Section of a Bundle
Definition. A smooth map $s: M \to E$ with $\pi \circ s = \mathrm{id}_M$ — a smooth choice of one point in each fiber. Local section: same, defined only on $U \subseteq M$.
Intuition. If the bundle is a field of fibers standing over $M$, a section is a "graph" threading through them. For a trivial bundle $M \times F$, sections = functions $M \to F$; for twisted bundles, sections are the correct generalization of functions.
The physics dictionary. - Matter field = section of an Associated Bundle (Vector Bundle). - Vector Field = section of the Tangent Bundle. - Gauge choice = local section of a Principal Bundle (pulling back the Connection 1-Form along it gives Landau's $A_i$).
Obstructions are topology. Vector bundles always have a global section (zero); it's nonvanishing sections that may not exist (hairy ball on $TS^2$). A principal bundle has a global section iff it is trivial — so the Hopf Fibration has none, and a global gauge for the Dirac Monopole cannot exist.
Related: Fiber Bundle · Vector Bundle · Principal Bundle · Local Trivialization · Module 06 - Vector Bundles and Sections
Sources: Sternberg Ch. VII.
Linked from
- Connection 1-Form
- Fiber Bundle
- Gauge Transformation
- Hopf Fibration
- Local Trivialization
- Möbius Band as a Bundle
- Module 01 — Smooth Manifolds Refresher
- Module 05 — Fiber Bundles: First Definitions
- Module 06 — Vector Bundles and Sections
- Principal Bundle
- Tangent Bundle
- Vector Bundle
- Vector Field
- Woit — Quantum Theory, Groups and Representations