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Frontiers in Number Theory, Physics, and Geometry II On Conformal Field Theories, Discrete Groups and Renormalization ABC... Volume I contains the contributions onthree broad topics: Ran

Frontiers in Number Theory, Physics, and Geometry II

Pierre Cartier Bernard Julia

Pierre Moussa Pierre Vanhove (Eds.)

Frontiers in Number Theory, Physics, and Geometry II

On Conformal Field Theories, Discrete Groups and Renormalization

ABC

F-91191 Gif-sur-Yvette France

e-mail: moussa@spht.saclay.cea.fr

Pierre VanhoveService de Physique Théorique CEA/Saclay

F-91191 Gif-sur-Yvette France

e-mail: pierre.vanhove@cern.ch

Cover photos:

Richard Feynman (courtesy of AIP Emilio Segre Visual Archives, Weber Collection);John von Neumann

Library of Congress Control Number: 2005936349

Mathematics Subject Classification (2000): 11F03, 11F06, 11G55, 11M06, 15A90,16W30, 57T05, 58B34, 81R60, 81T16, 81T17, 81T30, 81T40, 81T75, 81R05

ISBN-10 3-540-30307-3 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-30307-7 Springer Berlin Heidelberg New York

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The present book collects most of the courses and seminars delivered at themeeting entitled “ Frontiers in Number Theory, Physics and Geometry”, whichtook place at the Centre de Physique des Houches in the French Alps, March 9-

21, 2003 It is divided into two volumes Volume I contains the contributions onthree broad topics: Random matrices, Zeta functions and Dynamical systems.The present volume contains sixteen contributions on three themes: Conformalfield theories for strings and branes, Discrete groups and automorphic formsand finally, Hopf algebras and renormalization

The relation between Mathematics and Physics has a long history Let usmention only ordinary differential equations and mechanics, partial differentialequations in solid and fluid mechanics or electrodynamics, group theory is

essential in crystallography, elasticity or quantum mechanics

The role of number theory and of more abstract parts of mathematicssuch as topological, differential and algebraic geometry in physics has becomeprominent more recently Diverse instances of this trend appear in the works

of such scientists as V Arnold, M Atiyah, M Berry, F Dyson, L Faddeev,

D Hejhal, C Itzykson, V Kac, Y Manin, J Moser, W Nahm, A Polyakov,

D Ruelle, A Selberg, C Siegel, S Smale, E Witten and many others

In 1989 a first meeting took place at the Centre de Physique des Houches.The triggering idea was due at that time to the late Claude Itzykson (1938-1995) The meeting gathered physicists and mathematicians, and was theoccasion of long and passionate discussions

The seminars were published in a book entitled “Number Theory andPhysics”, J.-M Luck, P Moussa, and M Waldschmidt editors, Springer Pro-ceedings in Physics, Vol 47, 1990 The lectures were published as a second

The Scientific Committee for the meeting “Frontiers in Number ory, Physics and Geometry”, was composed of the following scientists: FritsBeukers, Jean-Benoˆıt Bost, Pierre Cartier, Predrag Cvitanovic, Michel Duflo,Giovanni Gallavotti, Patricio Leboeuf, Werner Nahm, Ivan Todorov, ClaireVoisin, Michel Waldschmidt, Jean-Christophe Yoccoz, and Jean-Bernard Zu-ber.

The-The Organizing Committee included:

Bernard Julia (LPTENS, Paris scientific coordinator),

Pierre Moussa (SPhT CEA-Saclay), and

Pierre Vanhove (CERN and SPhT CEA-Saclay)

During two weeks, five lectures or seminars were given every day to aboutseventy-five participants The topics belonged to three main domains:

1 Dynamical Systems, Number theory, and Random matrices,

with lectures by E Bogomolny on Quantum and arithmetical chaos, J Conrey

on L-functions and random matrix theory, J.-C Yoccoz on Interval exchangemaps, and A Zorich on Flat surfaces;

2 Polylogarithms and Perturbative Physics,

with lectures by P Cartier on Polylogarithms and motivic aspects, W Nahm

on Physics and dilogarithms, and D Zagier on Polylogarithms;

3 Symmetries and Non-perturbative Physics,

with lectures by A Connes on Galoisian symmetries, zeta function and malization, R Dijkgraaf on String duality and automorphic forms, P Di Vec-chia on Gauge theory and D-branes, E Frenkel on Vertex algebras, algebraiccurves and Langlands program, G Moore on String theory and number theory,

renor-C Soul´e on Arithmetic groups

In addition seminars were given by participants many of whom could havegiven full sets of lectures had time been available They were: Z Bern, A.Bondal, P Candelas, J Conway, P Cvitanovic, H Gangl, G Gentile, D.Kreimer, J Lagarias, M Marcolli, J Marklof, S Marmi, J McKay, B Pioline,

M Pollicott, H Then, E Vasserot, A Vershik, D Voiculescu, A Voros, S.Weinzierl, K Wendland and A Zabrodin

Preface VII

We have chosen to reorganize the written contributions in two halves cording to their subject This naturally led to two different volumes Thepresent one is the second volume, let us now briefly describe its contents.This volume is itself composed of three parts including each lectures andseminars covering one theme In the first part, we present the contributions onthe theme “Conformal Field Theories (CFT’s) for Strings and Branes” Theybegin with two intertwined sets of lectures by Don Zagier and by Werner Nahmwho have had a long personal interaction at the modular border betweenMathematics and Physics

ac-The presentation by Don Zagier starts with a review of the properties ofEuler’s dilogarithm and of its associated real Bloch-Wigner function Thesefunctions have generalizations to polylogarithms and to some real functionsdefined by Ramakrishnan respectively Their importance in Hyperbolic 3-

geometry, in Algebraic K 2m −1-theory (Bloch group) and their relation to

val-ues of Dedekind zeta functions (see volume I) at argument m are explained On

the other hand the modular group appears to be mysteriously related to theBloch-Wigner function and its first Ramakrishnan generalization The secondchapter of these lectures introduces yet more variants, in particular the Rogersdilogarithm and the enhanced dilogarithm which appear in W Nahm’s lec-tures, the quantum dilogarithm as well as the multiple (poly)logarithms whichdepend on more than one argument Their properties are reviewed, in par-ticular functional equations, relations with modular forms (see also the nextcontribution), special values and again (higher) K-theory

In his lectures on “CFT’s and torsion elements of the Bloch group” WernerNahm expresses the conformal dimensions of operators in the (discrete) series

of rational two dimensional (2d) Conformal Field Theories as the imaginarypart of the Rogers dilogarithm of torsion elements from algebraic K-theory

of the complex number field The lectures begin with a general tion to conformally invariant quantum field theories or more precisely with aphysicist’s conceptual presentation of Vertex operator algebras The ”ratio-nal” theories form a rare subset in the moduli space of CFT’s but one mayconsider perturbations thereof within the set of totally integrable quantumfield theories The following step is to present a bird’s eye view of totally in-tegrable two dimensional quantum field theories and to relate in simple casesthe scattering matrix to Cartan matrices of finite dimensional Lie algebras, inparticular integrality of the coefficients follows from Bose statistics and posi-tivity from the assumed convergence of partition functions, there are naturalextensions to arbitrary statistics

introduc-Then Nahm conjectures and illustrates on many examples that the ular” invariance of the chiral characters of a rational CFT admitting a totallyintegrable perturbation implies that all solutions to the integrability condi-tions (Bethe equations) define pure torsion elements in the (extended) Blochgroup of the complex field The perturbations that can be analyzed are de-fined by pairs of Cartan matrices of A D E or T type In fact Nahm gives

“mod-the general solution of “mod-the torsion equation for deformations of (A , A ) type

in-of perturbative quantum gauge theories This structure has been discussed

since by Deligne, Landsberg and Manivel It is different from and does

not seem related to similar magic triangles of dualities that contain also themagic square of Tits and Freudenthal in specific real forms and which appear

in supergravity and superstring models

The third series of lectures: “Gauge theories from D-branes”, were livered by Paolo Di Vecchia and written up with Antonella Liccardo Theyprovide an introduction to string models and the associated D(irichlet)-branes

de-on which open strings may end and they explain the emergence of Yang-Millsgauge theories on these extended objects They bridge the gap between 2dCFT’s and physical models in higher dimensions Perturbative string theoriesare particular conformal field theories on the string worldsheet Most nonper-turbative effects in string theory necessitate the inclusion of extended objects

of arbitrary spatial dimension p: the p-branes and in particular the Dirichlet

D p branes Branes allow the computation of the entropy of black holes andpermit new dualities between gauge and gravitational theories For instancethe celebrated AdS/CFT duality relates a closed string theory on the product

manifold S5× AdS5 to an open string theory ending on a D3 brane Theselectures start from the worldsheet description of perturbative superstring the-ory with its BRST invariant (string creation) vertex operators and proceed todescribe the “boundary state formalism” that describes the coupling of closedstrings to D branes Then the authors use the latter to compute the interac-tion between two D-branes, they discuss so-called BPS configurations whoseinteractions vanish and relate the low energy effective Born-Infeld interactions

of massless strings to their couplings to D branes

One seminar by Katrin Wendland concludes this part: “Superconformalfield theories associated to very attractive quartics” The terminology “at-tractive” was introduced by Greg Moore (see his lectures below) for thoseCalabi-Yau two-folds whose Picard group is of maximal rank, very attractive

is a further restriction on the transcendental lattice This is a review on thegeometrical realization of orbifold models on quartic surfaces and providessome motivation for reading the following chapters

