Differential Geometrical Methods in Theoretical Physics

474 Pages·1988·32 MB·English

by  Michael Atiyah (auth.)

Differential Geometrical Methods in
Theoretical Physics
NATO ASI Series
Advanced Science Institutes Series
A Series presenting the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The Series is published by an international board of publishers in conjunction with
the NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical Kluwer Academic Publishers
and Physical Sciences Dordrecht, Boston and London
D Behavioural and Social Sciences
E Applied Sciences
F Computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York, London,
H Cell Biology Paris and Tokyo
Series C: Mathematical and Physical Sciences -Vol. 250
Differential Geometrical Methods
in Theoretical Physics
edited by
K. Bleuler
and
M. Werner
Institute for Theoretical Nuclear Physics,
University of Bonn, Bonn, F.R.G.
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Research Workshop
and the 16th International Conference on
Differential Geometrical Methods in Theoretical Physics
Como, Italy
24-29 August 1987
Library of Congress Cataloging in Publication Data
International Conference on Differential Geometric Methods in
Theoretical Physics (16th: 1987 Como, Italy)
Proceedings of the 16th Internatlonal Conference on Differential
Geometrical Methods in Theoretical Physics (NATO Advanced Research
Workshop) Centra di Cultura Scientifica "Alessandro Volta" Villa
Olmo, Coma, Italy, 24-29 August 1987 I edited by K. Bleuler & M.
Werner.
p. cm. -- (NATO Advanced Science Institutes series. Series C,
Mathematical and physical sciences; 250)
Includes index.
1. Geometry, Differential--Congresses. 2. Gauge fields (Physics)
-Congresses. 3. String .models--Congresses. 4. Mathematical
phYSlcs--Congresses. 1. 'Blelller. Konrad. 1912- 11. Werner. M.
IIL Title. IV. Series: NATO ASI serles. Series C, Mathematical
and physical SClences ; no. 250.
OC20.7.D52I58 1987
530.1·5636--dc19 88-8262
CIP
ISBN 978-90-481-8459-0 ISBN 978-94-015-7809-7 (eBook)
DOI 10.1007/978-94-015-7809-7
All Rights Reserved
© 1988 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1988.
Softcover reprint of the hardcover 1st edition 1988
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying, recording
or by any information storage and retrieval system, without written permission from the
copyright owner.
TABLE OF CONTENTS
Introduction and Survey xi
The Impact of Physics on Geometry
M.Atiyah
I. MATHEMATICAL CONTRIBUTIONS
Weyl's Program and Modern Physics
G.W.Mackey 11
Elliptic Genera of Level N for Complex Manifolds
F.Hirzebruch 37
The PrinciplE of Triality and a Distinguished Unitary
Representation of SO(4,4)
B.Kostant 65
On E10
V.G.Kac, R.V.Moody, M.Wakimoto 109
II. SPECIAL ASPECTS OF STRING THEORY
Operator Methods in String Theory
L.Alvarez-Gaume 129
On Strings and J.Douglas Variational Principle
T.Regge 145
Groupoids and Lie Bigebras in Gauge and String TheorisE
M.E.Mayer 149
III. CONFORMAL INVARIANCE AND INTEGRABLE SYSTEMS
The Definition of Conformal Field Theory
G.B.Segal 165
Statistics and Monodromy in Two- and Three-Dimensional Quantum
Field Theory
J.Frohlich 173
vi
Quantum Groups (YBZF Algebras) and Integrable Theories:
An Overview
H.J.de Vega 187
Integrable Systems and Conformal Invariance
M.Karowski 211
Algebraic Aspects of Non-Perturbative Quantum Field Theories
B.Schroer 219
IV. TWO CLASSICAL DOMAINS: - SYMPLECTIC STRUCTURES IN PHYSICS
- GENERAL RELATIVITY
The Pairing Method and Bosonic Anomalies
S.Sternberg 261
A Multisymplectic Approach to the KdV Equation
M.J.Gotay 295
Singular Points in Level Sets of the Momentum Map
and Quantum Theory
H.Romer 307
Killing Spinors and Universality of the Hijazi Inequality
A.Lichenrowicz 317
Towards a Renormalizable Theory of Quantum Gravity
Y.Ne'eman, D.~ija~ki 333
Gauge Theory for Diffeomorphism Groups
P.W.Michor 345
Quantum Field Theory in Curved Space time
B.S.Kay
37~
V. SUPERSYMMETRIC STRUCTURES
Super theories
S.Ferrara
39~
The Integrability of N 16 Supergravity
H.Nicolai 40 :
Conformal Invariant Supersymmetric Theories in Four Dimensions:
A Practical Application of BRS Cohomology1,2
C.Lucchesi, O.Piguet, K.Sibold 40·
Supermanifolds, Supermanifold Cohomology, and Super Vector Bundles
U.Bruzzo 41·
vii
Representations of Lie Superalgebras:An Introduction
M.Scheunert 441
CONCLUDING REMARKS
M.Atiyah 461
Observations of Two of our Brightest Stars
L.Halpern 471
Organizers:
R.Cirelli, Milano, Italy
G.Casati, Milano, Italy
P.Cotta - Ramusino, Milano, Italy
K.Osterwalder, Zurich, Switzerland
K.Bleuler, Bonn, FRG
Advising Committee:
A.Lichnerowicz, Paris
A.Salam, Trieste
T.Regge, Torino
F.Hirzebruch, Bonn
INTRODUCTION AND SURVEY
The recent most impressive developments in theoretical
physics, i.e. gauge and string theories with their far
reaching relations to basic concepts of present day
mathematics brought a renewed and intense exchange between
mathematicians and physicists. In a way the past few years
might be compared to the great period of the mid-twenties
with its really decisive mathematical contributions to the
foundations of both quantum theory as well as general
relativity .. Among others, this far-reaching exchange led
H.Weyl in 1929 to the formulation of his (second) gauge
principle (concerning the phase of the quantum mechanical
wave function) which led, in turn (through the most
natural generalization by C.N.Yang and R.Mills in 1954) to
present day gauge theory which brought, first of all, a
real breakthrough with respect to a theoretical inter
pretation of an enormous realm of data in modern particle
physics. It led, in fact, to the well-known unification of
electromagnetism with weak interactions (Weinberg-Salam
theory) and, at the same time, to a most natural unifica
tion of particle and nuclear physics through QCD which
deals with the basic quark-gluon structure of heavy
matter, e.g. hadrons.
A very first and decisive step for this renewed exchange
came, as emphasized by M.Atiyah (cp. his introductory
article in this volume) through the insight that very much
the same mathematical or, better, geometrical concepts
(i.e. the most natural geometrical structure of gauge
theory) represent a decisive ingredient in recent topolo
gical research (e.g. Donaldson's results for 4-manifolds).
As an immediate consequence, however, of this new,
apparently most satisfactory situation one of the greatest
puzzles or, better, deepest quests of modern theory came
up again (this time in an even more stringent form): It is
the non-consistency between the two all-embracing basic
theories, i.e. general relativity and quantum theory.
Although gauge theory in its natural (i.e. pure) form
(e.g. QeD) leads according to t'Hooft to a so far
reasonable consistency (i.e. renormalizability), the
theory of gravitation, (although at the origin of the very
formulation of the gauge principle) still contradicts (in
view of its more refined structure) the quantisation
procedure (i.e. non-renormalizability in spite of all
efforts; in a minor extent the same problem holds also for
the combined electroweak theory). This most intriguing
situation calls, of course, for a deeper understanding and
naturally suggests - maybe in analogy to earlier situa-