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 defined 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 Poincaré-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 BValgebra 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. 14 3. Deformation theory of homotopy BV-algebras 27 4. Homotopy theory of homotopy BV-algebras 33 Appendix A. Koszul duality theory 35 Appendix B. Homotopy theory for algebras over a Koszul operad 43 Appendix C. Deformation and obstruction theory 49 References 54 1991 Mathematics Subject Classification. 18D50 (primary), 18G55, 55P48, 81T40, 17B69 (secondary).

Open-Closed Homotopy Algebras and Strong Homotopy Leibniz Pairs Through Koszul Operad Theory

“…Sections 6 and 7 are devoted to the study of OCHA and the zeroth homology of SC. One has that the operad H 0 (SC) is not quadratic but can be expressed with linear-quadratic relations, so that we can apply results of [6]. We prove in theorem 6.2.3 that H 0 (SC) is a Koszul operad.…”

“…In this section we follow closely the article by Imma Galvez-Carrillo, Andy Tonks and Bruno Vallette [6] in order to prove that the operad H 0 (SC) is Koszul. Indeed this operad is not quadratic, and we need first to give a description by generators and relations so that it is quadratic and linear.…”

“…By projecting the relations onto the quadratic part we obtain an operad qH 0 (SC) which turns out to be Koszul. By definition [6,Appendix A.3] one has that H 0 (SC) is Koszul.…”

“…Let us recall first the theory explained in [6] for quadratic-linear operads. A quadraticlinear operad is of the form F(E)/(R) with R ⊂ F…”

“…It is important to remark that though the case of SHLP is rather classical in the theory of operads (binary quadratic colored operads) the case OCHA is less classical because the operads involved are not quadratic. The technics used are the ones employed by Galvez-Carillo, Tonks and Vallette in the case of BV-algebras in [6]. In order to prove those theorems we use also the relation made by the second author between the swiss-cheese operad and OCHA in [12].…”