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 quaternionic unitary group $Sp(1)$) as structure group/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 (cf. fiber bundles in physics). For example, principal $SU(2)$-bundles over the 4-sphere $S^4$, which includes the quaternionic Hopf fibration, can be used to describe the Dirac charge 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.

Properties

Classification

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

$$\begin{array}{ccc} [X,B SU(2)] \cong [X,\mathbb{H}P^\infty] &\xrightarrow{\cong}& Prin_{SU(2)}(X), \\ [f] &\mapsto& f^* E SU(2) \cong f^*S^\infty \mathrlap{\,.} \end{array}$$

(Mitchell 2011, Theorem 7.4)

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

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

The quaternionic projective space $\mathbb{H}P^\infty$ is a CW complex, whose $n$-skeleton is $\mathbb{H}P^k$ with the largest natural $k\in\mathbb{N}$ fulfilling $4k\leq n$. (Hatcher 2001, p. 222).

For an $n$-dimensional CW complex $X$, the cellular approximation theorem (Hatcher 01, Theorem 4.8.) 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 six dimensions, one has $k=1$ with $\mathbb{H}P^1\cong S^4$. Hence there is a connection to cohomotopy:

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

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

Adjoint vector bundle

Proposition

For a principal $SU(2)$-bundle, the first Pontrjagin class of its adjoint bundle is given by:

$$p_1Ad(P) =-4c_2(P).$$

(Donaldson & Kronheimer 91, Eq. 2.1.37)

(This relation holds in general for principal $SU(n)$-bundles with $p_1 Ad(P)=-2nc_2(P)$.)

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:

  $$\pi_5BSU(2) \cong \pi_4SU(2) \cong \pi_4S^3 \cong \mathbb{Z}_2 \mathrlap{\,.}$$

  (Mitchell 2011, Corollary 11.2.) $SU(3)\twoheadrightarrow S^5$ is the unique non-trivial principal bundle, which can be detected by the fourth homotopy group:

  $$\pi_4 SU(3) \cong 1;$$
  $$\pi_4\big(S^5\times SU(2)\big) \cong \pi_4(S^5)\times\pi_4 S^3 \cong \mathbb{Z}_2 \mathrlap{\,.}$$

  (Mimura & Toda 63, Donaldson 1983, p. 295)

• One has $S^{4n+3}\cong Sp(n+1)/Sp(n)$, hence there is a principal Sp(1)-bundle $Sp(2)\twoheadrightarrow S^7$. See in particular Spin(5)/SU(2) is the 7-sphere. Such principal bundles are classified by:

  $$\pi_7 B SU(3) \cong \pi_6 SU(3) \cong \pi_6 S^3 \cong \mathbb{Z}_{12} \mathrlap{\,.}$$

  (Mitchell 2011, Corollary 11.2.)

Application in Yang-Mills theory

Proposition

Reducible anti self-dual Yang-Mills connections of a principal $SU(2)$-bundle $E$ over a compact, simply connected, oriented Riemannian 4-manifold $X$ are in bijective correspondence with non-zero pairs $\pm c\in H^2(X,\mathbb{Z})$ with $c^2=-c_2(E)$.

(Donaldson & Kronheimer 97, Prop. (4.2.15))

Proposition

Irreducible anti self-dual Yang-Mills connections of a principal $SU(2)$-bundle $E$ over a simply connected Riemannian 4-manifold $X$ are still irreducible under the restriction to any non-empty open subset.

(Donaldson & Kronheimer 97, Lem. (4.3.21))

Related entries

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

• Mamoru Mimura and Hiroshi Toda, Homotopy Groups of SU(3), SU(4) and Sp(2) (1963), Journal of Mathematics of Kyoto University 3 (2), p. 217–250, doi:10.1215/kjm/1250524818

• Simon Donaldson: An application of gauge theory to four-dimensional topology. In: Journal of Differential Geometry. 18. Jahrgang, Nr. 2 (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 (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]

• 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 SU(2)-bundle