Principal SU(2)-bundle

In mathematics, especially differential geometry, principal ${\displaystyle \operatorname {SU} (2)}$-bundles (or principal ${\displaystyle \operatorname {Sp} (1)}$-bundles) are special principal bundles with the second special unitary group ${\displaystyle \operatorname {SU} (2)}$ (isomorphic to the first symplectic group ${\displaystyle \operatorname {Sp} (1)}$) as structure group. Topologically, it has the structure of the three-dimensional sphere, hence principal ${\displaystyle \operatorname {SU} (2)}$-bundles without their group action are in particular sphere bundles. These are basically topological spaces with a sphere 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 ${\displaystyle \operatorname {SU} (2)}$-bundles are used in many areas of mathematics, for example for the Fields Medal winning proof of Donaldson's theorem or instanton Floer homology. Since ${\displaystyle \operatorname {SU} (2)}$ is the gauge group of the weak interaction, principal ${\displaystyle \operatorname {SU} (2)}$-bundles are also of interest in theoretical physics. In particular, principal ${\displaystyle \operatorname {SU} (2)}$-bundles over the four-dimensional sphere ${\displaystyle S^{4}}$, which include the quaternionic Hopf fibration, can be used to describe hypothetical magnetic monopoles in five dimensions, known as Wu–Yang monopoles, see also four-dimensional Yang–Mills theory.