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

Abstract

We introduce a subalgebra $\overline{F}$ of the Clifford vertex superalgebra ($bc$ system) 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 $\mathcal{S}\mathcal{F}(1)$. We show that $\mathcal{S}\mathcal{F}(1)$ and $\overline{F}$ are in an interesting duality, since $\overline{F}$ can be equipped with the structure of a $\mathcal{S}\mathcal{F}(1)$-module and vice versa. Using the decomposition of $\overline{F}$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(\mathfrak{osp}(1|2))$ at the critical level $k=-3/2$ as a module for $L_k(\mathfrak{sl}(2))$. The decomposition of $L_k(\mathfrak{osp}(1|2))$ 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 $\mathcal{S}\mathcal{F}(1)$, we prove that $L_k(\mathfrak{osp}(1|2))$ and ${\mathcal{V}}^{(2)}$ are in the duality of the same type. As an application, we construct and classify all irreducible $L_k(\mathfrak{osp}(1|2))$-modules in the category $\mathcal{O}$ and the category $\mathcal{R}$ which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra $N_{-3/2}(\mathfrak{osp}(1|2))$ as a $N_{-3/2}(\mathfrak{sl}(2))$-module. We extend this example, and for each $p\ge 2$, we introduce a non-conformal vertex algebra ${\mathcal{A}}_{new}^{(p)}$ and show that ${\mathcal{A}}_{new}^{(p)}$ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra ${\mathcal{V}}_{new}^{(p)}$ which is isomorphic to the logarithmic vertex algebra ${\mathcal{V}}^{(p)}$ as a module for $\widehat{\mathfrak{sl}}(2)$.

22 pages

References in corpus (1)

• On parafermion vertex algebras of $\mathfrak{sl}(2{)}_{-3/2}$ and $\mathfrak{sl}(3{)}_{-3/2}$