On the phase diagram of 2d Lorentzian Quantum Gravity

Recently a new model of 2d quantum gravity has been proposed [1]. It is defined using dynamical triangulations from a subclass of diagrams which can be given a causal structure. Such diagrams are generated by gluing together one dimensional time–slices or “universes” (in our case a set of vertices connected by space-like links forming a diagram with the topology of a circle) with time-like links such that they form a triangulated surface. Vertices connected by timelike links are causally related and a unique time can be assigned to the vertices of each time–slice. Such graphs can be given a Lorentzian metric by defining time-like links to have equal negative length squared and space-like to have positive. All triangles have equal area and the volume of spacetime is proportional to the number of triangles NT . The system has been found to have a non–trivial continuum limit only at an imaginary value of the cosmological constant λ. The geometry of space is maximally fluctuating but the system is much smoother than Liouville gravity: By defining the two point function to be