P.P. Cook's Tangent Space

Theoretical physics inspires art!

2011-04-20T21:57:00.000Z

No I'm not thinking of the appearance of Bagger-Lambert as a sandwich in Ian MacEwan's Solar. Yesterday Radiohead thanked those of us who bought their album "In Limbs" in mp3 format by giving away two extra songs for free, one of which is called "Supercollider". Unbeknownst to the theoretical physics community the group have been playing it live for a couple of years as you can see from the video link above. The lyrics are pasted below and are inspired, I presume, by some theoretical physics jargon... but how long do we have to wait for a song about supersymmetry? I know Muse have already sung about supermassive black holes, but that was not short for supersymmetric massive black hole as far as I know and there is a band called Slept On the Couch, but I do not think they have intentionally named themselves after a breed of superparticles :) Maybe some physicists out there could form a tribute band called SuSy and the Banshees. They could do a version of Peggy Sue, Peggy SuSy. Or "Killing (spinor) in the name of" by Rage Against the (LHC) Machine? We wait patiently with hope.

Super collider
Dust in a moment
Particles scatter
Parting from the soup

Swimming upstream
Before the heavens crack open
Thin pixelations
Coming out from the dust

In a blue light
In a green light
In a half light
In a work light

In a B-spin
Flip flopping
In a pulse wave
Outstepping

To put the shadows back into
The boxes

I am open
I am welcome
For a fraction
Of a second

I have jettisoned my illusions
I have dislodged my depressions

I put the shadows back into
The boxes

Spinorial Representations and Dynkin Diagrams

2011-02-08T13:08:00.002Z

I've been enjoying String Solitons and T-duality by Eric Bergshoeff and Fabio Riccioni today, which builds upon their work from last year D-Brane Wess-Zumino Terms and U-Duality. These are impressive papers and you can expect to hear more about them here in the not too distant future, in the meantime I thought I would try and manually resuscitate this old blog with some small comments on spinorial representations that this reading brought to mind. Picture, if you will, the Dynkin diagram for SO(d,d), oh all right here it is:

The $D_d$ Dynkin diagram otherwise known as SO(2d) one of whose real forms is SO(d,d).

Let there be d nodes to this Dynkin diagram and let them be numbered along the long leg from left to right 1 to (d-2), and for the two fish-tail nodes let the bottom one be number (d-1) and the top one node d.

One can form a spinorial representation of SO(d,d) by attaching an extra node, which we will number (d+1), to node (d-1) on the diagram above and considering all the roots associated to the extended Dynkain diagram such that the root $\alpha_{(d+1)}$ appears only once. This has the effect of constructing the representation of SO(d,d) with lowest weight $-\lambda_{(d-1)}$. Usually we work with highest weight representations, in this construction we work from the bottom up building on the lowest weight. This representation will be the spinorial representation.

So far not so much fun... But we may well wonder how big is this representation? To this end let us decompose the extended Dynkin diagram to tensor representations of SL(d,${\mathbb R}$) by deleting two nodes. Recall that in order to build the spinorial representation we added node (d+1), which is not shown and held it fixed - there was always one multiple $\alpha_{(d+1)}$ in any root of this representation - well now we will delete this and we will also delete node (d). Deleting node (d+1) gives the vector representation of SL(d, ${\mathbb R}$) of dimension $d$ while deleting node (d) gives the antisymmetric 2-index SL(d, ${\mathbb R}$) tensor of dimension $\frac{d(d-1)}{2}$. We can usefully denote these two representations by Young tableaux:

Generically the indices are denoted $a$ and $b$ but can range from 1 to (d). In fact upon deletion of the nodes these tableaux takes specific values $a=d$ and $b=(d-1)$. The dimensions of the representations mentioned above are clear when we let the values $(a,b)$ take all possible values allowed by the symmetry of the tableaux.

Now as we construct the spinorial representation more and more Young tableau will appear. How can we tell which ones will show up? As SO(d,d) has a finite-dimensional Lie algebra and as the Dynkin diagram is simply-laced, all roots have the same length and for a given (d) there are a finite number of them. Let each root have root length squared equal to 2. We can embed the roots into a vector space $V_{(d+1)}$ with basis elements $e_1,e_2,e_3\ldots e_{(d-1)}, e_d, e_{(d+1)}$. To do this we must preserve all the inner products encoded in the Dynkin diagram between the simple positive roots - this amounts to us being able to find an inner product which will achieve this. Let the simple positive roots in $V_{(d+1)}$ be

$$\alpha_{i}=e_i - e_{i+1} \qquad \qquad (1\leq i \leq (d-1))$$

$$\alpha_d = e_{(d-1)}+e_d+e_{(d+1)}$$

$$\alpha_{(d+1)}=e_d-e_{(d+1)}$$

Under the usual scalar product $\alpha_d^2=3$ while all the other roots have squared length 2, as desired. We therefore modify the inner product to be given by:

$$<\beta,\gamma>=\sum_{i=1}^{d+1}b_ic_i-(m_d)_\beta(m_d)_\gamma$$

Where $\beta=\sum_{i=1}^{d+1}b_ie_i=\sum_{i=1}{d+1}m_i\alpha_i$ and $\gamma=\sum_{i=1}^{d+1}c_ie_i$ and $(m_d)_\beta$ is the number of times the root $\alpha_d$ appears in the simple root expansion of $\beta$. So now all roots have length squared 2 and the inner products are those corresponding to the simple roots of our extended SO(d,d) diagram. For reference one can work out the fundamental weights of SO(d,d) in this vector space basis and for this inner product it is the vector with components $-\frac{1}{2}(1,1,\ldots 1,1,-1,(2-d))$ i.e. it is:
$$\lambda_{d+1}=-\frac{1}{2}(e_1+e_2+\ldots + e_{(d-2)}+e_{(d-1)})+\frac{1}{2}e_{(d)}+\frac{(d-2)}{2}e_{(d+1)}$$
This looks rather odd but then we are making a peculiar spinor construction by embedding the SO(2d) root lattice inside that of $E_{(d+1)}$. However you can see in the first $d$ entries the usual highest weight for the spinorial representation, see the more usual discussion (without the embedding in $E_d$) in Wikipedia for example (look at the section spin representations and their weights). The more familiar story and the connection to Clifford algebras follows from here. But we continue down our path less travelled...

We were building up the spinorial representation for which we held $m_d=1$ for all the roots in the representation. We may classify the roots that appear by the number of copies $\alpha_d$ they possess, and that number itself we call the level. So at level $m_d=0$ we find just

having dimension d. At level 1 we consider the tensor product and decompose it using the Littlewood-Hardy rules to find:

There are two possible Young tableau at level one but in fact only the first has root length squared equal to two, the second has length squared equal to four. As the roots in the representation all have length squared two only the first Young tableau can exist in the algebra. In fact as we keep going and construct the possible Young tableaux at level two, three, four... only the completely antisymmetric tableaux consisting of $2m_d+1$ boxes have length squared two - as a quick computation using the inner product shows.

Finally we have a closed statement that the spinorial representation of SO(d,d) can be represented by a sum of SL(d,$\mathbb R$) antisymmetric tensors of $2m_d+1$ indices. For a finite d this sum of tensors terminates when either $2m_d+1=d$ for odd d, or when $2m_d+1=d-1$ for even d:

For the even d case the dimensions of the $m_d+1=\frac{d}{2}$ Young tableaux may be summed to

$$d+\frac{d(d-1)(d-2)}{3!}+\frac{d(d-1)(d-2)(d-3)(d-4)}{5!}+...+d=\sum_{i=0}^{\frac{d}{2}-1}{ d\choose (2i+1)}=2^{(d-1)}$$

While for odd d there are $\frac{d+1}{2}$ tableaux giving:

$$d+{d\choose 3} + \ldots + {d\choose (d-2)}+{d\choose d}=2^{(d-1)}$$
N.B. for the sums one can use the neat trick of adding or subtracting $0=(1-1)^d$ to $2^d=(1+1)^d$ to prove the two sums above are the same. (Thanks to V. for pointing this out.)

Et voila! In both even and odd d we count $2^{(d-1)}$, if we happened to not be interested in SO(d,d) but rather SO(D), where D=2d is even we find the dimension of the spinorial representation is $2^{\frac{D}{2}-1}$. This is one of the two inequivalent Weyl spinor representations (there is one associated to each node in the fish tail of the Dynkin diagram) and together they give a Dirac spinor of dimension $2^{\frac{D}{2}}$. For odd D one has to start with the $B_n$ Dynkin diagram and that's another story...

Day Four of Eurostrings 2008

2008-07-06T13:01:00.006Z

The morning began breakfastless, and a little breathless, rushing from the shower to the conference. By day four conference fatigue was beginning to set in. It had absoulutely nothing to do with the Belgian-beer filled discussions. None whatsoever. However day four proved to give second wind to the meeting, filled with very interesting talks that I hope to give a flavour of here.

The morning review lecture was given by Samir D. Mathur, he does not like horizons, well at least those around black holes. One has to sympathise - is it really acceptable to cut a singularity out of a theory? Mathur prefers a fuzzball picture, where the black hole horizon becomes a statistical entity, emerging macrosopically - the canonical comparison is with temperature in the thermodynamical picture. In thermodynamics the temperature is a statistical quantity that can be measured over a large number of microscopic states, but if you sat on a hydrogen molecule (well maybe you already are, but what I mean is, if you had the molecule's view of a gas) you would be able to say a few things about your nearest neighbours relative velocities, and only with a large amount of time would you collect enough information to speak with confidence of the average molecular speed, or temperature. To a microscopic state the temperature is an odd concept, supposedly the black hole horizon is also an odd concept to a gravitational microstate. The fuzzball proposal functionally aims to reproduce the macrosopic black hole phenomena from collections of microstates. The brane microstates themselves do not have horizons in this setup, the horizon appears in the averaging over a large number of brane states. Old and familiar properties of black holes are reproduced in this picture, light can be trapped behind the horizon by an elaborate setup up of light deflecting states, Hawking temperatures can be reproduced and lately Hawking radiation can be produced by pair-production. For an introduction to the proposal you can read his papers here and here. The proposal offers a way to side-step Hawking's information paradox. Mathur's discussion of the information paradox can be read in this preprint, where he aims to make a review using pictures.

\begin{digression}
Kurt Vonnegut used to use a technique of repeating a small, catchy phrase when something of particular note happened in a sentence of his (e.g. in Cat's Cradle each reference to slipping off the mortal coil earns a: so it goes, or in Timequake ting-a-ling is the catchphrase). I think everytime someone tries to explain something with pictures I would like to insert a cowbell noise. So here's to Mathur: *cowbell*.
\end{digression}

Mathur's title this morning was "Lessons from resolving the information paradox". He threw out the notion two charge non-extremal black holes have a singular throat in the spacetime, the geometry may become complicated but not singular. We heard about tunnelling in fuzzball geometries, radiation and pair-creation, which you can read about in the links.

After coffee, we had talks from Eric D'Hoker ("Exact 1/2 BPS solutions in type IIB and M-theory"), Dario Francia ("Unconstrained higher spins and current exchanges") and Diego Chialva ("Chain inflation revisited").

Halfday Wednesday

2008-07-03T23:01:00.009Z

Well Wednesday of Eurostrings 2008 was a half day - the afternoon was left free to enjoy the pleasures of Amsterdam, or to work furiously on the latest Bagger-Lambert paper. So, of course, in honour of the half day here's a half-blog entry. Instead of writing only half sentences I will aim to halve my number of full sentences.

The weather in Amsterdam understood that it was a half day for our conference. Upto midday it was a balmy 27 degrees and sunny, but as I settled in for lunch an almighty, apocalyptic thunder storm came in, as you can perhaps see in the photo (starring Erik Tonni [left] and Diederik Roest [not left]). Erik and Diederik suggested that Bagger-Lambert theory may be getting too close to the truth for the almighty being's liking, and like the Tower of Babel, was about to be toppled by the ensuing thunderstorm. The storm passed, while I ate a very nice sandwich. I am not suggesting any causal connection between the weather and my digestion, but let's not rule it out.

Due to a lack of sleep here in Amsterdam, I all but missed the morning session (not a smart move on a half-day) so I am one of the worst people to tell you what was discussed. However let me put up the titles and one or two suggested papers. Perhaps a fellow Eurostring-ite who may stumble this way can let me know some more about the talks? The schedule was:

"Strongly coupled Quark-Gluon Plasma and AdS/CFT" by Edward Shuryak, see, perhaps, the paper here
"Is the AdS S-matrix simple?" by Romuald Janik (I was told that the short answer is: no)
Herman Verlinde gave a blackboard talk.
"Building a holographic superconductor" by Gary Horowitz.

I caught the final talk, on the relatively hot ;) topic of trying to undertand superconduction through the AdS/CFT correspondence. This was the same topic that Denef discussed on Tuesday, but today the role of the AdS background was emphasised in order to capture the charged scalar field around the black hole. The picture does not carry over to the Minkowksi background. Recall that the AdS/CFT correspondence has been invoked to describe heavy ion collisions and condensed matter physics, and even the quantum Hall effect has had a dual gravitational description given. The aim at present is to find the gravitational dual picture of superconductivity. The aim is to find a black hole solution that at some point grows hair. To read more about this see the preprint here.

The afternoon was filled with discussion and imported Coca-Cola.

The evening with discussion and beer.

Your humble correspondent flagellates himself gently for missing the talks.

Day Two of Eurostrings 2008

2008-07-01T12:11:00.012Z

Another day, another cup of soup and a sandwich for lunch. Today it was ham soup and a pineapple sandwich (my Dutch and my taste buds are not good enough to understand what the other ingredients were).

This morning we had a review lecture on the pure spinor formalism by Nathan Berkovits. If you want to learn this formalism, why not start with the reviews here (and here [or the blog article here]) and then end with the paper here. If you do this in one-and-a-half hours, but ensure you explain it to yourself very clearly, you will have your own simulation of this morning's nice review. Or, if you are feeling little tired, you could watch the video of Yaron Oz's l ectures to the CERN winter school.

Following Berkovits, Andreas Gustavsson, the third man of the present Bagger-Lambert multiple membranes revolution, spoke on..."Multiple M2's". He included his paper from last year and his more recent work on how the membrane triple product identity aids amplitude calculations. His talk was followed by thirty minutes from Frederik Denef, talking under the title of "the string landscape of quantum critical superconductors", which refers to work in progress with Sean Hartnoll. The central theme was that there are two landscapes in physics. The string theory landscape, constructed inside a unique fundamental theory (M-theory), with low energy excitations (gravitons, "3-formons" :) and superpartners) and where the intricate landscape is considered "party-spoiling". The second landscape is the condensed matter landscape, constructed from a unique theory (the standard model), with low energy excitations (neutrons, protons and electrons) and where the landscpe is still intricate but is useful. The heuristic message is that these two landscapes may be very similar. Denef gave us a toy model two dimensional array of spin one-half particles that illustrated the idea of quantum critical points - points in phase space where a second order phase transition occurs at zero temperature. The crucial features are all summed up in his graph:
A second example of criticality involved superconductors and whose features were given by a toy-modelin two dimensions: a Bose-Hubbard model. There is a phase transition between being an insulator and being a superconductor. This picture was to be compared with a charged scalar field in a Reisner-Nordstrom AdS background. The idea (due to Gubser) was that there is a quantum critical point here too that separates insulation from superconductivity. Namely when electrostatic repulsion of the charged scalar is larger than its gravitational attraction towards the singularity in the space-time, then a halo or cloud of charge forms around the black-hole. This is the superconducting picture. Otherwise the charge falls into the horizon and we have the insulating picture. We are to expect to hear more about this superconducting phase from Gary Horowitz tomorrow. Denef told us one could be optimistic that this picture could be constructed in string theory. Citing the "Gravity=Weakest force" paper of Arkani-Hamed, Motl, Nicolis and Vafa, Denef said that Reissner-Nordstrom black-holes should be able to decay and so there was an expectation that the electrostatic repulsion > gravitational attraction regime should exist. Perhaps microscopic physics and macroscopic physics are not so different after all?

In the last morning talk, Giulio Bonelli spoke under the title "On gauge/string correspondence and mirror symmetry" and you can read his preprint here. In the afternoon we heard an exuberant Vijay Balasubramanian talk about getting something from nothing. His title was "Statistical predictions from anarchic field theory landscapes". Out of chaos certain coarse-grained properties could become predictable he said, read more in the preprint. The final thirty minute talk of the day was given by Diederik Roest, who talked on my favourite subject: "The Kac-Moody algebras of supergravity". The talk covered decomposition of the algebra, the correspondence between de-forms, top forms and E(11) (preprint) and also his work with Axel Kleinschmidt on identifying the Kac-Moody algebras that are appropriate to three dimensional scalar theories with a quarter or less of the full supersymmetry (preprint). After coffee, we had a gong show for some researchers but unfortunately we had no gong. Poor Pierre Vanhove must have been kicking himself that he hadn't packed his legendary cowbell...

On my walk back home I encountered two mathematical omens in odd places, first a van that seemed like it could go to infinity and beyond:
And, second, I saw the hotel I should have been staying at:
Unfortunately there were no giraffes helping zebras to escape the circus... despite this bizarre story I'm not sure that truth is stranger than fiction. In fiction the same story could have happened but the giraffe might have been smoking a cuban cigar and saying that he loved it when a plan came together and all the while Pierre Vanhove skipping in front leading the animals with the merry din of his cowbell.

Eurostrings 2008

2008-06-30T20:41:00.008Z

The tram door closed viciously on Pierre Vanhove's rucksack and off it tootled away from Centraal Station (the tram, not the rucksack). My travelling companions were all on board, and only I was left behind with the rest of the amputated tram queue. The man behind me in the queue said "welcome to Amsterdam" in friendly English. We struck up a conversation and he asked what kind of conference I was attending. I told him it was physics, "serious" was his reply. I asked him what he recommended visiting while I was in the city, he said that for him it was all about wandering around and taking it all in. I pushed him and asked for one thing to see in particular, "the red light district". Or perhaps the upstairs floor of a cafe with a particularly good view over a canal that the tourists lack the energy to investigate. Even though my trip to Amsterdam was only beginning I wondered if this mixture of sites might not give a good impression of Amsterdam. From my walks today I am not disappointed. It is a beautiful city, the colourful boats that crowd the canal are laden with multicoloured bric-a-brac, the buildings on the banks appear disordered like the teeth of a friendly giant and yet each and every one appears spic and span upon inspection, even the cyclists speeding unstoppably down neat cycle paths carry their loved ones side-saddle on the back - a jumble of colourful clothes flying behind in the sunlight. For every ordered thing here there is a controlled disorder that is very pleasant to watch.

I am here for Eurostrings 2008, a smaller, quieter version of Strings, but which is packed with excellent speakers and an interesting crowd of participants. You're a String (thanks Per!) is being hosted this year by the University of Amsterdam, apparently it's in the same venue as Strings 1997. The organisation has been superb and we have had an excellent first day of talks which I will try and summarise here (and maybe expand upon later).

We began the day hearing Ashoke Sen talking about dyons in N=4 as discussed in his recent papers here and here. He described to us the protected index associated to BPS states, labelled d(Q,P). Here Q is the electric charge and P the magnetic charge. It is the number of BPS states weighted by , where is the helicity. That d(Q,P) is protected means that it does not change under a continuous variation of the coupling constant or moduli of the theory. In fact if the coupling constant is varied onl the BPS states remain and contribute to the counting. However d(Q,P) can make sudden jumps over "walls of marginal stability" - these are places where the BPS states may decay into BPS states. The domain wall itself is defined by four parameters which become discrete due to charge quantisation. Consequently d(Q,P) appears to depend not only on Q and P but also on the domain in which the moduli lie. One can calculate the partition function from d(Q,P) expressed as a function of T-duality invariant terms: , a discrete T-duality invariant and also the domain in which the protected index is calculated. It transpires that the partition function converges after analytic continuation of some of the variables but in "all known examples" the partition function ends up being invariant of the domain one started calculating in. What can one say about how the microscopic dyon partition function reproduces the macroscopic black hole entropy count? Well, first, within the domain of applicabilit of the partition function the entropy calculation is in agreement with the inclusion of the four derivative Gauss-Bonnet terms. So far, so good. But what about the phenomenon of discrete value changes in d(Q,P) as one jumps over domain walls? For the single black hole this microscopic property cannot be reproduced macroscopically, but for the multicentre black holes it agrees perfectly - one can see this is possible since for different values of moduli space multicentred balck holes may cease to exist as one crosses walls of marginal stability. At the end of his talk Sen focussed on how one could work towards a complete comparison between and (since a number of terms had been exponentially suppressed in the earlier comparison in order to compare like-for-like). The full picture would include both higher derivative corrections and quantum corrections, for the former one can use Wald's formula to make the calculations, for the latter Sen proposes a close scrutiny of duality. Starting with the near horizon geometry of a black hole and then analytcally continuing to the Euclidean solution one finds the metric. The partition function in this metric is the exponential of minus this Euclidean action, and is used together with a cut off to obtain:

By the AdS/CFT correspondence one can exactly calculate the partition function for the CFT:

Where is a rescaled ground state energy. Now the two expressions may be equated and the black hole entropy examined.

