Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves

Abstract

We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves ${\mathcal M}_{g,n}$ (which, according to Penner, is controlled by the ribbon graph complex) and the homotopy theory of $E_d$ operads (controlled by ordinary graph complexes with no ribbon structure, introduced first by Kontsevich). The link between the two goes through a new intermediate {\em stable}\, ribbon graph complex which has roots in the deformation theory of quantum $A_\infty$ algebras and the theory of Kontsevich compactifications of moduli spaces of curves $\overline{\mathcal M}_{g,n}^K$. Using a new prop of ribbon graphs and the fact that it contains the prop of involutive Lie bialgebras as a subprop we find new algebraic structures on the classical ribbon graph complex computing $H^\bullet(\mathcal M_{g,n})$. We use them to prove Comparison Theorems, and in particular to construct a non-trivial map from the ordinary to the ribbon graph cohomology. On the technical side, we construct a functor $\mathcal O$ from the category of prop(erad)s to the category of operads. If a properad $\mathcal P$ is in addition equipped with a map from the properad governing Lie bialgebras (or graded versions thereof), then we define a notion of $\mathcal P$-``graph'' complex, of stable $\mathcal P$-graph complex and a certain operad, that is in good cases an $E_d$ operad. In the ribbon case, this latter operad acts on the deformation complexes of any quantum $A_\infty$-algebra. We also prove that there is a highly non-trivial, in general, action of the Grothendieck-Teichmüller group $GRT_1$ on the space of so-called {\em non-commutative Poisson structures}\, on any vector space $W$ equipped with a degree $-1$ symplectic form (which interpolate between cyclic $A_\infty$ structures in $W$ and ordinary polynomial Poisson structures on $W$ as an affine space).