Abstract
The minimal model mathfrak{osp}left(1|2right) vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra hat{mathfrak{osp}}left(1left|2right.right) at certain rational values of the level k. We classify all isomorphism classes of ℤ2-graded simple relaxed highest weight modules over the minimal model mathfrak{osp}left(1|2right) vertex operator superalgebras in both the Neveu–Schwarz and Ramond sectors. To this end, we combine free field realisations, screening operators and the theory of symmetric functions in the Jack basis to compute explicit presentations for the Zhu algebras in both the Neveu–Schwarz and Ramond sectors. Two different free field realisations are used depending on the level. For k < −1, the free field realisation resembles the Wakimoto free field realisation of affine mathfrak{sl}(2) and is originally due to Bershadsky and Ooguri. It involves 1 free boson (or rank 1 Heisenberg vertex algebra), one βγ bosonic ghost system and one bc fermionic ghost system. For k > −1, the argument presented here requires the bosonisation of the βγ system by embedding it into an indefinite rank 2 lattice vertex algebra.
Highlights
The orthosymplectic Lie superalgebra osp(1|2) is the Lie superalgebra of endomorphisms of the vector superspace C1|2 that preserves the standard supersymmetric bilinear form on C1|2
The purpose of this article is to classify the simple relaxed highest weight modules over the minimal model osp(1|2) vertex operator superalgebras, that is, the simple quotient vertex operator superalgebras constructed from the affinisation of osp(1|2) at certain rational levels, called admissible levels
In the context of osp(1|2) this strategy was first used by Kac and Wang [5], as an application of their generalisation of Zhu algebras [15] to vertex operator superalgebras, to classify simple Neveu–Schwarz modules when v = 1
Summary
The orthosymplectic Lie superalgebra osp(1|2) is the Lie superalgebra of endomorphisms of the vector superspace C1|2 that preserves the standard supersymmetric bilinear form on C1|2. Every Z2-graded simple relaxed highest weight module over the minimal model osp(1|2) vertex operator superalgebra, B0|1(u, v), at level ku,v is isomorphic to one of the following or their parity reversals. In the context of osp(1|2) this strategy was first used by Kac and Wang [5], as an application of their generalisation of Zhu algebras [15] to vertex operator superalgebras, to classify simple Neveu–Schwarz modules when v = 1 These kinds of calculations require explicit formulae for the singular vector generating the ideal. The proof of these presentations is postponed to Section 4 These presentations are used to prove the main result of the article, Theorem 11, that is, the classification of Z2-graded simple relaxed highest weight modules over B0|1(u, v) and the rationality of B0|1(u, v) in category O. This necessitates splitting the calculation into two cases depending on whether k < −1 or k > −1 in order to assure that these bounds are saturated
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