A Complex Qutrit Inside an Octonionic One
Posted by John Baez
Dubois-Violette and Todorov noticed that the Standard Model gauge group is the intersection of two maximal subgroups of . I’m trying to understand these subgroups better.
Very roughly speaking, is the symmetry group of an octonionic qutrit. Of the two subgroups I’m talking about, one preserves a chosen octonionic qubit, while the other preserves a chosen complex qutrit.
A precise statement is here:
Over on Mathstodon I’m working with Paul Schwahn to improve this statement. He made a lot of progress on characterizing the first subgroup. is really the group of automorphisms of the Jordan algebra of self-adjoint octonion matrices, . He showed the first subgroup, the one I said “preserves a chosen octonionic qubit”, is really the subgroup that preserves any chosen Jordan subalgebra isomorphic to .
Now we want to show the second subgroup, the one I said “preserves a chosen complex qutrit”, is really the subgroup that preserves any chosen Jordan subalgebra isomorphic to .
I want to sketch out a proof strategy. So, I’ll often say “I hope” for a step that needs to be filled in.
Choose an inclusion of algebras All such choices are related by an automorphism of the octonions, so it won’t matter which one we choose.
There is then an obvious copy of sitting inside . I’ll call this the standard copy. To prove the desired result, it’s enough to show:
The subgroup of preserving the standard copy of in is a maximal subgroup of , namely .
All Jordan subalgebras of isomorphic to are related to the standard copy by an transformation.
Part 1. should be the easier one to show, but I don’t even know if this one is true! is a maximal subgroup of , and Yokota shows it preserves the standard copy of in . But he shows it also preserves more, seemingly: it preserves a complex structure on the orthogonal complement of that standard copy. Is this really ‘more’ or does it hold automatically for any element of that preserves the standard copy of ? I don’t know.
But I want to focus on part 2). Here’s what we’re trying to show: any Jordan subalgebra of isomorphic to can be obtained from the standard copy of by applying some element of .
So, pick a Jordan subalgebra A of isomorphic to . Pick an isomorphism A ≅ .
Consider the idempotents
in . Using our isomorphism they give idempotents in , which I’ll call . Since these are also idempotents in .
Hope 1: I hope there is an element of mapping to ⊂ .
Hope 2: Then I hope there is an element of that fixes and maps the subalgebra to the standard copy of in .
If so, we’re done: maps to the standard copy of .
Hope 1 seems to be known. The idempotents form a so-called ‘Jordan frame’ for , and so do . Faraut and Korányi say that “in the irreducible case, the group acts transitively on the set of all Jordan frames”, and I think that implies Hope 1.
As for Hope 2, I know the subgroup of that fixes contains . I bet it’s exactly . But to prove Hope 2 it may be enough to use /
Let me say a bit more about how we might realize Hope 2. It suffices to consider a Jordan subalgebra of that is isomorphic to and contains
and prove that there is an element of that fixes and maps the subalgebra to the standard copy of in . (In case you’re wondering, this is what I was calling .)
Hope 3: I hope that we can show consists of matrices
where are arbitrary real numbers and range over 2-dimensional subspaces of . This would already make it look fairly similar to the standard copy of , where the subspaces are all our chosen copy of in .
If Hope 3 is true, the subspaces don’t need to be the same, but I believe they do need to obey and cyclic permutations thereof, simply because is closed under the Jordan product.
So, we naturally want to know if such a triple of 2d subspaces of must be related to the ‘standard’ one (where they are all ) by an element of , where acts on the three copies of by the vector, left-handed spinor, and right-handed spinor representations, respectively — since this is how naturally acts on while fixing all the diagonal matrices.
This is a nice algebra question for those who have thought about triality, and more general ‘trialities’.
So, that’s where I am now: a bunch of hopes which might add up to a clarification of what I mean by “the subgroup of symmetries of an octonionic qutrit that preserve a complex qutrit”.