The moment map

Idea

A moment map is a dual incarnation of a Hamiltonian action of a Lie group (or Lie algebra) on a symplectic manifold.

An action of a Lie group $G$ on a symplectic manifold $X$ by (Hamiltonian) symplectomorphisms corresponds infinitesimally to a Lie algebra homomorphism from the Lie algebra $𝔤$ to the Hamiltonian vector fields on $X$. If this lifts to a coherent choice of Hamiltonians, hence to a Lie algebra homomorphism $𝔤\to (C^{\mathrm{\infty }}(X),\{-,-\})$ to the Poisson bracket, then, by dualization, this is equivalently a Poisson homomorphism

This is called the moment map or momentum map of the Hamiltonian action.

The name derives from the special and historically first case of angular momentum in the dynamics of rigid bodies, see Examples - Angular momentum below.

Definition

The Preliminaries below review some basics of Hamiltonian vector fields. The definition of the moment map itself is below in Hamiltonian action and the moment map.

Preliminaries

This section briefly reviews the notion of Hamiltonian vector fields on a symplectic manifold

The basic setup is the following: Let $(M,\omega )$ be a symplectic manifold with a Hamiltonian action of a Lie group $G$. In particular that means that there is an action $\nu :G\times M\to M$ via symplectomorphisms (diffeomorphisms ${\nu }_g$ such that ${\nu }_g^{*}(\omega )=\omega$). A vector field $X$ is symplectic if the corresponding flow preserves (again by pullbacks) $\omega$. The symplectic vector fields form a Lie subalgebra $\chi (M,\omega )$ of the Lie algebra of all smooth vector fields $\chi (M)$ on $M$ with respect to the Lie bracket.

By the Cartan homotopy formula and closedness $d\omega =0$

where ${ℒ}_X$ denotes the Lie derivative. Therefore a vector field $X$ is symplectic iff $\iota (X)\omega =dH$ for some function $H\in C^{\mathrm{\infty }}(M)$, usually called Hamiltonian (function) for $X$. Here $X$ is determined by $H$ up to a locally constant function. Such $X=X_H$ is called the Hamiltonian vector field corresponding to $H$. The Poisson structure on $M$ is the bracket $\{,\}$ on functions may be given by

where there is a Lie bracket of vector fields on the right hand side.

For $(M,\omega )$ a connected symplectic manifold, there is an exact sequence of Lie algebras

See at Hamiltonian vector field – Relation to Poisson bracket.

Hamiltonian action and moment map

Let $(X,\omega )$ be a symplectic manifold and let $𝔤$ be a Lie algebra. Write $(C^{\mathrm{\infty }}(X),\{-,-\})$ for the Poisson bracket Lie algebra underlying the corresponding Poisson algebra.

Examples

Angular moment

(…)

Properties

Relation to constrained mechanics

In the context of constrained mechanics? the components of the moment map (as the Lie algebra argument varies) are called first class constraints. See symplectic reduction for more.

Related concepts

The moment map is a crucial ingredient in the construction of Marsden–Weinstein symplectic quotients and in other variants of symplectic reduction.

References

General

Lecture notes and surveys include

• Joel W. Robbin, The moment map, an exposition, pdf

• Nicole Berline, Michèle Vergne, Hamiltonian manifolds and moment maps (pdf)

Original articles include

• Victor Guillemin, Shlomo Sternberg, Geometric asymptotics, AMS (1977) (online)

• Michael Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.

• Michael Atiyah, Raoul Bott, The moment map and equivariant cohomology, Topology, Vol 23, No. 1 (1984) (pdf)

Further developments are in

• M. Spera, On a generalized uncertainty principle, coherent states and the moment map, J. of Geometry and Physics 12 (1993) 165-182, MR94m:58097, doi

• Ctirad Klimčík, Pavol Ševera, T-duality and the moment map, IHES/P/96/70, hep-th/9610198; Poisson-Lie T-duality: open strings and D-branes, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, hep-th/9512124

• A. Cannas da Silva, Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Math. Lec. Notes Series, AMS 1999, (pdf)

• Friedrich Knop, Automorphisms of multiplicity free Hamiltonian manifolds, arxiv/1002.4256

• W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257-293.

See also

• wikipedia, moment map

Moment maps in higher geometry, Higher geometric prequantum theory, are discussed in

• Yael Fregier, Chris Rogers, Marco Zambon, Homotopy moment maps (arXiv:1304.2051)

Relation to symplectic reduction

Reviews include for instance

p. 26 of

• José Figueroa-O’Farrill, chapter 2+3 of PhD thesis (pdf)

Generalization: group-valued moment maps

• Anton Alekseev, Anton Malkin, Eckhard Meinrenken, Lie group valued moment maps, J. Differential Geom. Volume 48, Number 3 (1998), 445-495. euclid, MR1638045
• E. Meinrenken, Lectures on group-valued moment maps and Verlinde formulas, 35 pages, January 2012, pdf