Braids, circulants, and Jordan algebras

Sheppeard brings together a number of correspondences and formal schemes in her attempt to recreate the standard model. Here I will touch on Sundance Bilson-Thompson’s braids, Carl Brannen’s circulant matrices, and Michael Rios’s Jordan algebras of observables for quantum gravity. In a nutshell, the fermions of a single generation are represented by braids, the circulants encode information about the generations, and the Jordan algebras associate the braids with the circulants.

In Bilson-Thompson’s braid correspondence, the fermions of a single standard model generation are represented by braidings of three “ribbons”. A ribbon may itself contain a twist; the electric charge of a braid containing m twists and n antitwists is (mn)/3. The correspondence suggests a topological realization of the rishon model; but Bilson-Thompson’s correspondence still lacks a counterpart to the Harari-Seiberg hypercolor lagrangian. That is, no-one has put together a well-defined dynamical theory in which the braids interact as they should.

Carl Brannen is working on an algebraic preon model which seeks to explain the Koide mass formula. It turns out that masses satisfying the formula are eigenvalues of a 3×3 circulant matrix. Brannen and Sheppeard have worked on expressing all the standard model mass matrices in terms of circulants.

Finally, Michael Rios has written a series of papers in which he aspires to have M-theory emerge from a matrix model based on the exceptional Jordan algebra. Here he defines the algebra of observables in terms resembling loop quantum gravity; here he claims the result resembles an octonionic twistor string. Loop quantum gravity is the framework within which Bilson-Thompson’s correspondence is usually pursued, and it seems that Sheppeard hopes to motivate an association between braids and circulants via something like Rios’s constructions.

This effort has been going on since 2005 or so, and features in several of Sheppeard’s recent papers.




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