Formality theorems: from associators to a global formulation

Gilles Halbout

doi:10.5802/ambp.220

Abstract

Let M be a differential manifold. Let Φ be a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting from Φ. More precisely, following Tamarkin's proof, we construct a Lie homomorphism up to homotopy between the Lie algebra of Hochschild cochains on C ∞ (M) and its cohomology (r(M, ATM), [-, -]s). This paper is an extended version of a course given 8 - 12 March 2004 on Tamarkin's works. The reader will find explicit examples, recollections on G ∞ -structures, explanation of the Etingof-Kazhdan quantization-dequantization theorem, of Tamarkin's cohomological obstruction and of globalization process needed to get the formality theorem. Finally, we prove here that Tamarkin's formality maps can be globalized.

Highlights

Let M be a differential manifold and A = C∞(M ) the algebra of smooth differential functions over M
The cohomology H∗(Dpoly, b) of Dpoly with respect to b is isomorphic to the space Tpoly ([15])
We introduce an “external” bigrading on the cochain complex induced by the following bigrading on Λ·Tpoly⊗·: if x ∈ Tpoly⊗p1Λ · · · ΛTpoly⊗pn, |x|e = (p1−1+· · ·+pn−1, n−1). This grading gives a bicomplex structure on the vectorial space Hom(Λ·Tpoly⊗·, Tpoly), [d1T,1 + d2T, −] for Formality theorems which d1T,1 = [−, −]S is of bidegree (0, 1) and d2T = ∧ is of bidegree (1, 0)

Summary

Introduction

- The space Tpoly = Γ(M, ΛT M ) of multivector fields on M - The space Dpoly = C(A, A) = k≥0 Ck(A, A), of regular Hochschild cochains (generated by differential k-linear maps from Ak to A and support preserving). The cohomology H∗(Dpoly, b) of Dpoly with respect to b is isomorphic to the space Tpoly ([15])

Halbout

Globalization process

Existence of globalizable formality maps