Crystal Melting, Triality and Partition Functions for Toric Calabi-Yau Fourfolds

TL;DR

This paper explores crystal melting models for toric Calabi-Yau 4-folds, analyzing their structure, partition functions, and triality behavior, with the aim of understanding their mathematical and physical properties.

Contribution

It develops an efficient algorithm for constructing crystals, studies their evolution under triality, and introduces stable variables to understand partition function stabilization.

Findings

Partition functions stabilize along triality cascades.

Stable variables lead to consistent behavior of partition functions.

Crystal profiles exhibit interesting patterns in stable variables.

Abstract

We extend the study of the recently introduced crystal melting models associated to toric Calabi-Yau 4-folds in several directions. In particular, we investigate in greater detail the structure of these models for general toric CY 4-folds and flavor configurations, using the explicit example of $Q^{1,1,1}$ to illustrate our ideas. To this end, we develop an efficient algorithm for constructing crystals based on periodic quivers. A central goal of this work is to understand the behavior of crystals and their partition functions under triality. We analyze the evolution of crystals along periodic triality cascades and generate detailed data for these systems, including Hasse diagrams, partition functions, and the multiplicities of melting configurations. We introduce the notion of stable variables and show that they lead to the stabilization of the partition functions along cascades. Finally, we define the profile of the crystal partition function and observe that, when expressed in terms of stable variables, it displays interesting behavior. A further motivation for this work is to generate empirical data that may guide the search for a physically motivated generalization of cluster algebras associated with $2d$ (0,2) quiver theories and their triality transformations.