Arcadian Functor

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.

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)

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.

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.

As nosy snoopy noted, these crystal Calabi-Yau papers are really very interesting. I would like to know more about these random partitions. Okounkov has some notes here.

Recall that in Kapranov’s non-commutative Fourier transform for three coordinates $x$, $y$ and $z$, it is natural to represent monomials by cubical paths traced out on such melting crystal partitions. The two coordinate case goes back to Heisenberg’s original paper, as we have seen. In a modern guise, his sum rule arises in honeycombs, which look a bit like shadows of melting corners.

… the corpus of mathematics does resemble a biological entity which can only survive as a whole and would perish if separated into disjoint pieces.

Alain Connes

Actually, it’s freezing down here, but the northern summer is conference time! Later this month the northern hemisphere sees both Loops 07 in Mexico and Strings 07 in Spain. Closer to home in early July, we have GRG18 in Sydney (conference venue in photo). No doubt many other interesting conferences are also coming up soon.

In other news, I am quite saddened to see this mysterious message on Lieven Le Bruyn’s blog page. Good thing I printed out his helpful maths lessons. Bloomsday is June 16, which is only next week. I wonder what Homeric epic Lieven has to tell us. Also, keen followers of this blog should definitely take a look at the wonderful links on this Cafe post. Thanks, David.

As Tommaso points out, we must not forget Pascos 07 in London in early July. Have fun, Tommaso.

Whilst on the topic of AdS/CFT, Michael Rios has an interesting post on dimension altering weak coupling phase transitions for N=4 SUSY Yang-Mills.

A continuous change in dimension from six down to five is reminiscent of Thurston’s beautiful fractal 2-spheres, which are filled with a 1-dimensional curve. These arise in the study of 3-manifolds such as those with 1-punctured torus fibres over the circle. The punctures draw out a boundary for the manifold by tracing a knot. Now according to Matti, the fractional modular domains would fit into the domain for the once punctured torus moduli (the $n=2$ case) on the upper half plane. Perhaps the $n=5$ domain (or rather the theta functions) could be used to model a 5-sphere, much as the j-invariant Belyi map links the $n=2$ domain to $\mathbb{CP}^1$.

Note: For the new readers to this blog, our use of the term M Theory must not be confused with its more popular usage in string theory related papers. The letter M stands here possibly for Motive, or perhaps Monad. Although these terms do appear in the popular literature, they rarely correspond to the physical usage we would like to make of them.

It’s good to see Matti Pitkanen so happy about the fact that there are three generations in TGD. After all, it would be a poor theory that postdicted a number that contradicted observations, although not as poor as a theory that doesn’t postdict anything beyond established physics. From a category theory point of view, the extension of the modular group to a series indexed by n is most easily characterised by a groupoid on the objects n. Since there is an associated sequence of subgroups of the modular group, a group theorist may wish instead to study profinite completions.

Recall that Lieven Le Bruyn was discussing the fact that

$PSL_{2}(\mathbb{Z}) \simeq B_{3} \backslash \langle ( \sigma_1 \sigma_2 \sigma_1 )^{2} \rangle$

where $B_{3}$ is the braid group on three strands and the $\sigma_{i}$ are the usual generators of the group. This came up with regard to the work of Linas Vepstas on Minkowski’s devil’s staircase. Vepstas has studied the fractal symmetries of this continuous function from the interval to itself via mappings of binary trees. He observes that an infinite subtree is always isomorphic to the full tree. The Minkowski map arises as a mapping from the dyadic tree with root $\frac{1}{2}$ (this is like the bottom half of the positive surreal tree which we wanted to associate with Riemann zeta arguments) to the Farey tree. As Martin Huxley says, “A nice way of stating the Riemann hypothesis is that the Farey sequence is distributed as uniformly in the interval 0 to 1 as it possibly can be.”

By embedding a binary tree in the upper half plane, one naturally encounters fundamental domains for the modular group. Presumably one could play a similar game with n-ary trees in n dimensions (recall the tetractys), with categorified n-tuplet groupoids replacing the modular group. In terms of braid groups, one simply increases the number of generators. As we have seen, the restricted $B_{3}$ can describe the $n=2$ case of the (massless) fermions. Moreover, braid depth is naturally associated to the depth of MZV algebras. Note that only in the $n=2$ case does the Hurwitz doublet feature zeroes that appear to lie on the critical line, and the Hurwitz $\zeta_H (s)$ has an extra zero at $s=0$.

Aside: Now this looks cool!

Can anyone provide an update on Woit’s report on Witten’s new ideas? There seems to be no paper online, but one may materialise closer to Strings07.

From an M Theory point of view this $SO(2,2)$ theory represents a physical 2-Time domain, not merely a 2+1D model in a naive quantization scheme. Recall that for the $c=24$ case the partition function is exactly the function $J(q) = j(q) – 744$ where $j(q)$ is the famous q-expansion of the modular j-invariant (which we have been discussing) for $q = e^{2 \pi i \tau}$.

Now Matti Pitkanen has just been looking at how the Hurwitz zeta function (at the special point $a = 0.5$) naturally arises on imposing modular invariance on theta series. By definition, the j-invariant is invariant under modular transformations, so perhaps there is a simple relation between its theta function components and the components for the (let us call it) SUSY doublet of Hurwitz $\zeta (s,\frac{1}{2})$ and Riemann zeta functions. Starting with $\theta (0, \tau)$ we see that under $\tau \rightarrow \tau + 1$ this becomes $\theta_{01}(0, \tau)$. This in turn becomes $\theta (0, \tau)$ under a second transformation and both functions are part of the j-invariant triality. The third component $\theta_{10} (0, \tau)$ transforms to $(\sqrt{2}+ \sqrt{2} i) \theta_{10} (0, \tau)$ which is an 8th root of unity. This particular Hurwitz function is simply $\zeta_{H}(s) = (2^s – 1) \zeta (s)$, a simple multiple of the Riemann zeta function. According to Mathworld it has the interesting functional relation

$\zeta_{H}(s) = 2 (4 \pi)^{s-1} \Gamma (1-s) (sin \pi (1 + \frac{s}{2}). \zeta_{H}(1-s) + sin \pi (\frac{s}{2}). \zeta (1 – s))$

but this doesn’t seem quite right. This should be the usual functional relation for the Riemann zeta function. Anyway, the doublet $(\zeta, \zeta_H)$ describes the two term duality of the j-invariant.