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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

Pierre Cartier

I.H.E.S.

35 route de Chartres

F-91440 Bures-sur-Yvette

France

e-mail: [email protected]

Bernard Julia

LPTENS

24 rue Lhomond

F-75005 Paris

France

e-mail: [email protected]

Pierre Moussa

Service de Physique Théorique

CEA/Saclay

F-91191 Gif-sur-Yvette

France

e-mail: [email protected]

Pierre Vanhove

Service de Physique Théorique

CEA/Saclay

F-91191 Gif-sur-Yvette

France

e-mail: [email protected]

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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Preface

The present book collects most of the courses and seminars delivered at the

meeting entitled “ Frontiers in Number Theory, Physics and Geometry”, which

took 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 on

three broad topics: Random matrices, Zeta functions and Dynamical systems.

The present volume contains sixteen contributions on three themes: Conformal

field theories for strings and branes, Discrete groups and automorphic forms

and finally, Hopf algebras and renormalization.

The relation between Mathematics and Physics has a long history. Let us

mention only ordinary differential equations and mechanics, partial differential

equations 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 mathematics

such as topological, differential and algebraic geometry in physics has become

prominent 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 the

occasion of long and passionate discussions.

The seminars were published in a book entitled “Number Theory and

Physics”, J.-M. Luck, P. Moussa, and M. Waldschmidt editors, Springer Pro￾ceedings in Physics, Vol. 47, 1990. The lectures were published as a second

VI Preface

book entitled “From Number Theory to Physics”, with C. Itzykson joining

the editorial team (Springer, 2nd edition 1995).

Ten years later the evolution of the interface between theoretical physics

and mathematics prompted M. Waldschmidt, P. Cartier and B. Julia to re￾new the experience. However the emphasis was somewhat shifted to include in

particular selected chapters at the interface of physics and geometry, random

matrices or various zeta- and L- functions. Once the project of the new meet￾ing entitled “Frontiers in Number Theory, Physics and Geometry” received

support from the European Union this “High Level Scientific Conference” was

organized in Les Houches.

The Scientific Committee for the meeting “Frontiers in Number The￾ory, Physics and Geometry”, was composed of the following scientists: Frits

Beukers, Jean-Benoˆıt Bost, Pierre Cartier, Predrag Cvitanovic, Michel Duflo,

Giovanni Gallavotti, Patricio Leboeuf, Werner Nahm, Ivan Todorov, Claire

Voisin, Michel Waldschmidt, Jean-Christophe Yoccoz, and Jean-Bernard Zu￾ber.

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 about

seventy-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 exchange

maps, 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 renor￾malization, R. Dijkgraaf on String duality and automorphic forms, P. Di Vec￾chia on Gauge theory and D-branes, E. Frenkel on Vertex algebras, algebraic

curves and Langlands program, G. Moore on String theory and number theory,

C. Soul´e on Arithmetic groups.

In addition seminars were given by participants many of whom could have

given 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 ac￾cording to their subject. This naturally led to two different volumes. The

present one is the second volume, let us now briefly describe its contents.

This volume is itself composed of three parts including each lectures and

seminars covering one theme. In the first part, we present the contributions on

the theme “Conformal Field Theories (CFT’s) for Strings and Branes”. They

begin with two intertwined sets of lectures by Don Zagier and by Werner Nahm

who have had a long personal interaction at the modular border between

Mathematics and Physics.

The presentation by Don Zagier starts with a review of the properties of

Euler’s dilogarithm and of its associated real Bloch-Wigner function. These

functions have generalizations to polylogarithms and to some real functions

defined by Ramakrishnan respectively. Their importance in Hyperbolic 3-

geometry, in Algebraic K2m−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 the

Bloch-Wigner function and its first Ramakrishnan generalization. The second

chapter of these lectures introduces yet more variants, in particular the Rogers

dilogarithm and the enhanced dilogarithm which appear in W. Nahm’s lec￾tures, the quantum dilogarithm as well as the multiple (poly)logarithms which

depend on more than one argument. Their properties are reviewed, in par￾ticular functional equations, relations with modular forms (see also the next

contribution), special values and again (higher) K-theory.