In the second part: “Discrete groups and automorphic forms”, the theme isarithmetic groups and some of their applications Christophe Soul´e ’s lectures

“Introduction to arithmetic groups” set the stage in a more general contextthan was considered in the lectures by E Bogomolny in volume I of thisbook They begin with the classical reduction theory of linear groups of ma-trices with integral coefficients and the normal parameterization of quadraticforms Then follows the general (and intrinsic) theory of algebraic Lie groupsover the rationals and of their arithmetic subgroups; the finite covolume prop-erty in the semi-simple case at real points is derived, it may be familiar in the

Preface IXphysics of chaos The second chapter deals with presentations and finite ortorsion free and finite index subgroups The third chapter deals with rigidity:the congruence subgroup property in rank higher than one, Kazhdan’s prop-erty T about invariant vectors and results of Margulis in particular the proof

of the Selberg conjecture that arithmeticity follows from finite covolume formost simple non-compact Lie groups Automorphic forms are complex val-ued functions defined over symmetric domains and invariant under arithmeticgroups, they arise abundantly in string theory

Boris Pioline expanded his seminar with Andrew Waldron to give a cists’ introduction to “Automorphic forms and Theta series” It starts with thegroup theoretical and adelic expression of non holomorphic Eisenstein series

physi-like E 3/2 which has been extensively studied by M.B Green and his rators and also theta series From there one studies examples of applications

collabo-of the orbit method and collabo-of parabolic induction Among recent applicationsand beyond the discrete U-duality groups already considered in the previouslectures they discuss the minimal representation of SO(4,4) which arises also

in string theory, the E6 exceptional theta series expected to control the permembrane interactions after compactification from 11 to 8 dimensions on

su-a torus, new symmetries of chsu-aotic cosmology su-and lsu-ast but not lesu-ast work inprogress on the description of black hole degeneracies and entropy computa-tions M-theory is the name of the unifying, hypothetical and polymorphictheory that admits limits either in a flat classical background 11-dimensionalspacetime with membranes as fundamental excitations, in 10 dimensions with

strings and branes as building blocks etc .

Gregory Moore wrote up two of his seminars on “Strings and arithmetic”(the third one on the topological aspects of the M-theory 3-form still leads

to active research and new developments) The first topics he covers is the

Black hole’s Farey tail, namely an illustration of the AdS3× S3× K3 duality

with a two dimensional CFT on the boundary of three dimensional Sitter space One can compute the elliptic genus of that CFT as a Poincar´eseries that is interpretable on the AdS (i.e gravity or string) side as a sum

anti-de-of particle states and black hole contributions This can serve as a concreteintroduction to many important ideas on Jacobi modular forms, Rademacherexpansion and quantum corrections to the entropy of black holes

The second chapter of Moore’s lectures deals with the so called attractormechanism of supergravity After compactification on a Calabi-Yau 3-fold Xone knows that its complex structure moduli flow to a fixed point if one ap-proaches the horizon of a black hole solution This attractor depends on thecharges of the black hole which reach there a particular Hodge decomposition

In the special case of X = K3 × T2 one obtains the notion of attractive K3

already mentioned The main point here is that the attractors turn out to

be arithmetic varieties defined over number fields, their periods are in factvalued in quadratic imaginary fields Finally two more instances of the impor-tance of attractive varieties are presented Firstly the 12 dimensional so-called

“F-theory” compactified on a K3 surface is argued to be dual (equivalent) to

X Preface

heterotic string theory compactified modulo a two-dimensional CFT also down

to 8 dimensions It is striking that this CFT is rational if and only the K3surface is attractive Secondly string theory compactification with fluxes turnsout to be related to attractive Calabi-Yau 4-folds

The next contribution is a seminar talk by Matilde Marcolli on chaotic(mixmaster model) cosmology in which she relates a geodesics on the mod-

ular curve for the congruence subgroup Γ0(2) to a succession of Kasner fourdimensional spacetimes The moduli space of such universes is highly singularand amenable to description by noncommutative geometry andC∗ algebras.

John McKay and Abdellah Sebbar introduce the concept and six possibleapplications of “Replicable functions” These are generalizations of the elliptic

modular j function that transform under their Faber polynomials as

general-ized Hecke sums involving their “replicas” In any case they encompass alsothe monstrous moonshine functions and are deeply related to the Schwarzianderivative which appears in the central generator of the Virasoro algebra.Finally part II ends with the lectures by Edward Frenkel “On the Lang-lands program and Conformal field theory” As summarized by the authorhimself they have two purposes, first of all they should present primarily tophysicists the Langlands program and especially its “geometric” part but onthe other hand they should show how two-dimensional Conformal Field Theo-ries are relevant to the Langlands program This is becoming an important ac-tivity in Physics with the recognition that mathematical (Langlands-)duality

is deeply related to physical string theoretic S-duality in the recent works of

A Kapustin and E Witten, following results on magnetic monopoles fromthe middle seventies and the powerful tool of topological twists of supersym-metric theories which help to connect N = 4 super Yang-Mills theory in 4dimensions to virtually everything else The present work is actually aboutmirror symmetry (T-duality) of related 2d supersigma models

Specifically the lectures begin with the original Langlands program andcorrespondences in the cases of number fields and of function fields TheTaniyama-Shimura-Weil (modular) conjecture (actually a theorem now) isdiscussed there The geometric Langlands program is presented next in theabelian case first and then for an arbitrary reductive group G The goal is

to generalize T duality or Fourier-Mukai duality to the non abelian situation.Finally the conformal blocks are introduced for CFT’s, some theories of affineKac-Moody modules are introduced; at the negative critical level of the Kac-Moody central charge the induced conformal symmetry degenerates and thesemodels lead to the Hecke eigensheaves expected from the geometric Langlandscorrespondence

The third and last theme of this volume is “Hopf algebras and ization” It leads to promising results on renormalization of Quantum FieldTheories that can be illustrated by concrete perturbative diagrammatic com-putations but it also leads to the much more abstract and conceptual idea

renormal-of motives like a wonderful rainbow between the ground and the sky In thefirst set of lectures Pierre Cartier reviews the historical emergence of Hopf

Preface XIalgebras from topology and their structure theorems He then proceeds toHopf algebras defined from Lie groups or Lie algebras and the inverse struc-ture theorems He finally turns to combinatorics instances of Hopf algebrasand some applications, (quasi)-symmetric functions, multiple zeta values andfinally multiple polylogarithms This long and pedagogical introduction couldhave continued into motives so we may be heading towards a third les Houchesschool in this series.

