Frontiers in number theory, physics, and geometry I

637 Pages·2007·6.296 MB·English

by  Pierre Emile Cartier, Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove

FrontiersinNumberTheory,Physics,andGeometryI
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
PierreCartier PierreMoussa
I.H.E.S. ServicedePhysiqueThéorique
35routedeChartres CEA/Saclay
F-91440Bures-sur-Yvette F-91191Gif-sur-Yvette
France France
e-mail:[email protected] e-mail:[email protected]
BernardJulia PierreVanhove
LPTENS ServicedePhysiqueThéorique
24rueLhomond CEA/Saclay
75005Paris F-91191Gif-sur-Yvette
France France
e-mail:[email protected] e-mail:[email protected]
Coverphotos:
G.Pólya(courtesyofG.L.Alexanderson);EugeneP.Wigner(courtesyofM.Wigner).
LibraryofCongressControlNumber:2005936349
Mathematics Subject Classification (2000): 11A55, 11K50, 11M41, 15A52, 37C27,
37C30,58B34,81Q50,81R60
ISBN-10 3-540-23189-7SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-23189-9SpringerBerlinHeidelbergNewYork
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Preface
The present book contains fifteen contributions on various topics related to
NumberTheory,PhysicsandGeometry.Itpresents,togetherwithaforthcom-
ingsecondvolume,mostofthecoursesandseminarsdeliveredatthemeeting
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
mentiononlyordinarydifferentialequationsandmechanics,partialdifferential
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
suchastopological,differentialandalgebraicgeometryinphysicshasbecome
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 firstmeeting took place at the Centre dePhysique 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-BenoˆıtBost,PierreCartier,PredragCvitanovic, MichelDuflo,
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,
withlecturesbyE.BogomolnyonQuantumandarithmeticalchaos,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.
Inadditionseminarsweregivenbyparticipantsmanyofwhomcouldhave
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
onthetheme“Randommatrices:fromPhysicstoNumberTheory”.Itbegins
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
introductiontomathematicalmethodsofquantumchaos.Ageneralintroduc-
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,inrelationtoMaasswaveforms.NextP.WiegmannandA.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 propertiesof 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.
Inthethirdpart“Dynamicalsystems:intervalexchanges,flatsurfacesand
smalldivisors”,aregatheredalltheothercontributionsondynamicalsystems.
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 weconclude this volume with seminarsgiven by G.Gentile on Br-
junonumbersand dynamical systemsand by S.Marmion Realand 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;
E´coleNormaleSup´erieuredeParis;D´epartementdesSciencesdelamati`eredu
Commissariat`al’E´nergieAtomique;InstitutdesHautesEtudesScientifiques;
National Science Foundation; Minist`ere de la Recherche et de la Technolo-
gie and Minist`ere des Affaires E´trang`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
gratefultoMarcelleMartin,ThierryPaulandthestaffoflesHouchesfortheir
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,LaboratoiredePhysiqueTh´eoriqueetMod`elesStatistiques
Universit´e de Paris XI, Bˆat. 100, 91405 Orsay Cedex, France
• J.BrianConrey,AmericanInstituteofMathematics,PaloAlto,CA,USA
• Jens Marklof, School of Mathematics, University of Bristol, Bristol BS8
1TW, U.K.
• H.Then,AbteilungTheoretischePhysik,Universita¨tUlm,Albert-Einstein-
Allee 11, 89069 Ulm, Germany
• A. Zabrodin, Institute of Biochemical Physics, Kosygina str. 4, 119991
Moscow,RussiaandITEP,Bol.Cheremushkinskayastr.25,117259Moscow,
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–PlanckInstitutfu¨rMathematik,Vivatsgasse7,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,DipartimentodiMatematica,Universita`diRomaTre,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]
• PierreMoussa,ServicedePhysiqueTh´eorique, CEA/Saclay, F-91191Gif-
sur-Yvette, France, e-mail: [email protected]
• PierreVanhove,ServicedePhysiqueTh´eorique,CEA/Saclay,F-91191Gif-
sur-Yvette, France