Integrable Systems, Quantum Groups, and Quantum Field Theories

507 Pages·1993·12.265 MB·English

by  L. D. Faddeev (auth.), L. A. Ibort, M. A. Rodríguez (eds.)

Integrable Systems, Quantum Groups,
and Quantum Field Theories
NATO ASI Series
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Series C: Mathematical and Physical Sciences -Vol. 409
Integrable Systems,
Quantum Groups, and
Quantum Field Theories
edited by
L. A. Ibort
and
M. A. Rodrfguez
Departamento de Fisica Te6rica,
Universidad Complutense de Madrid,
Madrid, Spain
Springer Science+Business Media, B.V.
Proceedings of the NAro Advanced Study Institute and
XXIII GIFT International Seminar on
Recent Problems in Mathematical Physics
Salam anca, Spain
June 15-27,1992
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4874-3 ISBN 978-94-011-1980-1 (eBook)
00110.1007/978-94-011-1980-1
Printed on acid-free paper
AII Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproaucea or
utilized in any form or by any means, electronic or mechanical, including photo
copying, recording or by any information storage and retrieval system, without written
permission from the copyright owner.
PREFACE
This volume contains the proceedings of the NATO Advanced Institute
and the XXIII edition of the GIFT International Seminar on Theoretical
Physics, Recent Problems in Mathematical Physics, that was held at Sala
manca, Spain, from 15 to 27 June 1992. The Advanced Institute was or
ganized by the editors (L.A. Ibort and M.A. Rodriguez) and Professors A.
Bohm and P. Winternitz.
The Institute was conceived as a preparatory School for the XIX Interna
tional Colloquium on Group Theoretical Methods in Physics, which was held
in Salamanca the week after the Advanced Institute. That imposed that
the School covered a wide range of problems in Mathematical Physics, from
modern aspects of Integrability on Quantum Field Theories to Geometrical
Phases in Quantum Mechanics. Several courses were devoted to Conformal
Field Theory, integrability, Quantum Groups and Topological Field Theory.
Gravitation in low dimensions and Quantum Gravity were discussed in an
other series of lectures and finally, some other topics like, symmetries in
partial differential equations, geometric phases, etc. were addressed along
the School.
The level of the School was intended for advanced graduate students and
recent Ph. D's and the invited lecturers are leading experts in their fields.
The organizers would like to express their gratitude to the lecturers for their
beautiful and stimulating talks as well as for the careful preparation of their
contributions to these proceedings.
Apart from the organizers of the Advanced Institute, the editors want
to thank the invaluable help and support obtained from the rest of the Or
ganizing Committee of the XIX ICGTMP; Professors Aldaya, C.S.I.C. and
Universidad de Granada; J. A. de Azcarraga, Universidad de Valencia and
IFIC (C.S.I.C.); L. J. Boya, Universidad de Zaragoza; J. F. Cariiiena, Univer
sidad de Zaragoza; M. Lorente, Universidad de Oviedo; J. Mateos Guilarte,
Universidad de Salamanca; M. A. del Olmo and M. Santander, Universidad
de Valladolid.
The directors of the Advance Institute wish to thank the NATO Science
Committee for their support of the School and to the following Institutions
for their parijal financial contribution:
v
vi
Direcci6n General Interministerial de Ciencia y Tecnologia (DGI
CYT); Real SociedadEspaiiola de Fisica, G.I.F.T; Universidad Complutense
de Madrid, Universidad de Salamanca and Apple Espana.
The help of Mrs. Ascensi6n Iglesias during the School has been invaluable.
The Dean and the staff of the Facultad de Ciencias, where most of the sessions
of the School were held, kindly put many facilities at our disposal. We should
like to thank Prof. F. Fernandez and to the Professors of the Department of
Theoretical Physics, specially J. M. Cerver6 and J. Martin.
L.A. Ibort
M.A Rodriguez
CONTENTS
Preface. . . v
Participants IX
From Integrable Models to Conformal Field Theories
via Quantum Groups
L.D. Faddeev . . . . . . . . . . . . . . . . . . . . . . . . . 1
A brief History of hidden Quantum Symmetries
in Conformal Field Theories
C. Gomez and G. Sierra ....... . 25
q-Deformation of Semisimple and Non-semisimple Lie Algebras
H. Ruegg . ..................... . 45
Quantum Algebras and Quantum Groups in
q-special Function Theory
R. Floreanini and L Vinet. . . . . . . 83
Lectures in Topological Quantum Field Theory
D.S. Freed .............. . 95
Canonical Quantun Gravity and the problem of time
C.J. Isham ............... . 157
Higher Symmetries in Lower-dimensional Models
R. Jackiw . ............... . 289
Three Lectures on Supersymmetry and Extended Objects
P.K. Townsend . . . . . . . . . . . . . . . . . . 317
The Geometric Phase in Quantum Physics
A. Bohm ........... . 347
Quantum Mechanics on Phase Space
M. Gadella .......... . 417
Lie groups and solutions of nonlinear partial differential equations
P. Winternitz . . . . . . . . . . . . . . . . . . . . . . 429
Index . ... 497

