Frontiers In Quantum Field Theory 9789814530668, 9814530662

Table of contents :
Title
Copyright
Preface
Contents
Banquet Speech Honoring Prof. Keiji Kikkawa on his 60th Birthday
Introductory Remarks
Bunji Sakita
Antal Jevicki
Spenta Wadia
Lars Brink
Kazuo Fujikawa
Satoshi Matsuda
Part I: String Duality
M Theory Extensions of T Duality
1 Introduction
2 BPS --
Saturated p-Branes
3 M/IIB Duality
4 M/SO(32) Duality
5 Conclusion
U-Duality and Intersecting D-Branes
Duality and 4D String Dynamics
1 Introduction
2 N=2 Gauge Theory and String Compactifications
3 N=2 String-String Duality
4 N=1 Duality and Gaugino Condensation
Acknowledgements. 2 Dynamical Superalgebra and p-Branes3 Duality Groups
4 U-Duality and Non-Perturbative States
5 U-Duality and 11D
5.1 Perturbative and non-perturbative states
5.2 Dualities and non-perturbative spectrum
5.3 An example
6 Final Remarks
7 References
String Solitons and Singularities of K3
Acknowledgements
References
Collective Coordinate Quantization of Dirichlet Branes
1 Introduction
2 D-Brane and Collective Coordinates
3 Combinatorics of Perturbative Dirichlet String Theory
4 Semi-Classical Wave Function of D-Branes
5 D-Brane Eequation of Motion and Renormalization Group Flow. 6 Quantum Aspects of Macroscopically Charged D-BraneAcknowledgments
GKZ Hypergeometric Systems and Applications to Mirror Symmetry
1 Introduction
2 Mirror Symmetry and Quantum Cohomology Ring
3 Gauss-Manin System and Flat Coordinates
4 GKZ Hypergeometric System and the Flat Coordinate
4.1 Intersection ring
4.2 GKZ hypergeometric system
5 Summary
References
How Unstable are Fundamental Quantum Supermembernes?
1 Quantum Supermembranes
2 Zero Point Energy
3 Fermionic Variables
4 Wave Function
5 Discussion
Acknowledgments
References. Part II: Two Dimensional Strings and General RelativityR2 2D Quantum Gravity and Dually Weighted Graphs
1 Introduction
2 A Solvable Model of R2 2D Gravity
3 Sketch of Solution
3.1 The Itzykson-DiFrancesco formula for the DWG-model
3.2 The saddle point equation for the most probable representation in the planar limit
3.3 Calculation of characters in the large N limit
3.4 Solution of the lattice model of 2D R2 gravity
4 Physical Results and Conclusions
New Loop Equations in Ising Model Coupled to 2D Gravity and String Field Theory
1 Introduction.

Citation preview

FRONTIERS IN QUANTUM FIELD THEORY

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FRONTIERS IN QUANTUM FIELD THEORY 14-17 December 1995

Toyonaka, Osaka, Japan

Editors

H. Itoyama Osaka University, Japan

M.Kaku City College of New York, USA

H.Kunitomo Osaka University, Japan

M.Ninomiya Yukawa Institute, Japan

H. Shirokura Osaka University, Japan

1lh World Scientific •

Singapore •New Jersey• London •Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, FilTer Road, Singapore 912805

USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 51 Shelton Street, Covent Garden, U>ndon WC2H 9HE

British Library CataloguinK·ln-PubUcatlon Data A catalogue record for this book is available from the British Library.

FRONTIERS IN QUANTUM FIELD THEORY Copyright@ 1996 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-2817-1

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

v PREFACE This book is dedicated to the honor of Prof. Keiji Kikkawa, celebrating over three decades of his contributions to the world of theoretical high energy physics. His path-breaking contribution to a wide variety of critical areas in physics have greatly enriched our discipline and given us inspiration to follow in his footsteps. His seminal papers have taken root and given rise to entire disciplines that still generate intense theoretical interest. This volume contains the proceedings of an international physics conference convened in Osaka University on Dec. 14-17 , 1995, in his honor. Over 170 physicists from around the world attended this conference and more than 30 researchers presented talks on the latest developments in strings, quantum gravity, supersymmetry, and many other areas of physics. This book is divided into four parts, which in part overlap with his areas of interest. Part one concerns string duality, a subject of great importance, which he helped to initiate with his pioneering work on Tduality. Duality, for the first time, is giving us a wealth of information about the non-perturbative behavior of string theories. Part two concerns two dimensional strings, black holes, and general relativity. Prof. Kikkawa has made many significant discoveries in this field, including the first papers on string perturbation theory and string field theory. Part three concerns supersymmetry. Prof. Kikkawa published one of the first papers trying to make supersymmetry into a local gauge symmetry. Part four covers aspects of quantum field theory, to which he has contributed many important papers. It is our sincerest hope that this volume will not only help to advance our field by presenting the latest developments in research, but also serve to inspire new generations of physicists. We would also like to thank the many individuals whose tireless work has made this conference and this volume possible. These include the secretaries of our group, N. Shigenaga and E. Tanaka as well as our fellow graduate students M. Anazawa, Y. Arakane, K. Ezawa, K. Botta, M. Koike, T. Matsushita, K. Ohta, T. Oota, M. Sakaguchi, M. Sakamoto, A. Tokura, T. Yokono and M. Yoshida. Signed:

H. Itoyama, M. Kaku, H. Kunitomo, M. Ninomiya and H. Shirokura

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CONTENTS

Preface Banquet Speeches Michio Kaku Comments by A. Jevicki, S. Wadia, L. Brink, Kazuo Fujikawa, Satoshi Matsuda, H. Miyazawa, and K. /gi

v XlU

PART 1: STRING DUALITY

1

M Theory Extensions ofT Duality John H. Schwarz

3

U-Duality and Intersecting D-Branes Ashoke Sen

15

Duality and 4d String Dynamics Sham it K achru

21

Strings and Extreme Black Holes A. A. Tseytlin

35

Duality and Hidden Dimensions ltzhak Bars

52

String Solitons and Singularities of K3 Hirosi Ooguri

69

Collective Coordinate Quantization of Dirichlet Branes Soo-Jong Rey

74

GKZ Hypergeometric Systems and Applications to Mirror Symmetry S. Hosono and B. H. Lian

86

How Unstable are Fundamental Quantum Supermembranes?

96

Michio Kaku

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PART ll: TWO DIMENSIONAL STRINGS AND GENERAL RELATIVITY

111

R2 2D Quantum Gravity and Dually Weighted Graphs

113

Vladimir A. Kazakov

New Loop Equations in Ising Model Coupled to 2d Gravity and String Field Theory Ryuichi Nakayama and Toshiya Suzuki

126

Exact Solution of 1-Matrix Model Hiroshi Shirokura

136

Fusion Rules and Interactions for Macroscopic Loop Amplitudes M. Anazawa, A. Ishikawa, and H. Itoyama

141

Fractal Structure on Space-Time in Two-Dimensional Quantum Gravity Y. Watabiki

158

W00 Structures of 2D String Theory Ken-ji Hamada

168

String Theory and Matrix Models Y. Yoneya

178

Strong Cosmic Censorship in Quantum Gravity Akio Hosoya

188

String in Computer H. Kawai, N. Tsuda, and T. Yukawa

193

Renormalizability of Quantum Gravity in 2 + e Dimensions Hikaru Kawai, Yoshihisa Kitazawa, and Masao Ninomiya

205

W00 Gauge Theory B. Sakita

220

Multi-Plaquette Solutions for a Discretized Ashtekar Gravity K. Ezawa

227

Calogero-Sutherland Model and Singular Vectors of W Algebra Y. Matsuo

237

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Being Fascinated by Strings and Membranes: Is Kikkawa-Type Physics Possible at Ochanomizu A. Sugamoto

242

The Rubakov-Callan Effect and Black Holes T. Kubota

247

t/J 3 Cubed Graphs on a Random Surface N obuyuki Ishibashi

253

Wave Packet in Quantum Cosmology and Semiclassical Time Y. Ohkuwa and T. Kitazoe

257

PART III: SUPERSYMMETRY

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Exploring Gauge Theories with Supersymmetry Kenneth Intriligator

265

N = 2 Supersymmetric Gauge Theories and Soliton Equations Tohru Eguchi

277

Mass Hierarchies from Anomalies: A Peek Behind the Planck Curtain P. Ramond Integrability and Seiberg-Witten Theory H. Itoyama and A. Morozov

285

301

Integrable System and N = 2 Supersymmetric Yang-Mills Theory Toshio Nakatsu and K anehisa Takasaki

325

Picard-Fuchs Equations and Prepotentials inN= 2 Supersymmetric QCD K atsushi Ito and Sung-Kil Yang

331

Nonlinear Realization of the Superconformal Symmetry H. Kunitomo

343

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PART IV: ASPECTS OF QUANTUM FIELD THEORY

347

Gauge Independence of Gravitational Anomalies for Spin-3/2 Gauge Field Ryusuke En do and Masaki Sekine

349

Phase Operator Problem and an Index Theorem for Q-Deformed Oscillator K azuo Fujikawa

354

On Spinon Character Formulas A. Nakayashiki and Y. Yamada

367

Vortex Filament in a Three-Manifold and Its Classical Partition Function Waichi Ogura and Yukinori Yasui

372

Current Compensation and Edge Current in the Quantum Hall Effect K. Shizuya

379

Lowest Landau Level and Projection and Woo Algebra in Bilayer Quantum Hall Systems Zyun F. Ezawa

384

Decay Rate of Coherent Field Oscillation M. Yoshimura Wilson-Type Renormalization Group Study of Fermions Interacting with Gauge Field at Finite Temperature H. Takano, M. Onoda, /. lchinose, and T. Matsui

394

401

Field Theory on Von Neumann Lattice Kenzo Ishikawa and Nobuki Maeda

407

Program

411

List of Participants

413

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BANQUET SPEECH HONORlNG PROF. KEIJI KIKKAWA ON HIS 60TH BIRTHDAY MICHIO KAKU Phr16ics Dept., Citr1 College of the Citr1 Univ. of New York New York, N.Y. 10091 USA

Introductory Remarks It is a great honor tonight to be able to celebrate Keiji Kikkawa's 60th birthday

and his important contributions to theoretical physics. Many of the papers presented today are direct consequences of Kikkawa's original insight and ideas on the nature of elementary particle physics. I first heard of Kikkawa's path breaking work in 1968 when I was in the U.S. Army. I had just finished my undergraduate work at Harvard that year and, just a few days after graduation day, found myself in the Infantry. Not only was there turmoil in Southeast Asia, there was also turmoil in the world of physics when G. Veneziano and M. Suzuki discovered that the Euler Beta function had all the properties of a quantum S-matrix, except unitarity. I still remember that a flood of silly papers were hastily written trying to modify the Beta function in some trivial way to do phenomenology. I remember hearing David Jackson give a talk where he showed a circle graph showing the huge percentage of high energy papers devoted to this trendy subject. However, whenever the Beta function was altered even in the slightest way, its magical properties were destroyed. Today, we forget almost all these papers. But one of the handful of papers from that era which has stood the test of time was by Kikkawa, Sakita, and Virasoro, who used the Beta function as a Born term to introduce a unitary perturbation theory. This was the beginning of string perturbation theory. Because the U.S. Army is famous for grinding out any independent thought among its infantrymen, I brought along a copy of the KSV paper to keep my mind busy and read it religiously every night. I still remember crawling beneath a hail of live machine gun bullets, maneuvering in the thick, wet mud underneath barbed wire, and hearing the deafening blast of hand grenades exploding Iiear my ear, and then rushing back soaking wet to my barracks to read the KSV paper. The paper was startling. It proposed an entirely new geometrical language

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for physics, a language based on strange pictorial rules with rubber bands, Mobius strips, orientable and non-orientable diagrams, and dual diagrams. Luckily for me, these rules were entirely visual; since it was impossible for me to use paper or pencil while carrying a M-16 machine gun, I trained myself to manipulate them mentally, twisting and turning the most intricate N-loop non-planar, non-orientable diagrams inside-out in my head. (Today, things haven't changed much. Now I perform similar manipulations in my head while negotiating something equally hazardous: the New York subway system!) At that point, it became clear to me and others that the Veneziano amplitude was not a fluke; the KSV paper pointed to a very deep quantum theory which lay behind the amplitude. Their paper was not just another silly calculation: it was proposing an entirely new program and direction for the dual model. I got a temporary leave from the Infantry to go to Berkeley, and I quickly plunged into completing the KSV paper. Some of their rules were incomplete, and the measure (which controlled the singularity) was missing, but I found that I could complete their entire unitarization program using operators. My Ph.d. thesis, in fact, was to explicitly calculate all possible bosonic N-loop, M-point corrections to the Veneziano amplitude. Later at Berkeley, I was fortunate enough to meet a member of that team, M. Virasoro, who with B. Sakita was proposing a functional and "fishnet" approach to dual models. My goal was to incorporate fermions into the model by placing fermion propagators throughout that fishnet. Along with Virasoro and M. Yoshimura, we eventually published a paper on the subject, but we placed the fermions on the outer rim of the fishnet. A. Neveu arrived a few months after that, and excited explained to us his new model with J. Schwarz. (If we had stuck to the original idea of placing fermions throughout the interior of the fishnet, we might have gotten the Neveu-Schwarz model. And if we had continued to work on fishnets, we might have stumbled upon the 1/N expansion of matrix models. Oh well.) Later, while I was a post-doc at Princeton, I had long discussions with David Gross about the possibility of a true field theory behind the KSV program. However, it wasn't clear how to even start such a program. While visiting New York, I sought out Sakita and Kikkawa, who were recently hired by the City College of New York. This famous teaching institution was suddenly propelled into the front ranks of particle research when Bob Marshak, in his wisdom, hired Sakita and Kikkawa. I found, much to my surprise, that Kikkawa was interested in the same problem that I was working on: the field theory of strings. Both of us considered the KSV program unfinished. The KSV rules were simply postulated,

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without any action principle or underlying motivation. For example, how could you prove unitarity to all orders without a Hamiltonian? I noticed at that time one of Kikkawa's outstanding characteristics: he was always interesting in asking deep physical questions; he didn't particularly care what fad or fashion was prevailing at that time. This meant that he was always one step ahead of the pack. (This reminds me of a well-known story told about Rutherford. When a student asked him how come he never failed to be right there at each new wave of physics research, his reply was, "that's because I create the waves.") The prevailing wisdom at that time was that a field theory of strings was impossible because of the "double counting" problem, i.e. that adding s and t channel graphs, as in Feynman's theory, would overcount the Veneziano model, which was dual. Also, many people, citing the works of Heisenberg, Yukawa, and others, thought a string field theory, based on an extended object, would probably be non-local. This discouraged others from working on the problem. To Kikkawa, however, this was a challenge. We came at the problem from two different angles. Keiji's approach, as I recall, was to start with the free propagator in the light cone gauge. This meant that one didn't have to worry about ghosts, which were poorly understood. My approach was from the N-point of view, trying to solve the double counting problem and also show how to derive all possible diagrams from one action. We found that in the light cone gauge there was simple answer to this puzzle: each term in the the field theory explicitly broke duality, but that the final sum was always dual. Keiji and I could show, in fact, that the entire KSV unitarization program could be derived from a simple action. And the theory was still local, in the sense that strings joined at their endpoints, and the disturbance traveled down the string at less than the speed of light. That Christmas in 1973, I visited the home of S. Mandelstam, my former advisor, who had pioneered the light cone approach, and told him of our work. He pointed out a puzzle, however. He got a pair of scissors and paper, and carefully spliced together a bizarre concoction on the coffee table, four long flaps of paper colliding along a common line. I had to blink several times to see what he was doing. On the airplane trip back, I wrestled with this strange contraption. My rigorous training in the Army helped me to mentally unravel this thing, which I now understood was four strings interacting at an interior point. That winter, I was teaching freshmen electricity and magnetism laboratory at CCNY, drawing electrostatic potential lines. Suddenly, it occurred to me that the topology of these potential lines precisely traced out the evolution of all possible string interactions. By adjusting the charges, I could reproduce the four-string diagram, and in fact all possible string diagrams. In this way,

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we could visually demonstrate all conceivable string interactions, and find new interactions that were previously missed. It was a great loss, however, when Keiji decided to leave City College to go back to Japan. We struggled very hard to clear his immigration status. But in the end, he said to me that, among his many concerns, was the fact that his children were reaching that age where they could speak two languages, but eventually they would have to choose whether they would be Japanese or American. He wanted them to be Japanese. I understood, since my own parents faced the same problem after World War II, after leaving the camps in California. My parents decided their children would become Americans, and their kids have since (much to my regret) lost their ability to speak the Japanese language. It was also a great loss because we lost a great friend and companion. Keiji was famous for his generosity, patience, and kindness. We will never forget how he enriched our lives, both professionally but also personally. If we were in trouble, he was always there for us. But lurking behind that quiet, serene exterior was always a fierce determination to do fundamental physics. In this sense, he was the ideal collaborator, friend, and companion. However, our loss was the gain of others, because Keiji has continued to open new avenues for research, and many of these avenues are being explored in today's conference talks. He has persisted in going against the tide, always following his keen physical instincts. We are here today to honor these fine instincts. I would also like to read to you from telegrams and e-mails of people whose lives have been touched by Keiji. Also, many of Kikkawa's esteemed colleagues would like to make some personal comments.

Bunji Sakita I normally do not like to make a speech like this, because I am a pretty bad speaker and normally become awfully nervous. But this time I could not refuse, since on this occasion we cerebrate my dear friend Keiji Kikkawa's 60 birthday. Of course I am nervous, as you can see. I have known Keiji since the summer of 1965, when I went back to Japan for the first time after 9 years in U.S. In that summer I stayed at the University of Tokyo and I was introduced to Keiji. At that time he was young (well, he looks young still, but he was younger then) and a bachelor. But when I saw him three years later in Madison, Wisconsin in 1968, he was the father of a girl and he and his wife were about to have their second child. That was the year we worked together. Keiji, Miguel Virasoro and I worked together on dual models, which turned out to be the open string theory.

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Five years ago, on the occasion of my 60th year birthday celebration, Keiji talked about a story about me around that time. So I was waiting for this occasion for five years to be able to tell you about Keiji about the same time, in particular the KSV collaboration. In the fall of 1968, we were all interested in how to write down dual amplitudes, that is, string scattering amplitudes. By then the N-point tree amplitude had been constructed. I don't remember how we started the collaboration, but I remember vividly one incident. I suppose in the previous day we discussed what to do next and agreed to come to school the following day, on Saturday or Sunday, I don't remember, but anyway on a weekend. I remember I came to school a little bit before noon and I found Miguel. We chatted for a while, then Keiji joined us some time later. He had already finished lunch. We discussed things together before Miguel and I went off for lunch. By then we figured out more or less what kind of properties the one loop string amplitudes should have. But that was far from a concrete mathematical realization. After lunch Miguel and Keiji were discussing in Keiji's office that was across the hall from my office. Suddenly, I heard Miguel's loud voice, "Bunji, Bunji, Keiji has solved it!" Of course, there were a lots of twists and turns before finishing the work. But I remember that was an important first breakthrough we had. Another episode in that collaboration I remember is the following: We all felt that the expression we obtained was incomplete. I don't want to go into the technical details why we felt so. So, after we finished writing the paper, I held the paper for one or maybe two weeks, I don't remember the exact length. Then Keiji and Miguel got angry at me saying, "Well, you already have a tenured position. But we are research associates. We need papers published to get job." Well, I don't remember how I responded to that. But, certainly their strong driving force helped finish the paper and get it published. These are episodes which occurred twenty seven years ago. Since then, I have had close contacts with Keiji without interruption. We had a few years together in City College, but most of the time he stayed in Japan and I stayed in the U.S. Nevertheless he helped me great deal. On this occasion, therefore, I would like to express my sincere appreciation for his kind friendship.

