Hypercommutative operad as a homotopy quotient of BV

TL;DR

This paper provides an explicit formula for a quasi-isomorphism linking the Hycomm operad and the BV operad quotient, revealing their algebraic equivalence and explaining the Givental group action.

Contribution

It introduces explicit formulas for the quasi-isomorphism between Hycomm and BV/$ abla$, using Givental graphs and two different proof methods, including a homological explanation.

Findings

Explicit quasi-isomorphism formulas derived

Two different proof techniques demonstrated

Homological interpretation of Givental group action provided

Abstract

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/$\Delta$ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.