🔵Symplectic Geometry Unit 12 – Algebraic Geometry & Representation Links

Key Concepts and Definitions

• Symplectic manifold $(M, \omega)$ consists of a smooth manifold $M$ and 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 $dH = \omega(X_H, \cdot)$
  • Flow of $X_H$ preserves the symplectic form and is called a Hamiltonian flow
• Poisson bracket $\{f, g\}$ of smooth functions $f, g: M \to \mathbb{R}$ measures the failure of $f$ and $g$ to Poisson commute
  • Defined by $\{f, g\} = \omega(X_f, X_g)$
• Lagrangian submanifold $L \subset M$ is a submanifold on which the symplectic form vanishes ($\omega|_L = 0$) and has half the dimension of $M$
• Symplectomorphism $\phi: (M_1, \omega_1) \to (M_2, \omega_2)$ is a diffeomorphism that preserves the symplectic form ($\phi^* \omega_2 = \omega_1$)
• Moment map $\mu: M \to \mathfrak{g}^*$ associated with a Hamiltonian $G$-action on $(M, \omega)$ satisfies $d\mu^X = \iota_{X^\#} \omega$ for all $X \in \mathfrak{g}$

Historical Context and Development

• Symplectic geometry has its roots in classical mechanics and the study of phase spaces (cotangent bundles) of mechanical systems
• Lagrange and Poisson's work on celestial mechanics in the late 18th and early 19th centuries laid the foundation for the Poisson bracket and Hamilton's equations
• Sophus Lie's work on transformation groups in the late 19th century introduced the concept of a Lie group action, which plays a crucial role in equivariant symplectic geometry
• The modern formulation of symplectic geometry using differential forms was developed by Élie Cartan, Jean Leray, and André Weil in the first half of the 20th century
  • This formulation allowed for the application of powerful tools from differential geometry and topology to the study of symplectic manifolds
• The introduction of moment maps by Kostant, Souriau, and Marsden-Weinstein in the 1960s and 1970s provided a key link between symplectic geometry and representation theory
• Gromov's non-squeezing theorem (1985) and the development of symplectic capacities demonstrated the rigidity of symplectic embeddings and the existence of global invariants
• The study of symplectic quotients and reduction (Marsden-Weinstein reduction) has been a major theme in symplectic geometry since the 1970s

Algebraic Structures in Symplectic Geometry

• The Poisson algebra $C^\infty(M)$ of smooth functions on a symplectic manifold $(M, \omega)$ with the Poisson bracket $\{\cdot, \cdot\}$ forms a Lie algebra
  • The Poisson bracket satisfies antisymmetry, the Jacobi identity, and the Leibniz rule
• Hamiltonian vector fields $X_H$ form a Lie subalgebra of the Lie algebra of vector fields $\mathfrak{X}(M)$ under the Lie bracket of vector fields
  • The map $H \mapsto X_H$ is a Lie algebra homomorphism from $C^\infty(M)$ to $\mathfrak{X}(M)$
• The group of symplectomorphisms $\text{Symp}(M, \omega)$ is an infinite-dimensional Lie group with Lie algebra given by the space of Hamiltonian vector fields
• A Hamiltonian $G$-action on $(M, \omega)$ is a smooth action of a Lie group $G$ that preserves the symplectic form and admits a moment map $\mu: M \to \mathfrak{g}^*$
  • The moment map is equivariant with respect to the $G$-action on $M$ and the coadjoint action on $\mathfrak{g}^*$
• The Marsden-Weinstein quotient (symplectic reduction) $M /\!/ G := \mu^{-1}(0) / G$ of a Hamiltonian $G$-space $(M, \omega, \mu)$ is a symplectic manifold when $0$ is a regular value of $\mu$ and the action of $G$ on $\mu^{-1}(0)$ is free and proper

Representation Theory Basics

• A representation of a Lie group $G$ on a vector space $V$ is a smooth homomorphism $\rho: G \to \text{GL}(V)$
  • The representation is called faithful if $\rho$ is injective
• The dual representation $\rho^*$ of $G$ on the dual space $V^*$ is defined by $(\rho^*(g)\phi)(v) = \phi(\rho(g^{-1})v)$ for $g \in G$, $\phi \in V^*$, and $v \in V$
• A subrepresentation of $\rho$ is a $G$-invariant subspace $W \subset V$, i.e., $\rho(g)W \subset W$ for all $g \in G$
• An irreducible representation has no proper subrepresentations
• The adjoint representation $\text{Ad}: G \to \text{GL}(\mathfrak{g})$ is defined by $\text{Ad}(g) = d(c_g)_e$, where $c_g: G \to G$ is conjugation by $g$
• The coadjoint representation $\text{Ad}^*: G \to \text{GL}(\mathfrak{g}^*)$ is the dual of the adjoint representation
• The orbit method relates irreducible unitary representations of $G$ to coadjoint orbits in $\mathfrak{g}^*$, which carry a natural symplectic structure

Connections Between Algebra and Geometry

• The moment map $\mu: M \to \mathfrak{g}^*$ associated with a Hamiltonian $G$-action on $(M, \omega)$ intertwines the $G$-action on $M$ and the coadjoint representation on $\mathfrak{g}^*$
  • Coadjoint orbits in $\mathfrak{g}^*$ are symplectic manifolds, and the moment map pulls back the symplectic form on the orbit to $\omega$
• The Marsden-Weinstein quotient $M /\!/ G$ can be identified with the symplectic reduction of $M$ by the coadjoint orbit $\mathcal{O}_\mu := G \cdot \mu$ through $\mu$
• The Guillemin-Sternberg conjecture (now a theorem) states that the geometric quantization of $M /\!/ G$ is isomorphic to the $G$-invariant part of the geometric quantization of $M$
  • This relates the representation theory of $G$ to the geometry of the symplectic quotient
• The Atiyah-Guillemin-Sternberg convexity theorem states that the image of the moment map for a compact, connected Lie group action on a compact, connected symplectic manifold is a convex polytope
  • The vertices of the polytope correspond to fixed points of the action, and the edges correspond to one-dimensional orbits
• The Duistermaat-Heckman theorem expresses the pushforward of the Liouville measure on $M$ under the moment map in terms of the volumes of the reduced spaces $M_t := \mu^{-1}(t) / G_t$

Applications in Symplectic Geometry

• Hamiltonian mechanics classical mechanical systems are described by symplectic manifolds (phase spaces) and Hamiltonian flows
  • Symmetries of the system correspond to conserved quantities via Noether's theorem and the moment map
• Geometric quantization aims to construct a quantum-mechanical Hilbert space from a classical symplectic manifold
  • The Guillemin-Sternberg conjecture relates the quantization of a symplectic quotient to the representation theory of the symmetry group
• Symplectic topology studies global properties of symplectic manifolds using techniques from topology and analysis
  • Symplectic capacities (Gromov width, Hofer-Zehnder capacity) are invariants that measure the size of symplectic embeddings
• Mirror symmetry relates the symplectic geometry of a Calabi-Yau manifold to the complex geometry of its mirror manifold
  • This has led to powerful new insights in both symplectic geometry and algebraic geometry
• Floer theory is an infinite-dimensional analogue of Morse theory that studies the topology of the space of Lagrangian submanifolds in a symplectic manifold
  • Floer homology groups are symplectic invariants that have important applications in low-dimensional topology and Hamiltonian dynamics

Problem-Solving Techniques

• Identify the symplectic manifold $(M, \omega)$ and any relevant Lie group actions or symmetries
• Compute the moment map $\mu: M \to \mathfrak{g}^*$ associated with a Hamiltonian $G$-action using the defining property $d\mu^X = \iota_{X^\#} \omega$
  • Check equivariance of the moment map with respect to the $G$-action and the coadjoint action
• Analyze the level sets $\mu^{-1}(t)$ and the reduced spaces $M_t := \mu^{-1}(t) / G_t$ using the Marsden-Weinstein reduction theorem
• Apply the Atiyah-Guillemin-Sternberg convexity theorem to understand the image of the moment map for compact group actions
• Use the Duistermaat-Heckman theorem to compute integrals over reduced spaces in terms of the moment polytope
• Employ symplectic capacities and Gromov's non-squeezing theorem to study symplectic embeddings and rigidity phenomena
• Apply the Guillemin-Sternberg conjecture and geometric quantization to relate the representation theory of $G$ to the symplectic geometry of $M /\!/ G$

Advanced Topics and Current Research

• Symplectic field theory (SFT) is a generalization of Floer theory that studies the geometry of contact manifolds and their symplectizations
  • SFT has applications in the study of Reeb dynamics, symplectic fillings, and the topology of Legendrian knots
• Fukaya categories are $A_\infty$-categories that encode the symplectic topology of a manifold through the intersection theory of Lagrangian submanifolds
  • They play a central role in homological mirror symmetry and the study of D-branes in string theory
• Symplectic cohomology is an invariant of symplectic manifolds with contact-type boundary that captures information about periodic orbits and fillability
  • It is related to Hamiltonian dynamics, Reeb flows, and the topology of symplectic fillings
• Rabinowitz Floer homology is a variant of Floer homology that studies the existence and multiplicity of periodic orbits in Hamiltonian systems
  • It has applications in the study of magnetic flows, contact-type hypersurfaces, and the Weinstein conjecture
• Symplectic Khovanov homology is a categorification of the Khovanov homology of links that incorporates symplectic geometry and Floer theory
  • It has connections to low-dimensional topology, knot theory, and the representation theory of braid groups
• The symplectic geometry of moduli spaces (gauge theory, Higgs bundles, representation varieties) is an active area of research with connections to mathematical physics and geometric representation theory
  • These moduli spaces often admit natural symplectic structures and Hamiltonian group actions, which can be studied using the techniques of symplectic geometry and equivariant topology