M Theory Lesson 70

Since it seems to be Mad Moonshine blogging week, let’s take a quick look at Gannon’s paper The algebraic meaning of genus 0, mentioned back in Week 233.

After a nice review (for physicists) of the Moonshine theorem, Gannon gets around to discussing the braid group $B_{3}$. Recall that $B_{3}$ gives the modular group $PSL_{2}(\mathbb{Z})$ when quotiented by its centre. Now $B_{3}$ is the fundamental group for the space $SL_{2}(\mathbb{Z}) \backslash SL_{2}(\mathbb{R})$, which looks like the complement of the trefoil knot.

Even more amazing, $B_{3}$ is the mapping class group for an extended moduli space $M_{1,1}^{ext}$ of 1-punctured tori marked with a state $v$, which naturally appears in rational CFT. For conformal weight $k$, this group acts on (some convenient) characters $\chi (\tau, v)$ via

$\sigma_1 . \chi (\tau, v) = e^{\frac{-2 \pi i k}{12}} \chi (\tau + 1, v)$

$\sigma_2 . \chi (\tau, v) = e^{\frac{-2 \pi i k}{12}} \chi ( \frac{\tau}{1 – \tau} , \frac{v}{(1 – \tau)^{k}})$

where we recognise the usual action of modular generators $T$ and $S$ on $\tau$. Carl Brannen will just love those 12th roots. Naturally, we would like to compare all this to Loday’s trefoil on the K4 polytope, with crossings on the three squares. Since this polytope is dual to Mulase’s 6-valent ribbon vertex cell decomposition of $\mathbb{R}^{3}$, it must somehow describe the trefoil complement space, and the triality of the j-invariant would be lifted to a triality for these squares. Oh, perhaps we could use the torus that we made out of two such polytopes as cylinders based on honeycomb geometries. That is, draw the trefoil on this torus, which is two glued copies of the planar annulus (replacing the 2-sphere), each bounded inside by a central hexagon. On each annulus, the square tiles correspond to three principal directions in the honeycomb plane. Or maybe not!

M Theory Lesson 69

Recall that when replacing trees by dual polygons, one can distinguish the type of the associahedron face by the kind of diagonals for the polygon. For example, the K4 Stasheff polytope has 6 pentagonal faces and 3 squares. These are distinguished by the chorded hexagons where a diagonal that splits a hexagon in two corresponds to a square. This shows how pentagons may be paired, by taking dual diagonals, but squares are at best self dual. Labelled trees may be replaced by labelled polygons.

The description of trees as clusters of polygons, used by Devadoss in tiling moduli spaces, is better known to category theorists as the theory of 2-opetopes. The dimension 2 describes the planar nature of polygons, but this may be generalised. On that note, David Corfield points out a wonderful new paper on the arxiv.

Happy New Year

Lieven Le Bruyn has created a new blog, Moonshine Math (yes, the j-invariant!), born on Bloomsday as promised. Today also signals the coming of the new year, as Matariki rises in the sky. Maybe this year I’ll get less spam telling me I don’t need to be an average man any longer. Meanwhile, yesterday Mottle had an interesting post on a new paper by Dvali which considers a large number of copies of SM particle species as a route to explaining the heirarchy problem. Sounds a bit familiar.

Tommaso Dorigo posts about the D0 and CDF discovery of the $\Xi_{b}$ particle with a mass of $5.774 \pm 0.019$ GeV/$c^2$, which is about six times the proton mass. This particle is made of one down, one strange and one bottom quark.

M Theory Lesson 68

Matti Pitkanen now has a post about Farey sequences and the Riemann hypothesis in TGD. The idea that the hypothesis is not provable within standard mathematics appears to be gaining a foothold within physical constructions.

On the other hand, it is possible that the physical axioms could guide a concrete proof within a convenient model, such as the Jordan algebra M Theory, in which U duality is algebraically manifest. But the zeta function itself only enters here with the (operad) algebras associated to moduli integrals. So it is difficult to avoid the higher categorical framework in studying exact (eg. MHV) amplitudes, and this lands us back in the world of post ZF axioms.

After inhabiting this world for some time, it becomes difficult to look at zeta functions any other way. One simply can’t help looking at the Selberg axioms and thinking of closure under products, or factorisation, as topos-like axioms, even though these are radically different things. Recall that the interplay of + and x here is thought of as a higher distributive law for monads. This suggests that the Euler relation for zeta functions is about equating invariants based on monads, or rather that the distributivity $+ \times \rightarrow \times +$ is an identity. That is, that the distributivity of complex arithmetic is somehow more responsible for Euler’s product relation than the notion of primeness, which is used through the application of the fundamental theorem of arithmetic only after the product has been expanded.

This suggests that the higher dimensional versions of the Riemann zeta function should be thought of as non-commutative, non-associative and even non-distributive L-functions. Ah! So that’s why Goncharov likes Shimura varieties. Note that such considerations are necessary for understanding even the values of the Riemann function, since its arguments extend throughout the heirarchy.

Update: Khalkhali has a new post on Determinants and Traces in which he notes: “… Bost and Connes in their paper Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411–457, right in the beginning show that the above formula (5) gives the Euler product formula for the zeta function … In fact their paper starts by quantizing the set of prime numbers … Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality.”

