Finite-dimensional modules for the polynomial ring in one variable as a vertex algebra

TL;DR

This paper classifies finite-dimensional indecomposable modules for the polynomial ring in one variable viewed as a vertex algebra, highlighting differences from traditional associative algebra modules.

Contribution

It provides the first classification of finite-dimensional modules for this polynomial ring as a vertex algebra, clarifying their structure and distinctions from associative algebra modules.

Findings

Classification of finite-dimensional indecomposable modules

Distinction between vertex algebra modules and associative algebra modules

Insight into module structure for polynomial vertex algebras

Abstract

A commutative associative algebra $A$ over ${\mathbb C}$ with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for $A$ as a vertex algebra and the modules for $A$ as an associative algebra are not well understood. In this paper, I give the classification of finite-dimensional indecomposable untwisted or twisted modules for the polynomial ring in one variable over ${\mathbb C}$ as a vertex algebra.