nLab principal U(3)-bundle

Idea

Principal SU(3)-bundles are special principal bundles with the third special unitary group SU(3) as structure group (gauge group).

Characteristic classes

Proposition

A principal U(3)-bundle $P$ fulfills:

$$e(P) \equiv c_3(P).$$

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

(Hatcher 17, Prop. 3.13 c)

Associated vector bundle

For principal $U(3)$-bundles $P\twoheadrightarrow X$, there is an associated complex plane bundle $E=P\times_{U(3)}\mathbb{C}^3\twoheadrightarrow X$ using the balanced product. If $Q$ is the induced principal SU(4)-bundle (using the canonical inclusion $U(3)\hookrightarrow SU(4),U\mapsto diag(U,det(U)^{-1})$), then its adjoint bundle is given by:

$$Ad(P) \cong Ad(Q)\oplus(det(E)\otimes E)_\mathbb{R}.$$

If $P$ reduces to a principal SU(3)-bundle, this reduces to:

$$Ad(P) \cong Ad(Q)\oplus E_\mathbb{R}\oplus\underline{\mathbb{R}}.$$

Both relations hold in general for the maps $SU(n)\hookrightarrow U(n)\rightarrow SU(n+1)$.

(Donaldson & Kronheimer 91, p. 205)

Examples

• One has $S^{2n+1}\cong U(n+1)/U(n)$, hence there is a principal U(3)-bundle $U(4)\twoheadrightarrow S^7$. Such principal bundles are classified by:
  $$\pi_7B U(3) \cong\pi_6 U(3) \cong\mathbb{Z}_6.$$

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

• 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]

• Morgan Opie, A classification of complex rank 3 vector bundles on complex projective 5-space (2023) [arxiv:2301.04313 doi:10.48550/arXiv.2301.04313]