Vertex operator superalgebras and 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 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 Müger 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 $C_{V_{\bar{0}}^l}$ is minimal modular extension of $C_{V_{\bar{0}}}^0$. We prove that these modular tensor categories $C_{V_{\bar{0}}^l}$ are uniquely determined, up to equivalence, by the congruence class of $l$ modulo 16.

35 pages

Cited by in corpus (1)

• Orbifolds and minimal modular extensions