Quantum geometry of topological gravity

We study a c = -2 conformal field theory coupled to two-dimensional quantum gravity by means of dynamical triangulations. We define the geodesic distance r on the triangulated surface with N triangles, and show that dim[r(dH)] = dim[N], where the fractal dimension d(H) = 3.58 +/- 0.04. This result lends support to the conjecture d(H) = -2 alpha(1)/alpha(-1), where alpha(-n) is the gravitational dressing exponent of a spin-less primary field of conformal weight (n + 1, n + 1), and it disfavors the alternative prediction d(H) = -2/gamma(str). On the other hand, we find dim[l] = dim[r(2)] with good accuracy, where l is the length of one of the boundaries of a circle with (geodesic) radius r, i.e. the length I has an anomalous dimension relative to the area of the surface, It is further shown that the spectral dimension d(s) = 1.980 +/- 0.014 for the ensemble of (triangulated) manifolds used. The results are derived using finite size scaling and a very efficient recursive sampling technique known previously to work well for c = -2. (C) 1997 Elsevier Science B.V.