Topological Interpretations of Lattice Gauge Field Theory

Abstract:

We construct lattice gauge field theory based on a quantum group on a lattice of dimension one. Innovations include a coalgebra structure on the connections and an investigation of connections that are not distinguishable by observables. We prove that when the quantum group is a deformation of a connected algebraic group G (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of $G$-characters of the fundamental group of the lattice. Finally, we investigate lattice gauge field theory based on quantum SL ₂ℂ, and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice.