Renormalization and Motivic Galois Theory
نویسنده
چکیده
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. When working with formal Laurent series over Q, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.
منابع مشابه
The Motivic Thom Isomorphism
The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Deligne’s ideas on motivic Galois groups.
متن کاملFrom Physics to Number theory via Noncommutative Geometry, II
We give here a comprehensive treatment of the mathematical theory of per-turbative renormalization (in the minimal subtraction scheme with dimensional regularization), in the framework of the Riemann–Hilbert correspondence and motivic Galois theory. We give a detailed overview of the work of Connes– Kreimer [31], [32]. We also cover some background material on affine group schemes, Tannakian ca...
متن کاملGalois Equivariance and Stable Motivic Homotopy Theory
For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full a...
متن کاملA remark on the motivic Galois group and the quantum coadjoint action
It was suggested in [Kon 1999] that the Grothendieck-Teichmüller group GT should act on the Duflo isomorphism of su(2) but the corresponding realization of GT turned out to be trivial. We show that a solvable quotient of the motivic Galois group which is supposed to agree with GT is closely related to the quantum coadjoint action on Uq (sl2) for q a root of unity, i.e. in the quantum group case...
متن کامل