In his lectures on “CFT’s and torsion elements of the Bloch group” Werner

Nahm expresses the conformal dimensions of operators in the (discrete) series

of rational two dimensional (2d) Conformal Field Theories as the imaginary

part of the Rogers dilogarithm of torsion elements from algebraic K-theory

of the complex number field. The lectures begin with a general introduc￾tion to conformally invariant quantum field theories or more precisely with a

physicist’s conceptual presentation of Vertex operator algebras. The ”ratio￾nal” theories form a rare subset in the moduli space of CFT’s but one may

consider perturbations thereof within the set of totally integrable quantum

field 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 cases

the scattering matrix to Cartan matrices of finite dimensional Lie algebras, in

particular integrality of the coefficients follows from Bose statistics and posi￾tivity from the assumed convergence of partition functions, there are natural

extensions to arbitrary statistics.

Then Nahm conjectures and illustrates on many examples that the “mod￾ular” invariance of the chiral characters of a rational CFT admitting a totally

integrable perturbation implies that all solutions to the integrability condi￾tions (Bethe equations) define pure torsion elements in the (extended) Bloch

group 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

the general solution of the torsion equation for deformations of (Am,An) type

VIII Preface

with arbitrary ranks. These conjectures were analyzed mathematically at the

end of Zagier’s lectures.

After this background comes the seminar by Predrag Cvitanovic on in￾variant theory and a magic triangle of Lie groups he discovered in his studies

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 the

magic 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 de￾livered by Paolo Di Vecchia and written up with Antonella Liccardo. They

provide an introduction to string models and the associated D(irichlet)-branes

on which open strings may end and they explain the emergence of Yang-Mills

gauge theories on these extended objects. They bridge the gap between 2d

CFT’s and physical models in higher dimensions. Perturbative string theories

are 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

Dp branes. Branes allow the computation of the entropy of black holes and

permit new dualities between gauge and gravitational theories. For instance

the 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. These

lectures start from the worldsheet description of perturbative superstring the￾ory with its BRST invariant (string creation) vertex operators and proceed to

describe the “boundary state formalism” that describes the coupling of closed

strings to D branes. Then the authors use the latter to compute the interac￾tion between two D-branes, they discuss so-called BPS configurations whose

interactions 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: “Superconformal

field theories associated to very attractive quartics”. The terminology “at￾tractive” was introduced by Greg Moore (see his lectures below) for those

Calabi-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 the

geometrical realization of orbifold models on quartic surfaces and provides

some motivation for reading the following chapters.

In the second part: “Discrete groups and automorphic forms”, the theme is

arithmetic groups and some of their applications. Christophe Soul´e ’s lectures

“Introduction to arithmetic groups” set the stage in a more general context

than was considered in the lectures by E. Bogomolny in volume I of this

book. They begin with the classical reduction theory of linear groups of ma￾trices with integral coefficients and the normal parameterization of quadratic

forms. Then follows the general (and intrinsic) theory of algebraic Lie groups

over 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 IX

physics of chaos. The second chapter deals with presentations and finite or

torsion 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 for

most simple non-compact Lie groups. Automorphic forms are complex val￾ued functions defined over symmetric domains and invariant under arithmetic

groups, they arise abundantly in string theory.

