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**On some algebraic structures arising in string theory.**
*(English)*
Zbl 0871.17021

Penner, Robert (ed.) et al., Perspectives in mathematical physics. Proceedings of the conference on interface between mathematics and physics, held in Taiwan in summer 1992 and the special session on topics in geometry and physics, held in Los Angeles, CA, USA in winter of 1992. Boston, MA: International Press. Conf. Proc. Lect. Notes Math. Phys. 3, 219-227 (1994).

Fifteen years ago, I. Batalin and G. Vilkovisky constructed a quantization procedure whose operations define a commutative, associative algebra equipped with an additional anti-bracket structure which satisfies three basic identities: super anti-commutativity, super Jacobi identity, and the super derivation rule [Phys. Lett. B 102, 27-31 (1981)]. Such super algebras appeared already before, namely in M. Gerstenhaber’s work on the cohomology of associative rings [Ann. Math., II. Ser. 78, 267-288 (1963; Zbl 0131.27302)].

Very recently, in an unpublished preprint, B. Lian and G. Zuckerman pointed out that the homology of a topological chiral algebra can be also equipped with the structure of such a (so-called) Gerstenhaber algebra. Moreover, the algebras constructed by Batalin-Vilkovisky and by Lian-Zuckerman are equipped, in addition, with an odd operator \(\Delta\) satisfying \(\Delta^2=0\) and a variant of the Leibniz rule. Gerstenhaber algebras of this type are sometimes called BV-algebras (Batalin-Vilkovisky algebras).

The present paper provides a rather general construction principle for BV-algebras from data of topological conformal algebras arising in string theories. The results obtained in this general context are applied to the situation studied by Lian-Zuckerman (i.e., to the homology algebra of a topological chiral algebra) and lead to a simplification of their constructions and proofs. At the end of their paper, the authors briefly discuss the geometric meaning of their construction of BV-algebras, mainly with a view towards related constructions in string field theory.

Some closely related work on the Batalin-Vilkovisky formalism can be found in the papers of E. Getzler [Commun. Math. Phys. 159, 265-285 (1994; Zbl 0807.17026)] and of one of the authors, A. Schwarz himself [Commun. Math. Phys. 155, 249-260 (1993; Zbl 0786.58017)].

For the entire collection see [Zbl 0816.00027].

Very recently, in an unpublished preprint, B. Lian and G. Zuckerman pointed out that the homology of a topological chiral algebra can be also equipped with the structure of such a (so-called) Gerstenhaber algebra. Moreover, the algebras constructed by Batalin-Vilkovisky and by Lian-Zuckerman are equipped, in addition, with an odd operator \(\Delta\) satisfying \(\Delta^2=0\) and a variant of the Leibniz rule. Gerstenhaber algebras of this type are sometimes called BV-algebras (Batalin-Vilkovisky algebras).

The present paper provides a rather general construction principle for BV-algebras from data of topological conformal algebras arising in string theories. The results obtained in this general context are applied to the situation studied by Lian-Zuckerman (i.e., to the homology algebra of a topological chiral algebra) and lead to a simplification of their constructions and proofs. At the end of their paper, the authors briefly discuss the geometric meaning of their construction of BV-algebras, mainly with a view towards related constructions in string field theory.

Some closely related work on the Batalin-Vilkovisky formalism can be found in the papers of E. Getzler [Commun. Math. Phys. 159, 265-285 (1994; Zbl 0807.17026)] and of one of the authors, A. Schwarz himself [Commun. Math. Phys. 155, 249-260 (1993; Zbl 0786.58017)].

For the entire collection see [Zbl 0816.00027].

Reviewer: W.Kleinert (Berlin)

### MSC:

17B68 | Virasoro and related algebras |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |