Quantum geometry of topological gravity

Abstract

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 d H ] = dim[ N], where the fractal dimension d H = 3.58 ± 0.04. This result lends support to the conjecture d H = −2α 1 α −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 d H = −2 γ 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 l 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.