Chain level Koszul duality between the Gravity and Hypercommutative operads

TL;DR

This paper constructs a chain-level model demonstrating Koszul duality between the Gravity and Hypercommutative operads, refining previous results and linking topological and algebraic structures in operad theory.

Contribution

It provides a chain model for the Hypercommutative operad and proves its Koszul duality with the Gravity operad at the chain level, using topological methods.

Findings

Constructed a chain model for the Hypercommutative operad.

Proved the chain model is the linear dual of the bar construction of the Gravity operad.

Established Koszul duality between Gravity and Hypercommutative operads at the chain level.

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 $\overline{\mathcal{M}}_{0,n+1}$.