Classifying Rational Conformal Field Theories

Yesterday afternoon was quite a chilly day in London, the kind of day when being crammed into a packed and warm lecture room below ground level in the basement of Queen Mary college from where you can hear the tube rattle by was quite an attractive prospect. So at three in the afternoon yesterday that's where I and other London theoretical physicists gathered to hear Terry Gannon talk about "The classification of RCFTs".

First off, it gives me great pleasure to report to you that the "damn book" is finished :) after five years of hard slog Terry's book, Moonshine Beyond the Monster and available to buy from the 31st August, 2006. Hurrah. There's an excellent documentary by Ken Burns on the American Civil War that took longer to make than the war itself, I have no doubt that it will take me inestimably longer to understand this 538 page book than it took to write. Fortunately noone died in the making of the book, to the best of my knowledge. For some history of the Monster see Terry's Monstrous Moonshine: The First Twenty-Five Years.

Terry described his approach to trying to classify Rational Conformal Field Theories (you could look at Wikipedia for a brief definition of a RCFT, or a much better idea might be to start learning about CFT from scratch with Paul Ginsparg's Les Houches lectures, Applied Conformal Field Theories or Krzysztof Gawedzki's Lectures on Conformal Field Theory) by searching for invariants of the chiral algebra, or Frobenius algebra, that underlies the RCFT. By way of comparison, Terry said that the very succesful classification of the Lie algebras rested upon the invariant of the Dynkin diagram. But what invariants are worth considering, whose discovery will tell us most of the information about the algebra? Terry suggested two:

modular invariants (i.e. partition function on the torus)

NIM representations (i.e. partition function on the cylinder)

But he only had enough time to talk a little about the first and describe to us the modular functions that appear.

To commence one must settle upon a chiral algebra, or a vertex operator algebra, and Terry told us that some very nice choices are the affine Kac-Moody algebras (see Fuchs' Lectures on conformal field theory and Kac-Moody algebras section 16 for the definitions). A level, k, must also be picked. We were told that one way to imagine a chiral algebra is as a complexification, or 2-dimensionalisation, of a Lie algebra. If we denote all the objects appearing in a Lie algebra by a tree diagram, having all the properties of the Lie bracket at the branch (i.e. antisymmetric...) then the complexified version of the algebra turns each of the branches of the tree diagram into a cylinder: For more about this way of complexifying to get loop algebras we were referred to the work of Yi-Zhi Huang, in particular his book Two-Dimensional Conformal Geometry and Vertex Operator Algebras.

Returning to the CFT, the Hilbert space is described by irreducible representations of our affine algebra (left moving and right moving copies) which for a given level k, are paramaterised by highest weight labels. For the example of affine SU(2), the highest weights are characterised by two labels (, ) such that + = k. The Hilbert space may be written as:Where M is the multiplicity, and the one-loop partition function for this RCFT may be written in terms of the characters, : It turns out that the characters are modular functions, and are subject to the familiar S and T transformations: Furthermore, the partition function is modular invariant and characterised by its multiplicities, M.

At this point in the talk, Terry had about six minutes remaining and had arrived at what he thought of as the start of his talk, and defined the "modular invariant" he hoped to use to classify RCFTs:

Given some affine algebra at level k, a modular invariant is a matrix M of multiplicities describing the partition function, Z, such that,

Terry told us that these conditions gave rise to RCFTs that are "just barely" classifiable.

