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stub for cosmic Galois group
As I mentioned in the parallel discussion, Tamarkin sees the Hopf algebra of renormalization as the algebra of functions on the torsor of equivalences between different homotopic resolutions used to do the deformation quantization in a flavour of BV approach. The independece of resolution is discussed in his arXiv preprint from 2004.
But this “cosmic Galois group” is not the (the group corresponding to) the renormalization Hopf algebra, unless I am missing something.
Yes, Urs, you are absolutely right, it is quite different !
I have been adding a few more references to cosmic Galois group with (very) brief indications on what they are about.
In particular I have added to this entry and to other relevant entries the article by Kitchloo and by Kitchloo-Morava. (Many thanks indeed to Adeel Khan for reminding me of them.)
Just as a side remark, relating to the discussion in the other thread, on Synthetic Quantum Field Theory (schreiber):
there I am talking about “motivic quantization” being the step that takes spans of smooth groupoids in the slice over $\mathbf{B}^n U(1)_{conn}$
$\array{ && Z \\ & \swarrow && \searrow \\ X_1 && \swArrow && X_2 \\ & \searrow && \swarrow \\ && \mathbf{B}^n U(1)_{conn} }$to their image under the “motivic stabilization” of the $(\infty,n)$-category of correspondences.
In an upcoming master thesis, Joost Nuiten will discuss this in detail for $n = 2$ with “motivic stabilization” given by passage to KK-theory under forming twisted groupoid convolution algebras. For higher $n$ this is a proposal to be filled with life.
But notice that what Kitchloo-Morava consider is very close to this for $n = 1$. Their Lagrangian correspondences are correspondences in the slice over $\Omega^{2}_{cl}$, hence diagrams of the form
$\array{ && Z \\ & \swarrow && \searrow \\ X_1 && \swArrow && X_2 \\ & \searrow && \swarrow \\ && \Omega^2_{cl} }$(and they subject them to some nicety constraints). One obtains this second version from the first by “de-prequantizing”, namely by poscomposing (“dependent sum”) with the universal curvature map
$F_{(-)} \colon \mathbf{B}U(1)_{conn} \to \Omega^2_{cl} \,.$Now Kitchloo-Morava “motivically stabilize” by sending the space of Lagrangian correspondences to the spectrum of Lagrangian correspondences. Then they show that the resulting stable $\infty$-category of motivic such Lagrangian correspindences has a “cosmic” motivic Galois group.
To the extent that quantization is postcomposition with this motivic projection, the fact that the “cosmic Galois group” acs on the space of quantum field theories is then immediate…
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