Parallelism, Holonomy, Homotopy and (Co)homology

Abstract

This chapter is devoted to most topical and important applications of topology and geometry in physics: gauge field theory and the physics of geometric phases which vastly emerges from the notion of the Aharonov–Bohm phase and later more generally from the notion of a Berry phase and even penetrates chemistry and nuclear chemistry. The central notion in these applications is holonomy. Since holonomy is based on lifts of integral curves of tangent vector fields on the base manifold M of a bundle, and maximal integral curves may end in singular points of tangent vector fields, non-singularity of tangent vector fields plays its role. Non-zero tangent vector fields can be expressed as sections of the ‘punctured tangent bundle’ on M. This is a subject of the interrelation of holonomy with homotopy of fiber bundles, an important issue by itself. Therefore the chapter starts with two sections on homotopy of fiber bundles before gauge fields and finally geometric phases in general are considered. All these issues fall also into the vast realm of characteristic classes and index theory. In a first reading the first two sections may be skipped.