### Six Degrees

Whilst completely wasting my time on a postdoc application, I was musing over the example of Tamarkin. The 6-leaved 2-operad tree looks different as a stringy diagram. The blurring of vertices can turn the 2-operad tree into a 6-leaved 1-operad tree, associated with the 3D Stasheff associahedron, which regularly appears here. Any worldsheet will have the topological property that its boundary is just a collection of circles, so only by specifying a decomposition into punctured spheres (multi-pants) can one recover the internal circles corresponding to nodes of the 2-operad tree. Similarly, a higher level tree becomes a punctured sphere with a more complicated decomposition.

On the other hand, if we insist on associating surfaces with 2-operads, then the most symmetric choice for the 6-punctured sphere is the 2-level version of Tamarkin. Moreover, the pairing of punctures, marked by the three internal circles, is just like the Dehn map from the unpunctured genus 2 surface, whose moduli is also six dimensional (over $\mathbb{R}$).

Can we use all three twistor moduli to form a 3-operad triality? If the 6-punctured sphere is 1-operadic, and the pair just mentioned are somehow 2-operadic, is there a 3-level description including the moduli for the 3-punctured torus? Intriguingly, the 3-punctured torus is built from three copies of the 3-punctured sphere, which we can relate via Belyi maps to three different elliptic curves, the Cartesian product of which is again a nice 6-dimensional space. The genus 2 surface is usually decomposed into two 3-punctured spheres and two cylinders, whereas the 6-punctured sphere needs four 3-punctured spheres. So by gradually adding punctures to cylinders and re-gluing, we can turn the genus 2 surface into the torus into the sphere. Adding a puncture to a cylinder is the same as turning a single edge into a 2-level tree with 2 branches (the trivalent vertex), so this process can take us from 1-level trees to 2-level trees to 3-level trees. A typical 6-leaved 3-level tree (for leaves grouped as 1,1,2,2) corresponds to a polytope of dimension 12 - 3 - 1 = 8. The minimal dimension is 4, corresponding to the suspended Stasheff polytope, and an 11 dimensional polytope is obtained from the Tamarkin tree with added single edges on the top level.

On the other hand, if we insist on associating surfaces with 2-operads, then the most symmetric choice for the 6-punctured sphere is the 2-level version of Tamarkin. Moreover, the pairing of punctures, marked by the three internal circles, is just like the Dehn map from the unpunctured genus 2 surface, whose moduli is also six dimensional (over $\mathbb{R}$).

Can we use all three twistor moduli to form a 3-operad triality? If the 6-punctured sphere is 1-operadic, and the pair just mentioned are somehow 2-operadic, is there a 3-level description including the moduli for the 3-punctured torus? Intriguingly, the 3-punctured torus is built from three copies of the 3-punctured sphere, which we can relate via Belyi maps to three different elliptic curves, the Cartesian product of which is again a nice 6-dimensional space. The genus 2 surface is usually decomposed into two 3-punctured spheres and two cylinders, whereas the 6-punctured sphere needs four 3-punctured spheres. So by gradually adding punctures to cylinders and re-gluing, we can turn the genus 2 surface into the torus into the sphere. Adding a puncture to a cylinder is the same as turning a single edge into a 2-level tree with 2 branches (the trivalent vertex), so this process can take us from 1-level trees to 2-level trees to 3-level trees. A typical 6-leaved 3-level tree (for leaves grouped as 1,1,2,2) corresponds to a polytope of dimension 12 - 3 - 1 = 8. The minimal dimension is 4, corresponding to the suspended Stasheff polytope, and an 11 dimensional polytope is obtained from the Tamarkin tree with added single edges on the top level.

## 2 Comments:

Dear Kea,

this puncturing of sphere brings strongly in my mind the earlier discussion about the the decompositions of disks associated with planar algebras which in turn relate to inclusions of HFFs of type II_1. I wish I could remember more about it and about speculations it generated. Sad that my understanding of planar algebras is technically so poor.

[I added subtitles to the text below in an attempt to express the bird's eye of view.]

1. Decomposition of sphere to regions and number theoretic braids

A kind of puncturing and decomposition of sphere into regions takes place in the recent view about number theoretic braid strands as orbits of minima of Higgs vacuum expectation (in TGD sense of course): the fusion of minima corresponds to a reaction in which two strands fuse.

Higgs vacuum expectation is identified as a generalized eigenvalue of the modified Dirac operator depending on transversal coordinates. Higgs maxima (H=0) correspond rather literally to the tops of mountains on a 2-D landscape over 2-sphere and naturally to the punctures.

Higgs minima correspond to bottoms of valleys. Valleys are separated by saddle curves and the sphere decomposes into separate regions bounded separated by closed curves. 2-D landscape with Higgs modulus as height function provides a good visualization.

2. Conformal invariance and effective stringy behavior localizes to valleys

The induced spinor fields in different regions (coordinate patches) anticommute and along the separating saddle curves one has stringy anti-commutations. One has conformal field theory (stringy) behavior inside each region but conformal behavior fails globally and partonic 2-surface behaves in this discretized sense as 2-D object with each region defining its own conformal field theory.

3. Dimensional hierarchy in discretized sense

The picture generalizes to 3-D case: the light-like 3-surface behaves in discretized sense as 3-D object. This allows to understand the paradoxical sounding prediction of a dimensional hierarchy from discrete braids to light-like 3-surfaces. 3-D dynamics gives rise to generalized braid diagrams with strands identified as minima of Higgs.

4. What should one understand?

It would be interesting to try to understand following things.

*How this decomposition of sphere and its time evolution by generalized braiding relates to planar algebra decompositions of disk. The inclusions of hyper-finite factors are involved in both situations very concretely so that this kind of relationship is expected.

*Could one define time evolutions of planar algebra decomposition of disk and could they have a meaning as generalized braid diagrams with braid strand fusions defining particle reactions?

Matti

decompositions of disks associated with planar algebrasWell, those discs were associated with boundaries of pants trees, which are always taken here to be operadic: hence planar algebras.

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