Principal U(1)-bundle

In mathematics, especially differential geometry, principal ${\displaystyle \operatorname {U} (1)}$-bundles (or principal ${\displaystyle \operatorname {SO} (2)}$-bundles) are special principal bundles with the first unitary group ${\displaystyle \operatorname {U} (1)}$ (isomorphic to the second special orthogonal group ${\displaystyle \operatorname {SO} (2)}$) as structure group. Topologically, it has the structure of the one-dimensional sphere, hence principal ${\displaystyle \operatorname {U} (1)}$-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 ${\displaystyle \operatorname {U} (1)}$-bundles are used in many areas of mathematics, for example for the formulation of the Seiberg–Witten equations or monopole Floer homology. Since ${\displaystyle \operatorname {U} (1)}$ is the gauge group of the electromagnetic interaction, principal ${\displaystyle \operatorname {U} (1)}$-bundles are also of interest in theoretical physics. Concretely, the ${\displaystyle \operatorname {U} (1)}$-Yang–Mills equations are exactly Maxwell's equations. In particular, principal ${\displaystyle \operatorname {U} (1)}$-bundles over the two-dimensional sphere ${\displaystyle S^{2}}$, 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.