A hypergraph model for the cyclic BV operad and its applications

TL;DR

This paper introduces a hypergraph-based dg cyclic operad model for BV algebras, proving its cohomology matches the classical BV operad, and applies it to establish the cyclic formality of the operad of framed little disks.

Contribution

It constructs a new hypergraph model for the BV operad and demonstrates its cohomology and applications to formality of the framed little disks operad.

Findings

The hypergraph model has cohomology equal to the BV operad.

An explicit quasi-isomorphism from chains of FFM_2 to the hypergraph model is constructed.

Provides a new proof of the cyclic formality of FFM_2.

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 $FFM_2$.