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Stream: deprecated: algebraic geometry

Topic: physical tangent

Morgan Rogers (he/him) (May 10 2020 at 10:32):

John Baez said:

This is a huge story unto itself, but luckily I've written something about it already:

• The tenfold way.

You'll see that the "super-Brauer groups" of $\mathbb{R}$ and $\mathbb{C}$ show up here. These are used to classify matter in 10 kinds. These ten kinds have different properties with respect to time reversal (T) and switching particles and holes (C).

This is the most concrete explanation I've seen of the relationships between classes of algebra and symmetry groups. Thanks for sharing this! I have to admit that I was a little disappointed to see you falling into numerology in the closing remarks, though.

John Baez (May 10 2020 at 21:00):

If I were sure it were mere numerology I wouldn't have said it. But it's really true that:

1) the reason string theory works best in 10 dimensions is that

$\mathrm{dim}( \mathbb{O}) + \mathrm{dim} (\mathbb{C}) = 8 + 2 = 10$

2) the reason for the 10-fold way is that there are 8 real Clifford algebras and 2 complex ones; the 8 real ones come from real Bott periodicity, a period-8 phenomenon that's closely connected to $\mathrm{dim}(\mathbb{O}) = 8$, while the 2 complex ones come from complex Bott periodicity, a period-2 phenomenon that's closely connected to $\mathrm{dim}(\mathbb{C}) = 2$.

John Baez (May 10 2020 at 21:01):

So I don't think it's completely crazy. But as I said,

This may be more of a joke than a deep observation.

It's really too early to tell!

Morgan Rogers (he/him) (May 11 2020 at 09:21):

John Baez said:

1) the reason string theory works best in 10 dimensions is that
$\mathrm{dim}( \mathbb{O}) + \mathrm{dim} (\mathbb{C}) = 8 + 2 = 10$

Well... superstring theory uses ten dimensions. There are string theories with more. So you can see why I might say that an argument that ends with "...= 8 + 2 = 10" sounds like numerology. From what you've said and what I already understood beforehand, the number of dimensions changes as soon as we notice/incorporate a new symmetry into the theory, and perhaps you'll tell me that this just corresponds to adding some more algebras to this pile, but if that's the case, why should I stop and focus on some specific small collection of algebras when I can't be sure there aren't symmetries I hadn't yet noticed lurking around the corner? (sorry for the triple negative :joy: )

Morgan Rogers (he/him) (May 11 2020 at 10:49):

So I should be asking why we have to be working with division algebras, then?
I did a quick search, found one of John's blog posts on the octonions and am now investigating one of the references there, Feza Gürsey and Chia-Hsiung Tze's On the Role of Division, Jordan, and Related Algebras in Particle Physics.

Morgan Rogers (he/him) (May 11 2020 at 11:08):

This book seems to contain a lot about the mathematics of the quaternions and octonions without justifying the properties that uniquely define them (I admit I have only skimmed; maybe the justification comes later in the book than I have read?)
Having found another of John's blog posts, I find myself concerned by the quote:

Well, there are not too many places in physics yet where the octonions reach out and grab one with the force the reals, complexes, and quaternions do. But they are certainly out there, they have a certain beauty to them, and they are the natural stopping-point of a certain finite sequence of structures, so it is natural for people of a certain temperament to believe that they are there for a reason.

That sounds scarily like higher-dimensional, high-brow numerology to me (not necessarily on John's part, but certainly on the part of Dixon who is being referred to in the introduction of this blog post) :grimacing: But maybe there is some argument out there explaining why the algebras in physical theories should be division algebras! I would love to know.

Morgan Rogers (he/him) (May 11 2020 at 14:02):

Rongmin Lu said:

Morgan Rogers said:

So I should be asking why we have to be working with division algebras, then?

Because of the following correspondence:

• $\mathbb{R}$ "controls" motion in 1 dimension: multiplication by -1 reverses the direction of a 1-dim vector.
• $\mathbb{C}$ "controls" motion in 2 dimensions: multiplication by $i$ corresponds to a quarter turn anticlockwise.
• $\mathbb{H}$ "controls" motion in 3 dimensions. This is a discovery of Hamilton, and is used often in 3D graphics.

So the expectation is that $\mathbb{O}$ should say something about motion in 4 dimensions. Of course, the snag is we don't live in a universe with 4 spatial dimensions, it's 3 spatial + 1 time dimension, so it's possible that breaks down... or maybe the nonassociativity of the octonions will save the day. But that's the idea.

But that doesn't address my "why stop there?" question: the fact that these are division algebras doesn't seem fundamental to the argument you're giving. If the generalisation to the 'octonions controlling motion in 4 dimensions' works, why shouldn't the 16-dimensional algebra we get by applying Cayley-Dickson again 'control motion in 5 dimensions'?

John Baez (May 11 2020 at 16:09):

Morgan Rogers said:

John Baez said:

1) the reason string theory works best in 10 dimensions is that
$\mathrm{dim}( \mathbb{O}) + \mathrm{dim} (\mathbb{C}) = 8 + 2 = 10$

Well... superstring theory uses ten dimensions. There are string theories with more.

Right. I meant superstring theory. I'll make that clearer on my page.

Superstring theory relies on the worldsheet being a complex manifold (so, 2 dimensional) and the transversal directions having the property that spinors can be identified with vectors (so, 1, 2, 4, or 8-dimensional - the dimensions of the normed division algebras $\mathbb{R},\mathbb{C},\mathbb{H}$ and $\mathbb{O}$). So, classical superstring Lagrangians have supersymmetry in dimensions 3, 4, 6 and 10. But only the 10d octonionic case is anomaly-free.

My student John Huerta and I wrote some papers on this stuff:

• Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C-Algebras*, eds. Robert Doran, Greg Friedman and Jonathan Rosenberg, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80.

• Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011), 1373-1410.

John Baez (May 11 2020 at 16:12):

Then John went further on his own:

• John Huerta, Division algebras and supersymmetry III, Adv. Theor. Math. Phys. 16 (2012), 1485–1589.
• John Huerta, Division algebras and supersymmetry IV.

John Baez (May 11 2020 at 16:14):

The upshot is that using normed division algebras you can build Lie 2-supergroups governing superstring theories in dimensions 3, 4, 6 and 10, and Lie 3-supergroups governing super-2-brane theories in dimensions 4, 5, 7 and 11. The 10d case and the 11d case are the ones physicists believe can be consisently quantized.

John Baez (May 11 2020 at 16:15):

But I'd rather talk about Galois cohomology, since that's what I'm into these days.