Tài liệu Frontiers in Number Theory, Physics, and Geometry I ppt

Frontiers in Number Theory, Physics, and Geometry I

Pierre Cartier Bernard Julia

Pierre Moussa Pierre Vanhove (Eds.)

Frontiers in Number Theory,

Physics, and Geometry I

On Random Matrices, Zeta Functions,

and Dynamical Systems

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

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:

G. Pólya (courtesy of G.L. Alexanderson); Eugene P. Wigner (courtesy of M. Wigner).

Library of Congress Control Number: 2005936349

Mathematics Subject Classification (2000): 11A55, 11K50, 11M41, 15A52, 37C27,

37C30, 58B34, 81Q50, 81R60

ISBN-10 3-540-23189-7 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-23189-9 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is

concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,

reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication

or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,

1965, in its current version, and permission for use must always be obtained from Springer. Violations are

liable for prosecution under the German Copyright Law.

Springer is a part of Springer Science+Business Media

springer.com

c Springer-Verlag Berlin Heidelberg 2006

Printed in The Netherlands

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,

even in the absence of a specific statement, that such names are exempt from the relevant protective laws

and regulations and therefore free for general use.

Typesetting: by the authors and TechBooks using a Springer LATEX macro package

Cover design: Erich Kirchner, Heidelberg

Printed on acid-free paper SPIN: 10922873 41/TechBooks 543210

Preface

The present book contains fifteen contributions on various topics related to

Number Theory, Physics and Geometry. It presents, together with a forthcom￾ing second volume, 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.

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

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, ran￾dom matrices or various zeta- and L- functions. Once the project of the new

meeting entitled “Frontiers in Number Theory, Physics and Geometry” re￾ceived support from the European Union the High level scientific conference

was organized in Les Houches.

VI Preface

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-pertubative Physics, with lectures by

A. Connes on Galoisian symmetries, zeta function and renormalization,

R. Dijkgraaf on String duality and automorphic forms,

P. Di Vecchia 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, A. Zabrodin.

We have chosen to reorganize the written contributions in two parts ac￾cording to their subject. These naturally lead to two different volumes. The

present volume is the first one, 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 “Random matrices : from Physics to Number Theory”. It begins

with lectures by E. Bogomolny, which review three selected topics of quan￾tum chaos, namely trace formulas with or without chaos, the two-point spec￾tral correlation function of Riemann zeta function zeroes, and the two-point

spectral correlation functions of the Laplace-Beltrami operator for modular

Preface VII

domains leading to arithmetic chaos. The lectures can serve as a non-formal

introduction to mathematical methods of quantum chaos. A general introduc￾tion to arithmetic groups will appear in the second volume. There are then

lectures by J. Conrey who examines relations between random-matrix theory

and families of arithmetic L-functions (mostly in characteristics zero), that is

Dirichlet series satisfying functional equations similar to those obeyed by the

Riemann zeta-function. The relevant L-functions are those associated with

cusp-forms. The moments of L-functions are related to correlation functions

of eigenvalues of random matrices.

Then follow a number of seminar presentations: by J. Marklof on some

energy level statistics in relation with almost modular functions; by H. Then

on arithmetic quantum chaos in a particular three-dimensional hyperbolic

domain, in relation to Maass waveforms. Next P. Wiegmann and A. Zabrodin

study the large N expansion for normal and complex matrix ensembles. D.

Voiculescu reviews symmetries of free probability models. Finally A. Vershik

presents some random (resp. universal) graphs and metric spaces.

In the second part “Zeta functions: a transverse tool”, the theme is zeta￾functions and their applications.

First the lectures by A. Connes were written up in collaboration with M.

Marcolli and have been divided into two parts.

The second one will appear in the second volume as it relates to renor￾malization of quantum field theories. In their first chapter they introduce

the noncommutative space of commensurability classes of Q-lattices and the

arithmetic properties of KMS states in the corresponding quantum statistical

mechanical system. In the 1-dimensional case this space gives the spectral

realization of zeroes of zeta-functions. They give a description of the multiple

phase transitions and arithmetic spontaneous symmetry breaking in the case

of Q-lattices of dimension two. The system at zero temperature settles onto a

classical Shimura variety, which parametrizes the pure phases of the system.

