Quantization and topology: S L (2, R) -de Sitter invariant fields in two dimensions

We explore the interplay between quantization, local commutativity and the analyticity properties of the two-point functions of a quantum field in a non trivial topological cosmological background in the example of the two-dimensional de Sitter manifold and its double covering. The global topological differences make the many of the well-known features of de Sitter quantum field theory disappear. In particular there is nothing like a Bunch-Davies vacuum and there are no SL(2,R)-invariant fields whose mass is less than 1/2.