Homotopy, net-cohomology and superselection sectors in globally hyperbolic spacetimes

In this paper we show that the superselection sectors of a net of local observables, in arbitrary 4-dimensional globally hyperbolic spacetimes, define, as it happens in the Minkowski space, a C∗−category in which the charge structure manifests itself by the existence of a tensor product, a symmetry and a conjugation. The mathematical framework is that of the net-cohomology of posets according to J.E. Roberts. The net of local observables is indexed by a poset formed by a basis for the topology of the spacetime ordered under inclusion. The category of sectors, is equivalent to the category of 1-cocycles of the poset with values in the net. With respect to the papers [7, 14], we succeed to analyze the structure of this category because we show how topological properties of the spacetime are encoded in the poset used as index set: the first homotopy group of a poset is introduced and it is shown that the fundamental group of the poset and the one of the underlying spacetime are isomorphic; any 1-cocycle defines a unitary representation of these fundamental groups. Another important result is the invariance of the net-cohomology under a suitable change of index set of the net.