Moving Up III

Update: Woke up 7am first morning in heavenly warm bed at the observatory. Shovelled 2 feet of fresh snow from doorway before throwing myself down the hill through the beautiful snow draped forest, into town.

Origin of Species

Two weeks ago, Theoretical Atlas brought to our attention the work of Rivasseau, the constructive field theorist. Now kneemo, in a frenzy of blogging, points out a new paper by Rivasseau et al on rewriting QFT using trees.

The proposal rests on the use of combinatorial species, introduced long ago by Andre Joyal, the great category theorist. A lot of our playing with trees and funny infinite sums in M theory is about combinatorial species, although we haven't yet worried about the exact relation. A species is just a functor from the groupoid of finite sets (and bijections) to itself. Recall that this groupoid is a lot like the finite ordinals, which correspond to cardinalities of sets. An example that often comes up in topos theory is the functor that sends each set to its power set, namely the set of all subsets of the set. This example illustrates the general idea of sending a collection of objects to another collection, equipped with some structure related to the original.

LHC Rap

Thanks to Tommaso Dorigo for this must see link: the LHC rappers.

Moving Up II

Here's a photo from town. The Mackenzie basin is a very dry area of the country, with generally fine weather.

Moving Up

Tomorrow is my last day at work in Christchurch. In a few days I'll be sitting in the uncrowded outdoor spa at the base of Mt John, which is probably snow covered now, looking at this view. Tough to be poor? Not this week.

M Theory Lesson 212

The $3 \times 3$ democratic matrix is useful in many ways. Here we see it acts as a cyclic shift operator for three 1-circulants that additively act like modular 3 arithmetic. But observe that $(2 \cdot \textrm{id})^{2} = \textrm{id}$ and, similarly, the square of $(312) + (231)$ is twice $(312) + (231)$. That is, we have the relations

$A^{2} = A$
$B^{2} = 2A$
$C^{2} = \frac{1}{2} C = \textrm{id}$
$A + B + C = 0$

Exponentiating the third relation yields the multiplicative honeycomb rule $A \cdot B \cdot C = 1$. However, using ordinary matrix multiplication one obtains $ABC = 2 \cdot \textrm{id}$, so the correct normalisation factor for all four matrices is the reciprocal of the cubed root of 2, namely 1.25992. Alternatively, since 2 is really $-1$, $C^{2} = C$ and $B^{2} = - A$ and the correct normalisation is a cubed root of $-1$, that is a 6th root of unity.

M Theory Lesson 211

Generalising the matrix $K$ further still by adding phases to $(2)$, one can offset the exact Koide eigenvalues for a phase $\theta = \frac{2}{9} + \frac{2 \pi n}{3}$ in the component $(1)$ by a very small phase $\delta \simeq 10^{-7}$. Note that the cyclic $\frac{2 \pi n}{3}$ factors from the Fourier transform have not yet been accounted for in the basic $6 \times 6$ eigenvalue problem, so they are included in $\theta$. The charged lepton $K$ operator eigenvalues for the democratic 6-vector then take the form

$\lambda = a + b \textrm{cos} \theta + c \textrm{sin} \delta$

where the last term is considered a small electric field term. That is, by making the amplitude $c$ small, $\delta$ need not be small and one could take $\delta = \theta$. This indicates a correspondence between, on the one hand $(1)$ and magnetic fields, and on the other $(2)$ and electric fields, which holds even if the factor $(2)$ becomes a 2-circulant matrix. Electric magnetic duality then swaps the two triangles making up a basic $S_{3}$ hexagon, as previously discussed. For the cube this may be viewed as the duality of a triangle and trivalent vertex in the plane. That is, a duality for Space and Time.

M Theory Lesson 210

In the last lesson we saw how the 1-circulant eigenvalues $(312)$ and $(231)$ correspond to an eigenvector in modulo 7 arithmetic. The remaining $3 \times 3$ 1-circulant is the identity operator. Observe that for the operator $K$, the identity is an eigenvalue for any vector of the form $(\pm X, \pm X, \pm X, \pm X, \pm X, \pm X)$. For the cyclic group on 7 elements there are roughly $3 \times 2^{6} + 1 = 193$ such vectors, including zero. For elements of $\mathbb{R}$, or $\mathbb{Q}$, vectors of a fixed sign sequence also form an eigenline. In general we might call such a sign sequence an eigenpath for the identity. Other phase choices for the Kasteleyn matrix $K$ clearly alter the eigenspace structure. For example, the operator sends the vector $(X,X,X,X,X,X)$ to $\textrm{cos} \theta \cdot (X,X,X,X,X,X)$. Democratic matrices, with all entries equal to $X$, may also be considered eigenvectors.

