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[rdH] = dim[N], where the fractal dimension dH = 3.58 ± 0.04. This result lends support to the conjecture dH = −2α1α−1, where α−n is the gravitational dressling exponent of a spin-less primary field of conformal weight (n + 1, n + 1), and it disfavors the alternative prediction dH = −2γstr. On the other hand, we find dim[l] = dim[r2] with good accuracy, where l is the length of one of the boundaries of a circle with (geodesic) radius r, i.e. the length l has an anomalous dimension relative to the area of the surface. It is further shown that the spectral dimension ds = 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.