The minimal model \( \mathfrak{osp}\left(1|2\right) \) vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra \( \hat{\mathfrak{osp}}\left(1\left|2\right.\right) \) 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 \( \mathfrak{osp}\left(1|2\right) \) 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.

中文翻译：

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允许的水平osp 1 2 $$ \ mathfrak {osp} \ left（1 \ left | 2 \ right。\ right）$$最小模型及其相关的最高重量模块

最小模型\（\ mathfrak {osp} \ left（1 | 2 \ right）\）顶点算子超级代数是由仿射李超级代数\（\ hat {\ mathfrak {osp} } \ left（1 \ left | 2 \ right。\ right）\）在级别k的某些有理值处。我们分类ℤ的所有同构类₂ -graded简单轻松的权重最高的模块在最小模型\（\ mathfrak {OSP} \左（1个| 2 \右）\）Neveu–Schwarz和Ramond部门中的顶点算子超级代数。为此，我们结合了自由场实现，筛选算子和以Jack为基础的对称函数理论，以计算Neveu-Schwarz和Ramond区域的朱代数的显式表示。根据级别，使用两种不同的自由域实现。对于k <-1，自由场实现类似于仿射的Wakimoto自由场实现（\ mathfrak {sl}（2）\），最初是由于Bershadsky和Ooguri所致。它涉及1个自由玻色子（或1个Heisenberg顶点代数），1个βγ玻色子幻影系统和1个bc铁离子幻影系统。对于k> -1，此处提出的论点要求将βγ系统嵌入到不确定的2级晶格顶点代数中，以使其玻色化。