nLab principal SU(4)-bundle

Idea

Principal SU(4)-bundles (or principal Spin(6)-bundles) are special principal bundles with the fourth special unitary group $SU(4)$ (exceptionally isomorphic to the sixth spin group $Spin(6)$) as structure group (gauge group).

Principal $SU(4)$-bundles in particular induce principal SO(6)-bundles using the canonical projection $SU(4)\cong Spin(6)\twoheadrightarrow SO(6)$. Principal SU(4)-bundles also induce principal SU(2)-bundles and principal SU(3)-bundles using the canonical inclusions $SU(2)\hookrightarrow SU(3)\hookrightarrow SU(4)$.

Characteristic classes

Proposition

A principal SU(4)-bundle $P$ fulfills:

$$e(P) \equiv c_4(P).$$

(In general, a principal $SU(n)$-bundle $P$ fulfills $e(P)=c_n(P)$.)

(Hatcher 17, Prop. 3.13 c)

Adjoint vector bundle

Proposition

For a principal $SU(4)$-bundle, the first Pontrjagin class of its adjoint bundle is given by:

$$p_1Ad(P) =-8c_2(P).$$

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

Examples

• One has $S^{2n+1}\cong SU(n+1)/SU(n)$, hence there is a principal SU(4)-bundle $SU(5)\twoheadrightarrow S^9$. Such principal bundles are classified by:
  $$\pi_9B SU(4) \cong\pi_8 SU(4) \cong\mathbb{Z}_{24}.$$

(Mimura & Toda 63)

Related concepts

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, 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: Vector bundles and K-Theory, book draft (2017) [webpage, pdf]