Monday, May 23, 2005

I've Been Meme'd

Well this is a first for me, I have been tagged by Phil at Umbrae Canarum to participate in an interview about my reading habits. Joy! This is a meme. It's a thoroughly decent participate-if-you-want-to affair, and the way it works is that I get to target three other bloggers to answer these same questions on their blog, and the way they find out about it is if they read this. So it's non-invasive and, well, a little bit of self-indulgent fun. So I nominate Lubos Motl, Peter Woit and Lieven Le Bruyn. Here are my answers:

Total Number of Books I've Owned
Well this isn't easy. I have about 150 books that I have with me, these are mostly books that I have bought on an impulse and haven't liked enough to actually read, but yet still hope to. I think this is worryingly about a quarter of all the books I own! So I think I own about 600 books. The number I have owned and lost/sold/given away/destroyed charismatically is negligible (~15).

Last Book I Bought
I bought three books together, to ensure free delivery: Incompleteness: The Proof and Paradox of Kurt Godel by Professor Rebecca Goldstein (a professor of philosophy); Woken Furies by Richard Morgan; and Schismatrix Plus by Bruce Sterling.

Last Book I Read
I've been struggling with Cryptonomicon by Neal Stephenson for quite a while now, I'm nearly finished, but I guess this doesn't count. The last book I finished was The Little White Car by Danuta de Rhodes.

Five Books that Mean a Lot to Me
Permit me to say that I am fairly hedonistic when it comes to novels: for a book to mean a lot to me I simply have to have enjoyed it a lot. So without any literary heavyweights and in no particular order,

1. Lord of Light by Roger Zelazny
Without a doubt this is the book I have enjoyed reading the most. It's a sci-fi tale set on a planet where a few humans from a technologically advanced civilisation live with the indiginous population, masquerading as Gods from Hindu mythology, with the aid of their science. The story is about Mahasamatman "call me Sam", the Buddha, and his war with the other 'deities', and is told with rapid dialogue, a good sense of humour, plenty of action and is wonderfully imaginative.

2. The Fountainhead by Ayn Rand
I read this before going on to Atlas Shrugged, and while the latter has the more exciting political ideas in it, I simply enjoyed this more as a story. I loved the imperturbable hero Howard Roark, the architect whose ideology and muleish pursuit of his passions act as a rock against which the other characters bash and splinter. It has a plethora of deliciously mischevious characters, and has the larger than life tone of a soap opera. Pure fun and inspirational, especially if you have a stubborn streak and it certainly allowed me to enjoy architecture with fresh eyes.

3. Any Human Heart by William Boyd
This is the journal of a fictional man who happens to meet a whole cast of famous faces. I found it simultaneously funny and sad, and never lost interest. I include it here, because lately I keep wanting to reread it, so I must have enjoyed it more than I realised at the time. It's the only book that's made me want to know what the smoke from a fire built from the trees around my house would smell like. Also there's a scene where as an old man, the main character resorts to making a stew using dog food, which I particularly like, and always comes to mind when my cupboards are bare.

4. The Vintner's Luck by Elizabeth Knox
Just your average story of the life of a Vintner and his friend the angel, who meet fortuitously one year early in the life of the Vintner and agree to meet at the same time every year, with a few exceptions, until the end of the Vintner's life. A beatiful tale about loss, among other things.

5. Genius: Richard Feynman and Modern Physics by James Gleick
A non-technical biography of Richard Feynman. I love reading biographies of famous scientists, especially those that give insight about their life outside of their work, and Feynman is such a loveable character that it's hard not to enjoy this book.

So there goes the meme, or was that more "me me"?

Friday, May 13, 2005

The Spinorial Geometry Programme

Well it's been a quiet week on the seminar front. It also has seemed, at least to me, like it's been a quieter than usual last few months for new hep-th papers on the arxiv. There was one monday earlier this month, admittedly a holiday, when there was only one new paper, hep-th/0504234. It's not too easy to check at the hep-th level but you can look at the overall arxiv submission rate and see that since its beginning the submission rate has grown almost linearly. Unlike our exponential world population growth. Perhaps fewer scientists are being born! Also on the graph you can see that there is often a drop in submission rate going from December to January in consecutive years. Could there be a more objective reason than a desire to finish off work as the year ends, and then begin new research as the new year commences? There's also a frequent drop around July, when teaching slows down and summer is high, and presumably scientists travel to conferences, and other universities. Now if you put a straight edge along the graph the difference in month to month rate changes becomes most pronounced in the last few years, and perhaps, if anything, submission rate is beginning to level fall below the historical linear increase rate. We shall see in the next few years.

