Moduli of stable maps with fields

TL;DR

This paper constructs a new moduli space of stable maps with fields linked to a variety, a bundle, and a section, generalizing the Quantum Lefschetz principle and relating to existing comparison results.

Contribution

It introduces a novel moduli space framework for stable maps with fields, extending the Quantum Lefschetz principle and providing a new approach to comparison theorems.

Findings

The class matches the virtual fundamental class up to a sign.

The construction generalizes the Quantum Lefschetz hyperplane principle.

Provides a different method from previous comparison results.

Abstract

We construct a moduli space of stable maps with fields associated to a triple $(X,E,s)$ of a projective variety (or a DM stack with projective moduli space) a vector bundle and a section. We show the class coincides up to a sign with the virtual fundamental class on the moduli space of stable maps to $Z=Z(s)\subset X$. We show that this gives a generalization of the Quantum Lefschetz hyperplane principle. This generalizes similar comparison results of Chang--Li, Kim--Oh and Chang--Li and presents a different approach from Chen--Janda--Webb.