The End of the Road

Well we've reached the end of the LMS symposium in Durham, and I, for one, am really looking forward to going back home, and thinking about my own work problems. That said it has been a lot of fun to be here, and I may even try and learn something about derived categories off the back of all the talks related to them, and the enthusiasm of the categorists. I'm feeling pretty tired today, so I'm just going to list the last set of talks, starting with yesterday (a cold and miserably damp day).

In the morning the key-note speaker was Tom Bridgeland from the University of Sheffield, another of the enthusiastic categorists mentioned above, and he was talking under the title: "Spaces of stability conditions". He used words such as bounded coherent sheaf, topological B-model, elliptic curve, the Mukai transformation, T-duality, topological conformal field theory, Michael Douglas' Pi stability condition, Fukaya categories, special lagrangians (or slags), Richard Thomas and the McKay correspondence. Somehow all these words were related, and despite Tom Bridgeland being an excellent speaker (I do have a very good set of notes to work with from this talk), I'm afraid I know too little about algebraic geometry to be able to understand the topics. It has been suggested that a good place for me to start would be Serge Lang's book, "Introduction to Algebraic Geometry", and I'm giving it some consideration, although, at £364.27, I don't think I will be buying a copy any time soon. I believe many of the ideas discussed in the talk can be found in Tom Bridgeland's papers "Stability conditions on triangulated categories" and "Stability conditions on K3 surfaces".

In the afternoon we had a talk from Alexei Bondal entitled "Derived categories of toric varieties", and a second key-note speaker in the form of Ron Donagi who spoke about "Geometric transitions, Calabi-Yau integrable systems, and open GW invariants". Ron Donagi's talk was based on the papers "Geometric transitions and integrable systems" and "Geometric Transitions and Mixed Hodge Structures". After these talks we had a wine reception and a seven(!)-course banquet, which might explain some of my lethargy today :)

This morning we heard from Boris Dubrovin who gave the last talk of the symposium to an audience suffering from the morning-after effects of the banquet (principally the wine's effects) under the title "Frobenius manifolds and integrable hierarchies of the topological type". I refer the interested reader to his paper "Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants" for a flavour of the talk.

Now I have an afternoon to work in Durham and will catch my train home tomorrow, at which point I will catch up on my sleep. So ends my missives from the front here in Durham :)

Durham Talks Online Already!

A short comment to say that some of the pdf's of the talks so far at the LMS workshop in Durham are now available online here. I expect more will be added, along with copies of the student posters in due course, so keep checking back.

Generalised Geometry, Oscillating Integrals and Gauged WZ Terms

Today we have had a series of talks half of which were concerned with generalised geometry. Nigel Hitchin was due to speak this morning but was unable to due to illness, but he sent along his slides in his absence and Chris Hull and Richard Thomas filled the gap and made a very good presentation of his talk. The talk was entitled "B-fields, gerbes and generalized geometry" and the first half was a survey of the recent progress in the generalised geometry programme (see this related post about an introductory talk I saw given by Marco Gualtieri, a former student of Hitchin's). The second talk was concerned with looking at the different types of geometry classified by the programme, and was presented in by Richard Thomas. The slides from both parts are available online (part 1, part 2). Since much of the talk can be understood best by reading Gualtieri's thesis I will not say much apart from commenting on how well the two speakers did presenting someone else's slides. It really was impressive, bearing in mind they only had two days to prepare for it. Of course Richard Thomas did take advantage of the situation in order to add comedic value to his presentation, for example proclaiming that one of the slides was so straightforward that he was sure all of us in the audience understood it and so he wasn't going to spend any time talking about it (with a big grin), and more of the same throughout. In all it was a sterling effort from the two stand-in speakers.

After lunch, Claus Hertling, talked to us about "Oscillating integrals and nilpotent orbits of twistor structures", followed by Chris Hull's second talk of the day entitled "Generalised Geometry and Duality" based on his paper "A Geometry for Non-Geometric String Backgrounds". Hull's talk was probably the most accessible talk of the meeting so far for those of us with only an introductory background in physics.

