Preface

The 19th International Conference on Differential Geometric Methods in Theoret- ical Physics was held in Rapallo (Italy) from June 19th to 24th, 1990. Since its inception in 1971 in Bonn, this series of conferences has focused, broadly speak- ing, on those mathematical methods in physics which are of a differential-geometric nature. Among the various topics that formed the programmes of the last confer- ences, four themes seem to have played a central role, namely, non-commutative differential geometry, quantum groups, superalgebras and supermanifolds, and a complex of arguments embracing conformal field theory, integrable systems, and statistical mechanics. It was therefore decided to devote this 19th conference to these topics. The papers included in this proceedings volume are correspondingly divided into the following four sections:

- - non-commutative differential geometry; - - quantum groups and integrable systems; - - conformal field theory and related topics; - - superalgebras and supermanifolds.

A fifth section (problems in quantum field theory) contains two papers which do not really fit into any of the above divisions.

The conference programme included some thirty-five invited lectures. Since this 1990 meeting was intended to be particularly specialized, some "crash cours- es" covering basic materials were planned; they each consisted of two one-hour lectures, and were held by D. Kastler (non-commutative differential geometry), C. De Concini (quantum groups), and J. FrShlich (low-dimensional quantum field theory). To our regret, not all the lecturers submitted the text of their lectures for publication in these proceedings, so that only thirty-one papers have been in- cluded. In addition to these, eight short articles - - reflecting the contents of some of the posters presented during the conference - - make up a separate section.

A major feature of this conference was that, for the first time in this series, a fair number of the participants (about 30 out of a total of 160) came from Eastern Europe (including the German Democratic Republic and the USSR). This is of course another consequence of the remarkable political changes that have occurred in those countries in recent years. We hope this is a good omen for the future and that a closer and closer worldwide collaboration among physicists and mathematicians will take place.

A conference like this involves a considerable financial effort. It is therefore a pleasure to acknowledge financial help from those institutions which made the conference possible. These were the Committee for Mathematics of the Italian Re- search Council (CNR), the National Group for Mathematical Physics of CNR, the National Institute for Nuclear Physics (INFN), the City of Rapallo administra- tion, the Regione Liguria administration, the University of Genoa, and the "Cassa di Risparmio di Genova e Imperia." We also acknowledge clerical help from the

IV

Department of Mathematics of the University of Genoa, the City of Rapailo, and the Tourist Office of Rapallo.

The conference was opened by some welcome speeches, which were delivered by the Rector of the University of Genoa, Prof. Enrico Beltrametti, by the Mayor of the City of Rapailo, Dr. Mauro Cordano, by the Chairman of the Depart- ment of Mathematics of the University of Genoa, Prof. Giovanni Pistone, by the coordinator of the National Research Project "Metodi geometrici in relativith e teorie di campo" of the Italian Ministry for Universities, Research and Technol- ogy (MURST), Prof. Mauro Francaviglia, and, finally, by Prof. Konrad Bleuler. We would like to thank all these distinguished personalities for their contributions to the conference. Special thanks are due to Professor Bleuler for his exceptional and long-standing commitment to this series of conferences on the differential- geometric methods in theoretical physics and for the enthusiasm he is able to communicate to all of us.

Genoa, November 1990 The Editors

Foreword by Professor K. Bleuler

Institute for Theoretical Nuclear Physics University of Bonn

Recent years have witnessed most impressive and far-reaching developments in mathematical physics. This is to a large extent due to a renewed and deep-rooted discussion between mathematicians and physicists. In both domains there have been, in completely different directions, important developments (i.e. topology on the one hand and particles on the other) leading to a breathtaking confrontation. We are therefore reminded of that great earlier period of the 1920s with the simul- taneous creation and development of general relativity and quantum theory; that decisive step in the history of physics could not be conceived of without the contri- butions of the greatest mathematicians of that time, e.g. Hilbert, Weyl, Poincar6, and others. Their feeling of that profound mystery of "finding, or better, discover- ing basic mathematical structures hidden far behind empirical physical facts" has remained the leading idea for understanding nature ever since.

In the course of recent years such "idealistic" guidelines have determined the research projects to a large extent, too: generalized geometrical, topological and group theoretical principles have become the decisive tools for the interpretation and understanding of the vast mass of data resulting from the enormous experi- mental efforts of our time. This situation had, in a way, been foreseen in Plato's philosophy: his famous words, as emphasized and adapted by Heisenberg, "sym- metries axe more basic than particles" appear in fact to have been realized in a literal way by modern gauge theory.

Within the huge body of empirical data in hadron physics there are a practi- cally infinite number of different heavy fermions and bosons originally assumed to be elementary; one of the aims of thegange-theoretical approach to strong interac- tions is to reinterpret these different mass values as eigenvalues of the quark-gluon system which is singled out by the Yang-Mills local gauge principle and by a special choice of the gauge group, i.e. the group SU(3). Thus the invariances or symmetries of this system, in analogy to Einstein's gravitation theory and Salam and Weinberg's theory of electroweak processes, should determine all empirical masses.

In a rather problematic additional step the remaining elementary particles, i.e. leptons and quarks, were to be interpreted as the quantum states of a new and enlarged geometric structure: the so-called string. This "heroic" attempt, called "string theory," with its enormous hopes (it has been called "the theory of ev- erything") and its great disappointments, led, in any case, to an unprecedented impetus to mathematical research related to this geometric structure. The result was a far-reaching development of already known and new methods, even leading

Yl

to the creation of previously unknown domains of pure mathematics. These in turn allowed the discovery and understanding of interesting interrelations among var- ious conventional structures and physical problems of practical importance, such as statistical mechanics, superconductivity and the quantum Hall effect. As more general examples I might cite the sequence

knots and links ~ Jones polynomials ~ Yang-Baxter equations ~ statistical mechanics , conformal field theory

and the wonderful revival of Hopf's classical work triggered off by the present-day concept of "quantum groups".

