Tensor network formulation of two-dimensional gravity

Abstract

We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We show evidence that the lattice theory is purely topological in nature by formulating it on lattices with differing topologies and show that the partition function depends only on the Euler character of the triangulation.