While in Sydney I managed to pick up my notes from the Streetfest in 2005. Getzler spoke about open and closed modular operads.

Moduli spaces of genus $g$ curves with the Deligne-Mumford compactification form an example of a modular operad. In modular operad theory one replaces planar rooted trees by more general graphs, but we want to use surfaces with boundaries instead of graphs. For $n$ punctures (boundary circles) on a surface, $m$ boundary arcs and $h$ holes it turns out that the dimension of $M(g,n,h,m)$ is $6g – 6 + 3h + 2n + m$, and this compactified moduli is good enough to completely classify the TCFTs (meaning TFTs based on moduli spaces). Getzler ended with a reference to this paper…oh my, ribbon graphs again. Ribbon graphs form a modular operad under gluing of their edges. Note that this is orthogonal to the usual gluing of trees and more like the pasting of surfaces along boundary elements.

Note that in this paper, Costello carefully distinguishes the rooted trees from cyclic forests, which more correctly describe ribbon graphs. Operads are viewed as monoidal functors from the basic structure type: rooted trees, cyclic forests, or graphs.

Such musings led, as usual, to a google search, this time on the terms modular operad and 2-operad, which resulted in precisely one hit, namely a paper by Morava which I intend to take home and read.

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## kneemo said,

July 21, 2007 @ 6:48 pm

I particularly enjoyed page 5 of Morava’s paper. I’m thinking a possible application of the extended modular operad is in the description of string bits, where the full strings would be sequences of P^1s, laid end to end.

## Kea said,

July 21, 2007 @ 11:07 pm

Yes, this is interesting, kneemo. I will look at the Manin and Losev paper, although I suspect it will be difficult.