Wednesday, November 22, 2017

Exceptional Periodicity

As was hinted at in a previous post, it is possible to view the exceptional Lie algebras as the tip of an infinite algebraic spectrum.  This novel concept, we coined Exceptional Periodicity (EP), is now available for download on the arXiv: arXiv:1711.07881 [hep-th].

This EP structure was inspired by certain Yang-Mills-like gradings of the exceptional Lie algebras, as well as higher dimensional spin groups, used in approaches to unification.  It differs from the conventional infinite dimensional generalizations of e8 in that the Jacobi identity is not in general obeyed by these higher algebras, yet do retain structure similar to lattice vertex algebras.  Moreover, building on the "Magic Star" projection of e8, each of these higher algebras can be projected to higher Magic Stars, that generalize that of e8.  At the six inner vertices of the star, the cubic Jordan algebras are generalized to a cubic ternary algebra, first envisioned by Vinberg, dubbed T-algebras.  Such T-algebras are reminiscent of spin factors and Peirce decompositions of cubic Jordan algebras.

The e8 Magic Star thus encodes the exceptional Jordan algebra on its six star vertices, which exhibits triality, from its off-diagonal 8D components.  These higher stars do not retain this triality, as the bosonic off-diagonal parts do not grow as fast as the spinor part, which grows exponentially.

So what can be done with these higher EP Magic Stars?  The T-algebras appear to encode a rich matter sector, that generalize the 16, 32, 64 and 128 spinors found in the exceptional Lie algebras.  Such higher stars can be used to design higher mathematical universes, in a periodic, algebraic fashion.  More details will be given in a series of upcoming papers.  Stay tuned.

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