Homotopy Gerstenhaber Algebras

The purpose 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. In an unpublished paper of E. Getzler and J. D. S. Jones [GJ94], the notion of a homotopy n-algebra was introduced. Unfortunately the construction that justified the definition contained an error, which passed unnoticed in subsequent work, in spite of being heavily used in it. That work included the solution by Getzler and Jones [GJ94] of Deligne’s Conjecture, whose weak version had been proven in [VG95]; the construction by T. Kimura, G. Zuckerman, and the author [KVZ96] of a homotopy Gerstenhaber algebra structure (called a G∞-algebra therein) on the state space of a topological conformal field theory (TCTF); the extensions of the above work by Akman [Akm97, Akm99] and Gerstenhaber and the author [VG95]; a few papers delivered at the Workshop on Operads in Osnabrück in June 1998 [Vog98]. The purpose of this paper is to correct the error in the original construction of [GJ94], complete Getzler-Jones’ proof of Deligne’s Conjecture accordingly, and make appropriate corrections in [KVZ96]. First, let us describe the problem. A Gerstenhaber (G-) algebra is defined by two operations, a (dot) product ab and a bracket [a, b], on a graded vector space V over a ground field k of characteristic zero, so that the product defines a graded commutative algebra structure on V and the bracket a graded Lie algebra structure on V [1], the desuspension of the graded vector space V = ⊕ n V : V [1] = V . The bracket must be a graded derivation of the product in the following sense: [a, bc] = [a, b]c+ (−1)b[a, c], where |a| denotes the degree of an element a ∈ V . In other words, a G-algebra is a specific graded version of a Poisson algebra. Date: August 10, 1999.