Gromov-Witten Invariants

Definition

Gromov-Witten invariants are numerical values that count the number of curves of a certain class on a symplectic manifold, considering both their geometric properties and how they intersect. These invariants connect algebraic geometry and symplectic geometry, providing insights into the topology of manifolds and facilitating the study of their properties. They play a crucial role in understanding how different geometric structures can be represented and classified.

5 Must Know Facts For Your Next Test

1. Gromov-Witten invariants are defined for symplectic manifolds and are invariant under deformation, meaning they remain unchanged under small perturbations of the manifold.
2. They are calculated using techniques from both algebraic geometry and symplectic topology, linking these two fields in meaningful ways.
3. The invariants can capture information about the topology of the manifold, such as the number of holomorphic curves through a point.
4. Gromov-Witten invariants have applications in string theory, where they help describe the geometry of Calabi-Yau manifolds.
5. They also play a significant role in mirror symmetry, where they relate to the counting of holomorphic curves in dual spaces.

Review Questions

• How do Gromov-Witten invariants connect algebraic geometry with symplectic geometry, and why is this connection important?
  • Gromov-Witten invariants bridge algebraic geometry and symplectic geometry by counting holomorphic curves within symplectic manifolds and linking these counts to algebraic varieties. This connection is crucial because it allows mathematicians to apply tools and concepts from both fields, enriching their understanding of geometric structures. The interplay enhances our grasp of how different geometric forms can be studied through invariants that remain constant under various transformations.
• Discuss how Gromov-Witten invariants are computed and their significance in determining the properties of symplectic manifolds.
  • Gromov-Witten invariants are computed using techniques that involve analyzing moduli spaces of curves and their intersection theory on symplectic manifolds. Their significance lies in their ability to encapsulate geometric properties such as curve counts through points in the manifold. By understanding these invariants, researchers can derive crucial insights into the manifold's topological characteristics and explore relationships between various geometric entities.
• Evaluate the implications of Gromov-Witten invariants in string theory and mirror symmetry, particularly regarding Calabi-Yau manifolds.
  • Gromov-Witten invariants have profound implications in string theory as they aid in describing the geometric framework underlying Calabi-Yau manifolds, which are essential for compactification processes in string theory. In mirror symmetry, these invariants provide a link between two dual geometric spaces by establishing how counting holomorphic curves corresponds across these spaces. The results enhance our understanding of complex geometry's role in theoretical physics while fostering deeper mathematical connections.

Related terms

Symplectic Manifold: A symplectic manifold is a smooth, even-dimensional space equipped with a closed, non-degenerate 2-form that encodes the geometric structure needed for classical mechanics.

Curve Counting: Curve counting refers to the process of determining the number of curves that meet certain conditions within a given space, often used in the context of Gromov-Witten theory.

Quantum Cohomology: Quantum cohomology is a branch of mathematics that extends classical cohomology theories by incorporating Gromov-Witten invariants, helping to study the interplay between geometry and physics.