However submission rate is not in decline at KCL. That sole submission on the arxiv, that I referred to earlier, was written in part by Thomas Quella, a post-doc here at KCL. Also we are in the middle of a series of publications by George Papadopoulos et al. from KCL as part of his spinorial geometry programme aimed classifying all supersymmetric solutions, via his simplifying formalism.

The first spinorial geometry paper, "The spinorial geometry of supersymmetric backgrounds" by Joe Gillard, Ulf Gran and George Papadopoulos, appeared on the arxiv in October of last year. Since then there have been a series of three follow up papers, the latest one appearing on the arxiv this week:

  • "The spinorial geometry of supersymmetric IIB backgrounds" by U. Gran, J. Gutowski and G. Papadopoulos
  • "Systematics of M-theory spinorial geometry" by U. Gran, G. Papadopoulos and D. Roest
  • "The G_2 spinorial geometry of supersymmetric IIB backgrounds" by U. Gran, J. Gutowski and G. Papadopoulos

    The motivating idea is that the killing spinor equations of any supergravity theory (set \lambda in the general killing spinor equation, given in the last link, to zero) can be solved for an arbitrary number of killing spinors in a systematic way by making use of the fact that the supercovariant derivative has a Spin group gauge symmetry (e.g. a Spin(1,10) gauge symmetry for the case of eleven dimensional supergravity - in passing we refresh our memory and recall that the Spin(t,s) group is the double cover of SO(t,s)). We won't get to the Killing equations in this post, instead we will focus on the stability group of spinors. Spinors can be classified by their stability subgroup, which is the subgroup of the spin group that leaves the spinor invariant. This offers a systematic way to study killing spinors.

    At the start of the paper by Gillard et al. a recipe is given to write the spinors of supergravity, i.e. elements of a vector space which transform under Spin(1,10), as forms. The recipe commences by considering representations of Spin(10) instead of Spin(1,10). Complex/Dirac spinors in Spin(10) have 32-components and decompose into two 16-component irreducible representations. Mirroring this, the basis of our vector space, R^10 is split into two parts, each part spanning one copy of R^5. The two copies of R^5 are then related by some complex structure, identifying vectors in each R^5. So that the full vector space is determined from the orthonormal basis vectors of the "first" five coordinates, e.g. {e1,e2,e3,e4,e5}. Spinors are represented in the basis of forms, i.e. if S is a spinor, then it is expressed in the basis of 0-forms, 1-forms,...5-forms as S=lI+mA(eA)+[nAnB](eA^eB)+...[qAqBqCqDqE](eA^eB^eC^eD^eE). My apologies for the poor notation here, one day I will get mathplayer or some equivalent. This gives a canonical way to write Spin(10) spinors as forms. The action of the ten gamma matrices on the basis of forms is given in the paper and then gamma_0 is constructed (raising the field of play to Spin(1,10)) and the Majorana condition applied to the spinors.

    Having written a spinor in this manner, use is made of a theorem concerning representations of SU(n) carried by the basis forms. I don't know the theorem used but it results in a statement similar to "SU(n) irreducible representations are carried by p-forms on C^n".

    Additional comment: A theorem has that makes some inroads into understanding this step, albeit over a real vector space, is given in "Compact Manifolds with Special Holonomy" by Dominic Joyce, (Prop. 3.5.1) has the consequence that for a Lie subgroup, G, of GL(n,R), the bundle of p-forms over R^n splits into irreducible sub-bundles on which irreps of G act. [Thanks again Joe!]

