Homotopy Batalin-Vilkovisky algebras

Abstract

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defind by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Last version before publication. To appear in Journal of Noncommutative Geometry. 57 pages

References in corpus (4)

• Calabi-Yau algebras
• Formality of the chain operad of framed little disks
• Operads up to Homotopy and Deformations of Operad Maps
• Curved Koszul duality theory

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• Curved infinity-algebras and their characteristic classes
• On higher Poisson and Koszul--Schouten brackets
• Homotopy Relations for Topological VOA
• Cyclic operad formality for compactified moduli spaces of genus zero surfaces
• Formal formality of the hypercommutative algebras of low dimensional Calabi-Yau varieties
• Sigma-models and Homotopy Algebras
• A Free Frobenius Bialgebra Structure of Differential Forms
• A supergeometric approach to $L_{\infty }$-bialgebras