Gromov-Witten theory of root gerbes I: structure of genus $0$ moduli spaces

TL;DR

This paper studies the structure of genus 0 moduli spaces of twisted stable maps to root gerbes over a variety, providing explicit constructions and formulas relating their Gromov-Witten invariants to those of the base variety.

Contribution

It introduces explicit constructions of genus 0 twisted stable map moduli stacks to root gerbes and derives formulas linking their Gromov-Witten invariants to the base variety.

Findings

Explicit moduli stack constructions for genus 0 twisted stable maps.

Formulas relating Gromov-Witten invariants of root gerbes to those of the base variety.

Structural insights into genus 0 moduli spaces of twisted stable maps.

Abstract

Let $X$ be a smooth complex projective algebraic variety. Given a line bundle $\mathcal{L}$ over $X$ and an integer $r>1$ one defines the stack $\sqrt[r]{\mathcal{L}/X}$ of $r$-th roots of $\mathcal{L}$. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$. Our main results are explicit constructions of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$ starting from moduli stack of genus $0$ stable maps to $X$. As a consequence, we prove an exact formula expressing genus $0$ Gromov-Witten invariants of $\sqrt[r]{\mathcal{L}/X}$ in terms of those of $X$.