### M Theory Lesson 53

Chris Woodward made an intriguing comment on Tao's blog recently, on the topic of honeycombs. He said: I once had this crazy dream, that the quantum version of a honeycomb should be a honeycomb on a thrice-punctured sphere with hyperbolic metric.

Recall that the term quantum here refers to the multiplicative matrix problem. For us, a 3-punctured sphere is the Veneziano moduli space for 4-punctured spheres. Since such spaces rightly live in twistor land, as opposed to the Minkowski world of ordinary Feynman diagrams, a propagator is replaced by the bubble diagram analogue. In painting a 3x3 honeycomb on the pair-of-pants, three points are marked on each circle boundary. These are like the minimal bubbles of the $\mathbb{RP}^1$ diagrams, where marked points correspond to Feynman legs.

In the holographic logos 1-operad land, one projects down to two dimensional disc diagrams with marked points on the circles. Attaching lines between these points usually gives Jones' planar algebra diagrams. The problem in the 1-operad setting is that, in the 3x3 honeycomb case, it looks like there are a total of nine marked points, which one cannot pair up with internal lines. By allowing an internal honeycomb diagram, the nine external lines may be attached to the nine marked points. Each circle, with three marked points, may be viewed as a triangle. Note that nine additional internal vertices are required inside the large disc.

Recall that the term quantum here refers to the multiplicative matrix problem. For us, a 3-punctured sphere is the Veneziano moduli space for 4-punctured spheres. Since such spaces rightly live in twistor land, as opposed to the Minkowski world of ordinary Feynman diagrams, a propagator is replaced by the bubble diagram analogue. In painting a 3x3 honeycomb on the pair-of-pants, three points are marked on each circle boundary. These are like the minimal bubbles of the $\mathbb{RP}^1$ diagrams, where marked points correspond to Feynman legs.

In the holographic logos 1-operad land, one projects down to two dimensional disc diagrams with marked points on the circles. Attaching lines between these points usually gives Jones' planar algebra diagrams. The problem in the 1-operad setting is that, in the 3x3 honeycomb case, it looks like there are a total of nine marked points, which one cannot pair up with internal lines. By allowing an internal honeycomb diagram, the nine external lines may be attached to the nine marked points. Each circle, with three marked points, may be viewed as a triangle. Note that nine additional internal vertices are required inside the large disc.

## 2 Comments:

The honeycomb dream could lead to something! Your posts suggest that there is an underlying symnetry, like the eightfold way. Perhaps some sort of mathematical honeycomb lies behind quantum mechanics.

Hi Louise. Indeed! Honeycombs appear to make quantum mechanics so much easier to understand, just as Heisenberg pointed out in 1925.

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