A hypergraph model for the cyclic BV operad and its applications

Abstract

A dg cyclic operad $BVHgraphs$ of hypergraphs is introduced which comes equipped with an explicit quasi-isomorphism $BV\rightarrow BVHgraphs$ from the { cyclic} operad $BV$ of Batalin-Vilkovisky algebras. A proof that the cohomology of $BVHgraphs$ equals $BV$ occupies most of this paper. We use this model to construct an explicit quasi-isomorphism $Chains(FFM_2) \rightarrow BVHgraphs$ from the chain operad of the cyclic operad $FFM_2$ of the compactified moduli spaces of genus zero curves with marked framed points to the dg cyclic operad $BVHgraphs$ which, combined with the main result mentioned above, gives a new proof of the cyclic formality of $FFM_2$.