Simplifying 2-Bundle Descriptions of Invariants and Local Dynamics

Abstract

A string couples to a two-form gauge potential whose local descriptions need not fit together as a single global form. Principal 2-bundles provide a geometric description of this gluing. However, we recover surface amplitudes and quantum partition functions, as well as local classical field dynamics, without the 2-bundle formalism. For global phases, topological and index invariants, and twisted-sheaf moduli, established equivalences give alternative descriptions through cocycles, bundles, operators, or modules, with the required global compatibility data. For three-dimensional finite gauge theories with cohomological actions, Fourier transformation expresses the partition function as a sum of partition functions of ordinary finite-group gauge theories. An invertible local substitution reduces invariant classical actions for a class of 2-bundle connections represented by finite-dimensional crossed modules to ordinary gauge and tensor dynamics. These finite and local reductions explicitly eliminate field variables. A four-dimensional quadratic example instead admits a direct cochain evaluation even when an ordinary gauge-theory reduction fails. These results therefore do not require the additional 2-bundle presentation.