The Chern-Simons character of a lattice gauge field

by Anthony V. Phillips and David A. Stone


Let U be a generic SU(2)-valued latticegauge field on a triangulation of a manifold X of dimension at least 3. An algorithm is given for constructing from U a principal SU(2)-bundle over X, and in that bundle a connection with the following property. Its Chern-Simons character can be computed as the sum of local contributions (one for each 3-simplex of the triangulation), each one the sum of a one-dimensional integral and five other terms which are volumes of elementary regions in the 3-sphere. For any 3-cycle this sum, reduced mod Z, is gauge invariant. This character is related in the appropriate way to real and integral 4-cocyles representing the second Chern class of the bundle. If U has sufficiently small plaquette products, the isomorphism class of the bundle is shown to depend only on U, and not on details of the construction.

in Louis Kauffman and Randy Baadhio, eds., Quantum Topology, World Scientific 1993, 244-291