Connections and Parallel Transport

To differentiate a section of a bundle, one must compare fibers over different points — but there is no canonical way to do so. A connection is exactly the extra structure that provides one: a rule for parallel transport. In gauge theory the connection is the gauge potential $A_{\mu }$; the covariant derivative $\nabla =d+A$ is how it acts on matter. This page builds on principal-bundles.md and associated-bundles.md; its curvature is curvature.md.

The problem connections solve

Given a section $s$ of a vector bundle $E\to M$, the naive derivative ${\partial }_{\mu }s$ is not a section — it depends on the local trivialization and transforms inhomogeneously under a change of gauge (the derivative hits the transition function). One needs a covariant derivative $\nabla$ that produces a genuine section, so that $\nabla s=0$ has gauge-independent meaning ("$s$ is parallel").

Connection on a principal bundle

The clean definition lives upstairs on the principal bundle $P$. At each $u\in P$ the tangent space has a canonical vertical subspace $V_u=\ker d\pi$ (tangent to the fiber, $\cong \mathfrak{g}$). A connection is a smooth, $G$-equivariant choice of complementary horizontal subspace $H_u$, so that $$T_{u}P=V_u\oplus H_u.$$ Equivalently it is a $\mathfrak{g}$-valued connection 1-form $\omega \in {\Omega }^1(P;\mathfrak{g})$ that projects onto the vertical part: $\omega$ reproduces the generator of any vertical vector and annihilates horizontal ones, with the equivariance $R_g^{\ast }\omega ={\mathrm{Ad}}_{g^{-1}}\omega$. "Horizontal" is the infinitesimal notion of parallel: a curve is parallel if its tangent is everywhere horizontal.

The gauge potential (local form)

Pulling $\omega$ back along a local section (a gauge) ${\sigma }_{\alpha }:U_{\alpha }\to P$ gives the gauge potential, a $\mathfrak{g}$-valued 1-form on the base: $$A_{(\alpha )}={\sigma }_{\alpha }^{\ast }\omega =A_{\mu }^a{T}^{a}d{x}^{\mu }\in {\Omega }^1(U_{\alpha };\mathfrak{g}).$$ This is the physicist's $A_{\mu }$. On an overlap, the two local potentials are related by the transition function $g\equiv g_{\alpha \beta }$ via the gauge-transformation law $$\boxed{A_{(\beta )}=g^{-1}{A}_{(\alpha )}g+g^{-1}dg}$$ The inhomogeneous $g^{-1}dg$ term is precisely what a connection form must have to repair the non-tensorial derivative — and it is the familiar transformation of the gauge field. (For $U(1)$ it reduces to $A\to A+d\lambda$.) A connection is thus a global object even though its local representatives $A_{(\alpha )}$ are only defined per gauge.

The covariant derivative

On an associated bundle $E=P{\times }_{\rho }V$ (associated-bundles.md), the connection induces the covariant derivative $${\nabla }_{\mu }={\partial }_{\mu }+{\rho }_{\ast }(A_{\mu }),\nabla \psi =d\psi +{\rho }_{\ast }(A)\psi ,$$ where ${\rho }_{\ast }$ is the representation of $\mathfrak{g}$ on $V$. In the fundamental representation this is the minimal-coupling $D_{\mu }={\partial }_{\mu }+A_{\mu }$ (or ${\partial }_{\mu }-iq{A}_{\mu }$ with physics conventions). Its defining virtue: $\nabla \psi$ transforms homogeneously (like $\psi$ itself, tensorially), so covariant equations are gauge-invariant. The gauge principle of gauge/yang-mills.md is exactly the demand that derivatives be covariant — which forces the connection into existence.

Parallel transport

Integrating "horizontal" along a curve $\gamma$ from $p$ to $q$ gives parallel transport $P_{\gamma }:E_p\to E_q$, a linear isomorphism of fibers. It solves the transport equation ${\nabla }_{\dot{\gamma }}s=0$, whose solution is the path-ordered exponential $$P_{\gamma }=\mathcal{P}\exp (-{\int }_{\gamma }A).$$ Parallel transport depends on the path, not just its endpoints — and the failure to depend only on endpoints, i.e. the transport around a closed loop, is the holonomy, measured by the curvature. That path-dependence is the entire content of curvature.md and holonomy.md.

References

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