Terry finally asked us why bother classifying? Or, in his words, "who cares?" His answer was that the classification leads to interesting results. What more could you want? He gave us the example from Cappelli-Itzykson-Zuber from 1986 of the classification of affine su(2), which is completely classified for the levels, k, 4/k, k/2 is odd, k=10,16,28, and he told us a story he heard twice; once from Zuber about a correspondence he had with Victor Kac, and a second time the same story from Kac - so, he said, it must be a true story. It went like this: After having written down some of the classifications of affine su(2) in 1986, Zuber wrote to Kac about the results, who replied and pointed out the classification for k=10, which he said contained some exceptional numbers - literally numbers he thought came from the exceptional group E_6. Zuber said he didn't understand Kac nor pay it much heed until someone else repeated it years later and he dug out the letter, headed to the library and confirmed that all the numbers appearing in the classification do indeed have an intimate and mysterious (to this day...) relation with the groups A, D, E, and the symmetries of their Dynkin diagrams. At this point Terry bemoaned the fact that God was manifestly not benevolent since he insisted on making 2 a prime number...Terry's discomfort with 2 didn't seem justifiable until later on when he mentioned that his wife is expecting twins (excuse me for this weak pun) so I just put two and two together... :)

So the ADE-classification arises mysteriously from modular invariants, so that's why to classify RCFTs: because they might be interesting.

Does your ball roll at normal speed?

Flying in the face of recent efforts to redefine the scientist stereotype, described at cosmic variance, entropy bound and inkycircus*, comes the latest rebuttle from no less thanJose Mourinho, who manages to make his feelings known during an interview about Chelsea's pitch condition prior to their Champion's League fixture with Barcelona:

'Sometimes you see beautiful people with no brains. Sometimes you have ugly people who are intelligent, like scientists,' he said.

'Our pitch is a bit like that. From the top it's a disgrace but the ball rolls at normal speed.'

*By the way this is a great science news blog that I only just cottoned onto, which I heartily recommended to you all, if you haven't been there already.

Poncelet's Porism

A week ago last Friday John Silvester from KCL's very own maths department gave us a very entertaining geometry colloquium under the esoteric title "Pendulums, Pencils, and the Poristic Polygons of Poncelet".

John began with a couple of anecdotes. Having thanked the audience for his invitation to speak, he told us a story about an unnamed mathematician who was invited to talk on a BBC radio show and was told that the fee would be £100. The mathematician thought about this and then asked if they would prefer a cheque or cash. A second anecdote concerned a London Mathematical Society president who offered a cash prize to the speaker who gave the first talk in which he did not fall asleep, and then duly claimed the prize himself when he next gave a talk.

John's talk was on the subject of Poncelet's Porism so he showed us a wonderfully stern picture of Jean-Victor Poncelet who fought in the 1812 Napoleonic attack on Moscow, was imprisoned and there turned his mind to projective geometry (I wonder how many of those presently incarcerated at her majesty's pleasure on these shores are also making strident breakthroughs in mathematics). He also has a unit of power named after him in France. John then turned his attention to the word "porism" - what does it mean? Well according to the the free dictionary it has two meanings:

Po´rism
n. 1. (Geom.) A proposition affirming the possibility of finding such conditions as will render a certain determinate problem indeterminate or capable of innumerable solutions.
2. (Gr. Geom.) A corollary.

Poncelet's porism refers to the first case. So what is Poncelet's porism? Take two conic sections, now if you can draw one n-sided polygon (n>2) such that its sides all touch tangentially one conic section and its vertices all lie on the other one, then you can draw infinitely many. The infinite comes about because you are able to rotate the polygon (not held fixed though) so that its vertices all rotate around the outer curve. Phew. Let's look at some pictures and see if this is understandable:

a triangle (sort of) with vertices on a parabola and circumscribing a circle

a quadrilateral with vertices on a circle and sides tangent to a parabola

the classic triangle and two circles

There are plenty more of these animations to be found here.

John restricted our attention to the case of the triangle and the two circles. If you want to play with this set up yourself and convince yourself it really does work then there are some very nice interactive animations here (move the inner circles until the eyes open wide and then rotate the vertex on the outer circle) and here (move the pink line back and forth to change the inner circle's radius). This last link will be useful for describing John's talk since it includes a button for showing diagonals. The diagonals (for the triangle) are the lines that connect a vertex on the outer circle to the point where the polygon touches the inner circle opposite it. Push the button and see this. John was wondering about a line in the rather detailed page from mathworld concerning the porism. In particular, John was not convinced by the following line:

"For an even-sided polygon, the diagonals are concurrent at the limiting point of the two circles, whereas for an odd-sided polygon, the lines connecting the vertices to the opposite points of tangency are concurrent at the limiting point."