Hull commenced by motivating us that a special approach to the geometry of a spacetime containing strings was needed. He did this by reminding us that the string excitations contain gravity and so strings have the possibility of altering the background spacetime. From there Hull discussed T-dualities which interchange the winding(string)/wrapping(brane) modes with momentum modes, commenting that compactifications on mirror symmetric Calabi-Yau manifolds leads to the same physics but usually on different geometric backgrounds. The picture he wasnted to analyse was whether it would be possible to combine the two geometric backgrounds, so that the whole geometry under mirror symmetry is unaltered along with the physics. His proposal began by doubling the coordinate manifold and using one copy of the coordinates to express the physics in terms of the degrees of freedom coming from the momentum modes, and the other copy of the coordinates to express the same physics in terms of the winding modes. The extra degrees of freedom were to be halved using a set of duality conditions relating the fields in the theory, e.g. dA=*dA'+... and the actual spacetime coordinates used to specify physics were to be singled out by a choice that Hull called polarisation. Since this proposal doubles the tangent space and is a further doubling of Hitchin's generalised geometry, Hull suggested the procedure might be called generalised generalised geometry, but then rejected this for the pithier T-fold geometry. The two sets of coordinates were to be glued together and the transition functions would be T-duality functions. Hull asked the question whether one would be able to patch together the sets of coordinates in this way to give a spacetime manifold and then told us that in general it would not be possible. The end picture was that while local spacetime would be covered by a coordinate patch, globally the manifold structure would be lost, and Hull seemed to be suggesting that doing away with the global spacetime manifold was not such a bad thing, if it meant that there was a better equivalence between geometry and physics. It should go without saying that this was my understanding of the broad picture of Hull's talk, that I may have got the wrong end of the stick, that Hull's arguments can be found in his paper, and that if anyone (especially those who have heard this talk) have any comments/amendments then I would be grateful if they posted them as a comment :)

After the coffee break, where I drank my fifth coffee of the day and got hold of some of the precious chocolate chip cookies that the organisers have been supplying us with, José Figueroa-O'Farrill gave a talk on "Gauged WZ terms in sigma models with boundary" which was based on his paper with Noureddine Mohammedi, "Gauging the Wess-Zumino term of a sigma model with boundary". Before commencing his talk José told us about the new research partnership ERP, Edinburgh Research Partnership and asked us to visit their website, in particular for the vacancies currently available. So consider this information dutifully passed on.

As if five talks weren't enough for one day, we were treated to an extra evening talk after dinner at 8pm entitled "How to Knit a Scarf" presented by Alastair King. Little did I suspect, as I should have done, that this wouldn't be a workshop and there was no wool involved :) Indeed it was a gentle (by the symposium standards) category theory talk, giving the physicists in the audience a gentle appreciation of the "natural" approach of the category theorists. A scarf turns out to be a quiver diagram composed out of many copies of three nodes (A_3) arranged in a triangle and with arrows having an anticlockwise orientation on the basis diagram. It was very colourful, and looked like it would be fun to work with, but that is the limit of my appreciation at the moment :( Alastair King also modified Terry Gannon's comments earlier in the week drawing a comic parallel between categorists and beavers to a quote of his own that ran "Category theorists are not like beavers". The surreal debate trundles on, and I live in fear of what will happen when category theorists discover the jumper.

Stable Algebraic Varieties, Nearly Kahler Manifolds and Twistors

Today I have been on an excursion, for it was our day of rest. The organisers of the symposium arranged for us to leave Grey College at 9.15am and travel by coach to Bamburgh castle, which was the seat of the kings of Northumbria and also the setting for a peculiar worm legend, after that we were whisked off to Holy Island (at which point there followed a series of very poor "Holy Island X Batman!" jokes, where X was some local attraction on the island) and finally we made a brief aborted stop at Alnwick castle. Some outdoor scenes of Hogwart's from Harry Potter was filmed at Alnwick, but due to our sharp exit we didn't even see the outside of the castle. It rained for most of the day, still it was fun to get out of the lecture theatre for a bit.

Yesterday morning we listened to Richard Thomas talk to us about "Special metrics and stability of algebraic varieties". He told us about Geometric Invariant Theory (GIT) and how it may be used to understand a stability condition on both algebraic varieties and simultaneously on related Kahler and Kahler-Einstein metrics. The essential idea was that the stability of a vector bundle can be made into a well-defined notion even over non-compact groups by using a GIT :), and then extending the analogue of this idea to algebraic varieties, and Kahler-Einstein metrics. The detail about the different kinds of stability conditions that can exist can be read about in his paper with Ross entitled "A study of the Hilbert-Mumford criterion for the stability of projective varieties".

