Fiber Bundles, Connections, and Curvature

Fiber bundles, vector bundles, principal bundles, connections, curvature, characteristic classes, and Berry bundles.

  1. Home
  2. Notes
  3. Fiber Bundles, Connections, and Curvature

Mathematics / Fiber Bundles / Bundles, connections, and curvature

Fiber Bundles notes
Previous Next
Chern Classes and Bloch Bundles Fiber Bundles, Connections, and Curvature Principal Bundles and Connections Topological Fiber Bundles Vector Bundles, Frames, and Sections

The Big Picture

A fiber bundle is a mathematical device for describing the following situation:

At each point of some base space, another space is attached. Locally the whole object looks like a product, but globally the attachment may be twisted.

The basic notation is

π:E→B.\pi:E\to B.

Here:

  • BB is the base space. In physics this may be real space, spacetime, parameter space, or momentum space.

  • EE is the total space. It contains all the attached fibers.

  • π\pi is the projection map. It tells you which base point a point of EE lies above.

  • The fiber over b∈Bb\in B is

Eb:=π−1(b).E_b:=\pi^{-1}(b).

One-sentence version A fiber bundle is locally a product U×FU\times F, but it may fail to be globally a product B×FB\times F.

The main hierarchy is this:

topological fiber bundle⊃vector bundle\boxed{\text{topological fiber bundle}} \quad \supset \quad \boxed{\text{vector bundle}}

Vector bundles are fiber bundles whose fibers are vector spaces and whose transition functions are linear. Principal bundles are another important class of fiber bundles, where the fiber is a group GG and the bundle describes gauge frames.

Physical language Bundle language

Vector field on a manifold Section of the tangent bundle TM→MTM\to M One-form such as A=AμdxμA=A_\mu dx^\mu Section of the cotangent bundle T∗M→MT^*M\to M Electromagnetic vector potential Local expression of a connection on a U(1)U(1) bundle Berry connection Connection on a Bloch vector bundle Berry curvature Curvature of that connection Chern number Integral of a characteristic class Bloch occupied states {∣ua(k)⟩}\{|u_a(k)\rangle\} Local frame for a vector bundle over the Brillouin zone Gauge transformation Change of local trivialization/frame

The note is written to prevent a common problem: if one jumps directly to Berry bundles or Chern numbers, the words “base space,” “fiber,” “section,” and “connection” can feel like labels rather than definitions. We therefore build them in order.