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As an application of Roberts' cohomology (net cohomology), we prove thecompleteness of the DHR sectors of the local observables of the model in thetitle, detailed in [8]. This result is achieved via the triviality of the net1-cohomology, with values in the local fields, enhancing the Roberts' methodsto the case of anyonic Weyl nets, not satisfying the split property. We takeadvantage of using different causal index sets for the nets involved. Thepresence of anyonic commutation relations is treated introducing the notion ofnets graded by a generic group, and the related properties of graded localityand graded duality. As a further result, we obtain the description of twistedand untwisted sectors of the model as two symmetric subcategories of aW*-braided category, whose objects are the same as the dual category of thecompact Abelian group of the gauge symmetry. The work also furnish some hintsfor the analysis of the sector structure of generic models on differentspacetimes.

Massless scalar free Field in 1+1 dimensions, II: Net Cohomology and Completeness of Superselection Sectors