QFT and Topology in Two Dimensions: $$\mathrm{SL}(2, {{\mathbb {R}}})$$-Symmetry and the de Sitter Universe

Abstract

We study bosonic quantum field theory on the double covering \(\widetilde{dS}_{2}\) of the two-dimensional de Sitter universe, identified to a coset space of the group \(\mathrm{SL}(2, {{\mathbb {R}}})\). The latter acts effectively on \(\widetilde{dS}_{2}\) and can be interpreted as it relativity group. The manifold is locally identical to the standard the Sitter spacetime \({dS}_2\); 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 \(\mathrm{SL}(2, {{\mathbb {R}}})\)-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.