While General Relativity is not renormalizable with perturbation theory close to d=4, close to d=2 there exist an UV fixed point within the perturbative regime. The idea that such a fixed point still persist in d=4 in the non-perturbative regime was originally formulated by Weinberg in the 70s. Today, functional methods collected many evidences for the existence of such UV completion, however they do not allow for a clear investigation of the spectrum of the theory and a right definition of the observables. Reviving the original formulation of the conjecture, we employ perturbation theory in a scheme that allows for an explicit analytic continuation of the quantum theory to any value of d and inspect the limits d->2 and d->4. Specifically, we focus on the UV critical surface and how higher derivative operators contribute to it.
