Vertex operator superalgebras and 16-fold way

TL;DR

This paper studies the structure of vertex operator superalgebras with a focus on their module categories, revealing a classification pattern based on the integer parameter modulo 16 and constructing associated minimal modular extensions.

Contribution

It establishes that the module categories form fermionic and super-modular tensor categories, and constructs a family of vertex operator algebras with categories classified by integers mod 16.

Findings

The category $C_{V_{ar 0}}$ is a fermionic modular tensor category.

The centralizer $C_{V_{ar 0}}^0$ is generated by irreducible submodules and is super-modular.

The categories $C_{V^l_{ar 0}}$ are uniquely classified by $l mod 16$.

Abstract

Let $V$ be a vertex operator superalgebra with the natural order 2 automorphism $\sigma$. Under suitable conditions on $V$, the $\sigma$-fixed subspace $V_{\bar 0}$ is a vertex operator algebra and the category $C_{V_{\bar 0}}$ of $V_{\bar 0}$-modules is modular tensor category. In this paper, we prove that $C_{V_{\bar 0}}$ is a fermionic modular tensor category and the M\"uger centralizer $C_{V_{\bar 0}}^0$ of the fermion in $C_{V_{\bar 0}}$ is generated by the irreducible $V_{\bar 0}$-submodules of the $V$-modules. In particular, $C_{V_{\bar 0}}^0$ is a super-modular tensor category and $C_{V_{\bar 0}}$ is a minimal modular extension of $C_{V_{\bar 0}}^0$. We provide a construction of a vertex operator $V^l$ for each positive integer $l$ such that $C_{V^l_{\bar 0}}$ is minimal modular extension of $C_{V_{\bar 0}}^0$. We prove that these modular tensor categories $C_{V^l_{\bar 0}}$ are uniquely determined, up to equivalence, by the congruence class of $l$ modulo 16.