We derive explicit forms of the two-point correlation functions of the O(N) nonlinear sigma model at the critical point, in the large-N limit, on various three-dimensional manifolds of constant curvature. The two-point correlation function, G(x, y), is the only n-point correlation function which survives in this limit. We analyze the short distance and long distance behaviors of G(x, y). It is shown that G(x, y) decays exponentially with the Riemannian distance on the spaces R2×S1, S1×S1×R, S2×R, H2×R. The decay on R3 is of course a power law. We show that the scale for the correlation length is given by the geometry of the space and therefore the long distance behavior of the critical correlation function is not necessarily a power law even though the manifold is of infinite extent in all directions; this is the case of the hyperbolic space where the radius of curvature plays the role of a scale parameter. We also verify that the scalar field in this theory is a primary field with weight [Formula: see text]; we illustrate this using the example of the manifold S2×R whose metric is conformally equivalent to that of R3–{0} up to a reparametrization.
Power-law out of time order correlation functions in the SYK model
