A duality between vertex superalgebras \(L_{-3/2}(\mathfrak{osp}(1|2))\) and \(\mathcal{V}^{(2)}\) and generalizations to logarithmic vertex algebras (Q6169043)

The authors introduce a subalgebra \(\overline F\) of the Clifford vertex superalgebra which is completely reducible as a \(L^{Vir} (-2,0)\)-module, \(C_2\)-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra \(\mathcal{SF}(1)\). They show that \(\mathcal{SF}(1)\) and \(\overline{F}\) are in an interesting duality, since \(\overline{F}\) can be equipped with the structure of a \(\mathcal{SF}(1)\)-module and vice versa. Using the decomposition of \(\overline F\) and a free-field realization, the authors decompose \(L_k(\mathfrak{osp}(1\vert 2))\) at the critical level \(k=-3/2\) as a module for \(L_k(\mathfrak{sl}(2))\). The decomposition of \(L_k(\mathfrak{osp}(1\vert 2))\) is exactly the same as of the \(N=4\) superconformal vertex algebra with central charge \(c=-9\), denoted by \(\mathcal V^{(2)}\). Using the duality between \(\overline{F}\) and symplectic fermion algebra \(\mathcal{SF}(1)\), they prove that \(L_k(\mathfrak{osp}(1\vert 2))\) and \(\mathcal V^{(2)}\) are in the duality of the same type. The authors also construct and classify all irreducible \(L_k(\mathfrak{osp}(1\vert 2))\)-modules in the category \(\mathcal O\) and the category \(\mathcal R\) which includes relaxed highest weight modules.