Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants

Abstract

We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold. In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension <=4) and with supersymmetric gauge group (in any dimension). Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover. The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to TQFTs with or without embedded spin networks.