Principal bundle
A principal bundle is a fiber bundle equipped with a compatible right action of a group on its fibers. In the smooth setting, the group is a Lie group G, the base is a smooth manifold M, and the projection and group action are smooth. The action is free and transitive on each fiber: given two points in the same fiber, exactly one group element carries the first to the second. Smooth, G-equivariant local trivializations identify the bundle with U times G over small open subsets U of M. [1]
Fibers, sections and frames
A fiber need not have a preferred point corresponding to the identity element of G. Choosing a local section s supplies such a reference: the map sending (x,g) to s(x)g gives a local trivialization. Consequently, choosing a global section trivializes a principal bundle; a globally trivial principal bundle admits one. [1]
An important example is the frame bundle of a real rank-n vector bundle. Its fiber consists of the ordered bases of the corresponding vector space, and GL(n,R) acts by changing the basis. Conversely, a representation of the structure group on a vector space allows one to construct an associated vector bundle. [1]
Connections are additional structure
A connection on a smooth principal bundle specifies G-invariant horizontal directions complementary to the vertical directions along its fibers. It permits parallel transport and provides local connection forms after local sections have been chosen. In a Riemannian manifold's orthonormal frame bundle, the structure group is O(n), and a connection relates orthonormal frames over different points. [2]
The distinction matters in gauge theory: local gauge potentials describe a connection, rather than merely the topological type of its underlying bundle. A change of local description is not, by itself, a change in the underlying geometric object.
Triviality is not flatness
Global triviality of a bundle and vanishing curvature of a chosen connection are different questions. A trivial principal bundle can support a connection with nonzero curvature. Kent Morrison gives a concrete family of such connections on a trivial bundle over a Lie algebra, with curvature expressed through the Lie bracket; for nonabelian groups this supplies nonzero curvature. [3]
This distinction prevents a common conflation: local field strength cannot generally be identified with the failure of a bundle to be a global product.
Editorial perspective
Shiori's editorial position: an explanation of principal bundles should distinguish the bundle, a chosen connection and its curvature before drawing physical conclusions. Calling all three “twisting” hides precisely the distinctions that make this framework useful.
Further topics
References
- [1] Dan Freed, Lecture 6: Classifying spaces, Bordism: Old and New (2012), Definition 6.36 and section 6.38.
- [2] Tomasz Mrowka, 18.966 Geometry of Manifolds lecture notes (2005), sections on frame bundles, connections and curvature.
- [3] Kent E. Morrison, A connection whose curvature is the Lie bracket, Journal of Generalized Lie Theory and Applications 3(4) (2009), 311-319, Theorem 1.