Hyperkähler quotients

Definition

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.

5 Must Know Facts For Your Next Test

1. Hyperkähler quotients can be thought of as special cases of symplectic quotients where the underlying manifold has a hyperkähler structure, preserving rich geometric features.
2. When constructing a hyperkähler quotient, one usually requires an action of a compact Lie group, which plays a crucial role in ensuring that the quotient space inherits the hyperkähler property.
3. The process of taking hyperkähler quotients can be related to the notion of moduli spaces, providing a framework for studying families of hyperkähler manifolds parametrized by algebraic data.
4. Hyperkähler quotients have applications in theoretical physics, particularly in string theory and gauge theory, where they help understand various moduli spaces of solutions.
5. The study of hyperkähler quotients often involves techniques from both symplectic geometry and algebraic geometry, bridging these two areas through their shared connections with group actions.

Review Questions

• How do hyperkähler quotients relate to both symplectic geometry and algebraic geometry?
  • Hyperkähler quotients serve as an intersection between symplectic geometry and algebraic geometry by providing a method to construct new spaces from existing hyperkähler manifolds through group actions. The process involves taking symplectic quotients while ensuring that the hyperkähler structure is preserved. This connection is essential for understanding the geometric properties of these quotients and their applications in various mathematical contexts.
• What role does a compact Lie group play in the formation of hyperkähler quotients?
  • A compact Lie group is crucial in the formation of hyperkähler quotients as it governs the action on the hyperkähler manifold that leads to the quotient space. This group's action ensures that the resulting space maintains the hyperkähler structure. Without such an action, one cannot guarantee that the properties intrinsic to hyperkähler manifolds will persist in the quotient, highlighting the importance of group actions in this construction.
• Evaluate how hyperkähler quotients contribute to our understanding of moduli spaces in geometric contexts.
  • Hyperkähler quotients contribute significantly to our understanding of moduli spaces by allowing mathematicians to study families of hyperkähler manifolds through parameterization via algebraic data. By analyzing how these quotients change under varying group actions or parameters, researchers can classify different geometric structures and understand their relationships. This evaluation not only enriches our comprehension of hyperkähler manifolds but also sheds light on broader themes in both algebraic and differential geometry.

Related terms

hyperkähler manifold: A hyperkähler manifold is a Riemannian manifold that possesses three complex structures satisfying the quaternionic relations, making it a special case of a Kähler manifold with additional properties.

symplectic quotient: A symplectic quotient is obtained from a symplectic manifold by taking the orbit space of a Hamiltonian group action, leading to a new symplectic manifold with well-defined geometric properties.

Geometric Invariant Theory (GIT): GIT is a framework used to study geometric properties of algebraic varieties under group actions, allowing for the classification and construction of varieties via quotienting by group actions.