Cofiniteness and $P(z)$-tensor product bifunctors in orbifold theories associated to abelian but not-necessarily-finite groups

TL;DR

This paper develops $P(z)$-tensor product bifunctors for categories of $C_{n}$-cofinite twisted modules over a vertex algebra with an abelian automorphism group, including infinite and nonsemisimple automorphisms.

Contribution

It extends the construction of $P(z)$-tensor product bifunctors to cases with infinite, possibly nonsemisimple automorphism groups acting on vertex algebras.

Findings

Constructed $P(z)$-tensor bifunctors for infinite abelian automorphism groups.

Handled nonsemisimple automorphisms and infinite order automorphisms.

Extended tensor product theory to broader orbifold settings.

Abstract

Let $V$ be a M\"{o}bius vertex algebra and $G$ an abelian group of automorphisms of $V$. We construct $P(z)$-tensor product bifunctors for the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules (without $g$-actions) for $g\in G$ and the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules with $G$-actions for $g\in G$. In this paper, an automorphism $g$ of $V$ can be of infinite order and does not have to act semisimply on $V$, and the group $G$ can be an infinite abelian group containing nonsemisimple automorphisms of $V$.