Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras

Abstract

We prove that the categories of weight modules over the simple $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator algebras at fractional admissible levels and central charges are rigid (and hence the categories of weight modules are braided ribbon categories) and that the decomposition formulae of fusion products of simple projective modules conjectured by Thomas Creutzig, David Ridout and collaborators hold (including when the decomposition involves summands that are indecomposable yet not simple). In addition to solving this old open problem, we develop new techniques for the construction of intertwining operators by means of integrating screening currents over certain cycles, which are expected to be of independent interest, due to their applicability to many other algebras. In the example of $\mathfrak{sl}(2)$ these new techniques allow us to give explicit formulae for a logarithmic intertwining operator from a pair of simple projective modules to the projective cover of the tensor unit, namely, the vertex operator algebra as a module over itself.

62 pages, 3 figures; updated version: fixed typos and implemented referee comments; version to appear in Adv. Math

References in corpus (14)

• sl^(2)_{-1/2}: A Case Study
• Bosonic Ghosts at $c=2$ as a Logarithmic CFT
• Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Unitary and non-unitary $N=2$ minimal models
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Modularity of Bershadsky-Polyakov minimal models
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• An ${\mathfrak{sl}}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• $N=1$ super Virasoro tensor categories
• The regularization of Dotsenko--Fateev integrals
• The singularities of Selberg- and Dotsenko-Fateev-like integrals