Ionnis Papadimitriou then spoke to us about how to rigourously define an asymptotically flat spacetime and then considered its holographic description - you can read more about this here. Pierre Vanhove, minus his infamous cow bell, spoke next on the no-triangle hypothesis (update: why not read Lubos' analysis of the situation) for SuGra, which is, of course, just squared - or at least it has many remarkable similarities to make such a conjecture plausible. It turns out that the no triangle hypothesis should really be called the no-triangles, no bubbles and, in fact, just boxes in the one loop scattering amplitudes hypothesis - but that's not very catchy. For multiloop scattering diagrams, the no-triangle hypothesis informs us about the one-loop sub-terms that remain when one makes suitable cuts in the multiloop diagram. Pierre told us, without once ringing any kind of bell, not for cow, horse, nor wild mountain goat, that since the cancelltations in the gravity theory are due to the (colourless) gauge invariance the hypothesis can also be applied to other theories with less SuSy than . Pierre finished enigmatically by telling the audience that if is divergent he bets that it diverges at 9-loops. He didn't say how much he bets.

In the afternoon, following a sparse lunch of soup and a sandwich, Hirosi Ooguri talked under the title of Current Gauge Correlators for General Gauge Mediation - the idea was to extend the region of strong interactions from just the hidden sector to include the mediating sector that gives rise to the visible sector. You can read his paper with his collaborators on this subject here. After Ooguri, Marco Zagermann told us that is the pillow, and invited us to revisit D3/D7 brane inflation models. The inflaton is the separation distance between a D7 with flux turned on and a parallel D3. At the end of the period of inflation, cosmic strings condensed - the associated preprint is available here. Finally Ki-Myeong Lee talked about "New" Superconformal Chern-Simons Theories. Since this is work related to the increasingly popular multiple M2 work of Bagger-Lambert and Gustavsson, Lee told us that he had checked and he thought his models were still new at the time of talking and they would be published on the arxiv tomorrow (1st July, 2008 - the preprint can be found here). Lee showed us how to introduce a twisted hypermultiplet into Gaiotto-Witten theory in order to reproduce the 8 scalars of the Bagger-Lambert work. Hey presto, a new technique for building interesting theories was born. The last talk of the day was given by Niko Jokela from Helsinki on the interesting topic of N-Point Functions in the Rolling Tachyon Background, the arxiv preprint is here.

At the end of the day we had a reception hosted at the Academy of Arts and Sciences, of which Robert Dijkgraaf is the President. He told us that the academy was actually seven years older than the Netherlands and told a story of his predecessor who was approached by Vladimir Putin at a formal dinner and was greeted with the line "so you are a President too", Dijkgraaf's predecessor replied that "they came in all shapes and sizes".

After the reception I spent a nice hour wandering around Amsterdam in the sun.

So it goes...

2008-05-30T19:28:00.006Z

Well, that's how long a year-a-half is. What did I miss? The E8 genome was mapped with the fanfare of a press conference, the string wars continued, N=8 supergravity may be ultraviolet finite (also see the argument via string theory properties) if a no triangle hypothesis holds, funding was decimated/bleak/downsized, unparticles were cool, low dimension supergravity was maximally gauged here, here and here, the multiple M2 brane revolution began, an exceptionally simple theory of everything was the most popular paper on the arxiv, BeckerBeckerSchwarz appeared, Fields medallists blogged en masse (Tao, Connes, Borcherds), Mike Duff discussed string theory with Lee Smolin, Sidney Coleman gave lectures from the past, Stephen Hawking's acquired his own universe, the birth of string theory was chronicled, John Schwarz also shared his memories of early days, Murray Gell-Mann was videoed talking about beauty in physics, a Feynman video biography surfaced... and so on, and so on without end. Nothing stops while you're away.

Julius Wess, John A. Wheeler, Sidney Coleman, Jurgen Ehlers have all sadly passed away.

The fantasy author Robert Jordan died and won't complete his Wheel of Time epic and Kurt Vonnegut left his own world-line. So it goes...

So that's a year-and-a-half.

Large Volume Scenarios

2007-04-18T13:12:00.001Z

How does one hope to get most of the standard model out of string theory? Yesterday, Marcus Berg from the Albert Einstein Institute in Potsdam, gave a talk reviewing two scenarios, known by their abbreviations KKLT (for Kachrou, Kallosh, Linde, Trivedi) and LVS (for Large Volume Scenario) . His preference was for the LVS and his arguments were motivated mostly by the practicality of stabalising cycles inside the compact space.

In summary, The KKLT scenario is a particular set-up of IIB superstring theory, and was proposed as a way to obtain de Sitter vacua from string theory. This was progressive since it was already known that simply the lowest order terms of Sugra were not enough to construct a de Sitter background. KKLT showed that higher order string theory terms could give rise to such a background. In their model the compactified 6 (real) dimensional Calabi-Yau possesses "Klebanov-Strassler" throats and fluxes F_3, H_3 which are non-parallel. The fluxes stabilise the compact space and the addition of anti-D3 branes and associated instantons permits supersymmetry breaking and the emergence of a Minkowski or de-Sitter background. Due to the belief in a small positive cosmological constant the de-Sitter background existing in string theory is highly desirable.

The fluxes in KKLT are used to stabilise the dilaton and hence the string coupling constant, and to stabilise the moduli of the Calabi-Yau space. Marcus Berg gave us a review of how one goes about stabilising such moduli. One begins with ten-dimensional fluxes and a ten-dimensional metric, then upon reduction the part of the kinetic term that exists in the internal space gives rise to potential term in the Lagrangian associated with the moduli appearing through the reduction of the metric. To stabilise the moduli one can minimise the potential term and then find the moduli. The trouble with this, according to Marcus Berg, is that the potential is generically composed of exponentially suppressed terms and linear terms, so that the minimum of the potential occurs only over a very small range of parameters. What's the problem with that? Well simply maybe it is wrongheaded, the correct interpretation may be that a tree-level term is being cancelled against a non-perturbative term in the potential.

Gabriele Veneziano's Personal Recollection

2007-02-23T11:15:00.001Z

Did you know that string theory began in Pisa? So said Gabriele Veneziano in the first talk of his special course on string theory here in Pisa, entitled "The Birth of String Theory: A Personal Recollection". Perhaps he was indulging the audience. Perhaps. He told us that as a graduate student from the University of Florence he came in the Spring of 1966 to listen Sergio Fubini's talks in Pisa and really this was the beginning of the path towards a theory of strings. Of course, at that time string theory was not being proposed as a theory containing gravity but rather as a theory of the strong interactions. Veneziano described working on current algebras and superconvergence relations at the Weizmann Institute in 1966-67 and listening to Murray Gell-Mann's talk at Erice in July 1967 as being important in leading him to a dual resonance model.

From Gell-Mann, he learned about Geooffrey Chew's "expensive" bootstrap method for calculating scattering amplitudes. Chew' s bootstrap he said related objects of different types; baryons in the s-channel became mesons in the t-channel. And the rest of the story as they say is history.

JHEP Editorial Plea

2006-12-05T17:11:00.000Z

In my email today, a gentle call to scientists to support JHEP. One must wonder if JHEP is in trouble. Let's hope not. Read on:

Dear Colleague,

In the first half of 2006 our Journals have seen many important changes: a new instrumentation journal, JINST, has been launched, new scientific directors for JHEP and JCAP have been appointed to replace Hector Rubinstein, now Scientific Advisor to SISSA Medialab. We wish to remind you of the basic differences between our not-for-profit Journals and those published by commercial publishing companies.

The policy of the SISSA-IOP J-Journals is the following:
- to maintain the philosophy that publication of research results must be fully controlled by scientists, so as to ensure the highest scientific quality;
- to produce information efficiently at a reasonable cost, thereby minimizing the financial pressure on our libraries and grants.

We are convinced that it is unfair that publishing companies make huge profits exploiting the ingenuousness of scientists in the questions related with the publication of their own results in Scientific Journals. Although scientists voluntarily carry out all the publication-related work (starting with the actual writing of the paper to the peer-review), they are still requested to pay unwarranted
and outrageous subscription fees by commercial publishing companies for them to access these very journals as readers.

Here are some examples. The yearly subscription cost of our journals, which covers only necessary expenses unavoidably related with publication and marketing of all published scientific contributions, are the following:

JHEP: EUR 1,622
JCAP: EUR 1,174
JINST (free in 2006): 745 in 2007
(all institutional prices)

The sum of the subscriptions to Nuclear Physics B and Physics Letters B is more than fifteen times higher than that of JHEP (to which the combined NPB + PLB can be compared), i.e., 15,211 EUR (Institutional price) plus 10,301 EUR (Institutional price) = 25,512 EUR. In Instrumentation, JINST's main competitor, Nuclear Instruments and Methods A, charges as an annual subscription fee 12,191 EUR (Institutional price).

Exploiting this strategy, commercial publishing companies have managed to generate profits of the order of one billion euros a year(Elsevier), which are ultimately taken from research resources.

Besides being run and published entirely by electronic means, the other key features of our journals are:

1. The Editor-in-Charge is given full responsibility for acceptance or rejection of the paper. His word is final and cannot be questioned by the Editorial Office (on the other hand, authors can appeal against editorial decisions). This has proved to be very efficient in selecting papers of very high quality and consequently Thompson ISI's impact factors for JCAP and JHEP are amongst the highest in physics
(JINST started publication this year so it is not rated yet). Please see the data appended below.

2. Large companies misuse the copyright assignment, forbidding authors to use their own material when they need to do so, e.g., for publishing collected reprints. They have done it in the past, based on non-scientific considerations. We do not. Indeed unlike those of commercial publishers our policies are never in conflict with scientific interests because science is our only concern.

We very much rely on your support and we would appreciate it if you could contribute by conveying to colleagues the information above and encouraging those who have not yet done so to submit their results to our journals.

We do believe that there should not be any monopoly of publication. The existence of several journals (hopefully in the future all not-for-profit enterprises), protects the author against the possibility that if a mistake is made the paper cannot be
published. Furthermore, we see no reason why large companies involved in media, newspapers and other matters should have such control of scientific research to which they contribute nothing.

Is it up to all of us, and up to you as an author in particular, to stop this unacceptable state of affairs.

Sincerely yours,

Marc Henneaux - Scientific Director
Hector Rubinstein - Scientific Advisor

IF data

(We are fully aware that Impact Factors are far from being absolute measures of quality and can be, for instance, influenced by fashion effects. IFs give only a partial indication. The data below are thus to be taken with a grain of salt)

Journal IF 2003 IF 2004 IF 2005

JHEP 6.854 6.503 5.944
Physical Review D 4.358 5.156 4.852
Nuclear Physiscs B 5.409 5.819 5.522
Physics Letter B 4.298 4.619 5.301
Euro Phy J C 6.162 3.209

JCAP 7.914 6.793
A & A 3.781 3.694 4.223
Class and Quant Grav 2.107 2.941 2.938
Astrophysical Journal 6.187 6.237 6.308
inter J Mod Phys D 1.507 1.500 1.225

Lisa Randall Online!

2006-11-08T20:26:00.000Z

Just a short note to let you know that Lisa Randall, will be online tomorrow (Thursday 8th Novermber, 2006) for an open discussion about physics, strings, Warped Passages and how to create your own universe (presumably). The event is being run by Discover magazine, and to whet your appetite you can read an interview with Lisa from Discover earlier this year here. I don't know exactly what you have to do to be involved but presumably turn up at Discover magazine's site from 2pm until 3pm (in the reference frame of the eastern shore of the US) be dressed in your finest surfing gear (the web kind, anything else would be surreal wouldn't it?), bring a question and a bottle. Why not?

Thanks to Coco Ballantyne for the head's up.

A Penrose Universe

2006-09-12T19:54:00.000Z

First, there have been a number of introductory posts on the Barrett-Connes standard model spectral triple over at the n-category cafe, in particular see posts I, II, III and IV. Second John Barrett has been talking about his approach to finding the appropriate spectral triple in Cambridge yesterday, his paper, "A Lorentzian version of the non-commutative geometry of the standard model of particle physics" appeared on the arxiv on the same day as Connes' and suggested identical alterations, and while I have been feverishly attacking my thesis, Alejandro Rivero attended the talk and has made some comments about it on physics forums. He also has uploaded his notes from the talk, but these are a little hard to read. Also the Newton Institute have audio of all the talks from last week's workshop for you to enjoy here.

On a different note, a while ago I attended the Winter School on the Attractor Mechanism in Frascati but was unable to write up much about it due to being very busy. Well Per Kraus, who gave a set of talks at the school has helped me out by publishing his lecture notes on the archive as "Lectures on black holes and the AdS3/ CFT2 correspondence". Thank-you Per!

You have probably heard of Penrose tilings, Penrose diagrams, Penrose limits, Penrose triangles and even a Penrose staircase, well last week, during Roger Penrose's talk at the Noncommutative Geometry Workshop, I heard a little about a Penrose Universe. (In my picture it looks like Roger Penrose is keeping the audience entertained with his shadow puppet routine)It is a peculiar thing and represents Penrose's approach to understanding the second law of thermodynamics, that entropy increases, on a cosmological scale. Penrose points out that there is a contradiction in the entropy increase picture of the big bang, which is that the background radiation matches a model that is in thermal equilibrium. Such a state is just about the highest entropy state you can imagine for a system. Almost any other distribution besides uniform would result in a smaller entropy. The Big Bang picture requires the restriction of phase space at early time and the consequence that entropy ought to be tiny at the Big Bang. Penrose wonders how to resolve this contradiction, and seeks to resolve it by separating the entropy of the universe into that arising from matter (the eneergy momentum tensor) and that encoded in gravitational degrees of freedom (the Weyl curvature, see below).

In a nutshell it is a universe without a big crunch, and if taken to the extreme limit without a big bang, but which gives the features of energy density fluctuations in the background radiation. Penrose suggested his Weyl curvature hypothesis in 1979 as a physical origin of the increasing entropy of the universe with time. The Weyl curvature tensor is the traceless part of the Riemann curvature, i.e. the parts which when contracted upon two indices give zero for the Ricci (two-form) tensor. To quote Penrose's description,

"In Einstein’s theory the Ricci curvature R_{ab} is directly determined by the gravitational sources, via the energy-momentum tensor of matter (analogue of the charge-current vector J_{a} in Maxwell’s electromagnetic theory) and the remaining part of the space-time Riemann curvature, namely the Weyl curvature C_{abcd}, describes gravitational degrees of freedom (analogue of the field tensor F_{ab} of Maxwell’s theory)."

The Weyl curvature hypothesis is that the Weyl curvature is zero at the big bang but rises gradually as the universe ages. Consequently the Weyl curvature will not be zero at black hole singularities and we may use the Weyl curvature in this picture to distinguish between cosmological singularities and other singularities. As time passes, the Weyl curvature increases and gravitational masses attract each other more strongly forming a less-homogeneous universe, with clumped masses and higher entropy encoded in the dense packing massive bodies. So that early uniform universe may be explained by there being zero Weyl curvature. Penrose talks about the Weyl curvature's growth as freeing up gravitational degrees of freedom that may then be excited. It is the excitation of these gravitaional degrees of freedom that is the real measure of entropy. It is a nice picture. But just what drives the Weyl curvature's variance is a mystery to me. It does allow us to describe gravitational entropy increase with a tensor field, and of course to associate the arrow of time with such a field. So, at least, algebraically it is appealing. It also offers an alternative to a fast period of inflation in the early universe, which some might find equally as arbitrary as a varying curvature field.

The latest idea is built upon the findings of Paul Tod in his paper "Isotropic cosmological singularities: other matter models", where it is shown that even though the Ricci curvature blows up at the cosmological singularity the Weyl curvature remains finite. Penrose takes this finding and argues that near the big bang gravity becomes a conformal theory, so that he may rescale the metric to infinity and blow up the big bang singularity. The justification for this is that near the big bang, when temperatures are extremely high, there is little difference between the dynamics of massive and massless particles, all particles are treated as massless, and respect conformal equations of motion. Once the description of physics is conformally invariant, Penrose says that a sense of time is lost, tying in neatly with the low entropy ideas. Having blown-up the cosmological singularity, and beleiving that the Weyl curvature remains finite, has lead Penrose to ponder the smooth continuation of the Weyl curvature at the boundary. Perhaps, he suggests, in what he refers to as his "outrageous" proposal, there is a "conformal cyclic cosmology", in which one may knit the conformal geometry at the big bang to another conformal geometry prior to the big crunch, and thereby create a series of universes with a long-lived/eternal conformal geometry.

How can Penrose convince us that geometry may become conformal again at the end of the universe's lifetime? Well, he says, after most of the matter in the universe has been swallowed by black holes and has then been recycled back into the universe via massless Hawking radiation we are really only troubled by charged matter that escaped this process. Here we must presume that black holes can radiate away to pure radiation (which seems unlikely - no topology change, no unexcited microstates...) leaving a universe that may contain some unabsorbed charged matter (let's call all matter electrons) and photons. Now if we can come up with some way of doing away with the electrons, says Penrose, then we will be in business. For again without any massive particles left in the universe the scale of the metric has lost its meaning. This is the real weak point, since the mechanisms to get rid of electrons require either allowing their charge or their mass to dissipate over long time scales. But, of course, this may be possible. Once this position is arrived at one might imagine a conformal rescaling of the metric down to zero, so that a future infinite region is made finite and may be attached to the finite cosmological singularity of some other universe. Penrose argues that the appropriate conformally invariant verion of general relativity the spin-2 field picks up an inverse conformal factor when the conformal tranformation is applied to the metrc, while the Weyl curvature does not. Hence the matter density from the previous universe survives the conformal rescaling and passes over into the next universe. Penrose identifies this with the density fluctuations at the Big Bang - which is exceedingly appealing, and presumably testable.

The conformal rescaling marks the beginning of the "new" universe. In this picture there is also cosmological scale clock, whose ticks are the rescalings of the metric, so nothing to worry about on a local level. It is also imperitive that the conformal rescalings occur in the right way, i.e. to infinity at big bang singularity and to zero at late time. Effectively one must imagine that the previous universe occured at miniture scale comparatively, and the future universe will be built upon the swirling dust of ours at a gigantic scale. It doubles as a very nice picture for a science-fiction novel, as well as an exceedingly interesting proposal for the origin of the density fluctuations in the universe.

We have mentioned the assumptions, namely that black holes evaporate to pure radiation and that electron charge/mass dissipates. There are also questions about particle antiparticle pair creation, but which if we are able to argue in favour of some long term alteration of the properties of the electron, so that it eventually becomes pure radiation, this would not present a problem. Furthermore there seem to be mysterious forces driving the rescaling of the metric, for which it would seem some additional dilaton field may be necessary or some other argument presented.

You can hear Penrose talk on this in two places on the web, both of which took place at the Newton Institute. The first is from November 2005 at the Spitalfield's Day and the second occurred last
week (you will have to wait until the end to hear about this cosmological model). Penrose also has written up his description of wha he refers to as "conformal cyclic cosmology" in the proceedings of the EPAC, 2006, conference, and one can read the pdf here.

