On Charged Fields with Group Symmetry and Degeneracies of Verlinde's Matrix S

TL;DR

This paper proves that in certain quantum field theories with symmetry groups, the degeneracies of Verlinde's S-matrix correspond to orbifold theories, clarifying the structure of superselection sectors.

Contribution

It demonstrates that degenerate theories are orbifold theories and confirms Rehren's conjecture about the non-degeneracy of the enlarged theory.

Findings

The field net has no nontrivial DHR sectors if observables have finitely many sectors.

Degenerate theories are shown to be orbifold theories.

The symmetry in generic models factorizes into a group part and a quantum part.

Abstract

We consider the complete normal field net with compact symmetry group constructed by Doplicher and Roberts starting from a net of local observables in >=2+1 spacetime dimensions and its set of localized (DHR) representations. We prove that the field net does not possess nontrivial DHR sectors, provided the observables have only finitely many sectors. Whereas the superselection structure in 1+1 dimensions typically does not arise from a group, the DR construction is applicable to `degenerate sectors', the existence of which (in the rational case) is equivalent to non-invertibility of Verlinde's S-matrix. We prove Rehren's conjecture that the enlarged theory is non-degenerate, which implies that every degenerate theory is an `orbifold' theory. Thus, the symmetry of a generic model `factorizes' into a group part and a pure quantum part which still must be clarified.