is Spin(8) triality the missing ingredient for SUSY in Connes’ approach?

Thinking about a major puzzle in the NCG approach to the standard model, I remember that sci.physics.research, via Baez’ weeks, was very fond of triality (in the way of Evans?) to justify why some dimensions allow supersymmetry. And this pivots over SO(8), for which I asked a couple of abusive questions in mathoverflow:

Does ??(32)???8×?8 relate to some group theoretical fact?

Why SU(3) is not equal to SO(5)?

The second one included a nice ascii sequence of dynkin diagrams, using

o====o  SO(5), isometries of the sphere S4
o----o  SU(3) are the isometries of CP2
o    o  SU(2)xSU(2), isometries of  S2xS2. Also SO(4), so isometries of S3

I compared

        o                  o                         o
o----o    SO(8)    o----o     SU(3)xSO(4)    o====o     SO(5)xSO(4)
        o                  o                         o    

and I wonder if I should add Pati-Salam

o----o----o     SU(4)xSU(2)xSU(2)     

4 thoughts on “is Spin(8) triality the missing ingredient for SUSY in Connes’ approach?”

  1. This also takes me back to the “M-theory phenomenology homework” of trying to understand the superpartners of the 11d C-field.

    It must be possible to understand, visually and combinatorially, how and why the superpartners of an 11d spinor-vector (the 11d gravitino) organize themselves into an 11d symmetric second-rank tensor (the 11d metric) and an 11d antisymmetric 3-form (the C-field).

    I get these descriptions from “M(ysterious) Patterns in SO(9)” coauthored by Ramond:

    … And then to understand what happens to these fermions, when compactified e.g. on T^7 and S^7. (As a warmup for compactification on Witten spaces, perhaps.)

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