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

]]>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

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