Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Sunday, November 26, 2006

Hoppy Holiday

On a hot summer's day it is a relief to gain some altitude, albeit only a few hundred metres. Canberra, the capital of Australia, is situated south west of Sydney in the Southern Highlands.
Next week we'll be busy at the Morgan-Phoa Workshop at ANU. I'm looking forward to visiting my many friends there, the galahs and cockatoos. Recommended reading for the holidays includes Carl Brannen's wonderful new book and maybe a little topos theory.

Saturday, November 25, 2006

M Theory Lesson 5

Some days, when I look at the world, I think Nature's purpose was for us to breed a lot of dogs, so that when we all disappear, fighting each other for meagre food and water sources, the dogs will inherit the Earth.

The gallant kneemo and I have been thinking about (and writing about) this very interesting work of Brown. As Louise has discovered, equation 7.31 on page 96 is indeed very reminiscent of the Veneziano amplitude. It might also be worth noting that the other integrals of this type for n points are precisely the n point functions. Who would have guessed? So much physics just falling out of a bunch of associahedra.

Of course this is what we expect for gluons. Modular operads and other such goodies can be used to deal with higher loops, and higher operads are brought in to investigate other Standard Model particles. It's still early days for M-theory, but that vacuum seems to have disappeared in a puff of smoke. Perhaps a few more people will become interested in it soon, because one begins to suspect that the String picture, built with manifolds upon manifolds, is not actually relevant to real physics at all. Oh, dear.

Friday, November 24, 2006


And, as in uffish thought he stood,
The Jabberwock, with eyes of flame,
Came whiffling through the tulgey wood,
And burbled as it came!

One two! One two! And through and through
The vorpal blade went snicker-snack!
He left it dead, and with its head
He went galumphing back.

"And hast thou slain the Jabberwock?"

For the last few months I have looked everywhere for the squelchinos and squarkions. Under the bedsheets, high in the trees. Side of the toolshed, deep under seas. But alas, blind as I may be, they have failed to materialise, even though we now all tread the same path. Where have they gone?

Thursday, November 23, 2006

M Theory Lesson 4

Last time we looked at the MHV diagram technique in twistor String theory. Tree amplitudes in N=4 SUSY Yang-Mills might as well be thought of as QCD amplitudes. The real advantage of the MHV technique is that lower loop terms feed into the structure of higher loop ones, so the recursion is highly constructive. Where might this come from? I confess now that we have not been talking about operads without a purpose in mind.

But one thing at a time. The brilliant young Mahndisa has wisely shown enthusiasm for the work of Satyan Devadoss. Devadoss has spent a lot of time focusing on the moduli of punctured spheres M(0,n), or rather the real points of the compactified space. These spaces are tiled by associahedra. Those are the lovely polytopes of Stasheff that we have met a number of times before. So, the simplest case of real moduli for punctured spheres (which look a bit like trees, right?) can be described by a 1-operad.

What I might have neglected to mention before is that Brown has recently studied MZVs and integrals for such moduli, using Motivic Cohomology. In particular, any integral which we would like to associate with physical amplitudes is given in terms of MZVs. M-theory is so much fun, don't you think? Today's homework is to look through the Brown paper and find the Veneziano four point function.

Wednesday, November 22, 2006

What's On

Things will be fairly quiet around here next week. As mentioned on That Logic Blog, some of us will be going to Canberra for the Morgan-Phoa Mathematics Workshop. For anyone who's heavily into Operads, Categories or Topos Theory, and happens to be around this neck of the woods, it's a must do!

Need to Know Basics

Young OECD people must find it hard to imagine a world without computers and information technology. They have learnt the basics of logic in a way that even my generation cannot imagine. In my experience, they are not afraid of Category Theory, that is if they have any prediliction for mathematics. A Topos is a special kind of category which has a structure appropriate for doing intuitionistic logic. That funny word intuitionistic is actually a technical term, so don't worry too much about it. It includes ordinary classical (Boolean) logic, and other possibilities. All we need to keep in mind for now is that Topos Theory is the place where geometry meets logic.

There are now quite a few good books on topos theory. Rob Goldblatt's Topoi which is now available from Dover. For those with library access, there is the excellent book Sheaves in Geometry and Logic by the late Saunders Mac Lane and Ieke Moerdijk. Online texts include Triples, Toposes and Theories by Michael Barr and Charles Wells.

