Module 06 — Vector Bundles and Sections
Why this module: Matter fields in physics are sections of vector bundles. A charged scalar field is not a function $\psi: M \to \mathbb{C}$ — it's a section of a complex line bundle. This module makes that statement precise.
Core idea
A Vector Bundle is a fiber bundle whose fibers are vector spaces and whose Transition Functions act linearly ($G \subseteq GL(n)$). Sections of a vector bundle can be added and scaled pointwise — that's why fields (which we superpose) are sections of vector bundles specifically.
Physics anchor: A wavefunction with charge $n$ is a section of a line bundle $L^n$ over spacetime. When the bundle is trivial (Minkowski space, no monopoles), a section looks like an ordinary function — which is why you've never needed to know this. Nontrivial base or nontrivial bundle (monopole, AB ring) breaks that illusion.
Concepts
- Vector Bundle — definition, line bundles, rank
- Section of a Bundle — revisit; sections of vector bundles form a module over $C^\infty(M)$
- Tangent Bundle — the canonical rank-$n$ real example
- Frame Bundle — from a vector bundle to a principal bundle (preview of Module 07)
- Pushforward and Pullback — pulling back bundles along maps
Reading
- Sternberg - Lectures on Differential Geometry Ch. VII (as it treats associated/frame constructions)
Self-check before moving on
Why must transition functions of a vector bundle be linear on fibers? What extra structure does a line bundle over $M$ amount to, concretely, in terms of local functions and transition data?
Concepts in this module
Study tasks
- Read the concept notes
- Exercise: show the Möbius band is the nontrivial real line bundle over $S^1$, and that its every global section vanishes somewhere
- Exercise: sections of a trivial bundle $M \times \mathbb{C} \to M$ = functions $M \to \mathbb{C}$. Write the proof in one line, then meditate on it
- Exercise: over $S^2$, the tangent bundle has no nonvanishing section (hairy ball) — restate this as "$TS^2$ is nontrivial"