Parallel Transport and Holonomy
Definition. Given a connection and a curve $\gamma: [0,1] \to M$, each start point $p \in \pi^{-1}(\gamma(0))$ has a unique horizontal lift $\tilde\gamma$ (velocity always in the horizontal subspace). The end-point map $\pi^{-1}(\gamma(0)) \to \pi^{-1}(\gamma(1))$ is parallel transport. For a loop, transport returns to the same fiber, off by a group element: the holonomy $\mathrm{hol}(\gamma) \in G$.
Formula. $\mathrm{hol}(\gamma) = P\exp\left(-\oint_\gamma A\right)$ (path-ordered). For U(1), no ordering needed: $$\mathrm{hol}(\gamma) = \exp\left(i \oint_\gamma A_\mu\, dx^\mu\right).$$
Intuition. "Carry the internal quantity along the path, changing it as little as the connection allows." Path-dependence of the result is exactly curvature: infinitesimally, $\mathrm{hol}(\partial S) \approx \exp(-\int_S F)$ via Stokes Theorem.
Classical picture. Levi-Civita connection on $S^2$: parallel-transport a vector around a spherical triangle; it returns rotated by the enclosed area. Holonomy = curvature integrated.
Physics anchor. The phase a charged particle accumulates along a path, $e^{\frac{ie}{\hbar c}\int A_\mu dx^\mu}$ — the Wilson line/loop. Directly observable in the Aharonov-Bohm Effect: interference measures holonomy even where $F = 0$. Holonomy is the answer to "is $A$ mere bookkeeping?" — the gauge-invariant content of $A$ beyond $F$.
Related: Connection on a Principal Bundle · Curvature 2-Form · Aharonov-Bohm Effect · Exponential Map · Module 08 - Connections and Curvature
Sources: Sternberg Ch. VII.