Arcadian Functor

occasional meanderings in physics' brave new world

My Photo
Name:
Location: New Zealand

Marni D. Sheppeard

Friday, August 18, 2006

M theory: life, the universe and everything

My friends were keen that I should write something about M theory. Don't worry folks: I have not abandonned category theory for a boring 11 dimensional theory. I shall explain. Of course, I could redirect you to PF, but it seems to have crashed yet again (now I wonder why that is?). Or, I could redirect you to the new n-category Cafe (see blog roll), where lots of cool stuff will be happening soon.

Short story: we know how to do M theory rigorously. This is a fairly compelling case that our ideas about quantum gravity are, er, let's say, on the right track. Personally, I do not claim to understand much of it at all. I don't see how anyone could claim to, except maybe Ed Witten and the mysterious kneemo and a few other wizards.

Physically, M theory needs to formulate a Machian duality (Witten's favourite word) that looks like supersymmetry, but not in the sense of ordinary algebras or superpartners. Years ago, when I was trying to picture this, and stumbling clumsily with varying hbar or speed of light (while Louise was working it all out), I asked myself: if horizons are like boundaries, but the holographic mirror has to turn everything inside out, then what is a boundary? How do primordial black holes and the cosmological horizon fit together? Many of the most fundamental ideas in physics are about understanding boundaries. Consider Stokes' theorem, for instance.

After playing with a little mathematics I fell wildly in love with Algebraic Topology, because Stokes' theorem could be written pretty simply! The Machian principle also suggested an interconnectedness-of-all-things idea. It slowly became clear that this was impossible without category theory, because category theory is the mathematics of relationships (and people had already tried pretty well everything else).

But I won't bore you with diagrams, which I can't draw here anyway. It turns out that to understand M theory we need several pictures at once in our mind: twistor String theory, spin foam QG and matrix models.

It is a powerful mathematical theorem that the complex moduli of Riemann surfaces with punctures is closely related to a moduli of labelled (metric) ribbon graphs. A ribbon graph is a closed diagram (graph) of flat ribbons, which are allowed to cross over and under one another. By allowing twists, one can also study matrix models for the quaternionic ensemble. The expert on this is Mulase.

Anyway, since the Bilson-Thompson preons were made of ribbons, it made sense to think about these ribbon moduli. These spaces turn out to have cell decompositions that look a lot like the special polytopes that turn up in higher category theory. This is no accident, because it is all really about operads. Now, I knew that there was a first class maverick amateur physicist named Carl Brannen, who had already calculated the neutrino and charged lepton masses based entirely on ideas from the Geometric Algebra of Hestenes. What Carl did was associate preons (not quite the same as the Bilson-Thompson ones) with idempotent eigenmatrices in Clifford algebras.

A few days ago, people started talking about the Bilson-Thompson preons, yet again, and Carl briefly outlined his vastly superior version, which could explain the number of generations. I showed up, and so did Michael Rios (kneemo), who happens to be a young wizard expert in Jordan algrebras, and just about everything else it seems. The three of us got talking, and while I was struggling to understand all the algebra it dawned on me that, geometrically, the model for the idempotents were the special points in projective space from Mulase's ribbon graph theory. As soon as we realised that the orbifold Euler characteristic of the moduli space of the 6 punctured sphere was -6 we knew that was a derivation of the number of generations of String theoretic power. The gallant kneemo got working, and I tried hard too but mostly I just went into panic!

This is all about doing not only higher categorical cohomology (boundary theory), but also Poincare duality - the mother of all Poincare dualities. Mulase had already shown how to get T duality out of the real-symplectic Penner model. This used twisty single ribbons. The triple ribbons come in because the idempotents are 3x3. Carl's preons can be built out of a basis of 2 flat ribbons (particle, antiparticle) and two twisted ribbons (1 left, 1 right), each of which effectively has 4 labels. A left handed electron must pair with a right handed electron to generate mass.

