Direct limit completions of vertex tensor categories

TL;DR

This paper demonstrates that direct limit completions of vertex tensor categories preserve their structure, enabling the extension of vertex operator algebra theories to infinite orders and relating various module categories.

Contribution

It introduces conditions under which direct limit completions inherit vertex and braided tensor structures, extending the applicability of vertex operator algebra theory to infinite-order extensions.

Findings

Direct limit completions inherit vertex and braided tensor structures.

The theory applies to all known Virasoro and affine Lie algebra tensor categories.

Relates module categories of affine superalgebras and super Virasoro algebra to Virasoro modules.

Abstract

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.