Looking for a proper reference frame to navigate in an ambiguous quantum flow
Infinities of perturbative quantum field theories are blessing stars in the dark sky of the renormalizable physical world for the noncommutative navigator drawing a cosmic map from the miraculous catches of the spectral physicist fishing now in the roaring "Tera eVs".
We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group” U∗, which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup of U∗. The group U∗ arises through a Riemann–Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U∗ is a semi-direct product by the multiplicative group Gm of a pro-unipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes–Moscovici  ...
The natural appearance of the “motivic Galois group” U∗ in the context of renormalization confirms a suggestion made by Cartier in , that in the Connes–Kreimer theory of perturbative renormalization one should find a hidden “cosmic Galois group” closely related in structure to the Grothendieck–Teichmüller group. The question of relations between the work of Connes–Kreimer, motivic Galois theory, and deformation quantization was further emphasized by Kontsevich in . At the level of the Hopf algebra of rooted trees, relations between renormalization and motivic Galois theory were also investigated by Goncharov in . The “motivic Galois group” U acts on the set of dimensionless coupling constants of physical theories, through the map of the corresponding group G to formal diffeomorphisms constructed in . This also realizes the hope formulated in  of relating concretely the renormalization group to a Galois group...
These facts altogether indicate that the divergences of Quantum Field Theory, far from just being an unwanted nuisance, are a clear sign of the presence of totally unexpected symmetries of geometric origin. This shows, in particular, that one should understand how the universal singular frame “renormalizes” the geometry of space-time using the Dim-Reg scheme and the universal counterterms
Let us immerse once more in the last 23.16 minutes of this conference:
Conference Non-Commutative Geometry & Space-Time,KITP Program: Mathematical Structures in String TheoryAlain Connes, (Nov 17, 2005)