Then comes the series of lectures by Alain Connes; they were written up incollaboration with Matilde Marcolli The lectures contain the most up-to dateresearch work by the authors, including a lot of original material as well as thebasic material in this exciting subject They have been divided into two parts.Chapter one appeared in the first volume and covered: “Quantum statisticalmechanics ofQ-lattices” in dimensions 1 and 2 The important dilation opera-tor (scaling operator) that determined the dynamics there reappears naturally

as the renormalization group flow in their second chapter contained in thisvolume with the title: “Renormalization, the Riemann-Hilbert correspondenceand motivic Galois theory” It starts with a detailed review of the results ofConnes and Kreimer on perturbative renormalization in quantum field theoryviewed as a Riemann-Hilbert problem and presents the Hopf algebra of Feyn-man graphs which corresponds by the Milnor-Moore theorem to a graded Liealgebra spanned by 1PI graphs Singular cases lead to formal series and theconvergence aspects are briefly discussed towards the end

The whole program is reformulated using the language of categories, gebraic groups and differential Galois theory Possible connections to mixedTate motives are discussed The equivariance under the renormalization group

al-is reformulated in thal-is language Finally various tantalizing developments areproposed

Dirk Kreimer discusses then the problem of “Factorization in quantumfield theory: an exercise in Hopf algebras and local singularities” He actuallytreats a toy model of decorated rooted trees which captures the essence ofthe resolution of overlapping divergences One learns first how the Hochschildcohomology of the Hopf algebra permits the renormalization program with

“locality” Dyson-Schwinger equations are then defined irrespective of anyaction and should lead to a combinatorial factorization into primitives of thecorresponding Hopf algebra

Stefan Weinzierl in his seminar notes explains some properties of multiplepolylogarithms and of their finite truncations (nested sums called Z-sums)that occur in Feynman loop integrals: “Algebraic algorithms in perturbativecalculations” and their impact on searches for new physics Emphasis is onanalytical computability of some Feynman diagrams and on algebraic struc-tures on Z-sums They have a Hopf algebra structure as well as a conjugationand a convolution product, furthermore the multiple polylogarithms do have

a second Hopf algebra structure of their own with a shuffle product

Finally this collection ends with a pedagogical exposition by HerbertGangl, Alexander B Goncharov and Andrey Levin on “Multiple logarithms,

XII Preface

algebraic cycles and trees” This work has been extended to multiple arithms and to the world of motives by the same authors Here they relatethe three topics of their title among themselves, the last two are associated todifferential graded algebras of algebraic cycles and of decorated rooted treeswhereas the first one arises as an integral on hybrid cycles as a generalization

polylog-of the mixed Tate motives polylog-of Bloch and Kriz in the case polylog-of the (one-variable)(poly-)logarithms

We acknowledge most gratefully for their generous financial support to themeeting the following institutions:

D´epartement Sciences Physiques et Math´ematiques and the Service deFormation permanente of the Centre National de la Recherche Scientifique;

´

Ecole Normale Sup´erieure de Paris; D´epartement des Sciences de la mati`ere duCommissariat `a l’´Energie Atomique; Institut des Hautes Etudes Scientifiques;National Science Foundation; Minist`ere de la Recherche et de la Technolo-gie and Minist`ere des Affaires ´Etrang`eres; The International Association ofMathematics and Physics and most especially the Commission of the Euro-pean Communities

Three European excellence networks helped also in various ways Let

us start with the most closely involved “Mathematical aspects of Quantumchaos”, but the other two were “Superstrings” and “Quantum structure ofspacetime and the geometric nature of fundamental interactions”

On the practical side we thank CERN Theory division for allowing us

to use their computers for the webpage and registration process We are alsograteful to Marcelle Martin, Thierry Paul and the staff of les Houches for theirpatient help We had the privilege to have two distinguished participants:C´ecile de Witt-Morette (founder of the Les Houches School) and the lateBryce de Witt whose communicative and critical enthusiasm were greatlyappreciated

P Moussa

P Vanhove

Pr´ eface aux deux volumes du livre

“Fronti` eres entre Th´ eorie des Nombres,

Physique et G´ eom´ etrie”

Ce livre rassemble la plupart des cours et s´eminaires pr´esent´es pendant un stitut de printemps sur les: “Fronti`eres entre Th´eorie des Nombres, Physique

In-et G´eom´etrie” qui s’est tenu au Centre de Physique des Houches dans lesAlpes fran¸caises du 9 au 31 Mars 2003 Il comprend deux volumes Le pre-mier volume contient quinze contributions dans trois grands domaines: Ma-trices al´eatoires, Fonctions zˆeta puis Syst`emes dynamiques Ce second volumecontient, quant `a lui, seize contributions r´eparties ´egalement en trois th`emes:Th´eories conformes pour les Cordes et les Branes, Groupes discrets et Formesautomorphes et enfin Alg`ebres de Hopf et Renormalisation

Les relations entre Math´ematiques et Physique ont une longue histoire Ilsuffit de rappeler la m´ecanique et les ´equations diff´erentielles ordinaires, les

L Faddeev, D Hejhal, C Itzykson, V Kac, Y Manin, J Moser, W Nahm, A.Polyakov, D Ruelle, A Selberg, C Siegel, S Smale, E Witten et beaucoupd’autres

Une premi`ere conf´erence de ce type se tint en 1989 au Centre de Physiquedes Houches L’id´ee en ´etait venue alors `a Claude Itzykson (1938-1995) Cetterencontre qui rassembla math´ematiciens et physiciens th´eoriciens donna lieu

`

a des discussions longues et passionn´ees

Les s´eminaires parurent dans un volume intitul´e “Th´eorie des nombres

et Physique” ´edit´e par J.-M Luck, P Moussa et M Waldschmidt, SpringerProceedings in Physics, Vol 47, 1990 Quant aux cours, ils furent publi´es dans

un volume s´epar´e intitul´e, lui, “De la Th´eorie des nombres `a la Physique” C.Itzykson ayant alors rejoint l’´equipe ´editoriale, Springer (2`eme ´edition 1995)

Une fois acquis le soutien de la Communaut´e europ´eenne l’organisation decette “High Level Scientific Conference” aux Houches fut lanc´ee.

Le Comit´e Scientifique de la conf´erence “Fronti`eres entre Th´eorie des bres, Physique et G´eom´etrie” ´etait compos´e des scientifiques suivants: FritsBeukers, Jean-Benoˆıt Bost, Pierre Cartier, Predrag Cvitanovic, Michel Duflo,Giovanni Gallavotti, Patricio Leboeuf, Werner Nahm, Ivan Todorov, ClaireVoisin, Michel Waldschmidt, Jean-Christophe Yoccoz, et Jean-Bernard Zu-ber

Nom-Le Comit´e d’Organisation comprenait:

Bernard Julia (LPTENS, Paris - coordinateur scientifique),

Pierre Moussa (SPhT CEA-Saclay), et

Pierre Vanhove (CERN et SPhT CEA-Saclay)

Pendant deux semaines, cinq cours ou s´eminaires furent pr´esent´es chaquejour `a environ soixante-quinze participants Les sujets avaient ´et´e initialementordonn´es en trois groupes successifs avec comme pr´eoccupation essentielle decoupler autant que faire se pouvait les cours de math´ematiques et ceux dephysique:

1 Syst`emes Dynamiques, Th´eorie des Nombres et Matrices al´eatoires,avec des cours de E Bogomolny sur le Chaos quantique arithm´etique, de

B Conrey sur les fonctions L et la Th´eorie des matrices al´eatoires, de

J.-C Yoccoz sur les Echanges d’intervalles et de A Zorich sur les Surfaces plates;

2 Polylogarithmes et Physique perturbative,

avec des cours de P Cartier sur les Polylogarithmes et leurs aspects tiviques, de W Nahm sur la Physique et les Dilogarithmes, et de D Zagiersur les Polylogarithmes;

mo-3 Sym´etries et Physique non-perturbative,

avec des cours de A Connes sur les Sym´etries Galoisiennes, Fonction zˆeta etRenormalisation, R Dijkgraaf, Dualit´e en th´eorie des cordes et Formes auto-morphes, P Di Vecchia, Th´eories de jauge et D-branes, E Frenkel, Alg`ebres devertex, Courbes alg´ebriques et Programme de Langlands, G Moore, Th´eoriedes cordes et Th´eorie des nombres, C Soul´e, Groupes arithm´etiques

Nombreux sont les participants qui ont donn´e des s´eminaires et qui raient pu donner des cours si le temps n’avait manqu´e Ont donc parl´e: Z Bern,

au-A Bondal, P Candelas, J Conway, P Cvitanovic, H Gangl, G Gentile, D.Kreimer, J Lagarias, M Marcolli, J Marklof, S Marmi, J McKay, B Pio-line, M Pollicott, H Then, E Vasserot, A Vershik, D Voiculescu, A Voros,

S Weinzierl, K Wendland et A Zabrodin

Pr´eface XVNous avons d´ecid´e de r´earranger les contributions ´ecrites `a ces Actes endeux volumes dont voici le contenu.

