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

Autori
Adamović, Dražen ; Wang, Qing

Izvornik
Journal of algebra (0021-8693) 631 (2023); 72-105

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Vertex superalgebra ; Logarithmic vertex algebra ; Category O ; Relaxed highest weight module

Sažetak
We introduce a certain subalgebra $\overline F$ of the Clifford vertex superalgebra ($bc$ system), which is completely reducible (as a $L^{; ; ; Vir}; ; ; (-2, 0)$-- module) and $C_2$--cofinite, but not conformal (and not isomorphic to the symplectic fermion algebra $\mathcal{; ; ; SF}; ; ; (1)$). We show that there is an interesting duality between $\mathcal{; ; ; SF}; ; ; (1)$ and $\overline{; ; ; F}; ; ; $, given by the fact that $\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 from \cite{; ; ; A-2019}; ; ; , we 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 that of the $N=4$ superconformal vertex algebra with central charge $c=-9$, denoted by $\mathcal V^{; ; ; (2)}; ; ; $. Using the duality between $\overline{; ; ; F}; ; ; $ and $\mathcal{; ; ; SF}; ; ; (1)$, we prove that $L_k(\mathfrak{; ; ; osp}; ; ; (1\vert 2))$ and $\mathcal V^{; ; ; (2)}; ; ; $ are in a duality of the same type. As an application, we 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. We also describe the structure of the parafermion algebra $N_{; ; ; -3/2}; ; ; (\mathfrak{; ; ; osp}; ; ; (1\vert 2))$ as a $N_{; ; ; -3/2}; ; ; (\mathfrak{; ; ; sl}; ; ; (2))$-- module. Extending this example, we introduce for each $p \ge 2$ a non-conformal vertex algebra $\mathcal A^{; ; ; (p)}; ; ; _{; ; ; new}; ; ; $ and show that $\mathcal A^{; ; ; (p)}; ; ; _{; ; ; new}; ; ; $ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra $ \mathcal V^{; ; ; (p)}; ; ; _{; ; ; new}; ; ; $, which is isomorphic to the logarithmic vertex algebra $\mathcal V^{; ; ; (p)}; ; ; $ as a module for $\widehat{; ; ; \mathfrak{; ; ; sl}; ; ; }; ; ; (2)$. We also conjecture the existence of the conformal vertex algebra $\mathcal V(\mathfrak{; ; ; sp}; ; ; (2n))$ which is in a duality with the affine vertex algebra $L_{; ; ; -n-1/2}; ; ; (osp(1 \vert 2n))$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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