The structure of parafermion vertex operator algebras \(K(osp(1|2n),k)\) (Q2068157)

Fix positive integers \(n\) and \(k\). Consider the simple Lie superalgebra \(\mathfrak{g} = osp(1|2n)\). Let \(\mathfrak{h}\) be its Cartan subalgebra. Let \(V(k, 0)\) be the vertex operator algebra associated to the universal vacuum \(\hat{\mathfrak{g}}\)-module, \(L(k,0)\) be its simple quotient. Let \(M_{\hat{\mathfrak{h}}}(k, 0)\) be the Heisenberg vertex operator algebra. The paper studies the structure of three vertex operator algebras. \begin{itemize} \item[1.] The vertex operator subalgebra \(V(k, 0)(0) = \{v\in V(k, 0): h(0) v = 0, \forall h\in \mathfrak{h}\}\) of \(V(k, 0)\). \item[2.] The commutant vertex operator algebra \(N(osp(1|2n), k)\) of \(M_{\mathfrak{h}}(k, 0)\) in \(V(k, 0)\). \item[3.] The parafermion vertex operator algebra \(K(osp(1|2), k)\) associated to \(L(k, 0)\), which is the simple quotient of \(N(osp(1|2), k)\). \end{itemize} The paper gives explicit formulas for the generators of these three vertex operator algebras. Besides, the paper also gives explicit formulas for the generators of the unique maximal ideal in \(N(osp(1|2n), k)\). Remarkably, these formulas shows that \(K(osp(1|2), k)\) and \(K(sl_2, 2k)\) are the buidling blocks of \(K(osp(1|2n), k)\). They will be useful in the study of the representation theory of \(K(osp(1|2n),k)\).