Type: Preprint

The bundle of a lattice gauge field

Claudio Meneses, José A. Zapata;

Abstract:

Abstract. Given a smooth manifold M and a Lie group G, we consider parallel transport maps –groupoid homomorphisms from a path groupoid in M to G– as an alternative description of principal G-bundles with smooth connections on them. Using a cellular decomposition C of M, and a system of paths associated to C, we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains additional topological information of local nature, sufficient to reconstruct a principal G-bundle up to equivalence, in the spirit of Barrett [3]. Following a theorem of Pachner [15], we give a criterion to determine when two ELG fields over different cell decompositions C and C? define equivalent bundles. As a first concrete physical application, we define a simple operation that captures the geometric essence of the ’t Hooft loop operator in the theory of quantum gauge fields, and study its relation with the bundle structure of an ELG field.