Modularity of logarithmic parafermion vertex algebras

Abstract

The parafermionic cosets $C_k=\mathrm{Com}(H,L_k({\mathfrak{sl}}_2))$ are studied for negative admissible levels $k$, as are certain infinite-order simple current extensions $B_k$ of $C_k$. Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to $C_k$, all irreducible $C_k$- and $B_k$-modules are obtained from those of $L_k({\mathfrak{sl}}_2)$, as are the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible $B_k$-modules. The irreducible $C_k$- and $B_k$-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the $B_k$ are $C_2$-cofinite vertex operator algebras.

28 pages; v2 31 pages: many clarifications and improvements, especially to the example in Sec. 4.3

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Cited by in corpus (15)

• Tensor categories for vertex operator superalgebra extensions
• Braided tensor categories of admissible modules for affine Lie algebras
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Direct limit completions of vertex tensor categories
• Unitary and non-unitary $N=2$ minimal models
• Positive energy representations of affine vertex algebras
• Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
• Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$
• On parafermion vertex algebras of $\mathfrak{sl}(2{)}_{-3/2}$ and $\mathfrak{sl}(3{)}_{-3/2}$
• $N=1$ super Virasoro tensor categories
• New polar-finite forms of generalized Euler identities for $A_1^{(1)}$-string functions and mock theta conjecture-like identities
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
• Relaxed and logarithmic modules of $\widehat{{\mathfrak{sl}}_3}$
• Simple modules for Affine vertex algebras in the minimal nilpotent orbit