    This leads us not just to have irreps of SU(n), but also any Lie group. But this is the real case, so perhaps over the complex space, the reason for specialising to SU(n) irreps will become clear. Any explanations/corrections are welcome :)

    If this holds then all the basis p-forms, that have been constructed are over C^5, and so carry an irrep of SU(5). The dimension of these irreps is (5 choose p), so that the zero-form and the five-form both carry one-dimensional irreps of SU(5). The dimension tells us how the irrep transforms under SU(5), in particular how many indices it transforms under: each index gives five degrees of freedom of the irrep. So that one dimensional irreps have zero indices and transform trivially as scalars under SU(5), five-dimensional irreps have one index and transform as a vector and ten-dimensional irreps have two indices and transform as skew-symmetric matrices (as the indices are still coupled to the form indices, so the symmetric part is zero). Both the zero-forms and the five-forms of the spinor basis transform trivially under SU(5), so to find spinors with stability group SU(5) we form our spinors out of the 1 and e1^e2^e3^e4^e5 basis elements. By imposing the Majorana condition on this general spinor we learn that there are only two linearly independent spinors with stability group SU(5). Which is rather neat.

    I have to admit I had lots of difficulties even reading through the beginning of the first paper, but thanks to my good friend J. J. Gillard, who talked me through this stuff no less than five times, many of my difficulties have been overcome. If the increase rate of submissions to the archive is indeed slowing down perhaps it's because potentially prolific scientists everywhere are being distracted by people like me asking them to explain their work again, and again... :)
  • Friday, May 06, 2005

    E11 in the Reference Frame

    It is the election here in the UK tonight, I have poured myself a glass of wine and am watching Peter Snow leap joyously between his swingometer and his triangulated battleground (ITV are using ELVIS - their ELection VISualiser) and listening to pundits make extraordinary use of statistics (in one case representing the entire election result from a result in just one seat!). But, as the counting drags out, the gleam of the graphics fade and the studio discussions are recycled, I find myself needing a secondary activity to keep me occupied. So I have decided to write something in defense of poor old E11 after reading Lubos Motl despair that almost nothing is known about its properties. I should add that Lubos was making a passing comment in an article about Hermann Nicolai's promising cosmological billiards programme using E10, which certainly deserves attention and it is pleasing that it is being discussed in his Reference Frame.

    First let me quote Victor Kac who once wrote, "It is a well-kept secret that the theory of Kac-Moody algebras has been a disaster." Those familiar with E11 will know that it has a Kac-Moody algebra and by extension is a disaster, so Lubos is quite right in what he says, nevertheless there has been some progress in uncovering its association with eleven dimensional supergravity.

    E11 is the result of extending the E8 dynkin diagram by adding three nodes to it, giving:

    Dynkin diagram of E11
    The algebra that results is a Kac-Moody algebra, and infinite dimensional. This means that, as I overheard my supervisor in an echo of Kac, Peter West, telling a visiting speaker in the coffee room, "E11 is a complete mess". Since Peter has had me work on nothing but E11 since I started my PhD this is a favourite quote of mine, and perhaps I will put it on the front of my thesis, if I get there.

    There has been plenty of good news about E11 since Peter made his E11 conjecture (that E11 is the symmetry group giving rise to the M-theory dualities). But first some groundwork. The connection with 11-dimensional supergravity comes through decomposing its algebra into representations of A10 i.e. SU(11). The longest line of ten roots in the E11 Dynkin diagram is the A10 used, and is often called the gravity line. A restriction to the real form of SU(11) i.e. SL(11,R) is made, and then by including the eleventh generator from E11 (the one from the node that sticks out on the Dynkin diagram) this algebra is enlarged to GL(11,R). For the usual (1,10) signature of low energy M-theory/supergravity, the vielbein is an element of a coset of this group, namely of GL(11,R)/SO(1,10).

    The decomposition gives an infinite number of generators, classified by their Dynkin labels, and in particular the Dynkin label of the eleventh root which is called the level. Low level tables of these generators can be found here, back when E11 was known as E8+++. It was noted that a truncation of the algebra, to generators of level 3 and less, leads to eleven dimensional supergravity fields. Furthermore, and this is the nicest result I have seen so far, you can find the 10-dimensional theories from E11 too. In this case the vielbein is a member of GL(10,R)/SO(1,9), and the decomposition is of E11 into A9 representations. In this case there are two distinct ways to pick A9, which is a straight line of nine nodes on the Dynkin diagram:

    1. using the first nine nodes along the horizontal of the E11 diagram above
    2. using the first eight horizontal nodes and the one orthogonal node on the above diagram

    In the first choice the IIA theory is found, and in the second choice we find the IIB theory. More about this can also be read here. This is a bit more than one would expect from dimensional reduction, because chirality appears.

