Spin Foam Models of Quantum Gravity

We review the Ponzano-Regge model of 3-dimensional quantum gravity and explain how its infrared divergences are eliminated in the Turaev-Viro model by replacing the group SU(2) by the quantum group SU_q(2), which corresponds to introducing a nonzero cosmological constant. We also describe how these models and their generalizations to other groups and quantum groups fit into the framework of "higher-dimensional algebra" (or in other words, the theory of n-categories). Then we discuss recent extensions of these ideas to 4-dimensional quantum gravity and topological quantum field theories. All the models considered can be formulated either as state sum models in terms of triangulated manifolds as "spin foam models", in which the faces and edges of the dual 2-skeleton are labelled with representations and intertwiners, respectively. This dual formulation clarifies the relation to spin networks, which form a basis of states in loop quantum gravity.