Arcadian Functor

As recommended on NeverEndingBooks, the book Symmetry and the Monster by M. Ronan is a great read. I stood in a bookshop and read most of it, which didn’t take too long because it is relatively short.

The word Monster refers to the group of that name. Ronan goes into Borcherds’ proof of the Conway-Norton conjecture. Here is a nice 2 page article by Borcherds himself. It involves the j invariant, which came up in this recent PF thread. Physicists are interested in the Monster because of its connection with bosonic String theory. See for example the work of Tuite on genus 2 CFT.

Ronan doesn’t go into that much mathematics, but he does mention some really wonderful stuff, like the Leech lattice, or the fact that the sum of the first 24 squares is the same as 70 squared!

Yesterday we were treated to a double seminar by Robert Coquereaux on paths in graphs and quantum groupoids. Actually, it was more like a tour de force in computational magic and we struggled to keep up, but I will attempt to outline the idea.

A weak Hopf algebra A will have an algebra (B,.) and a dual algebra (Y,*), both of which are finite dimensional, ie. sums of matrix algebras. Given as input a simple Lie group L and a positive integer k, there will be a family of weak Hopf algebras associated to (L,k). Note that k is the level (affine rep theory), related to the deformation parameter q of quantum groups. The families consist of three types of algebra, Ak, Dk and Ek, associated to graphs.

The algebras (B,.) and (Y,*) are associative and compatible in the sense that the coproduct is a homomorphism. B is the direct sum of spaces Hn of a certain class of paths on the graph G of length n. The product turns out to be a vertical composition of basic >-< diagrams. In the case of SU(2) for k=1, the graph is the two node Dynkin diagram A2. A vertex (two leafed tree) in H0 is labelled by 0 or 1 at the inputs. There are two paths in H1, depending on the orientation of the edge. And for this example, that’s it. Dually, one has triangles instead of basic 3-valent vertices, with labelled edges. All these diagrams form a basis for vector spaces over C.

Similarly, the algebra (Y,*) deals with horizontal diagram composition. Both algebras have units, but it is not true that the coproduct on 1 gives 1 x 1. Also, the counit is not a homomorphism. This is the reason for the term ‘weak’. Given B and Y, there are two character theories, A and O respectively. O is a bimodule over A and a lot of the physical motivation for this type of weak Hopf algebra comes from the fact that the bimodule structure is given by certain matrices W(xy), where W(00) is a so-called modular invariant. The graph G is a Z+ module over A, and also a Z+ module over O.

The paths with fixed source or target generate two spaces Ds and Dt which appear when one looks at the coproduct on 1. The antipode switches labels on the >-< diagram and may have a coefficient, but for SU(2) this is 1. The natural pairing given by the basis elements acts on the matrix units to give a labelled tetrahedron. These are the Ocneanu cells.

Now let’s define creation and annihilation operators. The latter takes paths of length n to paths of length n-2 by removing backtracks. One must also insert a coefficient which is the square root of the ratio of the quantum numbers associated to the vertices at the source and target of the backtracking piece. To compute these quantum numbers on the graph, take the graph’s adjacency matrix, compute the eigenvalues/eigenvectors, normalise with respect to the leading eigenvector, and the scaled eigenvalues are the quantum numbers.

For example, for the Dynkin diagram for A3, there is a 4×4 matrix and the labels are (1,r,r,1) where r is the Golden ratio.

Taking En to be the number operator here and scaling by the highest eigenvalue b one recovers the relations for the Temperley-Lieb-Jones algebra. Cool stuff. The restriction of self-adjointness forces b to take on a certain range of well-known values.

In the Rosebrugh-Wood characterisation of Set there is a string of adjunctions between Set and [Setop,Set],
namely U-|V-|W-|X-|Y where Y is the Yoneda embedding. There is both a monad and a comonad hidden in here. To deal with annihilation and creation operators we are going to need both structures.

M-theorists would like to have a similar characterisation for the quantum topos of M-theory. By its higher dimensional nature, this is not simply a 2-category or 3-category. The dimension raising aspects of Gray type products are crucial to explaining the physical combination of systems. If the basic object is modelled on projective geometry, such as in twistor space, then one expects the combination of two particles to yield two copies of twistor space, more or less as noted long ago by Hughston and Hurd, who used the Kunneth formula to study mass generation in the twistor setting.

Witten’s study of N=4 SYM showed that certain amplitudes were localised on curves. Clearly something interesting is happening here. If one steps back and considers supersymmetry from a Machian perspective, then one finds it is related to the Real-Symplectic duality of Mulase et al in matrix models.

This manifests T-duality, but what about S-duality? Could this arise from including the octonion case? One shouldn’t be bothered by nonassociativity. After all, it comes up in the K-theory studies of T-duality, and in a simple monoidal categorical way. Moreover, the association of Jordan algebras and projective geometry means that one can make the connection to twistor string theory very explicit. In doing so, one ends up studying moduli spaces as orbifolds, with cell decompositions. It is well known that both these decompositions, and Deligne-Mumford type compactifications, can be described by operad like polytopes and their duals. In this higher categorical language, the web of dualities is a kind of Poincare duality in higher cohomology.