Antal J evicki I am very glad to send this note. My memory is very fresh ofthe first years at City College when Keiji Kikkawa was still there. He held a special status and was universally admired among all the graduate students. The word "special" I cannot explain more but people who met Kikkawa even for little will know the meaning. At that time we were especially, with Paul (Senjanovic), eagerly

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reading Keiji's (and Kaku's) latest work which was soon to become a classic. We (the students) felt a loss when Kikkawa decided to return. But I am very certain that many others have gained. I am most honored to wish Kikkawa a happy birthday. Spenta Wadia It is a pleasure for me to write a few words on the 60th birthday of Professor Keiji Kikkawa. Firstly when I received the message from Mike Kaku I was a bit surprised that Kikkawa is already 60! I first met Kikkawa when I was a graduate student at City College. He was the most patient and gentle teacher I have ever met till this day! I remember how he used to spend literally hours teaching students in his office. Most of these students were not very well prepared but he never showed any impatience with their endless questions. People, especially the Professor next door, used to wonder "what is Keiji doing next door for so long"! Around that time in City College there was major excitement ... those were the last days of the first epoch of the string model. There was the "functional integration " stuff of Sakita, and Kikkawa and Kaku were busy working long hours on string field theory. The styles of Kikkawa and Kaku were in vivid contrast .... Kikkawa was so calm and studied compared to the flamboyance of Kaku whose excitement was boundless! Regarding physics as a student I was very impressed with the work of Kikkawa and Hosoya on soliton quantization. In fact 20 years later it is clear that they were among the first people to discuss the Hamiltonian mechanics of collective coordinates ... or in modern parlance the symplectic structure on the moduli space of classical solutions. During the second epoch of string theory I had the opportunity to know about the truly seminal contribution of the Kikkawa, Sakita, Virasoro Dual Unitarization Program to the development of string theory, which is destined to be a fundamental theory of the real world. I wish Professor Keiji Kikkawa many more happy and productive years. Lars Brink I am sending you my warmest congratulations on your birthday from a cold Sweden. Some people do not seem to get older even though their passports say so and you are certainly one of those. I did know though of your age because when you visited us for the Nobel Symposium in 1986 you had sent us your data. We thought it must be wrong so we checked it discreetly. In science you have of course had an impeccable record. Like some other Japanese scientists you have sometimes been quite ahead of us. In string theory you followed your own line and I found myself often using your ideas

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many years after you had invented them. Already from the start you realized its connections to quantum field theory while most of us were bogged down in S-matrix theory and your work on light-cone string field theory was the base for many attempts of mine long after your pioneering work. Thinking back I seem to remember that you talked about membranes ten years ago. Now we know that they are important concepts in our theories. I also remember you writing about two-dimensional QCD which is also on some people's lips these days. I am sure that many people at the meeting will tell about your personal qualities and with your modesty you will just say no no, but let me add one quality that impresses me a lot and which has been very deer to me. Like very few Japanese {I hope the participants will not be too upset with me) you have the ability to think both in a Japanese way and in a western way. Many of our great friends of Japanese origin who have spent many years in the west are still very Japanese. You can be both. You have realistic views of western values. You know how we think and you can explain Japanese thinking so that we can understand. You have been most helpful to me in this respect both professionally and in my strive to learn about Japan. I wish you many more years offruitful scientific work. I cherish the memories of our meetings in the past and I look forward to seeing you in the spring. My very best wishes for the future. Kazuo Fujikawa I first met Prof. Kikkawa in Boulder, Colorado, in 1969 just after Kikkawa, Sakita and Virasoro initiated the program of dual unitarization. After Prof. Kikkawa and I came back to Japan in the middle 1970's, we often met at the annual meetings of the Physical Society of Japan and also at various workshops at the Yukawa Institute. In the first half of the 70's, Prof. Kikkawa spent much time teaching string field theory to Japanese physicists. In the latter 70's, Prof. Kikkawa was deeply involved in the applications of string theory to the hadron spectrum and hadronic reactions. To his great disappointment, however, his efforts received very little attention in Japan and abroad. He thus completely quit his work on string theory. His disappointment was enormous. So, after string theory was revived by the path integral formulation of Polyakov, I met Prof. Kikkawa at a workshop at the Yukawa Institute in the fall of 1981. At that workshop, I talked on a BRST formulation of the string path integral. After the workshop was over,

Prof. Kikkawa and myself went to a cafe near the well-known shrine, HeianJingu, in Kyoto. At the cafe, we enjoyed tea and much discussions of physics. I then suggested to Prof. Kikkawa that he might enjoy string theory again in

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view of new formulation of string theory. However, Prof. Kikkawa told me that he would never work on string theory again; this shows how he was hurt by the cool acceptance of his applications of string theory to hadron physics, not only in Japan but also abroad. Happily enough for us all, he did not stick to his promise at the cafe, and he soon· started to work on string theory again. This time, the perspective of string theory was completely different thanks to the works of Green and Schwarz and also of Witten. In the middle of this new surge of string theory, Prof. Kikkawa invented the notion of "duality," which was the main subject at this Workshop in Osaka to celebrate his 60th birthday. Prof. Kikkawa, who lived through the ups and downs of string theory, really deserves the credit given to his discovery of T duality. Personally, I spent much time learning string field theory by reading an introductory article in a Japanese Journal, Soryuusiron Kenkyuu, written by Prof. Kikkawa. I was also much impressed by his beautiful work on membrane theory. I think that if the late Prof. Tomonaga, one of the founders of modem .field theory, were alive and active in physics, he would be very much interested in the works of Prof. Kikkawa. In fact, I often wonder if Prof. Kikkawa might play the role of a second Tomonaga in Japan, in the sense that the style of theoretical physics of Prof. Kikkawa is oriented toward beauty and great perfection. On this occasion of the 60th birthday of Prof. Kikkawa, I wish him many happy returns and many more elegant discoveries in physics.

Satoshi Matsuda Good evening and how are you, Ladies and Gentlemen! Being together with all of you, it is my great pleasure and honor to be hear and speak for Professor Keiji Kikkawa on the occasion of his 60th birthday. To make my English speech fluent, please allow me to call you Kikkawa-san in Japanese style, or sometimes Keiji in western fashion. Like some of you in the audience, my first contact with Kikkawa-san goes back to my graduate student days, more than thirty years ago. In 1964, I finished my undergraduate work at the University of Tokyo and advanced to the Graduate School there. My advisor was Professor Hironari Miyazawa, one of the founders of Dispersion Relations. Under him there were already two bright graduate students, Hirotaka Sugawara and Mahiko Suzuki, three or four years senior to me. There was another young faculty member, Keiji Kikkawa, who had just become a fresh assistant in the research group. Keiji was a couple of years senior to Hirotaka and Mahiko. With some of my classmates, I started some calculations in SU(9) under

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the guidance of Professor Miyazawa and Sugawara-san. The group SU(6) to incorporate particle spin was already on the market, first proposed by Professor Sakita and others. Miyazawa and Sugawara's concern was to unify fermionic and bosonic particles in the same multiplet: a challenge toward unifying statistics and particle symmetry. This idea was really a prototype attempt at supersymmetry, which is nowadays referred to as the key symmetry in particle physics. I was given a problem of doing calculations with SU(9), but somehow our results were not published. One of the reasons might have been that Sugawarasan soon left for the United States to be a post-doc at Cornell. That was in 1965. So, we went to our young assistant, Kikkawa-san, and started another calculation using Gell-Mann's current algebras. Kikkawa-san was an excellent teacher, encouraging younger boys and leading us to some concrete results. Here is the copy of the first page of our first paper. The title is "CurrentGenerated Algebra and Mass Levels of the Hadrons." This is one of the papers Professor Nambu mentioned this morning. Among the authors are my classmates as well, in addition to Kikkawa-san and myself; Jiro Arafune, presently the Director of the Cosmic Ray Institute at the University of Tokyo, and Yoichi Iwasaki, now Professor at the University of Tsukuba. This paper was published in the Physical Review. Considering the economy of Japan in those days, the publication charge was astronomical. Several months' salaries of a young assistant would have been eaten up by the paper, so naturally we did not pay. In fact, in those days one dollar was worth 400 yen in the market although the official rate was 360 yen. I could testify to this if you should wish. Anyway, I guess that piece of work gave us fresh grad students the confidence to proceed further in the field of particle physics. My second big memory about Kikkawa-san is related to what was happening in 1969, a few years later, when I was at Caltech as a post-doc. Having enjoyed one summer at Berkeley, I went back to Caltech to spend another postdoc year there. Then, my colleagues at Caltech, Claudio Rebbi and Charles Chiu, hastened me to join them to read a new thick preprint by Kikkawa, Sakita and Virasoro sent from the University of Wisconsin. At that time, Kikkawa-san was a post-doc at Wisconsin, having moved from Rochester. It was a surprising paper, proposing the method of calculating the dual Feynman loop amplitudes in a systematic manner. So far before then, no convincing systematic formalism of practical use with the potential of evolving into a complete theory had been available in dealing with strong interactions. There it was. We had a possible candidate for it. We studied very hard the paper together. But our main concern was

xxii

gradually shifting from calculations of higher dual loops toward studies on the symmetry properties of the N-point dual amplitudes. Then with Claudio Rebbi and Charles Chiu I found the gauge operator, Lo - Lt, controlling the SU(1,1) projective symmetry of the Koba-Nielsen formula. At the same time, Charles Thorn, a Mandelstam student at Berkeley and Gliozzi, a Fubini student at Torino, found the same gauge operator. That was the beginning of the infinite dimensional Virasoro Algebra in our market. Our work at Caltech was triggered by the paper of the Wisconsin group. My real encounter with Keiji Kikkawa in the States, an encounter not through papers, but in face to face, was a little bit accidental. I did not see him since he had left Japan for the States in 1966. Kikkawa-san went back to Tokyo in '69 and left home again next year to join Sakita's group at the City College of New York. After Caltech, I was doing research at the Institute in Princeton. John Schwarz was at Princeton University, and I had many occasions to have contact with John. I remember well that John once came into my office which I was sharing with a young post-doc, Mike Green, and explained his findings of Neveu-Schwarz model on the blackboard. I don't think Mike was in the office then. At that time, Mike was working on the Pomeron business with Tony Zee and Bob Garlitz, and even didn't show any interest in dual models. Also, Pierre Ramond once visited us at Princeton from Fermi Lab. I remember he once said that he finally got the idea of the fermionic Dirac operator in Dual Resonance Models while walking in the woods at Princeton. Pierre found the Ramond model, being inspired by the artistic objects of the hanging trees in the woods of the Institute. What a story! Anyway, on one day from Princeton, my wife and I went to New York to meet our relative who was living in the Bronx. At one small Japanese supermarket in the Bronx, we happened to meet Kikkawa-san. I got an impression that he was a little depressed. Maybe Keiji had a hard time getting adjusted to his university job in the States. Well, I might have been wrong. A few years later Keiji, together with Mike Kaku here, the co-ordinator of this evening, produced an epoch making series of papers on strings. So, I would say that his life in the States was very productive and had made a great contribution to our field. But he came back to Japan in 1975 to take up a job at Osaka University. I had already been back home, taking a job at Kyoto University in late '72. Since then we have had lots of encounters here in Japan, talking together, drinking together, discussing politics and life together, and so on. Please allow me to express a few private emotional concerns of my own. To be honest, for Sugawara-san I have a sort of feeling for him as my close

xxiii

elder brother, while I have a similar close feeling for Kikkawa-san as another senior elder brother. This feeling had started in my graduate school days, and has grown since then till now. Some of you in the audience might as well share this sort of feeling toward them. Finally, to conclude, I would say that life is short. The truth is that we have spent more than half of our given life. I sincerely hope that Kikkawa-san should live long and enjoy many happy returns in your future, keeping your young looking features in healthy condition. Thank you very much for your attention.

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Part I String Duality

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3

M THEORY EXTENSIONS OF T DUALITY JOHN H. SCHWARZ

California Institute of Technology, Pasadena, CA 91125 USA T duality expresses the equivalence of a superstring theory compactified on a manifold K to _another (possibly the sam~ superstring theory compactified on a dual manifold K. The volumes of K and K are inversely proportional. In this talk we consider two M theory generalizations ofT duality: (i) M theory compactified on a torus is equivalent to type liB superstring theory compactified on a circle and (ii) M theory compactified on a cylinder is equivalent to 50(32) superstring theory compactified on a circle. In both cases the size of the circle is proportional to the -3/4 power of the area of the dual manifold.

1

Introduction

It is a pleasure for me to speak on the occasion of the 60th birthday of my good friend, Keiji Kikkawa. I met Keiji during my first visit to the Aspen Center for Physics in the summer of 1969. This was shortly after the appearance of the famous paper of Kikkawa, Sakita, and Virasoro,1 which introduced the idea of regarding the dual resonance model n-point functions as the tree approximation of a unitary quantum theory, rather than just as interesting functions for use in phenomenology. This seemed very novel and exciting to me at the time. In Aspen, Keiji and I collaborated in a study of the Regge asymptotic behavior of the KSV loop amplitudes.2 I found the experience so pleasant that I have returned to Aspen almost every summer since then. Many years later, after we had learned that the formulas of the earlier era described strings and were better suited to unfication than to hadrons, Kikkawa and Yamasaki discovered the "T duality" arising from compactification of closed string theories on a circle~ This marked the beginning of a new line of attack on string theory that has proved to be extremely fruitful:" T dualities are now recognized as a particular class of discrete dualities that are valid order-by-order in perturbation theory (i.e., they are perturbative), though T duality is non-perturbative on the world sheet. In my talk today I would like to describe two simple extensions of T duality in the context of "M theory." (M theory is a conjectured quantum theory in eleven dimensions whose low-energy effective description is eleven-dimensional supergravity.) An interesting aspect of these extensions is that they incorporate some of the more modern non-perturbative duality symmetries. The basic results that I wish to describe are represented pictorially in Fig. 1. This figure combines two well-known perturbative T dualities and

4

two non-perturbative identifications. The perturbative T duality on the left side of the figure is between type liB superstring theory on a circle and type IIA superstring theory on a circle of reciprocal radius.5 The perturbative T duality on the right side of the figure relates the Es x Es heterotic string on a circle and the S0(32) heterotic string on a circle of reciprocal radius~ One non-perturbative fact is that the IIA theory in ten dimensions actually has a circular eleventh dimension whose radius is proportional to the 2/3 power of the string coupling constant?• 8 Another is that the E8 x E8 heterotic theory in ten dimensions actually has an eleventh dimension that is a line segment I (or, equivalently, a Z 2 orbifold of a circle).9 The length of the line segment also scales as the 2/3 power of the string coupling constant. What I propose to do in this lecture is to by-pass the IIA and Es x Es theories and discuss the following two dualities: 1) the equivalence of M theory on a two-torus T 2 and liB superstring theory on a circle S 1 ; 2) the equivalence of M theory on a cylinder C and S0(32) superstring theory on a circle S 1 • In each case we will find that the area of the manifold T 2 and C (at fixed shape) scales as the -4/3 power of the size of the circle S 1 . The wrapping of all relevant p-branes on the compact spaces will be considered. By requiring a detailed matching of p-branes in each pair of dual theories we achieve numerous consistency tests of the overall picture and deduce relations among various pbrane tensions. The discussion of M/IIB duality is a review of results reported previously,10 whereas the M/S0(32) duality has not been presented previously.

2

BPS - Saturated p-Branes

A useful technique for obtaining non-perturbative information about superstring theories and M theory is to identify their BPS saturated p-branes. pbranes are p-dimensional objects that are characterized by a "tension" r,, which is the mass per unit p-volume (and thus has dimensions of (mass)P+l ), and a suitable (p + I)-dimensional world-volume theory. In theories with enough supersymmetry (all supersymmetric theories in ten or eleven dimensions, in particular) the tension of a p-brane carrying a suitable conserved charge has a strict lower bound proportional to that charge. When there is equality (T = Q), the p-brane is said to be BPS saturated. Such p-branes belong to "short" representations of the supersymmetry algebra. So long as the supersymmetry is not broken, the tension of such a p-brane cannot be changed by any quantum correction - perturbative or non-perturbative. This generalizes the well-known fact that the photon belongs to a "short" representation of the Poincare group and must remain massless so long as gauge invariance remains unbroken.

5

BPS saturated p-branes of superstring theories or M theory can be approximated by classical solutions of the corresponding effective supergravity theory that preserve half of the supersymmetry. Such solutions are not exact superstring solutions, of course, but they do demonstrate the existence of particular p-branes, give their tensions correctly, and exhibit other qualitatively correct properties. Generally these objects can be regarded as extremal black p-branes, i.e., extremal black holes (p = 0), extremal black strings (p = 1), etc. In solving the supergravity equations to obtain p-brane configurations it is sometimes necessary to include a source to match a delta function singularity in the equations. When the source is required, one often speaks of a "fundamental p-brane" and when it isn't of a "solitonic p-brane." It will not be necessary for us to keep track of this distinction here, however. The conserved charges carried by p-branes are associated with antisymmetric tensor gauge fields (1)

When these undergo gauge transformations c5An = dAn-1, the field strength Fn+l = dAn is invariant. A supergravity theory with such a gauge field typically has two kinds of BPS-saturated p-brane solutions. The electric p-brane has p = n - 1 and the dual magnetic p-brane has p = D - n - 3. When it is possible to reformulate the supergravity theory in terms of a dual poten•F plus possible interaction corrections) there tial A (satisfying F dA is no essential distinction between "electric" and "magnetic" p-branes. The two are interchanged in the dual formulation. In the case of an electric pbrane, the gauge field couples to the p-brane world volume generalizing the well-known j ·A interaction of charged point particles. It is also worth noting that the charges of dual electric and magnetic p-branes satisfy a generalized Dirac quantization condition QEQM E 21rZ.11 Let us now consider specific examples, beginning with type liB supergravity theory. This theory contains a complex scalar field p = x + ie-•, and so the specification of a vacuum is characterized by the modulus

=

=

OB

Po

i

= (p} = 21r + AB '

(2)

where AB is the string coupling constant. (Note that the analogous formula for N 4 theories in four dimensions has p0 8/21r + i/>-. 2 .) The S duality group of type liB superstring theory is an S£(2, Z) under which p transforms nonlinearly in the usual way. Therefore, p0 may be restricted to the usual fundamental region. The liB theory also has a pair of two-form potentials that transform as a doublet of the S£(2, Z) duality group. Therefore,

=

aW

=

6

the associated electric one-branes (strings) carry a pair of B charges. Suitably normalized, they can be chosen to be a pair of relatively prime integers (q1, q2). Setting (JB = 0 (to keep things simple), the tensions (in the liB string metric) arelo (3)

In the canonical metric 'liB) is a constant, and thus 'liB) ""' >:;/ 12 in the string metric. Note that in the string metric only 'TiB)(1, 0) is finite as ~B - 0. Therefore it is natural to regard it as the fundamental string and the others as solitons, though they are all mapped into one another by the duality group. The dual five-branes carry a pair of magnetic B charges. The liB theory also contains a four-form potential A4 with a self-dual field strength Fs. As a result, electric and magnetic charge are identified in this case and carried by a self-dual three-brane. The M theory story is somewhat simpler, since the only antisymmetric tensor gauge field in eleven-dimensional supergravity is a three-form A3. There are associated electric two-brane and magnetic five-brane solutions. It is apparently not possible to replace A3 by a dual six-form potential, so the electric-magnetic distinction is meaningful in this case. (The two-brane is "fundamental" and the five-brane is "solitonic" in the sense described earlier.) In the case of type I or heterotic strings the relevant low-energy theory is N 1D 10 supergravity coupled to an Es x E 8 or S0(32) super YangMills multiplet. In this case the relevant antisymmetric tensor is a two-form 8 1111 • The associated electric one-brane is the heterotic string. There is also a five-brane, which is the magnetic dual of the heterotic string. Type I strings do not carry a conserved charge, they are not BPS saturated, and therefore they can break. For these reasons, they can only be described reliably at weak coupling when they are metastable. Now that we have described the relevant theories and their p-branes we can turn to the duality analysis.

