Homotopy Gerstenhaber Algebras and Topological Field Theory

We prove that the BRST complex of a topological conformal field theory is a homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We also suggest a refinement of the original conjecture for topological vertex operator algebras. We illustrate the usefulness of our main tools, operads and “string vertices” by obtaining new results on Vassiliev invariants of knots and double loop spaces. Two-dimensional topological quantum field theory (TQFT) at its most elementary level is the theory of Z-graded commutative associative algebras (with some additional structure) [34]. Thus, it came as something of a surprise when several groups of mathematicians realized that the physical state space of a 2D TQFT has the structure of a Z-graded Lie algebra, relative to a new grading equal to the old grading minus one. Moreover, the commutative and Lie products fit together nicely to give the structure of a Gerstenhaber algebra (Galgebra), a Z-graded Poisson algebra for which the Poisson bracket has degree −1 (see Section 1). This G-algebra structure is best understood in the framework of 2D topological conformal field theories (TCFTs) (see Section 5.2) wherein operads of moduli spaces of Riemann surfaces play a fundamental role. G-algebras arose explicitly in M. Gerstenhaber’s work on the Hochschild cohomology theory for associative algebras (see Section 1 for this and several other contexts for the theory of G-algebras). Operads arose in the work of J. Stasheff, Gerstenhaber and later work of P. May on the recognition problem for iterated loop spaces. Eventually, F. Cohen discovered that the homology of a double loop space is naturally a G-algebra, see Section 1; in fact, a double loop space is naturally Date: February 24, 1996. Research of the first author was supported in part by an NSF postdoctoral research fellowship. Research of the second author was supported in part by NSF grant DMS9402076. Research of the third author was supported in part by NSF grant DMS-9307086.