Magic Motives

Speaking of orbifold Euler characteristics, let's put the magic formula

$f(n) = \prod_{m=0}^{n - 1} \frac{m!}{(m + n)!}$

in terms of Euler characteristics. First, let $m = 2g - 2$ be the Euler characteristic of a closed surface of genus $g$. This already suggests allowing non-orientable surfaces to account for odd values of $m$. Then consider moduli spaces $M_{m,n}$ for $(n + 1)$ punctured surfaces. The orbifold Euler characteristic of such a space will be denoted by $E_{m,n}$. Using Mulase's expression for $E_{m,n}$ and assuming it may be extended to the non-orientable case, one finds a natural definition for the moment coefficients of the form

$f(n) = \frac{1}{(n + 1)!} \prod_{m=0}^{n - 1} b_{m + 2} E_{m,n}^{-1}$

which is a product over surfaces of genus $g$ limited by $n$, and where $b_{i}$ is a Bernoulli number (for even $m$ these are defined in terms of zeta values for odd negative reals). One should take more care with the non-orientable factors, but this simple exercise shows that the zeta moments are naturally dependent on categorical invariants associated to complex moduli.

M Theory Lesson 62

Last November, in the pre maths blogger days, we started with Mulase's lectures on moduli spaces of Riemann surfaces. In particular, let us look once more at the $S_3$ action on the Riemann sphere $\mathbb{CP}^1$. The real axis is the equator, with the point $\frac{1}{2}$ sitting opposite the point at infinity. The dihedral action helps define a compactified form of the moduli space for the once punctured torus $M_{1,1}$ (elliptic curve), which was described by a glued region of the upper half plane, sitting above the unit circle. The j invariant gives the mapping from the 3-punctured Riemann sphere to the complex plane which respects the dihedral action on the equatorial triangle, and the torus orbifold is the quotient space.

The j invariant is used to obtain Grothendieck's ribbon diagram from the inverse image of the interval $[0,1]$, so both sphere and torus moduli are essential in understanding the ribbon for the 3-punctured sphere. Recall that these are the only moduli of real dimension 2. In the six dimensions of twistors there are three complex moduli, namely $M_{0,6}$, $M_{1,3}$ and $M_{2,0}$ which have (respectively) orbifold Euler characteristics of $-6$, $- \frac{1}{6}$ and $- \frac{1}{120}$. Octonion analogues of such moduli will be easier to understand using n-operad combinatorics, because non-commutative and non-associative geometry is a tricky business.

M Theory Lesson 61

The purpose of this post is mainly to invite Doug, one of our oldest companions, to comment here on the general direction of the Lesson series. Doug expressed an interest in doing so back in Lesson 59. So just this once, Doug can post a series of long comments here. I hope this will encourage him to set up his own blog. After all, anyone is free to do so!

In Lesson 59 we were looking at theta series for lattices. For the Leech lattice this involved the Ramanujan function, which is a Fourier series for the modular discriminant. Fourier coefficients for cusp forms were part of Conrey's motivation in looking at even moments for the Riemann zeta function.

Riemann Rave III

The aforementioned moment series (for the unitary ensemble) of Keating and Snaith is defined in their paper as

$f(n) = \prod_{j=0}^{n - 1} \frac{j!}{(j + n)!}$

for which one finds the first few terms

$1$, $\frac{2}{4!}$, $\frac{42}{9!}$, $\frac{24024}{16!}$, ...

It is fun to find alternative expressions with fewer factorials because the size of terms is then more immediately apparent. For example,

$f(n) = \frac{n^{n - 1}}{(2n - 1)!} \prod_{j=0}^{n-2} (n^{2} - j^{2})^{j-n+1}$

for which one can check, say the third term

$f(3) = \frac{3.3}{5.4.3.2} \frac{1}{9.9} \frac{1}{8} = \frac{6.7}{9!} = \frac{42}{9!}$

This shows up the dependence on the factorial of squares $n^2$, which is not so apparent in the original expression. Note that $24024=13.11.8.7.3$ is similarly made of pieces that are needed for a factor of $16!$ in the 4th term. The nth term provides the coefficient of the moment $\int_{0}^{T} | \zeta (0.5 + it) |^{2n} dt$ of the Riemann zeta function, as discussed by Conrey et al in the paper that appeared around the time of Keating's Vienna talk.

It seems that single prime factors miraculously appear in the numerators of $f(n)$. Moving on to $f(5)$ we find a numerator of $23.20.19.17.13.11$ (assuming I did the sum correctly). So if we ignore powers of 2, the first few numerators are built from single prime factors, covering mainly primes between $2n$ and $n^{2}$.

Gossip, Gossip

Those naughty experimentalists, such as Tommaso Dorigo and Charm, have been spreading rumours about a D0 discovery involving an excess of 4 bottom quark jet events. Of course, this has reignited the Higgs boson gossip, which must be starting to sound like a boy crying wolf even to those who actually (mistakenly) think that such a particle exists. One begins to suspect that these naughty children are just playing games to get a laugh out of all the crazy responses that pop up in the media in no time at all.