Le premier volume rassemble quinze contributions et se compose de troisparties regroupant chacune les cours et les s´eminaires relatifs `a un th`eme.Dans la premi`ere partie nous pr´esentons les contributions sur les: “Matri-ces al´eatoires: de la Physique `a la Th´eorie des nombres” Elle commencepar le cours d’Eug`ene Bogomolny qui passe en revue trois aspects du chaosquantique, `a savoir les formules de trace avec ou sans chaos, la fonction decorr´elation spectrale `a deux points des z´eros de la fonction zˆeta de Riemann etenfin les fonctions de corr´elation spectrales de l’op´erateur de Laplace-Beltramipour des domaines modulaires sujets au chaos arithm´etique Ces expos´esforment une introduction informelle aux m´ethodes math´ematiques du chaosquantique Une introduction plus g´en´erale aux groupes arithm´etiques est pro-pos´ee par Christophe Soul´e dans le deuxi`eme volume Suivent les le¸cons deBrian Conrey qui analyse les relations entre la th´eorie des matrices al´eatoires

et les familles de fonctions L (essentiellement en caract´eristique z´ero), doncdes s´eries de Dirichlet qui ob´eissent `a une ´equation fonctionnelle similaire `acelle que satisfait la fonction zˆeta de Riemann Les fonctions L consid´er´eessont celles qui sont associ´ees `a des formes paraboliques Les moments desfonctions L sont reli´es aux fonctions de corr´elation des valeurs propres dematrices al´eatoires

Nous avons rassembl´e ensuite les textes de plusieurs s´eminaires: celui deJens Marklof reliant la statistique de certains niveaux d’´energie `a des fonc-tions “presque modulaires”; celui de Holger Then sur le chaos quantiquearithm´etique dans un certain domaine hyperbolique `a trois dimensions etson lien avec des formes de Maass; puis Paul Wiegmann et Anton Zabrodin

´

etudient le d´eveloppement pour N grand d’ensembles de matrices complexesnormales; Dan Voiculescu passe en revue les sym´etries des mod`eles de Proba-bilit´es libres; finalement Anatoly Vershik pr´esente des graphes et des espacesm´etriques al´eatoires (universels)

Le th`eme de la deuxi`eme partie est: “‘Fonctions Zˆeta et applications” Lesexpos´es d’Alain Connes ont ´et´e distribu´es en deux chapitres, un par volume.Ils ont ´et´e r´edig´es avec Matilde Marcolli Ils contiennent les derniers r´esultats

de recherche des deux auteurs, de nombreux r´esultats originaux mais aussi lesbases de ce sujet excitant On trouve dans le volume II leur deuxi`eme chapitresur la Renormalisation des th´eories quantiques des champs Dans le premierchapitre A Connes et M Marcolli introduisent l’espace non commutatif desclasses de commensurabilit´e desQ-r´eseaux et les propri´et´es arithm´etiques des

´

etats KMS dans le syst`eme de M´ecanique statistique quantique correspondant.Pour les r´eseaux de dimension un cela conduit `a une r´ealisation spectrale desz´eros de fonctions zˆeta Dans le cas de dimension deux on peut d´ecrire les mul-tiples transitions de phase et la brisure spontan´ee de la sym´etrie arithm´etique

A temp´erature nulle le syst`eme tombe sur une vari´et´e classique (i.e tive) de Shimura qui param´etrise ses ´etats d’´equilibre L’espace non commu-tatif a une structure arithm´etique qui provient d’une sous-alg`ebre rationnelle

On trouvera dans cette partie le s´eminaire d’Andr´e Voros sur des fonctions

zˆeta construites `a l’aide des z´eros de la fonction zˆeta de Riemann, celui deJeffrey Lagarias sur les espaces de Hilbert de fonctions enti`eres attach´es auxfonctions L de Dirichlet Cette partie s’ach`eve avec l’expos´e de Mark Pollicottsur les fonctions zˆeta dynamiques et les orbites ferm´ees des flots g´eod´esiques

et hyperboliques

La troisi`eme partie s’intitule “Syst`emes dynamiques: Echanges d’intervalles,Surfaces plates et Petits diviseurs” Les le¸cons d”Anton Zorich donnent uneintroduction d´etaill´ee `a la g´eom´etrie des surfaces plates, celle-ci permet ded´ecrire les flots sur les surfaces de Riemann compactes de genre quelconquesans demander de connaissances pr´ealables Le cours de Jean-Christophe Yoc-coz analyse les applications ´echangeant des intervalles, par exemple, les appli-cations de premier retour de ces flots Les propri´et´es d’ergodicit´e des flots etdes applications sont reli´ees Ceci conduit `a ´etendre au cas de genre quelconqueles flots irrationnels du tore bidimensionnel Il faut commencer par g´en´eraliserdans cette situation un algorithme comme celui des fractions continues pouresp´erer ´etendre au genre quelconque les techniques de petits diviseurs.Enfin nous concluons ce volume par le s´eminaire de Guido Gentile sur lesNombres de Brjuno et les syst`emes dynamiques et celui de Marmi et al sur lesFonctions r´eelle et complexe de Brjuno Dans ces deux expos´es on perturbeles param`etres des rotations irrationnelles ou des applications de “twist”, on

´

etudie alors les conditions de stabilit´e des trajectoires qui sont donn´ees pardes conditions arithm´etiques subtiles (condition et nombres de Brjuno) ainsique la taille des domaines de stabilit´e dans l’espace des param`etres (fonctions

et de Werner Nahm Ces deux auteurs ont eu justement une longue tion scientifique `a la fronti`ere modulaire entre Math´ematiques et Physique

interac-La pr´esentation de Don Zagier commence par une revue des propri´et´es

du dilogarithme d’Euler et de sa fonction associ´ee r´eelle de Bloch-Wigner.Ces fonctions se g´en´eralisent respectivement aux polylogarithmes et `a desfonctions r´eelles d´efinies par Ramakrishnan Zagier explique leur impor-tance pour la g´eom´etrie hyperbolique `a trois dimensions, la K 2m −1-th´eorie

alg´ebrique et leur relation avec les valeurs de fonctions zˆeta de Dedekind

(voir le volume I) pour l’argument m Par ailleurs le groupe modulaire

sem-ble ˆetre reli´e myst´erieusement `a la fonction de Bloch-Wigner et `a sa premi`ere

Pr´eface XVIIg´en´eralisation de Ramakrishnan Dans le deuxi`eme chapitre de ce cours Zagierintroduit encore d’autres variantes en particulier le dilogarithme de Rogers

et le dilogarithme ”augment´e” qui apparaissent dans le cours de Nahm, ledilogarithme quantique et aussi les g´en´eralisations `a plusieurs arguments: les(poly)logarithmes “multiples” Suit une ´etude de leurs principales propri´et´es:

´

equations fonctionnelles, relation avec les formes modulaires (voir aussi lacontribution de Nahm qui suit), valeurs sp´eciales et, de nouveau, K-th´eoriealg´ebrique sup´erieure

Dans son cours sur les th´eories conformes et ´el´ements de torsion du groupe

de Bloch Werner Nahm exprime les dimensions conformes des op´erateurs

de certaines th´eories des champs conformes bidimensionnelles “rationnelles”d’une s´erie discr`ete (les th´eories minimales) comme partie imaginaire du dilog-arithme de Rogers d’´el´ements de torsion de la K-th´eorie alg´ebrique du corpsdes nombres complexes La pr´esentation commence par une introductiong´en´erale aux th´eories quantiques des champs invariantes conformes (CFT),plus pr´ecis´ement un expos´e physique et conceptuel des alg`ebres d’op´erateurs

de vertex Les th´eories rationnelles sont exceptionnelles dans l’espace des ules des CFT mais on peut consid´erer leurs perturbations (d´eformations) quirestent totalement int´egrables L’´etape suivante est un survol des th´eoriesquantiques bidimensionnelles totalement int´egrables et l’´etude d’une relationdans certains mod`eles simples entre la matrice S de diffusion et une matrice deCartan d’une alg`ebre de Lie de dimension finie; en particulier le fait que les co-efficients de cette matrice soient entiers est li´e `a la statistique de Bose et sa pos-itivit´e `a la convergence de la fonction de partition (cela peut se g´en´eraliser aucas de statistiques quelconques) Puis Nahm conjecture en s’appuyant sur denombreux exemples que l’invariance “modulaire” des caract`eres chiraux d’uneCFT rationnelle qui admet une perturbation totalement int´egrable impliqueque toutes les solutions des conditions d’int´egrabilit´e (´equations de Bethe)d´efinissent des ´el´ements de pure torsion du groupe de Bloch (´etendu) du corpsdes complexes Les perturbations qui sont ainsi analysables sont d´efinies pardes paires de matrices de Cartan de type A, D, E ou T Nahm donne enfin

mod-la solution g´en´erale des ´equations de torsion pour des d´eformations de type

(A m , A n) pour des rangs m et n arbitraires Ces conjectures sont analys´eesmath´ematiquement par Zagier `a la fin de son texte

Apr`es cette pr´eparation vient le s´eminaire de Predrag Cvitanovic sur

la th´eorie des invariants et un triangle magique de groupes de Lie qu’il ad´ecouvert lors de calculs perturbatifs pour des th´eories de jauge quantiques.Cette structure a ´et´e discut´ee depuis par Deligne, Landsberg et Manivel .

