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March 28, 2026

Geometry and the Exceptional Jordan Algebra

Posted by John Baez

I’m giving a talk online tomorrow at the 2026 Spring Southeastern Sectional Meeting of the American Mathematical Society, in the Special Session on Non-Associative Rings and Algebras. The organizers are Layla Sorkatti and Kenneth Price. I doubt the talk will be recorded, but you can see my slides.

Abstract. Dubois-Violette and Todorov noticed that the gauge group of theStandard Model of particle physics is the intersection of two maximal subgroups of F 4 \text{F}_4. which is the automorphism group of the exceptional Jordan algebra 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}). Here we conjecture that these can be taken to be any subgroups preserving copies of 𝔥 2(𝕆) \mathfrak{h}_2(\mathbb{O}) and 𝔥 3(ℂ)\mathfrak{h}_3(\mathbb{C}) that intersect in a copy of 𝔥 2(ℂ) \mathfrak{h}_2(\mathbb{C}). Given this, we show that the Standard Model gauge group consists of all isometries of the octonionic projective plane that preserve an octonionic projective line and a complex projective plane intersecting in a complex projective line. This is joint work with Paul Schwahn.

This is an introductory talk for mathematicians. Physicists may prefer the two talks here. Those go much further in some ways, but they don’t cover the new ideas that Paul Schwahn and I are in the midst of working on.

Posted at March 28, 2026 12:44 AM UTC

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Re: Geometry and the Exceptional Jordan Algebra

Indeed, and to include E 6E_6, SO(10)SO(10) and flipped SU(5)SU(5) GUT, take the F 4F_4 associator and add a complex translate, which gives E 6E_6 instead of E 6(−26)E_{6(-26)}.

Posted by: Metatron on April 20, 2026 2:47 PM | Permalink | Reply to this

Re: Geometry and the Exceptional Jordan Algebra

This is all very interesting from the viewpoint of reconstructing quantum theory. One much-discussed genre of such derivations go through the Euclidean Jordan algebras, and so the octonionic qutrit is an oddball that shows up quite naturally. It even has the nice feature that it supports a complete set of equiangular lines and thus a symmetric reference measurement. So, it would be nice in a way if it had physical relevance.

Posted by: Blake Stacey on June 10, 2026 5:16 AM | Permalink | Reply to this

Re: Geometry and the Exceptional Jordan Algebra

Yes, after a few beers I have grandiose hopes for obtaining the Standard Model from quantum information theory, with octonionic qutrits playing a fundamental role. Then I go to bed and wake up to the cold morning light feeling far less optimistic.

Luckily there are math problems in this subject that deserve to be solved and don’t require superhuman powers. Paul Schwahn and I have just about finished a paper proving the conjecture described in my talk, and an even better result! Stay tuned.

Posted by: John Baez on June 10, 2026 6:50 PM | Permalink | Reply to this

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