Modular $A_n(V)$ theory

TL;DR

This paper constructs a series of associative algebras $A_n(V)$ linked to a vertex operator algebra $V$, establishing a correspondence with irreducible modules and characterizing rationality through semisimplicity.

Contribution

It introduces a modular $A_n(V)$ theory for vertex operator algebras over any algebraically closed field, connecting algebraic properties to module classifications.

Findings

One-to-one correspondence between irreducible $A_n(V)$-modules and irreducible $V$-modules.

Rational $V$ iff all $A_n(V)$ are semisimple.

Finite dimensionality of homogeneous subspaces in irreducible modules for rational $V$.

Abstract

A series of associative algebras $A_n(V)$ for a vertex operator algebra $V$ over an arbitrary algebraically closed field and nonnegative integers $n$ are constructed such that there is a one to one correspondence between irreducible $A_n(V)$-modules which are not $A_{n-1}(V)$ modules and irreducible $V$-modules. Moreover, $V$ is rational if and only if $A_n(V)$ is semisimple for all $n.$ In particular, the homogeneous subspaces of any irreducible $V$-module are finite dimensional for rational vertex operator algebra $V.$