Refining the Shifted Topological Vertex

TL;DR

This paper derives explicit formulas for the shifted and refined topological vertices in topological string theory, enhancing the mathematical tools used in the field.

Contribution

It provides the first explicit expressions for the shifted topological vertex and its refined version, completing previous results in the literature.

Findings

Explicit formulas for $ ext{S}_{ ext{lambda mu nu}}(q)$ and $ ext{T}_{ ext{lambda mu nu}}(q,t)$ derived

Enhances mathematical understanding of topological string vertices

Completes existing literature on topological vertex functions

Abstract

We study aspects of the refining and shifting properties of the 3d MacMahon function $\mathcal{C}_{3}(q) $ used in topological string theory and BKP hierarchy. We derive the explicit expressions of the shifted topological vertex $\mathcal{S}_{\lambda \mu \nu}(q) $ and its refined version $\mathcal{T}_{\lambda \mu \nu}(q,t) $. These vertices complete results in literature.