Standard Model gauge group

Definition

In the finite-dimensional internal-symmetry convention, the Standard Model gauge group is often written

G_{\mathrm{SM}}^{\mathrm{prod}}=U(1)_Y\times SU(2)_L\times SU(3)_C.

On the known Standard Model particle multiplets, a central subgroup $\mathbb Z_6$ acts trivially. The corresponding effective group is

G_{\mathrm{SM}}^{\mathrm{eff}} =G_{\mathrm{SM}}^{\mathrm{prod}}/\mathbb Z_6 \cong S(U(2)\times U(3)).

Both the product and the quotient are called “the Standard Model gauge group”; statements sensitive to global topology must specify the convention.

The central kernel

The quotient is realized by

\phi:U(1)\times SU(2)\times SU(3)\longrightarrow S(U(2)\times U(3)), \qquad \phi(z,A,B)=(z^3A,z^{-2}B).

Its kernel is

\ker\phi=\{(z,z^{-3}I_2,z^2I_3):z^6=1\}\cong\mathbb Z_6.

The codomain is the special block unitary group. This finite central quotient leaves the Standard Model Lie algebra unchanged, but it restricts which group representations and principal bundles are globally allowed.

Physics labels

The factors encode weak hypercharge, weak isospin acting on left-handed doublets, and color. The subscripts $Y,L,C$ are physical labels, not additional mathematical operations. The matter representations factor through $G_{\mathrm{SM}}^{\mathrm{eff}}$ .

Warning: two meanings of gauge group

Here “gauge group” names a compact finite-dimensional structure group of internal symmetries. Given a principal $G_{\mathrm{SM}}$ -bundle $P\to M$ , its principal-bundle gauge group is instead the generally infinite-dimensional group

\mathcal G(P)=\operatorname{Aut}_{G_{\mathrm{SM}}}(P)

of bundle automorphisms covering the identity of spacetime. These groups should not be identified.

References

John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552. arXiv record. Relevant: §§2.3 and 3.1.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv record. Relevant: §§1 and 3.
John C. Baez, “Three Generations in $E_7$ ,” 2026. arXiv record. Relevant: §§1–2.