The main consequence of this undertaking is the insight that the conventional concept of space-time must undergo - - according to Riemann~s very first sugges- tion - - a thourough revision for the case of smallest dimensions. This led to the development of a "p-adic geometry," and to the "non-commutative differential ge- ometry" of A. Connes and D. Kastler. With this proposal a completely new and basic chapter in the history of physics has, in fact, been opened: according to a personal "message" of W. Pauli the most disturbing mathematical difficulties encountered in relativistic quantum field theory (the basis of theoretical physics for half a century) will appear in a new light and might lead to a new concept of elementary particles - - perhaps in a certain way analogous to the one suggested by string theory.

Thus, a novel and exciting stage of research has been initiated: new and ex- tended geometric structures cover simultaneously and in a most successful way very different domains of physics, from the lowest to the highest energies. This prompts an exchange between so far completely separate domains in physics and calls for a deeper understanding of the abstract but nonetheless intuitive structures in modem mathematics that are inherent in physical laws.

Enlarging this quest for human understanding and exchange to all nations of our world we are immediately reminded of the beautiful location of this meeting: RapaUo was, in fact, the place of the very first contacts and handshakes between the enemies of the first world war. It might thus in our days contribute to a deeper understanding and a real friendship between East and West. For this reason I heartily welcome our friends and participants from various eastern countries. In this connection I should not forget to express our great appreciation to the mayor of this city, Dr. Mauro Cordano, for this extremely kind hospitality, as well as - - speaking on behalf of us all - - to convey our heartiest thanks and greetings to the dedicated organizers of our meeting from the Department of Mathematics of the University of Genoa, Ugo Bruzzo~ Claudio Bartocci and Roberto Cianci: you did really a wonderful job, very much in the "spirit of Rapallo."

Bonn, October 1990

Foreword by Professor M. Francaviglia

Istituto di Fisica Matematica "J.-L. Lagrange" University of Turin

It is a great honour for me to be here in Rapallo at the official opening of this important and beautiful international conference and to have the opportunity of welcoming in Italy all the friends and colleagues who have come from all over the world to participate.

My present task is twofold. On the one hand I have the pleasant duty of sitting here and representing one of the major sponsors .who made this conference possible. I offer you a warm welcome from CNR (the Italian National Research Council); on behalf of its Scientific Committee I bring you in particular best wishes for a fruitful stay from GNFM (the National Group of Mathematical Physics). GNFM supports a large part of the scientific activity in Italy in the domain of mathematical physics, through a Visiting Professorship program and through the sponsorship of a limited number of conferences, among which this 19th DGM will certainly be one of the major events of 1990. On the other hand I also have the great honour of welcoming all participants on behalf of the National Research Project "Geometria e Fisica", of our MURST (Ministry of Universities, Research and Technology). This project was started in Italy ten years ago, with the aim of promoting and coordinating Italian research and international collaboration in the fields of interaction between physics and geometry. Besides being a relevant source of research funding in these years, this project has extensively helped the Italian scientific community to develop a number of coherent lines of research in this beautiful domain. Nowadays the project comprises over a hundred Italian investigators, including a large number of young researchers, belonging to 16 Universities scattered through the whole country. As the national coordinator of this project, it is my greatest pleasure to be able to participate in such an important task. This conference, which came into being entirely due to the active and strong will of the local group of this project working at the University of Genoa, will surely represent a milestone in the life of the project itself.

Although I can see many colleagues in the audience who would be much better than me in this job, I will nonetheless try here to stress in a few words the impor- tance of the subject we shall be discussing in Rapallo for these six days. The rela- tions between geometry and physics were already hidden in the celebrated treatise "M6chanique Analitique", written by Joseph-Louis Lagrange in 1788, where the basis of the modern approach to theoretical physics was laid down. These intimate relations were subtly envisaged as a means of understanding the very structure of our universe by the genius of Bernhard Riemann (1854) and fully developed by Albert Einstein in his famous and fundamental theory of general relativity

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(1916). The ideas and the methods embraced by Einstein in fact constituted one of the leading themes in the development of "classical" (non-quantum) theoretical physics, both in the direction of gravitational theories and, more recently, in the in- terpretation of gauge field theories in terms of principal connections. Soon after the formulation of general relativity, physics and mathematics seemed to have come to live in perpetual harmony, and for a few years it was believed that geometry would forever be the unifying language of physics. In the 1920s, however, the revolution of quantum mechanics began. With the advent of quantum mechanics, and later on with the extensive development of quantum field theory, physicists increasingly regarded analysis as the natural language for their discipline. Thus, theoretical physics and geometry, the latter oriented in those years towards the creation of more abstract conceptions, experienced a sort of repulsive force. Geometry, which in fact plays a fundamental role in the local formulation of classical physical laws, was long believed to be incapable of shedding light on quantum phenomena and hence relegated to be the language of just the "old" physics. Perhaps one of the reasons why a satisfactory solution to the problem of a coherent unification of gravity and quantum physics has so far remained elusive should be looked for in this divorce. This apparent dichotomy lasted for almost fifty years. However, the last two decades have seen a profound internal unification, both in mathe- matics and in physics, where deep-seated relations between seemingly unrelated fields have been discovered. Along with these internal revolutions, a renewed and stronger interaction between geometry and physics has taken place, and today we sense a great excitement as large portions of both disciplines are coming together. While until a few years ago the interaction of geometry and physics was mainly limited to the domains of differential geometry and to "classical" field theories, in the recent past other fundamental branches of geometry have found their way into physics and stimulated its development, often giving an enormous impetus to the investigations concerning the global behaviour of physical fields and their quan- tum properties, the structure of continua with their defects, the fascinating world of solitons and completely dynamical and quantum integrable systems. While the physicist of 20 years ago just spoke of groups, manifolds, tensors and perhaps prin- cipal connections, today we see the growing importance of subjects which perhaps are less familiar to physicists, like cohomology, supermanifolds, algebraic geometry, deformations, and even non-commutative geometry.

