Vertex algebras related to regular representations of SL₂

2025/02/03 by Dražen Adamović, Adamovic, Drazen, Antun Milas +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2502.01766

openalex publication_date 2025/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by C_p, p ≥ 2, which are closely related to the vertex algebra of chiral differential operators on SL(2) at level -2+(1)/(p). We prove that for p = 3, there is an isomorphism between C₃ and the affine vertex algebra L_-5/3(\mathfrakg₂) from Deligne's series. Moreover, we also establish isomorphisms between C₄ and C₅ and certain affine W-algebras of types F₄ and E₈, respectively. In this way, we resolve the problem of decomposing certain conformal embeddings of affine vertex algebras into affine W-algebras. An important feature is that C_p is (1)/(2) ℤ_{≥ 0}-graded with finite-dimensional graded subspaces and convergent characters. Therefore, for all p ≥ 2, we show that the characters of C_p exhibit modularity, supporting the conjectural quasi-lisse property.

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