Elle est diff´erente et ne semble pas reli´ee aux triangles magiques de groupes dedualit´e qui contiennent aussi le carr´e magique de Tits et Freudenthal dans desformes r´eelles bien choisies et qui sont bien connus en th´eorie de supergravit´e

et dans les mod`eles de supercordes

La troisi`eme s´erie de cours: Des D-branes aux th´eories de jauge, fut donn´eepar Paolo Di Vecchia et r´edig´ee avec Antonella Liccardo On y trouve uneintroduction aux mod`eles de cordes et aux D(irichlet)-branes associ´ees sur

XVIII Pr´eface

lesquelles les extr´emit´es des cordes ouvertes peuvent se d´eplacer, ils expliquent

aussi comment des D-branes multiples engendrent sur leur surface (volume )

des th´eories de jauge de Yang-Mills Ceci sert de pont entre les CFT sionnelles et des mod`eles physiques en dimension plus grande Les th´eories

bidimen-de corbidimen-des perturbatives admettent une bidimen-description bidimensionnelle commeCFT, c’est l’espace cible qui peut ´eventuellement avoir quatre dimensions Mais la plupart des effets non perturbatifs traduisent la pr´esence d’objets

de dimension spatiale quelconque p: ce sont les p-branes et en particulier les

D p-branes Les branes permettent de calculer l’entropie des trous noirs etsont `a l’origine de nouvelles ´equivalences (dualit´es) entre th´eories de jauge etth´eories de gravitation Par exemple la fameuse dualit´e AdS/CFT relie uneth´eorie des cordes ferm´ees sur la vari´et´e produit S5× AdS5et une th´eorie des

cordes ouvertes se terminant sur une D3-brane L’expos´e part de la tion CFT de la th´eorie des supercordes perturbatives avec ses op´erateurs devertex de cr´eation de cordes invariants BRST pour arriver `a leur descriptionpar le formalisme “des ´etats de bord” qui d´ecrit le couplage des cordes ferm´eesaux D-branes Ceci permet de calculer ensuite l’interaction entre D-branes, ondistingue le cas particulier BPS pour lequel les interactions se compensent DiVecchia relie ensuite les interactions effectives `a basse ´energie de type Born-Infeld des cordes de masse nulle `a leurs couplages aux D-branes

descrip-Cette premi`ere partie se termine par un s´eminaire de Katrin Wendland:Th´eories des champs superconformes associ´ees aux quartiques tr`es attrac-tives Le terme attractives fut introduit par Greg Moore (cf son cours ci-dessous) pour les vari´et´es de Calabi-Yau `a deux dimensions dont le groupe

de Picard est de rang maximum, tr`es attractives correspond `a une tion suppl´ementaire sur le r´eseau transcendant Il s’agit donc d’une revue desr´ealisations g´eom´etriques des orbifolds sur des surfaces quartiques et donnerasans doute envie de lire les chapitres suivants

restric-Le th`eme de la deuxi`eme partie “Groupes discrets et Formes automorphes”est la th´eorie des groupes arithm´etiques et certaines de leurs applications Led´ecor est plant´e par le cours de Christophe Soul´e d’Introduction aux groupesarithm´etiques, qui est plus g´en´eral que celui de Bogomolny au volume I Lecours commence par la th´eorie classique de la r´eduction des groupes lin´eaires

de matrices `a coefficients entiers et des formes normales des formes tiques Elle est suivie de la th´eorie g´en´erale (et intrins`eque) des groupes de Liealg´ebriques sur les rationnels et de leurs sous-groupes arithm´etiques; la pro-pri´et´e de covolume fini aux points r´eels dans le cas semi-simple est d´emontr´ee,elle est famili`ere aux physiciens du chaos Le deuxi`eme chapitre de ce courstraite des pr´esentations de ces groupes et de leurs sous-groupes finis ou biensans torsion et d’indice fini Le troisi`eme chapitre s’occupe de “rigidit´e”: lapropri´et´e du sous-groupe de congruence en rang plus grand que un, la pro-pri´et´e T de Kazhdan sur les vecteurs invariants et les r´esultats de Margulis,

quadra-en particulier la d´emonstration de la conjecture de Selberg que l’arithm´eticit´er´esulte de la propri´et´e de covolume fini pour la plupart des groupes de Liesimples non compacts

Pr´eface XIXLes formes automorphes sont des fonctions complexes d´efinies sur des do-maines sym´etriques et invariantes par des groupes arithm´etiques, elles ap-paraissent fr´equemment en th´eorie des cordes Boris Pioline a d´evelopp´e sons´eminaire avec la collaboration d’Andrew Waldron et propose ici une intro-duction aux Formes automorphes et aux s´eries Thˆeta par des physiciens Ellecommence par l’expression ad`elique des s´eries non holomorphes d’Eisensteinissue de la th´eorie des groupes; E 3/2a par exemple ´et´e ´etudi´ee en d´etail par lephysicien M.B Green et ses collaborateurs Apr`es avoir introduit ´egalementles s´eries thˆeta on arrive `a des applications de la m´ethode des orbites et del’induction parabolique Parmi les r´esultats r´ecents et au-del`a des groupes deU-dualit´e discrets consid´er´es plus haut (voir le cours pr´ec´edent par exemple),Pioline et Waldron discutent la repr´esentation minimale de SO(4, 4) que l’on

rencontre en th´eorie des cordes, la s´erie thˆeta exceptionnelle E6 qui est pos´ee d´ecrire les interactions des supermembranes (2-branes) apr`es compact-ification torique de 11 `a 8 dimensions, de nouvelles sym´etries des cosmologieschaotiques et enfin des travaux en cours sur la description des multiplicit´esd’´etats des trous noirs dont on veut calculer l’entropie

sup-La th´eorie M est le nom d’une th´eorie d’unification, hypoth´etique et morphe qui admet diverses limites, soit dans un espace ambiant `a 11 dimen-sions avec des membranes comme excitations fondamentales soit `a 10 dimen-sions comme des th´eories de supercordes avec des branes vari´ees Gregory

poly-Moore a r´edig´e deux de ses s´eminaires sous le titre Cordes et Arithm´etique (letroisi`eme sur les aspects topologiques de la 3-forme de la th´eorie M donne en-core lieu `a des recherches actives et des d´eveloppements nouveaux) Le premiersujet qu’il traite est intraduisible: “A black hole’s Farey tail”, il s’agit d’uneillustration de la dualit´e AdS3× S3× K3 avec une CFT bidimensionnelle sur

la fronti`ere de l’espace anti de Sitter `a 3 dimensions On peut calculer le genreelliptique de cette CFT comme une s´erie de Poincar´e qui s’interpr`ete du cˆot´eAdS (i.e cˆot´e gravit´e ou cordes) comme une somme de contributions des ´etatsparticulaires et des ´etats de cordes Ceci peut servir d’introduction concr`ete

`

a de nombreuses id´ees sur les formes modulaires de Jacobi, au d´eveloppement

de Rademacher et aux corrections quantiques `a l’entropie des trous noirs

Le deuxi`eme chapitre de Moore porte sur le m´ecanisme des attracteurs ensupergravit´e Apr`es compactification sur un espace de Calabi-Yau `a trois di-mensions X on sait que les modules de la structure complexe de X tendent vers

un point fixe lorsque l’on approche l’horizon d’une solution trou noir Cet tracteur d´epend des charges du trou noir qui y admettent une d´ecomposition

at-de Hodge sp´eciale Dans le cas particulier X = K3 × T2 on obtient la notion

de surface K3 attractive mentionn´ee ci-dessus Le fait essentiel ici est que lesattracteurs semblent devoir ˆetre des vari´et´es arithm´etiques d´efinies sur descorps de nombres, les p´eriodes prennent en fait leurs valeurs dans des corpsquadratiques imaginaires Le cours se termine par deux autres exemples del’importance des vari´et´es attractives Tout d’abord la “th´eorie F” `a 12 dimen-sions compactifi´ee sur une surface K3 doit ˆetre duale de (i.e ´equivalente `a)

XX Pr´eface

la th´eorie des cordes h´et´erotique effectivement `a 8 dimensions apr`es pactification” par une CFT bidimensionnelle; il est frappant de constater que

“com-cette CFT est rationnelle si et seulement si la surface K3 est attractive Le

deuxi`eme exemple est le lien entre compactifications avec flux des th´eories decordes et vari´et´es de Calabi-Yau attractives `a 4 dimensions