Naturally, we are all ears if D0 decides to tell us something Interesting and Statistically Significant, as they say.

In other gossip: the rumour that the Riemann hypothesis has been shown to be false has now extended beyond the blogosphere. One of my office mates heard it from a colleague in a casual conversation!

Light Metal Life

Universe Today reports on the new discovery from Texas of a system with two Jupiter like planets orbiting the metal poor star HD 155358. The metallicity suggests the star is about 10 billion years old.

The astronomers were surprised to find planets in such a system, because heavy elements are needed to assist core accretion in this model of planet formation. The alternative model suggests that instabilities cause a breakup of the disc, but the mechanism for this is unclear. "Having this process happen to form not just one, but two, planets around a star that had so little solid material available for planet-building is quite remarkable," Cochran said. This appears to be yet another confirmation of one of Louise Riofrio's predictions. Small dark matter objects may aid core accretion without the need for heavy elements.

Riemann Rave II

It may take a while to read through Gregory Moore's work on Arithmetic and Attractors, but it looks very interesting. He also has another paper on black hole entropy with S. D. Miller, who has a handy page on L-functions and apparently a keen interest in the Riemann zeroes.

Note that this work is from a string theory perspective and so does not consider, for example, a varying speed of light or an $\hbar \rightarrow 0$ which appears both as a cosmological horizon constraint in Riofrio's cosmology and also in the semiclassical analysis of random matrix ensembles done by Snaith and Keating.

Riemann Rave

The most intriguing physical anecdote in du Sautoy's The Music of the Primes is the one about Keating's talk, entitled "Random matrix theory and some zeta-function moments", at the Vienna meeting in 1998.

Many mathematicians were dubious that the physicists could tell them anything beyond their observations on the statistics of Riemann zeroes and quantum chaos on surfaces. Yet another fine bottle of wine had been put forward, which Keating collected in Vienna after his talk. Nina Snaith and Keating had found a formula for generating the numbers known as zeta moments. The sequence begins 1,2,42,24024. The last number had only just been found by mathematicians, and yet it fell out of the physicists' formula when they looked at it just before Keating's talk. Links to the papers by Keating and Snaith, and also to Snaith's thesis, are available on Watkin's page.

Later, Keating went to the library in Goettingen to look up Riemann's original notes on both the zeta zeroes and hydrodynamics. He made the two requests to the librarian, but was handed only one pile of papers. Riemann had been working on both problems at the same time.

More recently, Carl and Michael have been making solid progress on understanding quantum black holes in M Theory. It would be really very interesting if the quantum chaos of such black holes had something to do with the Riemann zeroes. Oh, wait. It already does, because one uses 3x3 Hermitean matrices in a matrix theory setting.

M Theory Lesson 60

I wish I had come across the work of Andrew Hodges before! Check out the papers:

1. Twistor diagram recursion
2. Helicity-independent formalism

Unbelievably, when I google MHV and operad, which I do periodically, I still get no hits (other than here). There are thousands of hits for MHV. What is everybody doing?

Riemann Revisited III

OK, even if the Hypothesis turns out to be false, that was hilarious, but seriously now ... one promising route to the Riemann Hypothesis is in fact to show that it is undecidable with the standard axioms. It is hard to imagine how this could be done. Even if the zeta zeroes were completely re-characterised in terms of higher categorical invariants, in a way that seemed utterly natural and compelling, that does not imply that we must look at the zeta function that way. Well, mathematically that is. Physically speaking, we only care about the zeta function in so much as it can describe measurable quantities.

But the zeta function soon gets swamped by its generalisations, inflating the difficulty of the problem. These are certainly physically relevant. Recall, for example, Brown's paper for MZVs, which contains a 1-operad computation of Veneziano amplitudes. M Theory cannot avoid considering this construction outside set theory.

Update: Thanks to K. L. Lange for further interesting remarks about the possibility that Pati has inadvertently made progress on showing that RH is unprovable within standard analysis.

Riemann Revisited II

I'm so excited by this claim of Tribikram Pati! I haven't read the paper yet, but quoting from the abstract: our analysis shows that the assumption of the truth of the Riemann Hypothesis leads to a contradiction. Maybe he's right! After all, we've seen that the zeta function really shouldn't be studied within the confines of Boolean logic. But then there is already a post by Julia Kuznetsova claiming to have found the (almost inevitably present) flaw. A reply by a K. L. Lange to this criticism, supporting the proof, states, "So the main idea of [Pati] was to show, that we need that delta to prove RH, but there is no delta, so we cannot prove RH after all ..." In other words, the paper may well show that there is no proof of the Hypothesis within standard analysis. That doesn't sound surprising at all.