Boris Pioline expanded his seminar with Andrew Waldron to give a physi￾cists’ introduction to “Automorphic forms and Theta series”. It starts with the

group theoretical and adelic expression of non holomorphic Eisenstein series

like E3/2 which has been extensively studied by M.B. Green and his collabo￾rators and also theta series. From there one studies examples of applications

of the orbit method and of parabolic induction. Among recent applications

and beyond the discrete U-duality groups already considered in the previous

lectures they discuss the minimal representation of SO(4,4) which arises also

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

a torus, new symmetries of chaotic cosmology and last but not least work in

progress on the description of black hole degeneracies and entropy computa￾tions. M-theory is the name of the unifying, hypothetical and polymorphic

theory that admits limits either in a flat classical background 11-dimensional

spacetime 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 anti-de￾Sitter space. One can compute the elliptic genus of that CFT as a Poincar´e

series that is interpretable on the AdS (i.e. gravity or string) side as a sum

of particle states and black hole contributions. This can serve as a concrete

introduction to many important ideas on Jacobi modular forms, Rademacher

expansion and quantum corrections to the entropy of black holes.

The second chapter of Moore’s lectures deals with the so called attractor

mechanism of supergravity. After compactification on a Calabi-Yau 3-fold X

one 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 the

charges 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 fact

valued 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 K3

surface is attractive. Secondly string theory compactification with fluxes turns

out 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 four

dimensional spacetimes. The moduli space of such universes is highly singular

and amenable to description by noncommutative geometry and C∗ algebras.

John McKay and Abdellah Sebbar introduce the concept and six possible

applications 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 also

the monstrous moonshine functions and are deeply related to the Schwarzian

derivative 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 author

himself they have two purposes, first of all they should present primarily to

physicists the Langlands program and especially its “geometric” part but on

the 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 from

the middle seventies and the powerful tool of topological twists of supersym￾metric theories which help to connect N = 4 super Yang-Mills theory in 4

dimensions to virtually everything else. The present work is actually about

mirror symmetry (T-duality) of related 2d supersigma models.

Specifically the lectures begin with the original Langlands program and

correspondences in the cases of number fields and of function fields. The

Taniyama-Shimura-Weil (modular) conjecture (actually a theorem now) is

discussed there. The geometric Langlands program is presented next in the

abelian 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 affine

Kac-Moody modules are introduced; at the negative critical level of the Kac￾Moody central charge the induced conformal symmetry degenerates and these

models lead to the Hecke eigensheaves expected from the geometric Langlands

correspondence.

The third and last theme of this volume is “Hopf algebras and renormal￾ization”. It leads to promising results on renormalization of Quantum Field

Theories that can be illustrated by concrete perturbative diagrammatic com￾putations but it also leads to the much more abstract and conceptual idea

of motives like a wonderful rainbow between the ground and the sky. In the

first set of lectures Pierre Cartier reviews the historical emergence of Hopf

Preface XI

algebras from topology and their structure theorems. He then proceeds to

Hopf algebras defined from Lie groups or Lie algebras and the inverse struc￾ture theorems. He finally turns to combinatorics instances of Hopf algebras

and some applications, (quasi)-symmetric functions, multiple zeta values and

finally multiple polylogarithms. This long and pedagogical introduction could

have continued into motives so we may be heading towards a third les Houches

school in this series.

Then comes the series of lectures by Alain Connes; they were written up in

collaboration with Matilde Marcolli. The lectures contain the most up-to date

research work by the authors, including a lot of original material as well as the

basic material in this exciting subject. They have been divided into two parts.

Chapter one appeared in the first volume and covered: “Quantum statistical

mechanics of Q-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 this

volume with the title: “Renormalization, the Riemann-Hilbert correspondence

and motivic Galois theory”. It starts with a detailed review of the results of

Connes and Kreimer on perturbative renormalization in quantum field theory

viewed as a Riemann-Hilbert problem and presents the Hopf algebra of Feyn￾man graphs which corresponds by the Milnor-Moore theorem to a graded Lie

algebra spanned by 1PI graphs. Singular cases lead to formal series and the

convergence aspects are briefly discussed towards the end.

The whole program is reformulated using the language of categories, al￾gebraic groups and differential Galois theory. Possible connections to mixed

Tate motives are discussed. The equivariance under the renormalization group

is reformulated in this language. Finally various tantalizing developments are

proposed.