La contribution suivante est un s´eminaire de Matilde Marcolli sur la mologie chaotique (mod`ele mixmaster) dans lequel elle fait correspondre `aune g´eod´esique sur la courbe modulaire du groupe de congruence Γ0(2) unesuccession d’espaces-temps de Kasner `a quatre dimensions L’espace des mod-ules de ces univers est tr`es singulier et doit ˆetre d´ecrit par la g´eom´etrie noncommutative et lesC∗-alg`ebres.

cos-John McKay et Abdellah Sebbar introduisent le concept et six tions possibles des fonctions “r´epliquables” Ce sont des g´en´eralisations de

applica-la fonction moduapplica-laire j qui se transforment par leurs polynˆomes de Fabercomme des sommes de Hecke g´en´eralis´ees faisant intervenir leurs r´epliques.Disons simplement qu’elles comprennent les fonctions du “Monstrous moon-shine” et qu’elles sont intrins`equement reli´ees `a la d´eriv´ee de Schwarz quiapparaˆcomme g´en´erateur de la charge centrale de l’alg`ebre de Virasoro.Enfin cette partie II se termine par les cours d’Edward Frenkel sur le pro-gramme de Langlands et les CFT Les deux buts de l’auteur ´etaient d’unepart de pr´esenter aux physiciens le programme de Langlands et en particulier

sa partie “g´eom´etrique” et d’autre part de montrer l’importance des th´eoriesconformes pour cette derni`ere Cette activit´e se d´eveloppe en Physique avec

la r´ealisation que la dualit´e math´ematique de Langlands est ment reli´ee `a la S-dualit´e de la physique des cordes Les travaux r´ecents de A.Kapustin et E Witten font suite `a des r´esultats du milieu des ann´ees 70 surles monopoles magn´etiques, l’outil surprenant des twists topologiques qu’ilsutilisent semble pouvoir relier la th´eorie de super-Yang-Mills N=4 `a 4 dimen-sions `a de nombreux probl`emes essentiels En l’occurence c’est la sym´etriemiroir (dualit´e T) de mod`eles sigma bidimensionnels qui se d´eduisent de lath´eorie `a quatre dimensions qui r´ealise cette dualit´e

fondamentale-Frenkel commence par le programme original de Langlands et les pondances pour les corps de nombres et les corps de fonctions Il pr´esente

corres-la conjecture (moducorres-laire) de Taniyama-Shimura-Weil (qui est maintenant unth´eor`eme) Puis il explique le programme de Langlands g´eom´etrique, d’aborddans le cas ab´elien puis pour un groupe r´eductif quelconque G Le but est

de g´en´eraliser la dualit´e T ou la dualit´e de Fourier-Mukai au cas non-ab´elien.Finalement il introduit les blocs conformes pour les CFT et des mod`eles as-soci´es `a des modules des alg`ebres de Kac-Moody affines Pour la valeur cri-tique (n´egative) du niveau de la charge centrale de Kac-Moody, la sym´etrieconforme induite d´eg´en`ere et ces mod`eles conduisent aux faisceaux invariants

de Hecke pr´edits par la correpondance de Langlands g´eom´etrique

Le troisi`eme et dernier th`eme de ce volume et donc des Actes est intitul´e

“Alg`ebres de Hopf et renormalisation” Il conduit `a des r´esultats prometteurs

Pr´eface XXIsur la renormalisation des th´eories quantiques des champs qui peuvent ˆetre il-lustr´es par des calculs diagrammatiques perturbatifs et concrets mais cela nousm`ene ´egalement comme un arc en ciel ´etrange entre ciel et terre au conceptabstrait de “motifs” La premi`ere s´erie de cours de cette partie est une revuehistorique par Pierre Cartier de l’apparition du concept d’alg`ebre de Hopf `apartir de la Topologie Il passe ensuite `a des exemples venant de la th´eorie desgroupes et alg`ebres de Lie, d´ecrit leur structure et donne des th´eor`emes destructure g´en´eraux inverses Il termine par des exemples d’alg`ebres de Hopf et

de leurs applications en Combinatoire: les fonctions (quasi)-sym´etriques, lesvaleurs de fonctions zˆeta multiples et enfin les polylogarithmes Cette intro-duction longue et p´edagogique aurait pu continuer dans le monde des motifs,est-ce le signe pr´ecurseur d’une troisi`eme ´ecole des Houches sur ce sujet? Vientensuite le deuxi`eme chapitre d’Alain Connes et de Matilde Marcolli Leur pre-mier chapitre sur la m´ecanique statistique quantique desQ-r´eseaux se trouvedans le premier volume L’op´erateur de dilatation qui y d´ecrivait la dynamiquer´eapparaˆıt ici comme le flot du groupe de renormalisation Ce texte sur Renor-malisation, correspondance de Riemann-Hilbert et th´eorie de Galois motiviquecommence par une revue d´etaill´ee des r´esultats de Connes et Kreimer sur larenormalisation perturbative en th´eorie quantique des champs vue comme unprobl`eme de Riemann-Hilbert On y trouve une pr´esentation de l’alg`ebre deHopf des graphes de Feynman qui correspond par le th´eor`eme de Milnor-Moore (voir le cours pr´ec´edent) `a une alg`ebre de Lie gradu´ee engendr´ee parles diagrammes 1PI Les cas singuliers conduisent `a des s´eries formelles maisles probl`emes de convergence sont bri`evement discut´es vers la fin L’ensemble

du programme est reformul´e dans le language des cat´egories, des groupesalg´ebriques et de la th´eorie de Galois diff´erentielle Des liens possibles avec lesmotifs de Tate mixtes sont envisag´es L’´equivariance sous le groupe de renor-malisation est reformul´ee dans ce language et des d´eveloppements fascinantssont propos´es

Trois s´eminaires correspondants concluent l’entreprise: Dirk Kreimer cute le probl`eme de la factorisation en th´eorie quantique des champs commeexercice sur les alg`ebres de Hopf et les singularit´es locales Il analyse en fait

dis-un mod`ele simplifi´e d’arbres “enracin´es” et d´ecor´es qui illustre la solution duprobl`eme des diagrammes divergents qui se recouvrent On apprend d’abordcomment la cohomologie de Hochschild des alg`ebres de Hopf permet un pro-gramme de renormalisation “local” Puis des ´equations de Dyson-Schwingersont d´efinies sans utiliser d’action, elles devraient permettre une factorisationcombinatoire avec des facteurs primitifs de l’alg`ebre de Hopf correspondante.Stefan Weinzierl explique dans son rapport quelques propri´et´es des poly-logarithmes multiples et de leurs troncations (des sommes emboˆıt´ees appel´eessommes Z) que l’on rencontre dans les diagrammes de boucles de Feynman

Il parle des algorithmes alg´ebriques dans les calculs perturbatifs et de leurimpact sur la recherche de physique (exp´erimentale) nouvelle Il met l’accentsur la calculabilit´e analytique de certains diagrammes de Feynman et sur les

XXII Pr´eface

structures alg´ebriques des sommes Z Celles-ci ont une structure d’alg`ebre deHopf mais aussi une conjugaison complexe et un produit de convolution Cecientraˆıne que les polylogarithmes ont une deuxi`eme structure de Hopf et unproduit “shuffle”

Le dernier s´eminaire est un expos´e p´edagogique de Herbert Gangl, der B Goncharov et Andrey Levin sur les logarithmes multiples, les cyclesalg´ebriques et les arbres Ce travail a ´et´e ´etendu depuis aux polylogarithmesmultiples et au monde des motifs par les mˆemes auteurs Ils relient ici lestrois parties de leur titre: les deux derni`eres sont associ´ees `a des alg`ebresdiff´erentielles gradu´ees de cycles alg´ebriques et d’arbres d´ecor´es enracin´es; latroisi`eme partie du titre exprime la relation entre les polylogarithmes multi-ples et des int´egrales sur des cycles hybrides ce qui g´en´eralise les motifs deTate mixtes de Bloch et Kriz dans le cas des (poly-)logarithmes ordinaires.C’est un plaisir de remercier ici pour leur soutien financier g´en´ereux `a cetteconf´erence les institutions suivantes:

Alexan-D´epartement Sciences Physiques et Math´ematiques et Service de tion permanente du Centre National de la Recherche Scientifique; ´Ecole Nor-male Sup´erieure de Paris; D´epartement des Sciences de la mati`ere du Com-missariat `a l’´Energie Atomique; Institut des Hautes Etudes Scientifiques; Na-tional Science Foundation; Minist`ere de la Recherche et de la Technologie etMinist`ere des Affaires Etrang`eres; International Association of Mathematicsand Physics et tout particuli`erement la Commission des Communaut´es Eu-rop´eennes

