Superstrings

337 Pages·1988·14.067 MB·English

by  Peter Goddard (auth.), Peter G. O. Freund, K. T. Mahanthappa (eds.)

Superstrings
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,
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Volume 175-Superstrings
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Volume 176-Nonlinear Evolution and Chaotic Phenomena
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Series B: Physics
Superstrings
Edited by
Peter G. O. Freund
University of Chicago
Chicago, Illinois
and
K. T. Mahanthappa
University of Colorado
Boulder, Colorado
Plenum Press
New York and London
Published in cooperation with NATO Scientific Affairs Division
Proceedings of a NATO Advanced Research Workshop on
Superstrings,
held July 27-August 1, 1987,
in Boulder, Colorado
Library of Congress Cataloging in Publication Data
NATO Advanced Research Workshop on Superstrings (1987: Boulder, Colo.)
Supersprings.
(NATO ASI series. Series B, Physics; v. 175)
"Proceedings of a NATO Advanced Research Workshop on Superstrings, held
July 27-August 1,1987, in Boulder, Colorado."-T.p. verso.
"Published in cooperation with NATO Scientific Affairs Division."
Includes bibliographical references and index.
1. Superstring theories-Congresses. I. Freund, Peter G. O. (Peter George
Oliver), date. II. Mahanthappa, K. T. III. North Atlantic Treaty Organization. Scien
tific Affairs Division. IV. Title. V. Series.
QC794.6.S85N37 1987 539.7121 88-6032
ISBN-13: 978-1-4612-8293-8 e-ISBN-13: 978-1-4613-1015-0
001: 10.1007/978-1-4613-1015-0
© 1988 Plenum Press, New York
Softcover reprint of the hardcover 1s t edition 1988
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying,
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PREFACE
The Advancea Kesearch Workshop on Superstrings was held on the
campus of the University of Colorado at Boulder from July 27th through
August 1, 1987.
Since the work of Green and Schwartz in the summer of 1984, string
theories have elicited tremendous amount of interest from both
theoretical physicists and mathematicians. The objective of the Workshop
was to bring together practitioners in the field to discuss the progress
and problems, and possible directions of future research.
There were ten talks of one hour each and twenty three talks of
one-half hour each. The talks covered new formulations and technical
developments. There were intense discussions both during and at the end
of the lectures; 'further discussions continued during lunch and dinner.
These proceedings contain all talks given at the Workshop except those by
Victor Kac, Darwin Chang and Doron Gepner.
The Workshop was sponsored by the North Atlantic Treaty
Organization, which provided generous financial support enabling many
young physicists from the U.S. and abroad to participate in the Workshop.
Additional co-sponsors were the U.S. Department of Energy and the
University of Colorado. The former offered further financial assistance
and the latter furnished clerical and technical services and its campus
facilities for the purpose of the organization and running of the
Workshop.
The International Organizing Committee consisted of John Ellis,
Francois Englert, Peter G.O. Freund (co-director), K. T. Mahanthappa
(co-director) and Abdus Salam.
Jon Bjorkman, Vasi1ios (Bill) Koures and Richard Loft assisted in
the day-to-day functioning of the Workshop. Pramee1a Mahanthappa and
Kathy Oliver organized the social programs. Without the assistance of
Linda Frueh it would not have been possible to organize and run the
Workshop so smoothly.
We thank the lecturers for their stimulating contributions and for
their cooperation in the preparation of this volume. We are grateful to
Linda Frueh for her editorial assitance.
reter G.O. Freund
K. T. Mahanthappa
Chicago and Boulder
November, 1987
v
CONTENTS
CHAPTER I: MATHEMATICAL TECHNIQUES
Modular Invariance and Infine-Dimensional Algebras ••• 3
P. Goddard
Off-Shell BRST Invariant Superjacobians • • • • • • • • • • • • • 17
S. J. Gates, Jr.
CHAPTER II: STRING FIELD THEORY
A Geometry Independent Field Theory of String • • • • 29
K. Kikkawa
The Gluing Theorem in the Operator Formulation of
String Field Theory •••••••••••• 39
C. R. Preitschopf
Conformal Field Theory Methods and Construction of