Formula (6) is $Tr_{s}(\Lambda A) Tr(SA) = 1$, a relation between trace and supertrace. Hmm. I would like to understand Koszul duality better because it applies to operads, and more generally PROPS.

M Theory Lesson 67

Terence Tao tells us about this paper by Guangming Pan and Wang Zhou on random $NxN$ complex matrices with entries of mean 0 and variance $\frac{1}{N}$. They claim to prove, under an assumption about moments, that the spectral distribution converges to the uniform distribution over the unit disc. This is called the circular law. In other words, the initial clustering of eigenvalues around the real line disappears as $N \rightarrow \infty$.

In would be interesting to see what this meant for honeycomb patterns in the limit of an infinite number of hexagons. Or perhaps it helps us understand the distribution of Farey numbers on the unit interval. Recall that successive terms $\frac{p}{q}$ and $\frac{r}{s}$ of a Farey sequence satisfy

$qr – ps = 1$

which is why the modular group appears when considering matrices $(r,p;s,q)$. Let $N$ be the number of terms in a Farey sequence. The Riemann hypothesis [1] is equivalent to the statement that the sum of differences between Farey terms and interval markers, namely

$\sum_{n=1}^{N} \delta_n \equiv | f_{n} – \frac{n}{N} |$

is bounded by $o (x^{\frac{1}{2} + \varepsilon})$ for all $\varepsilon > 0$ as the real number $x$ defining the sequence tends to infinity. The Farey sequences themselves are rational numbers less than 1, and fit onto the binary Farey tree described by Vepstas. The ends of the infinite tree fit onto the boundary of the Poincare disc, when the modular domain view is mapped there. Thus the interval markers above may be exchanged for roots of unity on the unit circle, and these compared to the leaves of the Farey tree.

Kauffman et al (p 51) show that this version of the Riemann hypothesis is equivalent to a question about messy unknots. They also look at DNA recombination. Unknots described by rational tangles are labelled by the pairs of adjacent rationals in a Farey sequence. So two tangles labelling two adjacent leaves of the tree at infinity can be used to construct unknots.

[1] H. M. Edwards, Riemann’s Zeta Function, Academic Press (1974)

Have a Nice Day

I apologise for moving off topic today, but I found this Smile Test very interesting. Apparently, most people are not as good at telling fake smiles as they think they are. The theory is that people are easily fooled because it is socially convenient not to know what people are thinking. Despite my expectations of doing badly, I actually did really well on this test (17/20). Now I realise that it’s easier not to care what people think when one’s ability to detect fakeness makes it impractical to take such things into account.

Ars Mathematica finally reports a retract of the claimed disproof of The Hypothesis. David Ben-Zvi has kindly provided notes on the recent Chicago conference, where Goncharov was talking about Motives, path integrals and trivalent graphs. Sounds intriguing. OK, I printed out the notes. Wow. OMG. Goncharov claims to have identified the category of mixed motives (a.k.a. the holy grail for ordinary real/complex geometry) in terms of path integrals for projective varieties. For instance, when the variety corresponds to modular subgroups indexed by $\hbar = \sqrt{N}^{-1}$, as in TGD or $N$-fold covers of moduli spaces, one gets Langlands from the cohomology. He concludes with a statement that Feynman integrals (with observables) are valued in motivic cohomology. Yeah, duh, the physicists know that. We just don’t know how we’re ever going to learn that much mathematics.

Ah! That means the S duality we need for the Riemann Hypothesis relies on the whole range of quantised $\hbar$, and is therefore necessarily omega-categorical. That was expected, because the surreal zeta arguments extend through the ordinals. It is fantastically exciting to have some confirmation of this link between $\hbar$ values and S duality. I wonder how string theory will deal with a variable $\hbar$. Oh, I see.

M Theory Lesson 66

Recall that Joan Birman et al studied knots in the Lorenz template with two generating holes X and Y. So knots are expressed as words in X and Y. In Robert Ghrist’s paper Branched two-manifolds supporting all links he shows that the template $\mathcal{V}_0$ on more letters contains an isotopic copy of every (tame) knot and link. More specifically, for a parameter range $\beta \in [6.5,10.5]$ every link appears as a periodic solution to the equation which is used to model an electric circuit. This is cool stuff. In M Theory we like ribbon diagrams which are twisted into loops like in the Lorenz template diagram. The universal template $\mathcal{V}_0$ can be embedded in an infinite sequence of more complicated templates, which in turn are embeddable in $\mathcal{V}_0$. Ghrist also considers flows arising from fibrations, such as the 1-punctured torus fibration for the figure 8 knot complement. This fibration flow is also an example of a universal flow.

I was quite intrigued when a mathematical biologist at a conference told me recently that no one really knew why DNA had four bases rather than two. Apparently it isn’t clear why self-replicating molecules fail to adopt a binary code in X and Y. Somebody else muttered something about hydrogen bonds and then, inspired and ignorant, I started rambling on about knot generation in templates. After all, DNA molecules need to know how to knot themselves.