What are Fiber Bundles and why do they matter in physics?
Short Answer
A fiber bundle is a space that locally looks like a product of two spaces, but globally may be “twisted.” In physics, fiber bundles provide the geometric framework for gauge theories: gauge fields are connections, matter fields are sections, and gauge transformations are bundle automorphisms.
The Basic Idea
Motivation: The Möbius Strip
Consider a Möbius strip. Locally, it looks like a rectangle I \times I. But globally, it’s twisted—you can’t untwist it into a cylinder without cutting.
This is a fiber bundle:
- Base space B = S^1 (the circle)
- Fiber F = I (an interval)
- Total space E = Möbius strip
Definition
A fiber bundle consists of:
- Total space E
- Base space B
- Fiber F
- Projection map \pi: E \to B
Such that locally E looks like B \times F:
For each point b \in B, there exists a neighborhood U and a homeomorphism: \phi: \pi^{-1}(U) \xrightarrow{\sim} U \times F
Types of Fiber Bundles
Vector Bundles
The fiber is a vector space V. Examples:
| Bundle | Base | Fiber | Physical Meaning |
|---|---|---|---|
| Tangent bundle TM | Manifold M | \mathbb{R}^n | Velocity vectors |
| Cotangent bundle T^*M | Manifold M | \mathbb{R}^n | Momentum, 1-forms |
| Line bundle | M | \mathbb{C} | Charged scalar field |
| Spinor bundle | M | \mathbb{C}^4 | Dirac fermions |
Principal Bundles
The fiber is a Lie group G, acting on itself by right multiplication.
A principal G-bundle P \xrightarrow{\pi} M has:
- Free right G-action on P
- M = P/G (base is the orbit space)
Examples in physics:
| Physical Theory | Structure Group | Bundle |
|---|---|---|
| Electromagnetism | U(1) | Circle bundle |
| Yang-Mills | SU(N) | SU(N)-bundle |
| General Relativity | SO(1,3) | Frame bundle |
| Spin geometry | Spin(1,3) | Spin bundle |
Connections
The Problem of Parallel Transport
Given a vector at one point, how do we “move” it to another point? On a curved space, there’s no canonical way to do this.
A connection provides a rule for parallel transport.
Connection on a Principal Bundle
A connection on P is a Lie algebra-valued 1-form:
A \in \Omega^1(P, \mathfrak{g})
satisfying compatibility conditions with the G-action.
Locally (in a trivialization), this becomes:
A = A_\mu dx^\mu, \quad A_\mu \in \mathfrak{g}
This is the gauge field!
Curvature
The curvature of a connection measures non-commutativity of parallel transport:
F = dA + A \wedge A
In components: F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]
This is the field strength!
Sections
Definition
A section of a bundle E \xrightarrow{\pi} B is a map s: B \to E such that \pi \circ s = \text{id}_B.
In other words: a section picks out one point in each fiber.
Physical Interpretation
| Bundle Type | Section | Physical Field |
|---|---|---|
| Line bundle | \psi: M \to E | Scalar field |
| Vector bundle | \psi: M \to E | Matter field |
| Spinor bundle | \psi: M \to S | Dirac fermion |
Covariant Derivative
A connection induces a derivative on sections:
D_\mu \psi = \partial_\mu \psi + A_\mu \psi
This is the covariant derivative—it transforms correctly under gauge transformations.
Associated Bundles
Given a principal bundle P and a representation \rho: G \to GL(V), we can form the associated vector bundle:
E = P \times_\rho V = (P \times V)/G
where G acts by (p, v) \sim (pg, \rho(g^{-1})v).
Example: Electromagnetism
- Principal bundle: U(1)-bundle over spacetime
- Representation: \rho_n: U(1) \to GL(\mathbb{C}), e^{i\theta} \mapsto e^{in\theta}
- Associated bundle: Line bundle (charge n matter field)
An electron (n = -1) is a section of the associated line bundle.
Gauge Transformations
A gauge transformation is an automorphism of the principal bundle:
g: P \to P, \quad g(p \cdot h) = g(p) \cdot h
Equivalently, a section g: M \to G.
Under gauge transformation:
- Connection: A \mapsto g^{-1}Ag + g^{-1}dg
- Curvature: F \mapsto g^{-1}Fg (covariant)
- Section: \psi \mapsto \rho(g^{-1})\psi
The physics is invariant under gauge transformations—this is gauge symmetry.
Topology and Bundles
Trivial vs Non-trivial Bundles
A bundle is trivial if E \cong B \times F globally.
Non-trivial bundles are topologically distinct from the product. Examples:
- Möbius strip (non-trivial line bundle over S^1)
- Hopf fibration S^3 \to S^2 (non-trivial circle bundle)
Characteristic Classes
Topological invariants that measure “how twisted” a bundle is:
| Class | Bundle Type | Physical Meaning |
|---|---|---|
| Chern classes c_n | Complex vector bundle | Magnetic charge, instanton number |
| Stiefel-Whitney w_n | Real vector bundle | Spin structure obstruction |
| Pontryagin p_n | Real vector bundle | Gravitational instantons |
Chern Number and Magnetic Monopoles
For a U(1) bundle over S^2:
c_1 = \frac{1}{2\pi} \int_{S^2} F \in \mathbb{Z}
This integer is the magnetic charge. Dirac’s quantization condition:
eg = \frac{n}{2}, \quad n \in \mathbb{Z}
follows from the requirement that the bundle be well-defined.
Summary: The Fiber Bundle Dictionary
| Physics | Mathematics |
|---|---|
| Gauge group | Structure group G |
| Gauge field | Connection on principal bundle |
| Field strength | Curvature of connection |
| Matter field | Section of associated bundle |
| Covariant derivative | Connection-induced derivative |
| Gauge transformation | Bundle automorphism |
| Magnetic charge | First Chern class |
| Instanton number | Second Chern class |
References
- M. Nakahara, Geometry, Topology and Physics, 2nd ed. (CRC Press, 2003), Chapters 9–11
- C. Nash and S. Sen, Topology and Geometry for Physicists (Academic Press, 1983)
- M. F. Atiyah, Geometry of Yang-Mills Fields (Scuola Normale Superiore, 1979)
- R. W. R. Darling, Differential Forms and Connections (Cambridge University Press, 1994)
- J. Baez and J. P. Muniain, Gauge Fields, Knots and Gravity (World Scientific, 1994)
- nLab: Principal bundle — Category-theoretic perspective