On Thursday of last week we also suffered a panel discussion on the nature of space-time, being organised by the sponsors the notorious Templeton foundation, I was a little wary. I think on the whole the event worked very well, it was simply not to my personal taste, but I went along to enjoy the views of (Rev. Dr.)John Polkinghorne(Eclesiastical physicist), (Prof.) Shahn Majid, (Rev. Dr.) Michael Heller (of the Vatican observatory), (Sir) Roger Penrose and (Prof.) Alain Connes. As you might imagine there was a very strong representation of the religious apprecatiation of spacetime, and even Alain Connes couldn't resist talking of his interpretation of three pages of "ancient text" by which he meant the Veltman Lagrangian of the standard model. I do not think it was a night of much scientific progress. But there were some anecdotal highlights. The evening was organised so that each panelist took five minutes to mention their conception of spacetime, there was an overhead projector and it appeared that the speakers were well organised having prepared detailed slides. Throughout the first two talks Alain Connes looked a little preoccupied, occasionally staring at the desk and sometmes laying his head upon it. Suddenly after the second speaker Connes sprang to life, borrowed some OHP slides and multicoloured pens from the others, and began to prepare his own slides there and then. Since the chairman was supposedly inviting the presentations at random this seemed a wonderfully carefree approach. It made for some nice theatre. We also heard an anecdote from Roger Penrose, in response to the first question from the audience which was along the lines of 'which came first quantum mechanics or general relativity?'. Penrose replied by telling of a time he had listened to a wonderfully animated lecture by John Wheeler and at the end there came a similar question from the audience, which came first G.R. or the quantum principle? Penrose said that a small voice in the front of the audience piped up and asked 'what is the quantum principle?' The small voice belonged to Dirac. A final amusing interchange involved Shahn Majid, the chairman (Jeremy Butterfield) and a mischievous Alain Connes. Shahn Majid was summing up his disenchantment with the present understanding of spacetime with the Shakespearean line "there is something rotten in the state of Denmark" (Penrose said later he thought Majid was referring to the Copenhagen interpretation), and Alain Connes responded with another Shakespeare quotation "Throw physics to the dogs; I'll none of it." It was left to Jeremy Butterfield to point out that the actual line from Macbeth is about "physic" (referring to medecine) and not "physics". So there was some enjoyment to be had from the evening after all.

To Commute or not to Commute...

2006-09-04T23:24:00.000Z

Sorry for the lack of posting this summer, but I have been trying to write up my thesis. In fact I still am trying, and for no sensible reason I am now doing this at the Noncommutative Geometry Workshop at The Isaac Newton Institute for Mathematical Sciences in Cambridge. The institute is a wonderful place, although I haven't looked around much I have already heard about the lawn on the roof (where you can sometimes see someone mowing, which must look very peculiar from the road) and seen the bust of Paul Dirac in the foyer. But I have been most impressed by the blackboard which is mounted in the lavatory, should you have a maths dispute in the bathroom - it is by far the geekiest thing I have ever seen. It is wonderful and then horrifying and finally wonderful again.

It is also very nice to be visting Cambridge again, and King's College looked especially pretty today in the sunshine. I refer you to my picture below (see there was sunshine today!)

The programme is available online and today was the first day of five days of talks. There's also a public debate on thursday at 8pm in Queen's Lecture Theatre at Emmanuel College entitled "The Nature of Space and Time: An Evening of Speculation" invoving a panel of Alain Connes, Roger Penrose, Shahn Mahjid, Michael Heller and John Polkinghorne which should be interesting. If you are in Cambridge and want to come along you might benefit from registering at the above link. Or maybe you won't benefit - it is not clear.

The first talk this morning was "The Quest for Non Commutative Field Theory" by Vincent Rivasseau and we heard about noncommutative field theory in review. The talk began with a reminder about why noncommutative geometry is an interesting approach to quantum gravity, it went like this:

Quamtum Mechanics (Non commutativity) + General Relativity (Geometry) = Non commutative geometry.

Noncommutative field theory is the generalisation of well-known quantum field theories such as the phi^4 theory to noncommutative spacetimes. The approach is to upgrade the normal scalar product to the simplest non-commutative product which is known as the Moyal product and denoted by an asterisk. One can read all about this in a review paper from 2001 by Michael Douglas and Nikita Nekrasov called, you guessed it, "Noncommutative Field Theory". Rivasseau described the problems of renormalisations of such a naive upgrade to noncommutative geometry, while the planar Feynman diagrams and their ultra-violet divergence remain renormaliazable the non-planar ones pick up an infra-red divergence. This goes by the name of UV/IR mixing and some more complicated terms are needed before the noncommutative version of the theory can be made renormalizable. See Rivasseau's paper with Gurau, Magnen and Vignes-Tourneret for the detail on the renormalizability of noncommuting phi^4 field theory. We were also introduced to the modifications of the Feynman diagrams resulting from the noncommutative promotion. In the commuting field theory one uses the heat kernel as the propagator, while in noncommutative geometry the Mehler Kernel (which is far more complicated than the heat kernel) is the starting point. Interactions, which we are used to describing by one spacetime point, become dependent upon four and a vertex is promoted to a box, the four points specifying the corners. Rivasseau et al also have a paper entitled "Propagators for Noncommutative Field Theories". The end of the talk was dedicated to the parametric space which is a new approach to noncommuative field theory described by Gurau and Rivasseau in their paper. Since I am trying to get a small understanding of the tools used in noncommutative geometry and the motivations I would like to mention a couple of recurrent topics, whose importance I was unable to understand during the talk. The first is that the quantum hall effect seems to be a very important physical example cited by the noncommutative geometers. The second tool that was apparently of great practical value is the so-called Langmann-Szabo duality, which I think was introduced in their paper "Duality in Scalar Field Theory on Noncommutative Phase Spaces".

At 11.35pm Albert Schwarz began talking to us under the title "Space and Time from Translation Symmetry". The talk followed very closely his paper of the same title. He did not talk about noncommutativity much but gave us an axiomatic description of quantum mechanics as a unital, associative algebra of observables, A, over the complex space. He described translations as acting as automorphisms of the algebra A, and soon generalized the idea of a tranlsation generator to a commutative subalgebra. He said he was not trying to give solutions but rather to formulate problems. Alain Connes was interacting with Schwarz from the front row and at one point Connes asked repeatedly about the observables of string theory, culminating with "...but what are the observables?" To which Schwarz replied "There is no question: 'what is observables?'". It was rather like a Jedi mind trick. Schwarz expressed a strong interest in the notion that all physical numbers should be rational, while anything else is just used for felicity. He advocated using p-adic numbers instead of real numbers and the functioning of this proposal can be read about in his recent papers with Kontsevich, Vologodsky and Shapiro [1 and 2].

After lunch, Samson Shatashvili talked under the title "Higgs bundles, gauge theories and quantum groups" who described his reasons for claiming that the so-called Yang-Mills-Higgs theories are dual to the nonlinear Schrodinger quantum system. The preprint (with A. Gevasinov) that the talk was based on is due to appear overnight at hep-th/0609024, but a fundamental paper in the literature, at almost ten years of age, is "Integrating Over Higgs Branches" by Greg Moore, Nikita Nekrasov and Shatashvili. At the end of his talk Shatashvili made the point that as far as he could tell his dual theories contained all the information required for geometric Langlands duality (although he also claimed to not know what geometric Langlands is) and both regimes of the duality are reasonably well understood. But I think we'll have to wait for the preprint...

Today's final talk was a big one. The speaker was the wonderful Alain Connes and he was talking about his recent short paper describing a theory of everything. Lubos Motl has commented extensively on this preprint which you can read by boosting to his Reference Frame. You can also read Alain Connes explanation of himself in the preprint, "Noncommutative Geometry and the Standard Model with Neutrino Mixing" but it will take a lot of work if you are of a more physical than mathematical constitution. Connes described his aim to encode the gravitational and the standard model Lagrangian in a purely geometric picture. The essence of the approach is not to use the metric to define the square of the line element, but rather to start with the line element, and not its square, by using the Dirac operator, D. In fact ds = 1/D. This approach was used to construct the standard model via the spectral action principle in work with Ali Chamseddine (see [1,2]. However the resulting theory was not able to match the standard model perfectly, it exhibited fermion doubling (as pointed out in the work of Lizzi, Mangano, Miele and Sparano) and the introduction of right-handed neutrinos caused Poincare duality to be violated. In his latest work Connes has fixed the problems and reproduced the standard model Lagrangian. This is no mean feat, at the beginning of his talk Connes bamboozled the audience by displaying the enormous Lagrangian of the standard model as written down by Veltman. It filled one page of A4 (single-spaced) and no-one in the audience could read it clearly. In Connes latest work one takes what is called the "finite space", F, of the standard model algebra which is 90-dimensional corresponding to 45 particles and 45 antiparticles. One then writes down the spectral action which has two terms, one for bosons and one for fermions, and one feeds in the spectral dimension....wait! What's the spectral dimension??? Well apparently this is the sequence of positive integers bounded above, and specified, by the metric dimension - and the metric dimension is our usual notion of dimension. There is also another type of dimension called the KO-dimension coming from K-theory, which I do not claim to understand, but Connes' fix of his theory involves allowing the metric dimension to take different values to the KO-dimension. In particular the conjugation properties of the relevant spinors and the necessity of removing his double fermions leads to picking the KO-dimension of the required space F, which Lubos has taken to calling the Connes manifold, to be 6mod8. From our experience of spacetime the metric dimension is 4 and in total the dimension of MxF becomes 10mod8 - which are dimensions that are exceedingly familiar from string theory. Connes strongly denied suggestions that his finite space F was anything like a Calabi-Yau manifold, but said that if someone showed that it was, then he would applaud. Having made these changes to the spectral dimension data that is fed into the spectral action formulation, Connes told us that he expanded out the explicit action and exactly reproduced the enormous Veltman Lagrangian. Due to the compactness of the notation this is an extremely elegent construction of the standard model, and while it may not answer the questions about why certain data are fed in, it is certainly a remarkable discovery. No doubt there is more to be uncovered along these lines. Connes told us that the preprint on the archive is a short version of a much more detailed paper to appear later on, again with Chamseddine. At the end of the talk Connes told the audience that the finiteness of the space F is really tantamount to there existing a basic unit of length, and it was revealed during the questions that it was really the Euclidean version of the standard model that had been constructed. Nevertheless the compact notation makes this approach worth some study.

It is clear to me after today that I wouldn't win the Krypton Factor challenge for observation: I have been surrounded by the words "noncommutative", "non commutative" and "non-commutative" and I still haven't worked out which is the officially endorsed spelling (see my non-renormalized spellings in the text). To hyphen or not to hyphen...that is really the question?

Physics: More damaging than drugs?

2006-08-05T23:24:00.000Z

I just had this advice entitled unequivocally "Don't Become a Scientist!" taken from Jonathan I. Katz's website forwarded to me by a friend (who quit physics to work in the financial sector) - I wouldn't have thought it to be of general interest, but apparently it is interesting enough to become a forwarded email in certain circles. It is, of course, of interest here, where all advice to young researchers from one's elders is welcome, no matter how terrifying :( - it has been discussed elsewhere over a year-and-a-half ago by Stephen Hsu (part 1 and part 2) at his blog, Information Processing, and also at Sean Carroll's former blog incarnation, Preposterous Universe, (see 5th January, 2005) and there were some comments on it by the Quantum Pontiff as well, but perhaps, like me, you missed this before:

"Are you thinking of becoming a scientist? Do you want to uncover the mysteries of nature, perform experiments or carry out calculations to learn how the world works? Forget it!

Science is fun and exciting. The thrill of discovery is unique. If you are smart, ambitious and hard working you should major in science as an undergraduate. But that is as far as you should take it. After graduation, you will have to deal with the real world. That means that you should not even consider going to graduate school in science. Do something else instead: medical school, law school, computers or engineering, or something else which appeals to you.

Why am I (a tenured professor of physics) trying to discourage you from following a career path which was successful for me? Because times have changed (I received my Ph.D. in 1973, and tenure in 1976). American science no longer offers a reasonable career path. If you go to graduate school in science it is in the expectation of spending your working life doing scientific research, using your ingenuity and curiosity to solve important and interesting problems. You will almost certainly be disappointed, probably when it is too late to choose another career.

American universities train roughly twice as many Ph.D.s as there are jobs for them. When something, or someone, is a glut on the market, the price drops. In the case of Ph.D. scientists, the reduction in price takes the form of many years spent in ``holding pattern'' postdoctoral jobs. Permanent jobs don't pay much less than they used to, but instead of obtaining a real job two years after the Ph.D. (as was typical 25 years ago) most young scientists spend five, ten, or more years as postdocs. They have no prospect of permanent employment and often must obtain a new postdoctoral position and move every two years. For many more details consult the Young Scientists' Network or read the account in the May, 2001 issue of the Washington Monthly.

As examples, consider two of the leading candidates for a recent Assistant Professorship in my department. One was 37, ten years out of graduate school (he didn't get the job). The leading candidate, whom everyone thinks is brilliant, was 35, seven years out of graduate school. Only then was he offered his first permanent job (that's not tenure, just the possibility of it six years later, and a step off the treadmill of looking for a new job every two years). The latest example is a 39 year old candidate for another Assistant Professorship; he has published 35 papers. In contrast, a doctor typically enters private practice at 29, a lawyer at 25 and makes partner at 31, and a computer scientist with a Ph.D. has a very good job at 27 (computer science and engineering are the few fields in which industrial demand makes it sensible to get a Ph.D.). Anyone with the intelligence, ambition and willingness to work hard to succeed in science can also succeed in any of these other professions.

Typical postdoctoral salaries begin at $27,000 annually in the biological sciences and about $35,000 in the physical sciences (graduate student stipends are less than half these figures). Can you support a family on that income? It suffices for a young couple in a small apartment, though I know of one physicist whose wife left him because she was tired of repeatedly moving with little prospect of settling down. When you are in your thirties you will need more: a house in a good school district and all the other necessities of ordinary middle class life. Science is a profession, not a religious vocation, and does not justify an oath of poverty or celibacy.

Of course, you don't go into science to get rich. So you choose not to go to medical or law school, even though a doctor or lawyer typically earns two to three times as much as a scientist (one lucky enough to have a good senior-level job). I made that choice too. I became a scientist in order to have the freedom to work on problems which interest me. But you probably won't get that freedom. As a postdoc you will work on someone else's ideas, and may be treated as a technician rather than as an independent collaborator. Eventually, you will probably be squeezed out of science entirely. You can get a fine job as a computer programmer, but why not do this at 22, rather than putting up with a decade of misery in the scientific job market first? The longer you spend in science the harder you will find it to leave, and the less attractive you will be to prospective employers in other fields.

Perhaps you are so talented that you can beat the postdoc trap; some university (there are hardly any industrial jobs in the physical sciences) will be so impressed with you that you will be hired into a tenure track position two years out of graduate school. Maybe. But the general cheapening of scientific labor means that even the most talented stay on the postdoctoral treadmill for a very long time; consider the job candidates described above. And many who appear to be very talented, with grades and recommendations to match, later find that the competition of research is more difficult, or at least different, and that they must struggle with the rest.

Suppose you do eventually obtain a permanent job, perhaps a tenured professorship. The struggle for a job is now replaced by a struggle for grant support, and again there is a glut of scientists. Now you spend your time writing proposals rather than doing research. Worse, because your proposals are judged by your competitors you cannot follow your curiosity, but must spend your effort and talents on anticipating and deflecting criticism rather than on solving the important scientific problems. They're not the same thing: you cannot put your past successes in a proposal, because they are finished work, and your new ideas, however original and clever, are still unproven. It is proverbial that original ideas are the kiss of death for a proposal; because they have not yet been proved to work (after all, that is what you are proposing to do) they can be, and will be, rated poorly. Having achieved the promised land, you find that it is not what you wanted after all.

What can be done? The first thing for any young person (which means anyone who does not have a permanent job in science) to do is to pursue another career. This will spare you the misery of disappointed expectations. Young Americans have generally woken up to the bad prospects and absence of a reasonable middle class career path in science and are deserting it. If you haven't yet, then join them. Leave graduate school to people from India and China, for whom the prospects at home are even worse. I have known more people whose lives have been ruined by getting a Ph.D. in physics than by drugs.

If you are in a position of leadership in science then you should try to persuade the funding agencies to train fewer Ph.D.s. The glut of scientists is entirely the consequence of funding policies (almost all graduate education is paid for by federal grants). The funding agencies are bemoaning the scarcity of young people interested in science when they themselves caused this scarcity by destroying science as a career. They could reverse this situation by matching the number trained to the demand, but they refuse to do so, or even to discuss the problem seriously (for many years the NSF propagated a dishonest prediction of a coming shortage of scientists, and most funding agencies still act as if this were true). The result is that the best young people, who should go into science, sensibly refuse to do so, and the graduate schools are filled with weak American students and with foreigners lured by the American student visa."

The Klein Four (A group)

2006-06-15T19:58:00.000Z

Perhaps The Klein Four have passed you by as well as me. Well fret not. They are an a capella group from the maths department of Northwestern Univeristy and they shot to fame last year with their love song Finite Simple Group of Order Two, which can be watched online:

They have an album out, full of more maths puns than you can shake a ~~stick~~ log at, which you can purchase via their website (where you can see some of their other performances) or even via iTunes.

The Klein four group, or Vierergruppe, is a direct product of two copies of Z_2, and allows us to solve the quartic.

When Art is Not Art

2006-06-15T11:53:00.000Z

Via the BBC, Empty plinth sidelines sculpture a very funny, non-physics story about a sculptor who packaged his work together with a plinth for it to stand upon in a gallery. The Royal Academy of Arts decided that they had received two separate entries into the competition to be exhibited and a panel of judges decided that the plinth was the better work of art and put it on display. Hilarious.

Cargese: The Lectures

2006-06-07T13:17:00.001Z

Well despite the beach life Cargese was a school and there were plenty of interesting lectures. The format for an average day was

0800-0900hrs Breakfast
0900-1030hrs Lecture 1
1030-1100hrs Coffee break
1100-1230hrs Lecture 2
1245-1630hrs Lunch and beach break
1630-1730hrs Lecture 3
1800-1900hrs Lecture 4

Which was very good and not too tiring. Most lecturers were given one morning and one afternoon slot, and frequently this wasn't enough time to bridge the gap between being completely pedagogical and also interesting to the experts in the audience. Let me give a list for posterity of all the talks we heard. Suggeseted preparatory literature for the talks can be found here.

BPS Black Holes by Bernard de Wit
Black Holes, Attractors and Topological Strings by Andrew Strominger
The Standard Model in String Theory from D-branes by A. Uranga
Time dependence and space-like singularities in String theory by M. Berkooz
Strings, Cosmology and Supersymmetry Breaking by S. Kachru
Multitrace deformations of vector and adjoint theories and their holographic duals by Rabinovici

Living in the Theorists' Paradise

2006-05-28T17:34:00.000Z

I find myself surrounded by the very pleasant scenery of Corsica, where I am attending the Cargese Summer School. I am sitting in a computer room, opposite the lecture theatre and there is a gentle mineral fragrance in the air carried by the rain. Fortunately this is the last day of the school and the first day an afternoon trip to the beach has been rained off. That's right: trips to the beach, and theoretical physics. Sometimes it is good to stop and appreciate your fortune.

The Cargese school commenced two weeks ago and covered a number of topics under the heading 'Strings and Branes: The present paradigm for gauge interactions and cosmology'. The school is located 20 minutes from the village of Cargese and is situated on the beach: at least it's a 2 minute walk to the beach from the institute, and views from the rooms on-site overlook a wonderful seascape, cliffs, beach and all. But, I gush... suffice it to say, it really is very nice here, and it is a pleasure to be here.

Not only is it nice it is steeped in physics history. For example, there is a peninsula called the t'Hooft peninsula where t'Hooft is supposed to have sat down and worked through the ideas that led to his Nobel prize on gauge theories and renormalization. I sat on the same peninsula, but I had forgotten my sunblock and had to retreat prior to having any great thoughts. In the garden of the institute is a tree, which is referred to as the wisdom tree, where students gather for discussion (in theory) and where it is said the lecturers have, in the past, climbed up into the branches of the tree to regail the students. Who can say how much truth there is in this. There was, in fact, some confusion as to which tree was indeed the one, true Wisdom Tree. All very worthy of Enid Blyton rather than the high energy physics community. From the garden it is possible to look out past the trees and locate a small island about a mile or so out in the sea. This is referred to as Polyakov Island after Polyakov swam out to it during one school. So, there is a sense of taking part in the continuation of physics lore while you are here. Perhaps the most astonishing feature of the school is it's two dogs: Calabi-Yau and Instanton. Calabi-Yau is seemingly quite an old dog, and saunters in and out of lectures at will (his world-weary presence, often asleep at the front is deemed a measure of respect for the lecturer, after all Calabi-Yau has probably listened to many more lectures in Cargese than anyone present - he probably already knows the full quantum theory of gravity and may be the most well-educated dog in the world). Furthermore on the nights spent in town he would invariably make the twenty minute journey and come and find us, even is we were stationed at house in town for a party, (where he would wait of his own volition patiently outside for our departure) and then he would join us for the journey home along the dark road. Although it is in truth hard to say who was leading whom. Not only is he probably the smartest dog in the world (if Carlsberg made dogs...) but he's also a wild party animal too. See picture above of Calabi-Yau working at full capacity.

