Principal bundle

A principal G-bundle over a base space M is a fiber bundle π: P → M with a continuous right action of a topological group G on P that is free and transitive on each fiber. This means that for each point x in M, the fiber π^-1(x) is a copy of G, and the action of G on the fiber is simply transitive. The quotient space P/G is naturally homeomorphic to M, and the bundle is locally trivial: every point of M has a neighborhood U such that π^-1(U) is G-equivariantly homeomorphic to U × G.

Principal bundles are often denoted by G → P → M, and they are classified by homotopy classes of maps from M to the classifying space BG. The existence of a global section is equivalent to the bundle being trivial, i.e., isomorphic to M × G.

A connection on a principal G-bundle is a choice of a G-invariant horizontal distribution on P, which allows parallel transport along curves in the base. The connection is described locally by a Lie-algebra-valued 1-form, and its curvature is a Lie-algebra-valued 2-form that measures the failure of the horizontal distribution to be integrable. The curvature form satisfies the Bianchi identity and plays a central role in the Chern–Weil theory, which constructs characteristic classes of the bundle from invariant polynomials on the Lie algebra.

Connections on principal bundles are the mathematical foundation of gauge theories in physics, where the structure group is typically a compact Lie group such as U(1), SU(2), or SU(3). The Yang–Mills equations, which govern the dynamics of gauge fields, are formulated in terms of the curvature of a connection on a principal bundle.

Given a principal G-bundle P and a left action of G on a space F, one can construct an associated fiber bundle P ×_G F with fiber F. For example, the tangent frame bundle of a smooth manifold is a principal GL(n,R)-bundle, and its associated bundle for the standard representation is the tangent bundle. Vector bundles with structure group G correspond to principal G-bundles via this construction.

Reduction of the structure group is a process of finding a principal H-bundle Q ⊂ P for a subgroup H ⊂ G such that P is isomorphic to Q ×_H G. For instance, a Riemannian metric on a manifold gives a reduction of the frame bundle from GL(n,R) to the orthogonal group O(n). The existence of a reduction to a discrete group corresponds to a flat connection.

Principal bundles appear in diverse areas beyond differential geometry. In algebraic geometry, a principal G-bundle (or G-torsor) is defined for a group scheme G, and they are classified by étale cohomology. In number theory, principal bundles over arithmetic schemes are related to Galois representations and the Langlands program.

A notable edge case is the universal principal bundle: the total space EG of the universal bundle over the classifying space BG is contractible, and the action of G on EG is free. This construction is fundamental in homotopy theory. Another subtlety is that the definition of a principal bundle can be extended to infinite-dimensional groups, such as the gauge group of a bundle, which is the group of automorphisms of the bundle.

Historically, the concept evolved from the notion of a fiber bundle introduced by Herbert Seifert and others in the 1930s, and it was formalized by Norman Steenrod in his 1951 book The Topology of Fiber Bundles. The term "principal bundle" itself was coined by Steenrod.

Fiber bundle

A space that locally looks like a product of a base space and a fiber.

Free action

A group action where the only element fixing any point is the identity.

Transitive action

A group action with only one orbit.

Connection

A choice of horizontal distribution on a principal bundle, enabling parallel transport.

Curvature

A 2-form measuring the non-integrability of the horizontal distribution.

Classifying space

A space BG such that principal G-bundles over M are classified by homotopy classes of maps M → BG.