Abstract :
Two-dimensional gravity with an R2-term is quantized around the R2-Liouville solution in the semiclassical way. Renormalization, regularization (infra-red, ultra-violet) and a topological term ∂ (ϕ∂ϕ)) ) are carefully treated. All (1-loop) divergences are renormalized by the cosmological constant (μ) and the R2-coupling-constant (β) for the case β > 0. The quantum meaning of the topological term is clarified. The renormalization group beta-functions of the couplings β and μ are obtained. It is found that the theory is conformal (i.e. the if beta-functions = 0) for w = (β/A) · (16π · 48π/(26 − cm)) ⩾ 1 (where A is a fixed area) exactly when the coupling constant ξ of the topological term takes the value of 1. As for 0 < w < 1, β is asymptotically free for cm < 26 and μ is asymptotically non-free.