I want to give you a feeling for some of the practical details of getting to Cargese just in case you are thinking of attending in the future. First off: the high energy physics school occurs every two years - if it's a World Cup or European Cup year then the school is on too and you have to apply early in the year. Registration this year closed in February. Cargese is on the south-west coast of Corsica, about an hours ride from Ajaccio airport, and you will almost certainly have to change flights somewhere in France to find a plane that will land in Ajaccio. The island has been invaded a number of times and this is reflected by the fact that you can get by speaking Italian here instead of French if you wish. Of course the modern invader is the tourist and so you can also survive using English, with a smattering of French. In fact, the locals do not like the people who buy a home here just for the holiday season and such houses have been known to burn down. Since you are likely to be taking a connecting flight you might want to be wary of one flight being delayed. This had significant financial implications for me since there were only two buses available from Ajaccio to Cargese upon arrival, and when my flight was delayed (resulting in 5 hours sitting in a Parisian cafe at the terminal in Orly, Paris - not quite 'living the dream') we had to hire a taxi at a cost of 130euros - this was subsidised by the School, and reduced to 100euros. You might think that arriving after midnight with no-one to meet you might be a problem but life at the Institute is very relaxed - so there was a poster on the wall and written in green ink was my name alongside the others who were late arrivals. Next to my name was a room number where I would be sleeping. The room was left open and keys were inside. The Institute is significantly remote for this calm attutude to security to be viable. But it is the little things like this that help to make Cargese a very peaceful place to be. The only other practical advice I can give you is that, just as in The Hitchhiker's Guide to the Galaxy, you should bring a towel.

The peaceful setting of the school and the emphasis of a healthy mixture of relaxation and work are wonderful. The mixture of mostly PhD students and young Postdocs was great for initiating collaborations and building relationships for future work and the school itself is the best I have been to during my PhD. Not only in terms of meeting fellow students but also in terms of the lecture quality. We heard lectures from De Wit, Strominger, Harvey, Douglas and Connes amongst others, and we even got to feel "the power of Nekrasov", on topics ranging from black hole entropy to noncommutative geometry with a healthy dose of lectures about realising the standard model in the string theory picture.

In the days of the cold war the school was funded by Nato and operated as a forum for bringing non-soviet scientists together. These days the event is quite global, but without a cold war the funding harder to come by. The school this year was supported by the European Science Foundation and CERN and a hearty thank-you is offered to the organisers of the school for the marvellous job they did to make this happen. So thank-you's to: Laurent Baulieu, Eliezer Rabinovici, Jan de Boer, Michael Douglas, Pierre Vanhove and Paul Windey. Without their organisation of funding, speakers, participants, schedule, support and ringing of the cowbell (although none had quite the enthusiastic glint in their eye as Pierre Vanhove when he got his hands on the bell) to get us into the lectures, Cargese quite simply would not have occurred, and it is hard to imagine it being organised any more successfully than this group managed. A special thank-you must also be reserved for Elena who took charge of all the school's administration and ensured everything ticked over nicely during the two weeks.

Some pictures and commentary on the lectures to follow.

I hate to trip but I gotta 'lope.

Back in Black

2006-05-22T00:32:00.000Z

Well it's been a while... I've often heard people wonder how researchers find the time to write a blog and do their work. Well while some bloggers are superhuman, this one is not. I've had a busy and not to mention stressful start to the year and really the blog only gets my attention when everything else is in good working order. What have I been up to? Well first of all I was applying for postdoc positions earlier in the year, the necessary finger-crossing meant that typing a blog became impossible for a short while. I was offered and accepted very happily a position at the Scuola Normale Superiore di Pisa where Augusto Sagnotti has recently moved. Much hurrahing all round. Second I have just been working hard on what will be the last part of my thesis. That's not to say the thesis is all written up and ready to submit, oh no I have left two months for that, and a spare third, just in case. Finally, a confession: I really haven't been to any seminars for ages now. It's peculiar but the seminar series at KCL has dried up for the last few weeks. So today I tidied up my papers, organised my room and put everything in its right place to begin writing up. But of course I better get my blog up to date first so I can give a running commentary of sorts on the ups and downs of submitting a thesis.

In my absence there have been a number of exciting papers on E10 and E11, in particular:

Enhanced Coset Symmetries and Higher Derivative Corrections by Neil Lambert and Peter West
Curvature corrections and Kac-Moody compatibility conditions by Thibault Damour, Amihay Hanany, Marc Henneaux, Axel Kleinschmidt and Hermann Nicolai
IIA and IIB spinors from K(E10) by Axel Kleinschmidt and Hermann Nicolai

The first two demonstrate the very exciting emergence of higher derivative terms very naturally from the large algebra approaches, in the first case for E11 and in the second case for E10. The third paper continues the success of the E10 research teams ability to find fermions in their approach, for which there is as yet no equivalent result for E11.

There have also been numerous great links from the other blogs, via Lubos we have the Horizon episode on Feynman, from which the stories will be very familiar, but it might be nice to see the man himself telling them. Thanks to Peter Woit we have links to all the talks at the recent Eurostings 2006 conference in Cambridge. Of particular interest to those predisposed to very large algebras are,

E11 and Ten Forms by Peter West
Hidden Symmetries and Fermions in M-Theory by Axel Kleinschmidt

Since videos and transparancies are available for all talks this conference site is highly recommended, also of interest will be the following talks:

The Quantum Structure of Black Holes by Samir Mathur
Singularities, Black Holes, and Attractor Explosions by Eva Silverstein

But there are plenty of good talks available here, so go and find out the latest from your favourite stringy research area.

Also I've noticed two review articles for the E11 approach to M-Theory are now available on the archive. They are both a couple of years old, but worth a look:

Algebraic structures in M-theory by Ling Bao
Hidden Symmetry Unmasked: Matrix Theory and E(11) by Shyamoli Chaudhuri

Now I have to make sure my thesis is nothing like these reviews...Ho-hum.

So let's see, things to do: 1. Learn Italian 2. Write-up thesis. So... the first thing I am going to do is fly off to Corsica tomorrow for the Cargese summer school (at much personal sacrifice to my better desires to start writing up!), and internet permitting I'll try and write some blog postcards from there.

I've just checked the weather and tomorrow it's supposed to be 31 Celcius and sunny, which sure beats the grey sheets of rain we had in Greenwich today.

Kallosh on Attractors

2006-03-27T17:34:00.000Z

Yesterday we heard the first of three different talks from Renata Kallosh. Her first chosen specialist subject was innocuously titled BPS and non-BPS Black Hole Attractors. This first talk really was for the back row of the audience at our school, where all the experts were sitting. Perhaps due to the time constraint, quantities were not defined and many ideas were assumed to be known by the audience. Unfortunately there is much work for me to do. At one point she paused and said to the audience:

"So far I was a bit sketchy...but this is something you can read. This is a known result."

Well this is true enough, so here are the references for Kallosh's first talk (just 1 hour):

Black holes and critical points in moduli space by S. Ferrara, G. W. Gibbons and R. Kallosh
Non-Supersymmetric Attractors in String Theory by P.K. Tripathy and S. P. Trivedi
The non-BPS black hole attractor equation by R. Kallosh, N. Sivanandam and M. Soroush
The Symplectic Structure of N=2 Supergravity and its central extension by A. Ceresole, R. D'Auria and S. Ferrera
E(7) Symmetric Area of the Black Hole Horizon by Renata Kallosh, Barak Kol
STU Black Holes and String Triality by Klaus Behrndt, Renata Kallosh, Joachim Rahmfeld, Marina Shmakova, Wing Kai Wong
Calabi-Yau Black Holes by Marina Shmakova

It would have been good to know these papers well before the talk began and as you can imagine the school degenerated to a workshop for the experts during this talk. However there were plenty of interesting things for us beginners to pick up. Such as that N=2 special geometry is useful and that symplectic invariants are useful. I'll try and reproduce my beginner's conception of special geometry in this post, mostly with the help of Christiann Hofman's masters' thesis, Dualities in N=2 String Theory (you can find the link near the bottom of the page).

Special geometry is the name given for the geometry associated to the scalar couplings of the vector and hypermultiplets of theories involving 8 supercharges, although the original use of the name was restricted to N=2, vector multiplets and four dimensions. Recall that the vector multiplet is an irreducible multiplet of super Yang-Mills theory, it is the enhancement of the gauge field to the supersymmetric setting, and has field content: Where X is a complex scalar, omega is a pair of spinors, Y is a triplet of scalars (arranged in an anitsymmetric 2 by 2 matrix) and A is a real gauge boson. For reference the hypermultiplet, when there are no central charges contains the fields: Here, A is a pair of scalar doublets, and zeta is a pair of spinors. When we include gravity in our supersymmetric gauge theory setting we find the metric is enhanced to the gravity multiplet, or Weyl multiplet, consisting of the metric and two fermionic fields of spin 3/2 called gravitini. These gravitational fields can couple to the content of the vector and hyper multiplets. Furthermore an additional vector multiplet is required if we wish to break auxillary gauge symmetries (see Lagrangians of N=2 supergravity-matter systems by de Wit, Lauwers and Van Proeyen). Only the vectors have physical significance, the remainder of the multiplets are auxillary fields. So if we commence with n vector multiplets from our super Yang-Mills theory, and then we include gravity to construct a sensible local theory, we find we require n+1 gauge fields. These gauge fields are the equivalents of our familiar Maxwell gauge field in electromagnetism, and including the dual fields we have 2(n+1) fields which are transformed into each other by the action of the symplectic group Sp(2n+2,R). The flux integrals of the field strengths and their duals give us electric, q, and magnetic, p, charges, and the symplectic transformation is interpreted as the generalization of electric-magnetic duality. So far, so good. But we neglected to mention that we also have n scalar fields which do not transform in such a well-mannered way under the symplectic group action. A suitable projective coordinate reparameterisation (giving us n+1 scalars) will, however, do the job, see Mohaupt's review Strings, higher curvature corrections and black holes for the overview. The scalars of the Lagrangian may be thought of as coordinates and, under the restrictions of supersymmetry, the geometry of the complex symplectic vector space C(2n+2)associated to the scalar coordinates is called special geometry. We end up with coordinates on our manifold, coming from the prepotential and the projective coordinate which do transform as a symplectic vector. However symplectic geometry is a little different from Riemannian geometry, for example symplectic manifolds have no local invariants like curvature.

If, like me, you have never come across any of this technology before you can see that there is plenty of work to do. Especially in picking up terminology and generic constructions. But don't despair! Take heart, all the experts at the school in Frascati presented some very pretty results (I was able to understand this from the joy in their eyes - inc omcing to this conclusion I have assumed the sanity of the speakers...) and it would seem the end result is worth the work.

Attractors

2006-03-22T16:49:00.001Z

Last Monday and Tuesday saw the start of the Winter school on Supersymmetric Attractor Mechanism here in Frascati. I have already described a little of the content of Per Kraus' talks, but we also have had a series of talks by his frequent collaborator Finn Larsen. Larsen has given three talks under the title Introduction to Attractors with applications to Black Rings and based upon his paper with Kraus: Attractors and Black Rings. At some point there will be a video online. At least one was made, and it seems there is a certain attractor mechanism for all real world content to eventually stabalise on the internet, so it seems fair to expect it will appear one day. I'll let you know if I hear anything.

If on a Winter's Night a Physicist...

2006-03-22T15:00:00.000Z

So, I find myself in Frascati, just 20km south-east of Rome attending SAM 2006 (School on the Attractor Mechanism). This is my first visit to Italy, and so very exciting for me (food, coffee, wine, olives, historic sites, art, physics...). There are 30 or so of us here at the Instituto Nazionale di Fisica Nucleare, where all the roads are named after famous theoretical physicists! The high energy bulding is on Via P. Dirac, which at some point turns into Via R. Feynman, a nice continuity. There are also roads for Pauli, Heisenberg, Schrodinger, Planck and others, but no Via Einstein! Of course the institute itself lies on the main road named after Enrico Fermi, so he doesn't appear on the campus map either, but that's okay.

The school was billed for beginners, and that is why I am here. Yesterday and today, we heard from Per Kraus and Finn Larsen. Kraus talked under the title "Black Hole Entropy and the AdS/CFT Correspondence" and I hope there will eventually be a video of the talk available online, but we shall see... For the impatient you can alreadywatch/listen to Kraus giving a talk based around his papers Microscopic Black Hole Entropy in Theories with Higher Derivatives and Holographic Gravitational Anomalies both with Fin Larsen. But I think the video of the three hour talk from our school will be much more elementary and welcoming. Hopefully I can make some comments about Fin Larsen's complementary talks in a later post.

Kraus began by telling us about the BTZ black hole (so-called for Banados, Teitelboim and Zanelli), emphasising the point that only for the BTZ black hole does a precise agreement occur between the microscopic and macroscopic counts of black hole entropy . The BTZ black hole is a 3-dimensional black hole similar to the Kerr solution, for a review see Carlip's The (2+1)-Dimensional Black Hole. It lives in three dimensional anti de Sitter space, AdS_3, a space with negative cosmological constant. Identifications in AdS_3 give rise to the BTZ black hole.

AdS_3 can be realised as a hyperboloid in a signature (+,+,-,-), i.e.. This is the Sl(2,R) group manifold. The BTZ black hole can be analysed by looking at the conjugacy classes of Sl(2,R). There are three conjugacy classes: hyperbolic, elliptic and parabolic, with the BTZ black hole sitting in the hyperbolic conjugacy class. Kraus, Samuli Hemming and Esko Keski-Vakkuri have written about this in Strings in the Extended BTZ Spacetime, see section 2. An identification is made with the conjugating elements and the left and right moving temperature, and we move into a thermodynamic setting. Mass, angular momentum, entropy formulae follow, and the equivalence of a thermal AdS_3 background with a BTZ upto various modular transformations in each case.

Kraus considered spacetimes whose near horizon geometry (when r approaches the event horizon, and considering only the dominant terms) factorises into AdS_3 x X x S^p, where X is an unspecified geometry (see Strominger's Black Hole Entropy from Near-Horizon Microstates for the motivation for looking at this geometry). Kraus demonstrated the equivalence of the Wald formula, for finding the entropy from a Lagrangian which includes higher-derivative corrections, with the Cardy density of states formula for a CFT for theories which have a general diffeomorphism invariance. Through this equivalence the exact entropy (i.e. including corrections) is derived solely from knowing the central charges of the theory. Furthermore Kraus presented a variational principle to give the central charge for some Lagrangian with higher derivative terms. In his final talk he looked at the use of gravitational anomalies for learning about the pictures on either side of AdS/CFT.

Kraus used two main examples to illustrate his talk:

D1-D5-P on T^4 x S^1 or K3 x S^1
M5 branes wrapped on 4-cycles in M-theory on CY_3 x S^1

He demonstrated how the BTZ black hole appears in each case and compared the entropy calculations in each case. The D1-D5-P entropy (Strominger and Vafa) is in exact agreement with the macroscopic Bekenstein-Hawking entropy, while the M5 branes' microscopic entropy (Maldacena, Strominger and Witten) gives a central charge consisting of two parts, the highest order part agreeing with the macroscopic count and the remainder being due to the presence of higher derivative terms in M-theory. I refer you to the two papers with Larsen linked to earlier to see the full application of the method with these examples in mind.

Classifying Rational Conformal Field Theories

2006-03-01T23:28:00.000Z

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?

2006-02-20T11:26:00.000Z

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

2006-02-06T23:29:00.000Z

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

2006-02-06T19:51:00.002Z

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

2006-02-02T10:36:00.000Z

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

2006-01-24T18:23:00.000Z

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 London for 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

2006-01-15T14:16:00.000Z

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.

UK Annual Theory Meeting

2005-12-29T18:30:00.000Z

The semester has ended, there are no weekly seminars to write about, I have had my yearly winter cold (but that was not interesting enough to say much about) and now Christmas has come and gone, but just before it began (I mean December 25th rather than the retail definition of mid-October) I started to write up the following...

For the last few days I have been getting up much earlier than normal, meeting many more friends, listening to more lectures and eating larger meals than normal. This is not some new life plan undertaken on my part but rather I have been to Durham to attend the annual Christmas UK theory meeting. The meeting takes place every year near Christmas and has a prestigous history of speakers. It is also a meeting that attempts to present an overview of the current state of the art in theory ranging from phenomenology and the construction of particle accelerators to string theory and tries to find a balance between them (which is very hard, and invariably different talks were appreciated to different degrees by disparate groups, save the final one).

I arrived at Grey College, where we were housed, at quarter-to-one on Monday, a feat which required a near superhuman effort to awaken at six-thirty that morning. It was my first benefit of daylight saving time, there was pristine blue sky and it was actually pleasurable to see the light reflected from all the unfamiliar angles of the buildings near my home. Despite the lack of sleep, the three hour train journey at a cost of £88.00(!) was very pleasant. However the rest of the afternoon was not so nice for me, since the talks were dedicated to experimental particle physics. While string theorists must not decouple from experiment, it is very easy for a PhD student in strings, who has specialised in group theory, differential geometry and gravitational physics to never have come into contact with much QCD or lattice simulations and certainly not at the level of the talks that first afternoon, alas. Of course the same is true within any large subject area, and from this blog you will know how often I have had trouble even following stringy seminars. From the number of sleepers in the various talks I feel there was a similar disappointment for some of the phenomenologists and others during some of the stringier talks. Ho-hum.

The first talk was "Results from the B factories" by Steve Playfer, where we heard about the latest data cuts and fittings comng out of the Belle and BaBar experiments. What are Belle and BaBar? Well they're the B meson factories and they look to see whether charge (particle-antiparticle) and parity (physical mirror) symmetries are violated in the decay of the B and the anti-B, or B-bar. Belle is located at KEK in Japan and BaBar is at the Stanford Linear Accelerator in California. The two experiments have confirmed (BaBar 2002) that the B-mesons do violate CP-symmetry. Steve is part of the BaBar team and his talk describing recent results, hinting at future results and improvements in error ranges displayed a healthy amount of competitive spirit with the Belle experiment.

The rest of the afternoon was filled with talks possessing a phenomenological nature, and again outside your humble blogger's limited range of semi-enlightenment. The roll-call reads: "Heavy Flavour Physics" by Matthias Neubert and "Lattice QCD and results from CLEO" by Paul MacKenzie. The evening was spent in the pub and then subsequently the "trendy" wine-bar Jimmy Allen's where all the cocktails had decidedly dismal names, but nevertheless a good time was had.

The tuesday began much earlier than most tuesdays, perhaps this was the effect of the drink the night before or perhaps it was due to the hour. The first talk, "Twistors and Perturbative Gravity" was given by Dave Dunbar from Swansea University and was of a similar format to many of the twistor talks I have seen this year in that there was a long review of the material for the skeptics in the audience who, like me, have so far shied away from twistors and MHV amplitudes. As ever the simplification of loop calculations coming from twistors seemed nothing short of miraculous, and, as ever, I have made a solemn pledge to myself to read Witten's twistor paper and work through some of the calculations myself.

The second talk was a report from the front line on the construction of the LHC at Cern given by Jos Engelen. We saw pictures of heavily-laden lorries shipping extraordinarily large structures back and forth, heard tales about the pitfalls of purchasing specially made detector crystals from a Russian manufacturer who "forgot to pay their electricity bills" and subsequently upped the price for the consignment, and we marvelled at the design ingenuity of the engineers having to form these perfectly symmetric detectors at such a large scale and underground too, in prticular Jos told us about one detector whose perfect circular cross-section was designed only to be achieved once the final segment was installed and its weight would bring about the perfect circle. We heard abut the 27km track (how perhaps only the rate of intallation of the 50m magnets needs to speed up), Altlas, CMS, LHCb and
Alice. In short all is well, Jos paraphrased this:

"We are going to do the first new physics in Summer 2007. This sentence does not define new physics, it defines Summer 2007."

