### M Theory Lesson 63

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!

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!

## 6 Comments:

With regard to Hurwitz versions of MZVs and polylogarithms I found an intriguing abstract, but nothing else. Further links would be appreciated (if they exist).

The product of 24 thetas appears also in elementary particle vacuum functionals from modular invariance: not so surprising after all because modular invariance is also present in bosonic string model. I remember also the eighth roots of unity bringing in in mind D=8 and octonions.

All these number theoretical functions should have versions corresponding to the hierarchy of subgroups of rational modular group, which one obtains by replacing Z with Z+q, q=k/n. Some familiarity with Langlands conjecture inspires the reductionistic dream that everything reduces to theta functions with characteristics [a,b] made fractional.

I understood why fermions must correspond to even integers n and bosons to odd (this implies Z_2 conformal symmetry for elementary fermions and 3 families without any further assumptions: a considerable improvement to earlier explanation: this process took something like 15 years!).

n-fold discretization corresponds to discretization for angular momentum eigenstates. Minimal discretization for 2j+1 states corresponds to n=2j+1. j=1/2 requires n=2 at least, j=1 requires n=3 at least, and so on. n=2j+1 allows spins j<=n-1/2. Very nice spin-quantum phase connection which very probably follows also from the representations of quantum SU(2).

The definition of q-functions given in Wikipedia is totally different from my proposal and probably based on quite different philosophy. My proposal might in fact work even without the assumption of Z_n symmetry for Riemann surface (restriction to subspace of moduli with this symmetry). If Z_n is relevant it is because the thetas must be sections in bundle (multivalued functions) with finite fiber: Z_n could define the fiber of this bundle.

For about year ago I found that also the q-variants of hydrogen atom radial wave functions were defined only for |q|<1 and physical picture led immediately to the definition for roots of unity and a prediction of exotic states for which n is replaced approximately with n/k, k =1,2,... The quantization Planck constant predicts exactly the claimed energies of these exotic states. There is experimental evidence for this kind of exotic states.

The fractality of Farey tree and its self inclusion brings in mind fractality of HFFs of type II_1 and their Jones inclusions.

The higher operad reductionist dream (an oxymoron in a sense) says this is the tip of the iceberg!

I understood why fermions must correspond to even integers n and bosons to oddExcellent. It makes sense: spin j should go up with categorical dim n as you say. And then BH entropies etc. should work out (kneemo?) with the biggest possible BHs coming from a 'universal' omega-category.

I'm always glad to hear about more experimental evidence. Can you put better numbers to it now?

What is clear is that the decomposition of upper plane to fundamental domains of extensions of modular group should

mean decomposition of the domains of modular group to subdomains invariant under larger group. The hierarchy of the modular subgroups of rational modular group therefore defines a hierarchy of smaller and smaller fundamental domains.

The minimum of three strands has also following interpretation. Z_1 does not fix quantization axis at all. Z_2 allows all quantization axes orthogonal to a line. Z_3 is the first group allowing unique fixing of quantization axes. This conforms with the interpretation of the dark matter hierarchy in terms of classical correlates for fixing quantization axes.

I found also extremely nice interpretation for the hierarchy of Z_n (and Planck constants) in terms of a hierarchy of increasing quantum criticality in modular degrees of freedom as the conformal symmetry Z_n increases. Note also the beautiful interpretation of divisors of n.

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