The noncommutative space has an arithmetic structure provided by a ratio￾nal subalgebra closely related to the modular Hecke algebra. The action of

the symmetry group involves the formalism of superselection sectors and the

full noncommutative system at positive temperature. It acts on values of the

ground states at the rational elements via the Galois group of the modular

field.

Then we report seminars given by A. Voros on zeta functions built on

Riemann zeroes; by J. Lagarias on Hilbert spaces of entire functions and

Dirichlet L-functions; and by M. Pollicott on Dynamical zeta functions and

closed orbits for geodesic and hyperbolic flows.

In the third part “ Dynamical systems: interval exchanges, flat surfaces and

small divisors”, are gathered all the other contributions on dynamical systems.

The lectures by A. Zorich provide an extensive self-contained introduction to

the geometry of Flat surfaces which allows a description of flows on compact

VIII Preface

Riemann surfaces of arbitrary genus. The course by J.-C. Yoccoz analyzes

Interval exchange maps such as the first return maps of these flows. Ergodic

properties of maps are connected with ergodic properties of flows. This leads

to a generalization to surfaces of higher genus of the irrational flows on the

two dimensional torus. The adaptation of a continued fraction like algorithm

to this situation is a prerequisite to extension of small divisors techniques to

higher genus cases.

Finally we conclude this volume with seminars given by G. Gentile on Br￾juno numbers and dynamical systems and by S. Marmi on Real and Complex

Brjuno functions. In both talks either perturbation of irrational rotations or

twist maps are considered, with fine details on arithmetic conditions (Brjuno

condition and Brjuno numbers) for stability of trajectories under perturba￾tions of parameters, and on the size of stability domains in the parametric

space (Brjuno functions).

The following institutions are most gratefully acknowledged for their gen￾erous financial support to the meeting:

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

mathematical physics and most especially the Commission of the European

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, July 2005 Bernard Julia

Pierre Cartier

Pierre Moussa

Pierre Vanhove

List of Contributors

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

• E. Bogomolny, Laboratoire de Physique Th´eorique et Mod`eles Statistiques

Universit´e de Paris XI, Bˆat. 100, 91405 Orsay Cedex, France

• J. Brian Conrey, American Institute of Mathematics, Palo Alto, CA, USA

• Jens Marklof, School of Mathematics, University of Bristol, Bristol BS8

1TW, U.K.

• H. Then, Abteilung Theoretische Physik, Universit¨at Ulm, Albert-Einstein￾Allee 11, 89069 Ulm, Germany

• A. Zabrodin, Institute of Biochemical Physics, Kosygina str. 4, 119991

Moscow, Russia and ITEP, Bol. Cheremushkinskaya str. 25, 117259 Moscow,

Russia

P. Wiegmann, James Frank Institute and Enrico Fermi Institute of the

University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA

Landau Institute for Theoretical Physics, Moscow, Russia

• D. Voiculescu, Department of Mathematics University of California at

Berkeley Berkeley, CA 94720-3840, USA

• A.M. Vershik, St.Petersburg Mathematical Institute of Russian Academy

of Science Fontanka 27 St.Petersburg, 191011, Russia

• 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

• A. Voros, CEA, Service de Physique Th´eorique de Saclay (CNRS URA

2306) F-91191 Gif-sur-Yvette Cedex, France

• J.C. Lagarias, Department of Mathematics, University of Michigan, Ann

Arbor,MI 48109-1109 USA

• M. Pollicott, Department of Mathematics, Manchester University, Oxford

Road, Manchester M13 9PL UK

• J.-C. Yoccoz, Coll`ege de France, 3 Rue d’Ulm, F-75005 Paris, France

• A. Zorich, IRMAR, Universit´e de Rennes 1, Campus de Beaulieu, 35042

Rennes, France

• G. Gentile, Dipartimento di Matematica, Universit`a di Roma Tre, I-00146

Roma, Italy

• S.Marmi, Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa,

Italy

X List of Contributors

P. Moussa, Service de Physique Th´eorique, CEA/Saclay, F-91191 Gif-sur￾Yvette, France

J.-C. Yoccoz, Coll`ege de France, 3 Rue d’Ulm, F-75005 Paris, France

Editors:

