Hypercommutative operad as a homotopy quotient of BV

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.

arXiv commentminor corrections added, to appear in Comm.Math.Phys

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Cited by in corpus (3)

• Toric varieties of Loday's associahedra and noncommutative cohomological field theories
• Generalized cohomological field theories in the higher order formalism
• Gravity prop and moduli spaces ${\mathcal{M}}_{g,n}$