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

TL;DR

This paper proves the rigidity and braided structure of weight module categories over certain affine superalgebras at fractional levels, confirming fusion decomposition conjectures and introducing new techniques for constructing intertwining operators.

Contribution

It establishes the rigidity of weight module categories for affine $rak{sl}(2)$ and $rak{N}=2$ superconformal algebras, confirming fusion product formulas and developing novel intertwining operator methods.

Findings

Categories of weight modules are rigid and braided ribbon.

Fusion product decomposition formulas are validated.

New techniques for intertwining operator construction are introduced.

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.