Conformal nets from minimal W-algebras

TL;DR

This paper establishes the strong graded locality of all unitary minimal W-algebras, constructing new super-Virasoro conformal nets and conjecturing a converse relation, with proofs for several superalgebra cases.

Contribution

It provides a uniform construction of conformal nets from unitary minimal W-algebras, including new N=3,4 super-Virasoro nets, and proves a conjecture relating rationality and W-algebras.

Findings

All unitary minimal W-algebras are strongly graded-local.

Constructed new N=3,4 super-Virasoro conformal nets.

Proved the conjecture for N=0,1,2,3,4 super-Virasoro cases.

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.