Vector Bundle
Definition. A Fiber Bundle whose fiber is a vector space $V$ and whose Transition Functions act by linear maps ($G \subseteq GL(V)$). Rank = $\dim V$; rank 1 = line bundle.
Intuition. A smoothly varying family of vector spaces. Because fibers are linear, sections can be added and multiplied by functions — sections form a $C^\infty(M)$-module. This linearity is why quantum fields, which superpose, must be sections of vector bundles.
Examples. Tangent Bundle $TM$; cotangent bundle $T^*M$ (whose sections are 1-forms); the Möbius line bundle over $S^1$; the charge-$n$ complex line bundles $L^n$ over $S^2$ (Dirac Monopole sectors).
Physics anchor. A charge-$n$ scalar field is a section of a complex line bundle, not a function. Over topologically trivial spacetime the bundle is trivializable and sections masquerade as functions $\psi(x)$ — Landau and standard QM live entirely in this disguise. The disguise fails around a monopole.
Two-way street with principal bundles. Vector bundle ⇒ Frame Bundle (principal). Principal bundle + representation ⇒ Associated Bundle (vector). Same information, two packagings.
Related: Section of a Bundle · Frame Bundle · Associated Bundle · Covariant Derivative · Module 06 - Vector Bundles and Sections
Sources: Sternberg Ch. VII.