Conformal nets from minimal W-algebras
Sebastiano Carpi, Tiziano Gaudio
TL;DR
The paper establishes that all unitary minimal $W$-algebras are strongly graded-local and hence define irreducible graded-local conformal nets, unifying constructions of Virasoro and $N=1,2,3,4$ super-Virasoro nets (up to free-fermion factors) and enabling the new $N=3$ and $N=4$ nets. It correlates strong graded locality with representation-theoretic properties, showing that strongly rational minimal $W$-algebras yield completely rational nets, while non-rational minimal $W$-algebras furnish non-completely rational nets via infinite families of irreducible representations. The representation theory for the resulting nets is developed across the Virasoro and various super-Virasoro cases, providing explicit unitary representations for $c$ in unitary ranges and demonstrating non-rationality outside those ranges. The results solidify the connection between VOA/W-algebra unitarity, graded-local nets, and complete rationality, and they deliver a suite of new conformal nets with precise representation-theoretic behavior, including the first rigorous constructions of $N=3$ and $N=4$ graded-local nets in this framework.
Abstract
We show the strong graded locality of all unitary minimal W-algebras, so that they give rise to irreducible graded-local conformal nets. Among these unitary vertex superalgebras, up to taking tensor products with free fermion vertex superalgebras, there are the unitary Virasoro vertex algebras (N=0) and the unitary N=1,2,3,4 super-Virasoro vertex superalgebras. Accordingly, we have a uniform construction that gives, besides the already known N=0,1,2 super-Virasoro nets, also the new N=3,4 super-Virasoro nets. All strongly rational unitary minimal W-algebras give rise to previously known completely rational graded-local conformal nets and we conjecture that the converse is also true. We prove this conjecture for all unitary W-algebras corresponding to the N=0,1,2,3,4 super-Virasoro vertex superalgebras.