Chamber structure for some equivariant relative Gromov-Witten invariants of $\mathbb{P}^1$ in genus $0$

TL;DR

This paper investigates genus 0 equivariant relative Gromov-Witten invariants of , revealing their piecewise polynomial nature across chambers and providing explicit formulas for differences between chambers, with connections to double Hurwitz numbers.

Contribution

It introduces a chamber structure for these invariants, characterizes polynomial differences between chambers, and simplifies formulas in a special negative chamber.

Findings

Invariants are piecewise polynomials in parameter space.

Differences between neighboring chambers are explicitly determined.

A simple expression is obtained in the totally negative chamber.

Abstract

In this paper, we study genus $0$ equivariant relative Gromov-Witten invariants of $\mathbb{P}^1$ whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighboring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber structure here shares some similarities to that of double Hurwitz numbers.