2021/09/14 by Dražen Adamović, Drazen Adamovic, Qing Wang +2
Mathematics · Physics and Astronomy · #17B69 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math-ph #math.MP #math.QA #math.RT #msc:17B69
paper · pdf · doi:10.48550/arxiv.2109.06475
22 pages
arxiv created 2021/09/14 · openalex publication_date 2021/09/14 · arxiv updated 2021/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a subalgebra F of the Clifford vertex superalgebra (bc system) which is completely reducible as a LVir (-2,0)-module, C2-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra SF(1). We show that SF(1) and F are in an interesting duality, since 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 arXiv:1711.11342, we decompose Lk(\mathfrakosp(1\vert 2)) at the critical level k=-3/2 as a module for Lk(\mathfraksl(2)). The decomposition of Lk(\mathfrakosp(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 F and SF(1), we prove that Lk(\mathfrakosp(1\vert 2)) and \mathcal V(2) are in the duality of the same type. As an application, we construct and classify all irreducible Lk(\mathfrakosp(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(\mathfrakosp(1\vert 2)) as a N-3/2(\mathfraksl(2))-module. We extend this example, and for each p ≥ 2, we introduce 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\mathfraksl(2).