The Standard Model gauge group and the exceptional Jordan algebra

Abstract: Recently, Todorov and Dubois-Violette have realized the gauge group of the Standard Model of particle physics as the intersection of two maximal connected subgroups in the exceptional Lie group F_4, which is the automorphism group of the exceptional Jordan algebra of octonionic Hermitian 3x3-matrices. We show how two understand this inclusion of groups in terms of substructures on the Jordan algebra: in fancy language, the gauge group is the group of symmetries of an "octonionic qutrit" that restrict to act as unitary operators on a complex qutrit and, within that, a qubit. This result can be phrased both in Jordan algebra and in projective geometry language, and we will elucidate this connection.

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