If you click on the aforementioned link showing the diagonals and move the pink line about I think you can see even there that it is not clear that the meeting point of the diagonals stays fixed. John demonstrated to us his expertise with both matlab and an excellent program called the Geometer's Sketchpad. Using matlab John took us through several pages of enormous calculations working out the locus of the meeting points of the diagonals and finally reduced the locus down to a sixth-order polynomial! Using the sketchpad John was able to convince us that the meeting point actually travels around a circle looped on top of itself three times.

John showed us much more, including the relation of three swinging pendula to Poncelet's porism (stagger the starts of three identical pendual and then draw straight lines between their bobs, these lines are tangent to a circle...) as well as the Encyclopedia of Triangle Centres (ETC but not et cetera) where one can look up famous triangles! The talk concluded with another example of gentle humour that had pervaded, with John borrowing the phrase of the late radio four presenter John Ebdon, "if you have been, thanks for listening."

Dark Matters

A quick pointer to the dark matter article that's on the BBC site at the moment as well as to the articles in The Guardian, The Telegraph, The Independent, Nature and New Scientist. The story concerns the findings of the team lead by Professor Gerry Gilmore at Cambridge who, by making use of the very large telescope array, constructed 3D maps of distant "dwarf galaxies" and have inferred from their motions certain properties of dark matter. Some rather exciting things too, via the BBC:

"The distribution of dark matter bears no relationship to anything you will have read in the literature up to now," explained Professor Gilmore.

"It comes in a 'magic volume' which happens to correspond to an amount which is 30 million times the mass of the Sun.

"It looks like you cannot ever pack it smaller than about 300 parsecs - 1,000 light-years; this stuff will not let you. That tells you a speed actually - about 9km/s - at which the dark matter particles are moving because they are moving too fast to be compressed into a smaller scale.

"These are the first properties other than existence that we've been able determine."

The BBC article notes that the research findings have yet to be submitted to a journal so hold your horses...a little.

Updates: Courtesy of Andrew Jaffe who has a link to the preprint The internal kinematics of dwarf spheroidal galaxies and to some discussion at Dynamics of Cats.

Blogtastic

A quick couple of links to two new articles about maths/physics blogging that both, coincidentally, came out this month. First Craig Laughton, of Gooseania fame, wrote an article about mathematics blogs for Mathematics Today; it's available online, so you can read it here. Second, physicsweb have a blogging editorial and a blogging article from Physics World available online.

I guess blog is the word of the month. Maybe the word "blog" is the blogosphere's equivalent of the word "smurf" for the smurfs. Perhaps in a few years all blog articles will be written using variations of "blog" as verbs and will be entirely about blogging. Blogging hell! I'm going to blog off now, and so on...

Automorphism Groups

Long time, no post eh? Well, if I said I met a girl as a consequence of a new year's resolution and got distracted but that she left last week for her home country you'd understand wouldn't you? Good. Oh, I was a little ill too.

So yesterday I headed over to City University Londonfor the first time ever, it's near Angel station in London and it actually has buildings that can compete with King's Strand campus for ugliness (see right for the cheery city logo attempting to liven up the Tait building where the mathematics department is). However, once inside, all was forgiven, not only for the simple reason that we can no longer see the architecture but also because I was looking forward to attending a rare group theory seminar in London. Infrequently do the theory group talks become group theory talks, but yesterday was just such a special day!

Robert Wilson from Queen Mary, University of London was talking under the title "Finite Groups with Small Automorphism Groups". Robert's homepage contains some links to the text of some of his previous talks, as well as a link to some lecture notes on finite groups which are on their way to becoming a nice looking book. The topic of today's talk was based on Robert's paper with John Bray, the pdf of which can be found here.