Aside: The difference between 192 and some other integers is the source of a very silly argument between Distler and Lisi.

The Naked Emperor

The blogosphere is abuzz with talk about this paper, by a non blacklisted physicist, entitled The Emperor's Last Clothes? The abstract begins:

We are in the middle of a remarkable paradigm shift in particle physics ...

I tend to agree with the paper up to this point. Unfortunately, the author seems to take a fairly traditional landscape point of view, reaching the conclusion that fundamental Standard Model parameters may not be computable for anthropic reasons. Dear me. A landscape of conservative vacua, based on the physics of a Model, is supposed to be a paradigm shift? Me thinks not.

The paper discusses the historical progression of demotions of humanity's special place in the universe. Until the early 20th century people thought the solar system sat at the centre of the Milky Way and that the Milky Way was the entire universe. On the other hand, landscape anthropomorphism claims a radical rethink of God's place in the universe, without even bothering to alter the status quo classical sea of galaxy superclusters. The more logical progression would be towards a physics which rethinks humanity's place in the universe by expanding the multiverse to include different observer types, not necessarily of human scale. The proponents of the landscape bog claim that, as difficult as it is to swallow, there are no alternatives and that an admission of this 'fact' is like the innocence of a child crying that all other physicists are stupid and the emperor is naked.

FQXi II

Having rudely queried the secretary about whether or not there were any women on the grants panel, I wasn't expecting any reply at all from FQXi, but today I received this email:

Dear Ms. Sheppeard

The makeup of the review panel is confidential. However, I can tell you (if these are the lines along which you are inquiring) that

(a) FQXi is committed to diversity in many aspects including by subject, and by geographical origin, race, gender, ethnicity, and career stage of the applicant, and

(b) none of these factors, or their lack of consideration, was by any plausible reading of the proceedings responsible for the negative outcome of your grant application. The panel had a lot of very tough decisions to make given the quality of the applications and the available funding. The panel was quite careful in their evaluations and ranking, and your proposal was not high-enough ranked to be funded, nor was it one of the unfunded proposals near the cutoff.

Thank you again for the effort that you put into your FQXi application, and I regret that the outcome was not positive for you.

Sincerely
Anthony Aguirre

Associate Scientific Director
Foundational Questions Institute

M Theory Lesson 209

Recall that the $6 \times 6$ operator $2K$, with basic permutations as eigenvalues, is of the form for circulants $(1)$ and $(2)$. What is the eigenvector? Let $K$ act on an object $(X,Y)$. Then one can solve the eigenvalue equation for $\lambda = (231)$ to obtain provided we do arithmetic mod 7. Try it yourself. The cyclic nature of the linear equations forces the eigenvector to live in such a ring. Choosing $K$ instead, rather than $2K$, we find that the same vector is an eigenvector for the other 1-circulant, $(312)$.

Ninja Prof

phdcomics

M Theory Lesson 208

Now consider the $6 \times 6$ Kasteleyn matrix given by This matrix is the unique such matrix with eigenvalues $(1) + (2) = (231)$ and $(1) - (2) = (312)$, the elements of $S_{3}$. It satisfies the relation $K^{2} = K + [0,(312) - (231)]$, using the same notation as the last post. This comes close to being idempotent, but the real idempotents are of course Carl's particle operators. A graph for this $K$ looks like the tiling by hexagons and triangles, which is a rectification of the hexagonal tiling of the plane. Observe that the six edges of the top left (1) form a hexagon within this graph, as do the other circulant components. The graph can be factored into two Hamiltonian circuits of length 12.

M Theory Lesson 207

A Kasteleyn matrix for a bipartite graph with $n$ vertices, drawn on the integer lattice in the plane, is an $n \times n$ adjacency matrix with non zero entries corresponding to edges $E_{ij}$, given by $K_{ij} = 1$ for horizontal edges and $K_{ij} = i$ for vertical edges. The square root of the determinant of $K$ counts the domino tilings of the checkerboard underlying the graph. An interesting paper by Stienstra includes examples of $6 \times 6$ generalised Kasteleyn matrices associated to $\mathbb{C}^{3} \backslash \mathbb{Z}_{6}$, where the row index corresponds to black vertices and the column index to white vertices, such as

1 -1 0 -1 0 0
0 1 -1 0 -1 0
-1 0 1 0 0 -1
-1 0 0 1 -1 0
0 -1 0 0 1 -1
0 0 -1 -1 0 1

which we observe is of the form

(1) (2)
(2) (1)

using $3 \times 3$ 1-circulants. It obeys the relation $K^{2} = 2K + [(312), 2(231)]$, where the final term is the obvious simple $6 \times 6$ matrix in terms of the permutation basis. Labelling edges with general complex units allows complex units as matrix entries. This example is derived from the hexagonal graph where the opposite sides of the hexagon are glued. That is, there are really only 18 edges, which is the number of non zero entries in $K$. If we did not glue edges there would be 24 non zero entries, based on 12 non zero entries for a pair of $3 \times 3$ circulants, just like the neutrino mixing matrix.