• Bernard Julia, LPTENS, 24 rue Lhomond 75005 Paris, France, e-mail:

[email protected]

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

France, e-mail: [email protected]

• Pierre Moussa, Service de Physique Th´eorique, CEA/Saclay, F-91191 Gif￾sur-Yvette, France, e-mail: [email protected]

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

Contents

Part I Random Matrices: from Physics to Number Theory

Quantum and Arithmetical Chaos

Eugene Bogomolny ............................................... 3

Notes on L-functions and Random Matrix Theory

J. Brian Conrey ................................................. 107

Energy Level Statistics, Lattice Point Problems, and Almost

Modular Functions

Jens Marklof .................................................... 163

Arithmetic Quantum Chaos of Maass Waveforms

H. Then ........................................................ 183

Large N Expansion for Normal and Complex Matrix

Ensembles

P. Wiegmann, A. Zabrodin ........................................ 213

Symmetries Arising from Free Probability Theory

Dan Voiculescu .................................................. 231

Universality and Randomness for the Graphs and Metric

Spaces

A. M. Vershik ................................................... 245

Part II Zeta Functions

From Physics to Number Theory via Noncommutative

Geometry

Alain Connes, Matilde Marcolli .................................... 269

XII Contents

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

Mark Pollicott ................................................... 381

Part III Dynamical Systems: interval exchange, flat surfaces, and

small divisors

Continued Fraction Algorithms for Interval Exchange Maps:

an Introduction

Jean-Christophe Yoccoz ........................................... 403

Flat Surfaces

Anton Zorich .................................................... 439

Brjuno Numbers and Dynamical Systems

Guido Gentile ................................................... 587

Some Properties of Real and Complex Brjuno Functions

Stefano Marmi, Pierre Moussa, Jean-Christophe Yoccoz ............... 603

Part IV Appendices

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

Part I

Random Matrices: from Physics to Number

Theory

Quantum and Arithmetical Chaos

Eugene Bogomolny

Laboratoire de Physique Th´eorique et Mod`eles Statistiques

Universit´e de Paris XI, Bˆat. 100, 91405 Orsay Cedex, France

[email protected]

Summary. The lectures are centered around three selected topics of quantum

chaos: the Selberg trace formula, the two-point spectral correlation functions of

Riemann zeta function zeros, and the Laplace–Beltrami operator for the modular

group. The lectures cover a wide range of quantum chaos applications and can serve

as a non-formal introduction to mathematical methods of quantum chaos.

Introduction .............................................. 5

I Trace Formulas ........................................... 7

1 Plane Rectangular Billiard ................................ 7

2 Billiards on Constant Negative Curvature Surfaces . . . . . . . . 15

2.1 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Quantum Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Construction of the Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Density of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Conjugated Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 Selberg Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.9 Density of Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 Selberg Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.11 Zeros of the Selberg Zeta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.12 Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Trace Formulas for Integrable Dynamical Systems . . . . . . . . 33

3.1 Smooth Part of the Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Oscillating Part of the Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Trace Formula for Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Semiclassical Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Eugene Bogomolny

4.2 Gutzwiller Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Trace Formula for the Riemann Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Chaotic Systems and the Riemann Zeta Function . . . . . . . . . . . . . . . . 46

6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

II Statistical Distribution of Quantum Eigenvalues . . . . . . . . . . . 49

1 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.1 Diagonal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.2 Criterion of Applicability of Diagonal Approximation . . . . . . . . . . . . 55

2 Beyond the Diagonal Approximation . . . . . . . . . . . . . . . . . . . . . . 58

2.1 The Hardy–Littlewood Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2 Two-Point Correlation Function of Riemann Zeros . . . . . . . . . . . . . . . 64

3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

III Arithmetic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

1 Modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2 Arithmetic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1 Algebraic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.2 Quaternion Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.3 Criterion of Arithmeticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.4 Multiplicities of Periodic Orbits for General Arithmetic Groups . . . 82

3 Diagonal Approximation for Arithmetic Systems . . . . . . . . . . 85

4 Exact Two-Point Correlation Function for the Modular

Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1 Basic Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Two-Point Correlation Function of Multiplicities . . . . . . . . . . . . . . . . 89

4.3 Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Two-Point Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Jacquet–Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . 98

7 Non-arithmetic Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103