Two-loop Renormalization in Quantum Gravity near Two Dimensions

We study two--loop renormalization in $(2+\epsilon)$--dimensional quantum gravity. As a first step towards the full calculation, we concentrate on the divergences which are proportional to the number of matter fields. We calculate the $\beta$ functions and show how the nonlocal divergences as well as the infrared divergences cancel among the diagrams. Although the formalism includes a subtlety concerning the general covariance due to the dynamics of the conformal mode, we find that the renormalization group allows the existence of a fixed point which possesses the general covariance. Our results strongly suggest that we can construct a consistent theory of quantum gravity by the $\epsilon$ expansion around two dimensions.