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On Deformation Quantization of Quadratic Poisson Structures

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Abstract

We study the deformation complex of the dg wheeled properad of \({{\mathbb {Z}}}\)-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck–Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all the universal quantizations of \({{\mathbb {Z}}}\)-graded quadratic Poisson structures (together with the underlying homogeneous  formality maps). In particular we show that two universal quantizations of Poisson structures are equivalent if the agree on generic quadratic Poisson structures.

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Notes

  1. We mean the completion with respect to the filtration of \(\textsf{dfGC}_d\) by the number of vertices.

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Acknowledgements

The research of A.Kh. was partially supported by the HSE University Basic Research Program. S.M. was partially supported by the University of Luxembourg RSB internal grant. We are grateful to the HSE and the UL for hospitality during our work on this project. We are very grateful to the referees of our paper for their critical remarks and useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, 31905, Haifa, Mount Carmel, Israel

    Anton Khoroshkin

  2. Department of Mathematics, Luxembourg University, Maison du Nombre, 6 Avenue de la Fonte, 4364, Esch-sur-Alzette, Grand Duchy of Luxembourg

    Sergei Merkulov

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  1. Anton Khoroshkin
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  2. Sergei Merkulov
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Correspondence to Anton Khoroshkin.

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Communicated by V. Pestun.

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Khoroshkin, A., Merkulov, S. On Deformation Quantization of Quadratic Poisson Structures. Commun. Math. Phys. 404, 597–628 (2023). https://doi.org/10.1007/s00220-023-04829-z

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