Gravity in $${\varvec{d=2+\epsilon }}$$ dimensions and realizations of the diffeomorphisms group

We discuss two distinct realizations of the diffeomorphism group for metric gravity, which give rise to theories that are classically equivalent, but quantum mechanically distinct. renormalize them in $d=2+\epsilon$ dimensions, developing a new procedure dimensional continuation and highlighting connections with constructions previously appeared literature. Our hope is frame candidates ultraviolet completions gravity $d>2$ some perturbative mean assess its existence $d=4$, also speculate on potential obstructions such finite values $\epsilon$. results suggest presence conformal window $d$ seems extend higher than four.