Table Of ContentQuantum Fields and
Quantum Space Time
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Recent Volumes In this Series:
Volume 362 - Low-Dimensional Applications of Quantum Field Theory
edited by Laurent Baulleu, Vladimir Kazakov, Marco Picco,
and Paul Windey
Volume 363 - Masses of Fundamental Particles: Cargese 1996
edited by Maurice Levy, Jean lIiopoulos, Raymond Gastmans,
and Jean-Marc Gerard
Volume 364 - Quantum Fields and Quantum Space Time
edited by Gerard 't Hooft, Arthur Jaffe, Gerhard Mack, Pronob K. Mitter,
and Raymond Stora
Series B: Physics
Quantum Fields and
Quantum Space Time
Edited by
Gerard 't Hooft
University of Utrecht
Utrecht. The Netherlands
Arthur Jaffe
Harvard University
Cambridge. Massachusetts
Gerhard Mack
University of Hamburg
Hamburg. Germany
Pronob K. Mitter
CNRS - University of Montpellier 2
Montpellier. France
and
Raymond Stora
Laboratory of Particle Physics. Annecy-Ie-Vieux
Annecy-Ie-Vieux. France
Springer Science+Business Media, LLC
Proceedings of a NATO Advanced Study Institute on
Quantum Fields and Quantum Space Time,
held July 22-August 3, 1996,
in Cargese, France
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Library of Congress Cataloging-in-Publication Data
On file
ISBN 978-1-4899-1803-1 ISBN 978-1-4899-1801-7 (eBook)
DOI 10.1007/978-1-4899-1801-7
© Springer Science+Business Media New York 1997
Originally published by Plenum Press, New York in 1997
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PREFACE
The 1996 NATO Advanced Study Institute (ASI) followed the international tradi
tion of the schools held in Cargese in 1976, 1979, 1983, 1987 and 1991.
Impressive progress in quantum field theory had been made since the last school in
1991. Much of it is connected with the interplay of quantum theory and the structure
of space time, including canonical gravity, black holes, string theory, application of
noncommutative differential geometry, and quantum symmetries. In addition there
had recently been important advances in quantum field theory which exploited the
electromagnetic duality in certain supersymmetric gauge theories. The school reviewed
these developments.
Lectures were included to explain how the "monopole equations" of Seiberg and
Witten can be exploited. They were presented by E. Rabinovici, and supplemented by
= =
an extra 2 hours of lectures by A. Bilal. Both the N 1 and N 2 supersymmetric
Yang Mills theory and resulting equivalences between field theories with different gauge
group were discussed in detail.
There are several roads to quantum space time and a unification of quantum theory
and gravity.
There is increasing evidence that canonical gravity might be a consistent theory
after all when treated in. a nonperturbative fashion. H. Nicolai presented a series of
introductory lectures. He dealt in detail with an integrable model which is obtained by
dimensional reduction in the presence of a symmetry.
The lectures by G. Mack started from the hypothesis that a truly fundamental
theory should start from an absolute minimum of a priori structure which merely reflects
the belief that the human mind thinks about relations between things. It was shown
how the fundamental equations of motion of physics, including a discretized version of
canonical general relativity in Ashtekar variables can be accommodated.
In black holes there is a particularly strong interplay between geometry and quan
tum effects. Black holes were a subject of the lectures by G. 't Hooft, and by L.
Susskind.
String theory was not a main topic at the school, but it was discussed in the context
of black hole physics in the lectures of Susskind. Recent evidence for the necessity of
going beyond strings as basic constituents was also addressed.
The question of how we can extract geometry from the quantum mechanics of
extended objects was also discussed from a more general point of view in the lectures
of J. Frohlich. He started from a Pauli-electron on a spin manifold and ended with a
model of quantum space time which has a quantum symmetry as an essential feature,
v
and is based on noncommutative differential geometry.
A comprehensive introduction to noncommutative differential geometry was pre
sented by A. Connes, and quantum field theories with quantum symmetry were exten
sively discussed in lectures by Alekseev, Faddeev, Pressley and Zumino. In this way the
connection was made with the topics which had been of main interest at the last school.
Alekseev showed that quantum symmetries could also appear as local gauge symme
tries. Faddeev constructed lattice models of current algebra by formulating them as
integrable models. Pressley gave an introduction to quantum groups and quantum
affine algebras with application to affine Toda theories.
