QFT and Topology in Two Dimensions: SL (2 , R) -Symmetry and the de Sitter Universe

We study bosonic quantum field theory on the double covering dS2 of the two-dimensional de Sitter universe, identified to a coset space of the group SL(2, R). The latter acts effectively on dS2 and can
be interpreted as it relativity group. The manifold is locally identical to the standard the Sitter spacetime dS2; it is globally hyperbolic, geodesically complete and an inertial observer sees exactly the same bifurcate Killing horizons as in the standard one-sheeted case. The different global
Lorentzian structure causes, however, drastic differences between the two models. We classify all the SL(2, R)-invariant two-point functions and show that: (1) there is no Hawking–Gibbons temperature; (2) there is no covariant field theory solving the Klein–Gordon equation with mass less than 1/2R , i.e., the complementary fields go away.