Vertex operator superalgebras and 16-fold way

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.