Conformal Invariance and String Theory

553 Pages·1989·9.382 MB·English

by  Petre Dita (Eds.)

C o n f o r m ai I n v a r i a n ce a nd
S t r i ng T h e o ry
Edited by
Ρ θ ί τθ D i tQ a nd V l a d i m ir G e o r g e s cu
Theoretical Physics Department
Central Institute of Physics
Bucharest Romania
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Library of Congress Cataloging-in-Publication Data
Conformai invariance and string theory/edited by Petre Dita and
Vladimir Georgescu.
p. cm. — (Perspectives in physics)
Lectures delivered at the Summer School on Conformai Invariance
and String Theory, held at Poiana Brasov, Romania, Sept. 1-12, 1987,
sponsored by the Central Institute of Physics-Bucharest.
Bibliography: p.
ISBN 0-12-218100-X (alk. paper)
1. Statistical mechanics—Congresses. 2. Conformai invariants-
Congresses. 3. String models—Congresses. I. Dita, P. (Petre),
Date- . II. Georgescu, V. (Vladimir), Date- . III. Summer
School on Conformai Invariance and String Theory (1987 : Poiana
Brasov, Romania) IV Institutul Central de Fizicà (Romania)
V Series
QC174.7.C66 1989
530.1'3-dc20 89-6941
CIP
Printed in the United States of America
89 90 91 92 9 8 7 6 5 4 3 2 1
C o n t r i b u t o rs
Numbers in parentheses refer to the pages on which the authors' contributions begin.
I. YA. AREF'EVA (165), Steklov Mathematical Institute, Ul. Vavilova 42, Moscow 117
333, USSR
PETER BOUWKNEGT (115), Institute for Theoretical Physics, University of Amsterdam,
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
P. CARTIER (443), Ecole Polytechnique, Centre de Mathématiques F-91128, Palaiseau,
France
P. DI FRANCESCO (63), Service de Physique Théorique, Institut de Recherche Fon-
damentale, CEA-CEN Saclay, 91191 Gif-sur Yvette Cédex, France
R. FLUME (89), Physikalisches Institut der Universität Bonn, Nussallee 12, 5300
Bonn 1, FR. Germany
JEAN-LOUP GERVAIS (205), Physique Théorique, Ecole Normale Supérieure, 24 rue
Lhomond, 75231 Paris Cédex 05, France
MALTE HENKEL (127), Fachbereich Physik, Universität Essen, Postfach 103764,
D-4300 Essen 1, FR. Germany
M. KAROWSKI (147), Theoretical Physics, ETH-Honggerberg, CH-8093 Zurich,
Switzerland
RICH*ARD KERNER (243), LPTPE, Université Pierre et Marie Curie, 4place Jussieu,
75005 Paris, France
W. LERCHE (367), CERN, 1211 Geneva 23, Switzerland
Yu. I. MANIN (285, 293), Steklov Mathematical Institute, Ul. Vavilova 42, Moscow
117333, USSR
HANS-PETER NILLES (305), CERN, 1211 Geneva 23, Switzerland
SYLVIE PAYCHA (335), Mathematics Department, Ruhr-University, Bochum, FR.
