Glossary

12.3 Symplectic quotients and GIT quotients

4 min read•july 30, 2024

Symplectic and GIT quotients are powerful tools for studying group actions in geometry. They arise from different fields but share deep connections, offering complementary perspectives on geometric structures and .

These quotient constructions play a crucial role in bridging symplectic and algebraic geometry. They provide a framework for understanding stability conditions, constructing moduli spaces, and exploring applications in mathematical physics and representation theory.

Symplectic vs GIT Quotients

Conceptual Foundations and Differences

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  • Symplectic quotients arise from symplectic geometry utilizing moment maps and Hamiltonian group actions on symplectic manifolds
  • GIT quotients stem from algebraic geometry employing techniques from to study group actions on algebraic varieties
  • Both quotient constructions produce spaces capturing group action geometry using different stability conditions and techniques
  • Symplectic quotients typically yield smooth manifolds under certain regularity conditions
  • GIT quotients may contain singularities
  • establishes a relationship between symplectic and GIT quotients under specific conditions bridging the two approaches

Mathematical Framework and Properties

  • Symplectic quotients map symplectic manifolds to the dual of the Lie algebra of a Lie group acting on the manifold
  • GIT quotients involve studying group actions on algebraic varieties (projective schemes)
  • Symplectic quotients inherit a symplectic structure from the original manifold under certain conditions (Marsden-Weinstein-Meyer theorem)
  • GIT quotients rely on the Hilbert-Mumford numerical criterion for stability involving one-parameter subgroups
  • Symplectic reduction serves as a geometric analog to symmetry reduction in Hamiltonian mechanics
  • GIT quotients provide a way to construct quotients in situations where the group action is not free or proper

Construction of Symplectic Quotients

Moment Maps and Their Properties

  • Moment maps serve as fundamental tools in symplectic geometry mapping symplectic manifolds to dual Lie algebras
  • Moment maps satisfy equivariance properties with respect to group action and coadjoint action on dual Lie algebra
  • Level sets of moment maps (μ−1(0)\mu^{-1}(0)) play crucial roles in construction
  • Zero often chosen as regular value for constructing symplectic quotients
  • Moment map μ:M→g∗\mu: M \rightarrow \mathfrak{g}^* where M denotes and g∗\mathfrak{g}^* dual of Lie algebra
  • Equivariance property expressed as μ(g⋅x)=Ad∗(g)(μ(x))\mu(g \cdot x) = Ad^*(g)(\mu(x)) for all g∈Gg \in G and x∈Mx \in M

Symplectic Reduction Process

  • Symplectic quotients constructed by taking preimage of regular value of moment map and quotienting by group action
  • Marsden-Weinstein-Meyer theorem guarantees inherited symplectic structure under certain conditions
  • Symplectic quotient dimension related to original manifold dimension and acting group dimension
  • Reduction process expressed as M//G=μ−1(0)/GM // G = \mu^{-1}(0) / G
  • Symplectic form on quotient denoted ωred\omega_{red} satisfying π∗ωred=i∗ω\pi^*\omega_{red} = i^*\omega where π\pi projection map and ii inclusion map
  • Examples include (symplectic reduction of Cn\mathbb{C}^n by torus action) and (symplectic reduction of cotangent bundles)

Stability Conditions in GIT and Symplectic Geometry

GIT Stability and Hilbert-Mumford Criterion

  • GIT stability defined using Hilbert-Mumford numerical criterion involving one-parameter subgroups
  • Numerical criterion for stability expressed as μ(x,λ)>0\mu(x, \lambda) > 0 for all non-trivial one-parameter subgroups λ\lambda
  • Semistable points correspond to μ(x,λ)≥0\mu(x, \lambda) \geq 0 for all λ\lambda
  • Polystability involves orbit closure intersecting stable locus
  • Examples include stability of points in projective space under linear group actions (configuration of points on projective line)

Symplectic Stability and Moment Map Behavior

  • Symplectic stability related to moment map behavior and group action on manifold
  • Kempf-Ness theorem establishes correspondence between GIT-stable points and certain points in zero level set of moment map
  • Semistable points in GIT correspond to points whose orbit closures intersect zero level set of moment map
  • Polystability in symplectic context related to gradient flow of norm-square of moment map
  • Examples include stability of polygons in R3\mathbb{R}^3 under rotations (polygon space)

Interplay Between GIT and Symplectic Stability

  • Stability conditions in both contexts identify "good" for constructing well-behaved quotient spaces
  • Kempf-Ness theorem provides bridge between algebraic and symplectic stability notions
  • Symplectic interpretation of GIT stability involves convexity properties of moment map image
  • Gradient flow of moment map norm-square connects symplectic and GIT perspectives
  • Applications include moduli spaces of vector bundles and representation varieties of surface groups