=

3

=

M/IID Duality

The duality described by the left side of Figure 1 relates M theory compactified on a two-torus and liB superstring theory compactified on a circle. The torus is described by its area AM (in the canonical eleven-dimensional metric) and a modular parameter T, which may be taken to lie in the fundamental region of the S£(2, 7l) modular group. The corresponding parameters of the liB theory are the modulus Po and the circumference of the circle LB (in the canonical ten-dimensional metric, which is the one that is invariant under

7

S£(2, 7l) transformations). To test the equivalence of these two constructions, we will examine the matching of all p-branes in nine dimensions. These include various wrappings of the ones identified in the preceding section as well as new p-branes that arise by Kaluza-Kiein mechanisms. The possible p-brane wrappings are depicted in Figure 2. As the figure shows, p-branes, for any p between 0 and 5, can be obtained by suitable wrapping of a p'-brane in either M theory or liB theory. The identifications are straightforward for p = 1, 2, 3, 4. They also work for p = 0, 5, but one must be careful to take account of Kaluza-Klein effects in these cases. (The Kaluza-Klein vector fields in nine dimensions - arising due to isometries - support additional electric zero-branes and magnetic five-branes.) When the matching is done correctly, one finds a one-to-one correspondence of p-branes and their tensions in nine dimensions. The details have been worked out previously.10 Here, I will simply state the results and discuss their implications. The most important result, perhaps, is that one must make the identification 10,12 (4) Po= r. Thus, the geometric S£(2, 7l) modular group of the torus is identified with the non-perturbative S-duality group of the liB theory! The canonical metrics g(M) and g(B) are related (after the compactifications) by (5)

71B)

Here, ~M) is the M theory two-brane tension. Both ~M) and (introduced earlier) are constants that define scales and can be set to unity without loss of generality, though I will not do that. In terms of these constants, the compactification scales AM and LB are related by (6)

A!i

14 . This means that if one compactifies the liB theory on a Thus, LB .... circle and lets the size of the circle vanish, while holding p0 fixed, one ends up with M theory in eleven dimensions! Conversely, if one compactifies the M theory on a torus and lets the torus shrink to zero at fixed shape, one ends up with a chiral theory - liB superstring theory - in ten dimensions. The matching of p-branes and their tensions also yields a number of relations among the tensions. Not only does one learn the relation between M and liB tensions in eq. (6), but also relations among the tensions of the M and liB theories separately. For the M theory, one learns that the five-brane tension is

8

proportional to the square of the two-brane tension

T~M) = _!_(~M))2. 211"

(7)

This formula implies that the product of electric and magnetic charges is the minimum value allowed by the quantization condition.13 For the p-branes of the liB theory one finds that all of their tensions can be expressed in terms of and the moduli .....(B)(q q ) _ A 1/2.....(8) (8)

TtB)

Ti

1, 2 - u. 9 1 i

TIB) = _!_(T(B))2 3 211" 1 T.(B)(

5

(9)

) 1 .6_1/2(T(B))3 q1,q2 =(211")2 9 1 ,

where q2(JB) 2 .6.q = ( q1- ~ AB

q~ + ).B ·

(10)

(11)

Equations (8) and (11) generalize eq. (3) to include (JB # 0. Note that the tension of all RR p-branes "" 1/>.B in the string metric, as expected for Dbranes.15 A number of amusing things are taking place in the p-brane matchings that gave these relations. Let me just mention one of them. In matching zero-branes there is a duality between the M theory two-brane and the liB strings that works as follows~ 0 The Kaluza-Klein excitations of the strings on a circle correspond to wrappings of the two-brane on the torus. Conversely, the wrappings of the SL(2, Z) family of strings on the circle correspond to the Kaluza-Klein excitations of the membrane on the torus. Let us pause for a moment to consider the physical meaning of what we have shown. The claim is that M theory on T 2 is the same thing as liB theory on S 1. If one considers the common nine-dimensional theory, one might imagine asking the question "How many compact dimensions are there?" This question has two correct answers - one and two - depending on whether one thinks of M theory or liB theory. This paradoxical situation has a simple resolution. Fields that describe "matter" in one picture can describe "metric" in the other and vice versa. This situation already occurs for more conventional T duality - say, for the heterotic string on a torus. In that case the duality mixes up internal components of the metric with those of the two-forms and U(1) gauge fields. What is new in the present case is that (i) the dual compact spaces have different dimensions, (ii) certain components of the liB theory

9

metric correspond to components of the three-form gauge field in M theory, (iii) certain components of the M theory metric correspond to the complex scalar p of the liB theory. Despite these differences of detail, the basic concept is the same. Another important distinction, of course, is that the generalization ofT duality considered here encodes non-perturbative features of the theory. 4

M/S0(32) Duality

Let us now consider the duality depicted in the right-hand portion of Figure 1. This is the equivalence of M theory compactified on a cylinder with 80(32) superstring theory compactified on a circle. Note that we do not specify whether the 80(32) theory is type I or heterotic. The reason, of course, is that they are different descriptions of a single theory~· 14 so it is both of them. The dilaton in one description is the negative of the dilaton in the other, and so the coupling constants are related by >.. = (.~~0 ))- 1 , where 0 represents 80(32), H represents heterotic, and I represents type I. Recall that the 80(32) theory has two BPS saturated p-branes: the heterotic string and its magnetic dual, which is a five-brane. As discussed earlier, the type I string is not BPS saturated and will not be considered in our analysis. The p-branes arising from compactification on a circle are straightforward to work out and are depicted in Figure 3. As in Section 3, one should also be careful to include p-branes of Kaluza-Klein origin. M theory, before compactification, has a two-brane and a five-brane, but new issues arise when one considers compactification on a manifold with boundaries, and so we must first get that straight. Consider first the Horava-Witten picture 9 - M theory on Ji10 x I, an eleven-dimensional space-time with two parallel ten-dimensional boundaries. This is the non-perturbative description of the Es x Es heterotic string, just as M theory on li 10 x 8 1 describes the type IIA superstring. We know that Es x Es theory, regarded as ten-dimensional has a one-brane (the heterotic string) and a dual five-brane. So we must ask what these look like in eleven dimensions. There is only one sensible possibility. An Es x Es heterotic "string" is really a cylindrical two-brane with one boundary attached to each boundary of the space time. Thus, an E8 x E8 heterotic string can be viewed as a ribbon with one E8 gauge group living on each boundary. This seems to be the only allowed two-brane configuration, since any other would give rise to a two-brane that remains two-dimensional in the weak coupling limit in which the space-time boundaries approach one

another. For the five-brane, the story is just the reverse. It must not terminate on the space-time boundaries, but can exist as a closed surface in the bulk. This is required so that it can give a five-brane and not a four-brane in the

10

weak coupling limit. Subsequent reduction on a circle to nine dimensions gives the wrapping possibilities depicted in Figure 3. The cylinder C has a height L1 and a circumference L2. These are convenently combined to give an area Ac = L1L2 and a shape ratio u = Ltf L2. The circumference of the circle for the S0(32) theory compactification is denoted Lo (in the heterotic string metric). Also, the p-brane tensions of the S0(32) theory are denoted 0 > and ~ 0 >. Now, guided by Figure 3, we again match p-branes in nine dimensions, just as we did in the previous section. One finds that the analog of the identification Po = r is

rf

u = >.r{j> = (>.~0 ))- 1 = Ltf L2.

(12)

This means that when the spatial cylinder is a thin ribbon (u 1), the type I string is weakly coupled. The analog of eq. (6) is ( T(O) L2 1 0

)-1 _ _ 1_T.(M) A3/2 -1/2 -

(211")2

2

c

0'

(13)

•

Aside from the factor of u- 112 the equations look the same. As before, for fixed u, Lo "" A(; 3/ 4 • Thus, the S0(32) theory in ten dimensions can be obtained by shrinking the cylinder to a point. As in Section 3, the p-brane matching in nine dimensions gives various tension relations. The only new one is 71o> = _1_ (L2) 2 (Tio))a s (21r)2 L1 1 .

( 14)

Combining eqs. (12) and (14), one learns that in the heterotic string metric, where 0 > is constant, T~O) "" (>.~>)- 2 , as expected for a soliton. On the other hand, in the type I string metric T~O)"" 1/>.~0 ) and 0 )"" 1/>.~0 ), as expected for D-branes.15 In the case of M/IIB duality we found that the SL(2, Z) modular group of the torus corresponded to the S-duality group of the liB theory. In the present case of M/S0(32) duality, the cylinder does not have an analogous modular group. However, the interchange L1 +-+ L2 (or >.1 +-+ >.H) corresponds to the strong/weak duality transformation of the S0(32) theory that relates the perturbative heterotic limit to the perturbative type I limit.

rf

5

Ts

Conclusion

On several occasions Keiji Kikkawa has pioneered concepts that have led to important advances in string theory. The lessons of T duality are still being

11

learned. In particular, the generalization ofT duality to M theory described here exhibits the non-perturbative equivalence of all known superstring theories, realizing a dream I have had for many years. Other generalizations of T duality have been found,1 6 and there may still be more to come. We owe Keiji a debt of gratitude for pointing us in this direction. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

K. Kikkawa, B. Sakita, and M.A. Virasoro, Phys. Rev. 184 (1969) 1701. K. Kikkawa and J.H. Schwarz, Phys. Rev. D1 (1970) 724. K. Kikkawa and M. Yamasaki, Phys. Lett. 1498 (1984) 357. A. Giveon, M. Porrati, and E. Rabinovici, Phys. Rept. 244 (1994) 77. M. Dine, P. Huet, and N. Seiberg, Nucl. Phys. 8322 (1989) 301; J. Dai, R.G. Leigh, and J. Polchinski, Mod. Phys. Lett. A4 (1989) 2073. K.S. Narain, Phys. Lett. 1698 (1986) 41; P. Ginsparg, Phys. Rev. D35 (1987) 648. P.K. Townsend, Phys. Lett. 8350 (1995) 184, hep-th/9501068. E. Witten, Nucl. Phys. 8443 (1995) 85, hep-th/9503124. P. Horava and E. Witten, hep-th/9510209. J. H. Schwarz, Phys. Lett. 8360 (1995) 13, hep-th/9508143; hepth/9509148;hep-th/9510086. R.I. Nepomechie, Phys. Rev. D31 (1985) 1921; C. Teitelboim, Phys. Lett. 1678 (1986) 69. P.S. Aspinwall, hep-th/9508154. M.J. Duff, J.T. Liu, and R. Minasian, hep-th/9506126. J. Polchinski and E. Witten, hep-th/9510169. J. Polchinski, hep-th/9510017. A. Sen, hep-th/9512203.

12

~

0

l\

0

13

14

u

II')

II')

15

U-DUALITY AND INTERSECTING D-BRANES

ASHOKE SEN Mehta Research Institute of Mathematics and Mathematical Physics 10 Kasturba Gandhi Mary, Allahabad ~1100~, INDIA

We derive some of the predictions of U -duality for degeneracy of solitonic states in six dimensional string theory, and verify these predictions by counting states of Dirichlet branes.

I shall begin this lecture by thanking the organisers of the conference for giving me the opportunity to come here and pay my tribute to Prof. Kikkawa on his 60th birthday. My talk will focus on some aspects of duality in string theory, - a subject that originated with the work of Prof. Kikkawa more than ten years agcl. By now there is overwhelming evidence for strong-weak coupling duality in various field theories and string theories. Most of the tests of duality in string theory, however, involve comparing properties of either massless, or nearly massless solitonic and elementary string states. In this talk I shall focus on a test of duality involving massive string states. The theory that we shall consider is type IIA theory compactified on a four dimensional torus, and for convenience we shall take this torus to be the product of four circles, each at self-dual radius. In ten dimensions, the massless fields of the type IIA theory consist of the metric GM N, anti-symmetric tensor field BMN and the dilaton CJ from the Neveu-Schwarz Neveu-Schwarz (NS) sector, and the gauge field AM and the rank three anti-symmetric tensor field CMNP from the Ramond-Ramond (RR) sector (0 ~ M, N ~ 9). Upon compactification on T4, we get a six dimensional theory whose gauge fields are Gm"' Bm" from the NS sector and Cmn"' A" and C" - the dual of Cpvp from the RR sector (0 ~ JJ, v ~ 5, 6 ~ m, n ~ 9). Elementary string states in string theory are charged under the NS sector gauge fields, but are neutral under RR sector gauge fields. Thus such a state can be characterized by an eight dimensional charge vector. In the basis where we arrange the vector fields

16

from the NS sector as

G6,.. B6,.. G1,.. B11-1

(1)

Gs,.. Bs,.. G9,.. B9,..

we denote the charge vector as P6 W6

P7 W7

Q=

(2)

Ps ws P9 W9

Thus Pm and Wm denote the momentum and winding along the mth direction respectively. We shall normalize the charges so that Pm and Wm are integer valued. The mass formula for an elementary string state is given by (3)

where N R and N L denote the oscillator levels on the right and the left moving sector of the world-sheet respectively, and Q~

1

= 2(Is ± L )Q , U1

(4)

= (~ ~) ·

(5)

The total number of supersymmetry generators in this theory is thirty two, sixteen of them coming from the left-moving sector of the world-sheet and sixteen coming from the right moving sector. A typical supermultiplet is not invariant under any of the supersymmetry generators, however if N R ( N L)

17

vanishes, then the corresponding supermultiplet is invariant under eight of the sixteen right (left) moving supersymmetry generators. Thus, for example, if NR NL 0, then the corresponding supermultiplet is invariant under 16 of the 32 supersymmetry generators. The other sixteen generators act nontrivially on the supermultiplet and generate a 16 dimensional Clifford algebra. Thus the super-multiplet, which forms a representation of this Clifford algebra, is 2 16 12 = 256 dimensional. This is known as the ultra-short multiplet. On the other hand, if NR = 0 but NL f. 0, then the supermultiplet is invariant under only eight of the 32 generators. The rest of the generators acting on the state form a 24 dimensional Clifford algebra giving a 224 / 2 = (16) 3 dimensional representation. This is known as the short multiplet. These states are known as BPS states and their degeneracies can be calculated using tree level string theory, as these degeneracies are not expected to change under quantum correction. If d(NL) denote the degeneracy of states with NR = 0, NL 2:: 0, then string theory gives

=

=

L

d( N L)qNL = 256

NL?_O

fr (~ ~ q: )

8

n=l

·

(6)

q

Thus, for example d(O)

= 256,

d(2)

= 9 . {16)3 .

(7)

Also note from eqs.(2)-(5) that for these states

NL

~(Qh- Q~J = ~QT LQ 2

2

9

LPmWm.

(8)

m=5

Let us now turn to predictions of duality symmetry in this theory. This theory has a conjectured duality symmetry group S0(5, 5; Zf. This of course contains the T-duality group SO( 4, 4; Z) as its subgroup, but also contains a Z 2 subgroup which converts the NS sector gauge fields to RR sector gauge fields and vice-versa: c67JJ G6JJ Cs9JJ B6JJ CssJJ G1JJ B1JJ +-+ c79JJ {9) c69JJ GsJJ C1s" Bs" AJJ G9JJ B9JJ cJJ

18

It also dualizes the three form field strength in six dimensions and changes the sign of the dilaton: Hp 11 p +-+ H 1111 p ~- -~. (10) Thus it acts as string-string strong-weak coupling duality in six dimensions. Note that since this duality transformation exchanges the NS sector gauge fields with RR sector gauge fields, acting on elementary string states that carry NS charge, it will produce a state carrying RR charge which does not exist in the elementary string spectrum. Thus these states must arise as solitons in string theory. Knowing the degeneracy of BPS saturated elementary string states carrying NS charge, we can also predict precisely the degeneracy of BPS saturated solitonic states carrying RR charge. Consider for example the states P6=m w6

=n

P7 = 0

0 Ps = 0 ws = 0 pg = 0 Wg = 0 W7=

(11)

As we can see from eq.(8), for BPS saturation, this state must have NL = mn, and hence degeneracy d(mn) given in eq.(6). Under the Z2 duality this is mapped to a state carrying m units of C67p charge and n units of Csg 11 charge. Let us denote this as (m, n) state. Thus if U-duality is correct, then this solitonic state must also have degeneracy d(mn). The question that we shall address is 'Is this prediction correct?' Before we proceed further, let us point out a problem. From eq.(3) we see that the mass of an (m, n) state will be given by (M(m,n)f =

~(m+n)2.

(12)

Thus M(m, n) = mM(1, 0) + nM(O, 1).

(13)

This shows that an (m, n) state can decay into m (1,0) states and n (0,1) states at rest. In other words, an (m, n) state is energetically indistinguishible from the end point of the continuum describing m copies of (1,0) states and n copies of (0,1) states. This makes it very difficult to study these states and calculate their degeneracy. We shall avoid this problem by compactifying the theory further on a circle 8 1 of radius Rand studying the Kaluza-Klein (KK)

19

modes of the ( m, n) state carrying momentum kf R along this new S 1 . This will correspond to a state in the five dimensional theory with mass

(14) It can be easily verified that this state has strictly less energy than any continuum with the same charge quantum numbers, as long as k and m + n are relatively prime. Thus the degeneracy of these states will be much easier to calculate and will give us the degeneracy of the (m, n) states in the six dimensional theory. This is the strategy we shall adopt. Before we proceed, we shall make one further modification of the problem. This involves a T-duality transformation which inverts the radius R of S 1 without affecting the original This has several effects:

r.

1. It converts the type IIA theory to type liB theory.

2. It converts a state carrying k units of momentum along S 1 to a state carrying k units of winding along S 1 • 3. It converts CmniJ to DmnliJ where m, n labels coordinates on T 4 , !labels the coordinate of S 1 on which this duality transformation is being performed and DMNPQ denotes the rank four antisymmetric tensor gauge field with self-dual field strength. Thus the prediction of the U -duality can be stated as follows: In type JIB theory compactified on T 5 = T 4 X S 1 ' the degeneracy of a soliton state carrying m units of Dsnl-' charge, n units of Ds911J charge, and k units of winding along the 1 direction is given by d(mn) given in eq.(6} fork and m + n relatively prime. We shall refer to a state carrying these quantum numbers as an (m, n; k) state.

The next question is: how can we construct these (m, n; k) states? It has become clear through the recent work of Polchinskr that the correct way to view these states is as Dirichlet branes (D-branes) wrapped around various cycles of the torus. In particular an (m, n; k) state will correspond to a bound state of m Dirichlet 3-branes wrapped around the 6- 7- 1 cycle of the torus, and n Dirichlet 3-branes wrapped around the 8-9-1 cycle of the torus. The quantum number k enters in the following way. It has been shown4 •5 that the dynamics of collective coordinates of such a configuration of D-branes is given by an 1 + 1 dimensional supersymmetric gauge theory, which can be obtained from the dimensional reduction of an N=2 supersymmetric U ( m) x U ( n) gauge theory in four dimensions, with a hypermultiplet in the adjoint representation

20

of the gauge group, and a hypermultiplet in the (m, n) representation of the group. The spatial dimension of this quantum field theory lies along the 8 1 labelled by the coordinate x 1 and the temporal direction is along the usual time direction. lfU(1)d denotes the diagonal U(1) subgroup, then the quantum number k corresponds to the electric flux associated with this diagonal U(1) gauge group along 8 1 • Finally, the BPS states of the theory correspond to supersymmetric ground states of this quantum field theory. Thus the prediction of U-duality can now be restated as a prediction for the degeneracy of ground states of this supersymmetric quantum field theory. This degeneracy, on the other hand, can be calculated using Witten index theorem. For example, in the case m = n = 1, we have a U(1)xU(1) gauge theory, and the prediction of U-duality is that this system will have a (16) 3 fold degenerate ground state in each sector carrying odd unit of U( 1)d electric flux along 8 1 • This was verified explicitly in ref~. The prediction for more general (m, n) was verified in rer.1. To conclude, we see that the D-brane technology enables us to make precise tests of string theory dualities involving massive string tests. This shows that the various dualities indeed have stringy origin and are not simply accident of low energy field theory. 1. K. Kikkawa and M. Yamasaki, Phys. Lett. B149 (1984) 357.

2. 3. 4. 5. 6. 7.

C. Hull and P. Townsend, Nucl. Phys. B438 (1995) 109. J. Polchinski, Phys. Rev. Lett. 75 {1995} 4124. E. Witten, Nucl. Phys. B460 {1996} 335. M. Bershadsky, V. Sadov and C. Vafa, hep-th/951 0225. A. Sen, Phys. Rev. D59 {1996) 2874. C. Vafa, hep-th/9511 088.

21

DUALITY AND 4D STRING DYNAMICS SHAMIT KACHRU Harvard University, Cambridge, MA 02198, USA We review some examples of heterotic/type II string duality which shed light on the infrared dynamics of string compactifications with N=2 and N=l supersymmetry in four dimensions.

1

Introduction

I am very happy to have the opportunity to speak about strong/weak coupling duality on this occasion honoring the 60th birthday of Professor Keiji Kikkawa. His own foundational work on T-duality 1 , the worldsheet analogue of S-duality, was in many ways instrumental in inspiring the recent developments in nonperturbative string theory. Strong-weak coupling dualities now allow us to determine the strong coupling dynamics of string vacua with N ~ 4 supersymmetry in four dimensions 2 • It is natural to ask if this progress in our understanding of string theory can be extended to the more physical vacua with less supersymmetry. For N=2 the01ies in four dimensions, quantum corrections significantly modify the mathematical structure of the moduli space of vacua, as well as the physical interpretation of its apparent singularities. This was beautifully demonstrated in the field theory case in 3 and it has more recently become possible to compute the exact quantum moduli spaces for N=2 string compactifications as well 4 •5 • This constitutes the subject of the first part of my talk. Of course, the case of most physical interest is N ~ 1 theories. In the second part of my talk, I discuss examples of dual heterotic/type II string pairs where the heterotic theory is expected to exhibit nonperturbative dynamics which may fix the dilaton and break supersymmetry 6 • The type II dual manages to reproduce the qualitative features expected of the heterotic side at tree level. It is to be hoped that further work along similar lines will result in a better understanding of supersymmetry breaking in string theory. ·

2

N=2 Gauge Theory and String Compactifications

Recall that the N=2 gauge theory with gauge group SU(2) is the theory of a single N=2 vector multiplet consisting of a vector A~', two Weyl fermions A and 1/J, and a complex scalar field , all in the adjoint representation of

22

SU(2). In N=1 language, this is a theory of an N=1 vector multiplet (~,A*') coupled to an N=1 chiral multiplet (4>, 1/J). The scalar potential of the theory is determined by supersymmetry to be

V(tf>) =

g~ !4>,4>+]2

(1)

We see that V vanishes as long as we take 4> = diag(a, -a), so there is a moduli space of classical vacua parameterized by the gauge invariant parameter u = tr(t/>2 ). At generic points in this moduli space M v of vacua, there is a massless N=2 U(1) vector multiplet A. The leading terms in its effective lagrangian are completely determined in terms of a single holomorphic function F(A), the prepotential:

(2) The first term determines, in N=1language, the Kahler potential (and hence the metric on M v) while the second term determines the gauge coupling as a function of moduli. In 3 the exact form of F including instanton corrections was determined. In addition, the masses of all of the BPS saturated particles were computed. This was reviewed in great detail in several other talks at this conference, so I will not repeat the solution here. It will suffice to say that the crucial insight is that the singular point u = tr(t/>2 ) = 0 where SU(2) gauge symmetry is restored in the classical theory splits, in the quantum theory, into two singular points u = ±A2 , where a monopole and a dyon become massless. In this talk our interest is not really in N = 2 gauge theories but in the string theories which reduce to N = 2 gauge theolies in the infrared. There are two particularly simple classes of d = 4, N = 2 supersymmetric string compactifications. One obtains such theories from Type II (A or B) strings on Calabi-Yau manifolds, and from heterotic strings on /(3 x T 2 (with appropriate choices of instantons on the K3). Here we briefly summarize some basic properties of these theories. Type IIA strings on a Calabi-Yau threefold M give rise to a four-dimensional effective theory with nv vector multiplets and nh hypermultiplets where (3)

The +1 in nh corresponds to the fact that for such type II string compactifications, the dilaton is in a hypermultiplet. The vector fields in such a theory are Ramond-Ramond U(1)s, so there are no charged states in the pertw·bative string spectrum. Furthermore, because

23

of the theorem of de Wit, Lauwers, and Van Proeyen 7 whicl1 forbids couplings of vector multiplets to neutral hypermultiplets in N=2 effective lagrangians, the dilaton does not couple to the vector moduli. This means that there are no perturbative or nonperturbative corrections to the moduli space of vector multiplets. On the other hand the moduli spaces of hypermultiplets are expected to receive highly nontrivial corrections, including "stringy" corrections with e- 1/9 strength 8 • One interesting feature of the moduli spaces of vector multiplets in such theories is the existence of conifold points at finite distance in the moduli space. At such points the low energy effective theory becomes singular (e.g., the prepotential develops a logarithmic singularity) 9 • This phenomenon is reminiscent of the singularities in the prepotential which occur at the "massless monopole" points in the Seiberg-Witten solution of N=2 gauge theory, singularities which are only present because one has integrated out a charged field which is becoming massless. In the case at hand, in fact, one can show that there are BPS saturated states (obtained by wrapping 2-branes around collapsing 2-cycles) which become massless and which are charged under (some of) the Ramond-Ramond U(l)s 10 • These explain the singularity in the prepotential. In fact at special such points, where enough charged fields (charged under few enough U ( 1)s) become massless, one can give them VEVs consistent with D and F flatness. This results in new "Higgs branches" of the moduli space. These new branches correspond to string compactifications on different Calabi-Yau manifolds, topologically distinct from M 11 , and there is evidence that all Calabi-Yau compactifications may be connected in this manner 12 •13 • The other simple way of obtaining an N=2 theory in four dimensions from string theory is to compactify the heterotic string (say E8 x E 8 ) on [(3 x T'l. Because of the Bianchi identity dH = Tr(R 1\ R)- Tr(F 1\ F)

(4)

one must embed 24 instantons in the E8 x E 8 in order to obtain a consistent theory. An SU(N) k-instanton on /(3 comes with Nk+ 1-N2 hypermultiplet moduli (where k 2:: 2N), and /(3 comes with 20 hypermultiplet moduli which determine its size and shape. Embedding an SU(N) instanton in E 8 breaks the observable low energy gauge group to the maximal subgroup of E 8 which commutes with SU(N) (Er for N=2, Ee for N=3, and so forth). In addition, there are three U(l) vector multiplets associated with the T 2 • Their scalar components are the dilaton S and the complex and kahler moduli T and p of the torus (both of which live on the upper half-plane H mod SL(2, Z)). At special points in the moduli space the U(1) 2 associated with T

24

and p is enhanced to a nonabelian gauge group:

= p = 1/2+i.f3/2-+ SU(3) (5) Because the dilaton lives in a vector multiplet in such compactifications, the moduli space of vectors is modified by quantum effects. On the other hand, the moduli space of hypermultiplets receives neither perturbative nor nonperturbative corrections. An interesting feature of the heterotic M v is the existence of special points where the classical theory exhibits an enhanced gauge symmetry (as described above for the compactification on T 2 ). Sometimes by appropriate passage to a Higgs or Coulomb phase, such enhanced gauge symmetry points link moduli spaces of N=2 heterotic theories which have different generic spectra (for some examples see 4 •14 ). It is natural to conjecture that such transitions connect all heterotic N=2 models, in much the same way that conifold transitions connect Calabi-Yau compactifications of type II strings. T

3

= p-+ SU(2)

X

U(1),

T

= p = i-+ SU(2) 2 ,

T

N =2 String-String Duality

From the brief description of heterotic and type II N=2 vacua in the previous section, it is clear that a duality relating the two classes of theories would be extremely powerful. If one were to find a model with dual descriptions as a compactification of the Type IIA string on M and the heterotic string on K3 x T 2 , one could compute the exact prepotential for Mv from the Type IIA side (summing up what from the heterotic perspective would be an infinite series of instanton corrections). Similarly, one would get exact results for Mh from the heterotic side- this would effectively compute the e-l/g corrections expected from the IIA perspective. In fact, such a duality has been found to occur in several examples in 4 •5 • One of the simplest examples is as follows. Consider the heterotic string compactified to eight dimensions on T 2 with r = p. Further compactify on a K3, satisfying the Bianchi identity for the H field by embedding c2 = 10 SU(2} instantons in each Es and a c2 = 4 SU(2} instanton into the "enhanced" SU(2) arising from the T = p torus. After Higgsing the remaining E1 gauge groups one is left with a generic spectrum of 129 hypermultiplets and 2 vector multiplets. The 2 vectors are T and the dilaton S- when T = i, one expects an SU(2) gauge symmetry to appear (the other SU(2) factor that would normally appear there has been broken in the compactification process). This tells us that if there is a type IIA dual compactification on a Calabi-

25 Yau M, then the Betti numbers of M must be h 11 (M)

= 2,

h21 (M)

= 128

(6}

There is a known candidate manifold with these Betti numbers - the degree 12 hypersurface in W Pt, 1 ,2 ,2 ,6 defined by the vanishing of p p-

z112

+ z212 + z36 + z46 + z52 + ••••

(7}

This manifold has in fact been studied intensively as a simple example of mirror symmetry in 15 •16 • The mirror manifold W has hu (W) 128, h21 (W) 2. The conjecture that IIA on M is equivalent to the heterotic string described above implies that liB on W is also equivalent to that heterotic string. The structure of the moduli space of vector multiplets of the heterotic string should be exactly given by the classical (in both sigma model and string perturbation theory) moduli space of complex structures of W. The mirror manifold can be obtained by orbifolding p = 0 by the maximal group of phase symmetries which preserves the holomorphic three-form 17 • Then the two vector moduli are represented by l/J and cP in the polynomial

=

=

(8)

It is also useful, following 15 , to introduce (9)

These are the convenient "large complex structure" coordinates on the moduli space of vector multiplets for the liB string. In order to test our duality conjecture, we should start by checking that the liB string reproduces some qualitative featmes that we expect of the heterotic Mv. For example, T = i for weak couplingS-+ oo is an SU(2} point. There should therefore be a singularity of Mv at this point which splits, as one turns on the string coupling, to two singular points (where monopolesfdyons become massless), as in the case of pure SU ( 2) gauge theory. The "discriminant locus" where the liB model becomes singular is given by (10) So we see that as a function of y for y :f. 0 there are two solutions for z and as y -+ 0 they merge to a single singular point z = 1. This encourages us to identify x = 1, y = 0 with T = i, S -+ oo of the heterotic string- the SU(2)

26

point. The metric on the moduli space for y at y = 0 and S at weak coupling also agree if one makes the identification y ""e- 5 . There is also a remarkable observation in 16 that the mirror map, restricted to y = 0, is given by

j(i) x=-j(rl)

(11)

where r 1 is one of the coordinates on the Kahler cone of M. Here j is the elliptic j-function mapping C onto H/ SL(2, Z). This tells us that the classical heterotic T moduli space, which is precisely H/SL(2,Z), is embedded in the moduli space of M at weak coupling precisely as expected from duality. In fact using the uniqueness of special coordinates up to rotations, one can find the exact formula expressing the IIB coordinates (x, y) in tet·ms of the heterotic coordinates (r, S). Of course with this map in hand there are now several additional things one can check. The tests which have been performed in 4 •18 •19 •20 include 1) A matching of the expected loop corrections to the heterotic prepotential with the form of the tree-level exact Calabi-Yau prepotential. 2} A test that the g-loop F-terms computed by the topological partition functions F9 on the type II side (which include e.g. R 2 and other higher derivative terms) are reproduced by appropriate (one-loop!) computations on the heterotic side. 3) A demonstration that in an appropriate double-scaling limit, approaching the T = i, S -t oo point of the heterotic string while taking a' -t 0, the IIB prepotential reproduces the exact prepotential of SU(2) gauge theory (including Yang-Mills instanton effects) computed in 3 • These tests give very strong evidence in favor of the conjectured duality. Given its veracity, what new physics does the duality bring into reach? • One now has examples of four-dimensional theories with exactly computable quantum gravity corrections. In the example discussed above, the SeibergWitten prepotential which one finds in an expansion about T = i, S -t oo receives gravitational corrections which are precisely computable as a power series in a'. • On a more conceptual level, the approximate duality of 3 between a microscopic SU(2) theory (at certain points in its moduli space) and a U(l) monopole/dyon theory is promoted to an exact duality, valid at all wavelengths, between heterotic and type II strings.

27 • There is evidence that at strong heterotic coupling, new gauge bosons and charged matter fields appear, sometimes giving· rise to new branches of the moduli space 21 •22 • • The e- 1 /g corrections to the hypermultiplet moduli space of type II strings are in principle exactly computable using duality (and may be of some mathematical interest). One might wonder what is special about the Calabi-Yau manifolds which are dual to weakly coupled heterotic strings. In fact it was soon realized that the examples of duality in 4 involve Calabi-Yau manifolds which are /(3 fibrations 23 • That is, locally the Calabi-Yau looks like CP 1 X /(3· In fact, one can prove that if the type IIA string on a Calabi-Yau M (at large radius) is dual to a weakly coupled heterotic string, then M must be a J(3 fibration 24 • To make this more concrete, in the example of the previous section, we saw M was defined by the vanishing of p --

12 z1

+ z212 + z38 + z48 + Zs2 + •••

(12)

in W J1, 1 ,2 ,2 ,6 • Set z1 = ..Xz2 and define y = z~ (which is an allowed change of variables since an identification on the W P 4 takes z1 -+ -z 1 without acting on z3 ,4 ,5 ). Then the polynomial becomes (after suitably rescaling to absorb .X)

(13) which defines a /{3 surfaces in WCP! 1 1 3 • The choice of .X in z1 = ..Xz2 is a point on C P 1 , and the /(3 for fixed ch~ic~ of .X is the fiber. It is not surprising that J(3 fibrations play a special role in 4d N=2 heterotic/type II duality. Indeed the most famous example of heterotic/type II duality is the 6d duality between heterotic strings on T 4 and type IIA strings on /(3 25 •26 • If one compactifies the type IIA string on a CY threefold which is a /(3 fibration, and simultaneously compactifies the heterotic string on a J(3 x T 2 where the J(3 is an elliptic fibration, then locally one can imagine taking the bases of both fibrations to be large and obtaining in six dimensions an example of the well-understood 6d string-string duality 27 • This picture is not quite precise because of the singularities in the J(3 fibration, but it does provide an intuitive understanding of the special role of J(3 fibrations.

4

N =1 Duality and Gaugino Condensation

Starting with an N = 2 dual pair of the sort discussed above, one can try to obtain an N = 1 dual pair by orbifolding both sides by freely acting symmetries. This strategy was used in 27 •28 where several examples with trivial

28 infrared dynamics we1·e obtained. Here we will find that examples with highly nontrivial infrared dynamics can also be constructed 6 • Our starting point is an N=2 dual pair (IIA on a Calabi-Yau M and heterotic on K 3 x T 2 ) where the heterotic gauge group takes the form

(14) H denotes the hidden sector and obs the observable sector. We will first discuss the technical details of the Z2 symmetry by which we can orbifold both sides to obtain an N 1 dual pair, and then we discuss the physics of the duality. Orbifold the heterotic side by the Enriques involution acting on [(3 and a total reflection on the T 2 • This acts on the base of the elliptic fibration (Zt, z2) by

=

{15) taking C P 1 -+ RP2 • In addition, we need to choose a lifting of the orbifold group to the gauge degrees of freedom. We do this as follows: • Put a modular invariant embedding into the "observable" part of the gauge group alone. • Embed the translations which generate the T 2 into Ef, constrained by maintaining level-matching and the relations of the space group. For example one could take Wilson lines At ,2 along the a and b cycle of the T 2 given by At=

1

2(0,0,0,0, 1, 1, 1, 1),

A2

1 = 2(-2,0,0,0,0,0,0,0)

(16)

Here At,2 = !Lt,2 where £1,2 are vectors in the Es root lattice. These Wilson lines break the Ef gauge symmetry to S0(8h ® 80(8)2. How does the Z2 map over to the type II side? i,From the action

(17) on the CP 1 base (which is common to both the heterotic and type II sides), we infer that the z2 must be an antiholomorphic, orientation-reversing symmetry of the Calabi-Yau manifold M. To make this a symmetry of the type IIA string theory, we must simultaneously flip the worldsheet orientation, giving us an "orientifold." In such a string theory, one only includes maps c) of the worldsheet E to spacetime M I z2 if they satisfy

(18) where Wt is the first Stieffel-Whitney class.

29

We know from 6d string-string duality that the Narain lattice r 20 •4 of heterotic string compactification on T' maps to the integral cohomology lattice of the dual K 3 •. This means that we can infer from the action of the Z2 on the heterotic gauge degrees of freedom, what the action of the z2 must be on the integral cohomology of the Ks fiber on the IIA side. Since we are frozen on the heterotic side at a point with 50(8) 2 gauge symmetry in the hidden sector, the dual J(3 must be frozen at its singular enhanced gauge symmetry locus. The 1(3 dual to heterotic enhanced gauge symmetry G has rational curves Ci, i = l, ... ,rank(G) shrinking to zero area (with the associated 8i = 0 too). It is easy to see, e.g. from Witten's gauged linear sigma model that in this situation the type II theory indeed exhibits an extra Z2 symmetry. The bosonic potential of the relevant gauged linear sigma model (for the case of a single shrinking curve) is given by

!~

V = 2

2

{ (

•

(L Qf(lt/>~1 2 -14>~1 2 )]- r?)

2

cr

+~~[I: Qi 2 (14>~1 2 + 14>~1 2 ))10";1 2 •

cr

Here the tf>s represent the [(3 coordinates while r parametrizes the size of the curve and tT is the Kahler modulus. Precisely when r ~ 0, the model has the Z2 symmetry t/> ~ -J>, tT ~ -0". Orbifolding by this Z2 then freezes the K 3 at its enhanced gauge symmetry locus, as expected. What is the physics of the dual pairs that one constructs in this manner? In the heterotic string, when there is a hidden sector pure gauge group

{19) one expects gaugino condensation to occur. This induces an effective superpotential {20) where Ab(S) "' e-cr.s and ab is related to the beta function for the running Gb coupling. It was realized early on 29 •30 that in such models (with more than one hidden factor) one might expect both stabilization of the dilaton and supersymmetry breaking. It has remained a formidable problem to determine

30

which (if any) such models actually do have a stable minimum at weak coupling with broken supersymmetry. For now, we will be content to simply understand how the qualitative structure of the heterotic theory (e.g. the gaugino-condensation induced effective superpotential) is reproduced by the type II side. This is mysterious because the type II N=2 theory we orientifolded had only abelian gauge fields in its spectrum, so we need to reproduce the strongly coupled nonabelian dynamics of the heterotic string with an abelian gauge theory on the type II side. The heterotic orbifold indicates the spectrum of the string theory as 9het -+ 0. The heterotic dilaton S maps to the radius R of the RP2 base of the type II orientifold (recall one obtains the RP2 by orbifolding the base P 1 of the 1 = ka'e 2cf>oo ka'g 2 and Qi 4GNQi 2 = ~a'g Q;, Ps =~a' A (cf.(6)). Thus one needs all four charges to be nonvanishing to get a non-zero area of the horizon, and thus a non-zero value of the analogue of the Bekenstein-Hawking entropy i A SBH:: 4GN .

(21)

glt is possible, of course, to consider more general (e.g. multi-center) choices for the harmonicfunctions F- 1 , K, f, k- 1 • In particular, taking all 4 functions to be 0 (3 )-symmetric but having centers at different points one finds the solution discussed in [26]. h If all charges are equal the dilaton is constant and the solution coincides with the extreme dyonic Reissner-Nordstrom black hole of the standard Einstein-Maxwell (a = 0) theory. The case of Q1 Q2 = P1, P2 = 0 corresponds to a solution in the case of dilatonic coupling constant a 1/.../3 [14, 27]. Note also that the small-distance structure of these D 4 backgrounds considered as string-theory solutions, should be actually analyzed from a higher dimensional point of view. 'Extremal black holes are expected to have zero thermodynamic entropy [28]. However, the proper definition of the entropy may be subtle in the present case. We shall avoid the discussion of this here, just making the assumption that the Bekenstein-Hawking formula still makes sense in the case of extreme dyonic black holes.

