Title:

Summary of discrete models for quantum gravity in three dimensions

We investigate discrete models for quantum gravity in three dimensions, based on topological quantum field theories. We begin by introducing the two main types of model which we shall investigate, namely Penrose's spin network model, and the PonzanoRegge and TuraevViro simplicial state sum models. We briefly review the work of Ponzano and Regge showing how, in a certain semiclassical limit of their model, they cover threedimensional Euclidean gravity. We go on to describe new work, by the author and J.W. Barrett, showing how the stationary points of a different semiclassical limit of the PonzanoRegge partition function may be mapped to flat threedimensional Lorentzian space, and consider how this partition function may be interpreted as a discrete version of a path integral for gravity in three dimensions. We describe the formalism of a topological quantum field theory. We examine the theory of Turaev and Viro, which is given by a simplicial state sum on a threedimensional manifold, based on representations of the quantum group U_{q}(sl(2)). We show that it reduces to the PonzanoRegge model in its q → 1 limit, and so it may be regarded as the naturally regularised version of that model. We describe new work investigating the relation between spin networks and simplicial state sum models. We show how the space of spin networks describes the state space for a twodimensional surface in the PonzanoRegge theory, and give a definition of the inner product on the state space which reproduces the topological inner product, defined by the union of two threemanifolds along their common boundary. We introduce Kauffman's qdeformed spin networks, and define the skein space, to which qspin networks belong. We conjecture that a quotient of the skein space of a surface is isomorphic to the state space of the TuraevViro theory.