In the afternoon yesterday we heard talks from Misha Verbitsky on "An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry" which was based on material from his paper; and Gabriele Travaglini who talked to us about the progress in understanding the calculation of loop diagrams coming from the twistor string programme. Travaglini's talk was entitled "From Twistors to Amplitudes" and was a review of the progress in understanding why some very complex loop Feynman diagrams turn out to have very simple contributions (i.e. zero) to amplitude calculations, as seen from the twistor viewpoint by making use of maximal helicity violating (MHV) diagrams to simplify otherwise horrendous calculations.

In the evening yesterday we had a pub quiz, where after some very suspicious recounting, victory was snatched from the grasp of two of the other (less suspicious) teams by means of a tie-break. The tie-break question was to guess the number of Durham's in the USA, it turns out there are 21, and after a dubiously large team (of the suspicious counting) claimed victory, Richard Thomas, whose team lost out in the 3-way tie-break, magnanamously conceded defeat by pointing out that the winning team had 19 participants, while most of the other teams had only 4 members. Good fun was had by all.

Days Three and Four...

Well there's only so long I can manage without a night of nine hours of sleep, consequently my mind is now mush, and there are only a few constructive comments I can make about the last few talks.

Let's be chronoloigical and go over yesterday's talks first. Werner Nahm greeted us in the morning with a talk entitled "Mirror symmetry for cohomology with values in vector bundles?" in which he described a symmetry of vanishing cohomology classes on the Hodge diamond similar to that of mirror symmetry (a "symmetry" in the Hodge diamond that predicts the existence of a Calabi-Yau manifold for cohomology groups H^{p,q} and H^{q,p} if either one is known to exist...in most cases, for some detail see Diego Matessi's notes in postscript). His aim was to try and reduce the non-vanishing cohomology groups down to a (reflectively) symmetric set of points on the Hodge diamond. The cohomology class H^{p,q}=H^{p,q}(X,V) where X is a compact Kahler manifold and V is a holonomic vector bundle. In the construction presented by Nahm V was taken to be an ample vector bundle (which was defined by its projection being an ample line bundle). For more detail see "Vanishing theorems for products of exterior and symmetric powers" by Laytimi and Nahm.

Since I am so very tired let me mention one of the lighter moments of the talk. At one point Nahm took up the comedic baton from Terry Gannon (more later, in response to comments from Clifford Johnson) by saying that really some of the theorems he was going to discuss were best thought about after two guinnesses: they were two guiness problems. This was a passing comment and no more was thought about it. However, during the interval his former student Katrin Wendland nipped out and purchased a fourpack of guinnesses and Werner was encouraged by the audience to drink them before giving the second-half of his talk - he seemed quite keen to do this, but held back until after the end of the talk when he could be seen supping from one of the cans, just before lunchtime. As a passing comment I should say that there are many of Werner's former students (as well as their students) here; there are at least three generations of the Nahm PhD-advisor/family-tree and certainly at least six members of the clan are here. Which makes this a very nice family reunion, for them.

The first afternoon talk yesterday was given Alessio Corti and was entitled "Examples of orbifold quantum cohomology" - I'm afraid due to my own lack of knowledge I wasn't able to get a lot out of this talk, however Alessio did tell us about stacks, and "stacky fans" so this vocabulary is a start at least (I am being very optimistic here - I have no idea how they might be useful to me, and unfortunately as a consequence I cannot get very excited about them). However I do expect some insights into Alessio's talk can be gained by reading (and understanding) his paper with Abramovich and Vistoli entitled "Twisted bundles and admissible covers".

The second afternoon talk was by Miles Reid and entitled "Orbifold RR and plurigenera", where RR stands for the Riemann-Roch theorem.

To end yesterday we were "treated" to an extra, unpublicised talk on a recent paper by Calin-Iuliu Lazaroiu entitled "Topological D-branes and noncommutative geometry". Unfortunately, to cap a demoralising afternoon, this talk was also outside of my comfort zone. My one piece of terminology I picked up was the definition of a necklace in a quiver diagram, which is, as you may guess, a closed loop on a quiver diagram. I suspect if one was inclined to make a serious investigation of noncommutative geometry one could do worse than looking through the work of Lieven Le Bruyn, who was cited a couple of times during the talk, or even consider buying his self-published textbook via NeverEndingBooks. At the moment Le Bruyn himself recommends reading the Lectures on Noncommutative Geometry by Victor Ginzburg.

