Theta functions, broken lines and 2-marked log Gromov-Witten invariants

TL;DR

This paper establishes a connection between theta functions, their multiplicative structures, and 2-marked log Gromov-Witten invariants for log Calabi-Yau surfaces, extending previous work on wall functions and 1-marked invariants.

Contribution

It extends the correspondence between theta functions and Gromov-Witten invariants to include 2-marked invariants for log Calabi-Yau surfaces, bypassing punctured invariants.

Findings

Relates theta functions to 2-marked log Gromov-Witten invariants.

Extends wall function and 1-marked invariant correspondence.

Provides a new approach for log Calabi-Yau surfaces with smooth very ample anticanonical divisor.

Abstract

Theta functions were defined for varieties with effective anticanonical divisor and are related to certain punctured Gromov-Witten invariants. In this paper we show that in the case of a log Calabi-Yau surface (X,D) with smooth very ample anticanonical divisor we can circumvent the notion of punctured Gromov-Witten invariants and relate theta functions and their multiplicative structure to certain 2-marked log Gromov-Witten invariants. This is a natural extension of the correspondence between wall functions and 1-marked log Gromov-Witten invariants.