Tuesday, November 21, 2006

M Theory Lesson 3

In Lesson 2 we looked at a theorem relating surface moduli to Ribbon Graph moduli. This takes the form of an equivalence between categories. The ribbon graphs are indeed supposed to remind one of 't Hooft's old diagrams for QCD, which appear in early String theory papers from the 1970s. But in Category Land one doesn't play with Feynman diagrams based on a Minkowski background. No, no. The lesson of Penrose's twistor theory is that sheaf cohomology is a good language for thinking about solutions to field equations. In Category Land and Machian physics this lesson takes on a whole new meaning, and one loses the desire to operate with Feynman diagrams at all. In twistor String theory one uses instead MHV diagrams, and things like gluon amplitudes become magically easier to compute. This is why a whole army of excited String theorists is currently busy calculating stuff for the LHC, quite convinced that they are doing QCD.

Monday, November 20, 2006

En Hyggelige By

Some colleagues have been writing a bit about cold places, with alarming naiveity. As a teenager I learnt to speak Danish whilst an exchange student in Odense. Danish is a useful second language in Greenland, because in 1721 European colonialism appeared when the Danish king sent Hans Egede on an expedition there.
Greenland was the setting for the recent film Eight Below, about sled dogs working in Antarctica. If you would like to know more about the old days in Antarctica, just ask some of my mates, like Grant Gillespie at Aoraki in NZ. Avalanche dogs are also fascinating to work with. In a proper rescue setting they can identify buried bodies faster than an expert with a transceiver. The analogy with professional theory is fully intended. Who knows what will happen the day after tomorrow? Speaking of the cold, the iceberg below was visible from the shore of the South Island just the other day. Nothing to do with recent global warming, because it takes many decades for them to drift so far north.

M Theory Lesson 2

Mulase and Penkava studied Ribbon Graphs and came up with a constructive proof of the correspondence between two kinds of moduli: a Riemann surface moduli space and a Ribbon Graph moduli space. The original theorem is due to Penner, Thurston and others, and relies on the study of Grothendieck's Children's Drawings.

One works with a category of Ribbon Graphs. An object is a collection of vertices and edges and an incidence map i. Arrows are pairs of arrows that form a commuting square with the two incidence maps. Vertices are always at least trivalent, but we then add bivalent vertices at the centre of each edge to create half edges. A cyclic ordering on half-edge vertices gives an orientation to the ribbon edges. By definition, a boundary of a graph is a sequence of directed edges which cycles back on itself. Then Euler's relation holds,

v - e + b = 2 - 2g

where g is the genus of the surface represented by the ribbon graph.

Sunday, November 19, 2006

M Theory Lesson 1

It's high time we began a set of lessons in M theory. Today we will begin to look at the combinatorial structure of moduli spaces of Riemann surfaces, as studied by Mulase.

Consider the following fractional transformations on the upper half plane. T is the map taking z to z + 1. A fundamental region for this map is a strip of width 1. We take the strip between -1/2 and 1/2 on the real line. The map S transforms the inside of the unit circle to the outside via z --> -1/z. Note that SS = 1 and (TS)^3 = 1. A fundamental region for the group so generated is the part of the selected strip above the unit circle. Maps compose via matrix multiplication for fractional transformations. The group generated by the map S fixes i and TS fixes the point w = exp(pi i/3), a root of unity.

By gluing this region into a cylinder with 2 singular points we obtain M(1,1), the orbifold moduli of the one punctured torus. The J invariant gives J(i) = 1 and J(w) = 0. We also take J(i oo) = oo. Then it is possible to describe the equivalence between elliptic curves by the relation J(tau) = J(tau'), where tau is the complex parameter which characterises the curve. So the moduli M(1,1) is parameterised by either tau in the upper half plane, or by z in the Riemann sphere CP^1 without the points 0,1,oo, which is the moduli M(0,4), obtained via a quotient of H using gamma(2).

Saturday, November 18, 2006


Make sure you check out the DE thread on CV. None other than George Ellis has been questioning Sean's assumptions. For some reason Sean has not banned him from the blog. In the mean time, Louise appears to be gaining supporters on her blog. NASA might just have done themselves in with the DE publicity stunt.