Vijay Balasubramanian spoke about his approach to "Gravitational Thermodynamics" which was quite different to all other talks on this subject that I have ever heard, for example there was almost no mention of charge ensembles (at least not until the questions at the end, when it was asked how his work relates to the more usual stringy appraoch to degrees of freedom). Before he began there was an advertisement for a postdoc position and a faculty position at the University of Plymouth, details to be found here. He spoke with great enthusiasm and volume about horizons and Unruh radiation and very large states. In gravitational thermodynamics the information loss paradox is side-stepped if black holes do theoretically contain all the degrees of freedom of the states that fall into them and that information could theoretically re-emerge from the black-hole, eventually. Pure states remain pure. Vijay was addressing the question of just what is a pure state and what would it take to be able to probe it. His arguments can be found in his paper with Vishnu Jejjala and Joan Simon, "The Library of Babel". But the main argument is that while states do remain pure and information is not lost, "almost no probes are able to differentiate the microstates of a black-hole". To find out what almost means you'll have to read the paper (7 pages), and I heartily encourage you to read the short story that the paper borrows its name from by Borges.

In the afternoon we had a very nice, pedagogical talk from Hans Peter Nilles, who amongst other things explained to us in detail exactly what an orbifold is. His title was "Grand Unification in Strings" and he said was based on work found here, here and href="http://www.arxiv.org/abs/hep-th/0504117">here.

Finally we ended the day with a talk by Keith Ellis, provocatively entitled "Marching Orders for QCD" which he said could be found online here, so I left. That evening also found me making further investigations into beverages of Jimmy Allen's.

The final morning began with Herman Verlinde building bridges between string theory and experimental particle physics. His title was "Geometry and the Standard Model" and his proposition was to decouple the landscape. He described the landscape as 10^500 doors with locks on, and suggested that we should start with the only key we have: the standard model. His aim was to start with the minimally supersymmetric standard model, to use this to find a moduli space for the SuSy vacua and to use this to find a configuration space of Calabi-Yau manifold singularities. The devil is in the detail, and he led us on a path that took in fractional branes, quiver gauge theory, Del Pezzo surfaces and how to draw a Moose. During the question session the attempt to build bridges between theory and experiment was welcomed, but only after the rivalry between string theorists and those hoping for a more readily tangible explanation of high-energy physics had once again flared up. A member of the audience had been willing to bet against string theory being the correct unifying theory, and Hermann Verlinde had steadfastly opposed his negative opinions by saying he was willing to take such a bet. This is the first time, beyond the internet, where I have witnessed any kind of head-on confrontation about the nature of string theory. I expect it will not be the last.

The final talk of the meeting was of a much more gentle nature and was enjoyed by a broad spectrum of the audience, regardless of their predisposition towards string theory. It was entitled "How did Einstein get the Nobel Prize?" given by the very entertaining Cecilia Jarlskog. She gave us a colourful review of the life of Alfred Nobel and the early history of the Nobel prize and we got to see all the nominations in the years leading up to Einstein's prize, and even to see some of the photocopies of handwritten letters of recommendation for the prize. All of which was very nice, and made for a fitting end to the meeting and also, for me, to the year of physics celebrating the centenary of Albert's miraculous year.

Dilogarithms

2005-12-07T05:06:00.000Z

Two noteworthy events occured at King's College last Friday. One of which involved "one-hundred drunk LSE students" rioting in the lower levels of our Strand campus and the other did not. The rioters flooded our building and caused havoc, some attempted to invade the maths classrooms no doubt eager to benefit from the wisdom of our faculty, and others began ripping ceiling tiles from the English department in an attempt to seemingly tear a building to bits whilst still inside it, causing £30,000 worth of damage. The second event of note last Friday was our mathematics department colloquium which was given by the impressive Don Zagier under the title "Dilogarithms and the Bloch group: from algebraic K-theory to modular forms to conformal field theory". The rioters caused our colloquium to start late, a fact that was noted by Professor Zagier when nearly out of time he suggested that perhaps he himself could riot in order to circumvent the need to vacate the lecture thatre at the prescribed time so he could get to the end of his talk.

Permit me to pass on some brief comments from the beginning of Zagier's talk about the dilogarithm with the caveat that most of this remains beyond my ken. Nevertheless the dilogarithm has been studied by some truly great mathematicians including Euler, Leibnitz, Abel, Lobachevsky, Borel, Ramanujan, and others so if only for historical interest it is nice to look at some of the properties of the dilogarithm in passing.

We are all on first name terms with the logarithm function (that's log to you and me):
But perhaps not so well known is it's relation the dilogarithm: There are plenty of others, which are specific values, s, of the polylogarithm, or Jonquiére's function, Zagier told us that there are only 8 closed forms for the dilogarithm, some of which include the golden ratio, The only irrationals on the left hand side are powers of the golden ratio. We might look for properties of the dilogarithm that are similar to those we know of the logarithm, for example we know that, Zagier told us that there was no simple equivalent of this for the dilogarithm, but that we did have a formula, named for Abel even though it had been discovered by others before him (Spence, 1809) and after him (Lobachevsky), Zagier told those of us taking notes in the audience to just write "junk" instead of the right-hand-side of the above, however, he stressed that the order of the left-hand-side terms was important. I think practitioners of this art tend to normalise their dilogarithm functions into the Rogers L-function, which gets rid of the junk. There are a number of useful identities for the dilogarithm two of which are,These lead to equivalent forms of the five-term equation via X ~ 1/X ~ 1-X ~ 1/(1-X) ~ 1-(1/X) ~ X/(X-1). Furthermore if you define a junk-removing functional equation: Then this equation has a modulo 5 symmetry,Try it and see!

Zagier went on to look at the Bloch-Wigner dilogarithm, whose five-term equation is equal to zero and to give an argument in terms of the volume of tetrahedra in a hyperbolic 3-space modulo PSL_2(C) about why it should be zero. It sounded very nice, but I can't say I understood. For those interested the argument is given in section four of sixth set of notes by Matilde Lalin from classes given by Fernando Rodriguez Villegas called "Topics in K-Theory and L-functions". Those interested in seeing many more five-term functional equations should look through Kirillov's Dilogarithm Identities. For an introduction to dilogarithms Maximon's article seems a good place to commence.

Zagier talked fluently, rapidly and energetically about many more topics, such as the Bloch group (if you google Bloch group you may not find much helpful information but you will come across Dr. Bloch's group page), the five-term equation related to the Schrodinger equation for a cubic potential, K-theory (in particular K_3) and also rational conformal field theory, where he spoke with great enthusiasm of the recent conjectures of Werner Nahm. He never once, to my knowledge, resorted to rioting to extend his talk.

Beyond Einstein

2005-12-01T15:23:00.000Z

Here's some very short (actually it's already started) notice of the physics equivalent of Live 8. To celebrate the world year of physics physicists, internet pioneers and others will be participating in a marathon 12-hour broadcast ~~performing covers of classic pop songs to raise awareness of physics~~ live on the web and will discuss Einstein's theory of special relativity and more. In the words of the BBC article,

Alongside E=mc², there will be discussion of gravitational waves, the search for the Higgs Boson and the mysteries that still linger behind Einstein's work.

So get the popcorn in, get broadband installed and go and watch (or put on in the background) the webcast at Beyond Einstein. Just so you know there are some well known names who will be participating later on today. Included, amongst others, on the list is (broadcast times are in brackets and are Central European Times):

Albert Einstein (1200-0000)

David Gross(1700)

Gerard't Hooft (1700)

Murray Gell-Mann(1700)

Paul Davies (2230)

Stephen Hawking (TBC)

Quite a list eh? Although there must be some code for the yellow and green stripes on the particpator list I presume they are all taking part. The full programme is available here.

Given that this is obviously a very-well prepared event it is amazing not to know anything about it until after it has begun! Clifford at cosmicvariance also found about it today, and I only heard about it thanks to the Emperor of our office, Peter McKeag - he's not Emperor for nothing!

Lions

2005-11-28T19:44:00.000Z

One of my fondest memories from my further maths lessons back in my school days occurred when one of my colleagues brought to our attention the many ways a mathematician may trap a lion. In many a surreal moment I have stared wanly into the middle distance trying to recall exactly what was in the list, and the best I could do was simply remember with fondness that it was funny. Fortunately for me I need stare wanly no longer and I may trap lions to my heart's content for Bjorn has compiled a list of many ways to do this. Thank-you Bjorn! He also shares an amusing exam answer where a student took up the ages old challenge of finding x with great success!

Also I've followed the advice of Mr Goose and started using The TeXer for making small gifs of tex online - it's very handy, but I wanted to draw your attention to the excellent host site, called The Art of Problem Solving, claiming to be the world's largest online maths community. In particular I wanted to share my pleasure at their little geometric animations in the bottom left hand corner of the site as I think they are wonderful, I particularly like this one (which I have stolen (!) the end result of from their excellent site ):

P.S. Peter Woit provides yet another excellent link to online lectures, this time by Penrose, Weinberg, Maldacena amongst others at the Perimeter Institute. Peter was referring, in particular, to the emergence of space-time talks which can be found at the bottom of the list on the left, and one of the talks is by Seth Lloyd, the views of whom have been recently commented on by Lubos Motl.

My first journal club

2005-11-28T19:41:00.000Z

The enthusiastic group of postdocs here at King's have organised a journal club for all of us students and interested faculty. I've not been involved in one before and so it will be interesting to see whether or not we make progress as a group or not. Our topic to guide our reading is gravitational thermodynamics and hopefully we will look at attractors as well as more recent alpha' corrections to the entropy formula. But to start off today we began with a background review. Here was our suggested reading list (totalling 315 pages)

Black hole thermodynamics by Simon F. Ross

TASI lectures on black holes in string theory by Amanda W. Peet

The Thermodynamics of Black Holes by Robert M. Wald

Black Holes by Paul K. Townsend
An abundance of middle initials, which bodes well. We really went over some definitions, such as surface gravity for the Schwarzschild black-hole, null hypersurfaces, temperature, Hawking radiation (which we hope to look at more closely next week), event horizon area increase (and we drew the analogy with entropy, but no more than that) and the formula at the heart of black hole thermodynamics:
So we didn't get too far into our review in an hour, and already there was some disagreement in our club between those who want to push on towards the string theory point of view and the recent papers and those who want to get the fundamental concepts under control before moving on. I foresee a rocky ride ahead, but it will be a fun endeavour. So, for now, our journal club rides on...

Healthy Scepticism in the Ranks

2005-11-26T21:11:00.000Z

Yesterday Andreas Recknagel gave our first mathematics colloquium of the year. We heard a very rapid overview of unification, string theory and conformal field theory that finished up with T-duality and mirror symmetry. He even showed us some "strings" he had in his pockets to describe the winding number. The talk was peppered with good humour the first occurred at the outset where he described "the string factory" to us, here are my notes on it:

Heterotic Geometries, all of them

2005-11-26T20:53:00.000Z

This week George Papadopoulos talked to us about the results from his latest paper, The spinorial geometry of supersymmetric heterotic string backgrounds, with Ulf Gran and Philipp Lohrmann, from my office. I used to keep a pink penguin on the site as a tribute to Philipp, that one could poke and push off some ice while reading the blog. Happy days. Now I have to treat him with much more respect since he and his collaborators have made use of the fundamental procedure of the spinorial geometry programme (which I will describe below) to produce a list of all the possible background geometries for the heterotic string, up to multiplication by a compact group in some cases.

The talk itself began a little ominously on Wednesday when it was realised that our electronic projector was not available, and instead of us all crowding around George's laptop, he managed to give us a blackboard talk which was impressive both for its clarity and in that he hardly ever looked to his laptop for guidance. However one consequence of this was that hardly any complex equations appeared even though there must have been many more intended, and also the speed of the talk was very good for those taking notes. I wonder if this policy of losing the projector is not something we should take up full time at King's...

George began by motivating the study of the Killing spinor equations by looking at the Euclidean instanton bound. In 4-dimensions,
Where we have completed the square. The bound is saturated when,
When we take the minus sign in the above we have the self-duality conditions on the field strength. Now suppose we wanted a spinorial version of this construction. We would start by coupling the spinor to the field strength forming:
Reproducing the equivalent of the Euclidean action above we have another formulation for the instanton bound, but now in terms of spinors,
Normalising the bound is attained if, These are the supergravity Killing spinor equations arrived at by considering the instanton bound.

Having motivated the Killing spinor equations, George moved on to describing the formalism he has developed for helping to solve them. He took us through a 4-dimensional example so we could see the general approach. In 4-dimensions the spin representation is Spin(4) which is the double cover of SO(4), the Euclidean rotation group. There is an isomorphism Spin(4)=SU(2)xSU(2) and there are two different Weyl spinors, with two components in each, which transform under only one of the SU(2)'s. George then described the setting up of a real vector space, whose components will act as a space of forms. In two-dimensions we have two basis elements in the vector space (e_1,e_2). Mimicking the splitting of the Weyl spinors to transform under two different copies of SU(2), we set up a second copy of this vector space by "complexifying" it. The basis of forms is:
We have a basis of two even and two odd forms which corresponds to two different chiralities. If we explicitly write down the gamma matrices using our basis we can get some work done,
I have used a vee for the inner derivative which acts in the opposite way to the exterior derivative; it's destructive while the wedge is constructive, e.g. . These are all we need to solve the Killing spinor equations. We first note that the equation is unchanged under spin(4) transformations, upto a Lorentz transformation on the field strength. That is we may orient the spinor how we wish, in particular we may pick the direction 1 (the "Clifford vacuum" if you like) in the basis:
Now we may go back and look hard at our basis for the gamma matrices and write down the combinations that annihilate the vacuum. We use these to define a new (solution) basis:
We also have a set of antiholomorphic gamma matrices in this basis which are the complex conjugates of the above. Finally expanding out the Killing spinor equation, by summing over holomorphic and antiholomorphic indices, in this basis leads to some simple expression,
Recalling that the n index gamma matrices are just the antisymmetric combination of the n individual gamma matrices, we see immediately that the first term annihilates the Clifford vacuum. Carrying on thinking about this in terms of a basis of states is fruitful since we also see that the remaining two terms must vanish independently, since they create different "Clifford states". That is the Killing Spinor equations reduce to,
So we see from this walkthrough just how simple the equations can become. In particular one can learn about the geometry of a number of setups. In the latest work, linked above, the authors make use of the fact that the heterotic string background resembles a Riemannian manifold, to find all the stability subgroups for the possible numbers and types of Killing spinors. From which the geometry of the background is determined. So impressively it is claimed that all the M-theory geometries corresponding to the heterotic string regime without alpha' corrections have been found. Of course it is not clear what happens to the Killing spinor equation when alpha' corrections are included, but it seems like they will be altered.

SuSy backgrounds and M-theory corrections

2005-11-18T19:43:00.000Z

It would be remiss of me not to point out the Vega Science Trust which has been recently highlighted on cosmicvariance by Mark Trodden; if nothing else go and watch the Feynman lectures available there. Elsewhere Seed magazine seems to be getting much blog support (see here, here and here). So jumping on the bandwagon I'd like to recommend the Seed magazine podcast so you can have "science is culture" articles wherever you take your mp3 player. In fact if you are keen on podcasts you should also listen to Berkley Groks.

This week Kellogg Stelle from Imperial College visited KCL to tell us about his work on corrections to M-theory at order {\alpha'}^3. In particular he described work which demonstrated how supersymmetry may be preserved by making use of a corrected killing spinor equation. Indeed one may work backwards and start with a corrected killing spinor equation and rediscover the corrections to the string background. The methods used for various supersymmetric backgrounds are based on the following papers written variously with Lu, Pope and Townsend:

Higher-Order Corrections to Non-Compact Calabi-Yau Manifolds in String Theory (Kahler manifolds)

Supersymmetric Deformations of G_2 Manifolds from Higher-Order Corrections to String and M-Theory (G_2 manifolds)

String and M-theory Deformations of Manifolds with Special Holonomy (Spin_7)

Generalised Holonomy for Higher-Order Corrections to Supersymmetric Backgrounds in String and M-Theory (Generalised holonomy)

If you want to learn more read through the first paper above, there are some surprising results. You might even be keen on thinking about the alpha' corrections and the group E10, or even their effect on entropy calculations. I'd write more but 'tis the season to be making postdoc applications and I really have to be seasonal, very quickly and as many times as possible. Ug, the drudgery.

A Silver Age in Black Hole Research?

2005-11-09T01:41:00.000Z

Yesterday afternoon Harvey Reall from Nottingham University came to talk at Imperial College. The talk entitled Black Holes and Extra Dimensions was aimed at masters level students, of which, there was only one in the audience :( However it was a nice talk, and somewhat of a black hole history lesson and worth describing here. Lubos Motl has a report of a very similar talk by Reall which you can find here.

Harvey began by describing the "Golden Age" (so named by Kip Thorne) of black hole research which began in 1963 with the discovery of the Kerr solution, and supposedly lasted until 1973 when the macroscopic black hole entropy formula was discovered by Hawking. The two main achievements of the "age" according to Reall were:

1. The black hole uniqueness theorems
2. Black hole thermodynamics

The uniqueness theorems (for more information you can read David Robinson's "Four decades of black hole uniqueness theorems")in 4 dimensions are twofold; there are separate proofs for the non-rotating and the rotating vacuum solutions. The non-rotating solution is due to Werner Israel (1967) and a more recent proof by Bunting and Masood-ul-Alam (1987) and shows that the only non-rotating, equilibrium solution is the Schwarzschild solution (1917). The only rotating, equilibrium solution is the Kerr solution (1963) as shown by Carter (1971), Hawking (1972) and Robinson (1975). These solutions are described by one (M) and two (M,J) parameters respectively. Generalisations to include the Maxwell field also exist (the Kerr-Newman solution).

Reall spent some time motivating the consideration of extra dimensions. For those interested in string theory he simply said that it is, in his opinion, the best candidate for a theory of quantum gravity. He justified this by saying that the entropy macroscopic entropy formula is the only experimental data that we have for quantum gravity and that microscopic count of entropic degrees of freedom gives matching numbers. For those not so convinced by string theory Reall offered the AdS/CFT (Maldacena 1997) or as he called it the Gauge/Gravity correspondence. The gauge/gravity correspondence was described for the case of a 5D gravity theory in the interior of the cylinder (shown right) being equivalent to a 4D gauge theory on the curved surface of the cylinder - the edge of the cylinder is at infinity by some clever choice of coordinates and three dimensions have been suppressed out of respect for our 4 dimensional universe. In short Reall pointed out that some 4D gauge theories are equivalent to 5D gravitational theories and so we may learn more about QCD calculations by looking at higher dimensional theories.

Reall then moved on to talk about constructing 5D solutions from the known 4D solutions. He began with the black string which was constructed by adding an extra flat dimension to the Ricci flat 4D Schwarzschild solution: This is the black string, it is infinitely long and so has infinite energy. To avoid this we compactify the z direction by identifying z~z+2/piR, the geometry is changed to 4D Minkowski space crossed with a circle. Reall told us that the Black string This is the black string, it is infinitely long and so has infinite energy. To avoid this we compactify the z direction by identifying z~z+2/piR, the geometry is changed to 4D Minkowski space crossed with a circle. Reall told us that the Black string is classically unstable when the radius of compactification, R, is greater than 2M. This is the so-called Gregory-Laflamme instability (1993) and it's endpoint is unknown. However Horowitz and Maeda (2001) conjectured that the endpoint was the non-uniform black string, which has a sine-wave distortion of the event horizon as it moves through its z coordinate. Toby Wiseman showed that these exist using a numerical approach in 2002. The downside is that they lead to decreasing entropy, which is unrealistic!

There is a generalisation of the Kerr solution to D dimensions called the Myers-Perry solution (1986). It has some familiar properties in that it's event horizon has topology S^{D-2} and it is uniquely specified by its mass and its angular momenta (there being [(D-1)/2] angular momenta, where [] means drop the fractions). Another familiar property is that the angular momenta are bounded. Having described this much Harvey Reall asked if there were any other types of higher dimensional black holes, i.e. does the rotating black hole uniqueness theorem still hold in higher dimensions?

