An isomorphism of motivic galois groups sciencedirect. 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 di. The motivic galois group, the grothendieckteichmuller group and the double shuffle group. All feynman integrals are examples of periods, and as such conjecturally carry an action of the motivic galois group, or dually the coaction of the hopf algebra of functions on the motivic galois group. My aim is to formulate a precise conjecture about the structure of the galois group gal m tf of the category m tf of mixed tate motivic. Beyond that, not much more is known see rabelaiss answer.
The motivic fundamental group should unify these extra structures they all should be shadows of an action of the motivic fundamental group. This allows us to perform computations in the galois group more simply. We go through the formalism of grothendiecks six operations for these categories. This conjecture gives a new point of view on algebraic k theory. What are the different theories that the motivic fundamental. Forschungsseminar fs2016 motivic galois group and periods organizers.
M has conjecturally an algebraic group attached to it, called the motivic galois group of m see del90, saa72. K field of characteristic 0, a rigid tensor k linear abelian category, l extension of k. On the decomposition of motivic multiple zeta values brown, francis c. From physics to number theory via noncommutative geometry, ii. Inspired by rogness homotopical generalization of the theory of galois extensions to ring spectra, we develop here an analogous theory for motivic ring spaces and spectra, establish a number of important properties of motivic galois extensions, and provide concrete examples of motivic galois extensions. The motivic galois group is to the theory of motives what the mumfordtate group is to hodge theory. 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. Motivic homotopy, arithmetic invariants and absolute galois. Historically, perturbative renormalization has always appeared as one of the most elaborate recipes created by modern physics, capable of producing numer. The motivic fundamental group of p 1 0 1 and the theorem. This paper develops an application of motives and the motivic galois group to periods in. This also realizes the hope formulated in 6 of relating concretely the renormalization group to a galois group.
Motivic galois coaction and oneloop feynman graphs arxiv. The book by yves andre referenced there is maybe a good next step. In my last post, i discussed how the existence of a polarisation implies an upper bound for the transcendence degree of the extended period matrix of an abelian variety, namely the dimension of the general symplectic group where is the. Galois theoryhomologymotivic galois groupmotives the motivic galois group of an algebraic variety x is gal motx n linear transformations of h which preserve all classes of algebraic cycles in the tensor algebra o1 m0 h m o. This cosmic galois group g g is noncanonically isomorphic to some motivic galois group. Forschungsseminar fs2016 motivic galois group and periods. 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 di eomorphisms constructed in 10. The purpose of this paper is to give an affirmative. The main result is that all quantum field theories share a common universal symmetry realized as a motivic galois group, whose action is dictated by the divergences and generalizes that of the. We then give a complete account of our results on renormalization and motivic galois theory announced in 35.
Again speaking in rough terms, the hodge and tate conjectures are types of invariant theory the spaces that are morally the algebraic cycles are picked out by invariance under a group, if one sets up the correct definitions. Tate motives qn n 2 z are tate objects of the category. 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 di eomorphisms constructed in 10. The motivic cohomology group in question is that related, by beilinsons conjecture, to the adjoint lfunction at s 1. Motivic homotopical galois extensions sciencedirect. Using his theory, voevodsky was able to prove the milnor conjecture, relating the milnor ktheory of a. Fol lowing beilinson, deligne, grothendieck among others, there should be a qlinear. The latter can be controlled by the uniform open image theorem obtained in 2009 by. Let kbe a sub eld of c and let kbe the algebraic closure of kin. There is also a motivic galois group of mixed motives. We also cover some background material on affine group schemes, tannakian categories, the riemannhilbert problem in the regular singular and irregular case, and a brief introduction to motives and motivic galois theory. Motivic cohomology of x ucla department of mathematics. Grothendiecks motivic galois theory now can be described as.
So far, this does not use the action of the motivic galois group, only a bound on the size of the motivic periods of mt z. Gk galkk is the galois group of the extension of k given by its algebraic closure k. In order to do so voevodsky, along with morel, also introduced the. The renormalization group can be identified canonically with a one parameter subgroup. The motivic galois group, the grothendieckteichmuller. Furthermore, there are these things called motivic galois groups which are on the cutting edge of maths research. Goncharov this paper is an enlarged version of the lecture given at the ams conference \motives in seattle, july 1991. What is called the cosmic galois group is a motivic galois group that naturally acts on structures in renormalization in quantum field theory. See delignes corvallis talk and milnes second seattle talk same conference as serres article. Let dmtqq be the triangulated subcategory of dmqq gen.
