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

## Tuesday, December 05, 2006

### JHEP Editorial Plea

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:

## Wednesday, November 08, 2006

### Lisa Randall Online!

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.

Thanks to Coco Ballantyne for the head's up.

## Tuesday, September 12, 2006

### A Penrose Universe

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,

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.

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.

## Monday, September 04, 2006

### To Commute or not to Commute...

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:

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?

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?

## Saturday, August 05, 2006

### Physics: More damaging than drugs?

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."

"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."

## Thursday, June 15, 2006

### The Klein Four (A group)

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.

They have an album out, full of more maths puns than you can shake a

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

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.

## Wednesday, June 07, 2006

### Cargese: The Lectures

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

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

0800-0900hrs BreakfastWhich 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.

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

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

## Sunday, May 28, 2006

### Living in the Theorists' Paradise

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.

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.

## Monday, May 22, 2006

### Back in Black

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:

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,

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:

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.

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

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.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

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,

Since videos and transparancies are available for all talks this conference site is highly recommended, also of interest will be the following talks:E11 and Ten Forms by Peter West Hidden Symmetries and Fermions in M-Theory by Axel Kleinschmidt

But there are plenty of good talks available here, so go and find out the latest from your favourite stringy research area.The Quantum Structure of Black Holes by Samir Mathur Singularities, Black Holes, and Attractor Explosions by Eva Silverstein

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:

Now I have to make sure my thesis is nothing like these reviews...Ho-hum.Algebraic structures in M-theory by Ling Bao Hidden Symmetry Unmasked: Matrix Theory and E(11) by Shyamoli Chaudhuri

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.

## Monday, March 27, 2006

### Kallosh on Attractors

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:

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.

"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):

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).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

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.

## Wednesday, March 22, 2006

### Attractors

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...

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:

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:

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.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

## Wednesday, March 01, 2006

### Classifying Rational Conformal Field Theories

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:

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:

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.

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:

But he only had enough time to talk a little about the first and describe to us the modular functions that appear.modular invariants (i.e. partition function on the torus) NIM representations (i.e. partition function on the cylinder)

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.

## Monday, February 20, 2006

### Does your ball roll at normal speed?

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.*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.

'Our pitch is a bit like that. From the top it's a disgrace but the ball rolls at normal speed.'

## Monday, February 06, 2006

### Poncelet's Porism

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 "

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:

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:

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."

*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´rismPoncelet'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:

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.

There are plenty more of these animations to be found here.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

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

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:

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.

"The distribution of dark matter bears no relationship to anything you will have read in the literature up to now," explained Professor Gilmore.The BBC article notes that the research findings have yet to be submitted to a journal so hold your horses...a little.

"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."

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.

## Thursday, February 02, 2006

### Blogtastic

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...

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...

## Tuesday, January 24, 2006

### Automorphism Groups

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 Londonfor 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 "

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.

So yesterday I headed over to City University Londonfor 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.

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