My question is simple:

How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?

I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling with the basics. All papers I've found just seem to jump into the deep-end or involve musings that are more inspirational than precise; so I'm eager to read what people here say.


1 Answer 1


A short answer would be: $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the profinite fundamental groupoïd of the operad of little discs.

If $X$ is an algebraic variety over $\mathbb{Q}$ we have an exact sequence $$ 1 \to\pi_1(X\otimes \overline{\mathbb{Q}},p) \to \pi_1(X,p) \to Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to 1 $$ Here $\pi_1(X\otimes \overline{\mathbb{Q}};p)$ is canonically identified with the profinite completion of the usual topological fundamental group $\pi_1(X(\mathbb{C}),p)$. If the basepoint is defined over $\mathbb{Q}$, this split and we have an action $$ Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to Aut(\widehat{\pi}_1(X(\mathbb{C}),p)). $$

The (profinite completion of the) fundamental groupoïds of the $C_2(n)$ inherit the operad structure. The trick is that all of it can be defined over $\mathbb{Q}$ as $C_2(n)$ is homotopy equivalent to the configuration space of points on the affine line $F(\mathbb{A}^1_{\mathbb{Q}},n)(\mathbb{C})$. One has to define rational "tangential base points" and check that the operad structure on the fundamental groupoïds is also defined over $\mathbb{Q}$. The resulting operad is described here. One can explicitly compute its automorphism group. This is the Grothendieck-Teichmuller group $\widehat{GT}$.

As everything is defined over $\mathbb{Q}$, $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ operates on the whole operad. So we have a morphism $$ Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \widehat{GT} $$ It follows from a theorem of Belyi that it is injective.

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    $\begingroup$ So once you have an action of the operad on the Hochschild complex (as in Deligne's conjecture) you have many, each conjugated by an element of $\mathrm{Gal}$. Is there an explicit way of seeing the difference between two of this conjugated actions on the complex? $\endgroup$ Sep 28, 2010 at 11:59
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    $\begingroup$ The Galois action only exists on the profinite completion of the fundamental group. So I don't see how Gal would act in your situation. I don't think the absolute Galois group really appears in the context of Deligne's conjecture. $\endgroup$
    – AFK
    Sep 28, 2010 at 12:15
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    $\begingroup$ Instead, what appears naturally is the motivic Galois group of mixed Tate motives over $\mathbb{Z}$ (in part because the $F(\mathbb{A}^1,n)$ are mixed Tate over $\mathbb{Z}$). It plays the same role as Gal but for the motivic prounipotent version of the fundamental groups. The two are related by a Zarsiki dense morphism $G_{\mathbb{Q}} \to G^{mot}_{\mathbb{Z}}(\mathbb{Q}_\ell)$ corresponding to the $\ell$-adic realization. See Kontsevich's "Operads and motives in deformation quantization". $\endgroup$
    – AFK
    Sep 28, 2010 at 12:18
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    $\begingroup$ PS: I think that writing an explicit action of the operad on the Hochschild complex of $k$-vector spaces usually requires the choice of an associator over $k$. In this case the $k$-points of the motivic Galois group $G_{\mathbb{Z}}(k)$ will act through the prounipotent $GT(k)$ on the coefficients of the associator. If $\mathbb{Q}_\ell \subset k$ you will get an action of $G_{\mathbb{Q}}$ through the $\ell$-adic realization morphism. But the coefficient of this action are very complicated, the simplest are given by the cyclotomic character and Soulé's characters. $\endgroup$
    – AFK
    Sep 28, 2010 at 12:34
  • $\begingroup$ Do you have a reference for this point of view on the Galois action on the little disk operad, ie the identification of the fundamental groupoids of the little disks with the one obtained from $F(\mathbb A^1,n)$ and certain tangental basepoints? It sounds like you're sketching a realization of the proposal of an algebro-geometric interpretation of the little disks described in Jack Morava's "The motivic Thom isomorphism". In any case this is certainly a "noncanonical" description of the Galois action; the fact that the Galois group sits in $\widehat{GT}$ is explained already in Drinfel'd... $\endgroup$ Jul 9, 2013 at 20:49

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