To complete my catch-up on what we've been listening to at Durham, I must mention this morning's very clear exposition on the topic "D-branes in Poisson sigma models" by Giovanni Felder. The talk was based on work completed with Alberto Cattaneo in the paper "Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model".

Finally, let me add my support to the notion expressed in previous comments that Tony Gannon is a comedy genius. It was pointed out that I failed to indicate this in my write-up of his talk earlier in the workshop, so permit me to correct this by relaying just one of the examples of his talent here. So, during his talk Gannon was explaining to us his ideas about category theorists, he was telling us (during an aside in the middle of his talk) that category theorists want to change the basis in which the logical structure of mathematics is framed. He said that they preferred not set theory but category theory, and he compared the two areas by saying set theorists like to describe maths using "nouns" while categorists prefer to use "verbs". He then made a leap, and told us his theory that category theorists are really like beavers. He said that the thing that makes beavers build dams is the sound of trickling water - they don't like it and they build a dam to make it stop. He said that if you took a tape recording of trickling water down to a riverbank where there were beavers and left the tape playing, that you could come back the next day andfind the tape recorder covered in pieces of wood. Somehow this idea reminded him of categorists, but having said all that he went on to praise the work and successes of category theory in a most appreciative way. Terry Gannon also owns a red t-shirt featuring a bear wearing green sunglasses, and he gets my thumbs-up for his very entertaining talking style.

Knots, Lattices and then Separation of Variables

The end of day two. In a change of plans I joined the tired-out club last night and lounged around in the JCR, so I was up bright-eyed and bushy-tailed this morning to hear the first of today's three talks. Terry Gannon was speaking under the title of "What knots can still teach R(ational)CFT", and although the talk commenced with the trefoil knot, for the mostpart the focus was on modular forms and the braid group. He did a good job of persuading the audience that it was a good idea to lift the action of modular forms from the upper half plane to a Lie group, and that when considering a modular form with a non-integer weight that it was useful to use B_3 (a braid group) as oppose to Sl_2(Z).

During the morning break and throughout the day we were given the opportunity to purchase some text books with a 20% discount, so after my switch card was not accepted (the salesman didn't have the facility to process it) I raised £28 pounds from my friends and purchased a copy of "Affine Lie Algebras and Quantum Groups..." by Jurgen Fuchs. There were very many other good books available too (including paperback copies of Polchinski vols 1 and 2) and one I hadn't heard about before, but which I will look up in the library, called "Gravity and Strings" by Tomas Ortin.

This afternoon's first talk was about lattice gauge theory and was given by Fedor Smirnov, his title was "Correlation functions for lattice exactly solvable models", and the second was by Evgeni Sklyanin and entitled "The Q operator, Bäcklund transformation and separation of variables".

It seems that all the talks so far have been recorded, so hopefully will appear online at some point in the future. Tonight's aims are nothing more complex than getting some sleep and having a pleasant conversation or two, although not in that order.

Matrix Factorizations

The first day of talks has finally come to an end. There were five talks in all today at a total time of four hours, with long coffee breaks (total: 6 coffees and 2 biscuits, it's amazing I can keep my hands steady enough to type). This afternoon's talks were all concerned with matrix factorisations. The first two were given by Matthias Gaberdiel under the title "Matrix factorisations and D-branes" (a good reference for this talk is here) and the final talk of the day was by Daniel Roggenkamp and entitled "Permutation branes and linear matrix factorisations". There was a significant overlap in these two talks, so I will only describe the first here.

Matthias Gaberdiel was interested in the string's viewpoint of the Calabi-Yau manifold. He argued that from a microscopic (C.F.T.) point of view that Calabi-Yau compactifictions are best understood at the Gepner point. A class of branes located at this point were found by Andreas Recknagel and Schomerus which preserved the full chiral symmetry of the theory. Gaberdiel called these RS branes (he said, at one point, that the abbreviation was an amalgum of Ramond-Ramond and Neveu-Schwarz but dropping an R and an N). However these RS branes do not account for all the D-brane (RR) charges. The case study used for the talk was the quintic superpotential: W=(x_1)^5+...(x_5)^5=0. In this case Gaberdiel told us that the RS branes only accounted for a 25-dimensional sublattice of the full RR charge lattice, and in particular that the D0 brane on the quintic was not described by any of the RS boundary states. The aim of his talk, he said, was to construct the fundamental branes using that give rise to the full RR charge lattice.

