Intersections on tropical moduli spaces

Abstract

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties ${\mathbb{P}}^1$, ${\mathbb{P}}^2$, ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and with Psi-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for ${\mathbb{P}}^2$ in Markwig-Rau-2008). Our approach uses tropical intersection theory and can unify and simplify some parts of the existing tropical enumerative geometry (for rational curves).

40 pages, 17 Postscript figures; updated to fit the published version

References in corpus (4)

• Tropical fans and the moduli spaces of tropical curves
• First Steps in Tropical Intersection Theory
• Tropical descendant Gromov-Witten invariants
• Intersecting Psi-classes on tropical M_{0,n}

Cited by in corpus (3)

• Intersection Theory on Tropicalizations of Toroidal Embeddings
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• On rational equivalence in tropical geometry