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Abstract. We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory. The physical importance of these structures is that they lead toward the classification of string theories at the tree level, because the structure constants of the algebras appear as all correlators of the theory. We suggest that an appropriate background for putting together those algebraic structures is the structure of an operad. On the one hand, as we point out, a conformal field theory at the tree level is equivalent to an algebra over the operad of Riemann spheres with punctures, cf. Huang and Lepowsky [22]. On the other hand, this one operad gives rise to several other operads creating these various algebraic structures. The relevance to physics is that theories such as conformal field theory or string-field theory provide a representation of the geometry of the moduli space of such punctured Riemann spheres in the category of differential graded vector spaces. This paper, one of a series, deals with a part of these algebraic structures, namely with the structure of a homotopy Lie algebra and the related structures of the gravity algebra and Batalin-Vilkovisky algebra. A richer structure, the moduli space of Riemann spheres, induces a homotopy version of a Gerstenhaber algebra, which contains

The goal of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B∞-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: One of them is a B∞-algebra, another, called a homotopy G-algebra, is a particular case of a B∞-algebra, the others, a G∞-algebra, an E 1 -algebra, and a weak G∞-algebra, arise from the geometry of configuration spaces. Corrections to the paper of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.

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