{*}}$

   be a coadjoint orbit. Then there exists a unique symplectic structure on

   ${displaystyle {mathcal {O}}(F)}$

   such that inclusion map

   ${displaystyle {mathcal {O}}(F)hookrightarrow {mathfrak {g}}^{*}}$

   is a momentum map.

2. Let
   ${displaystyle G}$

   act on a symplectic manifold

   ${displaystyle (M,omega )}$

   with

   ${displaystyle Phi _{G}:Mrightarrow {mathfrak {g}}^{*}}$

   a momentum map for the action, and

   ${displaystyle psi :Hrightarrow G}$

   be a Lie group homomorphism, inducing an action of

   ${displaystyle H}$

   on

   ${displaystyle M}$

   . Then the action of

   ${displaystyle H}$

   on

   ${displaystyle M}$

   is also Hamiltonian, with momentum map given by

   ${displaystyle (mathrm {d} psi )_{e}^{*}circ Phi _{G}}$

   , where

   ${displaystyle (mathrm {d} psi )_{e}^{*}:{mathfrak {g}}^{*}rightarrow {mathfrak {h}}^{*}}$

   is the dual map to

   ${displaystyle (mathrm {d} psi )_{e}:{mathfrak {h}}rightarrow {mathfrak {g}}}$

   (

   ${displaystyle e}$

   denotes the identity element of

   ${displaystyle H}$

   ). A case of special interest is when

   ${displaystyle H}$

   is a Lie subgroup of

   ${displaystyle G}$

   and

   ${displaystyle psi }$

   is the inclusion map.

3. Let
   ${displaystyle (M_{1},omega _{1})}$

   be a Hamiltonian

   ${displaystyle G}$

   -manifold and

   ${displaystyle (M_{2},omega _{2})}$

   a Hamiltonian

   ${displaystyle H}$

   -manifold. Then the natural action of

   ${displaystyle Gtimes H}$

   on

   ${displaystyle (M_{1}times M_{2},omega _{1}times omega _{2})}$

   is Hamiltonian, with momentum map the direct sum of the two momentum maps

   ${displaystyle Phi _{G}}$

   and

   ${displaystyle Phi _{H}}$

   . Here

   ${displaystyle omega _{1}times omega _{2}:=pi _{1}^{*}omega _{1}+pi _{2}^{*}omega _{2}}$

   , where

   ${displaystyle pi _{i}:M_{1}times M_{2}rightarrow M_{i}}$

   denotes the projection map.

4. Let
   ${displaystyle M}$

   be a Hamiltonian

   ${displaystyle G}$

   -manifold, and

   ${displaystyle N}$

   a submanifold of

   ${displaystyle M}$

   invariant under

   ${displaystyle G}$

   such that the restriction of the symplectic form on

   ${displaystyle M}$

   to

   ${displaystyle N}$

   is non-degenerate. This imparts a symplectic structure to

   ${displaystyle N}$

   in a natural way. Then the action of

   ${displaystyle G}$

   on

   ${displaystyle N}$

   is also Hamiltonian, with momentum map the composition of the inclusion map with

   ${displaystyle M}$

   ‘s momentum map.

Symplectic quotients[edit]

Suppose that the action of a Lie group G on the symplectic manifold (M, ω) is Hamiltonian, as defined above, with equivariant momentum map

${displaystyle mu :Mto {mathfrak {g}}^{*}}$

. From the Hamiltonian condition, it follows that

${displaystyle mu ^{-1}(0)}$

is invariant under G.

Assume now that G acts freely and properly on

${displaystyle mu ^{-1}(0)}$

. It follows that 0 is a regular value of

${displaystyle mu }$

, so

${displaystyle mu ^{-1}(0)}$

and its quotient

${displaystyle mu ^{-1}(0)/G}$

are both smooth manifolds. The quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to

${displaystyle mu ^{-1}(0)}$

equals the restriction of ω to

${displaystyle mu ^{-1}(0)}$

. Thus, the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, after (Marsden & Weinstein 1974), symplectic quotient, or symplectic reduction of M by G and is denoted

${displaystyle M/!!/G}$

. Its dimension equals the dimension of M minus twice the dimension of G.

More generally, if G does not act freely (but still properly), then (Sjamaar & Lerman 1991) showed that

${displaystyle M/!!/G=mu ^{-1}(0)/G}$

is a stratified symplectic space, i.e. a stratified space with compatible symplectic structures on the strata.

Flat connections on a surface[edit]

The space

${displaystyle Omega ^{1}(Sigma ,{mathfrak {g}})}$

of connections on the trivial bundle

${displaystyle Sigma times G}$

on a surface carries an infinite dimensional symplectic form

${displaystyle langle alpha ,beta rangle :=int _{Sigma }{text{tr}}(alpha wedge beta ).}$

The gauge group

${displaystyle {mathcal {G}}={text{Map}}(Sigma ,G)}$

acts on connections by conjugation

${displaystyle gcdot A:=g^{-1}(dg)+g^{-1}Ag}$

. Identify

${displaystyle {text{Lie}}({mathcal {G}})=Omega ^{0}(Sigma ,{mathfrak {g}})=Omega ^{2}(Sigma ,{mathfrak {g}})^{*}}$

via the integration pairing. Then the map

${displaystyle mu :Omega ^{1}(Sigma ,{mathfrak {g}})rightarrow Omega ^{2}(Sigma ,{mathfrak {g}}),qquad A;mapsto ;F:=dA+{frac {1}{2}}[Awedge A]}$

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

${displaystyle mu ^{-1}(0)/{mathcal {G}}=Omega ^{1}(Sigma ,{mathfrak {g}})/!!/{mathcal {G}}}$

is given by symplectic reduction.

See also[edit]

References[edit]

• 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
• Sjamaar, Reyer; Lerman, Eugene (1991), “Stratified symplectic spaces and reduction”, Annals of Mathematics, 134 (2): 375–422