Blogging Bounty

It appears that the new group blog Noncommutative Geometry really is the blog of Alain Connes, although to be fair it was set up by Arup Pal, according to Never Ending Books. There must be some sort of conspiracy going on here, because Terence Tao also has a new blog. And what are these great mathematicians blogging about? You guessed it: physics.

On Connes' blog, today's post by Goss is about the Bost-Connes reinterpretation of the Riemann zeta function as a statistical mechanical partition function.

Riemann Rumbling On

I've just finished reading the delightful book by Sabbagh, Dr Riemann's zeros. Although not a mathematician himself, Sabbagh competently launches into equations and diagrams, which he clearly explains for the lay reader. He spent a lot of time interviewing experts in the Riemann Hypothesis, to the point of attending their lectures, and he recorded the conversations.

The interaction of physics and the Riemann Hypothesis started with a memorable event, recounted in the book. Freeman Dyson was having afternoon tea as usual one day at Princeton in 1972. He was introduced to the visitor, Montgomery, who had been looking at the average gap in a long list of $\zeta$ zeroes. Montgomery mentioned his formula for the pair correlation, namely

$1 - ( \frac{\textrm{sin} \pi u}{\pi u} )^2$

at which point Dyson exclaimed that this was precisely the density of the pair correlation of eigenvalues of random matrices in the GUE.

Sabbagh was impressed by the awe that many mathematicians had for the Riemann Hypothesis. Conrey explained that the Riemann hypothesis is the most basic connection between addition and multiplication that there is, and Connes said: it is a basic primitive question about the adelic line which we don't understand. It is a question about the way addition is fitting with multiplication.

Reminds one of categorical distributive laws, heh? Recall that for us addition and multiplication are monads (please don't tell me you've forgotten about those). Anyway, a morphism $+ \times \rightarrow \times +$ which describes the commutativity of monads is a distributive law (wow - wikipedia is on to it). These are entities we need to think about in the context of quantum topos theory, because weak distributivity is the thing that separates quantum logic from the intuitionistic logic of an ordinary topos.

Riemann Rambling On

Amongst physicists, Riemann is best known for the concept of metric. But his one paper on number theory, where he defined the zeta function, was not the only work he did on the subject. Riemann actually computed some zeroes of $\zeta (s)$ himself. This was unknown for about 60 years, until Siegel went through some work of Riemann's and found a key to computing zeroes simply.

The Z function is defined by

$Z (t) = e^{i \theta (t)} \zeta (\frac{1}{2} + it)$

where

$\theta (t) = \textrm{arg} (\Gamma (\frac{2 i t + 1}{4})) - \frac{\textrm{log} \pi}{2} t$

The real zeroes of $Z (t)$ are the zeroes of $\zeta (s)$ on the critical line $s = \frac{1}{2} + it$. Positive real values of $t$ for which the $\zeta$ function is real are known as Gram points. By looking for pairs of Gram points, one can narrow down an interval where a zero of $\zeta$ must lie.

It is still rather impressive that Riemann managed to compute zeroes this way, with pen and paper, needing numbers such as the square root of 2 to something like 30 decimal places.

Motive Madness

There are two papers by A. B. Goncharov, namely

1. Multiple polylogarithms and mixed Tate motives
2. Periods and mixed motives,

which are referred to by Brown in his paper on multiple zeta values and period integrals, as an excellent study of the punctured sphere moduli M(0,4) $\simeq \mathbb{P}^1 \backslash \{ 0,1, \infty \}$ and its finite covers based on roots of unity.

Hmmm. Maybe this post should be called M Theory Lesson 18. These topics (motives, number theory, gluon amplitudes etc.) are getting awfully mixed up! No, never mind. I'd better get back to reading, I guess. I must be crazy. It's a gorgeous day outside...

M Theory Lesson 17

In the recent comments, Mahndisa wondered what I meant by mentioning Euler's relation

$e^{i \pi} = -1$

in M Theory Lesson 16. To be honest, it isn't entirely clear to me. But rather than going ahead and expanding the exponential in the usual way, I was asking why this function has the properties it does. We need to view Euler's relation anew, to see complex analysis in the light of diagrammatic reasoning. Non-standard analysis may take us a long way into topos methods, but it doesn't dissect the good old complex plane in the simple geometric way we require. Remember that those associahedra tilings were about real moduli. Now we need to understand the complex case, using 2-operads.

2-operads involve two-level trees. If we replace the upper level by the ordinal which counts the number of leaves, we obtain one-level trees with labels. It is known that such trees are appropriate for complex moduli. Now we would like to take the 2-operad polytopes, such as the hexagon, and tile spaces with them.

Parrot Smuggling Alert

Would the anonymous North American who found this blog by googling 'keas as pets' please desist with such disrespect for these rare birds. Unfortunately, your kind are all too common. Illegal capture and trading of kea are one of four main causes for their decline over the last few decades. The other causes are habitat degradation, introduced mammal nest predation and human killings (shooting or poisoning).

M Theory Lesson 16

Recall that Leinster's Euler characteristic for 1-categories could take on rational values. But we need more numbers than that. How might numbers be extended into the complex plane? We suspect that at some point higher categories will be involved.

For a finite directed graph, $\chi$ is simply given by $V - E$. For example, for the five arrow graph
we have that $\chi = -1$. Observe that this is the simplest way to obtain an Euler characteristic of -1 using circuit free graphs. An extra arrow is needed to make the diagram a category. What would it mean to take a square root of this diagram? Numbers must be represented by diagrams, so we must ask this question. One guess would be to call a triangle, with a 2-arrow in the interior, the number $i$. Let's think about it. Everybody remembers Euler's relation

$e^{i \pi} = -1$

Thinking of the logarithm instead, can we turn the multiplication (composition) into addition?

Luscious Langlands II

Since domains on the complex plane might really represent moduli, and we know everything should be about categories at the end of the day, it would be better to replace the eigenfunction $f$ with a more sheaf theoretic concept. As Kapustin and Witten say in their abstract: The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions.

Yes, N=4 SUSY Yang-Mills turned up when we were worrying about twistor string theory and calculating gluon amplitudes. Actually, these days the geometric Langlands conjecture is about an equivalence of categories, namely derived categories of sheaves. Schreiber has been blogging about such things. But where did the number theory go in all this String geometry? Isn't that what this is really about?

Hot Dog Stars

From Physics Web today we hear the news: magnetic heating could be playing a much more prominent role in the evolution of neutron stars than previously expected, claim astrophysicists in Spain and the US ... This could cause astrophysicists to rethink current theories of how neutron stars cool.

José Pons and colleagues used both satellite X-ray and ground based radio telescopes to investigate magnetic heating, which appears to be occurring in neutron stars. Louise Riofrio will be pleased to hear it. Another prediction comes true!

Luscious Langlands

There are very few papers that have remained in my possession, after years of travelling and moving about. One of these is Stephen Gelbart's An Elementary Introduction to the Langlands Program (Bull. Amer. Math. Soc. 10, 2 (1984) 177-215). These days there is no need to canvas mathematicians with a dim hope that they will see some connection of this to physics. I still understand very little of the Langlands program, but no longer for want of articles on the subject. Actually, I met Langlands once, back in the mid 90s. I had no idea who he was, much to the shock of the mathematicians I was socialising with at the time. Really cool guy - looks 20 years younger than he is.

Anyway, Gelbart's paper begins with a quote by Browder: that the possible number fields of degree n are restricted by the irreducible infinite dimensional representations of GL(n) was the visionary conjecture of R. P. Langlands. The basic concept needed is that of an L-function associated to a representation $\pi$. We can start with the Artin version, which depends on a morphism $\sigma: G \rightarrow GL_{n}(C)$ for a certain Galois group $G$, and takes the form

$L (s, \sigma) = \prod_p (\textrm{det} (I_n - \sigma (F_p) p^{-s}))^{-1}$

where $I_n$ is an identity and $F_p$ is a Frobenius map. Oh yes, this is supposed to look a bit like the Riemann zeta function. Mahndisa will be pleased to know that the p-adics soon show up, what with primes floating about all over the place.

Langlands set about finding a correspondence $\sigma \rightarrow \pi (\sigma)$. The $n = 2$ case ends up being about automorphic forms. Remember that pretty fundamental domain, related to the modular group, that came up when we were talking about the moduli space for the punctured torus? One considers nice functions

$f(z) = \sum_{1}^{\infty} a_n e^{2 \pi i n z}$

with respect to such domains, so that when the set of $a_n$ is multiplicative, $f$ is an eigenfunction for Hecke operators.

M Theory Lesson 15

Probably the first people to realise that the number of generations might have something to do with idempotents and Jordan algebras were Dray and Manogue in 1999, in The Exceptional Jordan Eigenvalue Problem, which was pointed out to me by kneemo. On pages 10 and 11 they discuss how the usual Dirac equation comes from the 9+1 dimensional one, which is written as a simple eigenvalue problem using a 2x2 octonion matrix, or again as a nilpotent equation using the Freudenthal product. The three generations fit into the Moufang plane, which are Jordan elements satisfying

$M \circ M = M$, tr$M = 1$

so the matrix components lie in a quaternion subalgebra of the octonions. These elements are primitive idempotents.

Naturally we should improve upon the reliance here on a higher dimensional Dirac equation, for which we see no real physical motivation. Brannen's idempotents are a big step forward in this regard. But we can also reinterpret the higher dimensions in a categorical context, where they are not naively taken to mean spatial dimension.

Quote of the Week

Inspired by Tommaso's regular column The Quote of The Week, I was amused by a comment on Carl Brannen's not-so-busy blog:

lots of us 10 year olds really do want to learn physics... Al

It brings to mind the old adage about science progressing one death at a time. If you write more, Carl, perhaps more 10 year olds will join the club. Actually, I must confess, you do write a fair amount, such as this recent comment on Clifford's post about Eureka moments:

My most recent such moment was when I realized that any non Hermitian projection operators in the Pauli algebra can be written in a unique way as a real multiple of a product of two Hermitian projection operators.

That's nice.

M Theory Lesson 14

The higher categorical parity cube is a broken pentagon, such as the one that appeared in operad theory. It is simply the diagram
The top face really looks like
if we use trees to label choices of bracketings. Carl Brannen's idempotents for lepton masses are also labelled by the parity cube, which describes three directions in space as well as three generations. It is nice to know that this agrees with the number of quark generations that we get by calculating an Euler characteristic for a gluon orbifold modelled by twistors.

No More Secrets II

Bernhard Riemann wrote only one 8 page paper in number theory [1]. It listed a number of conjectures, the most famous naturally being the Hypothesis. Let $\pi (N)$ be the number of primes less than or equal to $N$. Also recall the function

Li$(x) = \int_2^x \frac{dt}{\textrm{log} t}$

Another conjecture (later proved by von Mangoldt) that Riemann stated in his paper was that there should be a formula for $\pi (x)$ - Li$(x)$ valid for any $x > 1$. The only tricky term in the formula is a sum over the complex zeros $\rho$ of the $\zeta$ function,

$\sum_{\rho} \frac{x^{\rho}}{\rho}$

So the zeroes encode precise information about the distribution of the prime numbers. This imbues these points of the complex plane with a special significance, just as the ordinals along the real axis are also special because they represent ordinals, which we know are about counting elements of sets, or Euler characteristics for categories. Maybe the other numbers in the complex plane should be viewed this way as well. In other words, the complex plane is not given a priori as a boring set obeying the axioms of a field.

A topos theorist cannot escape such ideas, because number fields become heinously complicated. The ordinals make sense enough. Categories such as Set have an object of Natural Numbers. From this one can build the integers and the rationals, but then one has to worry about what one means by the real numbers because the set theoretic definitions don't all necessarily agree in other toposes. And why should we care more about the reals, from the prime at infinity, than the p-adics? The p-adic number fields can fit into the complex plane via Chistyakov's beautiful fractal patterns. From du Sautoy we also have some beautiful images of the $\zeta$ function:



[1] H. Davenport, Multiplicative Number Theory (LNM 74)

No More Secrets

Yesterday we were treated to a public lecture by the enigmatic Marcus du Sautoy on the music of the primes. It began with a discussion of the shirt numbers in European football teams, something that du Sautoy was particularly enthusiastic about. Fortunately, by the end of the hour we were well into the Riemann hypothesis.

He gave a wonderful explanation of what the zeroes mean. According to the hypothesis, the non-trivial zeros of the zeta function all lie on the critical line, the real part of $z$ equal to 1/2. Instead of plotting the axes of the complex plane, however, he showed the line on a plane marked by a vertical axis for frequency (of a Riemann harmonic component of the full prime counting function) and a horizontal axis for amplitude.

So there is a way to label the zeroes one by one, starting with the one lowest on the positive vertical axis at $y$ = 14.1347 (and remembering that there is a reflection symmetry across the horizontal axis). The label counts the number of nodes in a wave of some kind. All waves have the same amplitude. This fact must relate to the localisation we find with T-duality in twistor ribbon models. So a basic principle of quantum gravity forces the Riemann hypothesis to be true. What is this wave? The real world manifestation that du Sautoy mentioned was that of the energy levels for large atoms, such as uranium. But as we know, most analyses of the match between energy levels and Riemann zeroes involve the statistics of an ensemble. We would rather construct the zeroes one by one. The expectation is that the algorithm, if it exists at all, would become increasingly complex, so that the computation of all the primes would amount to a vast universal computation. But hang on a minute! Isn't that what our rigorous QFT/QG program is all about?

We learn in kindergarten that the energy levels of atoms correspond to the allowed shells for electrons. Thus there is a correspondence between $E_n$ and particle number. Now recall, we saw that the tricategorical aspects of QCD suggest a correspondence between particle number and categorical dimension. This is good, since we expect categorical dimension to provide a measure of complexity for a problem naturally couched in that context, such as calculating the nth zero.

Another clear aspect of du Sautoy's talk was his explanation of the logarithm function, as a method for converting multiplication into addition. The logarithm acts as the first mode of Riemann's wave decomposition. We have met the higher dimensional analogues before in operad combinatorics.

Finally, there were a few fun film clips from movies about mathematics. As Robert Redford says when the detective slots the codebreaker box into the aircraft control computer, No More Secrets.

David's No DE

As promised, David Wiltshire's long awaited No DE cosmology paper is coming out. This arxiv link should work tomorrow. A poor person here was heard complaining that someone was hogging the printer to print a book. But don't worry, it's only a 72 page paper.

Valentine's Day

I am the sort of person who has no awareness whatsoever of a day's cause for celebration. And so, quite unaware of the day, I found myself boulder hopping down the Waimakariri. But at the road end I came across a lovely old couple sipping champagne and eating Belgian chocolates, an enjoyable activity that I soon found myself invited to partake in! So, yes, the absense here has been due to another little wander in the hills. The remainder of the day turned out excellent as well. By dinner time I had completely forgotten it was Valentine's Day and couldn't figure out why the restaurant, where I went to treat myself to a well-deserved steak, was so busy - that is until a sweet old Kiwi French ski instructor chatted me up and reminded me. Anyway, back to work, folks. All those guys who sent me email cards - I'm afraid they must have been chewed up by my vicious spam filter. Oh well!

M Theory Revision

The modern understanding of Feynman diagrams comes from a beautiful body of mathematical work, such as that of Kreimer et al on the Hopf algebra structure of renormalisation. In a category theory setting, we know that such structures rely on the concept of operad. Moreover, higher dimensional operads appear essential in moving beyond CFT and addressing the problem of describing mass quantum numbers.

The emphasis on renormalisation is a mistake. This picture still works in a Minkowski background QFT, the idea being that the Standard Model in its rigorous guise will not be much altered. But we have seen that a twistor correspondence must be implemented, for it is only in this setting that the physical logic has a chance of being written in a topos like language. The twistor point of view changes our use of operads. The need for higher dimensional structures becomes even more apparent as we match the 1-operad associahedra to mere real moduli.

Victoria Canterbury II

It turned out to be a NZ-Korea Gravity Workshop, because we had a number of visitors from Korea. Sung-Won Kim, who was here for the Kerrfest, was visiting again, this time with greatly improved weather. He took many photographs which he has promised to send us. Yesterday we also heard excellent talks by Jong-Hyuk Yoon, Silke Weinfurtner, Alex Nielsen and Petarpa Boonserm.

I learnt that Yongmin Cho's Restricted Gravity decomposition has a very interesting application in lattice gauge theory. Yongmin Cho and Jong-Hyuk Yoon are off to Wanaka today to start the Rabbit Pass tramp, which they learnt about on the internet. I highly recommend it.

Victoria Canterbury

The joint UVW and UC Gravity Workshop kicks off today with a talk by Yongmin Cho from Korea entitled Restricted Gravity. David Wiltshire will then talk about his no DE cosmology, although this isn't clear from the title. Will report later.

It's happening

I hardly need to mention the latest United Nations climate change report. It has had quite a bit of press. But some people still believe it is all just sensationalism and there is no reason they cannot happily continue to drive their car around and buy heavily marketed goods with wasteful packaging and eat food, which magically appears on the supermarket shelves, until sated, for ever more. No wonder the rest of us worry.

In the local newspaper today there was a poll about climate change. It had three options: (1) it's happening and we're doomed (2) it's happening but we'll get by and (3) it's not happening. Option (1) gets the most votes at 39%, just ahead of option (2) at 38%, with option (3) not too far behind. Wow. Was public opinion this divided on, say, the Cold War?

Regular Irregular II

The double covering of the Klein quartic is described using the binary octahedral group, a subgroup of the quaternions. As Tony Smith points out, this group is given by the 48 elements

$\pm 1, \pm i, \pm j, \pm k$
$\frac{1}{2} (\pm 1 \pm i \pm j \pm k)$
$\frac{1}{2} (\pm 1 \pm i)$, $\frac{1}{2} (\pm 1 \pm j)$, $\frac{1}{2} (\pm 1 \pm k)$
$\frac{1}{2} (\pm i \pm j)$, $\frac{1}{2} (\pm j \pm k)$, $\frac{1}{2} (\pm k \pm i)$

of a 24 element binary tetrahedral group along with its dual. This is nice, because we were hoping that the genus 3 surface would have something to do with quaternions, since the real dimension of the moduli is 12, the same dimension as $\mathbb{HP}^3$. The binary tetrahedral group can be described by two generators $a$ and $b$ satisfying $a^3 = b^3 = (ab)^2$. There is a quaternion $q$ that satisfies $aq = qb$, namely $q = i \textrm{exp}(\frac{\pi j}{4})$. The binary octahedral group is described by generators $A, B$ and $q$ and relations $A^4 = B^3 = q^2 = ABq$.

This is all secretly about $SL(2,7)$, of order 336. Now it so happens that $PSL(2,7)$, of order 168 (recall the $24 \times 7$ from the last post), is about products for octonions. $PSL(2,7)$ is the group of linear fractional transformations of a heptagon, or the Fano plane. Remember that there are 7 imaginary octonions, just like there were 3 for the quaternions. That means $2^7 = 128$ possible sign combinations. The 480 octonion products come from
$480 = \frac{128 \times 7!}{8 \times 168}$

Regular Irregular

In the book The Eightfold Way (nothing to do with Gell-Mann) there is an article by Thurston on the beauty of the Klein quartic curve, which the book is about.

The Klein quartic is tiled regularly by irregular heptagons. Topologically it is a three holed (genus 3) oriented surface, and hence it has a hyperbolic geometry. It has interesting symmetries. Tiling the Poincare disc with the heptagons we see a central heptagon with seven surrounding ones. By following the tiling outwards we generate the sequence 7,7,14,21,35,56,91,147,238, ... which is precisely seven times the Fibonacci sequence! If the three holed surface is squished about it can be made to look like a tetrahedron with tubes for edges. Have fun playing.

M Theory Lesson 13

John Baez's TWF244 is really cool. He talks about a recent paper by Leinster on the Euler characteristic of a finite category. Consider the following. Order the objects in the category $1, 2, 3, \cdots ,n$. The integral adjacency matrix for the category sets $a_{ij}$ to be the number of arrows from $i$ to $j$.

For example, in the category

we would have a 3x3 matrix with all entries 1, since we must not forget the identity arrows. If the inverse to this matrix $A$ existed, the Euler characteristic would be the sum of entries in $A^{-1}$. Let's fill in a few more arrows. Imagine there were $k$ arrows from $1$ to $2$, $2$ to $3$ and $1$ to $3$ which forms a basic composition triangle. Similarly, imagine the dual triangle had $m$ arrows at each edge. Then the adjacency matrix would be a circulant based on $1,k$ and $m$. Such triangles look like idempotent equations $k^2 = k$, but here $k$ is an ordinal and we should wonder about composing one of the $k$ arrows from $1$ to $2$ with another one from $2$ to $3$, because the number of such compositions would naively be more than $k$. If $k^2 = k$ were true, and $k$ was ordinal, then it must be zero or one, which gives a particularly simple kind of category otherwise known as a poset.

Anyway, to cut a long story short, this characteristic works nicely for all sorts of things, such as orbifolds. In M Theory we counted the number of particle generations using an orbifold Euler characteristic, which might be a rational number in general. So we can think of this as a cardinality of a category! This is wonderful, because the physical result follows from the universality of $\chi$.

Moerdijk looked at Lie groupoids as a foundation for orbifolds, which seems like a logical thing to do from the Symmetry point of view. But remember that we encountered the orbifold Euler characteristic in the work of Mulase et al on ribbon matrix models.

Interludes

Alas, at this time of year the weather improves and the hills beckon. I had a bit of a cold and the best cure for such things is to launch oneself into some rough terrain and get the old ticker working.

Not content with some impressive analysis of the Higgs Bump, Tommaso has a new challenge: what will be the next discovery? This is a particle physics question. The options are Unexpected, SM Higgs, SUSY, Nothing, Other, extra dimensions, new Strong Dynamics, Substructure and LQ (say what?). Naturally, we have been called upon to vote. The first question is: what might Other mean? Since this seems like a cop out I'm going to ignore this option. As for Unexpected, history seems to show that we're not very good at spotting totally unexpected things, even if they are scratching our nostrils. And if it's entirely Unexpected, then what sort of theory is the prediction based on, anyway?

The Higgs, SUSY and extra dimensions are clearly out. Nothing must mean Nothing, as in Not Even some Interesting Rigorous SM stuff, despite there being NO Higgs boson. Hmmm. So that leaves Substructure, or new Strong Dynamics. These two options only managed a small piece of the pie in the grad student vote.

I'm going with new Strong Dynamics. If our understanding of the Standard Model changes radically when quantum gravity is considered, and in particular if quarks and gluons have something to say about the electroweak scale (and who knows?), then there is good reason to believe the LHC will determine this. All we need is to sort the theory out first!