A graphical interface for the Gromov--Witten theory of curves

TL;DR

This paper establishes a detailed connection between descendant Gromov--Witten invariants of curves, tropical curve counting, and Fock space operators, providing a new graphical approach to understanding these complex relationships.

Contribution

It proves a classical/tropical correspondence theorem and introduces an algorithm linking tropical Gromov--Witten invariants with Hurwitz numbers via Fock space operators.

Findings

Proves a classical/tropical correspondence theorem for descendant invariants.

Develops an algorithm for tropical Gromov--Witten/Hurwitz equivalence.

Relates tropical curve counting to Fock space operator algebra.

Abstract

We explore the explicit relationship between the descendant Gromov--Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov--Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient "graphical user interface" for Okounkov and Pandharipande's celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.