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

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.