A new approach to the operator formalism for Gromov-Witten invariants of the cap and tube

TL;DR

This paper introduces a novel operator formalism for Gromov-Witten invariants of the cap and tube, extending previous methods to include negative contact orders and non-equivariant cases.

Contribution

It develops a new approach based on Johnson's operator formula, expanding the operator formalism to broader settings in Gromov-Witten theory.

Findings

Extended operator formalism to non-equivariant cases.

Included negative contact orders in the formalism.

Provided new computational tools for Gromov-Witten invariants.

Abstract

Based on Johnson's operator formula for the equivariant Gromov-Witten theory of $\mathbb{P}^1$-orbifolds, we give a new approach to the operator formalism by Okounkov and Pandharipande regarding the $\mathbb{C}^*$-equivariant Gromov-Witten theory of $\mathbb{P}^1$ relative to one or two points. We also extend their operator formalism in the non-equivariant specialization to the case where we allow negative contact orders.