The book by yves andre referenced there is maybe a good next step. Grothendiecks motivic galois theory now can be described as. This paper develops an application of motives and the motivic galois group to periods in. Let dmtqq be the triangulated subcategory of dmqq gen. Since each automorphism in the galois group permutes the roots of 4. 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. Examples of galois groups and galois correspondences. 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. 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.
This allows us to perform computations in the galois group more simply. 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. 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. The relative picard group and suslins rigidity theorem 47 lecture 8. Vectk and consequently to an unconditional motivic galois group gal dr. Motivic homotopical galois extensions sciencedirect. Renormalization, the riemannhilbert correspondence, and. The motivic fundamental group should unify these extra structures they all should be shadows of an action of the motivic fundamental group. Here, the fundamental group refers to the unipotent motivic fundamental group in the sense of deligne 7. The motivic galois group, the grothendieckteichmuller group and the double shuffle group. The tensor product arises from the cartesian product of varieties. An unconditional theory of motives for motivated cycles was developed by andr e and96.
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. Motivic galois groups and periods martin orrs blog. Var k op mm k, associating to each variety x k its motive m x, the universal cohomological invariant of x. We recall the construction, following the method of morel and voevodsky, of the triangulated category of etale motivic sheaves over a base scheme. In terms of motivic homotopy theory, motivic cohomology is just the analogue of singular cohomology.
Tate motives qn n 2 z are tate objects of the category. Braids, galois groups, and some arithmetic functions. The theory of motives is an attempt to linearise the study of algebraic varieties. There is also a motivic galois group of mixed motives. A a generalization to several polynomials in several variables, that is to higher dimen. What are the different theories that the motivic fundamental. The galois group galmt f is by definition the semidirect prod uct gm. M has conjecturally an algebraic group attached to it, called the motivic galois group of m see del90, saa72. 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. Beyond that, not much more is known see rabelaiss answer.
In the sense of galois theory, that algebraic group is called the motivic galois group for pure motives. Motives, motivic sheaves, motivic cohomology, grothendiecks six operations, conservativity conjecture, motivic tstructures. This is the paper where the motivic hopf algebra and the motivic galois group are constructed using voevodskys motives. Motives with exceptional galois groups and the inverse galois. 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. The motivic galois group, the grothendieckteichmuller. The motivic galois group, the grothendieckteichm graduate. An isomorphism of motivic galois groups sciencedirect. Fol lowing beilinson, deligne, grothendieck among others, there should be a qlinear. Forschungsseminar fs2016 motivic galois group and periods organizers. Lecture notes on motivic cohomology carlo mazza, vladimir voevodsky, charles a.
K field of characteristic 0, a rigid tensor k linear abelian category, l extension of k. As far as the kernel and the cok ernel of the natural restriction map. Advanced studies in pure mathematics project euclid. Gk lim galois theory, that algebraic group is called the motivic galois group for pure motives. 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. 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. 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. 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. Goncharov this paper is an enlarged version of the lecture given at the ams conference \motives in seattle, july 1991. The motivic galois group is to the theory of motives what the mumfordtate group is to hodge theory. 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.
Polylogarithms and motivic galois groups yale math. Motivic cohomology of x ucla department of mathematics. 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 and motivic galois theory joint work with matilde marcolli 1. Motives, motivic galois groups and periods extended abstract. Maths abelian varieties periods of abelian varieties motivic galois groups and periods. We go through the formalism of grothendiecks six operations for these categories. The renormalization group can be identified canonically with a one parameter subgroup. 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. Historically, perturbative renormalization has always appeared as one of the most elaborate recipes created by modern physics, capable of producing numer. This conjecture gives a new point of view on algebraic k theory. See delignes corvallis talk and milnes second seattle talk same conference as serres article. 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.
However, as with any galois group of an algebraic extension, it is pro. 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. On the decomposition of motivic multiple zeta values brown, francis c. Although a rigorous construction of such an object for rational varieties has now. Renormalization and motivic galois theory request pdf. We present evidence for the conjecture using the theory of periods of automorphic. 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. 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. What is called the cosmic galois group is a motivic galois group that naturally acts on structures in renormalization in quantum field theory. Let kbe a sub eld of c and let kbe the algebraic closure of kin.
The latter can be controlled by the uniform open image theorem obtained in 2009 by. Motivic galois coaction and oneloop feynman graphs arxiv. 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. Therefore the motivic galois group coincides with the mumfordtate group. We then give a complete account of our results on renormalization and motivic galois theory announced in 35.
The purpose of this paper is to give an affirmative. The renormalization group can be identified canonically with a oneparameter subgroup of u. From physics to number theory via noncommutative geometry, ii. This also realizes the hope formulated in 6 of relating concretely the renormalization group to a galois group. Compiled from notes taken independently by don zagier and herbert gangl, quickly proofread by the speaker. Motivic galois coaction and oneloop feynman graphs. 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.
Glh x the conjugates of a period w r are then all periods. 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. In order to do so voevodsky, along with morel, also introduced the. In this paper, we study this structure in the case of feynman.
Motivic homotopy, arithmetic invariants and absolute. 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. The motivic cohomology group in question is that related, by beilinsons conjecture, to the adjoint lfunction at s 1. Nevertheless, it can shown 9 that the motivic galois group constructed above is isomorphic to noris motivic galois group. The motivic fundamental group of p 1 0 1 and the theorem. This cosmic galois group g g is noncanonically isomorphic to some motivic galois group. Furthermore, there are these things called motivic galois groups which are on the cutting edge of maths research. A remark on the motivic galois group and the quantum. Gk galkk is the galois group of the extension of k given by its algebraic closure k. 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. Motives with exceptional galois groups and the inverse. Renormalization and motivic galois theory joint work with. Using his theory, voevodsky was able to prove the milnor conjecture, relating the milnor ktheory of a.
819 1476 455 1481 37 956 811 533 932 824 989 1260 198 56 498 525 930 91 835 59 352 1353 330 394 1444 1084 411 350 1581 1248 1074 539 839 176 305 1154 52 601 1100