On Equivariant Gromov--Witten Invariants of Resolved Conifold with Diagonal and Anti-Diagonal Actions

TL;DR

This paper explores conjectural links between equivariant Gromov-Witten invariants of the resolved conifold under different symmetries and those of the projective line, verifying these in low genus cases.

Contribution

It introduces two conjectural relationships connecting invariants of the resolved conifold with those of alculus and provides partial verifications for genus zero, one, and two.

Findings

Validated conjectural relationships in genus zero

Provided evidence supporting validity in genus one and two

Suggests broader applicability of these relationships

Abstract

We propose two conjectural relationships between the equivariant Gromov-Witten invariants of the resolved conifold under diagonal and anti-diagonal actions and the Gromov-Witten invariants of $\mathbb{P}^1$, and verify their validity in genus zero approximation. We also provide evidences to support the validity of these relationships in genus one and genus two.