Moment map

Intuition [Beginner]

Moment map is a map whose components generate an infinitesimal group action. It gives a geometric rule for motion, constraint, or size without choosing ordinary distances as the main object.

The first picture is phase space: position and momentum are paired. A symplectic structure records how those pairs rotate into motion. It is less like a ruler and more like a turning rule.

This idea matters because Hamiltonian mechanics, reduction, and Floer theory all use the same pairing language.

Visual [Beginner]

The diagram shows a surface with arrows and level curves. It is a mnemonic for the way symplectic geometry ties motion to paired directions.

The picture is not a coordinate proof. It marks the objects that the formal definition makes precise.

Worked example [Beginner]

Use the plane with coordinates called position and momentum. A point records both where something is and how strongly it is moving.

For the energy rule "half position squared plus half momentum squared," the level curves are circles. The motion follows those circles instead of moving straight toward lower energy.

At the point with position 1 and momentum 0, the motion points in the momentum direction. After a quarter turn, the roles have exchanged.

What this tells us: symplectic geometry turns an energy rule into organized motion.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ denote a smooth manifold or $V$ a finite-dimensional vector space, according to the context. The prerequisites used here are 03.03.02, 05.02.01, 05.02.02. The concept moment map is the structure described by the following data: a symplectic form $\omega$, compatible maps or subspaces, and the equations preserving the relevant pairing.

For this unit, the sign convention is the geometric Hamiltonian convention

$$\omega (X_H,-)=dH.$$

When the unit is purely linear, replace differential forms by skew bilinear forms. When a group acts, infinitesimal generators are paired with Hamiltonian functions through the same equation ^{[Cannas da Silva §22]}.

A morphism between such structures is a smooth or linear map preserving the specified symplectic data. This preservation condition is the source of rigidity results absent from ordinary volume geometry.

Key theorem with proof [Intermediate+]

Theorem (three-condition characterisation of a moment map). Let a Lie group $G$ act on the symplectic manifold $(M,\omega )$ by symplectomorphisms, with infinitesimal action $\xi \mapsto X_{\xi }$ realised as a Lie-algebra homomorphism $\mathfrak{g}\to \mathfrak{X}(M)$, $[X_{\xi },X_{\eta }]=X_{[\xi ,\eta ]}$. For a smooth map $\mu : M \to \mathfrak{g}^$write$\mu^\xi(p) := \langle \mu(p), \xi\rangle$for$\xi \in \mathfrak{g}$. Consider the conditions:*

(C1) For every $\xi \in \mathfrak{g}$, ${\iota }_{X_{\xi }}\omega =d{\mu }^{\xi }$ (equivalently, $\omega (X_{\xi },-)=d{\mu }^{\xi }$).

(C2) $G$-equivariance: $\mu \circ g = \mathrm{Ad}^_g \circ \mu$forevery$g \in G$.*

(C3) The comoment is a Lie homomorphism: $\{{\mu }^{\xi },{\mu }^{\eta }\}={\mu }^{[\xi ,\eta ]}$ for every $\xi ,\eta \in \mathfrak{g}$.

Then (C1) makes ${\mu }^{\xi }$ a Hamiltonian function for $X_{\xi }$, and assuming (C1), the equivariance (C2) implies (C3); conversely, when $G$ is connected, (C3) implies (C2). When $G$ is disconnected, (C3) is equivalent to (C2) on the identity component $G^{\circ }$ and is the more general condition.

Proof. The sign convention is that of 05.02.01: ${\iota }_{X_H}\omega =dH$, with the induced Poisson bracket $\{f,g\}:=\omega (X_f,X_g)=X_g(f)$ from 05.02.02. By (C1), $X_{{\mu }^{\xi }}=X_{\xi }$ because $\omega$ is non-degenerate and both vector fields contract with $\omega$ to the same one-form $d{\mu }^{\xi }$. Hence each component ${\mu }^{\xi }$ is a Hamiltonian function for the fundamental field $X_{\xi }$.

(C2) ⇒ (C3). Fix $\xi ,\eta \in \mathfrak{g}$ and $p\in M$. Equivariance (C2) along the one-parameter subgroup $\exp (t\xi )$ reads

$${\mu }^{\eta }(\exp (t\xi )\cdot p)=⟨{\mathrm{Ad}}_{\exp (t\xi )}^{\ast }\mu (p),\eta ⟩=⟨\mu (p),{\mathrm{Ad}}_{\exp (-t\xi )}\eta ⟩.$$

