Standard Model gauge group as an F4 stabilizer intersection

Statement

Let $J=\mathfrak h_3(\mathbb O)$ be the exceptional Jordan algebra, and let the compact group $F_4$ act through its Jordan-algebra automorphisms. Suppose $X\subset B\subset J$ are Jordan subalgebras with

X\cong\mathfrak h_2(\mathbb C), \qquad B\cong\mathfrak h_3(\mathbb C).

Writing $\operatorname{Stab}_{F_4}(-)$ for setwise stabilizers and $(-)^0$ for the identity component,

\operatorname{Stab}_{F_4}(X)\cap \operatorname{Stab}_{F_4}(B)^0 \cong S(U(2)\times U(3)),

the effective Standard Model internal symmetry group. Here $\mathfrak h_2(\mathbb C)$ and $\mathfrak h_3(\mathbb C)$ are the complex-qubit and complex-qutrit Jordan algebras.

Equivalent larger-subalgebra formulation

The copy $X\cong\mathfrak h_2(\mathbb C)$ lies in a unique octonionic spin factor $A\cong\mathfrak h_2(\mathbb O)$ . Moreover $X=A\cap B$ , and

\operatorname{Stab}_{F_4}(A)\cap \operatorname{Stab}_{F_4}(B)^0 \cong S(U(2)\times U(3)).

The group on the right is the special block unitary group. For a standard pair,

\operatorname{Stab}_{F_4}(A)\cong\operatorname{Spin}(9), \qquad \operatorname{Stab}_{F_4}(B)^0 \cong (SU(3)\times SU(3))/\mathbb Z_3.

The proof uses the complex-qutrit stabilizer calculation and transitivity of $F_4$ on complex-qutrit subalgebras and compatible subalgebra pairs.

Why the identity component is essential

The full stabilizer $\operatorname{Stab}_{F_4}(B)$ is disconnected. Its extra component contains symmetries whose restriction to $B\cong\mathfrak h_3(\mathbb C)$ has the form $X\mapsto UXU^{-1}$ , where $U$ is antiunitary on $\mathbb C^3$ . Therefore

\operatorname{Stab}_{F_4}(X)\cap\operatorname{Stab}_{F_4}(B)

is strictly larger than the Standard Model group. The superscript $0$ is part of the theorem.

Interpretation and scope

The two complex Jordan algebras are observable algebras of a qubit and a qutrit, motivating an “octonionic qutrit” interpretation of $J$ . The theorem is a precise group-theoretic characterization; by itself it does not construct the Standard Model Lagrangian, select its fermion representation, or prove that $F_4$ is a physical gauge symmetry.

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv record. Relevant: Theorems 1–2 and §§3–5.
Ivan Todorov and Michel Dubois-Violette, “Deducing the Symmetry of the Standard Model from the Automorphism and Structure Groups of the Exceptional Jordan Algebra,” International Journal of Modern Physics A 33 (2018), 1850118. DOI record.
Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record. Relevant: Chapter IV.