If we built an L function on the surreals in the M Theory operadic landscape, would we care about the ordinary Riemann zeta function? Yes, of course we would, because it's very interesting! And the M Theory L function would contain the Riemann one anyway. I have reluctantly come to the conclusion that a decent definition for n-logoses really must sort out the meaning of Analysis in category theory. This can't be done the 1-topos way. At least surreals have infinitesimals. And now I should probably confess that I always had great difficulties with analysis. Well, I also have great difficulties with Geometry, Algebra and Logic, which is one good reason for studying category theory, because it rolls them all into one! But as Donaldson, Perelman and many other mathematicians have shown, beautiful proofs these days need analysis.

Riemann Revisited

On Lieven Le Bruyn's recommendation, I decided to read du Sautoy's book The Music of the Primes. It is exceedingly enjoyable and informative (except for the physics glitches).

The anecdotes range from Gauss's discovery of Ceres to Dyson's fortuitous meeting with Montgomery. I particularly liked the suspenseful introduction based on Bombieri's famous April Fool's Joke of 1997, stating that the Riemann hypothesis had been proved by a young physicist. He wrote, "the physics corresponding to a near absolute zero ensemble of a mixture of anyons and morons with opposite spins." Perhaps we should really name a particle the moron in honour of Bombieri's great humour!

Update: Wow! Lieven has pointed me to this post about a possible disproof of the Riemann hypothesis. Or rather, a proof that it cannot be proved within set theory? That would be just AMAZING!!! Here is the paper in question. Must go check it out...

Construction Site

What is an n-category? Something indexed by an ordinal n? But if we want to talk about higher toposes we can't begin by assuming that we know what an ordinal is! The 1-topos Set conveniently has an object of ordinals, characterised by a successor function. But n-logoses, being fundamental entities, cannot be based on diagrams in Set.

Another place where ordinals are found is in simplicial sets, which are Set valued functors from a category Ord of finite ordinals and order preserving functions. Think of real geometric n-simplices on n vertices, like the tetrahedron for the ordinal 3. But once again, in defining Ord, we assumed that we knew what an ordinal was. The only way out of this dilemma seems to be to construct ordinals as we define n-logoses. For an elementary topos, for example, we certainly need the concepts 0 and 1 (empty set and one point set), but the concept of the ordinal 2 could wait until we need it for the triangle of ternary logic. This suggests using this triangle whenever we might be tempted to use the ordinal 2. For example, a simplicial category Ord can't be defined until we have a notion of $\omega$-logos. Sounds like an awful lot of work!

The ancients knew that for a set theoretic cardinal n, there was always a decomposition into prime factors. This is the fundamental theorem of arithmetic, but it does not hold for general fields. Anyway, there's a problem with Euclid's original proof by contradiction: it's not constructive. Just because a statement is false, doesn't mean we can assume that its converse is true! And what is a converse? We've seen that 3-logoses require a notion of ternary complementation. Somewhere in the land of $\omega$-logoses there is an awfully complicated notion of primeness.

If any of this makes sense, it suggests that the infinite branchings of the surreal tree follow the process of logosification. Looking at the positive half only, we see a simple Y tree at the level of 2. At the branches, the number 2 has been defined along with the number 1/2. We conclude that the concept of 2-category must be accompanied by a T-dual concept of 1/2-category!

Maybe the appearance of associahedra as coherence laws for n-categories would be a little less surprising in this setup. That is, an (n+2)-leaved 1-tree polytope appears in the definition of n-category, as n goes to infinity.

M Theory Lesson 59

One of the most information packed 600+ page tomes in the library here is Sphere Packings, Lattices and Groups by J. H. Conway and N. J. A. Sloane. It looks at the 24 dimensional Leech lattice in numerous ways.

Chapter 1 covers the basics of lattice theory and sphere packings. On page 5 it is noted that for a hexagonal plane tiling, rather than basis vectors $(1,0)$ and $(0.5, 0.5 \sqrt{3})$ in two dimensions, it is useful to use the simple three dimensional coordinates $(1,-1,0)$ and $(0,1,-1)$, which lie on the plane $x + y + z = 0$. This basis gives a Dynkin diagram labelling of the $A_2$ root lattice. It corresponds to the densest sphere packing in two dimensions with a density of $\frac{\pi}{\sqrt{12}}$.

The theta function of an integral lattice is related to modular forms. For example, for the Leech lattice the series takes the form

$1 + 196560 q^4 + 16773120 q^6 + \cdots$

which follows from a term for the $E_8$ theta series minus another term which is 720 times the Ramanujan series

$q^2 - 24 q^4 + 252 q^6 -1472 q^8 + \cdots$

Defining $\theta (\tau) = \sum_{- \infty}^{\infty} e^{\pi i \tau n^{2}}$, one can use Riemann's observation that

$\theta (\frac{-1}{\tau}) = \sqrt{- i \tau} \theta (\tau)$

to prove the functional equation for the zeta function. Recall that in M theory this property of the zeta function is intimately related to the physical duality of Space and Time. In a logos style constructive approach to zeta functions it is therefore natural to view lattices and theta series as useful tools for the geometrization of operad polytopes.