Applications of Quotients to Moduli Spaces

Moduli Space Construction

  • Moduli spaces parameterize families of algebraic or geometric structures often constructed as quotients by group actions
  • Symplectic quotients construct moduli spaces of symplectic structures (flat connections on Riemann surfaces)
  • GIT quotients useful in algebraic geometry for constructing moduli spaces of algebraic varieties and vector bundles
  • combine aspects of symplectic and GIT quotients with applications in gauge theory
  • Examples include moduli space of stable vector bundles on algebraic curves and moduli space of instantons on 4-manifolds

Applications in Mathematical Physics

  • studied using both symplectic and techniques
  • moduli spaces constructed using hyperkähler quotients
  • Moduli spaces arising in string theory analyzed through quotient constructions
  • Examples include moduli space of flat connections on Riemann surfaces (related to Chern-Simons theory) and moduli space of vacua in supersymmetric gauge theories

Interplay Between Symplectic and Algebraic Geometry

  • Moment map techniques from symplectic geometry applied to study stability conditions in GIT and vice versa
  • Kähler quotients combine symplectic and GIT perspectives in complex geometry
  • Symplectic techniques provide tools for studying algebraic varieties with group actions
  • GIT methods offer algebraic approach to constructing symplectic quotients in certain cases
  • Examples include toric varieties (symplectic and algebraic perspectives) and quiver varieties (combining representation theory and symplectic geometry)

Key Terms to Review (23)

