Gromov-witten invariants

Definition

Gromov-Witten invariants are mathematical tools used to count the number of curves on a given symplectic manifold, providing important information about the geometry and topology of the space. These invariants relate to the intersection theory and have significant implications in both symplectic geometry and algebraic geometry, linking the behavior of curves to various geometric properties of the manifold.

5 Must Know Facts For Your Next Test

1. Gromov-Witten invariants can be seen as counts of holomorphic curves in a symplectic manifold, reflecting the manifold's topology.
2. These invariants are computed using techniques from both algebraic geometry and symplectic geometry, revealing deep connections between the two fields.
3. One important application of Gromov-Witten invariants is their role in mirror symmetry, where they help relate geometric properties of different manifolds.
4. They can also be used to derive relations in quantum cohomology, providing insights into quantum field theory and enumerative geometry.
5. The calculation of Gromov-Witten invariants often involves sophisticated mathematical tools like virtual fundamental classes and stable maps.

Review Questions

• How do Gromov-Witten invariants contribute to our understanding of curves in symplectic manifolds?
  • Gromov-Witten invariants help quantify the number of holomorphic curves that can be found within a symplectic manifold. By counting these curves, mathematicians can gain insights into the manifold's topological structure and geometric features. This counting is significant because it connects geometric properties with algebraic data, enriching our understanding of both fields.
• Discuss the relationship between Gromov-Witten invariants and mirror symmetry in the context of symplectic geometry.
  • In mirror symmetry, Gromov-Witten invariants play a critical role by establishing correspondences between pairs of manifolds. These pairs have 'mirror' relationships where the counting of curves on one manifold corresponds to geometric data on its mirror. This connection allows mathematicians to use Gromov-Witten invariants to derive properties and phenomena in seemingly unrelated areas, enhancing our grasp of complex geometrical concepts.
• Evaluate how Gromov-Witten invariants influence modern mathematical theories such as quantum cohomology and enumerative geometry.
  • Gromov-Witten invariants significantly impact modern mathematical theories by bridging classical geometry with quantum concepts. They enable calculations in quantum cohomology, where Gromov-Witten invariants serve as essential building blocks for understanding how different geometries behave under quantum transformations. This interplay enhances enumerative geometry by providing concrete numbers for curve counts and establishing deeper connections between physical theories and geometric structures.

Related terms

Symplectic Manifold: A symplectic manifold is a smooth manifold equipped with a closed, non-degenerate 2-form, which plays a central role in the study of Hamiltonian dynamics.

Moduli Space: The moduli space is a geometric space that parametrizes all possible curves or objects of a certain type, allowing for the classification and study of their properties.

Intersection Theory: Intersection theory studies the intersection of subvarieties within algebraic varieties, focusing on how they meet and the properties that arise from these intersections.