Exceptional quantum geometry and particle physics

Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space $\mathbb C\oplus\mathbb C^3$ is associated to the quark-lepton symmetry, (one complex for the lepton and 3 for the corresponding quark). More generally it is is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of "the algebra of real functions" on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras, (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.Comment: 37 pages ; some minor typo corrections. To appear in Nucl. Pays. B (2016), http://dx.doi.org/10.1016/j.nuclphysb.2016.04.01

Primitive axial algebras of Jordan type

An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-\eta)$ for fixed $0\neq\eta\neq 1$. Jordan algebras arise when $\eta=\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the $2$-generated examples. For $\eta \neq \frac{1}{2}$ this implies that the Miyamoto involutions are $3$-transpositions, leading to a classification.Comment: 41 pages; comments welcom