Four-dimensional Lattice Gauge Theory with ribbon categories and the Crane–Yetter state sum

Lattice Gauge Theory in 4-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 4-manifold. This definition encompasses both the standard formulation of d = 4 pure Yang–Mills theory on a lattice and the Crane–Yetter invariant of 4-manifolds. The construction also implies that a certain class of Spin Foam Models formulated using ribbon categories are well-defined even if they do not correspond to a Topological Quantum Field Theory. PACS: 11.15.Ha, 02.20.Uw, 04.60.Nc key words: lattice gauge theory, duality, spin foam model, ribbon category, quantum group, spin network