The meetings on Differential Geometrical Methods in Theoretical Physics have in these past 20 years been a fundamental forum where mathematicians and physi- cists could come together and work for the reconstruction of this important sym- biosis. Looking at the impressive program of this 19th conference we can be sure that this tradition is still alive and able to mantain the rapid evolution of the subject. Let us enjoy this conference and thank once more the organizers for their magnificent work!

Turin, October 1990

C o n t e n t s

1. N o n - c o m m u t a t i v e different ial g e o m e t r y

R. Coquereaux

M. Dubois-Violette

D. Kastler

Higgs fields and superconnections . . . . . . . . . . . . . 3

Noncommutative differential geometry, quantum mechanics and gauge theory . . . . . . . 13

Introduction to non-commmutative geometry and Yang-Mills model-building . . . . . . . . . . . . . . 25

2. Q u a n t u m groups and integrable sys tems

M. Batchelor

L.C. Biedenharn

R.K. BuUough, J. Timonen

F. Calogero

S. De Filippo, G. Landi, G. Marmo, G. Vilasi

H.J. de Vega

C. G6mez, G. Sierra

S. Majid

M.E. Mayer

J. Rawnsley

Measuring coaigebras, quantum group-like objects, and non-commutative geometry . . . . . . 47

Tensor operator structures in quantum unitary groups . . . . . . . . . . . . . . . . . . . . . 61

Quantum groups and quantum complete integrability: theory and experiment . . . . . . . . . . 71

Some ideas and results on integrable nonlinear evolution systems . . . . . . . . . . . . . . . . . . 91

An algebraic characterization of complete integrability for Hamiltonian systems . . . . . . . . . 96

Integrable lattice models and their scaling limits: QFT and CFT . . . . . . . . . . . . . . . . 107

Quantum groups, Riemann surfaces and conformal field theory . . . . . . . . . . . . . . . . . . . 120

Some physical applications of category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

From Poisson groupoids to quantum groupoids and back . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Quantization on Kghler manifolds . . . . . . . . . . . . . 155

M.V. Saveliev, A.M. Vershik A new class of infinite-dimensional Lie algebras (continuum Lie algebras) and associated nonlinear systems . . . . . . . . . . . . . . . . . 162

3. Conformal field theory and related topics

O. Babelon, L. Bonora

K.M. Bugajska

J.A. Dixon

G. Falqui, C. Reina

A. Floer

C. Itzykson

R.J. Lawrence

A. Rogers

A. Vaintrob

Exchange algebra in the conformal afflne sl2 Toda field theory . . . . . . . . . . . . . . . . . . . . . . . . . 173

Some properties of P-lines . . . . . . . . . . . . . . . . . . . . . 185

Breaking of supersymmetry through anomalies in composite spinor operators . . . . . . . 198

Conformal field theory and moduli spaces of vector bundles over variable Riemann surfaces . . . . . . . . . . . . . . . 209

Instanton homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

W-Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Connections between CFT and topology via knot theory ..................... 245

Stochastic calculus in superspace and supersymmetric Hamiltonians ............ 255

Geometric models and the moduli spaces for string theories ..................... 263

4. Superalgebras and supermanifolds

J.A. Domlnguez P6rez, ' D. Herugndez Ruip6rez, C. Sancho de Salas

D. Le~tes, V. Serganova, G. Vinel

Z. Oziewicz

I. Penkov, V. Serganova

3.M. Rabin

M. Rothstein

Supersymmetric products of SUSY-curves . . . . . 271

Classical superspaces and related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

Remarks on the differential identities in Schouten-Nijenhuis algebra . . . . . . . . . . . . . . . . 298

Generic irreducible representations of classical Lie superalgebras . . . . . . . . . . . . . . . . . 311

Krichever construction of solutions to the super KP hierarchies . . . . . . . . . . . . . . . . . . 320

The structure of supersymplectic supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

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5. Problems in quantum field theory

G. Dell'Antonio

Y. Ne'eman, C.-Y. Lee

Gauge fixing: geometric and probabilistic aspects of Yang-Mills gauge theories . . . . . . . . . . 347

A renormalizable theory of quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 359

6. Short contributions

S. Carillo

P. Furlan, A. Ganchev V. Petkova

A. Jalfe, O. Stoytchev

R. Loll

I.M. Mladenov

E. Poletaeva

T. Schmitt

P. Teofilatto

Third order nonlinear Hamiltonian systems: some remarks on the action-angle transformation . . . . . . . . . . . . . . . . . . 375

Tensor products of qP = 1 quantum groups and WZW fusion rules . . . . . . . . . . . . . . . . 379

