Principal Bundles and Gauge Theory

Abstract

In many areas of theoretical physics we encounter fields defined on a spacetime M taking values in some other space F. For instance, presume F as a real finite-dimensional vector space, then ∅ : M → F is a vector field. More generally, we can consider a family of spaces {F_x}xϵM varying over the points on M, that is ∅(x) ϵ F_x for each x ϵ M. A field ∅ is then understood as a section from the spacetime manifold into the bundle of spaces over M. This is exactly the idea encoded in the mathematical theory of fiber bundles. Namely, fiber bundles provide a tool to describe the global structure of physical fields.