Dirk Kreimer discusses then the problem of “Factorization in quantum

field theory: an exercise in Hopf algebras and local singularities”. He actually

treats a toy model of decorated rooted trees which captures the essence of

the resolution of overlapping divergences. One learns first how the Hochschild

cohomology of the Hopf algebra permits the renormalization program with

“locality”. Dyson-Schwinger equations are then defined irrespective of any

action and should lead to a combinatorial factorization into primitives of the

corresponding Hopf algebra.

Stefan Weinzierl in his seminar notes explains some properties of multiple

polylogarithms and of their finite truncations (nested sums called Z-sums)

that occur in Feynman loop integrals: “Algebraic algorithms in perturbative

calculations” and their impact on searches for new physics. Emphasis is on

analytical computability of some Feynman diagrams and on algebraic struc￾tures on Z-sums. They have a Hopf algebra structure as well as a conjugation

and 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 Herbert

Gangl, Alexander B. Goncharov and Andrey Levin on “Multiple logarithms,

XII Preface

algebraic cycles and trees”. This work has been extended to multiple polylog￾arithms and to the world of motives by the same authors. Here they relate

the three topics of their title among themselves, the last two are associated to

differential graded algebras of algebraic cycles and of decorated rooted trees

whereas the first one arises as an integral on hybrid cycles as a generalization

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

(poly-)logarithms.

We acknowledge most gratefully for their generous financial support to the

meeting the following institutions:

D´epartement Sciences Physiques et Math´ematiques and the Service de

Formation permanente of the Centre National de la Recherche Scientifique;

Ecole Normale Sup´ ´ erieure de Paris; D´epartement des Sciences de la mati`ere du

Commissariat `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 of ´

Mathematics 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 Quantum

chaos”, but the other two were “Superstrings” and “Quantum structure of

spacetime 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 also

grateful to Marcelle Martin, Thierry Paul and the staff of les Houches for their

patient help. We had the privilege to have two distinguished participants:

C´ecile de Witt-Morette (founder of the Les Houches School) and the late

Bryce de Witt whose communicative and critical enthusiasm were greatly

appreciated.

Paris B. Julia

July 2006 P. Cartier

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 In￾stitut de printemps sur les: “Fronti`eres entre Th´eorie des Nombres, Physique

et G´eom´etrie” qui s’est tenu au Centre de Physique des Houches dans les

Alpes 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 volume

contient, quant `a lui, seize contributions r´eparties ´egalement en trois th`emes:

Th´eories conformes pour les Cordes et les Branes, Groupes discrets et Formes

automorphes et enfin Alg`ebres de Hopf et Renormalisation.

Les relations entre Math´ematiques et Physique ont une longue histoire. Il

suffit de rappeler la m´ecanique et les ´equations diff´erentielles ordinaires, les

´equations aux d´eriv´ees partielles en m´ecanique des solides et des fluides ou en

´electromagn´etisme, la th´eorie des groupes qui est essentielle en cristallogra￾phie, en ´elasticit´e ou en m´ecanique quantique ...

La pr´e´eminence de la th´eorie des nombres et de parties plus abstraites des

math´ematiques comme les g´eom´etries topologique, diff´erentielle et alg´ebrique

s’est impos´ee plus r´ecemment. On en trouve des exemples divers dans les

travaux de scientifiques tels que: 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 et beaucoup

d’autres.

Une premi`ere conf´erence de ce type se tint en 1989 au Centre de Physique

des Houches. L’id´ee en ´etait venue alors `a Claude Itzykson (1938-1995). Cette

rencontre 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, Springer

Proceedings 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).

XIV Pr´eface

Dix ans apr`es, l’´evolution de l’interface entre physique th´eorique et math´e￾matiques poussa M. Waldschmidt, P. Cartier et B. Julia `a renouveler l’exp´erience.

Le choix des sujets changea donc quelque peu pour inclure cette fois-ci des

liens de la physique avec la g´eom´etrie, la th´eorie des matrices al´eatoires ou

des fonctions L et zˆeta vari´ees.