Germany
V. RITTENBERG (37), Physikalisches Institut der Universität Bonn, Nussallee 12,5300
Bonn 1, FR. Germany
H. SALEUR (63), Service de Physique Théorique, Institut de Recherche Fondamentale,
CEA-CEN Saclay, 91191 Gif-sur-Yvette Cédex, France
A.N. SCHELLEKENS (367), CERN, 1211 Geneva 23, Switzerland
LT. TODOROV (3), Institute for Nuclear Research and Nuclear Energy, Bulgarian
Academy of Sciences, P.O. Box 373, 1090 Sofia, Bulgaria
vii
LIST OF PARTICIPANTS
ANDR1CA, D. University of Cluj-Napoca
ANGHEL, V. INPR - Pitesti
ANTOHI, 0. I NPR - Ρ i test i
APOSTOL, M. Central Institute of Physics, Bucharest
AREF1 EVA, I.Ya. Steklov Institute of Mathematies,Mos cow
AVRAM CRISTEA, A. Central Institute of Physics, Bucharest
BADEA, M. Institute of Medicine, Bucharest
BANICA, C. INCREST - Bucharest
BATTLE ARNAU, C. University of Barcelona
BEC I U , M. Institute of Constructions, Bucharest
B0CI0RT, F. Central Institute of Physics, Bucharest
BOGATU, Ν. Central Institute of Physics, Bucharest
BOLDEA, Venera Central Institute of Physics, Bucharest
BOUTET de MONVEL, Anne Universite Paris VI
BOUWKN EGT, P. University of Amsterdam
BRANZANESCU, V. Po 1 ytechnica1 Institute Bucharest
BUNDARU, M. Central Institute of Physics, Bucharest
BUZATU, Daniela I C S I Τ - Bucharest
BUZATU, F. Central Institute of Physics, Bucharest
CAO, C.T.Y. Trinity College, Universi ty of Cambridge
CAPR INI, Irinel Central Institute of Phys ics, Bucha rest
CARTIER, P. Ecole Polytechnique, Pala î seau
CATA, G. Central Institute of Phys i es , Bucha rest
CEAUSESCU, M. Central Institute of Phys ics, Bucha rest
CEAUSESCU, V. Central Institute of Phys i es, Bucha rest
CI0NGA, Aurelia Central Institute of Phys ics, Bucharest
CI0NGA, V. Central Institute of Phys ics, Bucha rest
C I UBOTARU , Lum i η i ta Central Institute of Phys i es, Bucha rest
COSTACHE, G. Central Institute of Phys i es, Bucha rest
IX
Participante
χ
COSTIN, 0. Central Institute of Physics, Bucharest
COSTIN, Rodica INCREST - Bucharest
CRISTEA, 0. University of Cluj-Napoca
CULETU, H. ICECHIM - Bucharest
DEMUTH, M. Institute of Mathematics, Berlin
DI FRANCESCO, F CEΝ - Saclay
DI TA , P. Central Institute of Physics,Bucha rest
D I TA, Sanda Central Institute of Physics, Bucharest
DJAMO, V. I NPR - Pi testi
DOMITIAN, V. High School Ploîestî
DUDAS, E. University of Bucharest
DULEA, M. Central Institute of Physics, Bucharest
FAZAKAS, A. Central Institute of Physics, Bucharest
FLUME, R. University of Bonn
FÖRSTER, D. Freie Universität Berlin
FUCHS, J.
Inst, for Theoretical Physics, Heidelberg
GADIDOV, R.
INCREST - Bucharest
GEORGESCU, V.
Central Institute of Physics, Bucharest
GERVAIS, J.-L.
Ecole Normale Supérieure, Paris
GHEORGHE, A.C.
Central Institute of Physics, Bucharest
GHEORGHIU, H.
INPR - Pi testi
GRIGORE, R.D.
Central Institute of Physics, Bucharest
GUITTER, E.
CEN - Saclay
GUSS I , G.