Harvey answered that he and Emparan had showed that there existed a different type of black hole in 5 dimensions (2001) - the black ring. This solution is a rotating closed loop of black string, whose gravitational collapse is held at bay by one of its angular momenta. Heuristically the gravitational force is balanced by the centrifugal force. Of course one wonders why a similar rotating black cylinder couldn't exist in 4D...to project to 4D one could set r=0, corresponding to zero mass, hence no 4D equivalent solution to the black ring, at least not in this way. The 5D black ring's second angular momentum is zero and the solution has only 2-parameters with topology S^1 cross S^2.

It transpires that for a certain range of angular momentum there are two ring solutions, one small and one large. So together with the Myers-Perry solution there are some regions of parameters where 3 solutions exist for the given parameters. The uniqueness theorem for rotating black holes is well and truly lost for higher dimensions. The non-rotating uniqueness solution does still hold in higher dimensions (Gibbons, Ida & Shiromizu 2002).

Harvey finished up by telling us about supersymmetric black rings (also known in some circles as: "The Hairy, Tiny Black Hole Donut Theory") which were discovered by Elvang, Emparan, Mateos and Reall (2004); Gauntlett and Gutowski (2004); and Bena and Warner (2004). You can listen to Reall talk about black rings here. The solution is now stabilised by a non-zero second angular momentum (this one rotates the torus about its second circle). The solution has 7 parameters: 2 angular momenta, 3 electric charges, 3 magnetic dipoles and one relation between them all. Also noteworthy is the observation that the dipoles are not conserved, so there are only 5 conserved quantities amongst the seven parameters. The entropy formula is consequently much more complicated than usual but despite this it seems that string theory can still be used to give a correct microscopic count of the solutions degrees of freedom (Cyrier, Guica, Mateos and Strominger 2004), although Reall had some doubts about the count since it seemed to imply that one of the two angular momenta was zero, contrary to the macroscopic solution.

In his final remarks Reall reminded us that the golden age of black hole research was started off by the discovery of the Kerr solution, and he hoped that the silver age would be kicked off by the discovery of the black ring. Personally I'm already worried about naming the third age, since the bronze age is already taken.

A Bridge to Higher Spin Theory and AdS/CFT

2005-11-04T16:48:00.000Z

Last weekend I was pleasantly surprised when a friend phoned me and said he was in my neighbourhood and thought we could go for a drink. Unfortunately there were scheduled works on a bridge between where he was and where I live; I thought the bridge would be raised and he replied asking if that meant we were topologically disconnected... This was somewhat similar to how I felt yesterday at our weekly seminar, since the topic seemed to be "topologically disconnected" from my own small area of understanding. So in these comments I propose to try to build a small bridge to the beginnings of the subject matter discussed in the talk.

So Paul Heslop from DAMTP in the real Cambridge came to talk to the King's group under the title "On the higher spin/gauge theory correspondence" which is based on his work with M. Bianchi and F. Riccioni. Paul's talk was based mostly around their paper "More on La Grande Bouffe: towards higher spin symmetry breaking in AdS". Don't be put off by "La Grande Bouffe" it is the name of a film and it refers to an aspect of the theory where a higher spin field "eats" the entire chain of lower spin fields to acquire mass, these chains can be very long. The film of the same name, we were told by an anonymous member of our department, is about "three men and one woman" and the men all eat so much that they die! Via IMDB the plot is described as:

Four successful middle-aged men Marcello, a pilot; Michel, a television executive; Ugo, a chef; and, Philippe, a judge go to Philippe's villa to eat themselves to death.

It has 7.2 stars from 1225 reviews. So don't be put of by the term, which I believe is due to Massimo Bianchi.

The conjectured correspondence is that "the massless higher spin theory is holographically dual to a free gauge theory on the boundary". We can summarise it best using a picture given in Paul Heslop's talk:
If you wish to learn about the correspondence then you can read through the following papers of Bianchi et al:

"Higher Spin Symmetry and N=4 SYM" by Niklas Beisert, Massimo Bianchi, Jose F. Morales and Henning Samtleben

"Higher spin symmetry (breaking) in N=4 SYM theory and holography" by Massimo Bianchi

"Higher Spins and Stringy AdS5xS5" by Massimo Bianchi

However since there is a gap in my understanding, I thought it would be constructive to find a brief paragraph or two to express the formative ideas behind Vasiliev's higher spin theory, if I can. However a better option would be to listen and see pictures from a talk by Vasiliev here, where you will hear him commence by describing the totally symmetric massless free fields of Fronsdal (1978) and de Wit and Freedman (1980) where the bosonic case contains a field with s symmetrised indices which is double-traceless (i.e. if you contract two pairs of its indices it is zero). There is a uniquely associated action which is chosen by requiring that some action containing terms up to second order in derivatives of the field is gauge invariant. There are some familiar gauge invariance principles for s=1 (Maxwell) and s=2 (gravity) fields and the essence of higher spin theory is the question of whether there is a unifying gauge symmetry that exists for other higher spin fields as well.

What are the connections with strings and supergravity I hear you cry? Well Vasiliev goes on to say (in the video above) that while there is a limit on the spin of the massless particles in a d-dimensional theory (s=d-2) there is also a corresponding limit on the number of supersymmetries (e.g. he draws a parallel between the s<=2 and N<=8 in 4-dimensions). He says that this is "practically equivalent" to the limit on the number of dimensions of supergravity (d=11) and says that if one wishes to consider theories beyond supergravity then one might start by wondering what happens when one includes massless spin fields of spins higher than those in D=11 sugra. Further motivation comes from the Stueckelberg symmetries of the superstring which are similar to spontaneously broken symmetries of higher spin gauge symmetries. As well as from the work of Sundborg and Witten, arguing for a nonlinear theory with infinite higher spin fields in the bulk.

So there you have it a small bridge to travel over and go and study higher spin gauge theory. A couple of further papers to help you on your way towards the triangle above are:

"Higher Spin Gauge Theories"(be warned this is a link to a 205 page pdf), Volume 1 of the proceedings of the Solvay Workshop, a compilation of work in various areas related to higher spin theories.

"Notes On Higher Spin Symmetries" by Andrei Mikhailov

It transpired last weekend that the scheduled roadworks were not happening, the bridge was down, and just like this week's seminar experience my friend and I were topologically connected after all, and we went and had a merry afternoon in Greenwich.

Black Hole Attractors and Entropy

2005-10-16T21:14:00.000Z

On Friday, Atish Dabholkar from the Tata Institute of Fundamental Research, visited Imperial to talk about microscopic entropy counts, small black holes and the use of the attractor mechanism. This is a very interesting topic, and arguably the area where string theory has had its greatest success so far.

So let's recap: In 1973 Bekenstein suggested that the event horizon was proportional to the black hole entropy, shortly after Hawking became convinced of this and produced his famous equation relating the entropy of the black hole to a quarter of the event horizon. However, from statistical mechanics we have become accustomed to being able to understand entropy as a count of the degrees of freedom of a microscopic system, yet from general relativity we expect all material to fall towards a singularity, sometimes a point and sometimes a surface behind the event horizon. All degrees of freedom as we understand them classically are suppressed at the singularity so how to account for the entropy microscopically is quite a puzzle, we are led to believe that there are some hidden degrees of freedom. This is an ideal situation to turn to superstrings (at least if you are predisposed towards string theory), living in ten dimensions gives them six dimensions in which to hide such degrees of freedom from us four-dimensional beings. And indeed Vafa and Strominger were able to make a microscopic count for a five-dimensional black hole using string theory that agreed with the macroscopic entropy coming from the event horizon area. The idea is that we can consider in 10-dimensions coincident D5-branes, a D1-branes and strings going between the two types of brane, and the excitations of the strings account for the degrees of freedom (if you want to read more about this picture at an introductory level then chapter 16 of that wonderful purple book by Zwiebach is recommended). This is then compactified on a T^5 to give a point, corresponding to the singularity in the 5-dimensional spacetime. In fact this construction corresponds to the 5-dimensional Reissner-Nordstrom black hole, as can be seen from the algebraic approach taken by Clifford Johnson in his slightly less purple book D-branes. There are a number of equivalent dual pictures and one of the most illuminating in terms of finding a geometrical reason for picking this combination of branes and strings is described in a very readable paper by Samir Mathur, The fuzzball proposal for black holes: an elementary review. Mathur entices us to consider an M2-brane dimensionally reduced on a spacelike circle in the z-direction to give a NS string in the IIA theory and then further compactify our setting by wrapping the NS string around a circle in the y-direction. Now go back to the 11-dimensional picture and think about the event horizon of the M2, it is shrinking because the M2 tension is wrapped around two closed loops and pulling them tight. The horizon is shrinking to zero, and we find that we have zero macroscopic entropy. We aim to stabilise the 11-dimensional horizon and gain an entropy that isn't disappearing. This will mean that we have a stable extremal, or BPS, black hole, one that isn't radiating and shrinking. Again let's go back to the 11-dimensional M2 picture. The M2 is radiating, so that had there been any other compact dimensions transverse to the M2 it would try and excite them and blow them up. Aha! So let's compactify some of these other dimensions and wrap another brane around those and see if we can't balance the tension of the second brane with the expansion caused by the first brane and vice versa. We pick an M5 brane and place it transverse to z in 11-dimensions, giving us a NS5-brane in IIA, we wrap this around T^4 (transverse to the NS1) and S^1 (in the y-direction). Now this turns out to be enough to stop the shrinking of the z-direction in the 11-dimensional view, but both branes are wrapping the y-circle and it is shrinking. We may excite the y circle by adding momentum charges around the circle which have energy proportional to 1/R, so they have lower energy for larger R and keep the y-circle non-vanishing. Phew. Now we have three charges coming from the NS1-NS5-P system (which may be dualised to D1-D5-P) and a stable horizon. That this is a BPS state means that we can count the degrees of freedom for different values of the coupling constant g and still expect the count to stay the same. So this is a heuristic approach outlined by Mathur for picking this special system. For the actual counting I refer you to some of the literature here, here and here. And what about other black holes, in particular the Schwarzschild black holes: can we find a similar stringy construction for counting the microstates? Well, yes, we can see for example Englert and Rabinovici.

But what about that NS string we considered alone earlier, our arguments told us that it had zero entropy, and yet it still contains microscopic degrees of freedom, so what's going on? Atish Dabholkar started his talk by asking us whether the S(Q)=klog[\Omega(Q)] was absolutely correct and if we could compute corrections to both the macroscopic and microscopic counts of the form:

S=a_0A(Q)+a_1log[A(Q)]+a_2/A(Q)+....
klog[\Omega(Q)]=b_0A(Q)+b_1log[A(Q)]+b_2/A(Q)+....

He pondered whether we could compute the a's and the b's and did they agree, and then told us that for a class of BPS N=4, D=4 black holes this can be confirmed. He said that on the macroscopic side one must take into account higher derivative corrections to the action (i.e. graviton scattering) and work in the thermodynamic limit for the association between entropy and degrees of freedom to carried over exactly from statistical mechanics. If this approach is sensible, then we would find that our NS string would have contributions to the entropy but not at the first order.

Atish outlined his approach, or ingredients as he put it:

1. Action: N=2 sugra + topological string
2. Entropy: Bekenstein-Hawking-Wald formula
3. Solution: via the Attractor mechanism
4. An ensemble: some Ooguri-Strominger-Vafa mix of charges

He told us he would work with small black holes (= only two charges in the ensemble), where the counting can be done exactly and the classical area vanishes (as we saw above), and so corrections are essential. The approach is detailed in his 4-page paper Exact Counting of Black Hole Microstates and in his talk he commenced by telling us about how to regularise black hole backgrounds by using "stringy cloaks" and this is described in his 10-page paper with Renata Kallosh and Alexander Maloney entitled A Stringy Cloak for a Classical Singularity (you can watch a talk by Andrew Maloney on this paper here). Since the details of the talk are not suitable for blogging I will direct the interested reader to the other relevant and much longer papers written with Frederik Denef, Gregory W. Moore and Boris Pioline, the 35-page Exact and Asymptotic Degeneracies of Small Black Holes and the 103-page Precision Counting of Small Black Holes. Also of interest will be Ashoke Sen's Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity, and you can see the slides and listen to a related talk given by Sen here.

Sorry to trail off without describing the details but one they are tough, and two I am tired. All comments on this approach and joyous sonnets praising (and explaining)the usefulness of the attractor mechanism are welcome. There, and I didn't even mention supersymmetry once, oops.

Update: Check out Jacques Distler's post about David Shih's work on Ooguri-Vafa-Strominger constructions and see also his comments on Dabholkar et al's work and small black holes.

Hawking on Richard & Judy

2005-10-11T18:30:00.000Z

One day's notice for those of you in the UK that Stephen Hawking will be appearing on tomorrow's (12th October) Richard & Judy show, probably to talk about his new book. This is one of the most unlikely pairings I can imagine and should be fun to watch.

Nobel Prize 2005

2005-10-04T11:23:00.000Z

The Nobel prize for physics this year has been awarded to:

(1/2 of the prize) Roy J. Glauber for "for his contribution to the quantum theory of optical coherence",

(1/4 of the prize each) John L. Hall and Theodor W. Hänsch "for their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique."

Many hearty congratulations to them!

Update: the BBC have a short story on the award, and there are also comments from Lubos Motl and Peter Woit.

Topological Mass Generation

2005-10-01T03:00:00.000Z

So perhaps you thought the Higgs mechanism was the only mechanism for mass generation in four dimensional spacetime? Probably it is, but Roman Jackiw, co-discoverer of the axial chiral anomaly with Bell and Adler, spoke at Imperial College yesterday about an alternative and elegant method to generate a 4-dimensional mass term. Roman first described to us the three-dimensional model where a Chern-Simons term can be added to the Lagrangian to generate mass, but told us that his motivation would come from the Schwinger model in two-dimensions. In the Schwinger model massless Dirac fermions are added and then eliminated in order to generate a mass. With hindsight Topological Aspects of Gauge Theories by Jackiw, which is to appear in the Encylopaedia of Mathematical Physics would have been a good article to read before attending this seminar.

Roman went through the original model and then repeated the analysis using a number of dualised terms, he referred to this as going "towards the topological model". In particular he highlighted that the field acquires a mass due to the presence of a chiral anomaly in the axial vector current and leads to a massive pseudoscalar; the pseudoscalar being dual to the two-index field strength as well as being proportional to the divergence of the axial vector current.

Now Roman's aim was to take this two-dimensional model, made out of purely topological terms, and then write out the equivalent expression using the four dimensional topological objects. He said that he would call this topological mass generation since now he would refer to the terms we had before with their topological names.

In two dimensions, using the dualised terms, a pseudoscalar crops up that is the Chern-Pontryagin density, P, and the dual of the potential field, C^\mu=\epsilon^\mu\nu A_\nu, is the two-dimensional Chern-Simons current. These are suitable quantities to take across to four dimensions, however it turns out to be a requirement of the method that the dual of the axial current must be a conserved quantity, and this can be guaranteed to occur by adding two fields, added in the form of Lagrange multipliers to the dual Lagrangian (I omit the details here unfortunately because I still haven't opted for a way of putting TeX in these posts). Surprisingly when one does this in order to conserve the dual axial current, one obtains a gauge invariant dual Lagrangian - the two go hand in hand. The generalisation of the Schwinger model to four dimensions is now straightforward, and is carried out by using the four-dimensional topological terms. Roman finalised by mentioning two shortcomings of this approach, the first being that the anomaly producing dynamics has not been specified and as such this model presents a phenomenological model of mass generation. The second shortcoming was the resulting dual Lagrangian was a dimension eight operator, and this, I am told, presents difficulties for renormalization. However on the positive side the specific contribution needed for the anomaly appears in the expansion of the Born-Infeld action to quadratic terms. Furthermore Roman pondered whether it might not present a phenomenological description for the elusive \eta'. Roman reminded us that the \eta' is the ninth goldstone boson suspected to arise by promoting an SU(3)xSU(3) symmetry to a U(3)xU(3) symmetry. This topological mass generation if it were indeed applicable to the \eta' would give a numerical prediction of the \eta' mass. For some notes about the mysterious \eta' see here. Apologies for any mistakes, one day I will read the much-recommended book by Zee and learn some QCD. Now I've written it here I just have to do it...

Horizon on Hawking

2005-09-16T14:52:00.000Z

Last night Horizon was on the well chosen topic of information loss paradox in black holes. It chose to cover the story as Stephen Hawking's "greatest ever mistake", drawing unbidden parallels with Einstein's greatest mistake. It was hard not to get excited about a popularization of such a technical nature, and I felt somewhat let down when we were treated to the usual combination of ominous voice-over (why, oh why, must science be presented as if it's a horror movie?), violins playing purposeful music, dazzling graphics, and vague presentation of the story. In fact the story was shifted away from physics to one questioning Stephen Hawking's scientific reputation leading the Horizon team to entitle their programme not "The Information Loss Paradox" but "The Hawking Paradox".

I admit though that, before I watched the programme, I was excited about it all day; in the best of all worlds I was hoping to hear some commentary on Hawking's most recent paper, perhaps even some insights that might help me understand it. But alas not. The programme aired at 9pm on BBC2 yesterday, and my spirits immediately sank when the announcer introduced it as "the reputation of the world's most famous scientist at stake". The first images we see are from a beach and there's a wall with at least 6ft high graffiti on it, and the largest graffito of all is of Hawking's equation relating entropy and event-horizon area. Perhaps this was also meant to draw parallels with graffiti of E=mc^2, who knows? The voice over begins, and the camera ranges over the beach to an astrologer:

"What if the world were so strange we could never hope to understand it and science was wasting its time? It sounds like the sort of thing a mystic might say but this was a suggestion made three decades ago by the most famous scientist in the world, Stephen Hawking."

From then on one had the idea that the scientific story was going to lag behind the human story, but, for pity's sake, why?

The introduction focussed on Hawking's celebrity, with Kip Thorne saying of him:

"He's absolutely unique, and I think he has been a very important person in both the intellectual and the cultural life of the past century."

These fair comments were countered by the voice-over's,

"But recently doubts have been expressed by some physicists about Hawking's scientific reputation."

Thereby initiating the main story being addressed by Horizon, that perhaps Hawking ain't so great. Frankly this appeared as unfounded, unsupported and scurrilous journalism used to appeal to a wider audience, and at no stage were any of Hawking's conributions to physics not related to information loss discussed.

The information loss paradox was described as the result of a black-hole evaporating to nothing leaving behind only thermal radiation, i.e. carrying no information. That there would be a problem even if the black-hole didn't disappear was not made clear
(There is clarification about where this is a concern from Christophe Galfard in the comments). Without recourse to quantum mechanics this was described as a violation of one of the most fundamental principles of physics, that information is never destroyed. The voice over spookily summarised,

"Effectively bits of the universe are missing...nothing science knows not even our memories could be trusted."

Lawrence Krauss commented, probably to the delight of the Council for making Science Scary who seem to be in charge of Horizon,

"...at its most extreme scale what it means is everything you come to know and love would ultimately disappear."

This scary comment went without any guiding timescales.

Cue Leonard Susskind who was presented as Hawking's adversary in an immense intellectual battle. Susskind described how he felt a need to resolve how it was that one could watch someone cross into an event horizon and potentially be pulled apart, while the same person would feel no great change as they themselves fell across the horizon. Ah, a good old-fashioned change to Eddington-Finkelstein coordinates, at last some firm ground. The voice-over's interpretation of this coordinate change:

"The same equations were saying that someone could be both dead and alive."

Hmph. Susskind described his resolution that allowed both of these possibilities to coexist and resolved the information loss paradox: holography. That the black hole acts like an information projector and that anything that falls into it has its information "beamed" onto the lower-dimensional event-horizon, thus avoiding losing information inside the event horizon. Although quite what was going to happen to the information when the black hole evaporated was not addressed. Susskind's belief's were presented as being vindicated by Maldacena's paper, Eternal Black Holes in AdS. And finally the programme concluded after the Dublin conference where Hawking conceded that information was preserved. So, no chance of an explanation of the sum-over-topologies approach, ho-hum. There is some commentary by Lubos Motl on the paper Information Loss in Black Holes, and if you are interested in holography you could read TASI lectures on the Holographic PrincipleTASI lectures on holography by Bigatti and Susskind.

