Vertex operator superalgebras and the 16-fold way

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 $V_{\bar 0}$-module category $\mathcal {C}_{V_{\bar 0}}$ is a modular tensor category. In this paper, we prove that $\mathcal {C}_{V_{\bar 0}}$ is a fermionic modular tensor category and the Müger centralizer $\mathcal {C}_{V_{\bar 0}}^0$ of the fermion in $\mathcal {C}_{V_{\bar 0}}$ is generated by the irreducible $V_{\bar 0}$-submodules of the $V$-modules. In particular, $\mathcal {C}_{V_{\bar 0}}^0$ is a super-modular tensor category and $\mathcal {C}_{V_{\bar 0}}$ is a minimal modular extension of $\mathcal {C}_{V_{\bar 0}}^0$. We provide a construction of a vertex operator superalgebra $V^l$ for each positive integer $l$ such that $\mathcal {C}_{{(V^l)_{\bar 0}}}$ is a minimal modular extension of $\mathcal {C}_{V_{\bar 0}}^0$. We prove that these modular tensor categories $\mathcal {C}_{{(V^l)_{\bar 0}}}$ are uniquely determined, up to equivalence, by the congruence class of $l$ modulo 16.