Principal U(1)-bundle

Composition in the first unitary group

Template:Short description In mathematics, especially differential geometry, principal $\mathrm{U}(1)$-bundles (or principal $\mathrm{SO}(2)$-bundles) are special principal bundles with the first unitary group $\mathrm{U}(1)$ (isomorphic to the second special orthogonal group $\mathrm{SO}(2)$) as structure group. Topologically, it has the structure of the one-dimensional sphere, hence principal $\mathrm{U}(1)$-bundles without their group action are in particular circle bundles. These are basically topological spaces with a circle glued to every point, so that all of them are connected with each other, but globally aren't necessarily a product and can instead be twisted like a Möbius strip.

Principal $\mathrm{U}(1)$-bundles are used in many areas of mathematics, for example for the formulation of the Seiberg–Witten equations or monopole Floer homology. Since $\mathrm{U}(1)$ is the gauge group of the electromagnetic interaction, principal $\mathrm{U}(1)$-bundles are also of interest in theoretical physics. Concretely, the $\mathrm{U}(1)$-Yang–Mills equations are exactly Maxwell's equations. In particular, principal $\mathrm{U}(1)$-bundles over the two-dimensional sphere $S^2$, which include the complex Hopf fibration, can be used to describe hypothetical magnetic monopoles in three dimensions, known as Dirac monopoles, see also two-dimensional Yang–Mills theory.

Definition

Principal $\mathrm{U}(1)$-bundles are generalizations of canonical projections $B\times \mathrm{U}(1)\twoheadrightarrow B$ for topological spaces $B$, so that the source is not globally a product but only locally. More concretely, a continuous map $p:E\twoheadrightarrow B$ with a continuous right group action $E\times \mathrm{U}(1)\to E$, which preserves all preimages of points, hence $p(eg)=p(e)$ for all $e\in E$ and $g\in G$, and also acts free and transitive on all preimages of points, which makes all of them homeomorphic to $\mathrm{U}(1)$, is a principal $\mathrm{U}(1)$-bundle.

Since principal bundles are in particular fiber bundles with the group action missing, their nomenclature can be transferred. $E$ is also called the total space and $B$ is also called the base space. Preimages of points are then the fibers. Since $\mathrm{U}(1)$ is a Lie group, hence in particular a smooth manifold, the base space $B$ is often chosen to be a smooth manifold as well since this automatically makes the total space $E$ into a smooth manifold as well.

Classification

Principal $\mathrm{U}(1)$-bundles can be fully classified using the classifying space $\mathrm{BU}(1)$ of the first unitary group $\mathrm{U}(1)$, which is exactly the infinite complex projective space $ℂP^{\mathrm{\infty }}$. For a topological space $B$, let ${\mathrm{Prin}}_{\mathrm{U}(1)}(B)$ denote the set of equivalence classes of principal $\mathrm{U}(1)$-bundles over it, then there is a bijection with homotopy classes:

${\mathrm{Prin}}_{\mathrm{U}(1)}(B)\cong [B,\mathrm{BU}(1)]\cong [B,ℂP^{\mathrm{\infty }}].$

$ℂP^{\mathrm{\infty }}$ is a CW complex with its $n$-skeleton being $ℂP^k$ for the largest natural number $k\in \mathbb{N}$ with $2k\le n$. For a $n$-dimensional CW complex $B$, the cellular approximation theorem states that every continuous map $B\to ℂP^{\mathrm{\infty }}$ is homotopic to a cellular map factoring over the canonical inclusion $ℂP^k\hookrightarrow ℂP^{\mathrm{\infty }}$. As a result, the induced map $[B,ℂP^k]\hookrightarrow [B,ℂP^{\mathrm{\infty }}]$ is surjective, but not necessarily injective as higher cells of $ℂP^{\mathrm{\infty }}$ allow additional homotopies. In particular if $B$ is a CW complex of three or less dimensions, then $k=1$ and with $ℂP^1\cong S^2$, there is a connection to cohomotopy sets with a surjective map:

${\pi }^2(B)\to {\mathrm{Prin}}_{\mathrm{U}(1)}(B).$

$ℂP^{\mathrm{\infty }}$ is also the Eilenberg–MacLane space $K(\mathbb{Z},2)$, which represents singular cohomology, compare to Brown's representability theorem:

${\mathrm{Prin}}_{\mathrm{U}(1)}(B)\cong H^2(B,\mathbb{Z}).$

(The composition ${\pi }^2(B)\to H^2(B,\mathbb{Z})$ is the Hurewicz map.) A corresponding isomorphism is given by the first Chern class. Although characteristic classes are defined for vector bundles, it is possible to also define them for certain principal bundles.

Associated vector bundle

Given a principal $\mathrm{U}(1)$-bundle $E\twoheadrightarrow B$, there is an associated vector bundle $E{\times }_{\mathrm{U}(1)}ℂ\twoheadrightarrow B$. Intuitively, the spheres at every point are filled over the canonical inclusions $\mathrm{U}(1)\subset ℂ$. Due to the single rank, the vector bundle is only described by the first Chern class $c_1:{\mathrm{Vect}}_{ℂ}(B)\to H^2(B,\mathbb{Z})$, which is an isomorphism over CW complexes.

Principal bundles also have an adjoint vector bundle, which is trivial for principal $\mathrm{U}(1)$-bundles.

Examples

• By definition of complex projective space, the canonical projection $S^{2n+1}\twoheadrightarrow ℂP^n$ is a principal $\mathrm{U}(1)$-bundle. With $ℂP^1\cong S^2$, known as the Riemann sphere, the complex Hopf fibration $h_{ℂ}:S^3\twoheadrightarrow S^2$ is a special case. For the general case, the classifying map is the canonical inclusion:

$ℂP^n\hookrightarrow ℂP^{\mathrm{\infty }}\cong \mathrm{BU}(1).$

• One has $S^{2n+1}\cong \mathrm{U}(n+1)/\mathrm{U}(n)$, which means that there is a principal $\mathrm{U}(1)$-bundle $\mathrm{U}(2)\twoheadrightarrow S^3$. Such bundles are classified by:

${\pi }_3\mathrm{BU}(1)\cong {\pi }_2\mathrm{U}(1)\cong {\pi }_2{S}^1\cong 1.$

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

• One has $S^n\cong \mathrm{SO}(n+1)/\mathrm{SO}(n)$, which means that (using $\mathrm{U}(1)\cong \mathrm{SO}(2)$) there is a principal $\mathrm{U}(1)$-bundle $\mathrm{SO}(3)\twoheadrightarrow S^2$. Such bundles are classified by:

${\pi }_2\mathrm{BU}(1)\cong {\pi }_1\mathrm{U}(1)\cong {\pi }_1{S}^1\cong \mathbb{Z}.$

One has $\mathrm{SO}(3)\cong ℝP^3$ and the composition of the canonical double cover $S^3\twoheadrightarrow ℝP^3$ with the principal bundle $ℝP^3\twoheadrightarrow S^2$ is exactly the complex Hopf fibration $S^3\twoheadrightarrow S^2$. Since the first Chern class of the complex Hopf fibration is $-1$, the first Chern class of the principal bundle $ℝP^3\twoheadrightarrow S^2$ is $-2$.

See also

• Principal SU(2)-bundle

Literature

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External links

• principal U(1)-bundle at the nLab