Batalin-Vilkovisky algebras and two-dimensional topological field theories

By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator :A A ₊₁ such that ² = 0, and ,a]–a is a graded derivation ofA for allaA. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads.The author is partially supported by a fellowship of the Sloan Foundation and a research grant of the NSF