Principal BundleEdit

A principal bundle is a foundational structure in differential geometry that formalizes the idea of carrying a consistent group symmetry along a space. It consists of a total space P, a base space M, a Lie group G acting on P, and a projection π: P → M that ties the two spaces together. The action of G on P is required to be free and transitive on each fiber π−1(x) for x in M, so each fiber is a copy of the group G. Locally, the bundle looks like a product M × G, but globally the way these local pictures are glued together can encode rich twisting data. Principal bundles provide the language for gauge freedom in physics and for the systematic study of geometric structures tied to a symmetry group Lie group and its representations representation.

They are a special kind of fiber bundle fiber bundle, with a structure group G acting on the right on P. The projection π forgets the G-part and remembers only the base point in M, so a point p ∈ P sits over m = π(p) ∈ M, and the fiber over m is π−1(m) ≅ G under a chosen identification. The action R_g: P → P, given by p ↦ p·g, preserves the fibration and encodes the symmetry that travels along the fiber. This setup allows the study of geometric structures on M by looking at how these G-symmetric fibers are glued together across open sets.

Definition

Let M be a smooth manifold (or more generally a differentiable, topological, or complex manifold) and let G be a Lie group. A principal G-bundle is a quadruple (P, M, π, G) where:

π: P → M is a smooth surjective map, called the projection.
P carries a right action R: P × G → P, written p·g, which is smooth.
The action is free (p·g = p only if g is the identity) and each fiber π−1(m) is a G-torsor: the action of G on π−1(m) is simply transitive.
The action is compatible with the projection: π(p·g) = π(p) for all p ∈ P and g ∈ G.

Equivalently, P is a fiber bundle over M with typical fiber G, together with a smooth right G-action by bundle automorphisms that intertwines with the projection. The local triviality condition means that around every x ∈ M there exists a neighborhood U such that π−1(U) ≅ U × G in a G-equivariant way.

Local trivializations are described by transition functions gαβ: Uα ∩ Uβ → G on overlaps of a cover {Uα} of M, satisfying the usual cocycle condition on triple overlaps. The collection {gαβ} encodes the twisting of the bundle, and different cocycles can yield non-isomorphic principal bundles even when the base and group are fixed. See also classifying spaces classifying space and Čech cohomology Čech cohomology for a topological viewpoint on these twists.

Local trivializations and transition data

On an open cover {Uα} of M, a principal G-bundle is specified by diffeomorphisms Φα: π−1(Uα) → Uα × G that are G-equivariant, i.e., Φα(p·h) = (π(p), gα(p)·h) for a map gα: π−1(Uα) → G that, after composing with the projection to Uα, reduces to a map to G independent of the fiber coordinate. On overlaps, the two trivializations are related by transition functions gαβ: Uα ∩ Uβ → G via Φα ∘ Φβ−1(m, g) = (m, gαβ(m)·g). The cocycle condition gαβ(m)·gβγ(m)·gγα(m) = e holds on triple overlaps, ensuring consistency. The transition functions capture the essential twisting of the bundle and determine its isomorphism class.

Connections and curvature

A connection on a principal G-bundle provides a way to differentiate along the base M while respecting the G-symmetry. A connection is most succinctly described by a connection 1-form ω, a Lie(G)-valued 1-form on P, satisfying two properties:

Equivariance under the right G-action: R_g*ω = Ad_g−1 ω.
Reproducing the Lie algebra action along fundamental vector fields: ωξP = ξ for ξ in the Lie algebra 𝔤, where ξP is the fundamental vector field on P.

The connection determines horizontal subspaces HPp ⊂ TpP at each p ∈ P, consisting of vectors annihilated by ω. Locally, a connection is often described by a collection of local 1-forms Ai on Uα with values in 𝔤, related on overlaps by gauge transformations. The curvature 2-form F = dω + 1/2[ω, ω] measures the failure of horizontal transport to be path-independent and encodes topological and geometric information about the bundle. In physics, connections on principal bundles are central to gauge theories, where ω plays the role of a gauge potential and F its field strength.

Related concepts include the Maurer–Cartan form on G, which provides a canonical connection on the trivial bundle, and the idea of horizontal lifts of curves in M to P, yielding parallel transport along paths.

Associated bundles and representations

Given a representation ρ: G → GL(V) of G on a vector space V, one can form the associated vector bundle E = P ×G V, whose fiber over m ∈ M is the quotient P×V under the equivalence (p·g, v) ∼ (p, ρ(g)·v). Sections of E correspond to G-equivariant functions on P with values in V, and many geometric constructions on M can be described in terms of associated bundles. This framework unifies tangent bundles, cotangent bundles, and more specialized tensor bundles by choosing suitable representations of G and using corresponding principal bundles.

Common examples include the frame bundle FM of M with structure group GL(n, ℝ) or GL(n, ℂ), whose associated bundles recover the various tensor bundles on M. Special structure groups lead to natural reductions of the frame bundle, such as the orthonormal frame bundle with structure group O(n) or SO(n) for Riemannian manifolds, or unitary groups for Hermitian settings. See also frame bundle frame bundle and tangent bundle tangent bundle.

Classification and topology

Topologically, principal G-bundles over a fixed base M are classified up to isomorphism by homotopy classes of maps from M into the classifying space BG, provided M has suitable niceness properties. In the smooth category, the set of isomorphism classes of principal G-bundles corresponds to certain cohomological data on M, and, for many Lie groups G, to characteristic classes that live in the cohomology of M (for example, Chern classes for unitary groups). The classifying space BG serves as a universal parameter space for G-bundles: every G-bundle over M is the pullback of the universal bundle over BG via a classifying map φ: M → BG. See also classifying space classifying space and characteristic classes characteristic class.

The existence of global sections s: M → P is equivalent to the bundle being trivial, P ≅ M × G. Trivializability depends on both the topology of M and the chosen G; many manifolds admit nontrivial principal bundles, illustrating the potential for global twisting even when local triviality holds.

Examples and applications

The frame bundle FM of a smooth manifold M, with structure group GL(n, ℝ) or GL(n, ℂ), captures the linear-frame data at every point and underlies the formulation of tensor calculus on M. The tangent bundle TM is naturally realized as an associated bundle to FM via the standard representation of GL(n, ℝ) on ℝ^n.
For a Riemannian metric on M, one can reduce the structure group to O(n) or SO(n), yielding the orthonormal frame bundle and linking curvature to the Levi-Civita connection.
Spin geometry introduces the spin group Spin(n) and a corresponding principal Spin(n)-bundle whose associated vector bundles permit spinor fields. The existence of a spin structure is a topological condition tied to the manifold and its tangent bundle.
In gauge theory, principal bundles model the configuration space of gauge fields. Connections on these bundles give rise to gauge potentials and curvature forms that describe physical field strengths.