Note that the currently popular j invariant may be expressed easily in terms of theta series as

$j (\tau) = 32 \frac{(\theta (\tau)^8 + \theta_{01} (\tau)^8 + \theta_{10} (\tau)^8)^{3}}{(\theta (\tau) \theta_{01} (\tau) \theta_{10} (\tau))^{8}}$

which comes from basic elliptic function theory, but smells of triality, I think.

Blog Notice

What everybody was waiting for ... Borcherd's new blog! And the first post is on the uselessness of Planck units. He's already on my blogroll.

M Theory Lesson 58

A 5-edged 2-level tree can have at most 4 input leaves. These trees define two dimensional polytopes as shown. The pentagon is familiar from 1-operad tilings, or as the Mac Lane pentagon, and the hexagon from the braiding rule for braided monoidal categories. Only polygons of six sides or less are capable of tiling the plane regularly. Here are some purely pentagonal tilings.

Higher dimensional analogues of the hexagon polytope are known as permutohedra, because they describe permutations on $(d + 1)$ letters. Loday showed that by cutting hypercubes with hyperplanes one can naturally obtain both associahedra and then permutohedra in any dimension. Postnikov et al have done a lot of work on the combinatorics of generalised permutohedra.

As a shameless thief of pretty pictures, today I offer the reader the $6_3$ knot on the Klein surface! A six punctured Klein surface has a moduli space of real dimension 24, which is a number we like a lot.

M Theory Lesson 57

Recall that the Bilson-Thompson diagrams for left and right handed electrons are formed with three strands, each with a full negative twist representing a one third charge. Ignoring the twists, the simplest braid composition of the form $e_L e_R e_L$ looks like this: This knot is called a $6_3$ knot, according to the amazing online knot atlas, which lists the polynomial invariants for this knot. For example, the Jones polynomial takes the form

$-q^3 + 2 q^2 -2q + 3 -2 q^{-1} + 2 q^{-2} - q^{-3}$

Since all negative twists add up on going around the knot, there is a total ninefold twist. There is now a button to the knot atlas on the sidebar.

The Old Man

Given anonymous' interesting suggestion to use honeycombs to study the rings of Saturn, I couldn't resist posting this false colour NASA/JPL/SSI Cassini image, stolen from Universe Today. This view is from above the hot North pole (in shadow) where we recently saw a hexagon structure.

M Theory Lesson 56

Returning to Heisenberg's 1925 paper, we find the basic honeycomb sum rule (on page 3) as the quantum mechanical version of the classical relation

$\omega (n,a) + \omega (n,b) = \omega (n,a+b)$

To a category theorist this classical relation looks like a triangle, where two sides compose to give the third. Moreover, since $n$ is fixed, we may view $\omega(n,-)$ as a functor from a category containing the triangle $(a,b,a+b)$. This source category has addition as composition. There is only one object. In fact, it is an Abelian group in the form of a category on one object. The target category, containing the frequencies as arrows, is also an Abelian group, so the functor $\omega (n,-)$ is just a group homomorphism.

Following Heisenberg, the quantum relation takes the form

$\omega (n, n-a) + \omega (n-a, n-a-b) = \omega (n, n-a-b)$

Setting $a = 0$ we see that $\omega (n,n)$ acts as a (left) identity for frequency composition, replacing the classical $\omega (n,0)$. It is now necessary to consider a variable $n$ in the source category of the classical case, so frequencies are best expressed as 2-arrows between 1-arrow integers. Thus the honeycomb diagrams in the plane are associated with two dimensional categorical structures. Alternatively, and loosely speaking, a set theoretic real number classical physics suggests a categorical complex number quantum physics.

Updates II

Let's spell this out clearly: Matti Pitkanen has calculated the observed radius of the Dark Matter ring using a TGD Bohr orbit analysis. An absurdly obvious resulting prediction is that the next observed dark matter ring will also sit at a Bohr radius.

Gukov et al have a new paper entitled Link Homologies and the Refined Topological Vertex, in which they look specifically at invariants of the Hopf link. "Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes." Even cooler, check out the paper The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists.

Updates

Check out NASA's cool new Hubble image in the last post. Louise Riofrio has a post about it. The astro-ph paper is here. They use both weak and strong lensing to find the mass distribution of CL0024+17. Matti, I think this paper has enough information for you to do a Bohr orbit analysis. If this is a typical head-on view of a recent collision of cluster systems, there must be an awful lot of 'dark matter' in the cosmos. And they say one can't test quantum gravity with simple optical observations!

Meanwhile, Tommaso shows us how QCD triumphs once again with B hadron lifetimes. And Carl Brannen has updated his gravity equations at PF.

NASA News

NASA have announced a press conference for the discovery of a ring of Dark Matter. It is scheduled for 1 p.m. EDT on Tuesday May 15. For the Kiwis, that's 5am tomorrow morning (sigh, why can't the Americans get up in the middle of the night sometimes?).

