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Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory

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Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory Alain Connes1 and Matilde Marcolli2 1 Coll`egedeFrance 3, rue Ulm F-75005 Paris and I.H.E.S. 35 route de Chartres F-91440 Bures-sur-Yvette France [email protected] 2 Max–Planck Institut f¨ur Mathematik Vivatsgasse 7 D-53111 Bonn Germany [email protected] 1 Introduction ..............................................618 2 Renormalization in Quantum Field Theory ................624 2.1 BasicformulasofQFT.......................................625 2.2 Feynman diagrams . 627 2.3 Divergences and subdivergences . 629 3 Affine group schemes......................................633 3.1 Tannakiancategories ........................................634 3.2 The Lie algebra and the Milnor-Moore theorem . 635 4 The Hopf algebra of Feynman graphs and diffeographisms 636 5 The Lie algebra of graphs .................................640 6 Birkhoff decomposition and renormalization...............642 7UnitofMass..............................................648 8 Expansional ...............................................651 9 Renormalization group ....................................654 10 Diffeographisms and diffeomorphisms .....................663 11 Riemann–Hilbert problem ................................665 11.1 Regular-singular case . 666 11.2 Local Riemann–Hilbert problem and Birkhoff decomposition . 668 618 Alain Connes and Matilde Marcolli 11.3 Geometric formulation . 669 11.4 Irregular case . 670 12 Local equivalence of meromorphic connections ............673 13 Classification of equisingular flat connections ..............676 14 The universal singular frame ..............................680 15 Mixed Tate motives .......................................683 15.1 Motives and noncommutative geometry: analogies . 686 15.2 Motivic fundamental groupoid . 686 15.3 Expansional and multiple polylogarithms . 689 16 The “cosmic Galois group” of renormalization as a motivic Galois group .................................690 17 The wild fundamental group ..............................695 18 Questions and directions ..................................700 18.1 Renormalization of geometries . 700 18.2 Nonperturbative effects . 702 18.3 The field of physical constants . 703 18.4 Birkhoff decomposition and integrable systems . 704 19 Further developments .....................................706 References .....................................................707 1 Introduction We give here a comprehensive treatment of the mathematical theory of pertur- bative 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 categories, the Riemann–Hilbert problem in the regular singular and irregular case, and a brief introduction to motives and motivic Galois theory. We then give a complete account of our results on renormal- ization and motivic Galois theory announced in [35]. Our main goal is to show how the divergences of quantum field theory, which may at first appear as the undesired effect of a mathematically ill- formulated theory, in fact reveal the presence of a very rich deeper mathemat- ical structure, which manifests itself through the action of a hidden “cosmic Galois group”3, which is of an arithmetic nature, related to motivic Galois theory. 3 The idea of a “cosmic Galois group” underlying perturbative renormalization was proposed by Cartier in [15]. Renormalization, the Riemann–Hilbert Correspondence 619 Historically, perturbative renormalization has always appeared as one of the most elaborate recipes created by modern physics, capable of producing numerical quantities of great physical relevance out of a priori meaningless mathematical expressions. In this respect, it is fascinating for mathematicians and physicists alike. The depth of its origin in quantum field theory and the precision with which it is confirmed by experiments undoubtedly make it into one of the jewels of modern theoretical physics. For a mathematician in quest of “meaning” rather than heavy formalism, the attempts to cast the perturbative renormalization technique in a concep- tual framework were so far falling short of accounting for the main computa- tional aspects, used for instance in QED. These have to do with the subtleties involved in the subtraction of infinities in the evaluation of Feynman graphs and do not fall under the range of “asymptotically free theories” for which constructive quantum field theory can provide a mathematically satisfactory formulation. The situation recently changed through the work of Connes–Kreimer ([29], [30], [31], [32]), where the conceptual meaning of the detailed computational devices used in perturbative