Fermions, quantum gravity and holography in two dimensions

We study a model comprising $N$ flavors of K\"ahler Dirac fermion propagating on a triangulated two dimensional disk which is constrained to have a negative average bulk curvature. Dirichlet boundary conditions are chosen for the fermions. Quantum fluctuations of the geometry are included by summing over all possible triangulations consistent with these constraints. We show in the limit $N\to \infty$ that the partition function is dominated by a regular triangulation of two dimensional hyperbolic space. We use strong coupling expansions and Monte Carlo simulation to show that in this limit boundary correlators of the fermions have a power law dependence on boundary separation as one expects from holography. However we argue that this behavior breaks down for any finite number of massive fields in the thermodynamic limit and quantum fluctuations of the bulk geometry drive the theory into a non-holographic phase. In contrast, for massless fermions we find evidence that the boundary is conformal even for finite $N$. This is consistent with theoretical results in quantum Liouville theory.