Differentiating at $t=0$ on both sides: the left side gives $X_{\xi }({\mu }^{\eta })(p)$ by definition of the fundamental field, while the right side gives $⟨\mu (p),-[\xi ,\eta ]⟩=-{\mu }^{[\xi ,\eta ]}(p)$ from $\frac{d}{dt}{|}_0{\mathrm{Ad}}_{\exp (-t\xi )}\eta =-[\xi ,\eta ]$. The infinitesimal form of equivariance is therefore

$$X_{\xi }({\mu }^{\eta })=-{\mu }^{[\xi ,\eta ]}={\mu }^{[\eta ,\xi ]}.$$

Combine with (C1). Using $X_{{\mu }^{\eta }}=X_{\eta }$ and the Poisson-bracket convention $\{f,g\}=X_g(f)$:

$$\{{\mu }^{\xi },{\mu }^{\eta }\}=X_{{\mu }^{\eta }}({\mu }^{\xi })=X_{\eta }({\mu }^{\xi })=-{\mu }^{[\eta ,\xi ]}={\mu }^{[\xi ,\eta ]},$$

which is (C3). The same identity reads in symplectic-form language as $\{{\mu }^{\xi },{\mu }^{\eta }\}=\omega (X_{{\mu }^{\xi }},X_{{\mu }^{\eta }})=\omega (X_{\xi },X_{\eta })=({\iota }_{X_{\xi }}\omega )(X_{\eta })=d{\mu }^{\xi }(X_{\eta })=X_{\eta }({\mu }^{\xi })$, recovering the same chain.

(C3) ⇒ (C2) on $G^{\circ }$. Conversely, assume (C3) and (C1). The chain run above in reverse gives $X_{\xi }({\mu }^{\eta })=\{{\mu }^{\eta },{\mu }^{\xi }\}=-\{{\mu }^{\xi },{\mu }^{\eta }\}=-{\mu }^{[\xi ,\eta ]}$ for every $\xi ,\eta$. Read coordinate-free, this says

$$X_{\xi }(\mu )=-{\mathrm{ad}}_{\xi }^{\ast }(\mu )\text{as smooth maps }M\to {\mathfrak{g}}^{\ast },$$

where ${\mathrm{ad}}_{\xi }^{\ast }:{\mathfrak{g}}^{\ast }\to {\mathfrak{g}}^{\ast }$ is the coadjoint representation defined by $⟨{\mathrm{ad}}_{\xi }^{\ast }\nu ,\eta ⟩=⟨\nu ,[\xi ,\eta ]⟩$. Fix $\xi \in \mathfrak{g}$ and $p\in M$, and set

$$\nu (t):=\mu (\exp (t\xi )\cdot p),\rho (t):={\mathrm{Ad}}_{\exp (t\xi )}^{\ast }\mu (p).$$

Both are paths in ${\mathfrak{g}}^{\ast }$ with $\nu (0)=\rho (0)=\mu (p)$. The derivative of $\nu$ is ${\nu }^{\prime }(t)=X_{\xi }(\mu )(\exp (t\xi )\cdot p)=-{\mathrm{ad}}_{\xi }^{\ast }\nu (t)$. The derivative of $\rho$ comes from $\frac{d}{dt}{\mathrm{Ad}}_{\exp (t\xi )}^{\ast }={\mathrm{Ad}}_{\exp (t\xi )}^{\ast }\circ (-{\mathrm{ad}}_{\xi }^{\ast })=-{\mathrm{ad}}_{\xi }^{\ast }\circ {\mathrm{Ad}}_{\exp (t\xi )}^{\ast }$ (the coadjoint action commutes with itself in the obvious way), giving ${\rho }^{\prime }(t)=-{\mathrm{ad}}_{\xi }^{\ast }\rho (t)$. Both paths satisfy the same linear ODE $\dot{\zeta }=-{\mathrm{ad}}_{\xi }^{\ast }\zeta$ with the same initial condition; by uniqueness, $\nu (t)=\rho (t)$ for all $t$. This is (C2) for every group element $\exp (t\xi )$.

Every element of $G^{\circ }$ is a finite product of one-parameter-subgroup elements, and the equivariance condition is preserved under products because ${\mathrm{Ad}}^{\ast }$ is a homomorphism: if $\mu (g\cdot p)={\mathrm{Ad}}_g^{\ast }\mu (p)$ and $\mu (h\cdot q)={\mathrm{Ad}}_h^{\ast }\mu (q)$ for all $p,q$, then applying the first with $p$ replaced by $h\cdot p$ and using the second gives $\mu ((gh)\cdot p)={\mathrm{Ad}}_{gh}^{\ast }\mu (p)$. Hence (C2) holds on $G^{\circ }$. $\square$

Bridge. The three-condition characterisation is the analytic backbone for 05.04.02 (Marsden-Weinstein reduction): condition (C1) makes ${\mu }^{-1}(0)$ the locus where the fundamental fields $X_{\xi }$ are tangent to the level set, and condition (C3) is exactly the Lie-algebra homomorphism that makes the reduced bracket on $C^{\infty }(M//G)$ well-defined. The same package reappears in 05.04.03 (Atiyah-Guillemin-Sternberg convexity) and 05.03.01 (coadjoint orbits) as the Hamiltonian-action structure they consume; putting these together, the foundational insight is that a moment map is exactly a $G$-equivariant Lie homomorphism $\mathfrak{g}\to (C^{\infty }(M),\{\cdot ,\cdot \})$ valued in the symplectic Hamiltonians, and every downstream theorem in the strand reads this structure off.

