Landau Theory of Causal Dynamical Triangulations

Understanding the continuum limit of a theory of discrete random geometries is a beautiful but difficult challenge. In this optic, we review here the insights that can be obtained for Causal dynamical triangulations (CDT) by employing the Landau approach to critical phenomena. In particular, concentrating on the cases of two and three dimensions, we will make the case that the configuration of the volume of spatial slices effectively plays the role of an order parameter, helping us understand the phase structure of CDT. Moreover, consistency with numerical simulations of CDT provides hints that the effective field theory of the model lives in the space of theories invariant under foliation-preserving diffeomorphisms. Among such theories, Hořava-Lifshitz gravity has the special status of being a perturbatively renormalizable theory, while general relativity sits in a subspace with enhanced symmetry. In order to reach either of them, one would likely need to fine-tune some of the parameters in the CDT action or additional ones from some generalization thereof.