A duality between vertex superalgebras [formula omitted] and [formula omitted] and generalizations to logarithmic vertex algebras

Abstract

We introduce a certain subalgebra F‾ of the Clifford vertex superalgebra (bc system), which is completely reducible (as a LVir(−2,0)–module) and C2–cofinite, but not conformal (and not isomorphic to the symplectic fermion algebra SF(1)). We show that there is an interesting duality between SF(1) and F‾, given by the fact that F‾ can be equipped with the structure of a SF(1)–module and vice versa.Using the decomposition of F‾ and a free-field realization from [3], we decompose Lk(osp(1|2)) at the critical level k=−3/2 as a module for Lk(sl(2)). The decomposition of Lk(osp(1|2)) is exactly the same as that of the N=4 superconformal vertex algebra with central charge c=−9, denoted by V(2). Using the duality between F‾ and SF(1), we prove that Lk(osp(1|2)) and V(2) are in a duality of the same type. As an application, we construct and classify all irreducible Lk(osp(1|2))–modules in the category O and the category R, which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra N−3/2(osp(1|2)) as a N−3/2(sl(2))–module.Extending this example, we introduce for each p≥2 a non-conformal vertex algebra Anew(p) and show that Anew(p) is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra Vnew(p), which is isomorphic to the logarithmic vertex algebra V(p) as a module for slˆ(2).We also conjecture the existence of the conformal vertex algebra V(sp(2n)) which is in a duality with the affine vertex algebra L−n−1/2(osp(1|2n)).