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I have expanded the Idea-section at deformation quantization a little, and moved parts of the previous material there to the Properties-section.
I just added some rough notes from a lecture John Jones gave this morning (http://www.newton.ac.uk/programmes/GDO/gdow01). I didn't manage to get all the TeX right the first time, and the nLab seems down now so I will have to fix it later.
Thanks!! That’s great.
I’ll maybe go through it in a few minutes to fix the instiki-syntax errors (I guess you were copy-and pasting? )
Ah, I see you are editing right now. I was just about to look into it.
(It’s strange that the link to HKR theorem does not work. (?))
I am just not used to the instiki TeX so I usually have to do my best and then keep editing the page until I have fixed everything. For the HKR theorem, I fixed it by making "Theorem" lowercase.
I will add some remarks on the Deligne conjecture that Jones also made (which I was too tired to type yesterday).
Done, also added a page for John Jones.
Thanks!
I have added a few more links.
I am beginning to expand deformation quantization to include the discussion of deformation quantization of field theories by Costello-Gwilliam. So far I began to restructure the Definition-section accordingly. Will now fill in material, as time permits.
But meanwhile: in the course of this I slightly rearranged the material whose addition was announced in #5, #6 above. For instance the definition of Poisson manifolds and their deformation I moved out of the Properties-section into the Definition-section. This now makes Kontsevich’s theorem sit a bit lonely in a single subsection in the Properties-section. But I guess eventually we should expand there on its proof, which will justify a dedicated subsection after all.
I am thinking the relation discussed further below to Hochschild and cyclic cohomology deserves to be highlighted and expanded on much more, eventually. I’ll see what I can do. Will be forced offline in a short while, though.
I slightly re-arranged the references at deformation quantization. Igor Khavkine rightly amplified to me that Fedosov’s deformation quantization already applies also to (regular) Poisson manifolds, which was not well-reflected in the entry. So I moved that to the top of the list, where it seems to belong, so that Kontsevich’s result is now a little bit below.
Added the following quote from section 1.4
to the Idea-section at deformation quantization (with a tad of commentary):
Generally speaking, physics is based on $[$ strict $]$ quantization, rather than $[$ formal $]$ deformation quantization, although conventional quantization sometimes leads to problems that can be treated by deformation quantization.
added to the section Motivic Galois group action on the space of quantizations a pointer to the proof by Dolgushev that $\pi_0$ of the space of formal deformation quantizations of an $\mathbb{R}^n$ is indeed (a torsor over) the Grothendieck-Teichmüller group.
(Just heard a talk about this at GAP XI. Therefore just a brief pointer, don’t have much time)
there is an old MO discussion wondering about this
renaming this entry from “deformation quantization” to “formal deformation quantization” so that deformation quantization can serve as a disambiguation page with strict deformation quantization
I have brushed-up the two original bibitems (which had been both incomplete and broken):
François Bayen, Moshé Flato, Christian Fronsdal, , André Lichnerowicz, Daniel Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures., Annals of Physics 111 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]
François Bayen, Moshé Flato, Christian Fronsdal, André Lichnerowicz, Daniel Sternheimer, Deformation theory and quantization. II. Physical applications, Annals of Physics 111 1 (1978) 111-151 [doi:10.1016/0003-4916(78)90225-7]
added pointer to:
added pointer to today’s
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