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.