Forma-Trois r´eseaux europ´eens d’excellence nous ont aussi aid´e de diversesmani`eres Le r´eseau “Aspects math´ematiques du chaos quantique” fut le plusimpliqu´e, mais nous n’oublions pas les deux autres: “Supercordes” et “Struc-ture quantique de l’espace-temps et nature g´eom´etrique des interactions fon-damentales”

En ce qui concerne les aspects mat´eriels nous remercions la DivisionTh´eorie du CERN pour nous avoir permis d’utiliser leurs ordinateurs pour

le site web et l’enregistrement des inscriptions Nous tenons `a remercier aussiMarcelle Martin, Thierry Paul et le secr´etariat des Houches pour leur pa-tient travail Nous eˆumes le privil`ege d’accueillir deux participants de mar-que: C´ecile de Witt-Morette (fondatrice de l’Ecole des Houches) et son mariBryce de Witt dont l’enthousiasme critique mais communicatif fut grandementappr´eci´e

P Moussa

P Vanhove

List of Contributors

List of Authors: (following the order of appearance of the contributions)

• D Zagier, Max-Planck-Institut f¨ur Mathematik, Gottfried-Claren-Strape

26, 0-5300 Bonn, Germany

Department of Mathematics, University of Maryland, College Park, land 20742, USA

Mary-• W Nahm, Dublin Institute for Advanced Studies, Ireland

• P Cvitanovic, School of Physics, Georgia Institute of Technology, Atlanta,

• K Wendland, University of Warwick, Gibbet Hill, Coventry CV4-7AL, England

• Ch Soul´e, I.H.E.S., 35 Route de Chartres, F-91440 Bures sur Yvette, France

• B Pioline, LPTHE, Universit´es Paris VI et VII, 4 pl Jussieu, 75252 Paris cedex 05, France

A Waldron, Department of Mathematics, One Shields Avenue, University

of California, Davis, CA 95616, USA

• G Moore, Department of Physics, Rutgers University Piscataway, NJ 08854-8019, USA

• M Marcolli, Max–Planck Institut f¨ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

• J McKay, Department of Mathematics and CICMA Concordia University

1455 de Maisonneuve Blvd West, Montreal, Quebec H3G 1M8, Canada Abdellah Sebbar, Department of Mathematics and Statistics, University

of Ottawa, Ottawa, ON K1N 6N5, Canada

• E Frenkel, University of California, Berkeley, USA

XXIV List of Contributors

• P Cartier, I.H.E.S 35 route de Chartres F-91440 Bures-sur-Yvette, France

• A Connes, Coll`ege de France, 3, rue Ulm, F-75005 Paris, France

I.H.E.S 35 route de Chartres F-91440 Bures-sur-Yvette, France

M Marcolli, Max–Planck Institut f¨ ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

• D Kreimer, I.H.E.S 35 route de Chartres, F-91440 Bures-sur-Yvette, France

• S Weinzierl, Institut f¨ur Physik (ThEP) Universitat Mainz, D - 55099 Mainz, Germany

• H Gangl, MPI f¨ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany A.B Goncharov, Brown University, Box 1917, Providence, RI 02912, USA

A Levin, Institute of Oceanology, Moscow, Russia

Editors:

• Bernard Julia, LPTENS, 24 rue Lhomond 75005 Paris, France

• Pierre Cartier, I.H.E.S 35 route de Chartres F-91440 Bures-sur-Yvette, France

• Pierre Moussa, Service de Physique Th´eorique, CEA/Saclay, F-91191 sur-Yvette, France

• Pierre Vanhove, Service de Physique Th´eorique, CEA/Saclay, F-91191 sur-Yvette, France

Part I Conformal Field Theories for Strings and Branes

The Dilogarithm Function

Gauge Theories from D Branes

Paolo Di Vecchia, Antonella Liccardo 161

On Superconformal Field Theories Associated to Very

Attractive Quartics

Katrin Wendland 223

Part II Discrete Groups and Automorphic Forms

An Introduction to Arithmetic Groups

Christophe Soul´ e 247

Automorphic Forms: A Physicist’s Survey

Boris Pioline, Andrew Waldron 277

Strings and Arithmetic

Gregory Moore 303

Modular Curves, C∗-algebras, and Chaotic Cosmology

Matilde Marcolli 361

XXVI Contents

Replicable Functions: An Introduction

John McKay, Abdellah Sebbar 373

Lectures on the Langlands Program

and Conformal Field Theory

Edward Frenkel 387

Part III Hopf Algebras and Renormalization

A Primer of Hopf Algebras

Pierre Cartier 537

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

Alain Connes, Matilde Marcolli 617

Factorization in Quantum Field Theory: An Exercise in Hopf Algebras and Local Singularities

Dirk Kreimer 715

Algebraic Algorithms in Perturbative Calculations

Stefan Weinzierl 737

Multiple Logarithms, Algebraic Cycles and Trees

H Gangl, A.B Goncharov, A Levin 759

Part IV Appendices

List of Participants 777 Index 781

Table of Contents of Volume I

Part I Random matrices: from Physics to Number theory

Quantum and Arithmetical Chaos

XXVIII Contents

Part II Zeta functions

From Physics to Number theory via Noncommutative

Geometry

Alain Connes, Matilde Marcolli 269

More Zeta Functions for the Riemann Zeros

Andr´ e Voros 351

Hilbert Spaces of Entire Functions and Dirichlet L-Functions

Jeffrey C Lagarias 367

Dynamical Zeta Functions and Closed Orbits for Geodesic

and Hyperbolic Flows

Some Properties of Real and Complex Brjuno Functions

Stefano Marmi, Pierre Moussa, Jean-Christophe Yoccoz 603

Part I

Conformal Field Theories for Strings and Branes

The Dilogarithm Function

Don Zagier

zagier@mpim-bonn.mpg.de

I The dilogarithm function in geometry and number theory 5

1 Special values 6

2 Functional equations 8

3 The Bloch-Wigner function D(z) and its generalizations 10

4 Volumes of hyperbolic 3-manifolds 13

5 and values of Dedekind zeta functions 16

References 20 Notes on Chapter I 21

II Further aspects of the dilogarithm 22

1 Variants of the dilogarithm function 22

2 Dilogarithm identities 33

3 Dilogarithms and modular functions: the Nahm equation 41

4 Higher polylogarithms 58

References 63

The dilogarithm function, defined in the first sentence of Chapter I, is

a function which has been known for more than 250 years, but which for

a long time was familiar only to a few enthusiasts In recent years it hasbecome much better known, due to its appearance in hyperbolic geometry

and in algebraic K-theory on the one hand and in mathematical physics (in

particular, in conformal field theory) on the other I was therefore asked togive two lectures at the Les Houches meeting introducing this function andexplaining some of its most important properties and applications, and towrite up these lectures for the Proceedings

The first task was relatively straightforward, but the second posed a lem since I had already written and published an expository article on thedilogarithm some 15 years earlier (In fact, that paper, originally written as

prob-a lecture in honor of Friedrich Hirzebruch’s 60th birthdprob-ay, hprob-ad prob-appeprob-ared intwo different Indian publications during the Ramanujan centennial year—seefootnote to Chapter I) It seemed to make little sense to try to repeat in

4 Don Zagier

different words the contents of that earlier article On the other hand, justreprinting the original article would mean omitting several topics which wereeither developed since it was written or which were omitted then but are ofmore interest now in the context of the appearances of the dilogarithm inmathematical physics

The solution I finally decided on was to write a text consisting of twochapters of different natures The first is simply an unchanged copy of the

1988 article, with its original title, footnotes, and bibliography, reprinted by

permission from the book Number Theory and Related Topics (Tata Institute

of Fundamental Research, Bombay, January 1988) In this chapter we definethe dilogarithm function and describe some of its more striking properties:its known special values which can be expressed in terms of ordinary log-arithms, its many functional equations, its connection with the volumes of

ideal tetrahedra in hyperbolic 3-space and with the special values at s = 2

of the Dedekind zeta functions of algebraic number fields, and its appearance

in algebraic K-theory; the higher polylogarithms are also treated briefly The

second, new, chapter gives further information as well as some more recentdevelopments of the theory Four main topics are discussed here Three ofthem—functional equations, modifications of the dilogarithm function, andhigher polylogarithms—are continuations of themes which were already be-gun in Chapter I The fourth topic, Nahm’s conjectural connection between(torsion in) the Bloch group and modular functions, is new and especially fas-cinating We discuss only some elementary aspects concerning the asymptotic

properties of Nahm’s q-expansions, referring the reader for the deeper parts of the theory, concerning the (in general conjectural) interpretation of these q-

series as characters of rational conformal field theories, to the beautiful article

by Nahm in this volume

As well as the two original footnotes to Chapter I, which are indicated bynumbers in the text and placed at the bottom of the page in the traditionalmanner, there are also some further footnotes, indicated by boxed capitalletters in the margin and placed at the end of the chapter, which give updates

or comments on the text of the older article or else refer the reader to thesections of Chapter II where the topic in question is developed further Each ofthe two chapters has its own bibliography, that of Chapter I being a reprint ofthe original one and that of Chapter II giving some references to more recentliterature I apologize to the reader for this somewhat artificial construction,but hope that the two parts of the paper can still be read without too muchconfusion and perhaps even with some enjoyment My own enthusiasm for thismarvelous function as expressed in the 1988 paper has certainly not lessened

in the intervening years, and I hope that the reader will be able to share atleast some of it

