Principal and Associated Bundles

A principal bundle stores local basis freedom without choosing a preferred basis. Its fiber is a $G$-torsor: a space on which a Lie group $G$ acts freely and transitively, with no distinguished point playing the role of the group identity. Choosing one reference point identifies that fiber with $G$. A point of the fiber is therefore a frame-like choice rather than a vector. Representations convert the principal bundle into associated vector or tensor bundles. This construction gives a global meaning to frame changes and gauge transformations.

Required background. Tensor Fields and Natural Operations supplies tensor fields and change-of-frame laws used as associated-bundle examples; Vector Bundles and Transition Functions supplies frames and overlap cocycles. Exterior Algebra and Differential Forms supplies ordinary forms and wedge products, which the next page generalizes to Lie-algebra-valued connection forms.

Helpful background. Calculus and Linear Algebra supplies matrix groups and representations; The Riemann Curvature Tensor provides a familiar curvature example, while Parallel Transport along Curves supplies the transport and holonomy language generalized to principal bundles.

Principal bundles and local sections

A principal $G$-bundle is a manifold $P$ with projection $\pi:P\to M$ and a smooth right action $R_g(p)=pg$ such that the action is free and transitive on every fiber. Locally there are equivariant trivializations

$$\pi^{-1}(U_i)\cong U_i\times G, \qquad (x,h)g=(x,hg).$$

A local section $s_i:U_i\to P$ selects one point in each fiber and thereby identifies every $p\in\pi^{-1}(U_i)$ uniquely as $p=s_i(x)g$. On overlaps,

$$s_j(x)=s_i(x)g_{ij}(x),$$

and the group-valued transition functions satisfy the same cocycle condition as vector-bundle transitions. Gaiter gives this construction and the equivariance conditions in Gaiter 2024, §5.1, Definition 5.1, pp. 22–23 (PDF).

A principal bundle admits a global section if and only if it is trivial. Indeed, a section $s$ defines $M\times G\to P$ by $(x,g)\mapsto s(x)g$. Conversely, the constant identity section exists on a product bundle. This statement is special to principal bundles. A vector bundle always has its zero section, and triviality there requires a full global frame.

The frame bundle

For a rank-$r$ vector bundle $E\to M$, the frame bundle $\operatorname{Fr}(E)$ has fiber over $x$ consisting of ordered bases of $E_x$. The group $GL(r,\mathbb R)$ acts on the right by changing the basis. It acts freely because only the identity leaves a frame fixed, and transitively because any frame can be transformed into any other.

For $E=TM$, $\operatorname{Fr}(TM)$ is the linear frame bundle. A coordinate chart gives a local frame and hence a local section, but these sections need not combine globally. A Riemannian metric reduces the structure group to $O(n)$ by selecting orthonormal frames, and an orientation further reduces it to $SO(n)$. A Lorentzian metric instead reduces the group to $O(1,n-1)$. Spacetime orientation selects the determinant-positive subgroup $SO(1,n-1)$, which still has two components; time orientation is a separate choice of a continuous future cone. Choosing both gives the identity component $SO^+(1,n-1)$. These distinctions are developed further in Orientability, Spin Structures, and Global Obstructions. A reduction is extra global structure, not a change in the underlying tangent fibers.

Associated bundles and fields

Let $\rho:G\to GL(V)$ be a representation. The associated vector bundle is

$$P\times_\rho V=(P\times V)/\sim, \qquad (p,v)\sim(pg,\rho(g^{-1})v).$$

The inverse in the equivalence relation ensures that $[p,v]$ is unchanged when the frame is changed and the components transform oppositely. A local section $s_i$ of $P$ writes an associated-bundle section as $[s_i(x),v_i(x)]$. On an overlap the components obey

$$v_i=\rho(g_{ij})v_j.$$

Thus the familiar transformation law is the local shadow of a global associated bundle. Tensor bundles arise from tensor representations of the frame group; spinor bundles, when they exist, arise from representations of a spin lift rather than directly from the orthonormal frame bundle. Gaiter constructs associated bundles and proves the frame-bundle correspondence in Gaiter 2024, §5.1, Proposition 5.3 and subsequent construction, pp. 23–25 (PDF).

Equivariant maps $\Phi:P\to V$ satisfying $\Phi(pg)=\rho(g^{-1})\Phi(p)$ correspond one-to-one with sections of $P\times_\rho V$. This is often the cleanest global representation of a field with local frame components.

Gauge transformations and reductions

A gauge transformation of $P$ is a bundle automorphism covering the identity on $M$ and commuting with the right $G$-action. It is not generally just a map $M\to G$; that description requires a trivialization. Globally it corresponds to an equivariant map $u:P\to G$ obeying

$$u(pg)=g^{-1}u(p)g,$$

through $p\mapsto p\,u(p)$. For an abelian group the conjugation disappears, but topology can still prevent one global trivialization.

Reducing the structure group from $G$ to a closed Lie subgroup $H$ means finding a principal $H$-subbundle compatible with $P$. Equivalently, it is a global section of the bundle $P/H\to M$. To see the correspondence, an $H$-subbundle selects one $H$-orbit in every fiber of $P$; conversely, the inverse image in $P$ of a smooth section of $P/H$ is the required principal $H$-subbundle. Closedness ensures that $G/H$ is a smooth homogeneous space and the local bundle construction applies. A metric, orientation, and spin lift are different kinds of reductions or lifts and should not be conflated.

This page supplies kinematics. A connection is an additional choice of horizontal subspaces; a gauge field is not determined by the principal bundle alone. Likewise, a local gauge function cannot decide whether a transformation is redundant at a physical boundary.

Exercise: recovering a vector bundle

Let $P=\operatorname{Fr}(E)$ and let $G=GL(r,\mathbb R)$ act on $V=\mathbb R^r$ by the defining representation. Construct a natural map $P\times_GV\to E$ and show it is well defined.

Solution

A frame $p$ over $x$ is an isomorphism $p:\mathbb R^r\to E_x$. Define

$$\Psi([p,v])=p(v).$$

Under the equivalence $(p,v)\sim(pg,g^{-1}v)$,

$$\Psi([pg,g^{-1}v])=(pg)(g^{-1}v)=p(v),$$

so the map is well defined. It is fiberwise linear and bijective, and local frame sections make it smooth with a smooth inverse. Hence $P\times_G\mathbb R^r\cong E$.

References

• Gaiter, Jacob. Characteristic Classes, Principal Bundles, and Curvature. University of Texas at Austin notes, 2024, §5.1, pp. 22–26. Open PDF.