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----o SU(4)xSU(2)xSU(2)
o
Ah yep, Boya was a fan of the “M(ysterious) Patterns in SO(9)” thing.
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:
https://arxiv.org/abs/hep-th/9808190
… 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.)
Indeed. Baez did a lot of notes in his “this week…” about this.
I don’t know what you’re thinking but those triality representations show up in the massless states of the uncompactified Type IIA string (which is the one that grows an extra dimension and becomes the M2 brane of M theory at strong coupling)
https://ncatlab.org/nlab/files/MajerSuperstrings.pdf equation 1.30
https://en.wikipedia.org/wiki/Type_II_string_theory#Type_IIA_string_theory