Therefore the motivic galois group coincides with the mumfordtate group. In the sense of galois theory, that algebraic group is called the motivic galois group for pure motives. A guide to etale motivic sheaves joseph ayoub abstract. As far as the kernel and the cok ernel of the natural restriction map. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. A slightly weaker version of this question asks for a motive m such that the associated padic galois representations have algebraic monodromy group equal to the exceptional group in question.
In kitchloomorava 12 the cosmic galois group is related to the motivic tannakian group of a motivic stabilization of the symplectic category of symplectic manifolds and lagrangian correspondences between them, the stable symplectic category kitchloo 12. Motivic homotopy, arithmetic invariants and absolute. Lecture notes on motivic cohomology carlo mazza, vladimir voevodsky, charles a. Motives, motivic sheaves, motivic cohomology, grothendiecks six operations, conservativity conjecture, motivic tstructures. Compiled from notes taken independently by don zagier and herbert gangl, quickly proofread by the speaker. As a consequence, the motivic galois groups for andres motives are the maximal reductive quotients of the galois group that one obtains from noris categories.
That group is, or is closely related to, the group of algebraic periods, and as such is related to expressions appearing in deformation quantization and in renormalization in quantum field theory, whence it is also sometimes referred to. Glh x the conjugates of a period w r are then all periods. Motivic homotopy, arithmetic invariants and absolute galois groups one of the most spectacular works in mathematics in recent times is voevodskys proof of the milnor and blochkato conjectures, for which voevodsky was awarded the fields medal. Motivic galois groups and periods martin orrs blog. The relative picard group and suslins rigidity theorem 47 lecture 8. Braids, galois groups, and some arithmetic functions. However, as with any galois group of an algebraic extension, it is pro. That being said, the structure of the absolute galois group over the rationals which you can think of as a massive group comprised out of all galois groups of finite extensions of rationals, i.
Renormalization and motivic galois theory joint work with matilde marcolli 1. Renormalization and motivic galois theory joint work with. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Renormalization, the riemannhilbert correspondence, and. We recall the construction, following the method of morel and voevodsky, of the triangulated category of etale motivic sheaves over a base scheme. The absolute galois group over the rational number field q, denoted by gq galqq, is one of the classical mathematical objects which we really need to understand better. Polylogarithms and motivic galois groups yale math. The renormalization group can be identified canonically with a oneparameter subgroup of u. Motives, motivic galois groups and periods extended abstract. Gk lim galois theory, that algebraic group is called the motivic galois group for pure motives. Motivic galois coaction and oneloop feynman graphs. Following beilinson, deligne, grothendieck among others, there should be a qlinear abelian monoidal category mm k of mixed motives over k together with a monoidal functor m.
Since each automorphism in the galois group permutes the roots of 4. Here, the fundamental group refers to the unipotent motivic fundamental group in the sense of deligne 7. Examples of galois groups and galois correspondences. The galois group galmt f is by definition the semidirect prod uct gm. Most of the known and expected properties of motivic cohomology predicted inabs87andlic84canbedividedintotwofamilies. The motivic galois group, the grothendieckteichm graduate.
Our aim is to clarify the relation between galois groups and motivic galois groups in the context of andr es and noris categories of motives. Nevertheless, it can shown 9 that the motivic galois group constructed above is isomorphic to noris motivic galois group. The tensor product arises from the cartesian product of varieties. Jan 22, 2019 galois codescent for motivic tame kernels 3 as we will see in section 1, the signature map sgn f is trivial for i. Motives with exceptional galois groups and the inverse galois.
Although a rigorous construction of such an object for rational varieties has now. 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. An unconditional theory of motives for motivated cycles was developed by andr e and96. Motives with exceptional galois groups and the inverse. Var k op mm k, associating to each variety x k its motive m x, the universal cohomological invariant of x.
Maths abelian varieties periods of abelian varieties motivic galois groups and periods. The actual renormalization group is a 1parameter subgroup of the cosmic galois group in more detail, in connesmarcolli 04 the authors consider a differential equation satisfied by divergences appearing in the hopf algebra formulation of. Motivic galois coaction and oneloop feynman graphs matija tapuskovic abstract all feynman integrals are examples of periods, and as such conjecturally carry an action of the motivic galois group, or dually the coaction of the hopf algebra of functions on the motivic galois group. A remark on the motivic galois group and the quantum. The theory of motives is an attempt to linearise the study of algebraic varieties. In this paper, we study this structure in the case of feynman. In terms of motivic homotopy theory, motivic cohomology is just the analogue of singular cohomology.
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