Logarithms and deformation quantization

Anton Alekseev; Carlo A. Rossi; Charles Torossian; Thomas Willwacher

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arXiv · 1401.3200

Logarithms and deformation quantization

Anton Alekseev·Carlo A. Rossi·Charles Torossian·Thomas Willwacher

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Abstract

We prove the statement/conjecture of M. Kontsevich on the existence of the logarithmic formality morphism. 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. We also show that the logarithmic formality morphism admits a globalization from $\mathbb{R}^d$ to an arbitrary smooth manifold.

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Anton Alekseev, Carlo A. Rossi, Charles Torossian, Thomas Willwacher. 2014-01-14. Logarithms and deformation quantization. https://arxiv.org/abs/1401.3200

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