M Theory Lesson 75

So it seems the Morava paper belongs to a productive train of thought, leading to yet more incomprehensible papers by Manin et al. This is the honey that led Manin into a place hard to follow.

But the notion of extended modular operad is clearly a good one. Manin et al solve the problem of the missing 2-punctured moduli $M_{0,2}$ (which makes the operad messy) by replacing all the $M_{g,m}$ with the extended (compactified) spaces $L_{g,m,n}$. The index $n$ labels a second collection of $n$ marked points on the genus $g$ surface. There is a surjective morphism from $M_{0,m+2}$ to the $L_{0,2,m}$, which are toric varieties associated to permutahedra: phew, something we can understand!

So instead of looking for tilings of the ordinary complex moduli $M_{0,m}$ with 2-operad polytopes, we can tile the extended moduli spaces. Loday's geometric realizations of associahedra and permutohedra (from cubes) may come in handy at last. Recall that a 5-leaved labelling of the usual Stasheff associahedron in three dimensions becomes a 4-leaved labelling for permutations in a permutohedron, so Loday's shift in the number of marked points from $M$ to $L$ may clarify the abovementioned surjective mapping.