Vertex operator superalgebras and the 16-fold way

Abstract

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