🔵Symplectic Geometry Unit 8 – Moment Maps & Hamiltonian Actions

Key Concepts and Definitions

• Symplectic manifold $(M,\omega)$ consists of an even-dimensional smooth manifold $M$ equipped with a closed, non-degenerate 2-form $\omega$ called the symplectic form
• Hamiltonian vector field $X_H$ associated with a smooth function $H:M\to\mathbb{R}$ satisfies $dH=\iota_{X_H}\omega$, where $\iota$ denotes the interior product
• Poisson bracket $\{f,g\}$ of smooth functions $f,g:M\to\mathbb{R}$ measures the failure of $f$ and $g$ to Poisson commute and is defined by $\{f,g\}=\omega(X_f,X_g)$
  • Satisfies properties such as antisymmetry, bilinearity, and the Jacobi identity
• Moment map $\mu:M\to\mathfrak{g}^*$ associated with a Hamiltonian action of a Lie group $G$ on $(M,\omega)$ is a smooth map satisfying certain equivariance and compatibility conditions
  • Components of $\mu$ generate the action of $G$ on $M$ via Hamiltonian flows
• Coadjoint orbit $\mathcal{O}_\xi$ of $\xi\in\mathfrak{g}^*$ is the orbit of $\xi$ under the coadjoint action of $G$ on $\mathfrak{g}^*$ 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 $(M,\omega)$ under the action of a Lie group $G$
• Equivariance property $\mu(g\cdot x)=\mathrm{Ad}^*_{g^{-1}}\mu(x)$ relates the moment map to the coadjoint action of $G$ on $\mathfrak{g}^*$
• Components $\mu^\xi(x)=\langle\mu(x),\xi\rangle$ of the moment map, indexed by $\xi\in\mathfrak{g}$, are Hamiltonian functions generating the action of $G$
  • Hamiltonian vector field $X_{\mu^\xi}$ corresponds to the infinitesimal action of $\xi$ on $M$
• Moment map can be viewed as a collection of conserved quantities associated with the symmetries of the system
• Level sets $\mu^{-1}(c)$ 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 $H^2(M,\mathbb{R})$

Hamiltonian Actions Explained

• Hamiltonian action of a Lie group $G$ on a symplectic manifold $(M,\omega)$ preserves the symplectic form and admits a moment map $\mu:M\to\mathfrak{g}^*$
• Infinitesimal generators $X_\xi$ of the action, associated with $\xi\in\mathfrak{g}$, are Hamiltonian vector fields satisfying $dH_\xi=\iota_{X_\xi}\omega$
  • Hamiltonian functions $H_\xi$ are related to the moment map by $H_\xi=\mu^\xi$
• Orbits of a Hamiltonian action are isotropic submanifolds of $(M,\omega)$, meaning the symplectic form vanishes when restricted to the orbits
• Transitive Hamiltonian actions correspond to coadjoint orbits in $\mathfrak{g}^*$, which inherit a natural symplectic structure
• Momentum mapping $\mathbf{J}:T^*Q\to\mathfrak{g}^*$ for a cotangent lift of a $G$-action on a configuration manifold $Q$ is given by $\mathbf{J}(\alpha_q)(\xi)=\alpha_q(\xi_Q(q))$, where $\xi_Q$ 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 $\mathrm{Symp}(M,\omega)$ 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 $\mathfrak{g}^*$
• Symplectic quotients $M/\!/G=\mu^{-1}(0)/G$ 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 $\mathcal{O}_\xi\subset\mathfrak{g}^*$ carry a natural symplectic structure and are homogeneous Hamiltonian $G$-spaces
  • Kirillov-Kostant-Souriau symplectic form on $\mathcal{O}_\xi$ is given by $\omega_\xi(ad^*_\eta\xi,ad^*_\zeta\xi)=\langle\xi,[\eta,\zeta]\rangle$
• Cotangent bundles $T^*Q$ are symplectic manifolds with the canonical symplectic form $\omega=d\theta$, where $\theta$ is the Liouville 1-form
  • Cotangent lift of a $G$-action on $Q$ is Hamiltonian with moment map $\mathbf{J}:T^*Q\to\mathfrak{g}^*$
• Symplectic toric manifolds are compact connected symplectic manifolds equipped with an effective Hamiltonian action of a torus $\mathbb{T}^n$ of half the dimension
  • Classified by their moment polytopes, convex polytopes in $\mathbb{R}^n$ 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 $M/\!/G=\mu^{-1}(0)/G$ for Hamiltonian $G$-actions with proper moment maps
  • Reduced spaces are symplectic manifolds of dimension $\dim M-2\dim G$
• 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