The route taken was that initiated by Maxim Kontsevich who proposed a connection between Landau-Ginsburg models and supersymmetric B-type D-branes. It transpired that in order to make the appropriate action with an additional Lansdau-Ginsburg (superpotential, W) F-term invariant under a SuSy variation that an extra term proportional to E must be added such that W=EJ. I am told a clear exposition of this, and some actual definitions of these terms can be found in "Landau-Ginzburg Realization of Open String TFT" by Ilka Brunnera, Manfred Herbsta, Wolfgang Lerchea and Bernhard Scheunera. This is the matrix factorization referred to in the talk's title.

The argument proceeded that since Landau-Ginsburg models are related to N=2 minimal models and that these are the basic unit of the Gepner model description, that there should be a correspondence between them. Gaberdiel told us that this relationship is well known for the single minimal model, but in cases more general than this little is known. The correspondence suggested was then used to tease the D0 brane out of the quintic, the arguments leading to this can be read here. Furthermore the process could be generalised to other Gepner models and D-branes, thus allowing the full charge lattice to be uncovered from fundamental branes in some other cases, as well as the quintic. Gaberdiel took us through a "baby" example and talked to us about product theory and permutation branes (I would give a definition but I'm afraid I haven't quite understood their essence yet, but the literature in the links above should give a clear picture). However there are many areas still left to be investigated in this field, for while the quintic is an example of a Gepner model where the RS-branes generate a vector space of RR-charges and permutation branes generalise this and generate the full RR-charge lattice, there are cases amongst the (147) Gepner models where neither RS-branes nor permutation branes produce the full RR-charge lattice. Furthermore to identify the permutation branes one cornerstone involved considering certain preferred roots of unity. If the preferred roots (the notion of the preferred roots are derived from an index, m, associated with a U(1) charge in a certain N=2 bosonic coset model, again see the above links) of unity were all consecutive then the state being considered was judged to be a permutation brane, however in any other case the association between the model and the field theoretic content remains to be understood.

After dinner tonight they held a wine reception for us in the JCR of Grey College, where we are staying, and now we are thinking of hitting the town. If there are no posts tomorrow you can make a good guess that I hit the town a little too hard...

D-branes, Superpotentials and A-Infinity Algebras

The meeting in Durham is under way and the first two talks of the symposium have finished. They were both given (two 45 minute sessions, separated by one hour's worth of coffee) by Paul Aspinwall, hailing from the "other Durham", under the title "D-branes, Superpotentials and A-Infinity Algebras". The talk was based on work completed with Sheldon Katz in hep-th/0412209.

The aim of the talk was to find a generalised method for finding superpotentials through the medium of category theory looking at the topological field theory B-model (i.e. IIB string theory). It was all very new to me, and involved looking at chain complexes as brane coordinates in the Calabi-Yau 3-fold part of space-time. There were very many neat commutative diagrams relating the complex chains and the maps extending between them were identified as open strings extending between the branes in these coordinates. When it came to looking at decaying branes quivers turned up and there were some more neat diagrams showing the decay of a stack of D-branes down to other new stacks, and the resulting changes of gauge symmetry. So Paul Aspinwall did a good job of intertwining the mathematics of categories with results from string theory, but it would require a lot of work on my part to understand the talk well. Next is matrix factorisations, in fact in 8 minutes so I had better dash.

Off to Durham...

I've been invited to attend a symposium in Durham that starts this Friday. The difference between a symposium and a conference is that the attendees are expected to participate. As a PhD student this means I am expected to present a poster. I have known about it for some time but of course have only just finished my poster catchily entitled M-Theory Solutions in Multiple Signatures from E11 (click for larger version): I'm not really sure what is expected from a poster session. The few I have been to have rarely involved much participation from the person who made the poster; consequently I have tried to encourage as much self-reliance as possible from whomever may look at mine by including lots to read. Despite the purpose of the whole poster concept being a little vague, it has been fun making one, not least because I got to reacquaint myself with colours!

