nLab principal SU(2)-bundle

Idea

Principal SU(2)-bundles (or principal Sp(1)-bundles) are special principal bundles with the second special unitary group $SU(2)$ (isomorphic to the first symplectic group $Sp(1)$) as gauge group.

Principal SU(2)-bundles appear in multiple areas of mathematics, for example Donaldson's theorem or instanton Floer homology. Since $SU(2)$ is the gauge group of the weak interaction, principal SU(2)-bundles also appear in theoretical physics. For example, principal SU(2)-bundles over the four-dimensional sphere $S^4$, which includes the quaternionic Hopf fibration, can be used to describe the quantization of hypothetical five-dimensional ($\mathbb{R}^5\setminus\{0\}\simeq S^4$) magnetic monopoles, called Yang monopoles, compare also with D=4 Yang-Mills theory.

Classification

Principal SU(2)-bundles are classified by the classifying space BSU(2) of the second special unitary group $SU(2)$, which is the infinite quaternionic projective space $\mathbb{H}P^\infty$. ($ESU(2)\cong ESp(1)$ is then the infinite-dimensional sphere $S^\infty$.) For a topological space $X$, one has a bijection:

$$[X,BSU(2)] \cong[X,\mathbb{H}P^\infty] \xrightarrow\cong Prin_{SU(2)}(X), [f]\mapsto f^*ESU(2) \cong f^*S^\infty$$

Since $SU(2)$ rationalized is the rationalized Eilenberg-MacLane space $K(\mathbb{Z},3)_\mathbb{Q}$, one has that $BSU(2)$ rationalized is $K(\mathbb{Z},4)_\mathbb{Q}$. From the Postnikov tower, one even has a canonical map $c_2\colon BSU(2)\rightarrow K(\mathbb{Z},4)$, which is exactly the second Chern class and becomes an isomorphism under rationalization. Postcomposition then creates a map to singular cohomology:

$$c_2\colon Prin_{SU(2)}(X)\cong[X,BSU(2)]\rightarrow[X,K(\mathbb{Z},4)]\cong H^4(X,\mathbb{Z})$$

$\mathbb{H}P^\infty$ is a CW complex, whose $n$-skeleton is $\mathbb{H}P^k$ with the largest natural $k\in\mathbb{N}$ fulfilling $4k\leq n$. 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 $\mathbb{H}P^k\hookrightarrow\mathbb{H}P^\infty$. As a result, the postcomposition $[X,\mathbb{H}P^k]\rightarrow[X,\mathbb{H}P^\infty]$ is surjective. In particular for $X$ having no more than seven dimension, one has $k=1$ with $\mathbb{H}P^1\cong S^4$. Hence there is a connection to cohomotopy:

$$\pi^4(X)\rightarrow Prin_{SU(2)}(X)$$

Its composition with the second Chern class is exactly the Hurewicz map $\pi^4(X)\rightarrow H^4(X,\mathbb{Z})$.

Associated vector bundle

For principal SU(2)-bundles $P\twoheadrightarrow X$, there is an associated complex plane bundle $P\times_{SU(2)}\mathbb{C}^2\twoheadrightarrow X$ using the balanced product?.

Examples

• The canonical projection $S^{4n+3}\twoheadrightarrow\mathbb{H}P^n$ is a principal $SU(2)$-bundle. For $n=1$ using $\mathbb{H}P^1\cong S^4$, the quaternionic Hopf fibration $S^7\twoheadrightarrow S^4$ is a special case. In the general case, the classifying map is given by the canonical inclusion:
  $$\mathbb{H}P^n\hookrightarrow\mathbb{H}P^\infty \cong BSU(2).$$
• One has $S^{2n+1}\cong SU(n+1)/SU(n)$, hence there is a principal SU(2)-bundle $SU(3)\twoheadrightarrow S^5$. Such principal bundles are classified by:
  $$\pi_5BSU(3) \cong\pi_4SU(3) \cong\pi_4S^3 \cong\mathbb{Z}_2.$$

  $SU(3)\twoheadrightarrow S^5$ is the unique non-trivial principal bundle, which can be detected by the fourth homotopy group:

  $$\pi_4SU(3) \cong 1;$$
  $$\pi_4(S^5\times SU(2)) \cong\pi_4(S^5)\times\pi_4S^3 \cong\mathbb{Z}_2.$$
• One has $S^{4n+3}\cong Sp(n+1)/Sp(n)$, hence there is a principal Sp(1)-bundle $Sp(2)\twoheadrightarrow S^7$. Such principal bundles are classified by:
  $$\pi_7BSU(3) \cong\pi_5SU(3) \cong\pi_6S^3 \cong\mathbb{Z}_{12}.$$

Related entries

• principal U(1)-bundles

References

• Simon Donaldson. An application of gauge theory to four-dimensional topology. In: Journal of Differential Geometry. 18. Jahrgang, Nr. 2, 1. Januar 1983, doi:10.4310/jdg/1214437665
• Simon Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. In: Journal of Differential Geometry. 26. Jahrgang, Nr. 3, 1. Januar 1987, doi:10.4310/jdg/1214441485
• 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 SU(2)-bundle