Massless scalar free Field in 1+1 dimensions, II: Net Cohomology and   Completeness of Superselection Sectors
  As an application of Roberts' cohomology (net cohomology), we prove the completeness of the DHR sectors of the local observables of the model in the title, detailed in [8]. This result is achieved via the triviality of the net 1-cohomology, with values in the local fields, enhancing the Roberts' methods to the case of anyonic Weyl nets, not satisfying the split property. We take advantage of using different causal index sets for the nets involved. The presence of anyonic commutation relations is treated introducing the notion of nets graded by a generic group, and the related properties of graded locality and graded duality. As a further result, we obtain the description of twisted and untwisted sectors of the model as two symmetric subcategories of a W*-braided category, whose objects are the same as the dual category of the compact Abelian group of the gauge symmetry. The work also furnish some hints for the analysis of the sector structure of generic models on different spacetimes. 
Fabio Ciolli