Hopf Bundle

Prerequisites: Fiber Bundles, SU(2). Full path (8 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

The state of a spin-1/2 particle is described by two complex numbers, the amplitudes for spin up and spin down. After normalization these pairs fill out a three-dimensional sphere sitting in four real dimensions. Multiplying both amplitudes by the same phase has no measurable consequence. If we lump together all normalized states that differ only by such an overall phase, each lump is a circle, and what remains is the set of physically distinct spin states. That set is an ordinary two-dimensional sphere, the Bloch sphere: every spin state points in some direction in space.

The Hopf bundle is exactly this picture: the three-sphere is filled completely by circles that do not intersect, and each circle corresponds to one point of the two-sphere. Near any single point of the two-sphere the arrangement looks like a product of a small patch of the sphere with a circle. Globally it is not a product. There is no way to pick one point on every circle in a continuous way over the whole two-sphere, which in the spin example means that no smooth choice of phase convention covers all spin directions. Another sign of the twisting is that any two of the circles are linked once, like two rings of a chain.

The same structure appears for an electrically charged particle near a magnetic monopole. The phase of its wave function can be chosen consistently on patches around the monopole, but not on a whole sphere surrounding it. The Hopf bundle describes the monopole with the smallest allowed magnetic charge.

Concrete

"The Hopf fibration is a kind of projection from the three-sphere to the two-sphere. The two-sphere is the one you're likely to be familiar with—a beach ball is a good example. The two-sphere is formed by all points which are a constant distance from a center point. We write the two-sphere as to indicate that it is 2-dimensional." http://nilesjohnson.net/hopf.html

• The best explanation and especially how it is related to magnetic monopoles can be found in chapter 0 of the book Topology, Geometry and Gauge fields by G. Naber.
• See also http://math.ucr.edu/home/baez/week141.html
• For a nice visualization of the Hopf maps, see Niles Johnson : Hopf Fibration Video

Examples

Sources for these examples: Section 9.4 "Principal Bundles" in Geometry, Topology and Physics by Nakahara and http://math.ucr.edu/home/baez/week141.html

(The classification of the Hopf bundles as listed here is surprisingly similar to the classification of all simple Lie groups. Each Hurwitz algebra corresponds to one family of simple groups. The octonions play a special role, because they correspond to the exceptional family, which has only a finite number of members.)

Abstract

Write $S^3=\{(z_1,z_2)\in\mathbb{C}^2 : |z_1|^2+|z_2|^2=1\}$. The group $U(1)$ acts freely on $S^3$ by $(z_1,z_2)\mapsto(\lambda z_1,\lambda z_2)$, $|\lambda|=1$, and the quotient is the complex projective line $\mathbb{C}P^1\cong S^2$. The projection can be written explicitly as $$\pi(z_1,z_2) = \left(2\,\mathrm{Re}(\bar z_1 z_2),\; 2\,\mathrm{Im}(\bar z_1 z_2),\; |z_1|^2-|z_2|^2\right),$$ which lands on the unit sphere because $4|z_1|^2|z_2|^2+(|z_1|^2-|z_2|^2)^2=(|z_1|^2+|z_2|^2)^2=1$. This makes $\pi:S^3\to S^2$ a principal $U(1)$-bundle, the Hopf bundle.

Identifying $S^3$ with $SU(2)$ via $$U=\begin{pmatrix} z_1 & -\bar z_2\\ z_2 & \bar z_1\end{pmatrix},$$ the projection is $U\mapsto U\sigma_3U^\dagger = \vec x\cdot\vec\sigma$, with $\vec x=\pi(z_1,z_2)$, and the fibre action is right multiplication by the diagonal subgroup $e^{i\theta\sigma_3}$. So the Hopf bundle is the homogeneous bundle $SU(2)\to SU(2)/U(1)\cong S^2$, which is how it appears in the SU(2) description of spin.

The bundle is not trivial: a trivial bundle would have total space $S^2\times S^1$, which has $\pi_1\cong\mathbb{Z}$, while $S^3$ is simply connected. The form $\omega=\bar z_1dz_1+\bar z_2dz_2$, restricted to $S^3$, is purely imaginary and invariant under the $U(1)$ action, and it evaluates to $i$ on the generator of the action. It is therefore a $\mathfrak{u}(1)=i\mathbb{R}$-valued connection form. Its curvature descends to a constant multiple of the area form on $S^2$, and $\frac{i}{2\pi}\int_{S^2}F$ equals $\pm1$, the sign depending on orientation conventions. Principal $U(1)$-bundles over $S^2$ are classified by this first Chern number $c_1\in\mathbb{Z}$, and the Hopf bundle generates the group. In physics language, $c_1$ is the magnetic charge in units of the minimal Dirac charge, and the connection above is the vector potential of the unit monopole written without Dirac strings, since it lives on the total space.

As a map of spheres, $\pi$ has Hopf invariant one and generates $\pi_3(S^2)\cong\mathbb{Z}$. The Hopf invariant of a map $S^3\to S^2$ equals the linking number of the preimages of two distinct regular values, and for $\pi$ two fibres are great circles in $S^3$ with linking number one. The construction works with $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ and gives fibrations $S^0\to S^1\to S^1$, $S^1\to S^3\to S^2$, $S^3\to S^7\to S^4$ and $S^7\to S^{15}\to S^8$. By Adams' theorem on maps of Hopf invariant one, these are the only dimensions in which such a map exists. The quaternionic one is a principal $SU(2)$-bundle over $S^4$ with second Chern number $\pm1$, the bundle of the single BPST instanton.

Why is it interesting?

Hopf bundles are the correct mathematical tools that we need to describe the physics around a magnetic monopole or around instantons.

Yep. My advisor said that if you want to convince aliens that there's intelligent life on earth, tell them about 1) π and 2) the Hopf map.

https://twitter.com/math3ma/status/837075372617302016

It's famous because the map from the total space to the base was the first example of a topologically nontrivial map from a sphere to a sphere of lower dimension. In the lingo of homotopy theory, we say it's the generator of the group π3(S2).

http://math.ucr.edu/home/baez/week141.html

Hopf’s construction of P : S 3 → S 2 was motivated by his interest in what are called the “higher homotopy groups” of the spheres (see Section 2.5). Although this is not our major concern at the moment, we point out that P was the first example of a continuous map $S^m → S^n$ with $m > n$that is not “nullhomotopic” (Section 2.3). From this it follows that the homotopy group $π3 (S^2 )$ is not trivial and this came as quite a surprize in the 1930’s.

page 16 in Topology, Geometry and Gauge fields by Naber

"When line bundles are regarded as models for the topological structure underlying the electromagnetic field the Hopf fibration is often called “the magnetic monopole”."

https://ncatlab.org/nlab/show/Hopf+fibration

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Manifold
5. Fiber Bundles
6. Lie Groups
7. SO(3)
8. SU(2)