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

TL;DR

This paper explores two different ways to realize the diffeomorphism group in metric gravity, renormalizes them in dimensions slightly above two, and discusses implications for quantum gravity in higher dimensions.

Contribution

It introduces a new dimensional continuation procedure for metric theories and compares two realizations of the diffeomorphism group, highlighting their quantum differences.

Findings

Identification of two classically equivalent but quantum mechanically distinct realizations.

Development of a new method for dimensional continuation of metric theories.

Evidence suggesting a conformal window extending beyond four dimensions.

Abstract

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