The modular group and super-KMS functionals . . . . . . . . . . . . . . . . . . . . . . . 382

New quantum representation for gravity and Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . 385

Geometric quantization of the five-dimensional Kepler problem . . . . . . . . . . . . . . 387

Structure functions on the usual and exotic symplectic and periplectic supermanifolds . . . . 390

Symbols alias generating functionals - - a supergeometric point of view . . . . . . . . . . . . . . . 396

Sheaves of graded Lie algebras over variable Riemann surfaces and a paired Weil-Petersson inner product . . . . . . . . . . . . . . . . . 399

List of Participants

I. Aref 'eva, Steklov Mathematical Institute, U1. Vavilova 42, Moscow, GSP-1 117966 U.S.S.R.

I. Avramidi, Universit£t Karlsruhe, Institut fiir Theoretische Physik, Kaiserstrat3e 12, D-7500 Karlsruhe, F. R. Germany. E-Marl: BE12@DKAUNI2.BITNET

H. Bacry, C.N.R.S. - - Centre de Physique Thdorique, Case 907, F-13288 Marseille Luminy Cedex 8, France

C. Bartocci, Universit£ di Genova, Dipartimento di Matematica, Via L.B. Alberti 4, 1-16132 Genova, Italia.

M. Batchelor, Cambridge University, D.A.M.M.S., Mill Lane, Cambridge CB3 9EW, United Kingdom

C.M. Becchl, Universit£ di Genova, Dipartimento di Fisica, Via Dodecaneso 33, 1-16146 Genova, Italia

R. Bekhechl, King's College, Dept. of Mathematics, Strand, London WC2R 2LS, United Kingdom. E-Mail: RBEKHECHI@OAK.CC.KCL.AC.UK

L. Bettge, Universitiit Dortmund, FB Physik, Postfach 500500, D-4600 Dortmund 50, F. R. Germany

L.C. Biedenharn, Duke University, Dept. of Physics, Durham, NC 27706, U.S.A. E-Mail: JEC@PHY.DUKE.EDU

W. Bischoff, Albert-Ludwig-Universit£t, Fakult~t fiir Physik, Hermann-Herder- Strai3e 3, D-7800 Freiburg, F. R. Germany

A. Blasl, Universit£ di Genova, Dipartimento di Fisica, Via Dodecaneso 33, 1-16146 Genova, Italia

K. Bleuler, Institut f/ir Theoretische Kernphysik der Universit£t Bonn, Nussallee 14-16, D-5300 Bonn, F. R. Germany

F.J. Bloore, The University of Liverpool, D.A.M.T.P., P.O. Box 147, Liverpool L69 3BX, United Kingdom. E-Mail: SX35~LIVERPOOL.AC.UK

L. Bonora, S.I.S.S.A., Strada Costiera 11, 1-34014 Miramare-Grignano TS, Italia. E-Mail: BONORA@ITSSISSA.BITNET

V. Bonservizi, S.I.S.S.A., Strada Costiera 11, 1-34014 Miramare-Grignano TS, Italia. E-Mail: BONSERVI@ITSSISSA.BITNET

L.J. Boya, University of Texas at Austin, Physics Department, Austin, TX 78712 U.S.A. E-Marl: LUISJO~UTAPHY.BITNET

U. Bruzzo, Universit£ di Genova, Dipartimento di Matematica, Via L.B. Alberti 4, 1-16132 Genova, Italia. E-Maih BRUZZO~IGECUNIV.BITNET

K. Bugajska, York University, Dept. of Mathematics and Statistics, North York, Ontario, Canada M J3 1P3. E-Mail: BUGAJSKA~YORKVM1.BITNET

l:t.K. Bullough, U.M.I.S.T., Dept. of Mathematics, SackviUe Street, P.O. Box 88, Manchester M60 1QD United Kingdom. E-Mail: RKBULLOUGH@UMIST.AC.UK

N. Burroughs, Cambridge University, D.A.M.T.P., Silver Street, Cambridge CB3 9EW, United Kingdom. E-Mail: NJB16~PHX.CAM.AC.UK

C. Buzzanca, Universit£ di Palermo, Dipartimento di Matematica e Applicazioni, Via Arch]raft 34, 1-90123 Palermo, Italia.

E. Caccese, Dipartimento eli Matematica, Universit£ della Basilicata, Potenza, Italia

XIV

S. Carillo, Universit£ di Roma "La Sapienza", Dipart. Metodi e Modelli Matematici, Via A. Scarpa 10, 1-00161 Roma, Italia. E-Mail: CARILLO~ROMAI.INFN.IT

U. Carow-Watamura , Universit£t Karlsruhe, Institut f/Jr Theoretische Physik, Kaiserstrat3e 12, D-7500 Karlsruhe, F. R. Germany. E-Mail: BE05@DKAUNI2.- BITNET

R. Catenacci, Universit£ di Trieste, Dipartimento di Matematica, P.le Europa 1, 1-34017 Trieste, Italia. E-Mail: CATENACCI~PAVIA.INFN.IT

S. Catto, The City University of New York, Baruch College, Physics. Dept., 17 Lexington Ave., New York, NY 10010 U.S.A. E-Mail: SCBBB@CUNYVM.BITNET

R. Cianci, Universit£ di Genova, Dipartimento di Matematica, Via L.B. Alberti 4, 1-16132 Genova, Italia. E-Mail: CIANCI@IGECUNIV.BITNET

R. Collina, Universit£ di Genova, Dipartimento di Fisica, Via Dodecaneso 33, 1-16146 Genova, Italia

C. De Conclni, Scuola Normale Superiore, Piazza dei Cavalieri 7, 1-56100 Pisa, Italia A. De Pantz, Corticella Leoni 4, 1-37121 Verona, Italia H.J. De Vega, Universit6 de Paris VII, Labo. Phys. Theor. H. Energies, 4 P1. Jussieu,

Tour 16, let Et., F-75252 Paris Cedex 5, France. G. Dell 'Antonio, Universit£ di Roma "La Sapienza', Dipartimento di Matematica,

P.le A. Moro 2, 1-00185 Roma, Italia. E-Malh GIANFA@IRMUNISA.BITNET J.A. Dixon, University of Texas at Austin, Theory Group, Physics Department,

Austin, TX 78712 U.S.A. E-Mail: DIXON@UTAPHY.BITNET M. Djurdjevi~:, University of Belgrade, Dept. of Physics, P.O. Box 550, 19001 Bel-

grade, Yugoslavia. E-Mail: YUBGSS21~EPMFF41.BITNET V.K. Dobrev, Bulgarian Academy of Sciences, Institute of Nuclear Research, 72

Boul. Lenin, 1784 Sofia, Bulgaria. J.-A. Domfnguez P~rez, Universidad de Salamanca, Depto. de Matem£ticas, Plaza

de la Merced 1-4, E-37008 Salamanca, Espafia. B. Drabant , Universit£t Karlsruhe, Institut f/ir Theoretische Physik, Kaiserstra]3e

12, D-7500 Karlsruhe, F. R. Germany. E-Mail: BE08~DKAUNI2.BITNET M. Dubois-Violette, Universit6 de Paris-Sud, Labo. Phys. Theor. H. Energies, Bat.

211, F-91405 Orsay, France. E-Malh MADORE@FRCPNll.BITNET I.L. Egusquiza, University of Cambridge, D.A.M.T.P., Silver St., Cambridge CB3

9EW, United Kingdom. E-Mail: ILE10~PHX.CAM.AC.UK C. Emmrich, Albert-Ludwig-Universit£t, Fakult£t fiir Physik, H~rmann-Herder-

Stra~e 3, D-7800 Freiburg, F. R. Germany. E-Mail: CEMM~DFRRUF1.BITNET O. Eyal, Universit£t Karlsruhe, Institut fiir Theoretische Physik, Kaiserstrafle 12,

D-7500 Karlsruhe, F. R. Germany. E-Marl: BE01@DKAUNI2.BITNET G. Falqui, S.I.S.S.A., Strada Costiera 11, Miramare-Grignano, 1-34014 Trieste. E-

Mail: FALQUI@ITSSISSA.BITNET A. Fernandez Martlnez, Universidad de Salamanca, Depto. de Matemhticas, Plaza

de la Merced 1-4, E-37008 Salamanca, Espafia. F. Ferrari, Universit~.t Wien, Institut ffir Theoretische Physik, Boltzmanngasse 5,

A-1090 Wien, Austria. E-Maih FERRARI@AWIRAP.BITNET M. Ferraris, Universit£ di Cagliari, Dipartimento di Matematica, Via Ospedale 72,

1-09100 Cagliari, Italia. E-Mail: FERRARIS@VAXCA2.INFN.IT

XY

T. Filk, Universit£t Freiburg, Fakult£t f/jr Physik, Hermann-Herder-Strat~e 3, D-7800 Freiburg, F. R. Germany. E-Mail: FILK@DFRRUF1.BITNET

A. Floer, Fakult£t fiir Mathematik, Ruhr-Universit£t, Universit£tstraBe 150 NA 6/27, 4630 Bochum, F. R. Germany.

A. Folacci, Universit~ de Corse, Fac. des Sciences, 15 Quartier des 4 Fontalnes, F- 20250 Corte, France.

M. Forger, Universit/~t Freiburg, Fakult£t f/jr Physik, Hermann-Herder-Strafle 3, D-7800 Freiburg, F. R. Germany. E-Mall: FORGER@DFRRUF1.BITNET

M. Francaviglia, Universit£ di Torino, Istituto di Fisica Matematica, Via Carlo Alberto 10, 1-10123 Torino, Italia. E-Malh FRANCAVIGLIA~ASTRTO.INFN.IT

J. FrShlich, ETH-HSnggerberg, Theoretical Physics, CH-8093 Z/jrich, Schweiz. F_,- Mail: FROEHLICH@CZHETH5A.BITNET

A. Ganchev, Bulgarian Academy of Science, Institute for Nuclear Research, Boul. Lenin 72, 1784 Sofia, Bulgaria.

A. Gavrilik, Institute for Theoretical Physics, 252130 Kiev 130, U.S.S.R. F. Ghaboussi, Universit~it Konstanz, Fakult£t f/jr Physik, Postfach 5560, D-7750

Konstanz, F. R. Germany R.. Giachetti, Universit£ di Firenze, Istituto di Matematica Applicata, Via S. Marta

3, 1-50139 Firenze, Italia C. Gdmez, Universidad de Salamanca, Depto. de Fisica Te6rica, P1. de los Caidos,

E-37008 Salamanca, Espafia M. Gonz~ilez Le6n, Universidad de Salamanca, Depto. de Matemhticas, Plaza de la

Merced 1-4, E-37008 Salamanca, Espafia S. Gotzes, Universit£t Dortmund, FB Physik, Postfach 500500, D-4600 Dortmund

