nLab principal U(1)-bundle

Idea

Principal $U(1)$-bundles (or principal $SO(2)$-bundles) are principal bundles whose structure group is the unitary group $U(1)$ (equivalently: the circle group, and special orthogonal group $SO(2)$).

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 promiently in theoretical physics (cf. fiber bundles in physics). For example, principal $U(1)$-bundles over the 2-sphere $S^2$, which includes the complex Hopf fibration, can be used to describe the Dirac charge 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.

Properties

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 $\mathbb{C}P^\infty$. ($E U(1)\cong E SO(2)$ is then the infinite-dimensional sphere $S^\infty$.) For a CW complex $X$, one has a bijection:

$$\begin{array}{ccc} [X,BU(1)] \cong [X,\mathbb{C}P^\infty] &\xrightarrow{\cong}& Prin_{U(1)}(X) \\ [f] &\mapsto& f^*E U(1) \cong f^*S^\infty \end{array}$$

(Mitchell 2011, Theorem 7.4).

Since $U(1)$ is the Eilenberg-MacLane space $K(\mathbb{Z},1)$, one has that $B U(1)$ is $K(\mathbb{Z},2)$. (Hatcher 2001, Example 4.50.). Due to this, one can consider the identity $c_1\colon B U(1)\rightarrow K(\mathbb{Z},2)$, which is exactly the first Chern class and an isomorphism (Hatcher 2017, Theorem 3.10.). Postcomposition then creates a bijection to singular cohomology:

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

The infinite complex projective space $\mathbb{C}P^\infty$ 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 (Hatcher 2001, Theorem 4.8.) states that every map $X \rightarrow \mathbb{H}P^\infty$ 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 homomorphism $\pi^2(X)\rightarrow H^2(X,\mathbb{Z})$.

Associated vector bundle

For principal $U(1)$-bundles $P\twoheadrightarrow X$, there is an associated complex line bundle $L=P\times_{U(1)}\mathbb{C}\twoheadrightarrow X$ using the balanced product. If $Q$ is the induced principal SU(2)-bundle (using the canonical inclusion $U(1)\hookrightarrow SU(2),z\mapsto diag(z,z^{-1})$), then its adjoint bundle is given by:

$$Ad(Q) =L^2\oplus\underline{\mathbb{R}}.$$

(Donaldson & Kronheimer 91, Eq. (4.2.12))

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:

  $$\mathbb{C}P^n \hookrightarrow \mathbb{C}P^\infty \cong B U(1).$$

• 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:

  $$\pi_3 B U(1) \cong \pi_2 U(1) \cong 1\mathrlap{\,.}$$

  (Mitchell 2011, Corollary 11.2.)

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:

$$\pi_2 B U(1) \cong \pi_1 U(1) \cong \pi_1 S^1 \cong \mathbb{Z}.$$

(Mitchell 2011, Corollary 11.2.)

Related entries

• circle n-bundle with connection

• complex oriented cohomology

Particular principal bundles:

• double cover (principal O(1)-bundle)
• principal O(2)-bundle
• principal U(1)-bundle (principal SO(2)-bundle)
• principal U(2)-bundle (principal $Spin^\mathrm{c}(3)$-bundle, principal $Spin^\mathrm{h}(2)$-bundle)
• principal U(3)-bundle
• principal SO(3)-bundle
• principal SO(4)-bundle ($Spin^\mathrm{h}(3)$-bundle)
• principal SO(5)-bundle
• principal SO(6)-bundle
• principal SO(7)-bundle
• principal SO(8)-bundle
• principal SU(2)-bundle (principal Sp(1)-bundle, principal Spin(3)-bundle)
• principal SU(3)-bundle
• principal SU(4)-bundle (principal Spin(6)-bundle)
• principal Sp(2)-bundle (principal Spin(5)-bundle)

References

• Daniel Freed, Karen Uhlenbeck, Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer (1991) [doi:10.1007/978-1-4613-9703-8)]

• Simon Donaldson, Peter Kronheimer: The Geometry of Four-Manifolds (1990, revised 1997), Oxford University Press and Claredon Press, [oup:52942, doi:10.1093/oso/9780198535539.001.0001, ISBN:978-0198502692, ISSN:0964-9174]

• Allen Hatcher: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

• Stephen A. Mitchell, Notes on principal bundles (2011), Lecture Notes. University of Washington, 2011 [pdf, pdf]

• Allen Hatcher, Vector bundles and K-theory [web]

See also:

• Wikipedia: Principal U(1)-bundle