Gravity in d=2+ε dimensions and realizations of the diffeomorphisms group

We discuss two separate 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 for ultraviolet completions of quantum gravity in d > 2 and give some perturbative mean to assess their 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.

Automated summary

This study explores two separate realizations of the diffeomorphism group in metric gravity, showing classical equivalence but quantum distinction. By renormalizing them in specific dimensions, a new dimensional continuation procedure for metric theories is developed, shedding light on potential ultraviolet completions of quantum gravity candidates. The results hint at the presence of a conformal window in dimensions extending beyond four.