50, F. R. Germany. E-Malh UPH418@DDOHRZll.BITNET J.M. Guilarte, Universidad de Salamanca, Depto. de Fisica TeSrica, P1. de los Cal-

dos, E-37008 Salamanca, Espafia. E-Mall: ESANZ~USAL.ES P. Hajac, Mathematical Institute, 24-29 St Giles', Oxford OX1 3LB, United King-

dom. E-Malh PMH~VAX.OX.AC.UK K.M. Happle, Universit£t Freiburg, Fakult/it f/jr Physik, Hermann-Herder-Strage 3,

D-7800 Freiburg, F. R. Germany. E-Malh KLHA@DFRRUF1.BITNET M. Hayashi, Universit/it Karlsruhe, Institut f/Jr Theoretische Physik, Kalserstrage

12, D-7500 Karlsruhe 1, F. R. Germany. E-Malh BE08@DKAUNI2.BITNET F. Hegenbarth, II Universit£ di Roma, Dipartimento di Matematica, Via Fontanile

di Carcaricola, 1-00133 Roma, Italia M. Hellmund, Karl-Marx-Universit£t, Sektion Physik, Karl-Marx-Platz, DDR-7010

Leipzig, German D. R. D. Hern~indez Ruip~rez, Universidad de Salamanca, Depto. de Matem£ticas, P1.

de la Merced 1-4, E-37008 Salamanca, Espafia S. Huggett , Polytechnic SoUth West, Dept. of Mathematics and Statistics, Plymouth

PL4 8AA, United Kingdom. E-Mail: P07406@PA.PSW.AC.UK C. Hull, Queen Mary College, Dept. of Physics, Mile End Road, London E1 4NS,

United Kingdom. E-Mail: CMH@VI.PH.QMC.AC.UK C. Itzykson, Service de Physique Th~orique, CEN Saclay, F-91191 Gif-sur-Yvette

Cedex, France.

×Yl

B. Jensen, Universit~ de Corse, Fac. des Sciences, B.P. 52, F-20250 Corte, France. J. Kalkman, Rijksuniversiteit Utrecht, Mathematisch Instituut, Postbus 80 010, 3508

TA Utrecht, Nederland. E-Marl: KALKMAN~MATtt.RUU.NL D. Kastler , Universit6 Aix-Marseille II, Centre de Physique Th~orique, Case 907 -

Luminy, 13288 Marseille Cedex 9, France. A. Kellendonk, Physikalische Institut der Universitiit Bonn, 12 Nussallee, D-5300

Bonn 3, F. R. Germany A. Kempf, Universit£t Karlsruhe, Institut ffir Theoretische Physik, Kaiserstrai~e 12,

D-7500 Karlsruhe 1, F. R. Germany. E-Maih BE04@DKAUNI2.BITNET T. Kornhass, Universit£t Freiburg, Fakult£t ffir Physik, Hermann-Herder-Strat~e 3,

D-7800 Freiburg, F. R. Germany. E-Mail: KLHA~DFRRUF1.BITNET D. Krupka, Masaryk University, Dept. of Mathematics, Janackovo Nam. 2A, 66295

Brno, Czechoslovakia O. Krupkova, Masaryk University, Dept. of Mathematics, Janackovo Nam. 2A, 66295

Brno, Czechoslovakia H.P. Kilnzle, University of Alberta, Dept. of Mathematics, Edmonton, Alberta,

Canada T6G 2G1. E-Mail: HKUNZLE@UALTAVM.BITNET J. Laartz, Albert-Ludwig-Universit£t, Fakult~t f~r Physik, Hermann-Herder-Strag, e

3, D-7800 Freiburg, F. R. Germany N.P. Landsman, Cambridge University, D.A.M.T.P., Silver Street, Cambridge CB3

9EW, United Kingdom. E-Maih NPLll~PHX.CAM.AC.UK R.J . Lawrence, Harvard University, Mathematics Dept., 1 Oxford St., Cambridge,

MA 02138, U.S.A. E-Mail: LAWRENCE@ttUMA1 H.C. Lee, Chalk River Nuclear Lab, Theoretical Physics, Chalk River, Ontario,

Canada K0J 1J0. E-Malh 02153@AECLCR.BITNET D. LeYtes, University of Stockholm, Dept. of Mathematics, Box 6701, S-11385 Stock-

holm, Sweden. E-Maih MLEITES@NADA.KTH.SE A. Lichnerowicz, Coll~ge de France, 3 Rue d'Ulm, F-75005 Paris, France M. Lo Schiavo, Universit£ di Roma "La Sapienza", Dip. Metodi Modelli Matematici,

Via A. Scarpa 10, 1-00161 Roma, Italia It. Loll, Physikalische Institut der Universit~t Bonn, Nussallee 12, D-5300 Bonn 3,

F. R. Germany. E-Mail: LOLL@DBNPIBS.BITNET A. L6pez Almorox, Universidad de Salamanca, Depto. de Matem£ticas, Plaza de

la Merced 1-4, E-37008 Salamanca, Espafia L. Lusanna, Sezione I.N.F.N. di Firenze, Largo E. Fermi 2, 1-50125 Arcetri FI, Italia.

E-Mail: LUSANNA@VAXFI.INFN.IT G. Mackey, Harvard University, Mathematics Dept., 1 Oxford St., Cambridge, MA

02138, U.S.A. G. Magnano, S.I.S.S.A., Strada Costiera 11, Miramare-Grignano, 1-34014 Trieste.