= =

=

43

As in the purely electric case, there is a whole family of black hole solutions with the same values of electric and magnetic charges but different short-distance structure. They are 4-dimensional 'images' of certain (supersymmetric BPSsaturated) excited states of the solitonic string (16) which are described by a marginal 'deformation' of (16) which includes extra 'chiral' couplings (see (10) and the next subsection). The knowledge of underlying 2d conformal field theory makes possible to count the number d(N) of such states for a given set of (large) charges (Q1, Q2, Pt. P2) and to identify [17] the Bekenstein-Hawking entropy (21) with the statistical entropy lnd(N), supporting the proposal of [18]. This will be discussed in Section 3. 2.4

Generalisations and related models

The models (11),(16) are special cases of a chiral null model with curved transverse part,

L = F(x)ou [8v + K(u, x)Ou + 2..4;(u, x)Oxi]

+ ~'Rln F(x) + LJ. ,

LJ. = (G;j + B;j)(x)oxi8zi +'R(x).

(22) (23)

There exists a renormalisation scheme in which (22) is conformal to all orders in o:' provided the 'transverse' u-model (23) is conformal and the functions p-l, K, ..4;, ~ satisfy certain conformal invariance conditions. The simplest tractable case is when the transverse theory has at least (4, 0) extended worldsheet supersymmetry so that the conformal invariance conditions essentially preserve their 1-loop form [29], i.e. are the 'Laplace' equations in the 'transverse' background [21, 17], o;(e- 2•V?JGii8j)F- 1 = 0, etc. For example, the 8-dimensional transverse space may be chosen as a direct product of some 4-space M 4 and a 4-torus. If the functions defining M 4 model have S0(3) symmetry we are led back to the case of (16) with 1-center harmonic functions. Choosing A;(u, x)Oxi = A(x)[Oy1

+ a,(x)Ox'] ,

A= ~ . r r

+ ~(P1 + P2) r+ P1

'

(24)

we find a generalisation [17] of the 'dyonic' model (16) which describes a spherically symmetric D = 4 background with two extra electric charges (q, -q) and the metric function and the area of the event horizon being (the mass M remains the same as in (19))

44

(26) Further generalisations of the dyonic model (16) (responsible for the 'hair' of 4-dimensional dyonic backgrounds) are obtained, e.g., by switching on the u-dependence in K as in the fundamental string case (10) discussed above. Another possibility is to consider 6-dimensional models with M 4 -part having (locally) SO( 4)-invariant structure. In this case the transverse theory is the same as in the 5-brane model [24)

Hmnk = -2v'CJGP 1fmnkp01¢> =

-fmnktOt/,

where the indices m, n, ... run from 1 to 4. i One finds that the Laplace-type equations for p-l, K preserve their flat-space form (2) and thus (we assume that A;= 0; cf.(5),(14)) p-1- 1 + Q2 r2, 21>

e

r2

Ql K=1+2 , r

(28)

+p

= F f = r2 + Q2'

The special cases of this model are: (1) six-dimensional fundamental string (P 0) with momentum and winding numbers "' Q 1 ,2 [3]; (2) its D = 6 S-dual - soli tonic string solution [32) (Q 1 = Q2 = 0, P ::f. 0) which is a D = 6 reduction of D = 10 5-brane solution [24, 30]; (3) 'dyonic' D = 6 string of [33) (Ql 0, Q2 =I= 0, p =I= 0). Dimensionally reducing this D = 6 model to 5 dimensions along u (i.e. wrapping the 5-brane around S 1 x T 4 ) one finds the 3-charge (Q 1 , Q 2 , P) extreme dyonic D = 5 black hole background. k The 5-dimensional stringframe metric and the Einstein-frame metric are related by g,.. 11 = e-t(,>G,..v, e 4 4>(,) = P F K- 1 , e2 = e2 oo F /, so that

=

=

(29)

JThe model (22),(27) may thus be considered as an anisotropic generalisation of the 5brane model of [30, 24]: different isometric 5-brane coordinates are multiplied by different functions of the 4 orthogonal coordinates, cf. [31]. kin five dimensions an antisymmetric tensor is dual to a vector so that Hmnp is a magnetic dual of an electric field with charge P.

45 The mass of this D = 5 black hole, the 3-area of the r = 0 horizon and the Bekenstein-Hawking entropy are (cf.(19),(21)) 1 -G(s) GN= N

(30)

•

A

SBH

= Q2 = P

(31)

= 4GN.

= =

= =

we find .x- 1 f 1 + Pfr 2, ~ ~ 00 • Introducing + p~, p0 :: v'P we then finish with the D 5 extreme ReissnerNordstrom metric

If Q1 p2

=

r2

(32) This special case (Q1 = Q2 = P) was discussed in [19] in connection with reproducing the Bekenstein-Hawking entropy as a statistical entropy using the D-brane approach to count the number of corresponding microscopic BPS states. As in the case of the D = 4 dyonic black hole model, this counting can be done also in a direct way (for generic large values of (Ql, Q2, P)) using the fact that the small-scale ('throat') region of the corresponding D = 6 conformal model is described by a WZW model with level~t = Pfa' (see Section 3.3). 3

Dyonic black bole entropy from string theory

The D = 4 dyonic black holes (with four (17) or five (25) parameters) discussed above belong to the set of N = 1 supersymmetric BPS saturated extreme dyonic solutions of the leading-order effective equations of T 6 -compactified heterotic string which are parametrised by 28 electric and 28 magnetic charges [14, 17, 35]. m For fixed values of charges (Qi, Pi) one expects to find a subfamily of N = 1 supersymmetric BPS-saturated black hole backgrounds which all look the same at large distances but differ in their short-distance structure (i.e. at scales of order of compactification scale where their higher dimensional solitonic string origin becomes apparent). This classical 'fine structure' or 'degeneracy' is expected to be responsible for the black hole entropy [4, 5, 18].

=

1 In general, for a black hole in D dimensions with 9tt 1- IS/r 0 - 3 + ... the ADM mass is (see, e.g., [34]) M 1-'(D- 2)wo-2/16'1rG~) (w, 4'lr, w3 21r 2 , etc). The Bekenstein-Hawking entropy is expressed in terms of the volume A of (D- 2)-dimensional horizon surface by S BH = A/4a. mThese more general D 4 solutions can be constructed (17, 35) by applying special T- and S-duality transformations to the 5-parameter solution but not all of them directly correspond to exact string solutions, i.e. to manifestly conformal D 10 a-models.

=

=

=

=

=

46

To try to check the proposal [18] that one can indeed reproduce the Bekenstein-Hawking entropy as a statistical entropy of degenerate black hole configurations it is sufficient to consider the simplest non-trivial choice of the charges (Ql. Q2, Pt, P2). The aim is to explain the expression for the corresponding entropy {21) in terms of degeneracy of BPS states originating from possible small-scale oscillations of underlying six-dimensional string soliton (16),(5),(14). The relevant 'oscillating' D = 6 backgrounds are described by the general model (22) and can be thought of as special marginal deformations of (16). Since (16) defines a non-trivial CFT, the counting is not as simple as in the fundamental string ('pure electric') case, where, because of the matching onto string sources, one expects that BPS-saturated oscillating fundamental string configurations are in one-to-one correspondence with analogous BPS states in the free string spectrum [4, 5]. Still, it can be done at least approximately (for large charges) in a rather straightforward way. The key observation is that since the degeneracy is present only at small scales (all corresponding D = 4 black hole backgrounds look the same at large r) to find the number of different states one may first replace (16) by its shortdistance (r --+ 0) limit and then do the counting. In this 'throat' limit (16) reduces to a WZW-type model with level"= 4PtP2/ol. Then for large values of the level the counting of BPS states should proceed essentially as in the free string case, with only two (but crucial) differences. One is is that now the string tension of the 'transverse' part of the action is proportional to "· Another is that in contrast to what happens in the free string case, here the number of BPS oscillation states we should count is the same in the heterotic and type II theories. Indeed, the relevant marginal perturbations of the model (16) are only 'left', not 'right' (the functions in (22) can depend only on u, not on v to preserve the conformal invariance when both F and K are nontrivial). This is also related to the fact that the background in (16) is 'chiral' and has the same amount of space-time supersymmetry (N = 1, D = 4) in both theories (only 'left' perturbations will be supersymmetric). As a result, the statistical entropy will indeed be the same in heterotic and type II cases, in agreement with the fact that the corresponding black hole solution (and thus the Bekenstein-Hawking entropy) is the same in the two theories. This approach is legitimate if all four charges (Q1, Q2, Pt. P2) are large compared to the compactification scales (radii Rn of compact coordinates Yt, Y2 which we shall assume to be of order of #). The scale of the soliton is determined by .../P1 P2 and thus is large (i.e. the curvature at the throat is small) provided the magnetic charges are large. At the same time, if the largedistance value of string coupling is taken to be small (e•- < 1), to ensure that its short distance ('throat') value (e• 0 ) is also small one is to assume that the

47

electric charges are of the same order as the magnetic ones (see ( 18)). 3.1

'Throat' model and magnetic 'renormalisation' of string tension

The model (16),(5),(14) has a regular r--+ 0 limit described by (16, 17]

I=

1r~'

J

d2 uLr-+o = 4:

J

d2 u [(8z8z + 8u8u + e-zau8v)

+ (aY1 aY1 + atpatp + aoao -

= __!_, j 11'a +

4:

d 2 u (e-zauav

(33)

2 cos oaY1 atp)]

+ QtQ2 18u8u)

j d u (azaz + 8y18Y1 + 8tp8tp + 8088- 2cos88yt8V'). 2

Here -1 n---rOO, Q2----'Z=

(34)

r

u=

v = (QtQ2 1 PtP2)- 112v,

(Q1 1 Q2PtP2)- 112u,

Yt = P 1- 1 Yt

+ tp.

As already mentioned above, an important property of this model is that here (in contrast to, e.g., the 5-brane model (24]) the dilaton is constant in the wormhole region!" The throat region model (33) is equivalent to a direct product of the SL(2, R) and SU(2) WZW theories (divided by discrete subgroups) with both levels equal to "· 0 Since the level of SU(2) must be integer, we get the P quantisation condition for Pt P2 = For large values of charges, i.e. large "• the counting of states in this model should proceed essentially as in the flat space case (the fact that part of the action can be interpreted as a non-compact SL(2, R) WZW model then does not introduce any problem). The only (crucial) difference is that the transverse (z, y1 , tp, 8) part of the model has now 'renormalised' string tension

la' "·

1

P1P2

1

- +a-' --a-'2 a' .l

(35)

'

n Auuming that all 4 charges are of the IllUDe order we may ignore the difference between the values of the dilaton (string coupling) at r oo and at r 0 (see (18)). 0 Similar model (corresponding to the special caae of P1 ~. Q1 Q2 ) was previously discussed in (36]. ~'Moreover, each of P1 and P2 should also be quantized. The requirement of regularity of the 6-metric"' (dy1 + P1 (1- cos9}dcp] 2 + ... (the absence of Taub-NUT singularity) implies that one should beabletoidentifyy with period4wP,, which is possibleif2Pt/Rt k. By Tduality in !It-direction the IllUDe constraint should apply also to P2, i.e. 2P2 IRt Ia' In •.

=

= =

=

=

= =

48

If w and m are the winding and momentum numbers of a free heterotic string compactified on a circle of radius R, then the mass and oscillator level of BPSsaturated left-moving oscillation modes (NR 0, NL NL- 1) are given by (cf. (19),(20))

=

m

M

wR

=

1

= R +--;;--- = 4GN(Qt +Q 2) '

(36)

where we have used (6) to express m, w in terms of the 'space-time' charges Q1, Q2. The relations (36) are true in the case when both 5 1 and non-compact terms in the string action have the same overall coefficient (tension). The magnetic renormalisation (35) of the tension of the transverse part of the action implies that NL is to be rescaled by the ratio of the 'longitudinal' (1/2rra') and 'transverse' (1/2rra~) tensions (the oscillator level is proportional to the inverse of string tension)

NL--+ NL

a' P1P2Q1Q2 = -mw = ---;c-a~ 16GJ.,

(37)

Since the charges are large we ignore the quantum '-1' shift. 3.2

Statistical entropy

The number of BPS states in the free string spectrum with a given left-moving oscillator number NL » 1 is (see, e.g., [8]) d(NL)NL':>l ~ caN2 exp(4rr)N[} Then the statistical entropy of ensemble of states with the same charges but different left-moving oscillation modes is given by

(38) Using the expression (37) for N L we then find that, for large charges, coincides with the Bekenstein-Hawking entropy (21), s.ta!

r;:;- rr.,fQ1Q2PtP2 A lVL ~ G = -G = SnH . N 4 N

~ 4rrv

s.ta!

(39)

It should be emphasized that there is no ambiguity in the overall coefficients in these expressions for the entropy. For example, the coefficient in front of JNL in d( N L) is related to the fact that underlying degrees of freedom correspond to a ('half' of) 1-dimensional extended object, i.e. are described by a 2d field theory (see also (18]). Note also that the dependence on the Newton's constant effectively drops out~ qThe statistical entropy is given by a classical string-theory expression and does not depend on the string coupling. The constant G N needs to be introduced to express ( cf. (6))

49

3.3

D = 5 extreme dyonic black holes

The universality of the relation between the statistical and Bekenstein-Hawking entropies is confirmed by analogous consideration of the case of D = 5 extreme dyonic black holes described by {22),{27)-{31). Here the throat limit r -t 0 is described by a similar S£{2, R) x SU(2) WZW model {cf.(33)-{35)) r

I= -1-jd uLr-+O = - 1-jd u(e-zau8v+QIQ2 8u8u) 2

1

{40)

-;;;= 4a'2 . .l

{41)

2

'Tra'

'TrOt'

1

"'= -P, a'

1

Q2 -too, r

z =In 2

p

Its level K. = Pfa' again rescales the tension of the 'transverse' part of the action by a' fa~ = Pf4a'. The important factor of 4 difference compared to the expressions in {34),{35) is related to the increased dimensionality of the spatial sphere (here the harmonic functions depend on r 2 ). The analogues of the relations {36),{37) are {now we are using {6) for D 6 fundamental string, Q1 :aN ·jf, Q2 :aN· ~I!, where as in {30), GN::: G~))

=

=

=

{42)

a'

NL -t NL =a~ mw

7r2 p

= 64 ah Q1Q2.

{43)

The expression for the statistical entropy is then {cf.{31),{39))

r;;r S.tat:;::::: 47ry lVL:;:::::

1r

2v'Q1Q2P

2aN

A

= 4aN = SBH·

{44)

Notice again a remarkable consistency of numerical factors. In the special case of Q1 Q2 P the same conclusion was reached in [19] by first transforming the heterotic string solution into a type II string one

=

=

the string 'microscopic' quantum numbers in terms of 'space-time' charges (which appear in the effective field theory description). This is necessary in order to be able to compare with the Bekenstein-Hawking entropy. Since SBH itself originates from the on-shell value of the (euclidean) space-time effective action it also contains a factor of 1/GN. rThe 'transverse' (z, 1/J, 12. The 32 spinors Q~ may be classified as the spinor for SO( c + 1, 1) ® SO(d- 1, 1) with d + c + 2 = 12. Here c is interpreted as the number of compactified dimensions from the point of view of 100 string theory, and the extra 2 dimensions are considered hidden. This spinorx spinor classification is given in Table I for each dimension. The index a corresponds to the spinor of SO( c+ 1, 1). This group is not necessarily a symmetry, but it helps to keep track of the compactified dimensions, including the hidden ones. Furthermore, the same index a will be reclassified later under the maximal compact subgroup K of U-duality, thus providing a bridge between duality and higher hidden dimensions. Consider the maximally extended algebra of the 32 supercharges in various dimensions in the form p {Qaa• Q6{J } = 6a6..,,. ta{J I'

+ """ ..,,. •... ,.. L..., ta{J

zo6 ,., ... ,. •.

{1)

p:O,l, ..

Since the left side is the symmetric product of 32 supercharges, the right side can have at most t32 x 33 = 528 independent generators. The indices ab on z;~ are either symmetrized or antisymmetrized and have the same permu-

... ,..

54

...

tation symmetry as at{3 in')':;,···~~ •. The central extensions z;~ 11 are assumed to commute with Q~, P11 , but they are tensors of the Lorentz gr~up and hence do not commute with it. In (10,2) dimensions we will use M = 0', 0, 1, 2, · · ·, 10 for the space index instead of I'· In the a2xa2 representation (equivalent to chirally projected 64x64) only the 2- and 6- index gamma matrices ')'~1 M2 and -y~····Me are symmetric in et{3, and furthermore 'Y!~r·Me is self dual. Therefore, in 12 dimensions, on the right hand side of ( 1) there can be no PM, and the 528 generators consist of ZM1 M 2 , and the self dual zi;•... M.· The number of components in 462 respectively• Upon compactifix 11 x 1oxsxsx 7 each is 12 X2 11 66 and 12 121x2x3x4x5x6 cation to (10,1) we rewrite the 12D index M = (0', J.') where I'= 0, 1, 2, · · ·, 10 is an 11D index. Then we have (suppressing the 0' index)

=

=

z,.,2

-

P, $

-+

x, •... ,s

66 = 11 +55 462 = 462

(2)

which are the momenta and central charges in 11 dimensions pointed out in 8 • Continuing the compactification process to lower dimensions on R 4 - 1 •1 ® re+ 1 •1 , each eleven dimensional index I' decomposes into I' $ m where I' is in d dimensions and m is inc+ 1 = 11- d dimensions. Then each 11 dimensional tensor decomposes as follows P 11

-+

P11 $ Pm

z,ll

-+

z,ll $ z; $ zmn (a)

For example in d = 10 the type IIA superalgebra is recovered, with the 528 operators (P11 , P 1o, Z 1111 , Z 11 , X 111 ... 11 ., 115 ) where the± indicate self/antiself dual respectively. In Table I in each row labelled by (d- 1,1)/(c + 1,1) the numbers of each central extension of P, Z, X type with p Lorentz indices is indicated (these are the numbers that are not in bold type). As we go to lower dimensions one must use the duality between p indices and d - p indices to reclassify and count the central extensions z;~ 11 •• In the table a number in parenthesis means that it should be omitted from there and instead moved in the same row to the location where the same number appears in brackets . This corresponds to the equivalence of p indices and d - p indices. When p = d- p there are self-dual or anti-self-dual tensors. Their numbers are indicated with additional SUperscriptS ± in the form 1± 1 2± 1 a± 1 10± 1 as± Wherever they OCCUr •

x-;....

...

55

5!~:!

32Q! 50(c+1,1) ®L) 5~~~) 4-1,1

p.=!r

,....

P•,z"'"

P~.z;

X"'"'' ..

,~

p:T

p-=4

p:li

x•'•t" ,.

i,.. x'•r ..

x•r

x·· ,.1.,.,.

x,.~·-s

~

~~·

7\ r.t

(:l:,t6)

1+0 +O

1+1 +O

1 +0

0

1

...

(t.t6)

O+O +O

1+2 +O

.:o

1

0

(2,t6)

2+1 +O :3 .. 2+t

1+2 +O :3 .. 2+t

1+0 =t .. t

1

:3 .. 2 +t

((2,0),8+) ((o,2),8-)

3+3 +O :6 ..3+ +3-

1+1 =t+t .. t+t

3+[1) =(2,2) ..3+t

+3:6 .. 3+ +3-

~r1i

6

hl

(4,8)

4+6 +O :tO .. to

1+3 +0 =(2,2) .. 3+t 1+4 +1 =5+t =5+t

.!..!

(4,4") (4",4)

~

...

a.!

1+ +11+ +2+

~~

S0(1,1)

z.

SL(2,R)

-m.r

....

5£{2)411 50{1,1) 50{2)

...

5£(3) ®5L{2) 50{3) ®U{1)

{1)

,.

u

x

llllZ·

~

... ...

5+10 +1 =t+t5 •(4,4)

...

...