The symposium I am attending is actually a London Mathematical Society meeting, but is occurring in Durham, which is fine by me, I'm looking forward to some fresh air. The symposium title is Geometry, Conformal Field Theory and String Theory and judging from the programme they will be keeping us busy. That said, if possible, I will try and write up some notes from the talks, in my usual half-understood style. Although perhaps I am being optimistic to hope I will understand as much as one-half :)

Postscript: Thanks to Jenn See I have stopped being quite so pessimistic for a bit so I challenge you all to fight my giant battle monster, Mumrah:

mumrah is a Human-Sized Man-Eating Plant that eats Rocks, leaves a Trail of Goo, was brought back from a Distant Volcanic Island, Hovers Eerily, picks up Cars and Throws Them, and CANNOT BE STOPPED. Strength: 7 Agility: 5 Intelligence: 8 To see if your Giant Battle Monster can defeat mumrah, enter your name and choose an attack: fights mumrah using Strength Agility Intelligence

The Superblog is Born

Just when you thought you had enough physics blogs to keep you busy during the (short, of course!) time that occurs between productive thinking, along comes another one: Cosmic Variance. This is no ordinary blog though, this is the superblog. Just like musicians who take leave from their groups and form a supergroup, these physicists, amongst them the owners of the popular Preposterous Universe and Orange Quark, have thrown their voices together. The impressive list of contributers reads: Sean Carroll, JoAnne L. Hewett, Clifford Johnson, Mark Trodden and Risa H. Wechsler. While one wonders about the advantages of co-authoring a blog and despairs over the trail of dead blogs left behind, one wishes them all the best of luck with their endeavour, and is optimistic that they will be more entertaining than the average rock supergroup.

London Bomb Blasts

"This was not a terrorist attack against the mighty and the powerful; it is not aimed at presidents or prime ministers; it was aimed at ordinary working class Londoners, black and white, Muslim and Christians, Hindu and Jew, young and old, indiscriminate attempt at slaughter irrespective of any considerations, of age, of class, of religion, whatever, that isn't an ideology, it isn't even a perverted faith, it's just indiscriminate attempt at mass murder, and we know what the objective is, they seek to divide London. They seek to turn Londoners against each other and Londoners will not be divided by this cowardly attack." Ken Livingston, Mayor of London.

I have been rapidly making my roll-calls by email of friends and family checking that they are all okay (almost all outgoing mobile phone usage has been suspended for security reasons). It seems most of the people I know are fortunately only stuck in their offices because of the transport stoppage in central London.

Some commentary on today's tragic events can be found here.

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 ;)

Zeitgeist

The BBC are running a story on their newsplayer about a clinic that has opened in China that aims to cure people of their internet addiction. You can watch the short news footage via the BBC news website (click on "Watch BBC news in video", and then select the sci-tech index, and the title is "China opens clinic for internet addicts" - sorry I couldn't get a working link). I can't help but feel this is a crazy, luxurious use of a clinic, but perhaps I'm wrong, perhaps you can be medically addicted to the internet and can be "cured" by a two week stay at a clinic where you can play ping-pong. Perhaps.

Having poked a little fun at this piece of zeitgeist, I should say there is at least one young man featured in the newsclip who does need some help, for he thinks he is a potato. I quote the translation,

I can find myself again in computer games. In real life you are nothing but a small potato, but in computer games you can be a superman. I want to be a superman.

So, just in case this is a serious problem I want to do my duty as a responsible blogger, so sit back, take a deep breath and ask yourself if you are addicted to the internet, or if you are a potato. I would especially appreciate a response from potatoes :)

Vafa describes Topological String Theory

So earlier on this week, we, in London, were treated to a series of three talks by Cumrun Vafa describing Topological String Theory (TST). The interested throngs gathered in Imperial College on Monday for the first talk and included students from as far away as Cambridge and even Chicago. Now I have been doing my homework and making sure I read around and learn as much as I can about the language of string theory, and so I thought it would be a short pleasant step to TST. Alas not: there's always more to learn.