E-Mail: MAGNANO~ITSSISSA.BITNET S. Majid, Cambridge University, D.A.M.T.P., Silver St., Cambridge CD3 9EW,

United Kingdom. E-Maih SItM10@PttX.CAM.AC.UK M. Mamone Capria, Universit£ di Perugia, Dipartimento di Matematica, Via Van-

vitelli 1, 1-06100 Perugia, Italia

×VII

K.B. Marathe , The City University of New York, Brooklyn College, Brooklyn, NY 11210, U.S.A. E-Mail: KBM~BKLYN.BITNET

P.A. Marchet t i , Universit£ di Padova, Dipartimento di Fisica, Via Marzolo 8, 1- 35131 Padova, Italia. E-Mail: MARCHETTIQVAXFPD.INFN.IT

M. Marinkovi~:, University of Belgrade, Dept. of Physics, P.O. Box 550, 19001 Bel- grade, Yugoslavia. E-Mail: YUBGSS21@EPMFF41.BITNET

G. Marmo, Universit£ di Napoli, Dipartimento di Fisica, Mostra d'Oltremare Pad. 19, 1-80125 Napoli, Italia. E-Malh GIMARMO@CLUSNA.INFN.IT

L. Mart ina, Universit£ di Lecce, Dipartimento di Fisica, 1-73100 Lecce, Italia. E- Mail: MARTINA@VAXLE.INFN.IT

V.B. Matveev, Institute for Aviation Instrumentation, Dept. of Mathematics, Gertzena 67, 190000 Leningrad, U.S.S.R.

M.E. Mayer, University of California, Dept. of Physics, Irvine, CA 92717 U.S.A. E-Mail: MMAYER~UCIVMSA.BITNET

G. Mendella, Universit£ di Napoli, Dipartimento di Fisica, Mostra D'Oltremare Pad. 19, 1-80125 Napoli, Italia

I. Mladenov, Bulgarian Acad. Sciences, Central Biophysics Laboratory, U1. Acad. Boncnev 21, 1113 Sofia, Bulgaria

M. Modugno, Universit£ di Firenze, Istituto di Matematica Applicata, Via S. Marta 3, 1-50139 Firenze. E-Maih MODUGNO@IFIIDG.BITNET

V. Molotkov, Bulgarian Academy of Sciences, Institute of Nuclear Research, 72 Boul. Lenin, 1784 Sofia, Bulgaria

K. R. Miiller, Universit£t Karlsruhe, Institut fiir Logik, Postfach 6980, D-7500 Karl- sruhe, F. R. Germany. E-Mail: KLAUS~IRA.UKA.DE

R. Myers, McGill University, Physics Department., Rutheford Bldg., 3600 University Str., Montreal, Canada H3A 2T8. E-Mail: RCM@PHYSICS.MCGILL.CA

Y. Ne 'eman, Tel-Aviv University, Sackler Institute, Ramat-Aviv 69, 978 Tel Aviv, Israel. E-Malh B21A~TAUNOS.BITNET

H. Ocampo, Universit£t Karlsruhe, Institut fiir Theoretische Physik, Kaiserstrat3e 12, D-7500 Karlsruhe, F. R. Germany. E-Malh BE08@DKAUNI2.BITNET

O. Oglevetsky, Universit£t Karlsruhe, Institut fur Theoretische Physik, Kaiserstrat3e 12, D-7500 Karlsruhe 1, Postfach 6980, F. R. Germany. E-Mail: BE12~DKAUNI2. BITNET

P. Orland, The City University of New York, Baruch College, Physics. Dept., 17 Lexington Ave., New York, NY 10010 U.S.A.

Z. Oziewiez, University of Wroctaw, Institute of Theoretical Physics, Cybulskiego 36, Wroctaw, 50205 Poland.

S. Pasquero, Dipartimento di Matematica, Via L.B. Alberti 4, 1-16132 Genova, Italia E-Maih DOTMAT@ICNUCEVM.BITNET

O. Pekonen, University of Jyvaskyla, Dept. of Mathematics, Seminaarinkatu 15, 40100 Jyvaskyla, Finland

I. Penkov, University of California Dept. of Mathematics Berkeley, CA 94720, U.S.A. E-Mail: PENKOV~MATH.BERKELEY.EDU

A. Perelomov, Institute Theor. Exp. Physics, Moscow, U.S.S.R.

XVlll

E. Poletaeva, Pennsylvania State University, Dept. of Mathematics, 218 Mc Allister Bldg., University Park, PA 16802, U.S.A.

C. Procesi, Universit~ di P, oma "La Sapienza", Dipartimento di Matematica, P.le A. Moro 2, 1-00185, Roma, Italia

J .M. Rabln, University of California at San Diego, Mathematics Dept., La JoUa, CA 92093 U.S.A. E-Mail: JRABIN~UCSD.EDU

O. Ragnlsco, Universith di Roma "La Sapienza", Dipartimento di Fisica, P.le A. Moro 2, 1-00185 Roma, Italia. E-Mail: RAGNISCO~ROMAI.INFN.IT

P. Ramond, University of Florida, Dept. of Physics, Gainesville, FL 32611 U.S.A. E-Maih RAMOND~UFPINE

J. Rawnsley, University of Warwick, Mathematics Institute, Coventry CV4 7AL, United Kingdom. E-Mail: JHR~MATHS.WARWICK.AC.UK