LIST OF PARTICIPANTS
l. Ahluwalia, K.S. (Cambridge Univ .. , UK) 30. Fernandez Nunez, J. (Univ. de Valladolid,
Spain)
2. Alber, M. S. (Univ. of Notre Dame, USA)
3l. Figueroa GonZiilez, H. (Univ. de Zaragoza,
3. Aldaya, V. (Univ. de Granada, Spain)
Spain) .
4. Alvarez Gimenez, M. (Univ. de Santiago,
32. Freed, D. (Univ. of Texas at Austin, USA)
Spain)
33. Gadella, M. (Univ. de Valladolid, Spain)
5. Anderson, A. (McGill Univ., Canada)
34. Garcia Fuertes, W. (Univ. de Oviedo, Spain)
6. Azcarraga, A. de. (Univ. de Valencia, Spain)
35. Gingras, F (Univ. of Montreal, Canada)
7. Bamich, G. (Univ. Libre de Bruxelles,
Belgique) 36. G6mez Nicola, A. (Univ. Complutense de
Madrid, Spain)
8. Batlle Amau, C. (Univ. Politecnica de
Catalunya. Spain) 37. G6mez, C. (CSIC, Spain)
9. Benoit, L. (Univ. de Montreal, Canada) 38. GonziUez Ruiz, A. (Univ. Complutense de
Madrid, Spain)
10. Bergqvist, G. (Univ. Southampton, UK)
39. Giingor,F. (Istanbul Tech. Univ., Turkey)
11. Berkovich, A. (CSIC, Spain)
40. Hall, R. A. (Durham Univ., UK)
12. Bohm, A. (Univ. of Texas at Austin, USA)
13. Boya, LJ. (Univ. de Zaragoza, Spain) 4l. Harrington, R. J. (Univ. of Texas at Austin,
USA)
14. Cangemi, D. (MIT, USA)
42. Hernandez Heredero, R. (Univ. Complutense
15. Carinena, J.F. (Univ. de Zaragoza, Spain) de Madrid, Spain)
16. Cuemo Rejado, R. (CSIC, Spain) 43. Ibort, LA (Univ. Complutense de Madrid,
Spain)
17. Chruscinski, D. (Nicholas Copernicus Univ.,
Poland) 44. Illieva-Litova, N. (Bulgarian Acad. of
Sci., Bulgaria)
18. Debergh, N. (Univ. de Liege, Belgique)
45. Isham, C. (Imperial College, UK)
19. Deckmyn, A. (K. U. Leuven, Belgique)
46. Izquierdo Rodriguez, J.M. (Univ. de
20. Demichev, A. P. (Moscow State Univ.,
Valencia. Spain)
Russia)
47. Jackiw, R. (Columbia Univ., USA)
21. El Gmdechi, A. (Univ. de Montreal, Canada)
48. Jain, S. (Derbyshire College, UK)
22. Ellinas, D. (Univ. of Helsinki, Finland)
49. Jimenez Lorenw, F. (Univ. Politecnica de
23. Empanin, R. (Univ. del Pais Vasco, Spain)
Madrid, Spain)
24. Ezquerm Larrode, F. (Univ. de Zaragoza,
50. Jiracek, P. (Fac.of Nucl. Sci. and Phys. Eng.,
Spain)
Czechoslovakia)
25. Faddeev, L.D. (Steklov Mathematical
51. Kabat, D. (MIT, USA)
Institute, Russia)
52. Kausch, H. (Imperial College, UK)
26. Faucher, M. (Univ.ofMontreal, Canada)
53. Kimum, T. (Univ. of Texas at Austin, USA)
27. Fecko, M. (Comenius Univ. Bmtislava,
Czechoslovakia) 54. Kopecky, Z. (Masaryk Univ,Czechoslovakia)
28. Fernandez Cabrera, D. (Univ. de Valladolid, 55. Kucem, J. (Masaryk Univ., Czechoslovakia)
Spain)
56. Laartz, J. (Harvard Univ., USA)
29. Fernandez Jambrina, L. (Univ. Complutense
57. Larson, J. (College of William and Mary,
de Madrid, Spain)
USA)
x