It is said that the Hubble space telescope has observed a ghostly ring of dark matter in the cluster CL0024+17. Sounds interesting! An earlier paper on this cluster may be found here. There is a noteworthy remark on page 6: compared to the weak lensing mass ... and the virial mass ... the mass deduced here from the X-ray observations is lower by a factor of 4. Matti Pitkanen says, "Rings puts bells ringing!."

In other news, the Cassini mission has been observing organic molecules in the atmosphere of Titan.

Update: Here is the dark ring image from NASA: Observe that this is a LARGE scale image. The weak lensing circle is barely visible at the inner edge of the dark ring. That means a LOT of 'dark matter'.

Time from Being

In his Science of Logic, Hegel writes about the difficulties of knowing where to begin a philosophy. His analyses are so thorough that he finds he cannot solve this problem without understanding subjective time itself as an emergent phenomenon from a circular whole, where the Being of the beginning is also the end. He says:

But the modern perplexity about a beginning proceeds from a further requirement of which those who are concerned with the dogmatic demonstration of a principle or who are sceptical about finding a subjective criterion against dogmatic philosophising, are not yet aware, and which is completely denied by those who begin, like a shot from a pistol, from their inner revelation, from faith, intellectual Intuition, etc., and who would be exempt from method and logic. If earlier abstract thought was interested in the principle only as content, but in the course of philosophical development has been impelled to pay attention to the other side, to the behaviour of the cognitive process, this implies that the subjective act has also been grasped as an essential moment of objective truth, and this brings with it the need to unite the method with the content, the form with the principle. Thus the principle ought also to be the beginning, and what is the first for thought ought also to be the first in the process of thinking.

Hegel's concepts are organised in what he viewed as canonical triples, an idea later developed by C. S. Peirce, whose work on diagrammatic reasoning we often discuss. For example, the triple of Being is Quality, Quantity, Measure. One can't help thinking that the originators of quantum mechanics must have been fond of Hegel when one reads:

Everything that exists has a magnitude and this magnitude belongs to the nature of the something itself; it constitutes its specific nature and its being-within-self. Something is not indifferent to this magnitude, so that if this were altered it would continue to be what it is; on the contrary, an alteration of the magnitude would alter the quality of the something. Quantum, as measure, has ceased to be a limit which is no limit; it is now the determination of the thing, which is destroyed if it is increased or diminished beyond this quantum.

But the idea of number as reality is far older than Hegel's work. The ideal of number was at the core of the philosophy of the ancient Pythagoreans, for instance.

Tetractys

An anonymous commenter has mentioned the Pythagorean Tetractys, a triangle on 10 points, divided into 9 triangular pieces. Knutson, Tao and Woodward studied the correspondence between such tetractys diagrams and honeycombs in the plane. The vertices of the honeycomb become small triangles in the tetractys, as shown. This example is a three dimensional tetractys, for which each side has three small faces. In the NxN case, a puzzle diagram is built from a triangle of side length N. Puzzles use three kinds of pieces: small triangles with edge labels 1, small triangles with edge labels 0, and rhombi (joined triangles) with paired edges labelled 0 and 1.

We can view this as a ternary analogue of a top down view of a binary tree, for which a simple Y tree would look like an interval divided into two parts, with the central point marking the vertex. Recall that such viewpoints were used by Devadoss in his study of generalised associahedra. Further levels of the tree correspond to further subdivisions of subintervals. A well known instance of this game, which considers binary subdivisions in a 2D piece of the tetractys, is Sierpinski's triangle, of Hausdorff dimension log(3)/log(2).

M Theory Lesson 55

As Matti Pitkanen has commented, a major motivation for such an approach to elementary axioms for n-logoses is the possibility of easy number construction. In the 3-logos case, 3-adic numbers arise, much as the dyadic numbers appear on finite segments of the surreal tree for binary logic.

It follows that ordinary real numbers are difficult to understand, appearing both as infinite sequences in the dyadic case, and in the $\omega$-logos setting at the infinite prime. Models of the axioms that use real and complex geometry must therefore respect the p-adic heirarchy. In the case of the Jordan algebra M Theory, for instance, this is at least partly achieved via triality for the prime 3.

M Theory Lesson 54

Returning to lattice theory, recall that the order relation between objects is represented by an arrow $a \rightarrow b$. Thus the lattice of two elements (0 and 1) is the same as the category 2.

Now in the topos Set, truth and falsity arise as follows. Truth is a given arrow from the one point set into {0,1}. Falseness is a characterising arrow in a basic pullback square, such that the square source is the initial object in the category, namely the empty set. In other words, both true and false are arrows into a special object $\Omega$. The action of turning true into false introduces the concept of complementation. Grothendieck's great insight into topological spaces was to see that $\neg \neg U$ need not be $U$, and that this was a logical statement.

