Singular Structure in 4D Simplicial Gravity

We show that the phase transition previously observed in dynamical triangulation models of quantum gravity can be understood as being due to the creation of a singular link. The transition between singular and non-singular geometries as the gravitational coupling is varied appears to be first order. Dynamical triangulations (DT) furnish a powerful approach to the problem of defining and studying a non-perturbative theory of quantum gravity. Simply put, the functional integral over (euclidean) four-geometries is defined as some scaling limit of a sum over abstract simplicial manifolds. In two dimensions this approach has been very successful [1]. In addition to the calculation of gravitationally dressed anomalous dimensions, the DT approach, being discrete, allows for the use of nonperturbative methods such as computer simulation. In two dimensions this has yielded new results; the measurement of fractal dimensions characterizing the quantum geometry [2], power law behavior of matter field correlators on geodesic paths [3] and insight into the problem of formulating a renormalization group for quantum gravity [4]. The observation of a two phase structure for the four dimensional models was noticed early on [5]. This led to the speculation that a continuum theory of gravity could be constructed by taking an appropriate scaling limit near the phase transition. However, in [6], evidence was presented that the transition was discontinuous. Such a scenario would rule out arguably the simplest possibility – that a continuum theory could be obtained by tuning the theory to some critical (gravitational) coupling. Independent of these issues it was noted in [7] that the crumpled phase of this model possessed a singular structure – that the sum over triangulations was dominated by those with a single link common to a very large number of four-simplices. The endpoints of this link, termed singular vertices, were shared by a number of