### M Theory Lesson 205

The only closed bipartite graph on three edges is the theta graph, with two vertices. As a flat ribbon graph, the theta graph draws the 3 punctured Riemann sphere, but there is a version with a crossing that does something different. As explained in this paper (recommended by Lieven) any such graph embedded in a closed, oriented surface can be represented by a pair of permutations in $S_{n}$ where $n$ is the number of edges in the graph. For the theta graph, the orientation of the surface specifies different 3-cycles at each vertex, that is the two 1-circulants that are not the identity, namely $(231)$ and $(312)$. A 2-valent vertex in such a graph is associated with a 2-cycle in $S_{n}$, and so on.

Notice that one can interpret the alternating vertex structure as a 2-colouring of the child's drawing, say by black and white vertices. Every edge models the interval $(0,1)$ on the Riemann sphere. Now thanks to The Circle, we have an English translation of Grothendieck's classic paper, Sketch of a Program!

Notice that one can interpret the alternating vertex structure as a 2-colouring of the child's drawing, say by black and white vertices. Every edge models the interval $(0,1)$ on the Riemann sphere. Now thanks to The Circle, we have an English translation of Grothendieck's classic paper, Sketch of a Program!

## 2 Comments:

Kea,

it is interesting that "The only closed bipartite graph on three edges ..." is also the Dynkin Diagram of G2.

Further, in the paper you link about Dessin d'Enfant, a 2003 Notices AMS paper by Leonardo Zapponi, Figure 1 of "The dessins with three edges ..." shows 7 figures,

which are the Dynkin Diagrams (plus some addtional information) of

D4-Spin(8) and D4-Spin(8) and B3-Spin(7) and B3-Spin(7)

and

A4-SU(5) and G2 and G2

As you might expect, I would like to follow the Dynkin Diagram point of view to get to the E8 physics of Garrett Lisi and my version of it.

However, I find very interesting Grothendieck's 1984 Sketch of a Programme that you mentioned, especially:

"... I [Grothendieck] began to become interested in regular polyhedra, which then appeared to me as particularly concrete “geometric realizations” of combinatorial maps ...

M0,5 (for projective lines with five marked points) ... is a real jewel, with a very rich geometry closely related to the geometry of the icosahedron

...

The extension of the theory of regular polyhedra (and more generally, of all sorts of geometrico-combinatorial configurations, including root sys-

tems...) of the base field R or C to a general base ring, seems to me to have an importance ...

the theory of finite regular polyhedra, already in

the case of dimension n = 2, is infinitely richer, and in particular gives infinitely many more different combinatorial forms, in the case where we admit base fields of non-zero characteristic ...

we read off from the geometric regular polyhedron over the finite field (F4 and F5 for the icosahedron, F2 for the octahedron) a particularly elegant (and unexpected) description of the combinatorics of the polyhedron. It seems to me that I perceived there a principle of great generality, which I believed I found again for example in a later reflection on the combinatorics of the system of 27 lines on a cubic surface, and its relations with the root system E7

...

I [Grothendieck] have not up to now followed this call, nor met any other person willing to hear it, much less to follow it. ...

Will there be found one, some day, who will seize this opportunity? ...".

Kea,

maybe you could be the one Grothendieck asked about?

You also (except for age, but people age faster nowadays than they did in Grothendieck's public days) seem to fit his description:

"... at the university, I have learned everything I have to learn and taught everything I can teach there, and that it has ceased to be really useful, to myself and to others. To insist on continuing it under these circumstances would appear to me to be a waste ...

today I am no longer, as I used to be, the voluntary

prisoner of interminable tasks, which so often prevented me from springing into the unknown, mathematical or not. The time of tasks is over for me. If age has brought me something, it is lightness. ...".

Tony Smith

PS - A technical footnote:

If the E8/icosahedral physics that Garrett Lisi and I like could be seen as an accurate description of a local classical Lagrangian of the Standard Model and Gravity,

then

you might see the Grothendieck generalization to general base rings (such as F_p for prime number p) as describing sum-over-histories Path Integrals (i.e., prime 3 history diagrams might look like your "closed bipartite graph on three edges"),

and

such a physical interpretation might be closely related to the p-daic structures of Matti Pitkanen.

These ideas extend further in many interesting directions.

Wonderful comment, Tony. Yes, we do seem to be converging on a common point of view for QG with Matti's, your's, Garrett's, Kholodenko's, Carl's, Louise's, kneemo's, Witten's, strings etc. Gee, it's beginning to look like quite a crowd for The Empire ignore!

As you might expect, I would like to follow the Dynkin Diagram point of view to get to the E8 physics of Garrett Lisi and my version of it.Excellent. I'm not that interested in Lie groups myself, but all this does need tying together.

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