In replacing binary logic by ternary logic, it is natural to assume three arrows into an object $\Omega$. This suggests the replacement of pullback squares by cubes. Instead of a lattice $0 \rightarrow 1$ we begin with the next simplex, a triangle on the objects 0, 1 and 2. The triangle has a face, as do the square faces of a cube. Complementation must be replaced by a three-way swapping of basic truth values. This approach to 3-logos theory offers a new way of generalising topological spaces to higher categorical dimensions.

Observe that the faces (2-arrows) contribute to the weaker concept of complementation. In the trivalent corner of a spatial cube they play the role of dual time. It should not be surprising, therefore, to find that mass generation is linked to the sixth face of the tricategorical parity cube, the faces of which represent internalised edges of the Mac Lane pentagon.

M Theory Lesson 53

Chris Woodward made an intriguing comment on Tao's blog recently, on the topic of honeycombs. He said: I once had this crazy dream, that the quantum version of a honeycomb should be a honeycomb on a thrice-punctured sphere with hyperbolic metric.

Recall that the term quantum here refers to the multiplicative matrix problem. For us, a 3-punctured sphere is the Veneziano moduli space for 4-punctured spheres. Since such spaces rightly live in twistor land, as opposed to the Minkowski world of ordinary Feynman diagrams, a propagator is replaced by the bubble diagram analogue. In painting a 3x3 honeycomb on the pair-of-pants, three points are marked on each circle boundary. These are like the minimal bubbles of the $\mathbb{RP}^1$ diagrams, where marked points correspond to Feynman legs.

In the holographic logos 1-operad land, one projects down to two dimensional disc diagrams with marked points on the circles. Attaching lines between these points usually gives Jones' planar algebra diagrams. The problem in the 1-operad setting is that, in the 3x3 honeycomb case, it looks like there are a total of nine marked points, which one cannot pair up with internal lines. By allowing an internal honeycomb diagram, the nine external lines may be attached to the nine marked points. Each circle, with three marked points, may be viewed as a triangle. Note that nine additional internal vertices are required inside the large disc.

M Theory Lesson 52

The logical necessity of weakening distributivity in logos theory forces a study of pseudomonads, not just monads. Steve Lack has shown that a good theory for pseudomonads really requires Gray categories, our favourite tricategorical toys. This is the primary reason that a quantum analogue for a topos must go higher than 2-categorical structures.

The first kind of distributivity that we learn about is that of ordinary multiplication over addition. This is fully described by monads (in particular + and x) in a (causal) square involving the categories Set, Ring, Monoid (for multiplication) and Ab (for addition). The category of rings is where the numbers actually live. Now by characterising Set as a ground 2-logos, we begin to see that a very fundamental axiomatisation of M Theory should be possible, in terms of pseudomonads for 3-logoses.

Hopefully by now it has occurred to our readers that the term M Theory does not merely refer to an 11 dimensional supergravity.

M Theory Lesson 51

The gallant kneemo has pointed out that a locale is an important concept in topos theory. A locale is a simple generalisation of the lattice of open sets for a topological space. Localic toposes are discussed in the book of Mac Lane and Moerdijk.

By restricting attention to sober topological spaces, there is an equivalence of categories between spaces and a suitable collection of locales. Alternatively, there is a duality between spaces and frames, where a frame is just a locale in the opposite category. Now the initial object in the category of frames is a two point set {0,1}, because lattices always have a top element 1 and a bottom element 0.

When considering spaces, the most basic space is really the Sierpinski space $S$ and not a single point, which is often used to describe points in spaces via maps from the one point space. This space has the property of being both a space and a frame, the initial frame. An open set in a space $M$ is considered as a continuous map $M \rightarrow S$, whereby the inverse of 1 picks out the open set.

The existence of such a self-dual object in a categorical duality turns out to be very useful. Another example is the circle $U(1)$ in Pontrjagin duality for locally compact (Hausdorff) abelian groups. This is why kneemo's remark about a ternary analogue is interesting. We have seen that generalised Fourier transforms are important in M Theory. It is not enough, however, to simply replace the two element set with the three element set {0,1,2}. The inclusion of classical locales into the 'quantum' topos theory would require a higher categorical setting, for which the three possible two point subsets are perhaps rather subcategories of the triangle category on three points. This is suggestive of the need to consider three inclusions for 2-logoses into 3-logoses.

Aside: I have inserted a button to Carl's gravity simulator on the left sidebar.

Quote of the Month

"Now the elapsed time, even if it is really something different, is certainly measured most easily (if not most perfectly) by the plane area circumscribed by the planet's path", Johannes Kepler* (1571-1630)

Taking a value $e^{2 \pi i r}$ on the unit circle, for rational $r$, the area of the segment is $r \pi$. But taking an angle in rational radians, such as $e^{2 i r}$, one obtains an area of $r$. Just a thought.

