On Infinite Order Simple Current Extensions of Vertex Operator Algebras

TL;DR

This paper develops a categorical framework to rigorously construct and analyze infinite order simple current extensions of vertex operator algebras, exemplified by the lattice VOA $V_L$, and clarifies its module category structure.

Contribution

It introduces a direct sum completion of braided monoidal categories to handle infinite order simple current extensions of VOAs.

Findings

Constructed the category $ ext{C}_igoplus$ for infinite order extensions.

Applied the framework to the lattice VOA $V_L$ as an example.

Revealed the module category structure via categorical methods.

Abstract

We construct a direct sum completion $\mathcal{C}_{\oplus}$ of a given braided monoidal category $\mathcal{C}$ which allows for the rigorous treatment of infinite order simple current extensions of vertex operator algebras as seen in \cite{CKL}. As an example, we construct the vertex operator algebra $V_L$ associated to an even lattice $L$ as an infinite order simple current extension of the Heisenberg VOA and recover the structure of its module category through categorical considerations.