Of course despite my disappointment with the lack of theory, the human story was appealing. There were some very nice pieces of footage of Hawking working with his students. In one shot, prior to adopting his synthesiser, Hawking is seen talking to what looks like a seminar with the help of a student, Chris Hull. Chris Hull says that they happen to have a model of the universe with them and pulls out a cylinder and puts it on a table in front of the audience. Hawking makes some comments and grins, the student scratches his head and then turns the cylinder the other way up. It is identicle both ways.

Also, there were some encouraging comments about the trials of a student from Christophe Galfard, who, contrary to my earlier scandalous comments, has pointed out that he does not work in the signature (+---), described the start of his PhD with Hawking, saying:

"For the first year-and-a-half every sentence of Stephen's took me about six months to understand."

And of reading Maldacena's paper:

"I took a little while to read it, a little while being about a year-and-a-half."

Again: here, here!

The show ended with Hawking saying:

"I have no intention of stopping anytime soon. I want to understand the universe and answer the big questions, that is what keeps me going."

At no point did the programme mention the word string, nor topology! Is this really the best way to promote science to a popular audience? Could it not be done with at least a little humour, and less of the portentous voice over? Maybe even less of the human-interest story? After all the science is fascinating if communicated well. Oh for a more perfect world.

Not with a whimper...

2005-09-01T22:27:00.000Z

The BBC news website has a story entitled Black holes start with many bangs. Observation of multiple gamma ray bursts by the Swift observatory, designed to detect very short bursts, improves upon previous recordings of a single decaying burst. The bursts are expected to be associated with black hole formation, radiation from infalling material. But from the tone of the article it seems the astronomers are not clear about the causes yet. While you're thinking about black holes go and look at Jillian's Guide to Black Holes, if you haven't already done so, it is a beautiful site.

Also in the news recently is a new "three line" putative proof of Fermat's last theorem. Alexander Ilyin, a "doctor of technical science" who works in automated data processing in Omsk, unveiled his proof at a press conference on 23rd August, and according to this Pravda article

"colleagues in Omsk believe Alexander's proof is flawless and simple"

Furthermore the article continues,

"Omsk-based scientists and journalist have not found any errors so far"

Journalists are obviously of a much higher calibre in Russia. A follow-up article that fails to make it plain whether or not the proof has been withdrawn, but covers the popular history of Fermat's last theorem very nicely is here, and a discussion thread here.

Thanks to the Mighty Emperor of Room 102, Peter McKeag for pointing out this story.

The Joy of Ping

2005-08-27T02:33:00.000Z

I have been learning about trackbacking, and as the observant reader will note there is now a new trackback link beneath each post. So for those as uninformed as I was "trackbacking" is the name given to creating links to blog entries containing content relevant (hopefully) to the post you have just read. The purpose is really to keep a record of links that didn't make it into the post, principally because the links were created after the post. If you find that last sentence confusing, then don't think twice about going to see the movie Primer (incidentally, I watched it yesterday and found it very confusing, but I'm enjoying thinking thorugh its time-travel paradoxes, so forget my comments and go and see it and then tell me what happened). In short it is a blogger's duty to trackback to articles that have inspired comment.

How does one do this? Well, first you can only trackback to an article if you include a link to that article in your main post. And second you can only do this on blogs that support trackbacking - so standalone blogger blogs which do not support trackbacking have to use an external program, I am using haloscan. If trackback is supported then beneath a post will be a trackback link that gives a URL to ping. In essence, pinging means sending some information to another server to let it know you are there. One pings and is pinged, but one is never punged nor panged, one briefly can be pinging and one can certainly pong. If you keep a trackback-supported blog, then you can use your trackback software to ping other blog articles to let the world know you have something to say about that post. Some blogs have automated trackbacking (e.g. WordPress) where the software automatically pings every link in a post. This is almost enough to motivate a change of blog software...

Why all the sudden fuss about trackbacks and pings I hear you cry/sob? Well as you may have read here, here, here, here, here and here the arxiv is trying out trackbacking. On each abstract page there is now a trackback link, if any exist, so that it would seem that anyone in the blogosphere can comment on any paper. There is some manual checking of trackbacks so that supposedly only bloggers with legitimate comments about papers can add a trackback. It remains unclear how a comment may be judged legitimate, but probably a commen-sense test will suffice. Apart from this it would seem there are few safeguards and I can't decide if this is a good thing or a bad thing. For example, in the past I have attended seminars and attempted to understand a particular paper from the arxiv. Often I have understood a little of the beginning of a paper and have posted comments about that on this page. Now it seems I, along with others, will face the dilemma of whether sparse comments on a paper warrant a trackback to the abstract page of the paper. My feeling is that all legitimate discussion is positive and merits a trackback to the arxiv. I imagine that, if it takes off, trackbacking on the arxiv will offer a connection to debates about the papers content as well as earnest readers' descriptions of their attempts to understand (parts of) papers. To me it sounds utopian, but we'll have to see how it works out. Furthermore it may make the physics blogosphere lose its orbit, so to speak, after all will anyone who wants to make a comment about a paper and have it recorded on the archive also have to keep a blog? Recently there have been several bloggers sending posts from the midst of a conference, and at least one case of specific conference blog. Inclusion of comments from such blogs on the archive would seem to be a very exciting development since these are usually fairly technical, and hopefully useful. Perhaps this will motivate conferences to keep such blogs, and give an indirect link from the archive, via a blog entry, to footage or slides from relevent talks at a conference. Well let's hope so.

Elsewhere, the debate about the greatest physics paper ever trundles on, and still no-one has argued on grounds of simple beauty in favour of Kaluza's 1921 paper about the fifth dimension! Well except yours truly that is. Nevertheless similar to the BBC's series of greatest ever lists, cosmicvariance's lists have begun to hyperbole. Now the debates for the greatest ever physics textbook as well as the greatest popular science book are in full swing. Also Lubos has posted an updated link to the video footage of the talks from Sydneyfest. Also as reported here, here and here a second physics blog has become a book(let's not forget this trend was started by Lieven Le Bruyn)! Peter Woit's Not Even Wrong will be available in UK bookshops from 16th March, 2006. It promises to be an honest review of the toughest problems faced by string theory, and probably a critique of studying string theory with blinkers on. There has been some discussion about the merits of a non-string theorist writing a popular science book about string theory, but you can read them on Not Even Wrong the blog and, probably, the book in due course. Have no fear, I foresee no situation where this blog will become a book, although Tangent Space wouldn't be an awful title for a sci-fi novel (all suggestions for plot are welcome, in return for an earnest acknowledgement in the foreward)...

The 80th Best University in the Whole World!

2005-08-18T16:51:00.000Z

King's College London is the official 80th best university in the world at least according to the recent top 500 list. Peruse the top 100 here, this link has been gleefully hijacked from Goosania.

Also in blogland is Clifford Johnson's attempt to follow where the BBC lead by finding the top 5 greatest physics papers ever written as decided by the readers of Cosmic Variance. I've taken up the challenge and nominated my favourite(s), I'm really hoping I can get fifth place on the list for Kaluza's 1921 paper where he discovered the fifth dimension :) But I suspect a campaign for Kaluza enacted solely by me may not cut the mustard but we'll see... also there have been no string/supergravity papers so far, which ought to be rectified.

There are other lists on the web, but one of the most curiously titled is Forbes list of the best places to die. Perhaps also of interest are the top 200 sci-fi novels (there are plenty of lists like this though, such as this one), Physicsweb's readers' top 20 equations, and there's "The Top 100 Things I'd Do If I Ever Became An Evil Overlord".

It's Oh So Quiet...

2005-08-14T16:34:00.000Z

Things are quiet around here right now, no seminars, students and lecturers are away (avoiding the liquid sunshine) and nothing is stirring not even a mouse...so I have no notes to put up here.

However, the football season has commenced again, so it is a very exciting time, especially if, like me, you are a Queen's Park Ranger's fan as QPR have hit the headlines two weekends on the trot for non-football reasons. Last weekend, QPR fans were taunted by Hull fans about the London bombings and yesterday Gianni Paladini, a director at QPR, was forced at gunpoint to sign a piece of paper detailing his resignation from the board. Well this is sensational stuff and one really has to wonder who would do this and why, after all Paladini as a major stakeholder in the club has invested considerable amounts of money and also been pivotal in recently helping QPR to sign the defender Milanese. In short one would think that fans would appreciate Paladini's work, but also one would think that only very committed/insane fans would be driven to such rash actions. So something is amiss. Nevertheless, on the pitch, QPR have made a good start to the season, joint top in the Championship, and our legendary manager Ian Holloway is continuing to make football more fun, this week he summarised his pre-match team advice as "I told them to be awful." Let's hope he reaches the heady heights of this post-match interview again this season.

But even in the wilderness of summer in London there are plenty of physics links still that can be posted. First and most importantly the videos from LMS symposium Geometry, Conformal Field Theory and String Theory have started to appear online here. The organisers Patrick Dorey, Peter Bowcock and Katrin Wendland have done a fantastic job in collating all the videos so far and have even provided us with some photos of the symposium. In fact, for the curious, there's even a photo of me (I'm on the right) stuck in a castle window; at this point I am halfway through a tour of Bamburgh castle and am looking out the window and seeing the leader of our group (the one with the camera and the watch) a long way away and I'm wondering whether I'm being left behind... but it turns out, from the photos, that they were heading down to the beach and skipping the tour. There are also a number of other conferences that have provided media from their talks online (many of these come via Peter Woit's Not Even Wrong), so you can sit at home (un-jetlagged) and feel like you went to a lot of conferences, these include:

Strings 2005 - sound and slides.

Advanced Summer School on Modern Mathematical Physics in Dubna - A very impressive collection of introductory talks.

Mathematical Structures in String Theory at KITP - this runs from 1st August until 16th December and already there are videos available, and there is even a devoted (and secure!) weblog.

The SLAC Summer School: Gravity in the Quantum World and the Cosmos - videos are available under the titles for each talk.

Simons Workshop in Mathematics and Physics 2005 at Stonybrook - Vafa, Berkovits, Witten, Maldacena et al can all be listened to (slides are not generally available so you might have to listen very carefully...although some talks are close to those at Strings 2005, so some comparison is worthwhile)

A talk by Lee Smolin on Loop Quantum Gravity (from May 2005 with an astrophysics leaning - for the interested there are a host of astrophysics talks in the same directory)

The perfect quiet of this summer is well suited for the Ashes, however in case one thought that the genteel nature of a cricket test match might not be condusive to competition please see this violent yet thoroughly amusing game involving much Ozzie bashing on the Channel 4 website.

P.S. Please don't miss Lubos Motl's $3 challenge to disprove that Amazon, one of my favourite companies, are run by "crackpots" who are only allowing positive reviews of pseudo-science books which masquerade as science.

The End of the Road

2005-07-31T12:53:00.000Z

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!

2005-07-30T12:55:00.000Z

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

2005-07-29T17:34:00.000Z

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

2005-07-28T22:42:00.000Z

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...

2005-07-26T18:34:00.000Z

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

2005-07-24T21:29:00.000Z

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

2005-07-23T21:24:00.000Z

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

2005-07-23T14:07:00.000Z

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...

2005-07-21T02:01:00.000Z

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:

The Superblog is Born

2005-07-19T01:19:00.000Z

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

2005-07-07T15:14:00.000Z

"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

2005-07-06T19:45:00.000Z

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

2005-07-05T12:12:00.000Z

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

2005-07-01T01:13:00.000Z

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 :)

Cosmological Billiards talk online

2005-06-26T15:00:00.000Z

Well, moving house was much more tiring than I thought. Not only did I miss part II of the generalized complex geometry talk but also part III :( But I did get a lot of rest!

Just a short post to say that Thibault Damour's talk at Geometry and Physics after 100 years of Einstein's Relativity is available to listen to online here. His title was Symmetry and Chaos in Gravity and Supergravity and it's an excellent talk if you wish to start learning about cosmological billiards. All the talks from the meeting are available online as I found out thanks to Peter Woit's web research.

If only all talks were available online perhaps then I wouldn't miss so many of them...

Generalized Complex Geometry I

2005-06-20T19:31:00.000Z

This morning, a balmy, humid morning (they say we're getting plenty of hot air from France at the moment) I braved the excruciating temperatures of the tube and ventured over to Imperial College's Blackett Laboratory to hear Marco Gualtieri, a former student of Hitchin, give the first of three introductory talks on Generalized Geometry and Physics. Gualtieri's PhD thesis Generalized Complex Geometry is available online here. Generalised geometry is a popular seminar topic at the moment, and you can find Lubos Motl's description of the Harvard Postdoc Journal Club's discussion of it here.

The talk had been billed as an introduction to generalized geometry, and this appealed to such a large audience that room 1004 was packed to the gills. The first two-thirds of this first of three talks was devoted to setting up the linear algebra. As such it isn't really possible to write it up on the blog, as I still haven't investigated Techexplorer and Mathplayer enough yet :( Nevertheless we did learn some interesting things that it might be possible to express in words, without trying to repeat Lubos' post.

We commenced with a real 2n-dimensional vector space, E, with a non-degenerate inner-product < , >, an (n,n) bilinear form, which acts naturally on T + T*, the direct product of tangent and the cotangent bundles. From here one can form a Clifford algebra CL(E)~CL(n,n,R)=Mat(2^n,R), presumably as {CL(E),CL(F)}=-k< E,F >. If we consider an element of E, X+x, where X is in T and x is in T*, the natural product with a form r being defined as (X+x).r=i_X.r+x^r, where ^ is meant to be a wedge product. Then [(X+x)^2].r=< X+x,X+x >r, and the spin group is given by the even powers of (X+x), so that the parity of r is preserved under its action. The Lie algebra of SO(n,n) can be decomposed into three distinct parts, the endomorphims of T, and components in T* and T. These were labelled A, B and \beta by Gualtieri, with B:T->T* and \beta:V*->V and the specific action of the "B-field" was checked by exponentiating the 2-by-2 matrix whose bottom left entry was B and the remaining entries are zero, and acting with this matrix on the natural column vector formed by X+x. The vector is mapped to X+(x+i_X.B), so the action of the B-field is to translate points parallel to the T* direction, an amount determined by X.

Okay, I've looked back over this entry so far and it's more illegible than most of my other posts, so I'm going to advise any readers interested in the mathematical details, which is probably both of you :), to look at Gualtieri's thesis, for which there is a link above. From now on in, I will just describe the route taken in the talk.

We looked next at the exterior derivative, and wondered about its action on T+T*, and were motivated to use the "derived bracket construction", which is a nested set of two commutators, containing the exterior derivative, and two elements of T+T*. If this derived bracket construction was antisymmetrised with respect to the two T components of the elements of T+T* that it contained then we were told that we obtained the "Courant bracket". We then learned a few things about this bracket, including the fact that it doesn't satisfy the Jacobi identity, and so is not a Lie bracket, as we may have naively been thinking. So the question was raised by Gualtieri of just what the bracket might be and we were led down a path of constructing a "twisted Courant bracket" that would better suit our purposes.

This was motivated by first applying the exponentiated B-field (we recall that this is a group element of SO(n,n,R)) to two elements prior to putting them in the Courant bracket. It turns out that this gives us back the bracket, multiplied by exp(B), plus a 3-form term i_Xi_YdB. The next step is to form the twisted Courant bracket by adding a closed 3-form to the Courant bracket in such a way as to cancel out the dB contribution given by the new bracket's action on the elements of exp{B}.(T+T*).

We finished our first hour on generalized geometry with some geometry! A generalised Riemann structure was built on T+T*. This was started by picking a positive definite sub-bundle C+. This was pictured as the vertical axis on a usual Cartesian grid, where T* and T formed the null boundaries of the light cone. An equivalent negative definite bundle, C-, was picked to be orthgonal to C+. From there C+ was expressed as the usual Riemannian metric, g, plus a 2-form, b, and a general metric G was formed, which when restricted to T became the usual Riemannian metric, and in the general case a different symmetric volume form was found which Gualtieri has called the Born-Infeld volume to "coincide with physics terminology." From here we quickly made use of the Hodge dual and defined "Mukai pairing" and briefly thought about a twisted exterior derivative and its associated twisted Laplacian, before coming to the end.

Unfortunately I have to move house on Wednesday when part II is scheduled to take place, so I will miss the next installment. Fortunately for you poor readers it means you won't have to put up with another near-illegible post. Well at least not until after Wednesday :)

I hope it's less humid on Wednesday.

Solutions and Signatures

2005-06-16T02:23:00.000Z

Permit me to take a brief break from doing the new paper dance as first seen performed by The Quantum Pontiff, to make some comments. (By the way, make sure you check out Mr. Bacon's link to footage from the 1927 Solvay conference: Dirac, Bohr, Heisenberg, Pauli, Schrodinger, Lorentz, Einstein, Curie and many more all in two sparse minutes of fascinating footage.) For indeed I have a new paper out today, cowritten with Peter West:

Title: M-Theory Solutions in Multiple Signatures from E_11
Authors: P. P. Cook, P. C. West
Comments: 34 pages, LaTeX2e

I think "LaTeX2e" is a fairly uninteresting comment, but it isn't the done thing to make interesting comments in this section. I won't make many comments here either, as those who are interested can read the paper in their own time. In the paper we apply Keurentjes' observation's that Weyl reflections alter the signature of the local sub-algebra generators to the solution generating group element of E_11/H_11, mentioned here previously. Principally, we find that any solution of a (1,10) theory coexists with solutions in (2,9), (5,6) and their inverses. These are Hull's M* and M'-theories. It is a puzzle, beyond the scope of the paper, to understand how so many M-theories might all fit together.

Now I am free to think about other problems, and for a trainee theoretical physicist there are many, and to try and work out just how many ways there of constructing an N element string using only the two elements {A,B} such that no substring has a difference of more than three between the numbers of A's and B's in it.

Playing Catch-Up (Or some E11 related papers on hep-th)

2005-06-07T22:06:00.000Z

Summer is high, well it's not raining every day at least, and seminars are rare. I have just got back from Riga, in Latvia, where I went for a stag-do, and I hope this explains the recent lack of posting, not even a postcard I'm afraid. Riga is very pretty by the way.

In the past two weeks there have been no less than three papers with some relevance to E11 and my line of research, and I really ought to try and understand them all. In the meantime let me simply list them and make some comments:

1. Dualities and signatures of G++ invariant theories by Sophie de Buyl, Laurent Houart and Nassiba Tabti

The first paper makes use of Keurentjes' observation that Weyl reflections do not commute with the involution used to choose the local subalgebra. In an earlier paper one of the authors, Laurent Houart, together with Francois Englert and Marc Henneaux had applied the observation that a reduction from a very-extended theory, E11, to an over-extended theory, E10, by the deletion of an "end node" on the Dynkin diagram gives rise to two distinct theories. The two theories are arrived at by, in one case, applying a Weyl reflection in the deleted node's associated root before deleting the node, and in the other case by direct deletion of the node. The two resulting E10 theories have different signatures. This paper extends considerations of this idea to all the other G++ theories.

2. Hidden Symmetries and Dirac Fermions by Sophie de Buyl, Marc Henneaux and Louis Paulot

The second paper introduces spin one-half fermions into the G++ theories, but as I haven't read this thoroughly yet I will not say much. According to the introduction this results in the chaotic motion reported in the cosmological billiards picture being lost. Furthermore the null geodesic in E10/K(10), which encodes dynamics, becomes timelike once the spin one half fermions are present. I will add any further comments here later, if they come to me :)

3. IIB Supergravity Revisited by Eric A. Bergshoeff, Mees de Roo, Sven F. Kerstan and Fabio Riccioni

The third paper is also very interesting, and the idea is straightforward to describe. Back in 1983 when the IIB supersymmetry algebra was first written down, branes were not an important concept, so the only gauge fields that were considered were the two scalars, the two-form and the four-form. These couple to a string and a three-brane. Since then more emphasis has been placed on the five-brane, the seven-brane and the space-filling nine brane, which couple to a six-form, an eight-form and a ten-form respectively. The authors of this paper introduce these extra gauge fields to the IIB multiplet of fields without introducing any degrees of freedom, by asserting that a duality relation between the extra fields to the 1983 multiplet of fields. The supersymmetric variations of the new fields are given, and it is noted that the duality condition that was asserted is really a necessary condition for the algebra to close. They find that the ten-form can transform as a doublet and a quadruplet of SU(1,1) and the authors argue that no other independent ten-forms can be added. The findings for the ten-form multiplets match the previous deductions from E11 given in Very-extended Kac-Moody algebras and their interpretation at low levels by Axel Kleinschmidt, Igor Schnakenburg and Peter West. (Thanks to Sven Kerstan for kindly talking through the ideas of the paper with me, however, as ever, I proudly take responsibility for all mistakes in these notes :) )

I've Been Meme'd

2005-05-23T19:22:00.000Z

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"?

