Tropical Gromov-Witten Invariants

Definition

Tropical Gromov-Witten invariants are combinatorial invariants that count the number of certain types of tropical curves in a given tropical variety, serving as a tropical analog to classical Gromov-Witten invariants. These invariants help in understanding the geometry of moduli spaces and their compactifications, linking them to enumerative geometry through various structures like Schubert calculus and mirror symmetry.

5 Must Know Facts For Your Next Test

1. Tropical Gromov-Witten invariants arise from counting the number of rational tropical curves connecting specified points in a tropical variety.
2. These invariants can provide insights into classical Gromov-Witten invariants through degeneration techniques, revealing connections between algebraic and tropical geometry.
3. The computation of these invariants often involves combinatorial techniques and can be visualized using polyhedral geometry.
4. Tropical Gromov-Witten invariants relate to Schubert calculus by providing a way to count intersections of tropical curves with Schubert varieties.
5. They play a significant role in mirror symmetry by establishing correspondences between enumerative geometry on two dual varieties.

Review Questions

• How do Tropical Gromov-Witten invariants relate to classical Gromov-Witten invariants?
  • Tropical Gromov-Witten invariants serve as a combinatorial counterpart to classical Gromov-Witten invariants, allowing for the counting of rational curves in a tropical setting. They reveal insights into classical invariants through degeneration techniques, where one can relate the counts of tropical curves to those in algebraic geometry. This connection enhances our understanding of enumerative geometry across both settings.
• Discuss the significance of Tropical Gromov-Witten invariants in the context of Schubert calculus.
  • Tropical Gromov-Witten invariants significantly contribute to Schubert calculus by providing a framework to count the intersections of tropical curves with Schubert varieties. These counts help to establish relationships between different combinatorial problems and enumerative geometry. By understanding these intersections through tropical geometry, one can gain deeper insights into classical Schubert problems.
• Evaluate how Tropical Gromov-Witten invariants interact with mirror symmetry and what implications this has for enumerative geometry.
  • The interaction between Tropical Gromov-Witten invariants and mirror symmetry is profound as it reveals dualities between different geometric frameworks. The invariants allow for equivalences in enumerative geometry, linking counts of curves on one variety to those on its mirror. This relationship not only deepens our understanding of both tropical and classical invariants but also highlights the broader implications for dualities in algebraic geometry.

Related terms

Tropical Curves: These are piecewise linear maps from a one-dimensional domain into a tropical variety, which can be used to study the geometry of tropical varieties.

Moduli Spaces: These are geometric spaces that parameterize families of objects, such as curves, and are crucial for studying the deformation theory in algebraic geometry.

Mirror Symmetry: A duality between two different geometric frameworks, where properties of one space are reflected in another, often leading to equivalences in their enumerative invariants.