L.l

!.! 2,1

:5+t •5+t

1+5+ 5+[1) =2x6 •(0,5) +(5,0) +2(0,0)

1+10 +(5) =t+t5 =(4,4)

(8,4)

+[1) =7+2t .. 27+t

1+6 +15+[6) =7+2t .. 27+t

1+20 +[15) =t+35

(8+,(2,0)) (8- ,{0,2))

7+21 +21 +(7] :28+28 .. 28.

1+7 +35 +[21) =8+56 .. 63+t

1"' +35* =1* +35* =36 •

(16,2)

8+28 +56 +[28) :36+84 .. 120

ow

!.!

3

~~o~36

··~r1~5~ :t+9 +t26 sd35 . +t

Table I. Classification of Q~ and z:~

...

{1+56)

,.

+(4) :tO ... to

{1)

,.

-··· ..... ..... .....

5 ~ ®50{5)

...

us;m-

{4)

(1)

{5)

(1)

..... {15)

(6) ......

,.

{21)

...(7)

...

0

~•

{28) .......

0

0

~•

10+ +10=to+ +10.. (t0,1) +{1,t0)

-···

{1)

. ,.. under llD (or 12D) and K.

SL(S)

som-

8 t(O)

56

The total number of central extensions P, Z, X found according to this compactification procedure for each value of p are indicated in Table-I in bold characters. These totals are the same numbers found by counting the number of The bold numbers following the= sign correspond possibilities ab on z;~ to representations of SO(c + 1, 1) (making a connection to 12D) and those following the~ sign correspond to representations of K (to be discussed later in connection to duality). What is the meaning ofthe p-form central extension z;~ ? Since this is a charge in a global algebra, there ought to exist a (p+ 1)-form"local current 1 ... ,,. ( z) whose integral over a space-like surface embedded in d-dimensions gives

... ,,..

... ,

J;!,

J

z;~···p,. = d"- 1 E~' 0

1;!,1 ... ,,. (z).

(4)

The current couples to the fields of low energy physics (i.e. supergravity). In the case of usual central charges that are Lorentz singlets zab (i.e. p = 0) the current is associated with charged particles. Such a current may be constructed as usual from worldlines (or equivalently from local fields) as follows

J:" (z) = JdT 4=zf6bel (z- Xi (T))

OTX! (T).

(5)

I

The zf 6 are the charges of the particles labelled by i. This current couples in and it appears as the source in the equation of the action to a gauge field motion of the gauge field

A:,,

s- Ei I dT A:, (xi (T)) =I d"z A:, (z) J;" (z) O>. ()[AA:l (z) =

aTx! (T) z!l" I (6)

J:, (z).

The generalization to the higher values of p is straightforward: In order to have a charge that is a p-form we need a current 1 ... ,,. (z) that is a (p + 1)-form. This in turn requires a p-brane to construct the current,

J;!,

J;!,

1 ...

,,.(z) =

j dTdtrl ...du,~zf6 6"(z-Xi(T,trl,···u,))

(7)

I

xaTx{,o ... Ou,.X!,.J(T,trl,. ··u,), and its coupling to supergravity fields requires a (p + 1)-form gauge potential Ap 0 p 1 ···p,. (z) such that (8)

57

= ~ ~

jdd T

d

D'l··· D'p

A"o"• ... ,.. a6

(xi)

jl

VT

xi(p,o

•••

jl

V.sp

xil'p]

46

Zi

t

I

and jl

V.\

jl(~AI'ol'a···P,p) ( ) _ Jl'ol'a···P,p ( ) Z a6 Z . 46

v

(9)

As is well known by now there are perturbative as well as non-perturbative couplings of p-branes to supergravity in various dimensions. Hence the z;~ are present in the superalgebra and they correspond simply to the charges of p-branes. The classification of their ab indices under duality groups is the subject of the next section, but here we already see that there is a one to one correspondence between the p-forms z;~ and the (p + 1)-form gauge that appear as massless states in string theory in the potentials NS-NS or R-R sectors. The main message is that from the point of view of the superalgebra all p-branes appear to be at an equal footing. lsometries of the superalgebra that will be discussed below treats them equally and may mix them with each other in various compactifications. The theory in d dimensions has (p + 1)-forms which appear as massless vector particles in the string version of the fundamental theory. These act as gauge potentials and couple at low energies to charged p-branes. This generates a non-trivial central extension z;~ in the superalgebra. The number of such central extensions (ab indices) is in one-to-one correspondence with the number of the (p + 1)-forms and these numbers can be obtained by counting the possible combination of (symmetric/antisymmetric) indices ab associated with the supercharges.

... ,..

A::"• . .,..

... ,..

A::,.•.. ,..

... ,..

A::,.....,..,

3

Duality groups

In the discussion above we concentrated on the 110 (or 120) content of the supercharges and the central extensions. We now turn to duality. In string theory the T -duality group is directly related to the number of compactified left/right string dimensions. Therefore, in our notation, for a string of type II it is T = SO(c, c). Its maximal compact subgroup is k = SO(c)L ®SO(c)R where L, R denote left/right movers respectively. The index a on the supercharges Q~ corresponds precisely to the spinor index of SO(c)L ®SO(c)R (see table III in 12 ). Investigating the supercharges listed in Table I shows that the index a that was classified there under the hidden non-compact group SO(c+1, 1)hidden can be reclassified under the perturbatively explicit maximal compact subgroup k C TofT-duality, k = SO(c)L ®SO(c)R· These two groups are not subgroups of each other, but they do have a common subgroup SO( c). Recall that cis the

58

number of compactified dimensions (other than the two hidden dimensions), and SO(c) is the rotation group in these internal dimensions. Next we look for the compact group K that contains SO(c)L x SO(c)R, SO(c + 1) and that has an irreducible representation for the index a (total dimension N). By virtue of containing k C T the group K :J k must be related to a larger group of duality U that contains T. The groups K and U are listed in Table I. The subgroup hierarchy that emerges is as follows SO(c + 1, 1)

e eompoet + 2 hidden dim•.

® SO(d- 1, 1) •poeetime

on Q 0 do ab on Z IAt···IA,. a

l

l

SO( C + 1) 1 hidden dim SO(c)L ® SO(c)R

mo:cim1 ® 1R ®I: r~t> ® L r~~)

(211 + 2}1 ) ® (215 + 2}5 )

(12)

The 211 + 2}1 corresponds to the intermediate supermultiplet of UD supersymmetry. The structures Li r~'t,;> are listed in Table II up to level 5. Level

S0(9)L,R reps

ILR = 1 hR=2 ILR = 3 ILR =4

9B 44B + 16F

(Ei r~IL,R)) L R

1B

(9 + 36 + 156)B + 128F ( 1 + 36 + 44 + 84 ) +231+450 B h,R=5 + [16 + 128 + 576]J;' Table II. L/R oscillator states in 10D.

61

The S0(9)L,R representations in this table are reduced to representations of SO(d- 1)L,R ® SO(c)L,R· So, a general perturbative string state is identified by "index space" and "base space" in the form (13)

The base space are the quantum numbers coming through the ( rii, n) and the indices are given by the product of representations in (12) and Table II (which may be extended beyond level5). These are all the perturbative type II string states in d-dimensions. The spectrum of the non-perturbative states is much richer. There are many central charges in the supersymmetry algebra (see Table I) and those provide sources that couple to the NS-NS as well as R-R gauge potentials. Therefore one finds a bewildering variety of non-perturbative solutions of the low energy field equations as examples of non-perturbative states that carry the non-perturbative p-brane charges z;~···j.lp' We will take an algebraic approach to describe them, by imposing the stucture of the superalgebra discussed earlier. The base quantum numbers are now extended to include the non-perturbative charges that appear in the global superalgebra {here we concentrate on p =0-branes only, ignoring the higher p-branes in this paper).

Ivac,

- .... I > plA ;m,n,z

(14)

where the 0-brane charges are (rii, n, z1 ). From the point of view of string theory, the z 1 are non-perturbative charges that couple to the R-R sector, while (rii, n) are the perturbative charges that couple to the NS-NS sector. In the notation of Table I we identify the generators that correspond to (rii, n, z1 ) as follows

i,j

1,2, .. ·,c

rt=1,···,c,c+1

That is, rii corresponds to the Kaluza-Klein momenta excluding the extra hidden coordinate, the winding numbers ncorrespond to the last column or row of zr 1 r2 , while all remaining 0-brane charges are non-perturbative. Although there is a big asymmetry among these charges from the point of view of the string, they are on equal footing from the point of view of the superalgebra, and they are classified in higher multiplets of SO(c+ 1,1)hidden and of K C U as discussed before. The multiplets zo& = (rii, n, z 1 ) form the non-perturbative base in tPindicu (base) 0 • 11

According to the dimensions of representations in Table I, the 0-brane

zob

= (m, n, z1 )

62

There are two types of new non-perturbative states: those obtained by applying string oscillators on the non-perturbative base and those that cannot be obtained in this way, but which are required to be present to form a basis for U-duality transformations. The second kind require the extension of the indices such that complete K multiplets are obtained. These are needed as intermediate states in matrix elements of the superalgebra which is assumed to be valid in the full theory. So, a general state in the theory is identified at each IL,R as in (13). Both the base and the indices have non-perturbative extensions. The full set of states is required to form a basis for U-duality transformations at each fixed value of h,R· These states are not degenerate in mass, hence the idea of a multiplet is analogous to the multiplets in a theory with broken symmetry. The BPS saturated states are those with either h = 0 or IR = 0. Even for BPS saturated states there are the two types of non-perturbative states. Typically the non-perturbative indices occur for IL ~ 2, IR = 0. For the BPS saturated states one can derive an exact non-perturbative formula for the mass by using the supersymmetry algebra with central charges. For example for a state with non-trivial quantum numbers (m,n,pe+l) and IR 0, h m. n, the mass is (15)

=

=

The presence of the non-perturbative (quantized) Pe+l is a new piece in the mass formula usually written for BPS states. This formula is derived from the superalgebra with the usual methods, but allowing for the non-perturbative Pe+l· A special case is the uncompactified theory in 10D, for which the BPS states (called black holes in 3 ) have masses proportional to the 11th momentum. Further generalizations involving other non-trivial z1 will be given elsewhere. For non BPS saturated states we cannot give an exact mass formula. What about the hidden dimensions? In the uncompactified theory consider all the states, including their values of the non-perturbative 11th momentum . In Fourrier space the fields ~indiee• (zP, z 11 ) seem to be 11-dimensional. This is possible only if the indices also have an UD structure. At levels I = 0, 1 it has been known that this is true for a long time for the usual string states, and this is evident from Table II (Ievell = 1 is a singlet times the factor 2}: + 2}.5 which has UD content). At higher levels I~ 2 the string states by themselves do not seem to correspond to complete linear representaticms of U for all dimensions except for

...,..

tl = 3 (when U = Ea(a))· Similady, higher p-branes z:~ do not generally form linear repnsentations of U. Also they seem to form complete representations of SO(c, c) for all c:aaes except for (d 5, p 3),(d 3,4, p 2). We interpret these obaervatioaa to mean that the base is not generally a liB ear representation of either Tor U duality groups, but it is a liaearrepresentation of K or SO(c+ 1,1).

=

=

=

=

63

have the liD structure for the indices. The minimal structure of indices that would be needed in an liD theory was identified for all levels. This minimal structure has a definite pattern for massive states given by indicu =>

(2},' + 2},1') x Rf.'>.

(16)

The factor 2},' + 2},1' can be interpreted as the action of 32 supercharges on a set of S0(10) representations Rf. 1>at oecillator Ievell. For the minimal set of indices the factor Rf.1> is of the form of a sum of S0(9) representations that make up S0(10) representations.

RJ.I) =

t (E r~l')) t (E r~l'))

(17)

X

11 :1

i

L

1'=1

i

R

Each term in the sum over I' looks like the string states in Table II 11 • Only the highest term (I' = I) corresponds to the perturbative string states of level I. The remaining terms correspond to non-perturbative states with quantum numbers isomorphic to those listed in Table II at the given levels. The meaning of this pattern has not been understood so far. Furthermore, in the complete theory there may be more states beyond the minimal set displayed above.

5.e

Dualities and non-perturbative spectrum

The perturbative string states involved in the T-duality transformations are not all degenerate in mass. Therefore, T -duality must be regarded as the analog of a spontaneously broken symmetry, and the string states must come in complete multiplets despite the broken nature of the symmetry. It is well known that T = O(c, Cj Z) acts linearly on the the 2c dimensional vector (m, n). However it is important to realize that it also acts on the indices of 4Jindicu in definite representations 12• The action of T on the indices is an induced #:-transformation that depends not only on all the parameters in T but also on the background c X c matrices ( Gij I Bij) that define the tori rc . Since the states in the previous section are all in 1: = O(c)L x O(c)R multiplets, the Tduality transformations do not mix perturbative states with non-perturbative states. A U-multiplet contains both perturbative as well as non-perturbative Tmultiplets. Like the T-duality transformations, the U-duality transformations act separately on the base and the indices of the states described by (13) without miring indez and base spaces. The action of U on the base quantum numbers (m, n, zr) is a linear transformation in a representation of same

64

dimension as the representation of K listed in Table I 12 • The action on index space is an induced field-dependent gauge transformation in the maximal compact subgroup K, whose only free parameters are the global parameters in U. This (U, K) structure extends the situation with the (T, k) structure of the T-duality transformations described in the previous paragraph. The logical/mathematical basis for this structure is induced representation theory. The bottom line is that in order to have U-duality multiplets, in addition to the non-perturbative base, the "indices" on the fields in {13} must form complete K -multiplets. By knowing the structure of a U-multiplet we can therefore predict algebraically the quantum numbers of the non-perturbative states by extending the quantum numbers of the known perturbative states given in (12). The prediction of these non-perturbative quantum numbers is one of the immediate outcomes of our approach. 5.9

An example

It is very easy to analyze the case (d, c) = (6, 4) so we present it here as an illustration. In this case the spin group is S0(5) and there are 4 internal dimensions. The duality groups and index spaces follow from Tables 1,11 and {12). The relevant information is summarized by

U = S0(5, 5), T = S0(4, 4),

K =SO (5) ®SO (5) k =SO (4h ®SO {4)R

= 1 : ("'· r~IL,R)) L,R = 1L•R lL,R = 2 : (L:· r~IL,R)) L R = 9L 'R IL,R

i..JI

I

I

I

(18)

- 5'paee $ 4internal -

h,R = 3:

L,R

L,R

etc.

where the 9L,R have been reclassified according to their space and internal components. The reclassification is done also for the short (21, + 2} ), intermediate (2}l + 2}}) and long (2}f + 2}5 ) supermultiplet factors. It is clear from this form that the k =SO (4)L ®SO (4)R structure follows directly from the separate left/right internal components, while the spin of the state is to be obtained by combining left and right content of the space part. Here I will discuss an example involving BPS states which is very similar to another discussion on non-BPS states given in 12 . Let us consider the BPS saturated states (h :j:. 0, lR = 0). The base quantum numbers in q,~~L.iiO:u (base) form the 16 dimensional spinor representation of U = S0(5, 5) base=

(m,n,z 1 ) = 16 of S0(5,5)

(19)

65

Among these the eight quantum numbers ( m, n) are perturbative, while the remaining eight z 1 are non-perturbative. 0-branes that carry these quantum numbers provide the sources for the field equations of the 8 massless NS-NS vectors and the 8 R-R vectors respectively. The representation content of the . d"ICes .m Y'indicu .J.(I£ ,0) (b . 1n ase ) IS

+ 2}}) x ( "· LJs r(l~.)) L

indices = (2}1 X [

I

+non - perturbative

l

(20)

where (211 + 2}1 ) is interpreted as the SUSY factor. The full set of indices must form complete K = SO (5)L ®SO (5)R multiplets for consistency with the general U-duality transformation. It can be shown generally that the SUSY factor does satisfy this requirement because the supercharges themselves are representations of S0(5 ),pin x K 12 . Therefore, the remaining factor in brackets must be required to be complete S0(5),pin x K multiplets. At level IL = 1 the piece L; r~ 1 ) = 1 is a singlet, as seen in Table II. Hence no additional non-perturbative indices are needed at this level. At level IL = 2 the piece Li r~ 2 ) = 9L = 5i'ace ffi 4~nternal is classified under S0(5),pin X S0(4)L ®S0(4)R as (5, (0, 0)) + (0, (4, 0)).

(21)

Obviously, this is not a complete S0(5),pin x SO (5)L ®SO (5)R multiplet. Therefore, non-perturbative indices must be added just in such a way as to extend the (4,0) of k S0(4)L ® S0(4)R into the (5,0) of K S0(5h ® SO (5)R. That is (22)

=

=

This extension determines the required non-perturbative indices for this case. Note that this amounts to extending the 9L into a 10£, and similarly for rightmovers (23) This is precisely what was needed in section-1 in order to obtain consistency with an underlying liD theory 11 . At all higher levels IL,R the requirement for complete K -multiplets coincides precisely with the requirement of an underlying liD theory. Therefore the full set of indices are the same as those given in eq.( 17). The story is the same with the non-BPS-saturated states at arbitrary h,R· This result was found in 11 by assuming the presence of hidden 11-dimensional structure in the

66

non-perturbative type-IIA superstring theory in 10D. In ref.11 a justification for (23) could not be given. However, in 12 and in the present analysis U-duality demands (22) and therefore justifies (23), and similarly for all higher levels. Therefore for this particular compactification on R 6 ® I U-duality and UD Lorentz representations imply each other. A consistency check between U-duality and D-branes was reported in 14 and in this conference. It is of interest to compare that analysis to ours. We find complete agreement at levellL = 1. But at higher levels l ~ 2 our scheme requires more states than the D-brane degeneracy computed in 14• In his case the states corresponding to the non-perturbative indices were not considered, seemingly because the special U-duality transformation he considered (interchanging the two 8's in the 16 of (19)) has a trivial transformation on our index space (does not go outside ofthe 4f,;~). We have seen that under more general U-transformations the extra indices are needed both for U-duality multiplet& as well as for the UD interpretation. Thus, the D-brane or other interpretation of these extra states is currently unknown. For (d,c) = (10,0),(9,1),(8,2),(6,4) the analysis for h,R = 2,3,4,5 produces exactly the same conclusion as the llD analysis. That is, U-duality demands that the S0(9)L ®S0(9)R multiplet& Ei r~ 1 £,R) should be completed to S0(10)L ® S0(10)R multiplets. The minimal completion (17) is sufficient in this case. Hence, in these compactifications U-duality is consistent with a hidden llD structure, and in fact they imply each other. On the other hand for the other values (d,c) = (7,3),(5,5),(4,6),(3,7) the story is more complicated. At various low levels we found that the minimal index structure required to satisfy U-duality is different than the minimal structure of 11-dimensional supersymmetry multiplet& (17). If both U-duality and llD are true then there must exist an even larger set of states such that they can be regrouped either as llD multiplet& or as U-duality multiplet&. Exposing one structure may hide the other one. In fact we have shown how this works explicitly in an example in the case (7, 3) at low levels h,R 12 • However, it is quite difficult to see if the required set of states can be found at all levels.

r

6

Final remarks

The basic assumption that we made is that the superalgebra is valid in the sense of a (broken) dynamical symmetry for the full theory. By studying the isometries of the superalgebra, including the central extensions, many of the features of duality could be displayed while some new features became apparent, including the following:

1. The central extensions (and the supercharges) have a structure consistent with two hidden spacetime dimensions, with an overall signature (10,2). 2. As a consequence of central extensions of the superalgebra, p-branes naturally become part of the fundamental theory, and their interaction with p + 1 forms in supergravity are deduced. These p-branes contribute to the non-perturbative states demanded by U-duality and hidden higher dimensions on an equal footing. 3. The structure of U-duality in type II superstrings, the groups, the nonperturbative states and their classifications emerge naturally from the structure of the superalgebra. This is summarized by Table I. Furthermore, one may start with perturbative string states, but then add nonperturbative states that are needed in order to provide a basis for the underlying superalgebra and its isometries. This is a method of finding at least some of the a-priori unknown non-perturbative states. It would be interesting to find out if the complete theory has a 12D structure. The methods used to discuss the hidden 11th dimension may perhaps be extended to the 12th dimension. 1

References 1. 2. 3. 4.

5. 6. 7. 8. 9.

10. 11.

