Eulerian digraphs and toric Calabi-Yau varieties

TL;DR

This paper explores a class of affine toric Calabi-Yau varieties derived from eulerian digraphs, detailing their structure, generation, and implications for moduli spaces in superconformal theories.

Contribution

It introduces a method to generate eulerian digraphs using canonical moves and analyzes their impact on associated toric Calabi-Yau varieties, linking graph theory with string theory applications.

Findings

Eulerian digraphs can be generated by a few canonical moves.

The moves' effects on lattice polytopes are characterized.

Examples illustrate the construction and physical relevance.

Abstract

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can be generated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.