Logarithms and deformation quantization

Abstract

We prove the statement/conjecture of M. Kontsevich on the existence of the logarithmic formality morphism \(\mathcal {U}^{\mathrm{log}}\). This question was open since 1999, and the main obstacle was the presence of dr / r type singularities near the boundary \(r=0\) in the integrals over compactified configuration spaces. The novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half-plane. It gives rise to a version of Stokes’ formula for differential forms with singularities at the boundary which implies the formality property of \(\mathcal {U}^{\mathrm{log}}\). We also show that the logarithmic formality morphism admits a globalization from \({\mathbb {R}}^d\) to an arbitrary smooth manifold.