$C_n$-cofinite twisted modules for $C_2$-cofinite vertex operator algebras

TL;DR

This paper introduces a new concept of $C_n$-cofiniteness for twisted modules of vertex operator algebras and proves that all finitely-generated weak twisted modules are $C_n$-cofinite under certain conditions.

Contribution

It defines $C_n$-cofiniteness for weak $g$-twisted modules and proves their $C_n$-cofiniteness for all positive $n$ when the VOA is $C_2$-cofinite and of CFT type.

Findings

All finitely-generated weak $g$-twisted modules are $C_n$-cofinite for all positive $n$.

The notion of $C_n$-cofiniteness extends the understanding of module finiteness properties.

The results apply to $V$ with automorphisms $g$ in the context of $C_2$-cofinite VOAs.

Abstract

Given a vertex operator algebra $ V $ with a general automorphism $ g $ of $ V $, we introduce a notion of $ C_n $-cofiniteness for weak $ g $-twisted $ V $-modules. When $ V $ is $ C_2 $-cofinite and of CFT type, we show that all finitely-generated weak $ g $-twisted $ V $-modules are $ C_n $-cofinite for all $ n \in \mathbb{Z}_{>0} $.