The classification of 3-dimensional noetherian cubic Calabi-Yau algebras

TL;DR

This paper classifies all superpotentials leading to 3-dimensional noetherian cubic Calabi-Yau algebras, extending previous classifications from quadratic to cubic cases and providing formulas for automorphism determinants.

Contribution

It provides a complete classification of superpotentials for cubic Calabi-Yau algebras, building on prior work on quadratic cases and analyzing automorphism properties.

Findings

Classified all superpotentials for 3D noetherian cubic Calabi-Yau algebras.

Established a formula for the homological determinant of automorphisms.

Identified a unique exception in the determinant formula.

Abstract

It is known that every 3-dimensional noetherian Calabi-Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S. P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi-Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi-Yau algebras. As an application, we show that if $S$ is a 3-dimensional noetherian cubic Calabi-Yau algebra and $\sigma$ is a graded algebra automorphism of $S$, then the homological determinant of $\sigma$ can be calculated by the formula $\operatorname{hdet} \sigma=(\operatorname{det} \sigma)^2$ with one exception.