Exercises [Intermediate+]

Advanced results [Master]

The construction of moment map is invariant under symplectomorphism. In local Darboux coordinates, the form is modeled by

$${\omega }_0=\sum _{i}d{q}_i\wedge d{p}_i,$$

and global information is carried by the way these local models are glued. This separation between local normal form and global obstruction is a recurring feature of the subject ^{[Cannas da Silva §22]}.

For Hamiltonian group actions, the infinitesimal action, moment map, and Poisson bracket form one algebraic package. The identity $\omega (X_H,-)=dH$ converts functions into vector fields, and equivariance converts Lie brackets into Poisson brackets. Reduction, coadjoint orbits, and Floer complexes are built from this package.

Compactness and transversality questions enter when one counts trajectories or curves. In the finite-dimensional part of the strand, the essential inputs are closedness, nondegeneracy, and regular-value hypotheses. In Floer-theoretic units, analytic compactness replaces finite-dimensional regularity.

Synthesis. The moment map $\mu :M\to {\mathfrak{g}}^{\ast }$ for a Hamiltonian $G$-action on $(M,\omega )$ is the equivariant map whose components ${\mu }^{\xi }=H_{\xi }$ are the Hamiltonian functions generating the infinitesimal action of each $\xi \in \mathfrak{g}$, so the moment map records how the symmetry group acts on the phase space through conserved quantities. Convexity of the moment map image (Atiyah–Guillemin–Sternberg) forces the image of a compact Hamiltonian torus action to be a convex polytope, and this polytope completely determines $(M,\omega )$ up to equivariant symplectomorphism — symplectic geometry becomes combinatorics. The moment map is the organising tool for symplectic reduction (quotienting by the symmetry), the Duistermaat–Heckman formula (computing oscillatory integrals via polytope geometry), and the Gauges-theoretic construction of moduli spaces in gauge theory.

Full proof set [Master]

Proposition. Hamiltonian vector fields preserve the symplectic form.

Let $X_H$ be defined by ${\iota }_{X_H}\omega =dH$. Cartan's formula gives

$${\mathcal{L}}_{X_H}\omega =d{\iota }_{X_H}\omega +{\iota }_{X_H}d\omega =d(dH)+0=0.$$

Hence the derivative of ${\Phi }_t^{\ast }\omega$ along the Hamiltonian flow is zero, and ${\Phi }_t^{\ast }\omega =\omega$ whenever the flow is defined.

Proposition. A symplectic linear map preserves symplectic orthogonals.

Let $A:V\to V$ satisfy $\omega (Av,Aw)=\omega (v,w)$. If $x\in W^{\omega }$, then $\omega (Ax,Aw)=\omega (x,w)=0$ for every $w\in W$. Hence $Ax\in (AW{)}^{\omega }$. Applying the same argument to $A^{-1}$ gives equality.

Connections [Master]

• The smooth-manifold language comes from 03.02.01, and differential forms enter through 03.04.02.

• The closedness condition uses exterior derivative 03.04.04 and feeds de Rham cohomology 03.04.06.

• This unit connects directly to 03.03.02, 05.02.01, and 05.02.02 inside the symplectic strand.

• Kempf-Ness theorem and the GIT-symplectic dictionary 04.10.04. The moment map developed here is the symplectic-geometric input to the Kempf-Ness dictionary, which identifies the algebraic GIT quotient $X//G$ of a complex reductive group with the symplectic reduction ${\mu }^{-1}(0)/K$ at level zero. The moment-zero locus is the locus of GIT-minimum-norm representatives, and every concrete moduli space built by GIT acquires a Kähler structure through this identification.

• Hamiltonian action principles also connect to variational calculus 03.04.08.

Historical & philosophical context [Master]

Hamiltonian mechanics supplied the original phase-space formalism, with canonical coordinates and the pairing of position and momentum. Poincare's qualitative theory of dynamical systems and Arnold's geometric mechanics placed this formalism in the language of manifolds and differential forms ^[Arnold].

Gromov's 1985 introduction of pseudoholomorphic curves changed symplectic topology by producing global rigidity phenomena not visible from Darboux's local theorem ^{[Gromov 1985]}. Floer's work later adapted infinite-dimensional Morse theory to Hamiltonian fixed points and Lagrangian intersections ^{[Floer original papers]}.

Bibliography [Master]

@book{CannasDaSilvaSymplectic,
  author = {Cannas da Silva, Ana},
  title = {Lectures on Symplectic Geometry},
  publisher = {Springer},
  year = {2001}
}

@book{ArnoldMechanics,
  author = {Arnold, V. I.},
  title = {Mathematical Methods of Classical Mechanics},
  publisher = {Springer},
  year = {1989}
}

@article{Gromov1985PseudoHolomorphic,
  author = {Gromov, M.},
  title = {Pseudo holomorphic curves in symplectic manifolds},
  journal = {Inventiones Mathematicae},
  volume = {82},
  year = {1985},
  pages = {307--347}
}