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I'm interested in the relation between the two following constructions motivated by renormalization:

  • Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of an Hopf algebra $H$ of rooted trees. By Cartier-Milnor-Moore theorem, $H$ is the algebra of regular functions on some pro-algebraic group, which turns out to be the so-called Butcher group $B$.
  • Connes-Marcolli then considered a differential equation satisfied by divergences appearing in the above work, which leads to the introduction of a category of "equisingular flat connection" which, so far I understand, are more or less specific $B$-valued principal bundles equipped with a flat connection up to some equivalence relation. It turns out that this category is tannakian, meaning that it is equivalent to the category of modules over some pro-algebraic group $G$. They observe that $G$ acts on any renormalizable theory in a nice way, hence the name "cosmic Galois group" which was coined by Cartier. It is more than an analogy, since $G$ is (non-canonically) isomorphic to some motivic Galois group.

I do not claim to understand these things at all, especially the "physical" part, so my question is maybe naive. The vague question is:

Is there a "direct relation" between the group $G$ and the Hopf algebra $H$ ?

Of course there are some relations between the two, so maybe a more precise question is:

Is there a definition of $G$ purely in terms of combinatorics of Feynman graphs, or as the automorphism group of something (something else than a fiber functor) ?

In fact, if I understand Marcolli's survey correctly, $G$ action on physical theories factors through an action of $B$ (but it seems rather surprizing, so it's very likely that I misunderstood something). So for the sake of curiosity, another naive question is:

"Why" is it $G$, and not $B$ which plays the role of a cosmic Galois group ?

My motivation for this question comes from the (highly non-trivial) fact that (some group which is almost) $G$ embeds into the graded Grothendieck-Teichmuller group $GRT$, which can be defined as the automorphism group of a certain operad of Feynman graphs. Since $G$ is more or less by definition related to the combinatorics of these graphs, I'm wondering if there is a concrete "combinatorial" definition of it.

Edit: To be precise they also consider a more general Hopf algebra of Feynman diagrams (not only rooted trees). They call the corresponding pro-algebraic group the group of "diffeographism". My claim about the action of $G$ is related to this group and not to $B$, I guess.. So let's assume that I'm asking my question in this setting too.

Edit 2: I should have mentionned that there are in fact several Hopf algebras of Feynman graphs whose choice depends on the particular physical theory you're considering (i.e. you have to choose a specific "type" of Feynman graph), so there are in fact plenty of diffeographism groups, and $G$ is universal among them (which in fact answer my last question, although the Butcher group still seems to play a distinguished role). On the other hand it seems to me that mathematically it still make sense to consider the Hopf algebra of all Feynman graphs, but maybe what you get is precisely $\mathcal H _{\mathbb U}$ (see Gjergji's answer) ?

And just to expand what I'm saying in my comment to Gjergji's answer, it's rather clear for example that $\mathcal H _{\mathbb U}$ is isomorphic to the dual of the algebra of noncommutative symmetric functions, since the latter is also by definition a free graded noncommutative algebra with one primitive generator in each degree (which are analogs of power sum symmetric polynomials). In fact the abstract definition of $\mathcal H _{\mathbb U}$ itself is rather explicit and combinatorial, hence I'm really looking for a more conceptual link.

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This is more of a comment where I wanted to point out that there is a parallel treatment of this cosmic Galois group which avoids the Riemann-Hilbert approach and is more combinatorial in some sense.

The first paper to point out this point of view is, I think, "A Lie theoretic approach to renormalization" by K. Ebrahimi-Fard, J.M. Gracia-Bondia and F. Patras. There is also a nice summary in "Dynkin operators and renormalization group actions in pQFT".

First note that the cosmic Galois group $\mathbb U$ is defined as the affine group scheme associated to the commutative Hopf algebra $$\mathcal H _{\mathbb U}:=\mathcal U(\mathcal F(1,2,3,\dots) _{\bullet})^{\vee}$$ where $\mathcal F(1,2,3,\dots) _{\bullet}$ is the free graded Lie algebra with a generator in every degree $n > 0$.

The papers above explain, among other things, how the universality of the cosmic Galois group can be interpreted as saying that this Hopf algebra and the associated group scheme act on all Connes-Kreimer Hopf algebras of Feynman diagrams. However this Hopf algebra, also called the Descent algebra is known to act naturally on all connected graded commutative Hopf algebras, so it is universal in a broader sense.

Now, if you see the second paper, the descent algebra $\mathcal H _{\mathbb U}$ has a natural description in terms of generators defined from permutations with a fixed descent set, so perhaps this qualifies as a simple combinatorial definition. On the other hand the only property we needed from the algebras of Feynman diagrams was that they be graded connected commutative Hopf algebras, so I doubt that $\mathbb U$ can be described directly as automorphisms of such things...

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Thanks for the references. I found the first one by myself, and made the same objection as in your last sentence. Indeed, once you know that the cosmic Galois group is identified with the group you mention, you're left with (the dual of) some rather "familiar" Hopf algebra, and I've no doubt that people working on combinatorial Hopf algebras can give many different abstractly equivalent natural constructions of it, but without a natural link with Feynman graphs. That's why I'm hoping that there is something involving (an operad of) diagrams which would make it closer to GRT. – Adrien Apr 24 '12 at 9:07

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