The lessons from quantum field theory are important in the study of other complex
systems. This was demonstrated in K. Gawedzki's lectures on turbulence, which applied
renormalization group techniques borrowed from field theory. They are also important
for mathematics. A. Jaffe discussed elliptic genus and Stora lectured on equivariant
cohomology, both in a quantum field theory context.
This volume presents the lecture notes. We regret that the contributions of A.
Jaffe, E. Rabinovici and L. Susskind were not available in time for inclusion.
We wish to express our gratitude to NATO whose generous financial contribution
made it possible to organize the school. We thank Professor Elisabeth Dubois-Violette,
the director of the Institut d'Etudes Scientifique de Cargese, as well as the Universite
de Nice and Universite de Corte for making available to us the facilities of the Institute.
Grateful thanks are due to Annie Touchant for much help with the material aspects of
the organization, to the staff of the institute, and to Max GrieBl, York Xylander and
Volker Schomerus, who helped in Hamburg.
for the editors,
G. Mack,
director of the ASI
vi
CONTENTS
Lectures
Non-commutative gauge fields from quantum groups
A. Yu. Alekseev and V. Schomerus .......... . 1
Duality in N = 2 SUSY SU(2) Yang-Mills theory: a pedagogical
introduction
A. Bilal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Noncommutative differential geometry and the structure of space time
A. Connes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
Quantum integrable models on 1 + 1 discrete space time
L. Faddeev and A. Volkov . .................. . 73
Supersymmetry and non-commutative geometry
J. Frohlich, O. Grandjean, and A. Recknagel 93
Turbulence under a magnifying glass
K. Gaw~dzki . . . . . . . . . . . . . . . 123
Quantization of space and time in 3 and in 4 space-time dimensions
G. 't Hooft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Pushing Einstein's principles to the extreme
G. Mack . ................. , ................ 165
Integrable classical and quantum gravity
H. Nicolai, D. Korotkin, and H. Samtleben ........ 203
Quantum affine algebras and integrable quantum systems
V. Chari and A. Pressley . ........................... 245
Exercises in equivariant cohomology
R. Stora ............... . . ......... 265
Some complex quantum manifolds and their geometry
C. S. Chu, P. M. Ho, and B. Zumino .......... . 281
vii
Seminars
T-duality and the moment map
C. KlimiHk and P. Severa. . . . ...... 323
Symmetries of dimensionally reduced string effective action
J. Maharana ................................... 331
Non local observables and confinement in BF formulation of Yang-Mills
theory
F. Fucito, M. Martellini, and M. Zeni . . . . . . . . . . . . . . . . . . . . . 339
Disorder operators, quantum doubles, and Haag duality in 1 + 1
dimensions
M. Miiger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
The cohomology and homology of quantum field theory
J. E. Roberts. 357
Index. . . . . . . 369
viii
NON-COMMUTATIVE GAUGE FIELDS FROM QUANTUM GROUPS
Anton Yu. Alekseev 1, Volker Schomerus 2
1 Institute of Theoretical Physics,
Uppsala University,
Box 803 S-75108,
Uppsala, Sweden.
e-mail: alekseev@teorfys.uu.se
2 II. Institut fur Theoretische Physik,
Universitiit Hamburg,
Luruper Chaussee 149,
22761 Hamburg, Germany
e-mail: vschomer@x4u2.desy.de
ABSTRACT
We give a non-technical introduction into the theory of non-commutative lattice
gauge fields with quantum gauge group. The general construction is illustrated at the
example of the Hamiltonian Chern-Simons theory. We also review the counterpart of
the Noether's theorem for quantum groups.
INTRODUCTION
The aim of this contribution is to give a non-technical introduction into the theory
of R-matrix algebras and its applications to non-commutative lattice gauge theories
developed during last 5 years.
We start in Section 1 with the idea of introducing a quantum group as a gauge
group in some gauge theory. We treat quantum groups in the physicist-oriented for
malism due to Faddeev, Reshetikhin and Takhtajan (FRT) [1]. This idea can be im
plemented in the most simple way for lattice gauge theories [2]. The first example
of such a construction was given in [3]. We show that gauge invariance with respect
to a quantum gauge group leads to non-commutative gauge fields. This was realized
in [4]. The resulting theory is closely related to non-ultralocal integrable systems [5].
Lectures presented by A.Yu. Alekseev
1
