turkmath.org

Gauge fields describe parallel transport of point particles along curves, with respect to connections on principal bundles. This data is captured as a smooth functor from the smooth path groupoid of the base manifold into the delooping groupoid of the structure group, plus gluing data. Lattice gauge fields do this for discretized versions of a base manifold. A lattice gauge field is thus a functor from a discrete version of the path groupoid to a delooping groupoid. Lattice gauge fields are meant to serve as discrete approximations of regular gauge fields.

Higher gauge fields describe parallel transport of curves along surfaces, of surfaces along volumes, etc. Several versions of 2-dimensional gauge field have appeared in the literature. I will explain how to extend these ideas to lattice gauge fields on all dimensions, using Brown's cubical homotopy $\omega$-groupoid construction associated to filtered spaces, implementing a discrete notion of thin homotopy along the way.