nLab principal U(1)-bundle

Idea

Principal U(1)-bundles (or principal SO(2)-bundles) are special principal bundles with the first unitary group $U(1)$ (isomorphic to the second special orthogonal group $SO(2)$) as gauge group.

Principal U(1)-bundles appear in multiple areas of mathematics, for example the Seiberg-Witten equations or monopole Floer homology. Since $U(1)$ is the gauge group of electromagnetism, principal U(1)-bundles also appear in theoretical physics. For example, principal U(1)-bundles over the two-dimensional sphere $S^2$, which includes the complex Hopf fibration, can be used to describe the quantization of hypothetical three-dimensional ($\mathbb{R}^3\setminus\{0\}\simeq S^2$) magnetic monopoles, called Dirac monopoles, compare also with D=2 Yang-Mills theory.

Classification

Principal U(1)-bundles are classified by the classifying space BU(1) of the first unitary group $U(1)$, which is the infinite complex projective space $\mathbb{C}P^\infty$. ($EU(1)\cong ESO(2)$ is then the infinite-dimensional sphere $S^\infty$.) For a topological space $X$, one has a bijection:

$$[X,BU(1)] \cong[X,\mathbb{C}P^\infty] \xrightarrow\cong Prin_{U(1)}(X), [f]\mapsto f^*EU(1) \cong f^*S^\infty$$

Since $U(1)$ is the Eilenberg-MacLane space $K(\mathbb{Z},1)$, one has that $BU(1)$ is $K(\mathbb{Z},2)$. Due to this, one can consider the identity $c_1\colon BU(1)\rightarrow K(\mathbb{Z},2)$, which is exactly the first Chern class and an isomorphism. Postcomposition then creates a bijection to singular cohomology:

$$c_1\colon Prin_{U(1)}(X)\cong[X,BU(1)]\xrightarrow\cong [X,K(\mathbb{Z},2)]\cong H^2(X,\mathbb{Z})$$

$\mathbb{C}P^\infty$ is a CW complex, whose $n$-skeleton is $\mathbb{C}P^k$ with the largest natural $k\in\mathbb{N}$ fulfilling $2k\leq n$. For an $n$-dimensional CW complex $X$, the cellular approximation theorem states that every homotopy $X\rightarrow\mathbb{H}P^\infty$ is homotopic to a cellular map, which in particular factorizes over the canonical inclusion $\mathbb{C}P^k\hookrightarrow\mathbb{C}P^\infty$. As a result, the postcomposition $[X,\mathbb{C}P^k]\rightarrow[X,\mathbb{C}P^\infty]$ is surjective. In particular for $X$ having no more than three dimension, one has $k=1$ with $\mathbb{C}P^1\cong S^2$. Hence there is a connection to cohomotopy:

$$\pi^2(X)\rightarrow Prin_{U(1)}(X)$$

Its composition with the first Chern class is exactly the Hurewicz map $\pi^2(X)\rightarrow H^2(X,\mathbb{Z})$.

Associated vector bundle

For principal U(1)-bundles $P\twoheadrightarrow X$, there is an associated complex line bundle $P\times_{U(1)}\mathbb{C}\twoheadrightarrow X$ using the balanced product?.

Examples

• The canonical projection $S^{2n+1}\twoheadrightarrow\mathbb{C}P^n$ is a principal $U(1)$-bundle. For $n=1$ using $\mathbb{C}P^1\cong S^2$, the complex Hopf fibration $S^3\twoheadrightarrow S^2$ is a special case. In the general case, the classifying map is given by the canonical inclusion:
  $$\mathbb{C}P^n\hookrightarrow\mathbb{C}P^\infty \cong BU(1).$$
• One has $S^{2n+1}\cong U(n+1)/U(n)$, hence there is a principal U(1)-bundle $S^3\twoheadrightarrow U(2)$. Such principal bundles are classified by:
  $$\pi_3BU(1) \cong\pi_2U(1) \cong 1.$$

  Hence the principal bundle is trivial, which fits $SU(2)\cong S^3$ and $U(n)\cong\SU(n)\times U(1)$.

• One has $S^n\cong SO(n+1)/SO(n)$, hence there is a principal SO(2)-bundle $SO(3)\twoheadrightarrow S^2$. Such principal bundles are classified by:
  $$\pi_2BU(1) \cong\pi_1U(1) \cong\pi_1S^1 \cong\mathbb{Z}.$$

Related entries

• principal SU(2)-bundles

References

• Daniel Freed, Karen Uhlenbeck, Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)
• Allen Hatcher: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
• Allen Hatcher, Vector bundles and K-theory (web)

See also:

• Wikipedia, Principal U(1)-bundle