Vafa's plan for the three talks went:

I. What is TST?
II. Dualities & topological strings.
III. Black holes, topological M-theory & topological strings.

It all began well enough. Vafa told us that he knew the audience consisted of a mix between mathematicians and physicists and so his talks would of necessity be quite general. This sounded excellent. Vafa began by describing the concept of localisation in mathematics, where problems which are too hard to compute in general are reduced to an understandable subset. Of course in string theory the correlation functions are too hard to compute in general but a subset are exact. In string theory we map, X, from a Riemann surface into a 10-dimensional manifold, M^10 which is the product of a Minkowski spacetime R^4 which we concede exists and a missing 6-dimensional manifold M^6, which many are less sure of. The localisation that gives TST restricts the image of X to a point in R^4 crossed with M^6. So TST is the study of maps from the Riemann surface to M^6.

Vafa told us there were two types of TST:

- The A-model (the IIA superstring) where the localisation restriction is \bar{\partial X}=0, holomorphic maps.
- The B-model (the IIB superstring) where the localisation restriction is dX=0, constant maps.

Typically M^6 must be Ricci flat and Kahler, these two conditions give us a Calabi-Yau manifold (or complex 3-fold, as Vafa preferred). He went on to tell us that the moduli space of a CY n-fold naturally splits into the product of a complex manifold (with hodge number h^{1,n-1}) and a Kahler manifold (h^{1,1}). Now the A-model computations depend only on the Kahler manifold, while the B-model depends only on the complex manifold.

Then we had an aside about mirror symmetry. Vafa asked us to consider a map to an S^1 (the circle) target manifold of radius R. At a fixed time, the closed string is also a circle so in this case we are imagining a very simple map from one circle to a second which has radius R. In the spacetime image the energy of the unexcited string is determined by how many times it wraps the circular dimension, E~w.R, where w is an integer called the winding number. There is also the possibility of momentum states on the string which are subject to p~n/R, where n is the integer momentum number. Vafa now had us primed for T-duality. Since the string may have both winding states and momentum states, E+p is unaltered by T-duality, which exchanges R<->1/R, and w<->n.

The ante was then upped a little as we moved from an S^1 target space to a CY 1-fold (=T^2, the 2-dimensional torus). Defining the torus as a rectangle, of sides R_1 and R_2, with opposite edges being identified, Vafa defined two new quantities: \tau=iR_2/R_1 (complex structure parameter) and, the complexified area iA=iR_1R_2 (Kahler parameter). Now under a T-duality about the circle of length R_1, these quantities are mapped to each other i.e. \tau->iA, and iA->\tau. That is, T^2 is mirror to itself, and the complex and Kahler structures are exchanged. So we can see that A-model and B-model are interchanged, in this example and Vafa encouraged us to wonder whether or not this is true for a more general target manifold, or if T^2 is special.

It turned out that T^2 is special. Vafa told us that for a CY n-fold target space, T-duality changes the hodge numbers as h^{p,q}<->h^{n-p,q}, hence for T^2 (n=2, p=q=1) h^{1,1)<->h{1,1}. This was the turning point in the talk for me, because from here on in deductions were made from "facts" that I had not seen before. So I was very uncomfortable. For example, Vafa next considered the A-model and its Kahler manifold and gave an argument that the study of TST naturally leads one to be interested in CY 3-folds, without detailed knowledge of string theory. It went like this:

Fact: "dim(\bar{\partial X})"=(n-3)(1-g) + C_1(M)|(image)

where C_1(M)|(image) isthe first Chern class restricted to the image.

Deduction: If we are interested in the most straightforward maps, those with dim=0,then we could consider n=3, C_1(M)|(image)=0, which are CY 3-folds.

This is nice to know, but I don't feel like I've understood anything because I don't really know where the starting fact came from. Vafa looked at the B-model in a similar fashion and then wound up his first talk, before being lampooned with questions by the much-better-informed-than-I audience. So my notes will run dry here, but I'll give my wish list of terminology/facts so that a friendly reader can help me out, and in future I can remember what I don't know :)

- The Gromov-Witten invariant
- That the integral of the top Chern class defined over a vector bundle equals the GW invariant
- classical geometry is related to the periods of a holomorphic n-form
- Ray-Singer torsion
- Kodaira-Spencer theory of gravity

So that was the first talk, so despite the star billing I didn't attend the others, besides I'm just entering the final quarter of my second-year so it is time to start "working hard", as I was told by someone who clearly has my levels of work figured out.

If anyone reads this and went to any of the two other talks, comments on how they went are very welcome :)