nLab SU(5)

Idea

The special unitary group in complex dimension 5.

Embedding of the gauge group of the standard model

$SU(5)$ is of particular importence for the gauge group of the standard model. It is composed of the:

• first unitary group $U(1)$ for the electromagnetic interaction described by quantum electrodynamics (QED). $dim U(1)=1$ corresponds to the one photon.
• second special unitary group $SU(2)$ for the weak interaction described by quantum flavordynamics (QFD). $dim SU(2)=3$ corresponds to the three massive gauge bosons $W^-$, $W^+$ and $Z$.
• third special unitary group $SU(3)$ for the strong interaction described by quantum chromodynamics (QCD). $dim SU(3)=8$ corresponds to the eight types of gluons.

It is not the product $U(1)\times SU(2)\times SU(3)$, but instead its quotient group $G_SM=(U(1)\times SU(2)\times SU(3))/\mathbb{Z}_6$ (with all known particles invariant under this $\mathbb{Z}_6$ action). There is an embedding:

$$\Phi\colon U(1)\times SU(2)\times SU(3)\hookrightarrow SU(5), (e^{i\varphi},U,V)\mapsto\begin{pmatrix} e^{-3i\varphi}U & 0 \\ 0 & e^{2i\varphi}V \end{pmatrix}$$

It has $ker(\Phi)\cong\mathbb{Z}_6$ generated by $(\zeta_6,\zeta_3\mathbf{1}_2,\zeta_2\mathbf{1}_3)$ with the primitive roots of unity $\zeta_6,\zeta_2=\zeta_6^3,\zeta_3=\zeta_6^2\in U(1)$. Hence $G_SM\cong img(\Phi)\cong S(U(2)\times U(3))$.

Related concepts

• SU(2), SU(3), SU(4)

• GUT

References

• Howard Georgi, §18 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

  with an eye towards grand unified theory