renormalization is analysed. Their work shows that the recursive procedure used by physicists is in fact identical to a math- ematical method of extraction of finite values known as the Birkhoff decom- position, applied to a loop γ(z) with values in a complex pro-unipotent Lie group G. This result, and the close relation between the Birkhoff factorization of loops and the Riemann–Hilbert problem, suggested the existence of a geomet- ric interpretation of perturbative renormalization in terms of the Riemann– Hilbert correspondence. Our main result in this paper is to identify explicitly the Riemann–Hilbert correspondence underlying perturbative renormalization in the minimal subtraction scheme with dimensional regularization. Performing the Birkhoff (or Wiener-Hopf) decomposition of a loop γ(z) ∈ G consists of describing it as a product −1 γ (z)=γ−(z) γ+(z) z ∈ C, (1.1) of boundary values of holomorphic maps (which we still denote by the same symbol) γ± : C± → G. (1.2) defined on the connected components C± of the complement of the curve C in the Riemann sphere P1(C). The geometric meaning of this decomposition, for instance when G = GLn(C), comes directly from the theory of holomorphic bundles with structure group G on the Riemann sphere P1(C). The loop γ(z) describes the clutching data to construct the bundle from its local trivialization and the Birkhoff decomposition provides a global trivialization of this bundle. While in the case of GLn(C) the existence of a Birkhoff decomposition may be obstructed 620 Alain Connes and Matilde Marcolli ∞ C− C C + D Fig. 1. Birkhoff Decomposition by the non-triviality of the bundle, in the case of a pro-unipotent complex Lie group G, as considered in the CK theory of renormalization, it is always possible to obtain a factorization (1.1). In perturbative renormalization the points of P1(C) are “complex dimen- sions”, among which the dimension D of the relevant space-time is a preferred point. The little devil that conspires to make things interesting makes it im- possible to just evaluate the relevant physical observables at the point D,by letting them diverge precisely at that point. One can nevertheless encode all the evaluations at points z = D in the form of a loop γ(z) with values in the group G. The perturbative renormalization technique then acquires the following general meaning: while γ(D) is meaningless, the physical quantities are in fact obtained by evaluating γ+(D), where γ+ is the term that is holo- morphic at D for the Birkhoff decomposition relative to an infinitesimal circle with center D. Thus, renormalization appears as a special case of a general principle of extraction of finite results from divergent expressions based on the Birkhoff decomposition. The nature of the group G involved in perturbative renormalization was clarified in several steps in the work of Connes–Kreimer (CK). The first was Kreimer’s discovery [80] of a Hopf algebra structure underlying the recursive formulae of [7], [72], [112]. The resulting Hopf algebra of rooted trees depends on the physical theory T through the use of suitably decorated trees. The next important ingredient was the similarity between the Hopf algebra of rooted trees of [80] and the Hopf algebra governing the symmetry of transverse geometry in codimension one of [39], which was observed already in [29]. The particular features of a given physical theory were then better encoded by a Hopf algebra defined in [31] directly in terms of Feynman graphs. This Hopf algebra of Feynman graphs depends on the theory T by construction. It determines G as the associated affine group scheme, which is referred to as diffeographisms of the theory, G =Difg(T). Through the Milnor-Moore theorem [93], the Hopf algebra of Feynman graphs determines a Lie algebra, Renormalization, the Riemann–Hilbert Correspondence 621 whose corresponding infinite dimensional pro-unipotent Lie group is given by the complex points G(C) of the affine group scheme of diffeographisms. This group is related to the formal group of Taylor expansions of diffeo- morphisms. It is this infinitesimal feature of the expansion that accounts for the “perturbative” aspects inherent to the computations of Quantum Field Theory. The next step in the CK theory of renormalization is the construction of an action of Difg(T) on the coupling constants of the physical theory, which shows a close relation between Difg(T) and the group of diffeomorphisms of the space of Lagrangians. In particular, this allows one to lift the renormalization group to a one parameter subgroup of Difg, defined intrinsically from the independence of the term γ−(z) in the Birkhoff decomposition from the choice of an additional mass scale
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