K. Kikkawa-and M. Yamasaki, Phys.Lett. 149B (1984) 357. K. Kikkawa-and M. Yamasaki,, Prog.Theor.Phys.76 (1986) 1379. C. Hull and P. Townsend, Nucl. Phys. B438 (1995) 109. E. Witten, Nucl. Phys. B443 (1995) 85, and "Some comments on String Dynamics", hep-th/9507121, to appear in the proceedings of Strings '95. P.K. Townsend, Phys.Lett.B350 (1995) 184. ( = hep-th/9501068). J. Schwarz, Phys.Lett.B367 (1996) 97 (=hep-th/9510086); hepth/9509148; "M-theory extensions ofT duality", hep-th/9601077. P. Horava and E. Witten, Nucl. Phys. B460 (1996) 506 (= hepth/9510209). P. Townsend, "p-brane democracy", hep-th/9507048. J. Polchinski, Phys.Rev.Lett. 75 (1995) 4724 (= hep-th/9510017). See also, J. Polchinski, S. Chaudhuri, C.V. Johnson "NOtes on D-branes", hep-th/9602052. I. Bars, "Supersymmetry, p-brane duality and hidden spacetime dimensions", USC-96/HEP-B3 or hepth/9604139. I. Bars, Phys. Rev. D52 (1995) 3567 ( = hep-th/9503228 ).

68

12. I. Bars and S. Yankielowicz, Phys. Rev. D53 {1996) 4489 ( =hepth/9511098). I. Bars, "Consistency between 11D and U-duality", hepth/9601164 . 13. M. Blencowe and M. Duff, Nucl. Phys. 8310 {1988) 387. 14. A. Sen, Nucl. Phys. B450 (1995) 103. 15. R. Haag, J. Lopuszanski and M. Sohnius, Nucl.Phys.B88 (1975) 257. 16. See review, A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 {1994) 77.

69

String Solitons and Singularities of Ka Hirosi Ooguri

366 LeConte Hall, Department of Phy1ic1, Uniuer1ity of California at Berkeley Berkeley, CA 947!!0, USA. and Theory Group, Mail Stop SOA-5101, Lawrence Berkeley National Laboratory Berkeley, CA 941!0, USA.

It is my pleasure and honor to give a talk on occasion of Professor Kikkawa's 60th birthday. His various scientific achievements have been mentioned throughout the meeting, but I also want to add that he has been a great role model for generations of aspiring theoretical physicists. I myself frequented his group at Osaka during my graduate school years at Kyoto in order to discuss with him, and I am very grateful for his advice on scientific and other matters. I have to come back to one of his scientific contributions, the discovery of the T-duality in string theory 1 . Let me start with the T-duality in the context of the heterotic string. The heterotic string on R 9 x S 1 is invariant under transformation r - 2/r where r is the radius of the circle S 1 in the target space. (Once compactified on S 1 , the heterotic strings with Es x E 8 and S0(32) gauge groups are continuously connected, and we do not have to distinguish the two.) One can interpret 2 this transformation, called the Tduality, as a remnant of non-abelian gauge symmetry which manifests itself at the self-dual radius r = V2 and which is spontaneously broken for r i= V'i. To understand this, one only have to remember that the string on S 1 carries separate momenta for the left and the right movers parameterized by a pair of integers (n, m) as (PL, PR) = (nfr + rm/2, nfr- rm/2). In particular at r = 2, the integers (n, m) = (1, -1) and ( -1, 1) give (PL,PR) = (0, ±V'i). Vertex operators corresponding to them are e±iv'2XR, which together with iBXR generate the affine SU(2) symmetry on the worldsheet (the Frenkel-Kac construction). In the string theory, the worldsheet affine symmetry implies the target space gauge symmetry. Therefore the heterotic string acquires an extra SU(2) gauge symmetry at the self-dual radius r = V?.. Away from this radius, the SU(2) is spontaneously broken to U(1), which is the standard

Kaluza-Klein gauge symmetry generated by i8XR. Therefore the SU(2)/U(l) is non-linearly realized away from the self-dual point, and in fact one can show that it transforms r to 2/r.

70

I should mention that this argument does not apply to the type II superstring theory since the would-be gauge bosons generated by e±iv'2XR are removed by the GSO projection. In fact the T-duality transforms the type IIA string into the type liB string and vice versa 2 •3 • The story becomes more interesting if we combine this observation with the recent conjecture on dualities among string theories. It was suggested by Hull and Townsend 4 that the the heterotic string on R 6 x T 4 is equivalent to the type IIA string on R 6 x K 3 • Under this duality transformation, the value of the string coupling constant A is inverted to 1/ A. This conjecture implies the following interesting prediction on non-perturba.tive effects in the type IIA string theory 5 . As we saw in the above, the heterotic string gets an enhanced takes a special shape. This is non-abelian gauge symmetry when the torus the Frenkel-Kac mechanism on the worldsheet, and therefore is perturbative in A. One can show that the type IIA string dual to this heterotic string lives on K3 which has an orbifold-type singularity with vanishing 2-cycles. Since the string coupling constant is inverted, this enhancement of symmetry must take place non-perturba.tively in the type IIA side. Strominger noted that the type IIA string has a BPS-saturated solitonic state and that such a. state, if it wraps around the 2-cycle, may become massless as the cycle collapses to a point. He thus proposed that they could be candidates for the extra gauge bosons 6 • (Actually he proposed this scenario in a. slightly different situation, but it is equally applicable to this case.) In my talk, I discussed two issues: ( 1) How the type IIA string on R 6 x K3 behaves near the orbifold singularity where the 2-cycle degenerates. (2) How the enhancement of gauge symmetry occurs non-perturbatively along the line of Storminger's proposal. Since the details have already appeared elsewhere 7 •8 , here I only describe main points of these discussions. In [7), we have shown that the type IIA string in the neighborhood of the singularity on K3 is equivalent to the type liB string in a background of a soliton which is extended over 5 spacial dimensions. The equivalent can be established as follows. The geometry of K 3 in the neighborhood of the (resolved) orbifold singularity is modeled by one of the asymptotically locally Euclidean gravitational instantons. The asymptotic region of such a gravitational insta.nton has a. geometry of 8 3 jr where r is a. discrete subgroup of SU(2) and it fits nicely in the neighborhood of the orbifold singularity. Let us consider the case of r = Zn. The instanton can be constructed by considering 2-dimensional torus fibered over R 2 such that the torus degenerates at n points, x1. ... , Xn E R 2 . The total space is 4-dimensional and allows a. self-dual

r

71

metric. The metric is regular provided Zi ::j:. z; for i ::j:. j, and it becomes singular as these points approach to each other. If all the n points concide, there will be (n - 1) degenerating 2-cycles. This gives a model for the orbifold singularity on Ka. Now let us perform the T-duality on the torus. Before the T-duality, the special points Zi on R 2 have codimension 2 on the 4-dimensional space because the torus over z; is still of finite volume albeit degenerate. Since the T-duality on the torus interchanges its complex moduli and the Kahler moduli, the volume of the torus becomes zero at z; after the T-duality. This means that the codimension of the singularity is now 4. If one looks more closely, one finds that this configuration is exactly that of the solitonic 5-brane solution by Horowitz and Strominger 9 • It is called 5-brane since the string actually lives in (9 + 1) dimensions, so the codimension 4 means that the singularity is spread over (5 + 1) dimensions. One may regard this as a trajectory of a 5-dimensional extended object, i.e. 5-brane. One of the important properties of this solitonic 5-brane state is that each of them carries a unit charge with respect to the rank-2 anti-symmetric tensor field B,.., in the NS-NS sector of the string, which couples to the vertex operator 8X~-'aX". Thus we find that the type IIA string on in the neighborhood of the orbifold singularity of Zn type is equivalent to the type liB string with n solitonic 5branes each of which carries a unit charge with respect to the NS-NS B-field. Here we have moved from type IIA to type liB since the T-duality exchanges the two. As mentioned in the above, when all the points Zi coincide, Ka generates (n- 1) collapsing 2-cycles. Under the T-duality, this is mapped to the n solitonic 5-branes approaching to each other. This observation can be used to explain the non-perturbative enhancement of the gauge symmetry in the following way 8 . The type liB string bas two rank-2 anti-symmetric tensor fields, one comes from the NS-NS sector and the other from the R-R sector. The solitonic 5-brane state that appeared in the above carries the unite charge with respect to the NS-NS field, but no charge with respect to the R-R field. The type liB string is supposed to have the SL(2, Z) S-duality in 10 dimensions, as explained in the talk by J. Schwarz at this meeting. Under the S-duality, the charges with respect to the NS-NS and R-R fields transform as a doublet of SL(2, Z). Thus the S-duality in 10 dimensions predicts that there is a solitonic 5-brane which carries a unit R-R charge and zero NS-NS charge. Polchinski has shown that the Dirichlet brane carries the R-R charge and thus is a candidate for such a soli tonic state 10 • The Dirichlet brane is characterized by the fact that an elementary closed string may disappear on the trajectory of the brane. If we exchange the space and the time coordinates

72 on the worldsheet of the closed string disappearing on the brane, one finds that the Dirichlet brane introduces an open string whose end-point lies on the trajectory of the brane. If there is only one such brane, both ends of the open string should be on the same brane. The open string spectrum then contains the U(l) gauge boson on the (5 +I)-dimensional trajectory of the 5-brane. If there are more than one branes, an open string may stretch between two branes. If the branes are apart, the streched string is massive. However as the branes approach each other, the lowest energy states of the open string become massless. In the limit when all the n branes coalesce, the open string spectrum acquires the U(n) gauge boson, with each brane carrying the Chan-Paton factor that can be attached to the end-points of the open string. Since we have performed the T-duality and the S-duality to reach at this picture, it is useful to go back the process and unwind these duality transformations. We start with n coalescing Dirichlet 5-branes in the type liB theory, with elementary open strings stretching between them to generate the U ( n) gauge symmetry. If we do the S-duality transformation backward, one finds Dirichlet strings stretching between the solitonic 5-branes of Horowitz and Strominger. We then perform the T-duality back to go from type liB to type IIA. Under the T-duality, the Dirichlet strings stretching between the 5-branes is mapped to the Dirichlet 2-branes wrapping around the collapsing 2-cycles. Thus we have made a full circle and come back to the original proposal of Strominger. It is the Dirichlet 2-brane which generate the non-perturbative enhancement of gauge symmetry in the type IIA string on R 6 x Ka. Acknowledgements This work was supported in part by the National Science Foundation under grants PHY-9501018 and PHY-9514797 and in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098. References 1. 2. 3. 4. 5. 6. 7.

K. Kikkawa and M. Yamasai, Phys. Lett. 149B (1984) 357. M. Dine, P. Huet and N. Seiberg, Nucl. Phys. B322 (1989) 301. J. Dai, R. G. Leigh and J. Polchinski, Mod. Phys. Lett. A4 (1989) 2073. C. M. Hull and P. K. Townsend, Nucl. Phys. B438 (1995) 109. E. Witten, Nucl. Phys. B443 (1995) 85. A. Strominger, Nucl. Phys. B451 (1995) 96. H. Ooguri and C. Vafa, e-Print Archive: hep-th/9511164.

73

8. M. Bershadsky, C. Vafa and V. Sadov, e-Print Archive: hep-th/9510225. 9. G. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 197. 10. J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724.

74 COLLECTIVE COORDINATE QUANTIZATION OF DIRICHLET BRANES SOO-JONG REY Physics Department, Seoul National University, Seoul151-74! KOREA Collective coordinate quantization of Dirichlet branes is discussed. Utilizing Polchinski's combinatoric rule, semiclassical D-brane wave functional is given in proper-time formalism. D-brane equation of motion is then identified with renormalization group equation of defining Dirichlet open string theory. Quantum mechanical size of macroscopically charged D-brane is illustrated and striking difference from ordinary field theory BPS particle is emphasized.

1

Introduction

Recently nonperturbative string theory has given us many surprising results. There are now compiling evidences that all known perturbatively defined string theories are related each other by duality at nonperturbative level 1 • Central to this advance was progress to semiclassical string theory, in particular, deeper understanding of stringy topological solitons over the last few years 2 •3 •4 •5 •6 •7 • Previous study of string solitons, however, has been restricted mainly to lowenergy effective field theory approximation. While exact conformal field theories in a few cases has been known from the earliest days 3 •8 , further progress was hampered because of technical difficulties in dealing with geodesic motion interpolating between different conformal field theories, viz. space of nontrivial vacua of string field theory. In a recent remarkable work 9 , Polchinski has obtained an exact conformal field theory describing Ramond-Ramond charged solitons. These so-called D(irichlet)-branes are described in terms of Dirichlet open strings that are coupled to the underlying type II closed strings. Polchinski's work has cleared up many puzzling aspects that arose previously when string solitons were studied within low-energy effective field theory truncation. The 0-brane proposal has already passed many nontrivial consistency tests but all of them so far were mainly on static properties. With its simplicity and exactness it should now be possible to study quantum dynamics of string solitons in detail. In this talk I report my recent work 10 on several aspects ofD-brane dynamics: collective coordinates, semi-classical quantization, renormalization group interpretation of equation of motion and quantum mechanical size of macroscopically charged 0-brane.

75 2

D-Brane and Collective Coordinates

Consider a conformally invariant two-dimensional system. If a boundary is introduced at which the bulk system ends, then it is well-known that microscopic detail is renormalized into a set of conformally invariant boundary conditions. For a Gaussian model such boundary conditions are either Neumann or Dirichlet boundary conditions but not a combination of the two. Similarly, in closed string theory, it is possible to introduce worldsheet boundaries. At each boundaries, string coordinates X~-'(z, z) may be assigned to either Neumann (N) or Dirichlet (D) boundary conditions. Mixed boundary condition may seem break 10- or 26-dimensional Lorentz invariance. However, on toroidally compactified spacetime, target space duality R f+ o:' I R interchanges N and D boundary conditions. Hence Lorentz invariance is maintained up to target space T-duality. Denote N-coordinates as X;, i = 0, 1, · · ·, p and D-coordinates as ya, a = p + 1, · · ·, 9(25). Each worldsheet boundaries are mapped into spacetime extrinsic hypersurfaces of dimension (p + 1) spanned by X', viz. Dirichlet p-brane world-volume. Polchinski 9 has shown these D-branes are nonperturbative states of type II strings that carry RR-charges obeying Dirac quantization condition and that saturate BPS bounds. Worldsheet chiral symmetries restrict possible p-branes further. Type liB strings are worldsheet symmetric that even numbers of D-coordinates are possible, hence, contains p = -1, 1, 3 bra'nes and their magnetic duals. Similarly, for type IIA odd numbers of D-coordinates are allowed, viz. p = 0, 2, 4 branes and their magnetic duals. Since IIA and liB string theories are mapped into each other under target space duality R --+ o:' I R, one can build up all Dbranes from the oriented open string sector (p 9) in liB theory and cascade T-duality transformations. In type I string theory, because of worldsheet orbifold projection, only p = 1, 5, 9 branes are allowed. Worldsheet interaction of type II strings with D-branes are described by Dirichlet open string theory. Worldsheet interaction at each boundaries is deduced by cascade T-duality transformation of the known oriented 9-brane (open string) theory. For massless excitations, the worldsheet interactions at each boundaries are described by

=

9{25)

p

SB =

fdr["£A;(X 0 ,···,XP)8tX' i=O

+

"£

a(Xi)8n ya describes transverse translation of local D-brane world-volume Xi, hence, collective coordinates . Normally spacetime translations are redundant and correspond to null states. This is clear from rewriting v., = I d 2 z8z (tl>aO.zYa) + (c.c.), which decouples on a compact worldsheet. The decoupling fails precisely when D-boundaries are present and Va 's turn into genuine physical modes. This is consistent with spacetime point of view since, in the presence of a D-brane, translational symmetry is spontaneously broken and new Goldstone mode states should appear. The Va 's that fail to decouple and fail to be null are precisely those Goldstone states. Low-energy spacetime interactions type II strings with N independent D-branes are then described by massless modes of Dirichlet string theory: D = 10, N = 2B supergravity coupled to D = 10, N = 1 Dirac-Born-Infeld U(N) gauge theory on Ep+l dimensionally reduced and T-dualized onto Ep+l· Mismatch of spacetime supersymmetry and U(N) Chan-Paton gauge group does not cause problems since the D-brane excitations are confined only on Ep+l· Thus, dimensionally reducing and T-dualizing first and then making a 'nonrelativistic expansion' for small gauge and Goldstone field excitations 12 ,

Swv

-'l'rTp {

e-Jdet(GMN + BMN

}r;p+l -t

'l'rTp

+ FMN)

f e-va(1+~(F+B)~i+[D;,t/> 0 ] 2 +[t/> 0 ,t/>b] 2 ).

(2)

}r;p+l

Here Tpe- denotes p-brane static mass density obtained from T-dual transform of 9-brane dilaton tadpole amplitude on a disk 13 . The first and second terms give bare energy and Casimir energy of static D-branes. The second term also contains aforementioned Cremmer-Scherk coupling, hence, manifest gauge invariance of Kalb-Ramond BJ.Iv field is maintained. The third and fourth terms are kinetic and potential energy for moving D-branes.

n 3

Combinatorics of Perturbative Dirichlet String Theory

Consider stringS-matrix elements involving D-branes. Dirichlet string theory associates each D-brane to a Dirichlet boundary on which lives an independent 'Chan-Paton quark'. Interaction between D-branes and string states are then described by Riemann surfaces with handles and holes. This also implies that apparently disconnected worldsheet diagrams are in fact connected in spacetime as long as boundaries from disconnected worldsheet are mapped into common D-brane(s). This entails a new rule of string perturbation expansions for the S-matrix generating functional Z 14 • Let n = 1, · · · , N label N independent D-branes into which different species of 'Chan-Paton quarks' are mapped. Worldsheet perturbation theory for Z is then organized as oo

Znew

=

l

L

N!

N=O

J

N

oo l

4

n=1

h=O

h! mdm2!·· ·mN!