Lieven's Trinities II

Lieven Le Bruyn explains some of the mysteries of the monster, which is associated to a Riemann surface of genus

$g = 9619255057077534236743570297163223297687552000000001$

That's an awful lot of gluing of heptagon edges, which when halved define the ribbon graph for the surface. Lieven's construction involves our favourite modular group and its group algebra, basically all possible combinations of elements of the group. Let's start out with the much simpler M Theory group $S_{3}$, of permutations on three letters. The group algebra is the complex number combinations, such as the 1-circulants

$a \cdot 1 + b \cdot (312) + c \cdot (231) $

or mixtures of 1-circulants and 2-circulants. These algebras showed up in the Hopf algebra triples associated to operad polytopes like the permutohedra and associahedra, as investigated by Loday et al. In a physics variant on Lieven's challenge: can you match these numbers to something concrete, like particle spectra and mixing parameters? I'll buy the winner a few pints of good South Island beer.

Aside: The Kostant of Lisi fame has posted some interesting email correspondence on his door. It is available here.

Varying Alpha

I have been enjoying some of the talks at this week's PI conference on the variation of fundamental parameters. M. Kozlov gives a clear outline of current results, such as a single source analysis for the three basic parameters, yielding the result consistent with zero variation. In a new analysis of varying $\alpha$ using absorption lines from many quasar sources, M. Murphy et al (MNRAS 2008) conclude that there is a variation of

$\frac{\Delta \alpha}{\alpha} = -0.44 \pm 0.16 \times 10^{-5}$

in disagreement with other results that are consistent with zero change. In the plot below, the black points are binned data. He gave convincing arguments that their analysis of cloud dynamics etc was careful, which is also the impression I got from reading the papers a few years ago, but the analysis is very complex, requires model fitting and the theoretical implications are always glossed over. The question I have is, since one can only assume that a theoretical explanation of varying $\alpha$ would have wide implications across all of physics, how reliable is the molecular theory input? Murphy also briefly discussed recent varying $\mu$ (proton electron mass ratio) results, which are consistent with zero variation for a $z = 0.685$ source.

M Theory Lesson 206

Carl Brannen's new post on 1-circulant and 2-circulant operators extends his previous analysis to the remainder of the fundamental fermions and their quantum numbers. He works with $6 \times 6$ circulants of the form for $(1)$ a 1-circulant and $(2)$ a 2-circulant. Just as for the $2 \times 2$ case with numerical matrix entries, we can think of $(1) \pm (2)$ as the eigenvalues of the $6 \times 6$ operator. Notice that the idempotents obtained have simple 2-circulants $(2)$ of democratic form, which means that adding or subtracting them from $(1)$ results in another 1-circulant. For example, for the $e_{R}^{+}$ quantum numbers one finds that which is a unitary 1-circulant since all entries have norm $\frac{1}{3}$. The same matrix results from $(1) + (2)$ for $\overline{\nu}_{R}$. The democratic matrix with all values equal to $\frac{1}{3}$ comes from, for instance, the $\overline{d}_{L}$ quark idempotent. Tony Smith, who likes to think of the Higgs as a top quark condensate, might like this correspondence between Higgs numbers and quark operators.

M Theory Lesson 205

The only closed bipartite graph on three edges is the theta graph, with two vertices. As a flat ribbon graph, the theta graph draws the 3 punctured Riemann sphere, but there is a version with a crossing that does something different. As explained in this paper (recommended by Lieven) any such graph embedded in a closed, oriented surface can be represented by a pair of permutations in $S_{n}$ where $n$ is the number of edges in the graph. For the theta graph, the orientation of the surface specifies different 3-cycles at each vertex, that is the two 1-circulants that are not the identity, namely $(231)$ and $(312)$. A 2-valent vertex in such a graph is associated with a 2-cycle in $S_{n}$, and so on.

Notice that one can interpret the alternating vertex structure as a 2-colouring of the child's drawing, say by black and white vertices. Every edge models the interval $(0,1)$ on the Riemann sphere. Now thanks to The Circle, we have an English translation of Grothendieck's classic paper, Sketch of a Program!