Witten's Superstring Field Theory •••• 51
A. Jevicki
Gauge Invariance, BRST Invariance, and Ward
Identities in String Field Theory •• 61
S. P. de Alwis
Closed Strings and Witten's String Field Theory. • • • • • • • • 69
J. A. Shaprio
Geometric String Field Theory: Deriving String
Theory from First Principles • • • • • • 79
M. Kaku
On Higher Order Interactions for Superstrings • • • • • • • • • • 89
J. G. Taylor
CHAPTER III: STRING GEOMETRY AND ARITHMETIC
Uhler Geometry of String Theory •••••••••••••• 101
S. G. Rajeev
Numbers and Strings 109
P.G.O. Freund
vii
CHAPTER IV: COMPACTIFICATION AND PHENOMENOLOGY
Type II Superstrings in Four Dimensions • • • • • • • • • • • 117
J. H. Schwarz
Four-Dimensional Strings. • • • • • • • • • • • • • • • • • • • • 135
L. Dolan
String Thermodynamics and Cosmology • • • • • • • • • • • • • • • 147
F. Englert
Orbifold Compactification toward Standard Model • • • • • • • • • 159
J. E. Kim
CHAPTER V: POLYAKOV STRINGS
Liouville Strings 169
J.-L. Gervais
A Polyakov Path Integral with Ghosts ••••• 187
C. R. Ordonez, M. A. Rubin and R. Zucchini
String Path Integrals for Open or Unoriented
World Sheets • • • • • • • • • 199
S. Carlip, M. Clements and S. Della Pietra
CHAPTER VI: LOOPS
Two Loop Partition Function in (Compactified)
Heterotic String Vacua • • • • • • • • • • • • • • • • • • 207
J. J. Atick and A. Sen
String Perturbation Theory and Effective
Lagrangians. • • . • . • . . . . • . • • • • • • . . . • • 217
I. Klebanov
CHAPTER VII: ANOMALIES AND CHlRAL BOSONIZATION
Chiral Bosonization. 229
R. I. Nepomechie
Superstring Anomalies. 243
P. H. Frampton
Toplogy, Superspace, and Heterotic Gravitational
Anomalies • • • • . . • . • • • • 251
B. A. Ovrut
Anomaly-Free Theories Remain Anomaly-Free after
Compactification with No Isometries • • • • • • • • • • • 259
J. E. Bjorkman and Y. Tosa
Many New Anomaly-Free Theories in Higher Dimensions. • • • • • • 269
Y. Tosa and S. Okubo
viii
CHAPTER VIII: SUPERMEMBRANES AND OTHER ALTERNATIVES
Supermembranes • 287
K. S. Stelle
On the Massless Excitations of The 11d
Supermembrane 301
L. Mezincescu
Highlights of Parastring Theories. • • • • • • • • • • • • • • •• 313
F. Mansouri
Line Functional Theory for Strings • • • • • • • • • • • • • • •• 325
Y. Hosotani
Internal Symmetry for Type II Strings in
Oscillator Formalism. • ••••• 335
L. Clavelli
Participants 345
Index •••• 349
IX
Chapter I
Mathematical Techniques
MODULAR INVARIANCE AND
INFINITE-DIMENSIONAL ALGEBRAS
Peter Goddard
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Silver Street
Cambridge, CB3 9EW, U.K.
INTRODUCTION
This lecture is concerned with certain infinite-dimensional algebras, namely affine
Kac-Moody algebras and the Virasoro algebra, their representations and the modular
transformation properties of the characters of these representations. These algebras
and their properties are of central importance in the study of conformally invariant
two-dimensional systems, and hence in analysing the critical behaviour of suitable
two-dimensional statistical systems, and in string theories where they are of relevance
because of the two-dimensional nature of the string world-sheet. For a recent review
of these algebras in relation to their applications in physics see [1]. Our objective is to
show how the properties of representations of the Virasoro algebra can be understood
in terms of the corresponding properties of affine Kac-Moody representations [2-4].
v,
The Virasoro algebra [5], has a basis consisting of Ln, nEZ, and a central
element, c, satisfying the commutation relations,
[Lm' Ln] = (m - n)Lm+n + lc2m(m2 -1)c5m,-n, (la)
[Ln, c] = o. (lb)
v
In physical applications the representations of which are relevant are the unitary
ones, i.e. those which satisfy the hermiticity condition
L1 = L_n. (2)
In an irreducible unitary representation the central element c takes a fixed real value.
A further typical physical requirement is that the spectrum of Lo be bounded
below (because it corresponds to an energy or dilatation operator). Then, since
LoLn = Ln(Lo - n), (3)
the Ln, n > 0, act as lowering operators for the eigenvalues of Lo. Thus, if Ih) is an
eigenvector of Lo corresponding to the lowest eigenvalue h,
Lolh} = hlh), (4a)
Lnlh) = 0, n > o. (4b)
3