C. Relna, S.I.S.S.A., Strada Costiera 11, 1-34014 Miramare TS, Italia. E-Mail: REINA@ITSSISSA.BITNET

L. Richardson, 44 Fosseway, Clevedon, Avon BS21 5EQ, United Kingdom. S. Rodr iguez-Romo, UniversitKt Konstanz, Fakult~t ffir Physik (LS Dehnen), Post-

fach 5560 7750, Konstanz, F. R. Germany. E-Mail: PHEBNER~DKNKURZ1. BITNET

A. Rogers, University of London, King's College, Strand, London WC2R 2LS, United Kingdom. E-Mail: UDAH039~OAK.CC.KCL.AC.UK

M. Rothstein, University of Georgia, Dept. of Mathematics, Athens, GA 30602, USA. E-Mail: ROTHSTEI~JOE.MATH.UGA.EDU

M. Ruiz-Altaba, Universit~ de Geneve, Dept. de Physique Theorique, CH-1211 Gen~ve 4, Suisse. E-Maih RUIZALTB@CGEUGE52.BITNET

G. Sardanashvily, University of Moscow, Dept. of Theoretical Physics, Physics Fac- ulty, Moscow 117234, U.S.S.R.

M. Savellev, Institute for High Energy Physics, Theory Division, Protvino, Moscow Region 142284, U.S.S,R.

B. Sazdovic, Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia. E-Mail: EIPH004~YUBGSS21.BITNET

U. Sch~iper, Universit~t Freiburg, Fakult~t ffir Physik, Hermann-Herder-Strafle 3, D-7800 Freiburg, F. R. Germany

A. Sch | r rmacher , Universit~t Karlsruhe, Institut ffir Theoretische Physik, Kaiser- strafle 12, Karlsruhe D-7500, F. R. Germany. E-Maih BE04~DKAUNI2.BITNET

M. Schlieker, Institut flit Theoretische Physik, Kaiserstrafle 12, D-7500 Karlsruhe, F. R. Germany. E-Mail: BE02@DKAUNI2.BITNET

T. Schmitt , Karl-Weierstrass-Institut fiir Mathematik, Mohrenstrafle 39, Berlin 1086, Germany.

M. Schottenloher, Mathematische Institut der LMU, Theresienstrafle 39, D-8000 Mihlchen 2, F. R. Germany

J. Schwenk, Institut ffir Theoretische Physik, Kaiserstrafle 12, D-7500 Karlsruhe, F. R. Germany. E-Mail: BE02~DKAUNI2.BITNET

S. Shnider, Ben Gurion University, Dept. of Mathematics and Computer Science, P.O. Box 653, 84105 Beer Sheva, Israel. E-Mail: SHNIDER@BIMACS.BIU.AC.IL

Xl×

N. Sorace, UniversitY. di Firenze, Istituto di Matematica Applicata, Via S. Maria 3, 1-50139 Firenze, Italia

M. Spera, Universit£ di Padova, Dip. Metodi e Modelli Matematici, Via Belzoni 7, 1-35131 Padova, Italia E-Malh SPERA@PDMSA1.UNIPD.IT

A. Stahlhofen, Universit£t Stuttgart, Institut fiir Theoretische u. Ang. Physik, Pf- effenwaldring 57, D-7000 Stuttgart 80, F. R. Germany

O.T. Stoytchev, Institute for Nuclear Research, Elementary Particle Division, Boul. Lenin 72, 1184 Sofia, Bulgaria

R. Tammelo, Estonian Academy of Sciences, Institute of Physics, Riia 142, 202400 Tartu, Estonia, U.S.S.R.

P. Teofilatto, II Universit£ di Roma, Dipartimento di Fisica, Via Orazio Ralmondo, 1-00173 Roma, Italia.

P. Truini, Universit£ di Genova, Dipartimento di Fisica, Via Dodecaneso 33, 1-16146 Genova, Italia. E-Maih TRUINI@GENOVA.INFN.IT

V. Tsanov, University of Sofia, Dept. of Mathematics and Informatics, A. Ivanov Str. 5, Sofia 1126, Bulgaria ]

S.T. Tsou, Mathematical Institute, 24-29 St Giles', Oxford OX1 3LB, United King- dom. E-Maih TSOU~VAX.OX.AC.UK

A. Vaintrob, U1. Akademika Tchelomeya 1 Kv. 328, Moscow 117630, U.S.S.R. S. Vidussi, Universit£ di Trieste, Dipartimento di Matematica, P.le Europa 1, 1-34017

Trieste, Italia G. Vilasi, Dipartimento di Fisica Teorica, Universit• di Salerno, Via S. Allende,

84081 Baronissi SA, Italy P. Vitale, I.N.F.N Sezione di Napoli, Mostra D'Oltremare, 80125 Napoli, Italia I. V. Volovich, Steklov Mathematical Institute, U1. Vavilova 42, Moscow, GSP-1

117966 U.S.S.R. S. Watamura , Institut ffir Theoretische Physik, Kaiserstrafle 12, 7500 Karlsruhe, F.

R. Germany. E-Malh BE05@DKAUNI2.BITNET S.L. Woronowicz, University of Warsaw, Dept. of Mathematical Methods in Physics,

Hoza 74, 00682 Warsaw, Poland C.S. Xiong, S.I.S.S.A.; Strada Costiera 11, 1-34014 Miramare TS, Italia. E-Mail:

XIONG@ITSSISSA.BITNET C.Z. Zha, Xinjiang University, Physics Dept., Urumqi, Xinjiang, China D. Zwanziger, New York University, Department of Physics, Washington Sq., New

York, NY 10003, U.S.A. P. Zweydlnger, Universit~.t Karlsruhe, Institut fiir Theoretische Physik, Kaiserstra~e

12, D-7500 Karlsruhe 1, Postfach 6980, F. R. Germany