M Theory Lesson 204

An octahedron is also known as a rectified tetrahedron, where rectification is the truncation of corners from midpoints along each edge. This construction paints the faces of the octahedron in two colours, depending on whether the face arises from a tetrahedral face or an interior surface. The group of 24 symmetries of this object is isomorphic to the permutation group $S_{4}$. The 48 element quaternionic octahedral group is associated to a double cover of the genus 3 Klein curve. The special quaternion $q = i \textrm{exp}(\frac{\pi j}{4})$ is used to give the relations for this group in terms of the two generators $a$ and $b$. Kneemo pointed out that one can use this representation, along with octonions, to describe the units of the $E8$ lattice.

Now let's have fun rectifying the other polytopes that arise in ternary geometry. A rectified cube has four square and four triangular faces. The dual to a cube, an octahedron, is a birectified cube. A rectified dodecahedron is a icosidodecahedron. An example in the plane turns a heptagon tiling into a tiling with heptagons and triangles.

FQXi Proposal

Well, Schreiber et al on the FQXi panel of seven judges for large grants have rejected my proposal for a postdoctoral stipend of 22000 dollars per year. Since this was expected, I have already resigned from my waitressing job and I will be moving to Wanaka at the end of the month to be closer to the mountains. When the contaminated, rotting flesh starts falling off academia, I will be far away.

Mermin Magic

This week's PIRSA lectures include an enjoyable talk by M. Skotiniotis on his 2007 paper about epistemic models for hidden variable versions of Spekkens' toy quantum mechanics. In particular, the Mermin-Peres magic square is introduced. This is a $3 \times 3$ square of tensor products of Pauli operators of the form

$X^1$, $X^2$, $X^1 X^2$
$Y^2$, $Y^1$, $Y^1 Y^2$
$X^1 Y^2$, $X^2 Y^1$, $Z^1 Z^2$

corresponding to two qubits in three directions, which is related to the 2-direction three qubit Mermin pentagram of the form The number theoretic nature of these objects is discussed in the arxiv link. M theorists will notice the likeness of the magic square to certain mixing matrices in HEP phenomenology.

Carbon Beauty IV

Recall that the genus 0 Euler structure $(V,E,F)=(24,36,14)$ had two interesting models, namely the permutohedron (truncated octahedron) and the truncated cube. Similarly, the Euler structure $(12,18,8)$ has the two models of the truncated tetrahedron and the ternary polytope with four pentagons and four squares. Is there a dual for the truncated icosahedron? Yes, in fact the truncated dodecahedron shares the Euler structure $(60,90,32)$ with its 12 10-sided faces and 20 triangles. In summary, the three pairs of polytopes have the same dual decomposition into two types of face polygon. Note also that the dodecahedron itself is a so called fullerene graph because it has 12 pentagonal faces. Recall that this Platonic trinity is but one of many trinities matching the quaternionic $(\mathbb{R},\mathbb{C},\mathbb{H})$ triple, which appeared for instance in the ribbon graph matrix theory of Mulase et al. Observe that the transformation which takes the truncated tetrahedron to the other 8 sided polytope acts on two edges of a tetrahedron via a string type duality, deforming the hexagons on either side of the edge into pentagons and the triangles into squares. This is the self dual complex number case of flat ribbon graphs. In future M Theory lessons, we will look more carefully at twisted ribbon graphs associated to $\mathbb{R}$ and $\mathbb{H}$ and other triples related to ternary geometry. As Louise Riofrio would say, M Theory can be taught in kindergarten!

Winter Weekend

Christchurch lies at sea level, but sometimes in winter it snows here. It wasn't the nicest day to be out yesterday, but a long walk home was unavoidable when the buses stopped running up the hill where I live. This morning the skies cleared and I could only sigh as I viewed the glistening distant alps on my way to work.

Alia Sabur

Ah, if I had a penny for every time someone has told me that men have a wider range of mathematical ability and that's why the tail of the distribution is naturally filled with men. So who recently broke the 300 year old record to become the world's youngest professor? Alia Sabur, a lowly woman. Thanks, Women in Science.

Carbon Beauty III

Observe how the buckyball trinity builds the prime $p = 11$ from lower primes, particularly the prime 5. Not only do we use the Hecke group $H5$, but the 11 buckyballs (truncated icosahedra) defining the genus 70 curve each have 60 vertices, which come from 5 copies of the 12 vertex truncated tetrahedron, which has 4 hexagonal and 4 triangular faces. Another choice for the Euler structure $(V,E,F)=(12,18,8)$ is the 4 pentagon and 4 square faces of the third ball in the ternary geometry of the cube. And since buckyballs are mixtures of pentagons and hexagons, the buckyball trinity averages these geometries.