The Spinorial Geometry Programme

2005-05-13T18:15:00.000Z

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... :)

E11 in the Reference Frame

2005-05-06T00:12:00.000Z

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:

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 :)

Triangle Seminar Troubles

2005-05-05T20:01:00.000Z

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.

Giuseppe describes Maldacena's Long String

2005-04-30T21:56:00.000Z

This is a delayed write-up of a similarly postponed group meeting. It seemed appropriate. The meeting was scheduled for monday but started last wednesday at 1500hrs. The informative email from Sylvain told us that Giuseppe d'Appollonio would talk about Maldacena's paper on the long string in two-dimensional string theory and non-singlets in the matrix model, hep-th/0503112. For the definition of long strings as classical string trajectories in a non compact space, we were referred to hep-th/0001053. Naturally I managed to get to grips with all of the first four pages of the main paper, and I printed off the second. I had also read Lubos Motl's description of a talk on this very paper (I would encourage you to read this article for your general education, and then come back and read this article to satisfy your curiosity to see just how little I manage to get out of a group meeting, in fact just read Lubos' blog). This, I felt, put me in a strong position, to just maybe get something out of a group meeting for once. However, I know nothing of the matrix model.

Fortunately, and kindly, Giuseppe began with a brief overview of singlets in the matrix model, beginning by telling us that the matrix model is a way of describing two-dimensional gravity, and that regularisation of the theory comes from the triangulation of the two-dimensional surface. If we take M to be an N by N matrix then we have can write an action of the form

\Int dM exp{-\beta \Int[ Trace {(dM/dt)^2} + V(M) ]}

Where "Trace {(dM/dt)^2}" is a kinetic term and "V(M)" is a generic potential term defined on the matrix. \beta was added after the rest when a short discussion of the continuum, or double scaling, limit occurred, see later. GdA asked us to consider the case when M is a matrix of constants, correpsonding to a flat geometry, so that we could get a beginner's feel for the model. We note that the kinetic term vanishes in the action in this case.

GdA next wrote an explicit formula for the potential:

V(M)=\sum_N[ { g_N \over N } Trace {M^N} ]

GdA described quadratic potential as a harmonic oscillator and drew two parallel lines of interaction on the board, the cubic potential and drew three interactions, the quartic and drew four interactions. For triangulation, he said in passing, a cubic potential was used. Next the double scaling limit was defined as the continuum limit coming from sending N to infinity along with appropriate limits on \beta, such that the string coupling constant g_s, (not shown here, but perhaps hidden in \beta, or maybe a notational error on my part: perhaps g_N should be g_s, any comments are welcomed) which is proportional to 1/N, remains finite. I should emphasise the potential for my notational errors in this write-up, so be wary.

GdA then lead us through a diagonalisation of the M, commencing with a Lagrangian, L:

L = 1/2 Trace {dM/dt}^2 - V{M}

M(t)=U'(t)L(t)U(t) where U' is the complex-conjugate of U, a unitary matrix; and L is the diagonal matrix of eigenvalues of M. We note that V(M) is unchanged, but the kinetic term is rewritten:

Trace (dM/dt)^2 = Trace (dL/dt)^2 + Trace [L, (dU/dt)U']^2

To see this judicious use is made of (dU/dt)U'=-U(dU'/dt), which comes from d/dt{UU'}=0. The Hamiltonian may be written as

H=-1\over D(L)\sum d^2/dL_i^2 D(L) +\sum V(L_i) +\sum_{i lt j} [\Pi_{ij}^2 + \Pi'_{ij}^2]\over [L_i - L_j]^2

Where D(L)=\Pi_{i lt j}(L_i-L_j), and "i lt j" means "i less than j". The case where the difficult last term vanishes is considered, i.e. when \Pi_{ij}\Psi = 0, so that \Psi transforms trivially under SU(n). Under this condition this is the Hamiltonian for the matrix model in the singlet sector. D(L)\Psi gives an antisymmetry condition in the eigenvalues so we believe this is really a fermionic theory in an upside down harmonic oscillator potential. States are filled up on only one side of this potential. So that's my beginner's knowledge of the matrix model, in a nutshell.

GdA moved onto the long string paper, which is a proposal for the non-singlet sector in the matrix model. Maldacena considers an action,

S{\phi}=\int\sqrt{-g}d^2x[g^{ab}\partial_a\phi\partial_b\phi + QR^(2)\phi + 4\pi\mu exp{2b\phi}] - \int d\tau d\sigma [\partial_+X^0\partial_-X^0]

(Yes, I know, I need to start using mathplayer or come up with some trick for displaying formulae.) Where we have the usual k.e. term, a potential poportional to the dilaton and a Liouville potential term. The ansatz X^0=\tau has been used. GdA told us that Maldacena first considers the case where \mu=0, only the linear dilaton potential remains. The solution is described by a path in the worldsheet (\tau,\phi) that commences at -\tau at minus \phi infinity with a gentle positive gradient. This increases quickly at positive \phi and crosses the \tau=0 in this region; the rest of the solution is the reflection in the \phi axis, giving an overall path like the contour of the end of a cigar. Despite this being a worldsheet picture, the target space, time-evolution picture matches the intuition that the string is stretched all the way in from infinity. This is called the long string, as it extends back to minus infinity twice, so is very long; in fact it is so long it can even be seen in London all the way from Princeton where it originates :) One considers the tip of the string as the centre of mass, \phi_m. This can be thought of as a massive particle moving under a force that pulls the centre of mass back towards negative infinity. In terms of string tension, T, the energy, E, is E=2T(\phi_m - \phi_c). The "2" comes because the string doubles back on itself and the tension acts on the centre of mass twice in the same direction. N.B. this is the energy required to stretch the string a length 2(\phi_m-\phi_c). \phi_c is a cutoff in the energy, and is therefore a little contentious, but we can understand that if we didn't make a finite cutoff then the change in energy in our calculation would be divergent, due to the stretch being from minus infinity.

Now comes a neat suggestion, Maldacena introduces an FZZT brane that is extended back to minus infinity in \phi. The long string is then considered as an open string attached to the FZZT and stretched to its limit in positive \phi. The cutoff at \phi_c is now seen to cut the FZZT brane out of the picture and also subtracts the divergent energy associated to the excitations of the brane, which extends back to infinity.

This took us up to page 9 of the paper. The remainder was covered much more quickly and no meaningful commentary on it can be made here by me, alas. So in all, I gained 5 pages on my preparation through the meeting, this was very satisfying, and perhaps a record.

Freund-Rubin Superpotentials Revisited

2005-04-27T16:50:00.000Z

Neil Lambert gave a seminar today on his paper from earlier this year entitled "Flux and Freund-Rubin Superpotentials in M-Theory", which is a continuation of work begun here. As usual my desire for an efficient arrival, i.e. get there at start time +/-delta where delta is minimized by my journey resulted in a small additive delta, and so I missed the very beginning. I'm not sure this is crucial as most of the material was unfamiliar to me.

Anyway Neil was asking 2 key questions as I got there, about the vacua in the landscape or haystack:

-Is the standard model there?
-How special or generic is it, if it is indeed there?

At this point NL emphasised that Douglas et al. who are working on the landscape simply aim to count the number of vacua there are and not the probability density of each vacuum existing (which may or may not be a delta function). NL summarised that "the counting statistics of the vacua is maths, the probability density is the physics." In the coffee break afterwards he pointed out the potential usefulness of the counting statistics, in that one may be able to find for some cases (i.e. specific vacua criteria) an absolute count that is less than one, in which case there is no such vacuum. This tells us about the probability density of the vacua and in such cases we learn some physics. It seems, at least in London, that the landscape is not as quite the rage here as it is in North America at the moment, so this elaboration was needed.

In the introduction NL stated that in the talk we would be concentrating on M-theory vacua M_4 x X, and that for N=1 Minkowski vacua X is the group G_2. He then told us that over twenty years ago Freund and Rubin introduced a class of M-theory (D=11 supergravity back then) "compactifications" with spacetimes AdS_4xX. To obtain N=1 supersymmetry in this case X is weak G_2. Weak G_2 was defined as a pair of conditions on the three-form \phi and its dual *\phi contsructed in the theory:

d\phi=4\lamdba *\phi, d*\phi=0

At a later point NL mentioned that while G_2 is restored under the condition \lambda=0, it is not true to think of weak G_2 being close to G_2 for small values of \lamda. In the paper it is pointed out that this is expected as \lambda can be made small by increasing the volume of the manifold, but this clearly doesn't change the properties of the manifold in a non-trivial way. NL said that his paper was essentially redoing the paper of Beasley and Witten but with weak G_2 rather than G_2.

The main body of the talk closely followed the layout of the paper, but without giving the detail of the calculations, and was a lively talk, however since so much of this was alien to me I don't feel comfortable trying to regurgitate it here. So I will just give three of NL's summary points:

1. the consistent construction of the the potential and superpotential of the Freund-Rubin compactications in the presence of topological fluxes.
2. that when the fluxes were turned on the F-R terms were driven to zero, resulting in a non-supersymmetric minimum
3. there are no supersymmetric vacua other than F-R or pure G_2

A major concern of NL's was that the Kaluza-Klein modes arising from the compactification are of the same order as the cosmological constant, and he was particularly interested in looking for ways to lift a decently small, positive cosmological constant from the theory using a "KKLT mechanism". For NL this means not fine-tuning the KK terms to get the desired cosmological constant but rather seeing if it was even a possibility. One scheme he considered involved the wrapping of a 9-cycle at which point he stopped and admitted, with spirit, that he didn't think there was a single person who beleived in a 9-brane. This was perhaps the second-most humerous comment of NL's upbeat seminar, the first being upon a specialisation to the bosonic case when he said "since I'm a supersymmetry guy, I never talk about fermions." Quite right too - in truth, I only wish that I could.

After one question, Nicolas announced the usual "coffee and cookies" in the coffee room, which was greeted by a very silent joy and gratitude.

Duff Talk Missed

2005-04-26T12:15:00.000Z

Due to the relatively early hour of Duff's talk, 1300hrs, and the need to crusade across London from East to West to get there, well, in short, I missed his layman's guide to M-theory. Any second-hand news of this talk will be reported here at a later date, and sources will be given.

Donaldson on Curvature and Physics

2005-04-25T23:15:00.000Z

I trotted out of Drury Lane and off to Imperial College's Clore Lecture Theatre this afternoon to see Field's medallist Simon Donaldson talk on "Curvature and Geometry in Physics". The talk is the first of a series this week to mark the launch of IC's very own Institute for Mathematical Sciences, of which Donaldson will be the first president. Tomorrow Michael Duff, soon to be Principal of the Faculty of Physical Sciences at IC, will give a talk entitled "A Layman's Guide to M-Theory" - hopefully we will hear his favourite meaning for the M. But the rest of the week's talks are less relevant to theoretical physics, it's almost as if there are other mathematical sciences beyond high energy physics :)

Michael Duff introduced Lord May of Oxford, head of the Royal Society, who said hello and then told us that "Simon Donaldson needs no introduction", but, of course, we had already had two. With a slight buzz of interference the microphone system was switched to SD who began by telling us that he was there to fight the pure mathematician's corner, but had in deference to the name of the new institute included a science, namely physics, in his title. This was to be expected. SD used a series of very clear slides, making three or less points per slide, and began the talk by asking the audience to consider the manner in which a line field gives rise to integral curves and whether or not there was an analog for plane fields giving rise to integral surfaces. The answer in general was no. Then to captivate the mind of the physicists in the audience SD exhibited the example of a trolley moving on a flat surface, and then transported it to a plane field. In the flat case the trolley moved around a quadrilateral returns to its starting point, but in the plane field example the trolley is generally displaced along the normal to the plane field at the start/end point. The curvature was defined at this point as a skew-symmetric map from the plane field, P, to the normal to the plane field P': PxP->P'. And from this starting point four relations to curvature were summarised as

1. Deviation from the flat model
2. Intrinsic/Extrinsic
3. Parallel transport
4. Integrability conditions for an overdetermined set of variables

No. 2 was elaborated on and the difference between the two types of curvature was described as whether or not a fly (presumably a mathematician constrained to the manifold [the mathematican having by supposition the dimension of the manifold, without any harm coming to him or her] and not really a fly) living on the surface would be able to deduce if it was living on a plane or not. If the 'fly' could tell, e.g. if it lived on a sphere, then the curvature is intrinsic. If the 'fly' could not tell, e.g. if it lived on the curved side of a cylinder, then the curvature is extrinsic. This was good to know.

SD then made some comments about general relativity because it is the most common place curvature turns up in physics, arising from the pseudo-Riemannian metric, g, on the space-time manifold. But SD decided to focus on curvature in electromagnetism instead. The geometric interpretation of EM comes through complex line bundles, L, over flat space-time, R(3,1); so that L~CxR(3,1), but not as he emphasised canonically. There is an association between the scalar and vector potentials of EM and the field of subspaces inside L (connection) and furthermore that the Maxwell field strength, F, is asssociated to the curvature of L. SD continued that the complex line bundles were only apparent in quantum mechanics, where the wavefunction was really a section of L, and the Hilbert space of wavefunctions is really a set of sections of L. Gamma(L) was defined, here for use later, to be the set of sections of L. This part of the talk was rounded off by a comment that in string theory there is a more complicated bundle.

The focus was then shifted to curvature and geometry, commencing with topology and the Poincare conjecture in three dimensions and SD's quiet endorsement of Grigori Perelman's work. The Ricci flow of the metric with respect to a parameter, theta, was given in a short equation as partial{g}/partial{theta}=-Ric(g_theta) (here Ric means the Ricci curvature defined from the metric) and the question of whether one could deform an arbitrary three-dimensional metric to Ric=constant of the three sphere was described.

Next, complex algebraic geometry (CAG) was looked at. SD commenced by saying that CAG involved making an association between a set and a ring. For example the set of points {x,y} making a unit circle is associated to the ring {Set of all polynomials on x, y with complex coefficients}/{}. Projective geometry was briefly described as the association between a vector z~Kz where K is a non-zero real(?) coefficient (as an example of the results of a projection was the equivalency of the parabola, ellipse and the hyperbola, and this motivated the definition). An example set was given using the projection as X={z: f(z)=0}/{z~Kz}, where f is a homogeneous polynomial. SD then declared that in projective geometry one has sections of line bundles as oppose to functions, such as L -> X and that a ring can be constructed as R= Direct Sum from k=0 to infinity of Gamma(kth order tensor product of L). At this point the example of toric varieties was discussed and entirely lost on me but it looked interesting nonetheless, and can be added to my list of interesting things to look-up one day, which is frankly far too long. When I came to, the basic principle of the talk was being given inside a red box, so it must be important. It is that there is an association between positive curvature (high g) and Gamma(L) being large (many sections).

Some more was said. Then a recap was given, which consisted of a picture with a line bundle, L, over a set then from the left hand side of L emerged an arrow to algebraic geometry and the ring defined as the direct product sum of Gamma(L)'s, and from the right came an arrow to differential geometry, and a line of arrows from curvature of L = F, to g (suppose positive definite), to curvature of g (Ricci curvature). Finally the question why was asked and the answer was given in the example of Calabi-Yau metrics, where the set was given as X={a^5+b^5+c^5+d^5+e^5=0}/{Some equivalency relation, not given}.

Afterwards a reception was held on the eight floor and some very nice food was enjoyed by Vid and myself and I had a glass of wine. We didn't talk about curvature.

New Blog Promise and a False Start

2005-04-25T13:51:00.000Z

Ok, so this has become one of the worst kept blogs online. But, hopefully, no more! The new semester is just beginning here at King's so I hereby vow to attempt to publish the happenings of the seminars and group meetings each week. However today's group meeting has been postponed till wednesday. However, Donaldson is talking at Imperial later today, so all being well, I will record the goings on here later.

Political Argument Dimensionally Reduced

2005-03-15T15:34:00.000Z

Thanks to thegreenfairy who has inadvertently helped me better locate my political self by pointing me to the political compass, as well as being a very entertaining read. The compass test asks you six pages of questions and then returns your political bias in two dimensions: one being an economic left-right measure, the second being an authoritarian-libertarian axis best described on their site. So in future I hope to forestall political debate by quoting my numbers at would-be agitators: -0.88 economic left/right and -5.18 libertarian/authoritarian. Of course everyone knows that a full theory of politics is 11-dimensional, but I think there is much to be gained from a working two-dimensional theory.

Titan and Rat-Attack Bombs

2005-01-15T16:13:00.000Z

First, my apologies for not updating more frequently but that is what comes from trying to keep too high a tone to the blog. So on a lighter note...
Today we are priveliged to see the first pictures of the surface of Titan, and they are amazing.

This one, from the BBC news website, shows an image taken while the Huygens probe was 5 miles above the surface. One has to remember that before this it was not even known whether Titan even had any land, it was suggested that its composition may have been liquid. So that these pictures show land was perhaps the most interesting outcome that could have been hoped for. One very interesting aspect of this discovery and the rapid dissemination of these pictures was that for a few hours astronomers and the layperson were levelled in their knowledge of Titan. For example, early analysis from "experts" of pictures like the one above was that the flat, dark, featureless area was probably a liquid ocean and that the dark lines were a river delta. Which is, of course, as much as any non-expert would have assumed. The probe was fortunate enough to land on a solid surface, and some of its final images were of very flat smooth objects, many of which were almost spherical. These were interpreted as rocks, the deductions then probably run that the smooth upper parts of the rocks could be explained as weathering from Titan's atmosphere. However, the curvature at the base could not be explained in the same way, instead the next most likely hypothesis is that the rocks are as they are due to fluid wearing, hence the deduction that the lucky Huygens probe not only landed on dry land but on a flood plain, when the flood was out. It's an exciting and rare thing for us laypeople to be able to think about cutting edge discoveries, perhaps this is how it will be when the LHC starts producing results, but I doubt it will be tangible enough for the evening news, no matter the import.

Also in the news today is the unveling of some incredible US military chemical warfare proposals from 1994 and earlier. Suggestions for development included a bomb to deploy an aphrodisiac that would turn the massed ranks of the enemy into a raging homosexual orgy: brothers in arms indeed. It is most incredible that these weapons, although not developed, are possibilities. There were also proposals for chemicals that would cause wasps to sting and angry rats to attack; can you imagine being the person who decides when this weapon has been most effective: "are the rats angry enough yet?". Even more unfortunate, for fans of slapstick at least, was that another weapon to make the enemy flatulent was suggested, but, ahem, there was no follow through :)

Simple Groups Post on Lievin Le Bruyn's Blog

2004-11-01T22:49:00.000Z

Scooting around the set of blogs seeding from the String Coffee Table to try and filter the most appropriate links from the many good links, I came across a simple groups post on Lieven Le Bruyn's blog @matrix which is required reading if you are here. I am keeping the best links so far on The Planet Thundera. All recommendations are welcome via comments.

Formerly Promised Notes...

2004-10-30T01:01:00.000Z

I have uploaded the notes I made on Cahn's book and used to give a talk to my esteemed fellow students at KCL in room 102 (All hail the Emperor!). These are a work in progress and are intended to give a speedy, rather than detailed, overview of the first few chapters of Cahn. http://www.mth.kcl.ac.uk/~ppcook/

Review of E11 and M-Theory

2004-10-17T20:08:00.000Z

Those interested in the emerging relations between E11 and M-theory could do worse than read Ling Bao's impressive masters' thesis "Algebraic Structures in M-Theory" which is in postscript format. Topics include nonlinear realisations, Kac-Moody algebras and symmetries in M-theory (principally E11), the conformal group and more... http://fy.chalmers.se/~tfebn/LingMasterthesis.ps

Semisimple Lie Algebras, Roots, Dynkin Diagrams and All That...

2004-10-17T14:14:00.000Z

Robert Cahn's excellent book Semisimple Lie Algebras is available for free at http://www-physics.lbl.gov/~rncahn/book.html soon I will post a quick summary of it's first few chapters.