Paul Schwahn on Nostr: Pursuing the question "what structures in the Cayley plane are preserved by ...

Pursuing the question "what structures in the Cayley plane are preserved by (SU(3)×SU(3))/ℤ₃ or the standard model gauge group" is the same as giving a geometric interpretation to the homogeneous space F₄/SU(3)SU(3) and F₄/S(U(3)U(2)) (I'll be a bit lax with notation here and treat groups with the same Lie algebra as the same). Now these spaces show up in several fibrations:

First of all, F₄/S(U(3)U(2)) fibers over both F₄/SU(3)SU(3) and 𝕆ℙ²=F₄/Spin(9), with fiber ℂℙ² and Spin(9)/S(U(3)U(2)), respectively. (What does the latter mean?)

F₄/S(U(3)U(2)) is also a flag manifold of F₄ (i.e. a homogeneous projective variety), and can be viewed as the space of complex two-dimensional isotropic subspaces of the tracesless part 𝔥₃⁰(𝕆⊗ℂ) ⊂ 𝔥₃(𝕆⊗ℂ). See Prop. 3.3 in [Correia, Pacheco, Svensson: Harmonic surfaces in the Cayley plane, https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12376] or Prop. 6.7 in [Landsberg, Manivel: On the projective geometry of rational homogeneous varieties, https://ems.press/journals/cmh/articles/382]. (How is this related to structures inside the 𝑟𝑒𝑎𝑙 Cayley plane?)

Next, there's the symmetric space F₄/Sp(3)SU(2) of all ℍℙ²s in 𝕆ℙ². Over it fibers the space F₄/SU(3)SU(2) of all chains ℂℙ²⊂ℍℙ² in 𝕆ℙ² (with fiber Sp(3)/SU(3), the space of all ℂℙ²s in ℍℙ²). But F₄/SU(3)SU(2) is also a circle bundle over the flag manifold F₄/S(U(3)U(2)). (What does the circle mean here?)

So instead of providing answers, I just multiplied the questions. Oops...