Don't worry: spin foam models are in there, too. Louis Crane's geometrization of matter proposal was about keeping the spin foam topological and relaxing the restraint on using manifolds. Matter should somehow be related to the singularities, where the nature of a point (vertex of the spin foam) is given by the surrounding 3-space. It was a lot of fun to play around with 3 dimensional hyperbolic geometry and knots, but it wasn't clear how to make the higher genus surfaces (boundaries of the 3-spaces) look exactly like black holes (or 'dark matter'). Now we can do it. The trick is to think not of an actual surface, but the whole moduli space being modelled by projective geometry (twistors).

More on this later. If anyone happens to be in Sydney, I'll be talking about ribbon graphs to some category theory guys this coming week, at 2pm on Wednesday August 23 in the Maths department at Macquarie.

8 Comments:

Blogger L. Riofrio said...

Wonderful post! It is great that you can write about your work as it happens, where someone won't delete it. It sounds like you and the rest of this group are on the right track. When you come up with testable predictions, it can't be ignored forever.

It is fascinating how much one can come up with from Mach's Principle and consideration of boundaries. I hope to hear more from you and your group soon.

August 19, 2006 7:16 AM  
Blogger Christine said...

Kea,

I just superficially understand what you are saying. But one thing at least: Mach's Principle and boundaries (*and* the cosmological constant) do have a deep, DEEEP, connection.

See the paper by Gilman [Gilman, R.C., 1970, Phys. Rev. D, 2, 1400]. Write Einstein's equations (EE) in integral form. A cosmological model is considered Machian from Gilman's scheme if it does not contain source-free contributions, that is, a surface term in the integral formulation of EE must vanish, the so-called "Gilman condition". (I guess a lot of people are anaware about this old paper. I was. But search engines are great.) It's a wonderful paper because it's so vague to simply claim "Mach's Principle", there are so many versions of MP everywhere. I like Gilman's approach to set up things in a very precise manner.

That is the purely classical part of the story.

I've started working a bit on this considering braneworlds ( see here). But since that paper (a very short note in fact) I had no time to explore more. There is a lot to do. A quantum theory of gravity ultimately must explain all this.

Cheers,
Christine

August 19, 2006 2:19 PM  
Blogger Kea said...

Hi Christine

Indeed, I have never seen the Gilman paper. Thanks for the reference! I was impressed by your paper when I first came across your blog. Yes, a theory of QG must get this right at the quantum mechanical level - and the claim is that this one will. Of course, it isn't all written out, and we haven't done piles of calculations, but it's obviously right, if you know what I mean.

August 19, 2006 2:37 PM  
Blogger Christine said...

Hi Kea,

Very nice to know that your work is bringing forth new ideas and results! I'm very interested! I hope someday I can get an understanding of all this.

I have a deep internal certainty that only when we *understand* inertia we will have quantum gravity.

Best of luck,

Christine

August 19, 2006 2:47 PM  
Blogger QUASAR9 said...

kea!
Nope Dick Cheney, Donald Rumsfeld, Son of Bush & Tony Blair are bidding for that dubious title.

I'm thinking more like
"four ladies and Quantum"
flaq for short.

August 21, 2006 12:52 AM  
Blogger ick Thompson said...

Congrats on the blog. Very Nifty!

I don't know if I should "out" kneemo, but he is a graduate student (I don't think he has his doctorate yet) named Michael Rios, out in California. We have corresponded in the past.

Dick Thompson (aka selfAdjoint)

August 21, 2006 12:43 PM  
Blogger Kea said...

selfAdjoint!

So happy to meet you at last! I did actually figure out who kneemo was, but couldn't find out anything about him other than papers and affiliation.

I wonder what he's up to? Hee, hee.

August 21, 2006 5:57 PM  
Blogger kneemo said...

Hi Kea

Nice blog! It seems like an ideal haven for Keas and ribbon graphs alike. :)

August 21, 2006 8:05 PM  

Post a Comment

<< Home