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We discuss the renormalization of Einstein-Hilbert's gravity in$d=2+\epsilon$ dimensions. We show that the application of the path-integralapproach leads naturally to scheme- and gauge-independent results on-shell, butalso gives a natural notion of quantum metric off-shell, which is the naturalargument of the effective action, even at the leading order in perturbationtheory. The renormalization group of Newton's constant is consistent with theasymptotic safety scenario for quantum gravity in that it has an ultravioletrelevant fixed point. We extend the approach to the analysis of curvaturesquare operators, understood as composites operators, which allows for thedetermination of the spectrum of scaling operators at the scale invariant fixedpoint. The analysis suggests that there is one operator that becomes relevantclose to $d=4$ dimensions, while other operators previously found in theliterature are either marginal or trivial on-shell.

Composite higher derivative operators in $d=2+\epsilon$ dimensions and the spectrum of asymptotically safe gravity