I should point out that Ellis is actually discussing the GR inhomogeneity idea, as studied by quite a number of relativists. This is quite different from the correct explanation of a varying speed of light, but it does the job of eliminating the Cosmic Blunder. Moreover, this explanation has been considered in the context of Machian principles, which we know underlie M theory.


Back in the mid 1980s I was in the lab pounding the ingredients for a ceramic high Tc superconductor with a pestle and mortar. We would do this for a while with each sample to get the powder nice and homogeneous. In 1989 Fisher suggested the possibility of a new phase in superconductors, a disordered vortex-glass phase. Nowadays condensed matter physicists do all sorts of things with superconductors, such as construct precision clocks using niobium microwave cavities.

One of the most exciting developments in CMP at present is the theory of topological quantum computation. Particles with knotty statistics, called anyons, can be potentially used to build a fault tolerant quantum computer. And it also seems that materials science and nanotechnology is moving ahead in leaps and bounds.

Now imagine what we could do with a new physical theory, a theory that does not merely assume that the universe is FRW homogeneous and isotropic. After all, it's the matter that is interesting, not the inferred background in which it appears. Imagine a theory where anyon knots live instead in the pure world of categorical logic, where we are no longer constrained by the limitations of analytical geometry.

Friday, November 17, 2006


The NASA press release on the new supernovae type IA measurements is out. Louise Riofrio has already blogged about the report. Personally, I was surprised that the initial questions from science journalists included nothing about modified gravity theories. I guess they didn't speak to the right theorists. We were told the precise amount of energy contained in a centimetre cubed of space. Say what? They can't be serious. I hope someone is taking track of the fact that some theories are predicting such Hubble results.

Thursday, November 16, 2006


You probably know this already, but we have another NASA press conference coming up, so I'd better let you all know...

Friday Nov 17 at 7am NZ time

The mysterious header says:
NASA will host a media teleconference with Hubble Space Telescope astronomers at 1 p.m. EST Thursday, Nov. 16, to announce the discovery that dark energy has been an ever-present constituent of space for most of the universe's history.

George Jones at PF points out that this could mean no Dark Energy. However, it appears more likely that the respectable scientists will simply rule out weird and uninteresting quintessence alternatives, without considering better possibilities. We'll just have to wait and see. For me, that's a fresh 5am start.

Tuesday, November 14, 2006

For Children

Two interesting new papers on the arxiv:

Frenkel on Langlands' and a Unified Theory of Maths


Children's Drawings From Seiberg-Witten Curves
Authors: Sujay K. Ashok, Freddy Cachazo, Eleonora Dell'Aquila

I haven't had a chance to look at this one, but the Children's Drawings are those nice ribbon pictures that we've been talking about. From page 7: The Grothendieck correspondence is a bijection between classes of dessins and special classes of maps on punctured Riemann surfaces called Belyi maps. At first sight this might seem far removed from gauge theory physics...

You don't say!

Monday, November 13, 2006

For Rae Ann

Rae Ann made a comment on Motl's blog in response to a posting about this book, which was mentioned in the last post here. Rae Ann appeared to think that the mind connection sounded anthropic. Matti Pitkanen corrected this misconception on his blog, but I thought I might make a brief remark on this point myself.

Rae Ann, the problem with String Theory (and the Standard Muddle) was that it used a universal observer kind of mathematics, just like when you were at school and they made you graph functions using Cartesian coordinates, which you pictured as labelling some space Out There. Humans were thinking of the cosmos as if they were such universal observers.

Fortunately, we now know how to get around this problem and define the notion of observer properly, so that humans can be both within and without reality.

Sunday, November 12, 2006

Changing Times

There are two words in ancient Greek for time. One is kairos, a personal time, an opportune moment. The other is chronos, the time of clocks and the outer world. We are all familiar with the feeling of shifting time, when we focus on an interesting problem. In GR time also shifts, but with the gravitational field. Dirac thought about two kinds of time, an atomic and a celestial. The philosopher Hegel developed a theory of the world in which the world's consciousness develops through the history of ideas, which defines a timeline for the world.

Be not afraid when new hope is put before you. Check out this really cool new book

Friday, November 10, 2006


The majority of Quantum Gravitists agree that the unified theory will get rid of space and time. But what does this mean? Does it mean letting a classical spacetime emerge from a summation over quantum geometries, such as spin foam graphs? No, of course not. It means removing space from one's conception of the cosmos. When we observe binary pulsars for example, we should not imagine this object occupying a concrete objective classical spacetime out there. Even a traditional dynamical spacetime is completely inadequate. Quantum physics has taught us that a good way to approach a physical problem is to consider what we measure. One must be much more careful about the constraints one puts upon different classes of observation. If we wish to 'observe' the universe, we must define what we mean by this so that we are able to write down the correct geometry for this experiment.

People speak of holography in terms of data on a boundary. Where is this boundary? With respect to what may I position it? The best I can do is try to imagine that it is within myself.

Thursday, November 09, 2006

TWP Duality

TWP is an excellent popular account of the current status of physics, provided the reader appreciates that the alternative ideas mentioned in later chapters are by no stretch of the imagination adequate to meet the experimental challenges that currently face our field. For example, Category Theory is not even mentioned. This is a serious omission in the present climate of ideas. On the other hand, the personal anecdotes about pivotal events in the development of String Theory are particularly enjoyable. Smolin mentions, for example, his great surprise at the non-discovery of proton decay. I come from a slightly younger generation: the real DWAT (Don't Worry About That) generation. The current form of the Standard Model was well established before I became an undergraduate in the late 1980s.

The most common response to questions that our generation asked was: DWAT. We haven't seen proton decay? DWAT. No Higgs? DWAT. No Gravitational Waves? DWAT. Needless to say, after having thought quite a bit about physics over the years, by the time it came to the observations of type IA supernovae my generation greeted the supposedly obvious explanation with some skepticism.

On page 210 of TWP, Smolin mentions Milgrom's Law, which is the observation that the apparent breakdown of Newton's law in galaxies occurs at a scale whose value of acceleration matches the apparent acceleration of the cosmological constant. Rather than follow the r^(-2) law, outlying bodies appear to obey a r^(-1) law.

Now it was shown by Bertrand [1] in 1873 that central forces for closed orbits can only obey one of two possible laws: either the r^(-2) law or the r law (Hooke's law). The latter is the inverse of the law that we would like to obtain. Is there a simple way to rescue this situation? Yes, of course. When there is a correlation between a cosmological scale and a local scale we should not be afraid to apply T duality and/or S duality to the problem. M theory dictates that this is indeed the way to look at things.

We can then remove ourselves from the empirical world of MOND by viewing the effect as Dark Matter in the form of black holes. This is still a form of modified gravity, so both of the popular views may be seen to be correct, in some weak sense. The black hole picture can explain why we should expect a correspondence of scales, because the cosmic horizon is naturally correlated to Dark Matter in this form.

[1] Goldstein Classical Mechanics

Dark Energy Epicycles

Over at CV Sean has a new post entitled Out-Einsteining Einstein. Alas, on trying to post a technical remark regarding Dark Matter I discovered that I have been banned from CV. This blog will just have to post a bit more on the subject now.

As is well known, Einstein thought the cosmological constant was a big mistake. The early solutions to his equations, without the constant, described expanding universes, but no one at that time had made any observations indicating that the universe was in fact expanding, and Einstein was horrified by a vision of the cosmos that went against all previous ideas. Eventually the constant was forgotten, except by a few die hard General Relativists who liked playing with the equations.

In 1998 an apparent (note this word) acceleration in the expansion was observed using Type IA supernovae, which act as standard candles due to the uniformity in their nature. In the Standard Cosmology, much other data appeared to confirm the assessment that the expansion was in fact accelerating. But in the last year or two it has become clear that the standard picture cannot fit the supernovae data. The word on the street has been that our understanding of the standard candles must be flawed. Funny then that they fit the data so well at the low z part of the curve.

Anyway, the Standard Cosmology is wrong. It is quite easy to explain away the apparent acceleration. In fact, some General Relativists have been doing just that using simple modifications of the standard picture. The true cosmology, however, is radically different, as is well known by the great cosmologist Louise Riofrio. It means thinking about a varying speed of light, at the very least, and I say that because it gets much, much more radical the more you look at it.

Amongst Quantum Gravitists it is well understood that the correct Background Independent unified theory will radically alter our conceptions of space and time. We are only in the early days of M theory, but we can see that a varying speed of light cosmology is probably inevitable. It also has the advantage of actually agreeing with the data.

Wednesday, November 08, 2006

The Gauntlet III

Over at PF, the poster E.L. and I have agreed on a bet at 1000:1 odds. At 1c invested this is a huge win of $10 (is that $US 10 E.L.?), to be collected after decent data analysis from the full beam LHC. The bet is based on my claim that at this time, probably in 2008 or 2009, that of any new particles discovered at the LHC the vast majority of physicists will call none of them KK modes.

Any other takers?

The Gauntlet II

Christine Dantas has kindly explained her reasons for leaving the blogosphere, namely a dislike for the battleground which has been interfering with her efforts to conduct a peaceable blog on QG. We are very sad to see her go, and wish her all the best. It is true that the arguments currently raging are in parts very bitter and unacademic.

But there is a reason for all this. This blog has no intention of closing down. The fewer QG blogs there are, the louder this one will become, even if only one lonely soul out there is listening. Or even if there are none, because in desperation one could always hope that the Internet had become aware. Make no mistake - we are fighting for something. I have no quarrels with any individual. By and large they all mean well. And perhaps one could say that the right ideas will win out in the end, and so why not let it slide? Because I cannot stand by and watch such great lies being foisted upon humanity. I thank Blogger from the bottom of my heart, for this is the only option I have to speak. I have edited my blogroll today. The sites of TGD Diary (Matti Pitkanen) and Tony Smith are highly recommended. These physicists are both banned by the arxiv preprint server.

On a lighter note, Scott Aaronson appears to be making headway in his argument on CV, with Rob Knop conceding

If you keep saying stuff like that, I’m gonna have to be forced to re-evaluate my idea that computer science is a subdiscipline of math, and start wondering if it’s physics instead. And you know that the last thing a good red-blooded American ever wants is to be forced to re-evaluate his preconceptions.

Today I have received a copy of Lee Smolin's The Trouble with Physics, so I leave you to curl up in a warm corner and read it.

Update Thursday: I now appear to have been banned by CV. Yeah!

Tuesday, November 07, 2006

Scott's Gauntlet

Scott Aaronson famously posted the remark

[Theoretical Computer Science], which by any objective standard has contributed at least as much over the last 30 years as (say) particle physics or cosmology to humankind's basic picture of the universe

on his blog, to which the respectable Sean Carroll found it necessary to reply at CV. Since my comments at CV have a tendency to be deleted (mysteriously whenever they refer to alternative cosmologies) I thought I would post a few comments here. Mahndisa has kindly rescued some of them from the garbage can (see previous post here).

So Sean, if I might repeat my question: when you claim we have gained the insight of emergent spacetime, I would like to know which approach to QG you have in mind. The most promising approaches (if that means Strings, then consider the black holes of Kallosh, for example) use categorical ideas which lie at the very core of quantum computation. Since emergent spacetime is being touted as a huge advance in our understanding of the universe, I believe this strongly supports Scott’s argument.

It has also been pointed out that the Standard Model, in its original formulation, was a huge leap forward in our understanding. This is no doubt true. However, should it turn out that a rigorous form of the Standard Model relies heavily on categorical formalism, and not so much on the gauge concept, then even this part of physics (which it is an act of desperation to have to call upon) favours Scott's point of view.

Sean's latest reply to the crackpots (which I enthusiastically assure you includes yours truly) was: and will our more sensible commenters please remember that replying to crackpots is as crucial a part of the Crackpot Dynamic as the original crackpottery itself? Resist the temptation! I have kept the comment lest Sean suddenly decides to censor himself at some later date.

Today we should also mourn the passing of Christine Dantas' wonderful Quantum Gravity blog, which appears to have gone to the great blogbone in the sky.

How art thou fallen from heaven, O Lucifer, son of the morning! How art thou cut down to the ground, which didst weaken the nations!

Monday, November 06, 2006


Many modern Grail stories have a root in the early romances of von Eschenbach, who lived from 1170 to 1220. According to the official Templar website, in 1216 von Eschenbach wrote:

They live from a Stone whose essence is most pure. If you have never heard of it I shall name it for you here. It is called Lapsit exillis. By virtue of the Stone the Pheonix is burned to ashes, in which he is reborn.

To von Eschenbach, the Grail was never really a material cup, but a jewel like the jewel in the lotus, a symbol of enlightenment, of something intangible and always beyond reach.

I must confess that the reason I know this is because I'm a bit of an opera fan. The work of von Eschenbach was a source for Wagner. Just the other day I was thinking how nice it would be to hear das Rheingold again, and the beautiful arias of the Faustian Alberich as he curses the gods.

Saturday, November 04, 2006

Going, Going, Gone

A large fraction of the world's human population is supported by a few glacier fed rivers, such as the Ganga and Brahmaputra. The Himalaya is currently home to 33000 square km of glaciers, but these are rapidly retreating in line with global warming. The rate of glacial retreat depends critically on temperature. A small increase in current temperatures will increase the rate of glacial melt. Some forecasts state that up to a quarter of the world's mountain glacier mass could disappear by 2050. That's a lot of drinking and irrigation water.

The home of the Kea is near some of the world's few advancing glaciers. These steep mountain glaciers advance partly because heavier than usual snows at high altitude build up the neve ice mass. It is often pointed out by the naysayers that the ice thickness on the great Greenland ice sheet is currently growing. This is a natural result of the increased precipitation that results from climate change. These people also like to point out that sea levels have not changed that much. That's right. The ice hasn't melted yet. But it is melting.

glaciology links

Friday, November 03, 2006

Carnivale I

Summer begins with a happy physics blogging day. Christine Dantas is celebrating her blog's first birthday with guest posts from Daniele Oriti (spin foams), our dear Quasar9 and yours truly. Over on PF people continue to discuss the new heavy baryons and Carl Brannen is blogging from the Particle Physics conference in Hawaii, which Woit also reports on. Cheerful as always is Louise Riofrio, who today has another interesting post on our future in space. And we must not forget our respected colleague Motl from Harvard, who has an entertaining post on the anthropic principle.

Thursday, November 02, 2006

Categorical Ceilidh

This blog may have been a little neglectful lately of pure categorical adventures. Yesterday's seminar by Ross Street was on Kan extensions in group representation theory. This came about because Elango Panchadcharam has been writing a thesis here on Mackey functors and Ross Street is planning a basic text on category theory from a 2-categorical point of view.

Initially one allows G and H to be arbitrary small categories with an arrow i from H to G, and M a module category which is cocomplete. The restriction functor based on i between the functor (representation) categories has a left adjoint, which for those who know is where the left Kan extensions come in. One has categorical formulae for Lan(V) where V is a particular representation. In general, these use the comma category (i,A) for A an object of G, which is like a weak pullback of i and the arrow representing A in Cat. But in Mackey theory one really wants to consider two subgroups H,K in G. Then one ends up looking at the comma category (i,j) where j is now an inclusion for K.

Group theorists try to avoid the (i,j) because it's not a group, but a groupoid. But as Ross Street pointed out, the (i,j) contains a lot of important information. One gets the Mackey Decomposition theorem for instance, in terms of Res and Ind functors.

Dorette Pronk gave a wonderful talk about Conformal Field Theory and nuclear functors. In Segal's original definition of a CFT the source category of disjoint unions of oriented circles is not really a category. One adds in the familiar cylinders to get identities. But cylinders alter volumes, so one should be more careful about defining the source and target categories to get the conformal structure right. The categories used are always monoidal *-categories. The arrows are chosen according to the notion of nuclear ideal. This is a subset N(A,B) of each hom set which is closed under arbitrary composition, tensor, star and conjugation, and such that there exists a natural bijection from N(A,B) to Hom(I,A*oB). An example is Hilb with only the Hilbert-Schmidt maps.

In the CFT source category Pants (the name caused some amusement amongst the mathematicians) the nuclear ideal eliminates the cylinder problem. The target category is taken to be CLR, the category of correct linear relations, which I won't define here. Finally, a CFT is just a nuclear functor from Pants to CLR.