A graphical interface for the Gromov-Witten theory of curves

A graphical interface for the Gromov-Witten theory of curves

Renzo Cavalieri, Paul Johnson, Hannah Markwig, and Dhruv Ranganathan
July 16, 2019

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 , as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.

1. Introduction

The first goal of this article is to fill in the following square of correspondences:

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