🔵Symplectic Geometry Unit 8 – Moment Maps & Hamiltonian Actions
Moment maps and Hamiltonian actions are powerful tools in symplectic geometry, linking symmetries to conserved quantities. They provide a framework for understanding the dynamics of physical systems and the geometry of symplectic manifolds.
These concepts have wide-ranging applications, from classical mechanics to quantum physics and beyond. By encoding symmetries and conservation laws, moment maps and Hamiltonian actions enable the study of reduced systems and the construction of new symplectic manifolds.
Symplectic manifold (M,ω) consists of an even-dimensional smooth manifold M equipped with a closed, non-degenerate 2-form ω called the symplectic form
Hamiltonian vector field XH associated with a smooth function H:M→R satisfies dH=ιXHω, where ι denotes the interior product
Poisson bracket {f,g} of smooth functions f,g:M→R measures the failure of f and g to Poisson commute and is defined by {f,g}=ω(Xf,Xg)
Satisfies properties such as antisymmetry, bilinearity, and the Jacobi identity
Moment map μ:M→g∗ associated with a Hamiltonian action of a Lie group G on (M,ω) is a smooth map satisfying certain equivariance and compatibility conditions
Components of μ generate the action of G on M via Hamiltonian flows
Coadjoint orbit Oξ of ξ∈g∗ is the orbit of ξ under the coadjoint action of G on g∗ and carries a natural symplectic structure
Marsden-Weinstein reduction allows for the construction of reduced symplectic manifolds by quotienting out the symmetries of a Hamiltonian G-action
Moment Maps: Fundamentals
Moment maps encode symmetries of a symplectic manifold (M,ω) under the action of a Lie group G
Equivariance property μ(g⋅x)=Adg−1∗μ(x) relates the moment map to the coadjoint action of G on g∗
Components μξ(x)=⟨μ(x),ξ⟩ of the moment map, indexed by ξ∈g, are Hamiltonian functions generating the action of G
Hamiltonian vector field Xμξ corresponds to the infinitesimal action of ξ on M
Moment map can be viewed as a collection of conserved quantities associated with the symmetries of the system
Level sets μ−1(c) of the moment map are invariant under the G-action and play a crucial role in symplectic reduction
Existence and uniqueness of moment maps depend on topological conditions such as the triviality of the G-action on H2(M,R)
Hamiltonian Actions Explained
Hamiltonian action of a Lie group G on a symplectic manifold (M,ω) preserves the symplectic form and admits a moment map μ:M→g∗
Infinitesimal generators Xξ of the action, associated with ξ∈g, are Hamiltonian vector fields satisfying dHξ=ιXξω
Hamiltonian functions Hξ are related to the moment map by Hξ=μξ
Orbits of a Hamiltonian action are isotropic submanifolds of (M,ω), meaning the symplectic form vanishes when restricted to the orbits
Transitive Hamiltonian actions correspond to coadjoint orbits in g∗, which inherit a natural symplectic structure
Momentum mapping J:T∗Q→g∗ for a cotangent lift of a G-action on a configuration manifold Q is given by J(αq)(ξ)=αq(ξQ(q)), where ξQ is the infinitesimal generator of the G-action on Q
Symplectic Manifolds and Group Actions
Symplectic manifolds provide a natural framework for studying Hamiltonian systems in classical mechanics
Symplectic form encodes the Poisson bracket structure and the dynamics of the system
Symplectomorphisms are diffeomorphisms that preserve the symplectic form and form a group Symp(M,ω) under composition
Lie group actions on symplectic manifolds come in various flavors: symplectic, Hamiltonian, or multiplicity-free
Symplectic actions preserve the symplectic form but may not admit a moment map
Multiplicity-free actions have orbits of maximal dimension and are closely related to integrable systems
Moment map, when it exists, intertwines the G-action on M with the coadjoint action on g∗
Symplectic quotients M//G=μ−1(0)/G by a Hamiltonian G-action inherit a reduced symplectic structure
Provide a way to construct new symplectic manifolds with reduced symmetry
Examples and Applications
Coadjoint orbits Oξ⊂g∗ carry a natural symplectic structure and are homogeneous Hamiltonian G-spaces
Kirillov-Kostant-Souriau symplectic form on Oξ is given by ωξ(adη∗ξ,adζ∗ξ)=⟨ξ,[η,ζ]⟩
Cotangent bundles T∗Q are symplectic manifolds with the canonical symplectic form ω=dθ, where θ is the Liouville 1-form
Cotangent lift of a G-action on Q is Hamiltonian with moment map J:T∗Q→g∗
Symplectic toric manifolds are compact connected symplectic manifolds equipped with an effective Hamiltonian action of a torus Tn of half the dimension
Classified by their moment polytopes, convex polytopes in Rn obtained as the image of the moment map
Hamiltonian actions and moment maps play a key role in the study of integrable systems, such as the Calogero-Moser system and the Toda lattice
Applications in geometric quantization, where the moment map helps define a prequantum line bundle and a polarization
Theoretical Results and Proofs
Atiyah-Guillemin-Sternberg convexity theorem states that the image of the moment map for a compact connected Hamiltonian G-space is a convex polytope, the convex hull of the images of fixed points
Marsden-Weinstein reduction theorem constructs symplectic quotients M//G=μ−1(0)/G for Hamiltonian G-actions with proper moment maps
Reduced spaces are symplectic manifolds of dimension dimM−2dimG
Kirwan surjectivity theorem asserts that the moment map for a compact connected Hamiltonian G-space is surjective onto a convex polytope
Delzant's theorem characterizes compact connected symplectic toric manifolds in terms of their moment polytopes, which are Delzant polytopes
Duistermaat-Heckman theorem relates the pushforward of the Liouville measure under the moment map to the stationary phase approximation of certain oscillatory integrals
Provides a way to compute volumes of reduced spaces and intersection pairings on symplectic quotients
Computational Techniques
Numerical methods for computing moment maps and their images, such as discretization of the symplectic manifold and approximation of the group action
Algorithms for constructing symplectic quotients and reduced spaces, such as the Marle-Guillemin-Sternberg normal form and the Meyer-Marsden-Weinstein reduction
Computational algebraic geometry techniques for studying moment polytopes and their combinatorial properties
Software packages like polymake and Macaulay2 for manipulating polytopes and solving systems of polynomial equations
Finite-dimensional approximation schemes for infinite-dimensional Hamiltonian systems, such as the Rayleigh-Ritz method and the Galerkin method
Numerical integration of Hamiltonian systems using symplectic integrators that preserve the symplectic structure and the symmetries of the system
Examples include the Störmer-Verlet method and the Lobatto IIIA-IIIB pair
Connections to Other Areas
Relation to geometric invariant theory (GIT) and the Kempf-Ness theorem, which establishes a correspondence between symplectic quotients and GIT quotients
Moment maps in Kähler geometry, where they are related to the scalar curvature and the Futaki invariant
Calabi conjecture and the existence of constant scalar curvature Kähler metrics
Role in the geometric Langlands program, where moment maps are used to construct the Hitchin system and the Hitchin fibration
Applications in mathematical physics, such as gauge theory, string theory, and the study of moduli spaces of flat connections
Connections to representation theory through the orbit method and the geometric quantization of coadjoint orbits
Kirillov's character formula and the Duistermaat-Heckman formula for the Fourier transform of the pushforward of the Liouville measure
Relation to integrable systems and the Arnold-Liouville theorem, which characterizes completely integrable Hamiltonian systems in terms of their moment maps