Principal Bundles

A principal bundle is a fiber bundle whose fiber is its structure group, acting on itself. It is the cleanest carrier of a symmetry group over a manifold, and it is exactly the object underlying a gauge theory: the gauge group lives in the fiber, gauge transformations are its vertical symmetries, and gauge fields are connections on it (connections.md). This page builds on fiber-bundles.md and uses the Lie groups of group-theory/.

Definition

A principal $G$-bundle is a fiber bundle $\pi :P\to M$ with a free, fiber- preserving right action of a Lie group $G$ on $P$, $$P\times G\to P,(u,g)\mapsto u\cdot g,$$ such that $G$ acts simply transitively on each fiber. Concretely each fiber ${\pi }^{-1}(p)$ is a $G$-torsor: a copy of $G$ with no preferred identity — points can be compared ($u_2=u_1\cdot g$ for a unique $g$) but there is no canonical "$e$". The base is the quotient $M=P/G$. The structure group and the fiber coincide, and the transition functions of fiber-bundles.md take values in $G$ acting on itself by left multiplication.

Trivializations and gauge

A local trivialization ${\pi }^{-1}(U)\cong U\times G$ is equivalent to a local section $\sigma :U\to P$ (take $\sigma (p)=(p,e)$). In physics a choice of local section is a choice of gauge. The transition functions $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G$ relate overlapping sections by ${\sigma }_{\beta }={\sigma }_{\alpha }\cdot g_{\alpha \beta }$ and satisfy the cocycle condition. A global section exists iff the bundle is trivial — so the existence of a global gauge is a topological question, and its obstruction is why gauge fixing can fail globally (Gribov ambiguity, gauge/faddeev-popov.md).

Gauge transformations

A gauge transformation is a bundle automorphism $\Psi :P\to P$ covering the identity on $M$ and commuting with the $G$-action ($\Psi (u\cdot g)=\Psi (u)\cdot g$). Such maps form the gauge group $\mathcal{G}$. Each is equivalent to a $G$-valued function on $P$ (equivariant), or locally to a function $g:U\to G$ — the familiar $x\mapsto g(x)$ of gauge theory. These are the vertical automorphisms (they move points within fibers); they are an infinite-dimensional group, not to be confused with the finite-dimensional structure group $G$ itself.

Classification of bundles

How many principal $G$-bundles does a given base carry? The cocycle data classifies them:

Classification theorem. Isomorphism classes of principal $G$-bundles over $M$ are in bijection with homotopy classes of maps $[M,BG]$ into the classifying space $BG$; equivalently with ${\check{H}}^1(M;G)$ (Čech cohomology with coefficients in $G$).

Practical consequences via homotopy:

• $G$-bundles over $S^n$ are classified by ${\pi }_{n-1}(G)$ (clutching two hemispheres by an equatorial map $S^{n-1}\to G$).
• $U(1)$-bundles over $M$ ↔ $H^2(M;\mathbb{Z})$ — the first Chern class; over $S^2$ this is ${\pi }_1(U(1))=\mathbb{Z}$, the Dirac monopole charge and flux quantization.
• $SU(2)$-bundles over $S^4$ ↔ ${\pi }_3(SU(2))=\mathbb{Z}$ — the instanton number (characteristic-classes.md, advanced/topological.md).

Why principal, not just vector

Everything about a gauge theory is cleanest on the principal bundle:

• the connection (gauge potential) is a single global object there, even when no global gauge exists (connections.md);
• matter fields in any representation $\rho$ are sections of the associated vector bundle $P{\times }_{\rho }V$ (associated-bundles.md), all built from the one principal bundle;
• global/topological features (monopoles, instantons) are properties of the principal bundle, independent of which matter representation one looks at.

References

• Kobayashi & Nomizu, Foundations of Differential Geometry, Vol. 1, Ch. 1–2.
• Nakahara, Geometry, Topology and Physics, Ch. 9–10.
• Baez & Muniain, Gauge Fields, Knots and Gravity, Part II.