On the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex

TL;DR

This paper introduces a new approach to compute the 1-leg Donaldson-Thomas $bZ_2 imes bZ_2$-vertex using restricted pyramid configurations, establishing connections with the $bZ_4$-vertex and deriving explicit formulas.

Contribution

It develops a novel method involving restricted pyramid configurations and symmetric interlacing properties to explicitly compute the 1-leg Donaldson-Thomas $bZ_2 imes bZ_2$-vertex, linking it to the $bZ_4$-vertex.

Findings

Explicit formula for the 1-leg Donaldson-Thomas $bZ_2 imes bZ_2$-vertex.

Identification of a unique class of symmetric interlacing pyramid configurations.

Connection established between $bZ_2 imes bZ_2$-vertex and $bZ_4$-vertex.

Abstract

We introduce a notion of restricted pyramid configurations for computing the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex. We study a special type of restricted pyramid configurations with the prescribed 1-leg partitions, and find one unique class of them satisfying the symmetric interlacing property. This leads us to obtain an explicit formula for a class of 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex through establishing its connection with 1-leg Donaldson-Thomas $\mathbb{Z}_4$-vertex using the vertex operator methods of Okounkov-Reshetikhin-Vafa and Bryan-Young.