The reader interested in knowing more about dilogarithms should alsoconsult the long article [22] of A.N Kirillov, which is both a survey papertreating most or all of the topics discussed here and also contains many newresults of interest from the point of view of both mathematics and physics

The Dilogarithm Function 5

Chapter I The dilogarithm function

in geometry and number theory1

The dilogarithm function is the function defined by the power series

is obvious and leads by induction to the extension of the domain of definition

of Limto the cut planeC
[1, ∞); in particular, the analytic continuation of

the dilogarithm is given by

6 Don Zagier

Thus the dilogarithm is one of the simplest non-elementary functions onecan imagine It is also one of the strangest It occurs not quite often enough,and in not quite an important enough way, to be included in the Valhalla ofthe great transcendental functions—the gamma function, Bessel and Legen-dre - functions, hypergeometric series, or Riemann’s zeta function And yet

it occurs too often, and in far too varied contexts, to be dismissed as a merecuriosity First defined by Euler, it has been studied by some of the greatmathematicians of the past—Abel, Lobachevsky, Kummer, and Ramanujan,

to name just a few—and there is a whole book devoted to it [4] Almost all

of its appearances in mathematics, and almost all the formulas relating to it,have something of the fantastical in them, as if this function alone among allothers possessed a sense of humor In this paper we wish to discuss some ofthese appearances and some of these formulas, to give at least an idea of thisremarkable and too little-known function

A

1 Special values

Let us start with the question of special values Most functions have either

no exactly computable special values (Bessel functions, for instance) or else acountable, easily describable set of them; thus, for the gamma function

values of z for which z and Li2(z) can both be given in closed form:

The Dilogarithm Function 7

Li2



3− √52



.

Let me describe a recent experience where these special values figured, andwhich admirably illustrates what I said about the bizarreness of the occur-rences of the dilogarithm in mathematics From Bruce Berndt via Henri Cohen

I learned of a still unproved assertion in the Notebooks of Srinivasa

Ramanu-jan (Vol 2, p 289, formula (3.3)): RamanuRamanu-jan says that, for q and x between

8 Don Zagier

The quantitative interpretation turned out as follows [9] : The difference

between the left and right sides of Ramanujan’s equation is O

expπ2/5

log q

for

x = 1, q → 1 (The proof of this used the identities

which are consequences of the Rogers-Ramanujan identities and are surely

among the most beautiful formulas in mathematics.) For x → 0 and q → 1 the difference in question is O

+ log21 +√5

In contrast to the paucity of special values, the dilogarithm function satisfies

a plethora of functional equations To begin with, there are the two reflectionproperties

The Dilogarithm Function 9Together they say that the six functions

five arguments: if these are numbered cyclically as z n with n ∈ Z/5Z, then

Li2



− xy z

+ Li2



− yz x

+ Li2



− zx y

discovered by Kummer (1840) and Newman (1892) Finally, there is thestrange many-variable equation

All of the functional equations of Li2 are easily proved by differentiation, C

while the special values given in the previous section are obtained by ing suitable functional equations See [4]

combin-10 Don Zagier

3 The Bloch-Wigner function D(z)

and its generalizations

D

The function Li2(z), extended as above to C
[1, ∞), jumps by 2πi log |z| as

z crosses the cut Thus the function Li2(z) + i arg(1 − z) log |z|, where arg

denotes the branch of the argument lying between −π and π, is continuous.

Surprisingly, its imaginary part

D(z) = (Li2(z)) + arg(1 − z) log |z|

is not only continuous, but satisfies

(I) D(z) is real analytic on C except at the two points 0 and 1, where it is

continuous but not differentiable (it has singularities of type r log r there).

The above graph shows the behaviour of D(z) We have plotted the level curves D(z) = 0, 0.2, 0.4, 0.6, 0.8, 0.9, 1.0 in the upper half-plane The val- ues in the lower half-plane are obtained from D(¯ z) = −D(z) The maximum

of D is 1.0149 , attained at the point (1 + i √

3)/2.

The function D(z), which was discovered by D Wigner and S Bloch

(cf [1]), has many other beautiful properties In particular:

(II) D(z), which is a real-valued function onC, can be expressed in terms of

a function of a single real variable, namely



(2)

The Dilogarithm Function 11

which expresses D(z) for arbitrary complex z in terms of the function

D(e iθ) =[Li2(e iθ)] =

4 for 0≤ θ ≤ 2π.) Formula (2) is due to Kummer.

(III) All of the functional equations satisfied by Li2(z) lose the elementary

correction terms (constants and products of logarithms) when expressed

in terms of D(z) In particular, one has the 6-fold symmetry

D(z) = D



1−1z



= D

1

1− z



=−D

1

con-The functional equations become even cleaner if we think of D as being

a function not of a single complex number but of the cross-ratio of four suchnumbers, i.e., if we define

Then the symmetry properties (3) say that D is invariant under even and

anti-invariant under odd permutations of its four variables, the five-term relation(4) takes on the attractive form

12 Don Zagier

where f : P1(C) → P1(C) is a function of degree n and a0 = f (z0)

(Equa-tion (1) is the special case when f is a polynomial, so f −1(∞) is ∞ with multiplicity n.)

Finally, we mention that a real-analytic function on P1(C)
{0, 1, ∞},

built up out of the polylogarithms in the same way as D(z) was constructed

from the dilogarithm, has been defined by Ramakrishnan [6] His function(slightly modified) is given by

However, it does not seem to have analogues of the properties (II) and (III):

for example, it is apparently impossible to express D3(z) for arbitrary complex

z in terms of only the function D3(e iθ) = ∞ n=1 (cos nθ)/n3, and passing from

Li3 to D3removes many but not all of the numerous lower-order terms in thevarious functional equations of the trilogarithm, e.g.:



− 2D3



x(1 − y) y(1 − x)

satisfying ∆ z1G n = ∆ z2G n = n(n − 1)G n , where ∆ z = y2(∂2/∂x2+ ∂2/∂y2)

is the hyperbolic Laplace operator with respect to z = x + iy ∈ H They are

The Dilogarithm Function 13obtained as

|z1− ¯z2|2 + 2

In between GH

n and G H/Γ n are the functions G H/ n Z = r ∈Z GH

n (z1, z2+ r) It turns out [10] that they are expressible in terms of the D m (m = 1, 3, , 2n − 1), e.g.,

I do not know the reason for this connection

4 Volumes of hyperbolic 3-manifolds

The dilogarithm occurs in connection with measurement of volumes in ean, spherical, and hyperbolic geometry We will be concerned with the last ofthese Let H3be the Lobachevsky space (space of non-euclidean solid geome-try) We will use the half-space model, in which H3 is represented byC × R+

euclid-with the standard hyperbolic metric in which the geodesics are either cal lines or semicircles in vertical planes with endpoints inC × {0} and the

verti-geodesic planes are either vertical planes or else hemispheres with boundary

in C × {0} An ideal tetrahedron is a tetrahedron whose vertices are all in

∂H3=C ∪ {∞} = P1(C) Let ∆ be such a tetrahedron Although the verticesare at infinity, the (hyperbolic) volume is finite It is given by

Vol(∆) =  D(z0, z1, z2, z3), (7)

where z0, , z3∈ C are the vertices of ∆ and  D is the function defined in (5).

In the special case that three of the vertices of ∆ are ∞, 0, and 1, equation

(7) reduces to the formula (due essentially to Lobachevsky)

Surprisingly, its imaginary part

D(z) = (Li2(z)) + arg(1 − z) log |z|

is not only continuous,... given in the previous section are obtained by ing suitable functional equations See [4]

combin-10... 0.6, 0.8, 0.9, 1.0 in the upper half-plane The val- ues in the lower half-plane are obtained from D(¯ z) = −D(z) The maximum

of D is 1.0149 , attained at the point (1 + i √