INCREST - Bucharest
HARAB0R, Ana University of Craiova
HARABOR, V. University of Craiova
HENKEL, M. University of Bonn
HRISTEA, M.R . ICECHIM - Bucharest
IANCU, E. University of Bucharest
IMBROANE, A. Po1 ytechnica1 Institute, Cluj-Napoca
INCULET, V. University of Bucharest
IONISE! , G. University of I a s i
IORDANESCU , R. Central Institute of Physics, Bucharest
ISAR, A. Central Institute of Physics, Bucharest
Participants xi
KALITZIN, S. INRNE - Sofia
KAROWSKI, S. Freie Universität Berlin
KERNER, P. Un i ver s i te Paris VI
KISS-TOTH, T. Osîjek University
LUDU, A. University of Bucharest
LU Κ I Ε RS KY, J. University of Wroclaw
LUPU, C. ICPE - Bucharest
MAGNON, Anne Université Clermont II
MANDA, H. Central Institute of Physics, Bucharest
MAN I Ν, Yu. I. Steklov Institute of Mathematics, Moscow
MANOLIU, Mîhaela Central Institute of Physics, Bucharest
MARCHETTI, P. University of Padova
MARTIN, M. INCREST - Bucharest
MAZILU, D. Central Institute of Physics, Bucharest
MANDESCU, R. IPMP - Buzau
MANTOIU, M. High School Brasov
MINNAERT, P. Universite de Bordeaux
MINTI , H. ICPE - Bucharest
MOCANU, Raluca Central Institute of Physics, Bucharest
MUNOZ, C. University of Mad r i d
MUNTEANU, V. University of Bucharest
NAIDIN, Andreea IPGG - Bucharest
NICULA, M. University of Bucharest
NILLES, H.P. CERN - Geneva
NYKANEN, E. NORDITA - Copenhagen
PANTEA, D. Central Institute of Physics, Bucharest
PASCU, Ε. INCREST - Bucharest
PATEOPOL, L. I PA - Bucharest
PAYCHA, SMvie Ruhr Universität Bochum
PISO, M. ICPE - Bucharest
POENARU, D. Central Institute of Physics, Bucharest
POPESCU, S. University of Bucharest
POPP, O.T. Central Institute of Physics, Bucharest
PRIPOAE, G. I CS I Τ - Bucharest
xii Participante
PURICE, R. Central Institute of Physics, Bucharest
P U Τ IN A R , Μ. INCREST - Bucharest
RADULESCU, F. INCREST - Bucharest
RICHERT, J. CRN - Strasbourg
RITTENBERG, V. University of Bonn
ROSU, H. Central Institute of Physics, Bucharest
SALEUR, H. CEN - Saclay
SALIU, L. University of Craiova
SARU, D. ICPE - Bucharest
SASU, G. Central Institute of Physics, Bucha rest
SANDULESCU, A. Central Institute of Physics, Bucha rest
SANDULESCU, N. Central Institute of Physics, Bucharest
SCHELLEKENS, A.N. CERN - Geneva
SOFONEA, M. INCREST - Bucharest
SPINEANU, F. Central Institute of Physics, Bucharest
STAN-SION, C. Central Institute of Physics, Bucharest
STEINBRECHER, G. University of Craiova
STRATAN, G. Central Institute of Physics, Bucharest
TARINA, M. University of Cluj-Napoca
TATARU, L. University of Craiova
T0D0R0V, I.T. INRNE - Sofia
T0P0R-P0P, V. Central Institute of Physics, Bucharest
T0P0R-P0P, Rodîca High School Bucharest
VISINESCU, Anca Central Institute of Physics, Bucharest
VLAD, Mada1 i na Central Institute of Physics, Bucharest
VONSOVICI, A. University of Bucharest
WE IGT, G. I HEP Ber1 i n-Zeu then
WOLLENBERG, M. Institute of Mathematics, Berlin
P r e f a ce
This book contains the lectures delivered at the Summer School on Conformai
Invariance and String Theory, held at Poiana Brasov, Romania, 1-12 Septem-
ber 1987. The aim of the School was to review some of the most important
problems and results in these two strongly interrelated areas of theoretical
physics, which are currently extremely active and in which much progress is
being made. Relations between the two domains were emphasized for the ben-
efit of students in each field, who were thereby enabled to interact and get
accustomed to the concepts and methods of the other field. The first part of the
volume presents the conferences dealing with the implications of conformai
invariance in the study of two-dimensional systems. The second part contains
several lectures on recent advances in string theory and related topics.
The School was sponsored by the Central Institute of Physics-Bucharest,
and acknowledges a grant from UNESCO.
We owe special thanks to all the invited lecturers for their contributions to
the success of the School; Professor Vladimir Rittenberg's assistance was
decisive in the early stage of the School's organization.