    Further results relating D=11, IIA and IIB, which are "central to string theory" are given in hep-th/0407088. My single piece of work has been concerned with a group element of E11(although the group element is also applicable to all the oxidised supergravity theories built from any of the very-extensions of the semisimple Lie groups) that encodes the vielbein for the half BPS cases, so that a generator resulting from a decomposition is associated with a brane solution. For the low level generators which coincide with 11-dimensional supergravity and M-theory, the M2, M5 and the pp-wave are found, as you would expect. The real question is what role do the other generators, which extend off to infinity in number, play?

    I have presented a slightly skewed presentation of E11 research, so let me redress this and point out that significant research has also been carried out on E11 by Englert, Houart, Keurentjes and the Mkrtchyans, amongst others.

    So, while almost nothing is known about E11, the little that is known is very promising and quite exciting. Of course the next challenge is to find some fermions...

    Footnote: This was posted a few days after starting it, as my graphics card broke. The very last recounts of the votes in the election have been completed, and the results almost perfectly matched the exit polls. Nevertheless the whole counting process, and crazy graphics that goes with it, was quite good fun, and I think we should keep doing it :)

    Thursday, May 05, 2005

    Triangle Seminar Troubles

    Being a second year PhD student can be demoralising. You have learnt enough to have quite a good idea of how vast the string theory literature is, but also to know just how much of it you don't have a chance of reading before you finish your PhD. As if this wasn't enough to spoil your mood, you also have to be reminded of your lack of knowledge at the weekly seminars. However compared to my first year, the amount of useful information I can get from seminars is significantly improved. Last year the best I could hope to get from the weekly meetings was a list of vocabulary, indeed at times, it wasn't so different from French lessons, although I didn't keep a vocabulary book. But this year, mostly, seminars have been much more illuminating. Until yesterday.

    Yesterday I caught the tube with a number of the post-docs and members of the faculty and we travelled to Imperial to hear Jose Barbon (Madrid) and Katrin Wendland talk as part of the London Triangle seminars which happen occasionally, and are shared between Imperial, KCL and QMW. These are usually the most technical of the different seminars I have attended, and the topics have a tendency to be quite specialist. Consequently I'm not going to feel too disappointed because I didn't understand much, and instead of describing the talks in any detail I'm going to give the titles and where the important papers can be found, and mention any vocab that caused me to lose my way.

    Jose Barbon talked first on "QCD Chiral Dynamics and AdS/CFT Models", and it seemed he gave a very good talk and was very familiar with the various techniques of doing calculations using the AdS/CFT correspondence. Unfortunately I'm not so familiar with it and was lost quite quickly. The talk was based upon his paper with Hoyos, Mateos and Myers entitled "The Holographic Life of the \eta'" and new vocabulary was light on the ground, although I did hear the word "bolt" used for the first time, in this case it was described as being equivalent to a "wall repelling Wilson loops" (which was equally alien terminology for me), but the word itself does not turn up in the relevant paper.

    After no questions, which I feel is often a sign that the speaker has bamboozled the audience, we enjoyed some very welcome coffee and sticky cakes.

    The second talk was given by Katrin Wendland, who is based at Warwick and Chapel Hill, and was a very decent algebraic geometry lesson. Her talk was entitled "How to construct SCFTs associated to a family of smooth K3 surfaces" - first piece of vocabulary thanks to wikipedia:

  • K3 surface - a hyperkahler manifold in four dimensions, having SU(2) holonomy

  • KW's talk was based around her paper "On Superconformal Field Theories Associated to Very Attractive Quartics" and was concerned with expressing SCFTs in a neat form classified by a sum of quartic polynomials on CP^3. Incidently, if you look at the wikipedia entry on complex projective spaces you find it carries the health warning:
    This article may be too technical for most readers to understand. Please expand it to make it more accessible to non-experts — without removing the technical details — and remove this notice once so done.
    I think whoever has the job of putting this quote in place should start doing the same job on the arxiv :) Anyway KW gave us a very specific recipe for characterising SCFTs as a Z_4 orbifold, from the tensor product of two Z_2 orbifolds. The example of the tensor product of two Gepner models was used and her paper with Nahm cited. The talk was very lively, and as with the Barbon, it was disappointing that I didn't know enough to benefit from it :(

    I went back to Drury Lane with my tail between my legs and felt discouraged.