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.
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Notes
In fact, there are five properties in [4] to be satisfied. However, the others are trivially true for a formality morphism given by universal formulas such that \(\mathcal {U}_1\) is the Hochschild–Kostant–Rosenberg morphism.
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Acknowledgments
We are grateful to G. Felder and S. Merkulov for inspiring discussions and for the interest in our work. We would like to thank J. Löffer who participated in the earlier stages of the project. The work of A.A. was supported in part by the grant MODFLAT of the European Research Council (ERC) and by the Grants 140985 and 141329 of the Swiss National Science Foundation. T.W. was partially supported by the Swiss National Science Foundation, Grant 200021_150012. Research of A.A. and T.W. was supported in part by the NCCR SwissMAP of the Swiss National Science Foundation.
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Alekseev, A., Rossi, C.A., Torossian, C. et al. Logarithms and deformation quantization. Invent. math. 206, 1–28 (2016). https://doi.org/10.1007/s00222-016-0647-7
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DOI: https://doi.org/10.1007/s00222-016-0647-7