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In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an associative algebra. Existence and classification of quantizations of arbitrary Poisson manifolds come out as a by-product of his formality theorem. Specifically, a quantization of a Poisson manifold is defined as star product on the manifold i.e. a formal deformation of the associative algebra of functions on the manifold such that the first order term coincides with the Poisson bracket up to gauge transformations (i.e. the Poisson bracket is the semi-classical data of the star product).

Poisson manifolds can alternatively be characterised as symplectic Lie 1-algebroid (e.g. in the terminology of Roytenberg's arXiv:math/0203110).

Symplectic Lie 2-algebroids identify with Courant algebroids.

Questions:

1) Is there a known concept of quantization of Courant algebroids analogue as the one for Poisson manifolds (i.e. as a deformation of a specific structure such that the semi-classical data is a Courant algebroid)?

2) Is there an analogue of the formality theorem that would imply existence and classifications of such quantizations of Courant algebroids?

Thank you for any reference or comment!

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  • $\begingroup$ In a paper with Frank Keller there is a suggestion of what a quantization of a Courant algebroid could be. But I fear that there are many other options which might be more interesting? Here the ref: Deformation Theory of Courant Algebroids via the Rothstein Algebra. J. Pure and Appl. Alg. 219 (2015), 3391–3426. $\endgroup$ – Stefan Waldmann Mar 2 '17 at 8:25
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    $\begingroup$ Re 1) - don't know much of this, so just a comment - according to Bressler Courant algebroids are classical limits of vertex algebroids, so presumably the latter are quantizations of the former $\endgroup$ – მამუკა ჯიბლაძე Mar 4 '17 at 17:49

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