Robert's talk was wonderfully pedagogical, beginning with the title he made sure everyone in the room new what a group is and what an automorphism is. He said it was an article of faith for him that all groups are finite :) The only real question was what small meant. So an automorphism of a group, G, is an isomorphism, A, of G: G->G. Robert took the time to show us that set of all automorphisms of a group is also a group, denoted Aut(G), i.e. take A, B, C to be elements of Aut(G), then clearly AB is in Aut(G) and A(BC)=(AB)C, while since the Id and the inverse maps are all isomorphisms too, Aut(G) is a group. We were then introduced to a subgroup of Aut(G), called the group of inner automorphisms, or Inn(G), so-called because these automorphism actions are constructed from elements "in" G. Take g in G, then define an automorphism: This is clearly a map from G to G and it's an isomorphism since: It turns out that inner sutomorphisms form a normal subgroup of G, as, for B in Aut(G), i.e. So Inn(G) is a normal subgroup of Aut(G).

Robert also showed us the outer automorphism group, Out(G), which is defined as a quotient group, Next we had a proposition, that if the inner automorphism was the identity automorphism, then the group element, g, that it was constructed out of would be in the centre of the group, Z(G), So that, Then we came to the crucial examples. First let the group be a cyclic group of order n, i.e. integers under addition modulo n, which is generated by the element a. That is, Now suppose B is an automorphism of this group, so that, Now since B is an automorphism then the whole of the cyclic group must be reproduced by its action. This places the restriction on the possible values k can take; k must be coprime with n (their greatest common divisor must be one). So the number of automorphisms of G, the order of Aut(G), is simply the number of integers less than n which are coprime with n. The function that counts this number is yet another of Euler's and is called the Euler totient function denoted . For the cyclic group the order of the automorphism group, given by the Euler totient function, is less than the order of the group.

Now consider a second example, taking G to be a non-abelian simple group, i.e. it has no normal subgroups besides the trivial ones of the identity and the group G itself. In this case the centre of G is just the identity element, so that, So that, in this case, So now we find out what was meant by "small" in the title; if the order of Aut(G) is greater than the order of G we say that the automorphism group is large, and if it is smaller than the order of G then we say that it is small.

Robert showed us some properties of the totient function, which are really useful and can all be found in the wikipedia article, before describing his work with Bray. He told us about Kourovka notebooks, which contain open problems in group theory, where it was in volume 15, question 43 by Deaconescu whether, might be true for all finite groups and whether equality might be attained only for cyclic groups. Together with Bray he found a simple group for which these properties didn't hold. For what it's worth, the group is the twelve-fold cover of , which is a simple group of order 22.21.20.16.3=443520. How did they do this? Well Robert said he picked up his copy of the Atlas of Finite Groups (online atlas) and when he got to page three he found the counterexample he needed using the formulae he told us in the talk, as well as the properties of the Euler totient function, and hey presto! They also began to wonder just how small can these small (by which I mean the ratio of the orders of Aut(G) to G) automorphism groups be, and they discovered that they can be arbitrarily small.

So there you go. A very nice pedagogical talk, which perhaps felt like the beginning of two talks, and a lesson in the art of proactive mathematics.

Cosmic Variance's GOAT

Well I thought I would just write a reminder to all of you interested in GOATs (greatest of all time) that Clifford at Cosmic Variance has been running his vote for the top five physics papers of all time for just under a week. To make your vote just leave a comment under your favourite paper's post on the site, any noise will do (but only one noise is counted). The current standings are:

Newton's Principia 23 votes

Dirac's Quantum Theory of the Electron 13 votes

Noether on Symmetry and Conservation Laws 12 votes

The EPR Paper 7 votes

Einstein's General Relativity 5 votes

So it's time to reinvigorate the votes! Newton's walking it, but General Relativity can't finish bottom - that's a disaster. So come on, if you haven't voted go and do so and help avoid this travesty. I've used my one vote but am tempted to try and beat the system just because General Relativity ought to finish higher than EPR. Surely.