Skein Modules and Lattice Gauge Field Theory
Doug Bullock, Charles Frohman, and Joanna Kania-Bartoszyńska
We construct lattice gauge field theory based on a quantum group on a lattice of dimension 1. 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 (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of characters of the fundamental group of the lattice with respect to the corresponding algebraic group. Finally, we investigate lattice gauge field theory based on quantum $SL_2{\mathbb C}$, and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice.