Archive for November, 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.

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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.

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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?

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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.

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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!

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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.

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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.

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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.

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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.

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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).

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