Quantum gravity and non-linear sigma models

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My research focuses on the problem of constructing a quantum theory which describes the interaction of fields and particles with the gravitational field. This remains THE unsolved problem in fundamental theoretical physics. My particular research program in this area involves constructing toy quantum gravity models; that is, theories which address the basic issue but are less encumbered by the profound technical difficulties associated with this problem. Typically, these toy models involve physics in which`space' has less than three dimensions. One such class of theories, quantum dilaton gravity, has only one spatial dimension. These models can be pushed quite far, and in them one can address some of the difficult issues in quantum gravity- for instance, the problem of time, the occurrence of singularities and the decay of black holes. My current work in this area proceeds from a novel procedure for constructing quantum theories, and predicts a discrete, lattice-like structure for space. Another problem that I work on concerns non-linear sigma models. String theories are a type of non-linear sigma model in which the background spacetime geometry is flat, as in the special theory of relativity. If the background is not flat, then the self-consistency of the theory requires that certain equations, called the RG flow, have fixed points and these fixed points are the permissible backgrounds. I am looking at RG flows and addressing questions concerning the relationship between these equations and the dynamical changes (that is, changes in time) of geometries and other fields. Finally, I am looking at theories which generalize conformal field theories. The latter are field theories over a two dimensional spacetime, characterized by an infinite dimensional symmetry algebra, called the Kac-Moody algebra. I am looking at generalizations which are field theories in a four dimensional spacetime, and are characterized by a certain generalizations of the Kac-Moody algebras, called 2-Toroidal Lie algebras.