Extensions of Conformal Nets and Superselection Structures

TL;DR

This paper explores the structure of conformal nets and their duals, revealing new symmetry groups, sector extensions, and examples of non-Moebius covariant sectors, advancing the understanding of superselection sectors in conformal QFT.

Contribution

It demonstrates that the dual net of a conformal quantum field theory remains conformal, introduces an infinite-dimensional symmetry group, and constructs examples of non-Moebius covariant sectors.

Findings

Dual nets are conformal and exhibit normal and conormal properties.

The net and dual net generate an infinite-dimensional symmetry group.

Examples of solitonic sectors and non-Moebius covariant sectors are provided.

Abstract

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Moebius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2,R), showing that they violate 3-regularity for n>2. When n>1, we obtain examples of non Moebius-covariant sectors of a 3-regular (non 4-regular) net.