What is a twisted modular operad?

Question

I find Getzler and Kapranov's article Modular Operads difficult to understand. Can anyone explain what a (twisted) modular operad is conceptually, or what the underlying idea behind the concept of a modular operad is? For example, how can we write down the explicit structure map in a twisted modular operad?

Answer 1

Here is a long and belated answer, mostly written so that I can work out some details I should have worked out long ago. The short version of it is the following: when $g > 0$, one needs the notion of a twisted modular operad in order to describe the precise sense in which the "gravity operad" is an operad. I do hope all signs below are correct.

The "standard" motivation for introducing twisting is that it is needed for the higher genus generalization of the bar transform (the "Feynman transform"): twisted modular operads appear naturally as "Feynman duals" of ordinary operads. The following is a different perspective.

Remark: Since the gravity operad was introduced by Getzler, who also proved it dual to the hypercommutative operad, and since the gravity operad is the only twisted modular operad discussed in any detail in Getzler-Kapranov's paper, it seems plausible that this was one of their main motivating example for introducing the notion of twisting.

The starting point is the situation in genus zero, and Getzler's paper "Operads and moduli of genus 0 Riemann surfaces". We consider the collection of spaces $\newcommand{\M}{\overline{M}} \M_{0,n}$, which form a cyclic operad in the category of algebraic varieties. Taking cohomology we get a cyclic co-operad $H^\bullet(\M_{0,n})$. So far nothing surprising. The surprise is that even though the spaces $M_{0,n}$ do not form an operad in any natural sense, their cohomologies still do - this is the "gravity" operad. Moreover, the two cohomology operads are dual in a precise sense (which is purely a genus zero phenomenon), but I will not talk about that.

The magic words that make the cohomology of $M_{0,n}$ an operad are "Poincaré residue". For this read Deligne, "Hodge II", pp 31-32, or Peters-Steenbrink, pp 92-93. The short version is that the Poincaré residue is defined whenever you have an open subvariety $U \subset X$ where $X$ is smooth and the complement is a simple normal crossing divisor. Let the divisor be $D_1 \cap \cdots \cap D_N$, let $D_I = \bigcap_{i\in I} D_i$, and let $D_I^\circ$ denote the interior of the intersection (the complement in $D_I$ of all other components $D_j$). Then it is a map $H^\bullet(U) \to H^{\bullet-|I|}(D_I^\circ)$. Apply this to $X = \M_{0,n}$ and $U = M_{0,n}$: the set of possible intersections $D_I$ is exactly the set of all stable trees with $n$ legs, and $D_I^\circ$ is the corresponding product of open moduli spaces $M_0(\Gamma) = \prod_{v \in \mathrm{Vert}(\Gamma)} M_{0,n(v)}$ (where $n(v)$ = valence of the vertex).

Hence we have co-composition maps $H^{\bullet+|\mathrm{Edge}(\Gamma)|}(M_{0,n}) \to H^\bullet(M_{0}(\Gamma))$, in particular $$ H^{\bullet+1}(M_{0,n+n'}) \to H^\bullet(M_{0,n+1}) \otimes H^\bullet(M_{0,n'+1}),$$ and we would like to say that this makes the cohomology of $M_{0,n}$ a co-operad. There is one obvious problem here, which is that there is a degree shift in the definition. This is not such a big deal: we can get rid of it by declaring instead that the collection $$ \{ H^{\bullet-1}(M_{0,n})\}$$ should form a co-operad of graded vector spaces.

But even this does not give us a co-operad. The issue is in the very definition of Poincaré residue: it is only defined up to an ordering of the boundary divisors, which potentially introduces a sign ambiguity. This is dealt with explicitly in Deligne who twists everything by a kind of orientation sheaf $\varepsilon^n$ to get things defined canonically. However, there is a simple operadic solution also to this problem: the correct statement is instead that the collection $$ \{ H^{\bullet-1}(M_{0,n}) \otimes \mathrm{sgn}_n\}$$ does form a cyclic co-operad of graded vector spaces. Here $\mathrm{sgn}_n$ is the sign representation of the symmetric group.

So far so good. Now we wish to generalize this to higher genera. Again the spaces $\M_{g,n}$ form a modular operad, and their cohomologies $H^\bullet(\M_{g,n})$ a co-operad. We would like to play the same game again to get a co-operad structure on the cohomology of $M_{g,n}$. The self-glung gives us maps $$ H^{\bullet+1}(M_{g+1,n-2}) \to H^{\bullet}(M_{g,n})$$ that we would like to use to define the modular co-operad structure. Hence we are led to introducing a second degree shift, now depending on the genus: one might hope that the spaces $$ \{ H^{\bullet+g-1}(M_{g,n}) \otimes \mathrm{sgn}_n\}$$ will form a suitable modular co-operad. This is unfortunately not true! The self-gluing does not work with this definition, as the $\mathbb{S}_2$-action on the two points that get identified is not the right one. In fact no strategy like the one used in genus zero will work in this situation.

What one should then do is to define one's way out of the situation. Instead of an "ordinary" operad $\newcommand{\P}{\mathcal{P}}\P$ with structure maps $$ \P(\Gamma) \to \P(n)$$ where $\Gamma$ is a rooted tree with $n$ inputs, or an "ordinary" modular operad with structure maps $\P(\Gamma) \to \P(g,n)$ where $\Gamma$ now has genus $g$ and $n$ legs, one introduces a "twisting" $\newcommand{\D}{\mathfrak{D}}\D$ such that there are maps $$ \D(\Gamma) \otimes \P(\Gamma) \to \P(g,n).$$ Then $\D(\Gamma)$ is required to depend functorially on $\Gamma$ in certain ways making it meaningful to talk about associativity and equivariance conditions. One has defined the notion of a twisted modular operad.

The game we played in genus zero can now be re-interpreted: instead of saying that $ \{ H^{\bullet-1}(M_{0,n}) \otimes \mathrm{sgn}_n\}$ is a cyclic co-operad, we can say that the usual cohomology groups $ \{ H^{\bullet}(M_{0,n})\}$ form a cyclic $\D$-co-operad, where $\D(\Gamma)$ is given by a suspension for each edge on the graph, and tensoring with the sign representation of the symmetric group acting on the set of edges: in other words, $\D(\Gamma) = \mathrm{Det}(\mathrm{Edge}(\Gamma))^{-1}$, using the terminology introduced in Getzler-Kapranov's paper. This more abstract (but in a sense much more natural) definition now works without any changes also in higher genus!!

In this framework one can also give a nice explanation of why these suspensions and sign changes worked in genus zero. The correct cocycle to twist with was $\D(\Gamma)$ defined in the preceding paragraph. But $\D$ is cohomologous, in an appropriate sense, to the cocycle $D(\Gamma) = \mathrm{Det}(H_1(\Gamma))^{-1}$, which is trivial when restricted to trees. In fact $\D$ and $D$ differ by a coboundary, and this coboundary is precisely given by putting a suspended copy of the sign representation in each spot $(g,n)$. So we have recovered exactly the recipe for making $H^\bullet(M_{0,n})$ into a co-operad that we wrote down in an ad hoc way above.

Answer 2

In my understanding, combinatorially, a modular operad is what Dror Bar-Natan calls a circuit algebra, which is a generalization of a planar algebra in which virtual crossings are allowed. In particular, as Jim Conant points out in the comments, a modular operad can be plugged into itself. A description may be found, for example, in these slides of a talk by Jana Archibald.

Virtual tangles, a-fortiori usual tangles, form an algebra over a modular operad. This is the intuition I have for why modular operads might be natural or useful- that they capture the obviously natural and useful structure of virtual knotted objects.