M Theory Lesson 12

One of Devadoss's many papers on the real points of the moduli of punctured spheres is Cellular Structures determined by Polygons and Trees. In this paper he considers how the real case may be extended to the compactification of the complex moduli of $n$ points.

We saw that labels on trees were not necessary for the real case, which is described by associahedra. Devadoss points out that by labelling the leaves of trees with the numbers $1,2, \cdots n$ one can describe the complex moduli. Points on $\mathbb{CP}^1$ determine non-planar trees, because diffeomorphisms of the sphere can permute any two points without collision (recall that collisions were the basis for understanding the real case). This cyclicity is allowed on branch labellings, because the leaves should really be considered identical.

Now consider that we really want 2-dimensional operads to describe complex moduli. As it happens, 2-ordinal trees are exactly like 1-level trees with an integral number of branches on each leaf of the base 1-tree. However, 2-ordinals may have any collection of numbers $a_1, a_2, a_3, \cdots a_m$ of branches. So the simple labelled trees needed for the complex moduli are only a subset of the 2-ordinals. It remains to understand, therefore, in what sense the complex moduli is really a substructure of something, in this higher dimensional setting. Recall that in topos theory thinking, number fields are never to be considered fundamental in themselves.