A Furey's superalgebra and the standard model

[mitchell porter]

TL;DR Summary

A new take on standard model "algebrology"

Nichol Furey (best known for trying to get the standard model from the octonions) has now contributed to the genre of "getting the standard model, except for the top quark, from a nonstandard supersymmetric structure":

"A Superalgebra Within: representations of lightest standard model particles form a Z_2^5-graded algebra"

This should be compared with ideas in a 2009 research note by @arivero, "Unbroken supersymmetry without new particles", page 4-5, where he talks about two ways in which the 96 fermionic degrees of freedom of the standard model, can be truncated to just 84: by considering only the "charged fermions" (and leaving out the the three flavors of neutrino), or by considering only the "lightest fermions" (and leaving out the three colors of top quark). Furey is taking the latter path, and then including the gauge bosons of the standard model as well.

@arivero was motivated by the idea that the "84 fermions" might be superpartners of the 84-component "C-field" of d=11 N=1 supergravity; we talked about it on this forum.

 

[arivero]

Yep, as I mentioned there, the "84 fermions" idea comes from a counting of degrees of freedom of my advisor, LJ Boya, in some of his last drafts on representation theory and strings. The "duality" view, on the other hand, is my own half baking, the idea being that we could look both to "protected dirac mass" and "protected majorana mass" somewhere in two dual objects of size 84. So not only the C-field, also its dual.

 

[mitchell porter]

I've only scraped the surface of the paper, and still don't see a way to bring it into contact with supergravity. But there's a picture on page 12 (figure 5) that is useful for general understanding.

At some level, Furey is talking about an algebra of 16x16 matrices. As this diagram shows, there is a block decomposition, and various physical operators are then associated with the blocks. You can see that there are grey blocks, dark blue blocks, and light blue blocks. The grey blocks are the first two generations, the dark blue blocks are gauge bosons and "Weyl momentum operators", and the light blue blocks are the third generation.

The interpretation of the light blue blocks is discussed on pages 10-11. It turns out that in the interpretation where the top quark is missing, the other third-generation states (tau neutrino and right-handed bottom quark) occur twice; and that there is an alternative interpretation of the light blue blocks in which all the third-generation states occur, but are realized "nonlinearly".

OK. What's interesting, is that the block decomposition is manifestly some kind of outer product, or is the square of a fivefold partition. There's five, let's say, projection operators, and each possible outer product of a pair of projection operators, corresponds to one of the blocks. (I apologize for using terminology that may not be strictly correct, I have not grasped the algebraic nuances.)

So it would actually be easier to compare Furey's concept to the sBootstrap! (For the general reader: the sBootstrap is also described in @arivero's 2009 paper above, and is the subject of this PF mega-thread.) There will necessarily be differences, e.g. gauge bosons are not part of the sBootstrap's output, and I don't think that Furey's five idempotents (equation 40) correspond to the five flavors of the sBootstrap. But there's probably some overlap, e.g. the way that a 10 of SU(5) is an antisymmetric square of the 5-bar. It also suggests looking for an "algebrological" formulation of the sBootstrap.

 

[arivero]

Furey's current work is continuation of her lectures in youtube, if someone comes here going for the "algebrological" path (happens once each ten years), go to her youtube channel.

 

[kodama]

TL;DR Summary: A new take on standard model "algebrology"

Nichol Furey (best known for trying to get the standard model from the octonions) has now contributed to the genre of "getting the standard model, except for the top quark, from a nonstandard supersymmetric structure":

why's except for the top quark?

are the objects fermions?