Friday, March 09, 2012

Is this a simulation?

There is reason to suppose that this, all this, is a simulation. Not the least enticing reason, but perhaps a misleading one, is that quantum mechanics tells us that at a small scale the world is described by probabilities and statistics. Not only is it described by a statistical distribution but unless we probe closely the statistical distribution is not rendered for us to notice. This is efficient and puts us in mind of a simulation or video-game where an algorithm decides what will be displayed on screen: if we zoom up close the texture of the simulation breaks down, but how it breaks down into pixels is determined by the rules of the program.

Heisenberg took seriously the notion that only what we can measure exists. Not so long ago I enjoyed Quantum: Einstein, Bohr and the Great Debate about Reality by Manjit Kumar and learnt that Heisenberg had been inspired, long before he knew much about matrices or anti-commuting variables, by the track of a charged particle through a cloud chamber. As the particle passes through the vapour of the cloud chamber it ionises molecules around which other molecule condense. Thus a track of condensed vapour is formed following the path of the charged particle/atom/molecule. But, reasoned Heisenberg, although the path appears to be continuous it is actually only a sequence of points which occur where each ion in the vapour is formed - unless it is measured the particle's postion is not known for certain. Of course faced with this thought Heisenberg opted for the wild solution that when the particle's position is not measured it does not exist. In the film version of Michael Frayn's play Copenhagen, Heisenberg is shown being inspired while walking beneath a series of pools of light and then vanishing in the shadows between. It all leads very temptingly to the idea that, just like in a computer game, wheresoever we do not look is not rendered and that the fine-grain detail could be displayed by according to an algorithm. So is this a simulation?

Let me take a different tack and wonder whether it might be possible one day (let t tend towards infinity...) to construct a simulator. We might dream of something into which we may project all the information in our most advanced brain-state and where all its functions can be replicated as if in the real world, or perhaps I should say the world we conceive as real presently. It will be informative to think about this notion of the  real world. To do so permit me the assumption that there is some grand unified theory possessing a large symmetry and let the simulator we are considering be a perfect simulation. That is there are no flaws in the simulation that would allow the insider, the brain in the vat, to deduce that he/she/it exists inside the simulator (the cat from The Matrix is back in the bag, so to speak, and there are no rounding error problems inside the simulator for a messianic figure to take advantage of...). Note that I am taking this to mean this means that machine must encode all there is to know about M-theory in one form or another. Finally let us introduce a new constraint, let the machine be localised, i.e. it is not infinite in any direction - it can fit in a bounded volume, nutshell or even teapot, a big enough teapot. All the assumptions that have been made can be recast as saying we have a quantum gravity simulator in a box, let us call it M-box.

M-box buzzes, fizzes, gurgles and pops, and by assumption recreates all the information content of a universe. It must be one hell of a machine. If we suppose it could exist what are the corollaries for the physical description of the universe it exists within and simulates perfectly? Since we assumed it is finite in extent and yet contains all of M-theory must we throw out of M-theory any non-local effects that occur? After all the machine is localised and can recreate perfectly M-theory. By localised I had better emphasise that I mean that its internal workings do not rely on any non-local physical effects. This point is worth some more words. Our usual picture of a computer simulation involves information encoded in 1's or 0's in bits of information, more ambitious quantum computers would build information upon data units that can be in a superposition of two states - these are called qubits. Qubits could be built on electron spin for example. An electron is a relatively simple solution of quantum field theory. For the M-box we would like to build its circuitry on solutions of M-theory; we would like the M-box to be able to excite membranes extending into the compact higher-dimensional space. Membranes can be spun and vibrated, but membranes can also be U-dualised into fivebranes. To take this to its limit the clever M-box we imagine will encode data by U-dualising membranes. We try hard not to think about how it might do this. But we emphasise that the M-box is imagined to encode the full set of M-theory solutions by U-dual rotations of a single membrane. Mathematically the object which encodes all these solutions is the coset $\frac{E_{11}}{{\cal R}(E_{11})}$ where ${\cal R}(E_{11})$ is a real-form of $E_{11}$. Now we may ask the question is there a single solution to M-theory which itself possesses the full symmetries of M-theory. The closest we know of is the BKL cosmological singularity which is expected to carry the symmetries of $\frac{E_{10}}{{\cal R}(E_{10})}$. But awe-inspiring as a cosmological singularity may be it is still not enough to be part of the circuit board of our conjectured M-box. If we did construct such an $E_{10}$ M-box then once we embedded our brain-state inside the simulation we would be able to deduce that some parts of the full $E_{11}$ symmetry are missing (we would probably need some simulated psychiatric help after our simulated selves came to this conclusion). So if one could identify a solution to M-theory which itself possessed a full $E_{11}$ symmetry then we would be in business and M-boxes could start to be rolled off the manufacturing line. However as things stand if we deduce that there is an $E_{11}$ symmetry to M-theory and we wondered if we were simulated we might imagine that the simulator is not an M-box but rather an M'-box (surely this is a catchier name than M-box 360?) which is running in a universe which is governed by M'-theory - which has a larger symmetry than $E_{11}$. The argument is then repeated so that M-box $\subset$  M'-box $\subset$  M''-box $\subset \ldots $.  One would agree that this is not an enticing picture for someone looking for a closed description of the universe and hoping that we do live in a simulation. One of the attractions of a Kac-Moody algebra is that it has infinite generators and one might hope yet that a clever way to embed $E_{11}$ inside itself could be found and associated with an M-theory (bound-state) solution. Alternatively keen simulationists might prefer to spend their time in Skyrim instead.

Wednesday, April 20, 2011

Theoretical physics inspires art!

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

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 

Tuesday, February 08, 2011

Spinorial Representations and Dynkin Diagrams

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


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-Richardson 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...

Sunday, July 06, 2008

Day Four of Eurostrings 2008

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.

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*.

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").

Thursday, July 03, 2008

Halfday Wednesday

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.

Tuesday, July 01, 2008

Day Two of Eurostrings 2008

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 lectures 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.

Monday, June 30, 2008

Eurostrings 2008

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.