* New Astronomy, translated by W. H. Donohue (C.U.P. 1992)

Mach Speed

In the first paragraph of Principles of the Theory of Heat [1], Mach states, "The collection of these instances of the physical behaviour of a body, which are connected with the mark of our sensations of heat - the collection of reactions - is termed its thermal state, or state with respect to heat."

Mach was the first great modern relationalist. In his writing, the observable properties of bodies are considered to define physical states, and this pragmatism is taken far more seriously than the concept of universal (theoretical) law. Later, Mach discusses causality: "Where we assign a cause, we only express a relation of connection, an existing of fact; that is to say, we describe. ... It is far better to regard the conceptual determinative elements of a fact as dependent upon one another in exactly the same sense as the mathematician, for example the geometer, does."

This is very much in the spirit of relational set theory, the axioms of which are properly formulated in topos theory. In particular, the axiom of comprehension states that there exists a set (contained in a chosen universal set) containing all elements $x$ such that $x \in \phi$, where $\phi$ is a property that may be stated logically. Recall that it was Gray who first considered extending this axiom to category theory, whereby he found himself developing the Gray tensor product for bicategories.

[1] in English, ed. B. McGuinness (Reidel 1986)

Witten's News

Here is a bookmark on the new work of Witten on exact black hole solutions and drinking moonshine. Matti Pitkanen has an informative post on the connection with TGD, and of course our colleague Motl also has a post.

An interesting fact about the Leech lattice is discussed by Tony Smith: the neighbours of the origin in 24 dimensional surreal number space form a Leech lattice. Recall that the surreal numbers give a tree structure to the reals, and as we know, trees are more fundamental than real numbers.

Speaking of canonical constructions for the reals, recall that in using associahedra, we came across integral arguments for the Riemann zeta function. But the full Riemann hypothesis requires us to understand the function on the complex plane. The surreal tree is a binary logic construction of the reals. A Space Time complexification could give us a notion of complex number.

M Theory Lesson 50

Fundamental to an n-logos is the concept of n-ality. Dualities belong to 2-logos structures, which are built upon the binary logic of the line element [0,1] and the parity square. In modern language, a duality expresses an equivalence of categories, which we take to be a bicategorical concept. In M Theory we see that triality is about the interplay of three parity cubes, based on ternary logic for the values 0,1 and 2.

Observe that connections between the logic values form a basic n-simplex, which may be labelled with directed faces as in Street's orientals, which describe strict n-categories. Unsurprisingly then, Kapranov's non-commutative Fourier transform is built upon simplices and cubes. This kind of Fourier transform should be a basic construction in M Theory, as it was for Heisenberg and Dirac.

M Theory Lesson 49

Recall that internalisation turns one square into four. This occurs in the category of (stable) trees, discussed in Leinster's book Higher Operads, Higher Categories from page 230. For example, consider the square in the diagram. This square is one of the three squares on the 9 faced Stasheff polytope, which one obtains by reducing the 5-leaved tree quadruple squares to ordinary faces, yielding a kind of classifying space. This diagram makes the contraction and expansion moves on trees more explicit. Recall that the entire polytope is labelled by a simple 1-level tree as befits a 1-operad polytope.

Observe that the total number of squares describing the Stasheff polytope is 6x5 (for the pentagons) plus 3x4 (for the squares) which comes to 42!

Blogging Bone II

Unfortunately, Carl's excellent PF thread on Schwarzschild orbits has been locked by the moderators after some negative reports from GR experts. A number of particle physicists have expressed some appreciation for Carl's work, but of course they couldn't possibly know anything. Don't forget to look at Carl's new gravity applet.

On a positive note, Louise tells us that her paper about $GM = t c^3$ has finally been accepted for publication by a major journal. It will be nice to see it in print at last. Maybe she will talk about it at GRG18 in Sydney in July. I wish Carl could make this conference as well. Is there anybody that would be willing to buy him an airline ticket?

M Theory Lesson 48

Urs Schreiber at The Cafe has developed a sudden interest in cubes, with an interesting post called The First Edge of the Cube. He says, "all that is required is playing Lego with these squares: two of them paste together to give a transition function. Six to give a triangle. Twelve a tetrahedron – these shapes are the differential cocycle representing the differential cohomology class of the n-transport....One aspect of the power of this formulation of differential cohomology is that I haven’t even had to mention the word 'differentiable' yet. Nor the word 'continuous'." Meanwhile Tao has a post on triple L series, which I would like to understand, but I'm dreaming.

In the realm of gauge theory this is all very complicated, which is why we like to do things with n-logoses instead. An n-logos should be defined long before one starts worrying about gauge symmetry, out of the simple combinatorics of cubes and Street's orientals and also n-operad polytopes. For example, we saw the 9 faced Stasheff polytope arise in $\mathbb{R}^3$ when tiling the (real) moduli space of the 6-punctured sphere. An example of this reduction of 6 dimensions to 3 dimensions appeared yesterday when Doug mentioned that typical Calabi-Yau spaces are fibred manifolds with 3-torus fibres, and he associated the 3-torus with the three Time dimensions of twistor triality. A 1-dimensional torus is just a loop, like the magnetic field loops around a current line, mused over long ago by Maxwell.

Mirror symmetry on such a Calabi-Yau space is a kind of T duality. So we see the interplay of T duality and S duality in the world of the prime 3. The Langlands picture is thereby related to the evenness issue (2 is for matrix), mentioned in Schreiber's post. The true primes are the odd primes, starting with three. An investigation of pure mass in this context leads one to the simple formula for Baryon fraction, as found by Louise Riofrio, who correctly predicted, amongst other things, the molten core of Mercury (a modern day perihelion).

Meanwhile, Carl's work on Schwarzschild orbits has progressed. Here is his Gravity applet. Carl says, "As an aside, I almost wasted my time programming up the usual geodesic equations. In fact, I actually sat down to do it. What stopped me was the realization of how much more difficult the usual geodesic equations are to use than the simple equations of motion I've found here".

Blogging Bones

Around the blogosphere: Tommaso Dorigo just can't help rambling on about searches for fairy (ie. non existent) bosons, and we always enjoy his posts. Above is an artist's image of the newly discovered molten core of Mercury. Matti Pitkanen finds cold fusion is hot news, and for those who thought blogging was just a fad, my brother tells me his dog Barney now has a blog at a dog blogging site.

M Theory Lesson 47

As Louise Riofrio likes to point out, Einstein was fond of discussing imaginary Time, so that the metric could be written with a bunch of plus signs. In octonionic M Theory, it can be useful to think about 3 Times and 3 dual Space directions. The reduction to complex numbers gives one piece of the triality, and so we would like to consider just one Space and one Time dimension as a complex plane, with a duality that interchanges them.

Recalling that the complex (unitary ensemble) ribbon matrix models are self-dual under T duality, it is natural to consider S duality, or rather electric-magnetic duality, as a candidate operation for the study of this Space Time interchange.

In the 19th century, J. C. Maxwell unified electricity and magnetism by observing the apparent duality in the known phenomena of (1) magnetic field loops about a current line and (2) electric fields emanating from a pointlike charge. He used hexagons in his description of the aether through which electromagnetic waves travel.

Today we are interested in problems such as the Riemann hypothesis for the zeta function, which has zeroes on both the $x$ axis of $\mathbb{C}$ and the line $x = \frac{1}{2}$, and most probably nowhere else. Recall that the (upper half of the) vertical line appears in the natural description of the moduli space for the 1-punctured torus (an elliptic curve), as the boundary of a fundamental domain for the modular group, which just happens to leave invariant some BPS black hole masses when one considers a transformation for the complex parameter $\tau = \frac{\theta}{2 \pi} + i \frac{4 \pi}{e.e}$, where $e$ is electric charge and $\theta$ is an instanton parameter. Chalmers has considered the connection between the four point amplitude in N=4 SUSY YM and the Riemann hypothesis.

M Theory Lesson 46

Henry Cohn studies sphere packing in different dimensions. In 2004, along with A. Kumar, he proved that the Leech lattice is the unique densest lattice in $\mathbb{R}^{24}$.

If $V$ is the volume of a fundamental polytope for a lattice, and $r$ is the minimal length of a basis vector for the lattice, then with spheres of diameter $r$ the packing density in dimension $n$ is

$\rho = \frac{\pi^{k}}{2^{n} (k)!} V^{-1} r^n$

where $k$ is $\frac{n}{2}$ and for odd $n$, $k! = \Gamma (k + 1)$. Cohn and Kumar solve the Leech problem by finding an $r = 2$ basis under the normalisation $V = 1$, which saturates a known upper bound on $\rho$. It turns out that the Leech lattice has 196560 vectors of minimal length equal to 2. The next smallest length for vectors is about $\sqrt{6}$.

One scales the minimal vectors to fit on a unit sphere $S^{23}$. The minimal angle satisfies $\textrm{cos}\phi = 0.5$. Looking at points on spheres is something one does in coding theory. The connection with coding theory is a good way to look at energy minimisation problems. Think of the selected points as satisfying some potential. Cohn and Kumar have a concept of universally optimal distribution for points on spheres.

The W Factor

Tommaso Dorigo has a wonderful post today on the W boson width, which is measured to be $2.032 \pm 0.071$ GeV. Since he could not post the picture showing the full collection of these results, I will post it here. Observe that the result agrees very well with 'Standard Model' predictions, based on nine simple decay channels of the W boson into pairs. Don't ask me about the LEP-2 error bars.

GR Revisited

Sly old Carl Brannen has been busy discussing Schwarzschild black holes in flat space at PF. It seems that particle physics just wasn't interesting enough for him.

Regulars at PF often use cool signatures on their posts. For example, turbo-1 quotes Einstein (1924):

"The ether of general relativity therefore differs from that of classical mechanics or the special theory of relativity respectively, in so far as it is not 'absolute', but is determined in its locally variable properties by ponderable matter."