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

A commutative associative algebra A over C with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for vertex algebra A and the modules for associative algebra A 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 C as a vertex algebra.