A little news: I’ve been contacted by a UK production company who wish to make a documentary about the time that Sonja Rendell and I were trapped on an exposed ledge on Mt Isobel for 8 days in bad weather, in late 2003. Of course, they might also want to talk to me about the time that I fell in a crevasse whilst soloing on the Grosser Aletsch glacier in Switzerland, or maybe even the time that I crossed Fyfe pass on an epic traverse of the divide near Mt Cook.

We’ll wait and see, I guess!

…and the mind creates reality…

Consider a few examples. In Turok’s universe there was no Big Bang and god’s time runs through Brahmian cycles. Other people think the world was created only a few thousand years ago. Most physicists believe it sprung magically into being, spacetime and all, about 13 billion years ago, Earth time.

Einstein knew that all of these ideas were physically wrong. In a quantum universe, the passage of time and the meaning of mass are things that we measure. For example, when we look at an electron we find that its mass = 0.5109989 MeV. Now why is that?

Well, PF seems to have crashed permanently. I just listened to the Lee Smolin and Brain Greene radio interview, which I thought was very good, although perhaps a little tame under the circumstances. The following is just a story. I apologise for the lack of links.

The history of the discovery of DNA is quite fascinating. It was heavily influenced by a little book that Schroedinger (yes, the physicist) wrote in the 1940s in Dublin, entitled simply What is life? It bothered Schroedinger a lot that the biologists had shown conclusively that the transfer of genetic information was dictated entirely by a certain mysterious molecular structure, because he knew that the rules that governed this information transfer were quantum mechanical, and yet he also knew that these complex molecules could not be explained by quantum mechanics. He argued that to truly understand it, we would need even newer physics.

Nowadays we know that DNA (and related) molecules like to knot themselves, and of course they are made of helical ribbons. But, hang on a minute, information transfer in quantum computation is also about knots and ribbons. Computer scientists know this very well. Could these two things be linked?

It shouldn’t really be any surprise that a quantum theory of gravity has something to say about biology. After all, look how quantum mechanics affected chemistry. A few years ago I started to notice that a lot of neuroscientists and geneticists were talking about a notion of perception that was highly non-trivial in its vision of space. Somehow the mind creates its own reality. It sees what it wants to see.

In a decent Machian theory of quantum gravity one must likewise make sure that space does not really exist. The arrow of time is not laid out upon a general relativistic manifold, like an a priori worm, eating up any hope we might have of understanding the origin of mass.

So what is fundamental? Well, at least information transfer is clearly more fundamental than space. But what does transfer mean? Do I mean then to now? This is where category theory comes in. By simply drawing an arrow one does not presume that its interpretation should be as a movement in some god-given time. A diagram should represent precisely an experimental question, even a question such as what is the rest mass of the electron? We are quite used to this idea from Feynman diagrams in QFT. And indeed, operad theory is really a kind of advanced Feynman diagram calculus; a calculus that does algebra, geometry and logic all together.

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 3×3. 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.

And I saw when the Lamb opened one of the seals, and I heard, as it were the noise of thunder, one of the four beasts saying, Come and see. Apocalypse of John (Rev 6.)

Now that we have QGC (M-theory) via the operadic ribbon graph calculus and the duality principle, what next?

I’m not a mathematician, but one cannot help wondering about the Riemann hypothesis. People have been saying for a long time that all one needs to prove the Riemann hypothesis is a cleverly constructed operator whose spectrum gives the zeros of the zeta function. Moreover, people know that the linearity of the trace is the real problem. Well, QGC fixes this problem!

Think of localisation on a line in the MHV diagram technique!

Update: the gallant kneemo has established beyond doubt that the M-theory works! See the thread below.

I found out only today (when I went into a newsagency) that the Sundance octopi have made the cover of New Scientist. The article, seemingly based on an interview with Lee Smolin, makes no mention of category theory, although it does quote John Baez. One can read a little for free.

When I mentioned octopi in a talk recently I was immediately asked by a member of the audience how something with three, or maybe six, legs could be called an octopus. Moreover, apparently octopi isn’t even the correct pluralisation.

But what is a good term? Sundance preons is really too crass, and the word preon is very misleading when translating the diagrams into a QG context. I finally decided upon the boring and very ubiquitous ribbon diagrams. Not very exciting, is it? Maybe we should use Grothendieck’s term, children’s drawings. Imagine trying to sell that one to the phenomenologists.

Update: There is now a PF discussion about octopi, with a great introduction by CarlB to his theory, not to mention a confirmation of the correctness of the lepton mass derivation … and so the jigsaw falls slowly into place, like the collapse of a terrifying Hegelian wave function, heralding a new time.

I’m not much of a blogchatter, or any kind of chatter, but I do want to be a part of the Future Mind, whose memory we are collectively building. Today I thought I would do a little housework on the haystack of Physics Forums threads.

In particular, The Third Road thread, which has become a little messy. Links to recent papers and discussions appear on page 12. The original argument with Careful starts around page 7. The Baratin-Freidel thread is still growing, as is the de Sitter one. The preons also appeared in this nuclear thread, and I’m quite fond of this old thread.

Away from PF we have beautiful pearls like the Anthropic Debate thread at Not Even Wrong, or the Anthropic Brain thread at Cosmic Variance.

Happy reading. I’m taking the dog (not my dog) for a walk.