Infrared behavior at negative curvature

It is argued that constant negative curvature provides a very convenient infrared regulator for interacting euclidean quantum field theories. The two-dimensional XY model on a space with such curvature is disordered at arbitrarily small non-zero temperatures by topologically non-trivial fluctuations (vortices), even though the effect of perturbative fluctuations (spin-waves) is negligible at large distances. Similarly, constant negative curvature brings the infrared behavior of QCD (as well as the ultraviolet behavior, of course) under analytic control without any conflict with gauge invariance. In particular, for weak coupling there is a unique single instanton solution and the instanton gas is truly dilute.