Une fois acquis le soutien de la Communaut´e europ´eenne l’organisation de

cette “High Level Scientific Conference” aux Houches fut lanc´ee.

Le Comit´e Scientifique de la conf´erence “Fronti`eres entre Th´eorie des Nom￾bres, Physique et G´eom´etrie” ´etait compos´e des scientifiques suivants: Frits

Beukers, Jean-Benoˆıt Bost, Pierre Cartier, Predrag Cvitanovic, Michel Duflo,

Giovanni Gallavotti, Patricio Leboeuf, Werner Nahm, Ivan Todorov, Claire

Voisin, Michel Waldschmidt, Jean-Christophe Yoccoz, et Jean-Bernard Zu￾ber.

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 chaque

jour `a environ soixante-quinze participants. Les sujets avaient ´et´e initialement

ordonn´es en trois groupes successifs avec comme pr´eoccupation essentielle de

coupler autant que faire se pouvait les cours de math´ematiques et ceux de

physique:

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 mo￾tiviques, de W. Nahm sur la Physique et les Dilogarithmes, et de D. Zagier

sur les Polylogarithmes;

3. Sym´etries et Physique non-perturbative,

avec des cours de A. Connes sur les Sym´etries Galoisiennes, Fonction zˆeta et

Renormalisation, 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 de

vertex, Courbes alg´ebriques et Programme de Langlands, G. Moore, Th´eorie

des 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 au￾raient pu donner des cours si le temps n’avait manqu´e. Ont donc parl´e: 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. Pio￾line, M. Pollicott, H. Then, E. Vasserot, A. Vershik, D. Voiculescu, A. Voros,

S. Weinzierl, K. Wendland et A. Zabrodin.

Pr´eface XV

Nous avons d´ecid´e de r´earranger les contributions ´ecrites `a ces Actes en

deux volumes dont voici le contenu.

Le premier volume rassemble quinze contributions et se compose de trois

parties 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 commence

par le cours d’Eug`ene Bogomolny qui passe en revue trois aspects du chaos

quantique, `a savoir les formules de trace avec ou sans chaos, la fonction de

corr´elation spectrale `a deux points des z´eros de la fonction zˆeta de Riemann et

enfin les fonctions de corr´elation spectrales de l’op´erateur de Laplace-Beltrami

pour des domaines modulaires sujets au chaos arithm´etique. Ces expos´es

forment une introduction informelle aux m´ethodes math´ematiques du chaos

quantique. 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 de

Brian 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), donc

des s´eries de Dirichlet qui ob´eissent `a une ´equation fonctionnelle similaire `a

celle que satisfait la fonction zˆeta de Riemann. Les fonctions L consid´er´ees

sont celles qui sont associ´ees `a des formes paraboliques. Les moments des

fonctions L sont reli´es aux fonctions de corr´elation des valeurs propres de

matrices al´eatoires.

Nous avons rassembl´e ensuite les textes de plusieurs s´eminaires: celui de

Jens Marklof reliant la statistique de certains niveaux d’´energie `a des fonc￾tions “presque modulaires”; celui de Holger Then sur le chaos quantique

arithm´etique dans un certain domaine hyperbolique `a trois dimensions et

son lien avec des formes de Maass; puis Paul Wiegmann et Anton Zabrodin

´etudient le d´eveloppement pour N grand d’ensembles de matrices complexes

normales; 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 espaces

m´etriques al´eatoires (universels).

Le th`eme de la deuxi`eme partie est: “‘Fonctions Zˆeta et applications”. Les

expos´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 les

bases de ce sujet excitant. On trouve dans le volume II leur deuxi`eme chapitre

sur la Renormalisation des th´eories quantiques des champs. Dans le premier

chapitre A. Connes et M. Marcolli introduisent l’espace non commutatif des

classes de commensurabilit´e des Q-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 des

z´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. commuta￾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