Chain level Koszul duality between the Gravity and Hypercommutative operads

Abstract: Let $\overline{\mathcal{M}}{0,n+1}$ be the moduli space of genus zero stable curves with $(n+1)$-marked points. The collection $\overline{\mathcal{M}}={\overline{\mathcal{M}}{0,n+1}}{n\geq 2}$ forms an operad in the category of complex projective varieties; its homology $Hycom= H(\overline{\mathcal{M}})$ is called the Hypercommutative operad. In this paper we construct a chain model for the hypercommutative operad, i.e. an operad of chain complexes $C_^{dual}(\overline{\mathcal{M}})$ which is weakly equivalent to the operad of singular chains $C_(\overline{\mathcal{M}})$. We prove that $C_^{dual}(\overline{\mathcal{M}})$ is the linear dual of the bar construction $B(grav)$, where $grav$ is a chain model of the gravity operad based on cacti without basepoint. This shows that the Gravity and Hypercommutative operad are Koszul dual also at the chain level, refining a previous result of Getzler. The construction is topological, since $C_*^{dual}(\overline{\mathcal{M}})(n)$ is the cellular complex associated to a regular CW-decomposition of $\overline{\mathcal{M}}_{0,n+1}$.