Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where one could read the proof of theorem 6 by M. Nori?) For example I'm interested in the action of GT on deformation quantization mentioned in ch. 4.6., and an update on the issues of chapter 5. Where one could read more about all that?

Edit: The conj.s I'd like to know more about are: That (the algebra generated by) periods of mixed Tate motives come from Drinfeld associators (p.30); That GT comes from the motivic Galois group's action on the Periods (p.30); That GT is $Aut(Chains(C_2))$ (p.31); What is the "universal map $P_{\mathbb{Z},Tate}$$\longrightarrow$$P_{\mathbb{Z},Tate}$" on p. 32?; Where can one read more about the action of GT on deformation quantization (p.32,33)?

Edit: "Theorem 6 (M. Nori)" asked about above is a sheaf version of a theorem by Beilinson. Nori's article about it with the proof (the statement in question is Basic lemma (first form)) and the reference to Beilinson's paper. An article by Morava contains many new ideas, some remarks on a motivic version of the little disk operad, and a possible algebraic topology frame for Kontsevich's ideas on motives and deformation quantization, and how that would fit to Connes' & Kreimer's Galois theory of renormalization.

Edit: Conc. "GT is $Aut(Chains(C_2))$" here and here new articles (communicated by B.V., thanks!).

Edit: Mathilde Marcolli mentions in this article, on her and Connes' work relating renormalization and a "cosmic Galois group", that Kontsevich developed a renormalization theory continuing his article above and relating "in a natural setting" to Connes'/Marcolli's motivic galois action. Her article ends with some remarks on how Beilinson's conjectures may be viewed "extremly suggestive" as something like renormalization, hinting at geometric interpretations of L-values at non-integer points. It would be great if someone knew where to read more about both issues.

Edit: Spencer Bloch's thoughts (video lecture) on motives and renormalization relate to the theme too.

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    $\begingroup$ I'm not sure that Nori's work has ever been written up. He has given a course (or maybe courses) on it over the years at University of Chicago, and so notes may exist. Probably the easiest way to find out about possible course notes, papers, or other presentations of his theory is just to write to him directly. $\endgroup$
    – Emerton
    Commented Oct 14, 2010 at 14:36
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    $\begingroup$ What do the conjectures you are wondering about say? $\endgroup$
    – Romeo
    Commented Oct 14, 2010 at 15:17
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    $\begingroup$ I'm no expert but I heard a talk by Thomas Willwacher about some of these topics. He has an article together with Severa. Maybe you can find more there. $\endgroup$ Commented Oct 15, 2010 at 10:14
  • $\begingroup$ Maybe some newer information can be found in a series of articles and lectures of Leila Schneps, see ncatlab.org/nlab/show/Leila%20Schneps for links. $\endgroup$ Commented Apr 26, 2011 at 9:17
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    $\begingroup$ A new archive paper related to Nori's work: Annette Huber, Stefan Müller-Stach, On the relation between Nori motives and Kontsevich periods, 1105.0865 $\endgroup$ Commented May 6, 2011 at 13:22

2 Answers 2


The action of GT on deformation quantization has been developed in http://arxiv.org/abs/1009.1654 (Willwacher) and before in http://arxiv.org/abs/math/0202039 (Tamarkin).

The fact that GT is Aut(Chain(C2)) is true or not ,depending if you work in the unstable homotopy category (Fresse: http://math.univ-lille1.fr/~fresse/E2RationalAutomorphisms.html) or in the stable one (Willwacher: http://arxiv.org/abs/1009.1654).

For new developments concerning the relation between GT and the Galois group of periods of mixed Tate motives over Z, you can take a look at the recent work of Francis Brown.


  • $\begingroup$ Thanks! You mean this by Francis Brown?: people.math.jussieu.fr/~brown/MTZ.pdf $\endgroup$ Commented Feb 25, 2011 at 19:38
  • $\begingroup$ Yes this is what I mean. He proves is that the Galois group of mixed Tate motives is generated by the motivic fundamental group of $\mathcal{M}_{0,4}$ (which is $\mathbb{P}^1$ minus three points). As far as I understand, what remains to be proved is that there are no more relations than the one coming from $\mathcal M_{0,5}$. $\endgroup$
    – DamienC
    Commented Feb 27, 2011 at 14:47
  • $\begingroup$ Also, about Nori's approach to periods, as quoted by Kontsevich: isn't there something written about this in "Une introduction aux motifs" (Yves Andre, Panoramas et synthese, SMF, 2004 - I haven't heard about an english translation, sorry). $\endgroup$
    – DamienC
    Commented Apr 26, 2011 at 8:25
  • $\begingroup$ I think that in one of the articles in the Handbook of K-theory there is a reasonable sketch of Nori's approach. $\endgroup$ Commented Apr 26, 2011 at 9:14
  • $\begingroup$ About Nori and Kontsevich, there is a recent preprint that was posted on the arXiv recently: arxiv.org/abs/1105.0865 (this is for information; I haven't read it). $\endgroup$
    – DamienC
    Commented May 11, 2011 at 11:13

I am late to this discussion, but it seems that briefly after it died out, Vasily Dolgushev established part of what you are looking for (I just heard him advertize this result at GAP XI in Pittsburgh):

that the connected components of the space of (stable) deformation quantizations of a Euclidean space is indeed a torsor over the Grothendieck-Teichmüller group is shown here

  • Vasily Dolgushev, Stable Formality Quasi-isomorphisms for Hochschild Cochains I (arXiv:1109.6031)

with a review in

  • Vasily Dolgushev, Exhausting formal quantization procedures (arXiv:1111.2797),

see theorem 3.1 there.

Aspects of the generalization to spaces other than Euclidean are then discussed in

  • Vasily Dolgushev, Christopher Rogers, Thomas Willwacher, Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields (arXiv:1211.4230)

I should collect some more pointer on the nLab at deformation quantization -- Motivic Galois group action on the space of quantizations


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