Four-dimensional lattice gauge theory with ribbon categories and the Crane–Yetter state sum

Abstract

Lattice Gauge Theory in four-dimensional Euclidean space–time is generalized to ribbon categories which replace the category of representations of the gauge group. This provides a framework in which the gauge group becomes a quantum group while space–time is still given by the “classical” lattice. At the technical level, this construction generalizes the spin foam model dual to lattice gauge theory and defines the partition function for a given triangulation of a closed and oriented piecewise-linear four-manifold. This definition encompasses both the standard formulation of d=4 pure Yang–Mills theory on a lattice and the Crane–Yetter invariant of four-manifolds. The construction also implies that certain classes of spin foam models formulated using ribbon categories are well-defined even if they do not correspond to a topological quantum field theory.