Gromov–witten invariants

Definition

Gromov-Witten invariants are mathematical objects used in algebraic geometry and symplectic geometry that count the number of curves of a given degree that can be mapped to a target space, such as a projective variety. They provide crucial information about the geometry of the target space, connecting it to enumerative geometry and the study of moduli spaces of curves.

5 Must Know Facts For Your Next Test

1. Gromov-Witten invariants generalize classical intersection numbers by counting not just points but entire curves in a given class.
2. These invariants can be computed using techniques from quantum field theory, particularly in relation to string theory, which leads to rich interconnections between physics and mathematics.
3. The invariants depend on the geometry of both the target space and the underlying curve, making them sensitive to variations in the structure of these spaces.
4. In particular cases, Gromov-Witten invariants can be computed explicitly for specific target varieties, leading to important results like mirror symmetry.
5. They play an essential role in the study of quantum cohomology, providing insights into how classical geometric properties transform under quantum effects.

Review Questions

• How do Gromov-Witten invariants relate to the study of moduli spaces and what significance do they hold in enumerative geometry?
  • Gromov-Witten invariants provide a way to count curves within moduli spaces, offering insights into how these spaces are structured. Their ability to count curves translates directly into enumerative geometry, where they help to solve problems regarding how many curves of a given degree exist under certain conditions. This relationship enhances our understanding of both moduli spaces and the geometric properties represented by Gromov-Witten invariants.
• Discuss how Gromov-Witten invariants are computed and what role quantum field theory plays in this process.
  • The computation of Gromov-Witten invariants often utilizes techniques from quantum field theory, particularly through the lens of string theory. By using a combination of algebraic geometry and physical principles, mathematicians can derive these invariants from generating functions that encapsulate curve counts. This interplay between mathematics and physics reveals deeper connections in understanding complex geometrical structures through the lens of quantum effects.
• Evaluate the impact of Gromov-Witten invariants on the development of modern algebraic geometry and its intersection with physics.
  • Gromov-Witten invariants have significantly influenced modern algebraic geometry by providing powerful tools for understanding the shape and structure of varieties. Their applications extend beyond pure mathematics into theoretical physics, particularly string theory, where they help illuminate phenomena such as mirror symmetry and quantum cohomology. This synergy has led to new mathematical discoveries and has transformed how researchers approach problems in both fields, making Gromov-Witten invariants a pivotal concept bridging diverse areas.

Related terms

Moduli Space: A moduli space is a geometric space that parametrizes a family of objects, such as algebraic curves or vector bundles, allowing for the study of their deformation and classification.

Enumerative Geometry: Enumerative geometry is the branch of mathematics that deals with counting geometric figures satisfying certain conditions, often leading to deep connections with other areas like physics and string theory.

Symplectic Geometry: Symplectic geometry is a branch of differential geometry that studies symplectic manifolds, which are even-dimensional spaces equipped with a closed non-degenerate 2-form, playing a key role in classical and modern physics.