Quantizing Poisson Manifolds

This paper extends Kontsevich’s ideas on quantizing Poisson manifolds. A new differential is added to the Hodge decomposition of the Hochschild complex, so that it becomes a bicomplex, even more similar to the classical Hodge theory for complex manifolds. These notes grew out of the author’s attempt to understand Kontsevich’s ideas [Kon95a] on quantizing Poisson manifolds. We introduce a new differential on the Hochschild complex, so that it becomes a bicomplex, see Theorem 2.1. This differential respects the Hodge decomposition of the Hochschild complex of a commutative algebra discovered by Gerstenhaber-Schack [GS87]. Thus, the Hochschild complex becomes similar to the ∂-∂̄-complex in complex geometry. Hopefully, Hodge-theoretic ideas à la Deligne-Griffiths-Morgan-Sullivan [DGMS75, Sul77] will eventually result in proving Kontsevich’s Formality Conjecture, which implies local quantization of an arbitrary Poisson manifold, a hard problem that has been around for almost twenty years [BFF78], see [Wei95] for the most state-of-the-art survey of this subject. Acknowledgment. I am very grateful to Maxim Kontsevich, from whom I learned at least two thirds of what is discussed in this paper. I also thank J.-L. Brylinski, M. Flato, M. Gerstenhaber, J.-L. Loday, J. Stasheff, J. Millson, D. Sullivan, B. Tsygan, and A. Weinstein for helpful conversations. I express my gratitude to Lew Coburn for his invitation to participate in the 1996 Joint Summer Research Conference on Quantization at Mount Holyoke, where this paper was delivered. 1. Kontsevich’s Formality Conjecture 1.1. Some formalities. Let A = C(X) be the algebra of smooth functions on a smooth real manifold X . Let C(A,A) be the (local) Hochschild complex of the algebra A over X , i.e., C(A,A) = {φ ∈ Hom(A, A) | φ(f1, . . . , fn) is a differential operator in each entry f1, . . . , fn}. The Hochschild-Kostant-Rosenberg Theorem [HKR62] provides the computation of the corresponding Hochschild cohomology