Alan Weinstein: Alan Weinstein is a prominent mathematician known for his significant contributions to symplectic geometry and mathematical physics. His work has had a profound impact on the understanding of symplectic quotients, which arise in various mathematical contexts, including geometric invariant theory. Weinstein's ideas have shaped the study of symplectic manifolds and their applications in areas such as mechanics and quantization.
Character Varieties: Character varieties are algebraic varieties that classify the representations of a given group into a fixed structure, typically focusing on the characters associated with these representations. They provide a geometric framework to study the representations of groups and have strong ties to both symplectic geometry and geometric invariant theory, showcasing how algebraic and geometric methods can be intertwined to understand representation spaces.
Classical Mechanics: Classical mechanics is a branch of physics that deals with the motion of objects and the forces acting on them, typically described by Newton's laws. It serves as the foundation for understanding physical systems, providing insight into energy conservation, the dynamics of motion, and the relationships between different physical quantities.
Coadjoint orbits: Coadjoint orbits are geometric objects that arise in the representation theory of Lie groups and symplectic geometry, specifically representing the action of a Lie group on the dual space of its Lie algebra. They serve as a crucial structure for understanding symplectic manifolds, especially in the context of Hamiltonian dynamics and the reduction of symplectic manifolds under group actions.
Complex projective space: Complex projective space, denoted as $$ ext{CP}^n$$, is a mathematical construct that represents the set of lines through the origin in $$ ext{C}^{n+1}$$. It captures the idea of equivalence classes of non-zero vectors in $$ ext{C}^{n+1}$$ under scalar multiplication, allowing us to study geometric and topological properties of complex manifolds and play a crucial role in symplectic geometry and GIT quotients.
Convexity of the momentum map: The convexity of the momentum map refers to the property of a momentum map being a convex function, which means that its image forms a convex subset in the dual space of the Lie algebra associated with a symplectic manifold. This concept is crucial in understanding the relationship between symplectic geometry and geometric invariant theory, particularly in how it relates to symplectic quotients and GIT quotients, where convexity helps determine the stability of orbits and the structure of these quotient spaces.
Cotangent Bundle: The cotangent bundle of a manifold is the vector bundle that consists of all the cotangent spaces at each point of the manifold, effectively capturing the linear functionals on the tangent spaces. This construction plays a crucial role in symplectic geometry as it provides a natural setting for defining symplectic structures and studying Hamiltonian dynamics.
Geometric Invariant Theory: Geometric Invariant Theory (GIT) is a framework that studies the action of groups on algebraic varieties and classifies objects up to equivalence by analyzing their invariant properties. It plays a vital role in understanding how symmetries can affect the structure of geometric objects, especially within the context of symplectic geometry and the interplay with complex algebraic varieties. This theory enables us to create quotients that reflect the geometric structures influenced by group actions.
GIT quotient: The GIT quotient, or Geometric Invariant Theory quotient, is a construction that allows one to form a new space from a given space with group action by identifying points that are equivalent under this action. This process is crucial in both algebraic geometry and symplectic geometry, as it helps to simplify complex geometric structures by partitioning them based on symmetry. The GIT quotient creates a new space that reflects the orbits of the original space under the group action, enabling researchers to study properties of these orbits effectively.
Hamiltonian group action: A Hamiltonian group action is a smooth action of a Lie group on a symplectic manifold that preserves the symplectic structure and is generated by a Hamiltonian function. This concept connects the dynamics of the system with geometric properties, allowing for the analysis of symplectic manifolds in the context of group actions. It plays a crucial role in understanding how symmetries influence the geometry of complex algebraic varieties and in defining symplectic quotients.
Hamiltonian Reduction: Hamiltonian reduction is a process in symplectic geometry that simplifies the study of Hamiltonian systems by reducing the number of degrees of freedom through symplectic quotients. This method allows one to analyze the behavior of dynamical systems while preserving their essential characteristics, leading to a deeper understanding of their geometric structure and behavior under symmetry.
Higgs bundles: Higgs bundles are pairs consisting of a holomorphic vector bundle and a Higgs field, which is a section of a certain endomorphism bundle. They arise in the study of moduli spaces and play a significant role in both algebraic geometry and mathematical physics, particularly in the context of integrable systems and gauge theory. The interplay between Higgs bundles and symplectic geometry is evident through their connection to geometric invariant theory and symplectic quotients.
Hyperkähler quotients: Hyperkähler quotients are a special type of symplectic quotient that arise in the study of hyperkähler manifolds, which are a particular class of manifolds equipped with a rich geometric structure. These quotients provide a way to construct new hyperkähler manifolds by taking a hyperkähler manifold and dividing it by the action of a group, typically a compact Lie group, which preserves the hyperkähler structure. This construction is essential for understanding the geometry and topology of the resulting spaces and their relation to both symplectic geometry and geometric invariant theory.
Kempf-ness Theorem: The Kempf-ness Theorem provides a powerful criterion for determining when a point in a variety is stable or semi-stable under the action of a reductive group, using the concept of moment maps. This theorem connects algebraic geometry and symplectic geometry by showing how the properties of moment maps can be used to classify points based on their stability, which is essential for understanding the geometric structure of quotients formed by these group actions.
Marsden-Weinstein Theorem: The Marsden-Weinstein Theorem provides a way to construct symplectic manifolds by reducing the symplectic structure of a Hamiltonian system with a symmetry, utilizing moment maps. This theorem connects the concepts of symplectic reduction and the geometry of orbits in the presence of group actions, facilitating the study of reduced spaces in symplectic geometry.
Moduli spaces: Moduli spaces are geometric structures that parametrize families of mathematical objects, allowing us to understand the various shapes and configurations these objects can take. These spaces provide a way to classify objects up to certain equivalences, like symplectic manifolds or curves, and they play a crucial role in understanding the relationships between different geometric entities and their properties.
Orbits: In the context of symplectic geometry and geometric invariant theory, orbits refer to the paths traced by points in a symplectic manifold as they are acted upon by a symplectic group or a Lie group. These orbits encapsulate the notion of symmetry within the manifold and are fundamental in understanding how the structure of the manifold interacts with these groups, particularly when considering quotients formed from these actions.
Reduced Space: Reduced space refers to the simplified space obtained after applying a symplectic reduction process, where symplectic manifolds are modified to account for symmetry through the action of a group. This concept helps in understanding the essential structure of a Hamiltonian system by effectively reducing the dimensions and eliminating extraneous variables associated with symmetries, making it easier to analyze the system's dynamics.
Stability condition: A stability condition is a mathematical framework that determines when a point in a symplectic manifold is considered stable under the action of a group. This concept plays a crucial role in the context of symplectic quotients and GIT (Geometric Invariant Theory) quotients, as it helps identify which orbits remain in a desirable position for constructing quotient spaces. By establishing stability criteria, one can analyze how symplectic structures behave under group actions and make informed decisions about the resulting geometric properties.
Symplectic Manifold: A symplectic manifold is a smooth, even-dimensional differentiable manifold equipped with a closed, non-degenerate differential 2-form called the symplectic form. This structure allows for a rich interplay between geometry and physics, especially in the formulation of Hamiltonian mechanics and the study of dynamical systems.
Symplectic quotient: A symplectic quotient is a construction in symplectic geometry that arises from the process of taking a symplectic manifold and applying a group action to it, leading to a reduction of dimensions while preserving symplectic structure. This concept is closely related to notions in algebraic geometry and involves the use of moment maps, which encode the way a symplectic manifold interacts with symmetry. By forming the quotient with respect to a group action, one can analyze the geometric and topological properties of the resulting space, often yielding insights into both symplectic and algebraic structures.
Toric Varieties: Toric varieties are a special class of algebraic varieties that can be described using combinatorial data associated with fans, which are collections of cones in a lattice. They provide a bridge between algebraic geometry and combinatorics, allowing for the study of geometric objects through their combinatorial structures, often arising in the context of torus actions and symplectic geometry.
Vladimir Rokhlin: Vladimir Rokhlin was a prominent mathematician known for his contributions to symplectic geometry and the study of symplectic quotients. His work established foundational results that connect symplectic geometry with geometric invariant theory (GIT), shedding light on how symplectic quotients can be understood through algebraic and geometric frameworks.
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