Topological Interpretations of Lattice Gauge Field Theory

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,C), and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice.Comment: 35 pages, amslatex, epsfig, many figures; email addresses: [email protected], [email protected], [email protected]

Shadow world evaluation of the Yang-Mills measure

A new state-sum formula for the evaluation of the Yang-Mills measure in the Kauffman bracket skein algebra of a closed surface is derived. The formula extends the Kauffman bracket to diagrams that lie in surfaces other than the plane. It also extends Turaev's shadow world invariant of links in a circle bundle over a surface away from roots of unity. The limiting behavior of the Yang-Mills measure when the complex parameter approaches -1 is studied. The formula is applied to compute integrals of simple closed curves over the character variety of the surface against Goldman's symplectic measure.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-17.abs.htm