Note also that the $p = 7$ truncated cube has $(V,E,F)=(24,36,14)$, which is the same Euler structure as the permutohedron. Thus all three truncated Platonic geometries have a double Euler structure. The $3 \times 8$ splitting of the 24 vertices of the permutohedron is like the pairing of squares to form three cubes in M Theory, or the three squares of the associahedron, which was associated to the crossings defining the trefoil knot in $\mathbb{R}^{3}$, the complement of which has the cover of the modular group, namely the braid group on three strands, as fundamental group. Gee, if I repeat that a lot it's because I find it very, very interesting!

Riemann Again

Sorry Carl, I can't resist. Recall Connes' remark regarding the Riemann Hypothesis that

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.

In this light, the very simple use of the adeles in Li's paper comes across almost as an insult to the spirit of Connes' approach.

One gets the eerie feeling that Li has a strong subconscious sense of a promising line of attack, but that this has led him into the labyrinth of murky delusion, a frightening place of which there is no need to say more. But if nothing else, the paper has caused a lot of bright people to ponder the mystery that is the rational adeles. In M Theory, since a prime $p$ is pretty well always associated (as logos building blocks) to categorical dimension, even if only to count sets, the adeles must be an $\omega$-categorical construction. There is no problem defining rational numbers or appropriate limits in this setting, so the mystery lies in what it means to tack the real numbers on the end, at the infinite prime.

One nice property of the adeles as an Abelian group is that it is isomorphic to its Pontrjagin (Fourier) dual. Somehow this is analogous to the schizophrenic property of the group $U(1)$ in the full Stone duality setting, enriched to the $n$-category hierarchy setting. Or, as kneemo would put it, a string is secretly a necklace of pearls.

Carbon Beauty II

In the buckyball paper by Singerman and Martin, the genus 70 buckyball curve appears as the $p = 11$ analogue of the Klein surface for $p = 7$. The construction relies on the Hecke group $H5$, generated by

$S: z \mapsto - \frac{1}{z}$
$T: z \mapsto - \frac{1}{z + \phi}$

where $\phi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio. The golden ratio turns up in many places in noncommutative geometry, for example as weights for a quantum groupoid. Note that the modular group is also a Hecke group for $\phi = 1$. By a theorem of Hecke, $H5$ is discrete precisely because $\phi = 2 \textrm{cos} \frac{\pi}{5}$ where 5 is an ordinal. Note that the special phase $\frac{\pi}{5}$ (or double this) also has nice properties in relation to the Jones polynomial, which is universal for quantum computation at a 5th root of unity.

Riemann Hypothesis

OK, I suppose I must provide a link to the new paper by Xian-Jin Li with the simple title A proof of the Riemann hypothesis. Like many around the blogosphere, I have no intention of trying to decipher it, although the claimed proof, by a well known number theorist, relies heavily on Fourier analysis and ideas from Connes NCG. In other words, it sounds highly promising. Despite the expertise of the author however, my hunch is that there's a flaw, because the claimed proof is mostly standard analysis. However, perhaps flaws in the proof will be easy to iron out using techniques from quantum information theory.

Update: Terence Tao believes there is an error in equation (6.9) on page 20. He comments that the Fourier transform really ought not to be this powerful. Given the standard analytical form of Fourier transform used in the claimed proof this would seem a reasonable statement, but perhaps in an $\omega$ categorical framework (where as usual we associate primes $p$ with categorical dimension) this could be modified to obtain a decomposition of the form (6.9).

Carbon Beauty

In a series of posts on the buckyball trinity, Lieven Le Bruyn gives a link to this paper by P. Martin and D. Singerman on the genus 70 buckyball curve. On page 8 they discuss the familiar Fano geometry of seven points and seven lines, described by a 7 dimensional circulant

1110010
0111001
1011100
0101110
0010111
1001011
1100101

This geometry is embedded in the genus 3 Klein surface. The buckyball curve appears as an embedding space for the $p = 11$ analogue, described by an 11 dimensional circulant

10110100011
11011010001
11101101000
01110110100
00111011010
00011101101
10001110110
01000111011
10100011101
11010001110
01101000111

One guesses that this must somehow be associated to the algebra $E11$, the mathematics of which give the likes of Woit so much confidence that they know more than a lot of very smart string theorists. But in M Theory, we don't care so much about the continuum mathematics, because the physics is actually much simpler than that.