Many of our colleagues from the Central Institute of Physics helped us in
various ways. We are especially grateful to Drs. Valentin Ceausescu and
Marin Ivascu for their inestimable help. Dr. Gabriel Costache's experience
was a great benefit and we thank him for his permanent support.
Ms. Geta Uglai, the technical secretary of the School, handled most of the
tedious work related to the School with high competence and efficiency, for
which we owe our warm gratitude.
THE EDITORS
Bucharest
xiii
FINITE TEMPERATURE 2-DIMENSIONAL QFT MODELS OF CONFORMAL
CURRENT ALGEBRA*^
I.T. Todorov
1. Introduction
Synopsis on Gibbs states and KMS boundary condition
2. Two Dimensional Conformai QFT and CCA on the Circle
2A. Conserved chiral currents. Compact picture. CCA commu-
tation relations
2B. Lowest weight (LW) unitary representations (UEs) of CCA.
The Sugawara formula
3. KMS States on the CCA
SA. KMS correlation functions in terms of charge distribution
3B. Local QFT model associated with a free complex Weyl
3
field
4. Classification of 2-Dimensional Conformai Models of
Chiral U(l) Current Algebra
4A. Construction of primary charged fields from localized
morphisms of the CCA
4B. Field algebras with single and double valued represen-
tations on the circle
5. Concluding Remarks
References
*)_,
'The present notes are a preview of a work in progress by D.Buchholz,
G.Mack and the author and are based on the lectures presented at
the International School on Group Theoretical Methods, Varna,Bu 1 garia
June 1987 and Poiana Brasov Summer School, September 1987.
Conformai Invariance and String Theory 3 Copyright © 1989 by Academic Press, Inc.
All rights of reproduction in any form reserved
ISBN 0-12-218100-X
1. Introduction
Conformai quantum field theory (QFT) models in 1+1 dimensions give
rise to infinite dimensional Lie algebras of observables whose lowest
weight (positive energy) representations have been classified. They yield
explicit Gibbs states and provide a new method of computing Virasoro cha-
racters and partition functions. A discrete series of models associated
with the chiral U(l) conformai current algebra (CCA) l/U(l) + ^U(l), is
singled out by a "univalence" requirement (related to the Bose-Fermi al-
ternative) for the basic charged fields. The first terms of this series
correspond to previously studied special points (characterized by a higher
symmetry) of the critical line of the Ashkin-Teller model (see, e.g.,
[Ya 1, 3] and references to earlier work cited there).
The rigorous approach to finite temperature equilibrium QFT (see
e.g. [Ha 1, Ka 5] as well as the book [Br 1] and further references cited
there) and the work on algebraic methods in the study of 2-dimensional
conformai models (see [De 1 Fe 2, Fu 1, Lu 1, Ca 5, Be l Wi 1, Kn 1,
5 9
Go 2,3, To 1,2,3 , Ge 3, Fu 2] among others) have been pursued apart from
each other. One objective of these lectures is to show how the Kubo-Martin-
Schwinger (KMS) boundary condition can be used to evaluate finite tempera-
ture correlation functions of fields satisfying Lie (super)algebraic (anti)
commutation relations. Combined with small distance operator product
expansions (OPEs) they allow to compute the internal energy and hence the
partition function of the system.
These computations appear as a part of a wider program: the construc-
tion and classification of 2-dimensional local QFT models with a given al-
gebra of observables, the CCA, each theory having a discrete additive set
of superselection sectors.
Thus one obtains a (comparatively) simple example in which the "alge-
braic QFT" program of Haag et al. ([Ha 2, Do 1, ..., Do 2]) is realized.
In particular, we present what appears to us a neat construction of the
charged fields as localized morphisms of the CCA - a construction which
has appeared under many disguises (as "bosonization", "vertex operator",
"Coulomb gas" representation ...) and at various levels of rigour (for a
small sample of references, see [Sk 1, St 1,2, Ma 3, Fr 1,2, Ca 3,4, Di 2]).
4