HOMOTOPY OF POSETS, NET-COHOMOLOGY AND SUPERSELECTION SECTORS IN GLOBALLY HYPERBOLIC SPACE-TIMES

Abstract

We study sharply localized sectors, known as sectors of DHR-type, of a net of local observables, in arbitrary globally hyperbolic space-times with dimension ≥ 3. We show that these sectors define, as it happens in Minkowski space, a C*-category in which the charge structure manifests itself by the existence of a tensor product, a permutation 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 space-time ordered under inclusion. The category of sectors, is equivalent to the category of 1-cocycles of the poset with values in the net. We succeed in analyzing the structure of this category because we show how topological properties of the space-time 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 one of the underlying space-time 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.