Non-Crystallographic Coxeter Groups

Today, prior to setting off for our afternoon seminar, I was in my office in Drury Lane listening to the surprise news that London had won the bid for the 2012 olympics, coming through the radio (as the internet video was too busy to work). The rather biased commentator reported the news that people gathered in Trafalgar Square, ten minutes away, were celebrating and being bulstered by office workers who had run onto the streets to celebrate. So, of course, we ran from our office to join the imagined street party, but as you can see on the moblog on the right, there was no-one else there.

Nevertheless, as I pottered down to the KCL campus on The Strand there were more than the average number of smiling faces and when I reached the Strand itself I was just in time for the Red Arrows to fly overhead releasing a stream of red, white and blue smoke behind them, which looked suspiciously like the French Tricolour.

The seminar today was on the topic of "Affine Toda field theories related to Coxeter groups of non-crystallographic type" and was delivered by Andreas Fring, based on work he completed with Christian Korff. For the mostpart the talk focussed on the non-crystallographic groups and their embedding in the simply-laced semisimple Lie algebras, i.e. A_n, D_n and E_n groups, and not so much on the affine Toda field theories, so I will focus on the embedding, but of course the relation to ATFT's can be read about in the paper.

Fring commenced with some comments about the golden ratio, which we denote by c=(A^2)-1=(1+\sqrt{5})/2=1.6180339887... Among the examples of the ratio occurring in the world, was one from the financial world: that the sides of a credit card are in the golden ratio.
The Ising model, which is an integrable model, can be realised as an (E_8*E_8)/E_8 coset model. Even after the conformal symmetry is lost, the E_8 symmetry remains in the form of an E_8 ATFT. The primary result of the paper being presented was that there is an even more fundamental symmetry than E_8 underlying this model based around the non-crystallographic group H_4. Indeed the mass spectrum of the E_8 ATFT is dependent on only four masses, the remaining four masses are multiples of the first four. The new masses are in fact c, the golden ratio, times the initial four masses. A similar relation was also reported to appear for the Sine-Gordon model, but this time involving D_6 instead of E_8. The claim was that an explanation of the appearance of the golden ratio would come from embedding the non-crystallographic into crystallographic Coxeter groups, as H_2+H_2->A_4, H_3+H_3->D_6, H_4+H_4->E_8.

The results were presented as I have indicated up-front, with the speaker's intention being to present an explanation afterwards. However things became a little turbulent when a little further into the description of the results, some members of the audience began asking for explanations, clearly not content with such a presentation of results without justification. At one point it occurred to me that the seminar might end abruptly due to the number of pointed comments being exchanged. Still it was good to see such a keen interest being taken in the detail and by the end of the seminar everyone was on good terms. Consequently the presentation of the result became a little haphazard as the speaker dealt with the questions, and if that haphazard nature comes across in this post then I will have done my job well :)

The question of whether or not it was possible to formulate an ATFT for non-crystallographic groups was posed by Fring. A model was given, not based on a Lie algebra, and it was wondered whether or not it was integrable. The answer was that integrability problems could be avoided if the non-crystallographic group could be embedded in a crystallographic group. The fact that the theory based on the crystallographic groups was known to be integrable (apparently this is something we know from finding Lax pairs based on Lie algebraic quantities, about which I know nothing), meant that the theory based on the embedded non-crytsallographic group would also be integrable.

Having described the solution, Fring defined the Coxeter group, as the set of Weyl reflections, {S_i}, such that (S_iS_j)^h=1, for some integer h. The embedding of the roots of H_4 in E_8 was explicitly given, the roots corresponding to the two H_4 were chosen so that there were no connections on the E_8 Dynkin diagram between \alpha_n and \alpha_{n+4} where n=1...4, and the first four roots belonged to one copy of H_4, and the remainder to a second copy. This had the advantage that the Weyl reflections between S_i and S_{i+4} would commute and aided in the computation found in the paper. In order to achieve the embedding a map, w, was used which took a set of roots of E_8 and split them as indicated above into the union of the sets of roots of H_4 and c.H_4. The origin of the golden ratio, c, and its necessity in what was being achieved was not clear but it was shown that it worked by considering the orbits of the simple roots under the successive action of S_1...S_n, and then that the orbits are mapped into each other, upto the golden ratio, as expected by w. Fring told us that the original work detailing the embedding was done by Shcherbak (Wavefronts and reflection groups, Russ. Math. Surveys 43, 149-194, 1988) and Moody and Patera (Quasicrystals and Icosians J.Phys. A26, 2829-2853, 1993), but attempts to find these papers on the internet has proved fruitless.

Very similar schemes for embedding H_3 in D_6 and H_2 in A_4 are included in the paper. The fact that this can be acheived explains the pairing between masses in certain ATFT's, however one has to wonder why the examples of E_8, D_6 and A_4 ATFT's weren't all described in terms of a H_2 embedding, as at first glance this would seem possible, and would give a unifying framework. Of course, the motivation was to explain the mass pairings and this has been achieved, and probably there is a reason not to explain the ATFT's in terms of H_2 variables.

This is a question to be addressed by more intelligent people than me. I outdid myself at the end of the seminar by asking the speaker if he had measured the ratio of credit card sides himself. He said he hadn't, and that he had looked it up on the internet, but that it was experimentally possible! However he was able to quote the size of the sides in mm, so maybe my question wasn't so stupid ;)