N

®(II (dYnJ) ®L h! ® L

(3)

a 1 ,.··,a,.=1

where m; ~ 0, Ea m, = h. For fixed N, we sum over the number h of worldsheet boundaries and sum each of the h 'Chan-Paton quarks' (independent D-branes) from 1 to N. Then we integrate over the transverse positions of each D-branes and finally sum over the total number N of D-branes. Since summing over the number of holes and the 'Chan-Paton quarks' amounts to summing over the number of holes mapped into a given D-brane, the above combinatorics can be organized as Znew

=

oo

L

oo

N:O

=

n:1

oo

L

liNII[dY:J® L J L II (

® N!

l

oo

[dY:J

N=O

n:1

L

1

h2! ...

h~=O

h1=0

N

® N!

oo

1

h1! 1 hn!)

oo

L

1 hN!

hN:O

(4)

h,.:O

Exponentiation of disconnected worldsheets then generate a completeS-matrix generating functional Z. In old perturbation theory 15 each Dirichlet boundaries are mapped into independent spacetime points, subsequently integrated over Zold

=

Loo

N=O

1 N!

N

II

n:1

J

[dY:J

Loo

l

n/5n.,1

(5)

h,.:O

The difference arises because the D-branes are extrinsic structure to spacetime. Combinatorics for the S-matrix generating functional in Dirichlet string perturbation theory may be rephrased as follows. Prepare for m disconnected,

78

compact Riemann surfaces, create n~a holes arbitrarily distributed among the m Riemann surfaces. Map each holes to the world-volume of N independent D-branes allowing duplications. Finally sum over m, n~a, N independently with appropriate combinatoric factors Sms of n~a boundaries into m Riemann surfaces and Sst of n~a boundaries into N D-branes

We note that a single exponentiation maps each Dirichlet boundaries doubly into disconnected Riemann surfaces and into independent D-brane worldvolumes in a symmetric manner. 4

Semi-Classical Wave Function of D-Branes

Consider Dirichlet string partition function in the background of the type II string fields in which all worldsheet boundaries are mapped into a single Dbrane world-volume Ep+l· The partition function serves as a generating functional, hence, S-matrix elements between D-brane and string states are derived from local variation of background string fields. The partition function is also related to the (Euclidean) wave functional 'lilt of the D-brane. The new combinatoric rule relates the wave functional to the partition function (7) Here S 1 sums up all one-particle irreducible connected worldsheet diagrams, whose boundaries are mapped to the D-brane world-volume. Integration over world-volume gauge field is already made for 'lilt to ensure type II winding quantum number conservation. Dirichlet string perturbation theory yields 00

sl

[Y(J] =

L e~(h- )

2 s(h)

(8)

h=l

in which S(ta) denotes amplitude with h-holes. Sum over handles is implicitly assumed in the definition of S(ta). Higher order contributions S(~a;::: 2 ) come from annulus, torus with a hole etc or sphere with three holes etc. They amount to D-brane mass renormalization. For type II string all except the disk diagram (h = 1) vanishes identically because of spacetime supersymmetry nonrenormalization theorem. For D-instanton, this is consistent with known results that the RR instantons

79

are exact to all orders in string perturbation theory. The disk amplitude S(l) for type II superstring is easily obtained from 9-brane boundary state 16 after appropriate T-duality transformations. For simplicity, keeping only the transverse fluctuation of the D-brane world-volume S1

J J

= Tp

dEp+le-3

i:O

.p, C) '

(2)

H3-i,i(W.

where t/J = (1/Jl, · · ·, tPh2,l(W)) are parameters in the polynomial deformation of the mirror W, and the right hand side is given the structure of the Jacobian ring J .p above. In their original work, Candelas et J determined the quantum cohomology ring starting from the Jacobian ring J.p for the mirror of a quintic hypersurface in P 4 • In general we may define the quantum cohomology ring through a specific basis (flat coordinate) of the Jacobian ring, {1,0,u0 11 ,0< 3>} (a,b = 1, · · ·, h2 •1 (W)), where 0 0 , O" and 0< 3 > represent the elements with charge one, two and three, respectively, in the Jacobian ring. The flat coordinate is characterized by the properties that OaO" 6!0< 3 > , 0 0 0( 3 ) 0 in the Jacobian ring. Then the relations OaOb Ec K 1• 1.tc(t(t/J))Oc determines the coupling as a function in flat coordinates. We may compute the quantum cohomology ring at the so-called large radius limit where we have non-trivial q-expansion for the coupling:

=

Kt.t,,.(t)"'

=

=

JM ha A h, A he+ L Ni ,,q; q; 1 ; 2 •••

1

2 •• •

q~" .

(3)

The first term in the expansion are the classical intersection numbers for the elements ha E H 1•1(M, Z) (a= 1, · · ·, h 1•1(M)) and qi := e2 "'' 1' (i = 1, · · ·, n = h2 •1 (W)). It has been verified in numerous examples that the numbers Nith···i, are integers, possibly negative, which "count" the instantons appearing in the quantum correctionl -lo.

3

Gauss-Manin system and flat coordinates

In this section we will characterize the flat coordinates through the analysis of the Gauss-Manin system. Let us consider the quintic hypersurface M in P 4 and its mirror W obtained by orbifoldizing M by G = (Zs )3 • We take the defining equation for W .p as P.p = ~zf + · · · + ~zg- t/Jt/1 with ¢1 = z1z2Z3Z4Zs. The deformed Jacobian ring J .p is given by (4)

89

We fix a basis of J.p as {tp< 0 >,tp(l>,tp< 2 >,tp< 3>} := {1,cJ>,c/>2 ,cJ>3 } indicating the charges by the superscripts(, tp(i) refers to the element with charge i or homogeneous degree 5i). Then the Gauss-Manin system is a set of first order differential equations satisfied by the period integrals. They are given by w := (w4 L ~ 1-t/J 511>4

r' - ~r = 0

+ rr 2r'2 -

~

3r") r

(9)

s=0 .

The differential equation for r('¢) coincides with the Picard-Fuchs equation. The relation (7) determines the coupling and the flat coordinate t t( '¢) as follows; 1 C ({}'1/J) 3 I!Jt s( '¢) (10) Kttt(t) = r21- tjJ5 I!Jt ' 8'¢ = r('¢) .

=

The Picard-Fuchs equation satisfied by r(t/J) in (9) can be arranged to {9!- 5z(59z + 4)(59z + 3)(59: + 2)(59z + 1)}w(z) = 0,

(11)

!)s

where z = (5 and w(z) = -5'1/Jr('¢). It is evident that the indices of the Picard-Fuchs equation are zero at the large radius limit z = 0, and the monodromy becomes maximally unipotent. All the solutions can be determined by the standard Frobenius method starting from the series w0 (z,p) := 5 n+p + 1 n+p 1 wt~ ~ LJ rr n+p+ z . 0 ne can ven·fy th a t the rat•10 t( z) = 21ri wo z , wh ere 1 wo(z) := wo(z,p)lp=O and w1(z) := gPwo(z,p)lp=O• coincides with the flat coordinate. - 5~wo(z) and s('¢) = - 5~wo~ solve the In fact the functions r('¢) equations (9). Because of the behavior t "" 2 ~;log(z) near the large radius limit, we obtain the desired q-expansion (3) with q = e2 "'it and C = (2!:)3 • We can read off the prepotential F(t) for the quantum coupling Kut from the form of the Gauss-Manin system in the flat coordinate (7). To see this note that the first order system (7) is equivalent to I!Jl Ke:e(t) 8lv< 0> = 0, where

=

v< 0 > is the first row of the periods v = (v
.t,

t.=~. t6=~~ ~2 ,

...

t2,=~~ (~2 )(9- 2 ).

{8)

With these weights, it is easy to prove, using Euler's theorem, that the partition sum 1 becomes Z(t,~,Jj)

L

= t•

~A p2(#v 2 -4) 1

{9)

G

where A is the number of plaquets of the graph G and #v 2 the number of positive curvature defects. Note that the latter are balanced by a gas of negative curvature defects, whose individual probabilities are given in ( 8). After tuning the bare cosmological constant ~ (controlling the number of plaquets) to some critical value ~c(P), we expect this model to describe pure gravity in a large interval of Jj. On the other hand, for ~ fixed and Jj = 0 we entirely suppress curvature defects except for the four positive defects needed to close the regular lattice into a sphere. It is thus clear that Jj is the precise lattice analog of the bare curvature coupling Po in ( 3). The phase Jj = 0 of "almost flat" lattices - very different from pure gravity- was discussed in detail in

3 3.1

Sketch of Solution

The Itzyluon-DiFrance•co formula for the DWG-motlel

The basic fact which allows to solve the DWG-model is the representation of the matrix integral ( 5) in terms of the character expansion with respect to the irreducible representations R of the group GL(N), given by Itzykson and DiFrancesco 12 :

•

"' TI,(hi-

Z(t, t ) = c LJ

{h",h"}

TI·

1)!!ht!!

.(h~- hi!) X{h}(A) X{h}(B).

'•'

'

(10)

1

The sum runs here over all representations R{ h} characterized by the Young tableaux with h,; -i+ 1 boxes in the i-th row. The non-negative integers h,; obey the inequalities: (11) ht:+t > h,;. Only the representations with equal number ofeven(odd) weights h11 (h 0 ) enter the sum in ( 10). The Weyl-Schur characters are defined in the standard way:

117

(12) as a determinant of Schur polynomials Pn(9) defined through

eE~~h, =

Ez"

Pn(9) with 9,

= ~ tr[At

(13)

n=O 3.~

The S4tltlle Point Equ4tion for tke Mo•t Prob4ble Repre•ent4tion in the Pl4n4r Limit

With the Itzykson-DiFrancesco formula we achieve what is necessary to make the large N limit analytically treatable: the drastic reduction of the number of degrees of freedom. Instead of N 2 integrations over the hermitean matrix M,; we are left with only N summations over the highest weight components h 1 < h2 < ... < hN. If we assume that the the characteristic h' • are of the order N (which will be confirmed by the solution) we can see that the summation factor in ( 10) is of the order expN, and as usually one can apply in this situation the saddle point approximation. For our particular model ( 8- 9) the saddle point equation (SPE) is obtained by equating to zero the logarithmic derivative of the weight of summation in ( 10) with respect to he or h0 (we assume identical distributions for h4 and h 0 ). The SPE reads:

r dh, h-h' p(h') =-lnh.

2F(h)+P Jo

(14)

where

(15) and p( h) is the density of the highest weight components in the representation R. It is related to the resolvent H(h) of the highest weight components:

H(h)

= ~ _1_ = 1~~ dh' LJ h - h,.

•=1

0

p(h') . h - h'

(16)

Although the details of the derivation of the SPE contain some subtleties, we can roughly say that the term 2F(h) comes from the two characters in ( 10) normalized by the Van-der-Monde determinant Ll{h) = fli>;(ho - h;) (one takes A= B for the model dually equivalent to our model), the second

118

n.,; (

term COme& from log deriVatiVe Of .6_ (h) and Of the prodUct h: - hj) 1 and 0 logh comes from the Sterling asymptotic& of (he- 1)!! (or h !!). As has been discussed in detail in 4 , this equation actually does not hold on the entire interval [0, a], but only on an interval [b, a] with 0 ~ b ~ 1 ~ a: Assuming the equation to bold on [0, a] would violate the implicit constraint p(h) ~ 1 following from the restriction ( 11). The density is in fact exactly saturated at its maximum value p(h) = 1 on the interval [0, b]. A similar saddle point method for the representation expansion was first successfully used in 7 for the calculation of the partition function of the multicolour Young-Mills theory on the 2D sphere (see also 8 , 9 ). 3.3

Ca.lcultdion of Cha.ra.cter11 in the La.rge N Limit

To solve the SPD ( 14) for H(h) we have to find a method to calculate effectively the characters, i.e. the function F(h). In this section we describe a method to do it, which is based on the identities for Schur characters, based on the basic properties of Schur polynomials. Leaving aside the details of the derivation (which can be found in our basic paper 1 ), we summarize these identities in one equation already in the large N limit. Introducing the function: G(h) = eH(h)+F(h) (17) we have:

Q

h-1 =

L

00

t 39

t=l Gt

+L

t=l

2q_!_ln(x{o\!.}(a)) G 9 , N atl, 2

(18)

where the coefficients of the positive powers of G in ( 18) are directly related to the correlators of the matrix model dual to the model defined by the eqs. ( 8- 9), i.e. the model with potential vA.(MA) = (MA) 4 :

t

(19)

We have also assumed for the moment that only a finite number Q of couplings are non-zero (i.e. t 39 = 0 for q > Q). Furthermore, we were able to show in our work 5 that ( 18) implies the functional equation e

H(h)

= ( -1)(Q-l)h IIQ G9 (h), tq

(20)

t=l

where the G 9 (h) are the first Q branches of the multivalued function G(h) defined through ( 18) which map the point h = oo toG= 0. The saddlepoint

119

equation ( 14), together with ( 20), defines a well-posed Riemann-Hilbert problem. It was solved exactly and in explicit detail in 5 for the case Q = 2, where the Riemann-Hilbert problem is succinctly written in the form

2ReF(h) + H(h) 2F(h) + ReH(h)

-In(-~) -lnh,

(21)

where ReH(h) denotes the real part of H(h) on the cut (b, a] and ReF( h) denotes the real part of F(h) on a cut (-oo, c] with c < b. This case corresponds to an ensemble ofsquares being able to meet in groups offour (i.e. flat points) or two (i.e. positive curvature points). We termed the resulting surfaces "almost flat". It turned out that all the introduced functions could be found explicitly in terms of elliptic functions. The method can be easily generalized to any Q. However, the solution cannot be in general expressed in terms of some known special functions. In the next section we will show that the model of our interest can be solved in terms of elliptic parametrization, more complicated than its particular case, Q=2, of almost flat planar graphs. Let us also note that the equation ( 20) can be explicitly solved for F(h) in terms of a contour integral for some wide class of big Young tableaux. Hence we can find the character rather explicitly knowing H(h) and a set of Schur constants (t1, t2, ... , tq) (see

3.4

Solution of the Lattice Model of2D R 2 Gra11ity

Let us notice that for the model defined by ( 8, 9) which we chose as a lattice realization of the 2D R 2 gravity, the equation ( 18) contains an infinite number of both negative and positive powers of the G-expansion. Labeling the first two weights as in 5 : t 2 , t4 1 we now include all the even t 29 with q 2::: 3, assigning them the following weights: t 29 t 4£f- 2. Equation ( 18) can then be written compactly as:

=

t2

h- 1 = G

t4

+ G (G _E)+

positive powers of G.

(22)

Dropping the positive powers of G (for G small enough) and inverting this equation, we see that there are two sheets connected together by a square root cut running between two finite cut points, d and c. On the physical sheet a further cut (running from b to a), corresponding to eH(h), connects to further sheets.

120

The equations ( 21) now read: 2ReF(h) + H(h) 2F(h) + ReH(h)

-=

-In(-"'-) th-t. -lnh,

(23)

the first coming from the large N limit of the character ( 20) and the second, in view of ( 16), being the saddlepoint equation ( 14). These two equations tell us about the behaviour of the function 2F(h) + H(h) on the cuts of F(h) and H(h), respectively. We have introduced the notation ReF(h) to denote the real part on the cut of F(h), and similarly for ReH(h). The principal part integral in ( 14) is thus denoted in ( 23) by ReH(h). Our object is to now reconstruct the analytic function 2F(h) + H(h) = 2ln G( h) - H (h) from its behaviour on the cuts. To do this we first need to understand the complete structure of cuts. We already know the structure of cuts of H(h); it has a logarithmic cut running from h = 0 to h = b, corresponding to the portion of the density which is saturated at its maximal value of one, and a cut from b to a corresponding to the "excited" part of the density, where the density is less than one. It thus remains for us to understand the cut structure of In G(h). The function G( h) has two cuts on the physical sheet. The first cut, running from b to a, corresponds to the cut of eH(h.), the second cut, running from cut point c to cut point d, corresponds to the cut of eF(h.). To see whether In G(h) has any logarithmic cuts, we first notice from ( 22) that G(h) is non zero everywhere in the complex h plane except possibly at infinity. Thus for In G(h) the only finite logarithmic cut points are at h = b, defined to be the end of the fiat part of the density (this corresponds to the end of the cut of eH(h.)), and possibly the cut point c, defined to be at the end of the cut of eF(h.). The only remaining question is whether this logarithmic cut starting at h = b goes off to infinity or terminates at c. For large h we see from ( 22) that In G( h) = In E + 0( i.e. there is no logarithmic cut at infinity. We conclude that the cut structure of the function In G( h) consists of the cuts corresponding to eF(h) and eH(h.) connected together by a logarithmic cut, whose cut points are band c. We thus understand the behaviour of 2F(h) + H(h) on all of its cuts. Standard methods then allow us to generate the full analytic function 2F( h)+ H(h). The four cut points, a and b defining the cut of H(h), and c and d defining the cut of F(h) are fixed later by boundary conditions. Without going into the details of the solution let us present at least one final result: the density of weights p(h) of the most probable representation R{h}. It can be obtained from 2F(h)+H(h) by using the saddle point equation 2F(h) + ReH(h) =-In hand the fact that the resolvent ( 16) for the Young

k),

121

tableau can be written as H(h) = ReH(h) =f i1rp(h). To obtain the result one has to use all the analytical information listed above. After some tedious calculations we obtain: p(h) =

~- i K

In [Bt(aic(u- iv), q)

l

8t( 2~(u+iv),q) '

1r

(24)

where v and u are defined by

~ ') v =6n -1( v~·lc

lc

and

(a- h)(b- d) (a- b)(h- d)' lc),

, (a- b)(c- d) - (a- c)(b- d)"

2 _

(25)

(26)

K and K' are the complete elliptic integrals of the first kind with respective moduli lc and lc' = ~- E is the complete elliptic integral of the second kind with modulus lc, E(v, lc') is the incomplete elliptic integral of the second kind with argument v and moduluslc' and 6n 1 en and dn are the Jacobi Elliptic functions. The nome q is defined by (27)

To fix the constants a, b, c and d, we expand i1rp(h) = 2F(h) + H(h) +In h for large h and compare the resulting power series expansion to that obtained from inverting ( 18): 2tt )1 2F(h)+H(h)=2lnt+ ( ~-1 h. The terms of

o(h12 ).

+

(28)

o( ,:2 ) depend on the as yet unknown positive powers of G in

( 22). Afinal boundary condition is fixed by the normalization of the density. Using this information we can calculate the physical quantities of interest, for example, the derivative of the free energy with respect to the cosmological constant. Denoting the free energy by Z(t, ~. /3), Z = e-N 2 z, we have

a

a~ Z(t, ~./3) =

1

1

2

1

4 ~ ({ NtrM } - 1). = 2 +{h)

(29)

where (h) = J dh p(h) h. The result of the last integration, as well as the conditions for a, b, c, d are too bulky to be presented here (see 1 for the details). In the next section we will present already the physical consequences of it, namely, the universal expression for the free energy near the flattening limit.

122

4

Physical Results and Conclusions

One needs to do quite tedious calculations to extract the physical results from our solution summerized in the equations ( 24- 29). The details can be found in 1 and here we present only the essential results: 1. We found the equation for the critical curve of pure gravity in the parameter space (>.,,B). It did not show any "flattening" phase transition in the whole interval 0 < {3 < oo. Pure gravity appears to be the only possible critical behaviour for the large scale fluctuations of the metric (the only infrared stable critical point) within our model. This should be true for a big class of similar lattice models of pure R 2 gravity in 2 dimensions. For example, if we would allow only the deficits of angle being 0, ±1r in the vertices of our quadrangulation, this physical conclusion should not change. On the other hand, the multicritical behaviours can occur for a more complex parameter space (more t•, t couplings to vary) and we could expect some new phase structures. Sums over graphs with the sign-changing Boltzmann weights might well be necessary for this. 2. Once we realize that nothing interesting happens in our model for finite values of {3 we should concentrate our attention on the double limit .B -+ 0, >. -+ >.c(f3) 1- -Jif3 + 0({32 ) with the double limit parameter :z: 1 + fl(>.c- >.) fixed. Note that this is a very natural, dimensionless scaling parameter; Ac- >. is the continuum cosmological constant with dimension of inverse area and {3 controls the number of curvature defects per unit area (and thus also has dimension of inverse area). We arrive at the free energy in the vicinity of the (locally) flat metric:

=

=

4 [ 6 4t5 4 Z(t >. {3) = :z: - -:z: I I 15[32 2

15 2 + -:z: 8

-5 - :z: ( :z: 2 - 1)li/2] . 16

(30)

This scaling function exhibits two different behaviours in the two opposite limits: 1. :z:c -+ 1 - the pure 2D gravity regime for any finite positive .B and ), -+

Ac(.B): Z(t, >.,{3) "'(>.c- >.) 5 / 2

(31)

This corresponds to the standard value of the string susceptibility exponent = -1/2. This results mean that even for very big but finite R 2-coupling p- 1 we will always find on big enough distances the metric fluctuations obeying the pure gravity scaling. In some sense one can say, that the R 2 2D gravity does not exist as a special phase of the 2D gravity. The R 2 term can be always

'Y.tr

123

dropped from the action in the long wave limit. Our results demonstrate this nonperturbatively, starting from a solvable lattice model. 2. In the opposite limit z - oo, i.e., {3 - 0 with ~ fixed, we obtain: 11"2 p4 Z(t, ~./3) = 48 (1- ~)2

(32)

This limit corresponds to a sum over metrics flat everywhere except of a finite number of curvature defects. One can easily see from the purely combinatorial calculations that ( 32) corresponds to the sum over all flat metrics with only four defects with the deficit of angle 11". Expanding ( 30) as a power series in {3 (the first term of which is given by ( 31) we see that the second term in the series expansion corresponds to the the sum over all flat metrics with five 1r defects plus one -11" defect, the third to six 1r defects plus two -11" defects, etc. This limit does not appear to describe a new phase of smooth metrics. It corresponds instead to the statistics of a dilute gas of curvature defects introduced in an otherwise flat metric. It has nothing to do with the 2D gravity but it is an interesting statistical mechanical model in itself. More than that, one can prove that the dominating graphs are not lattice artifacts: in this limit the manifold ••forgets" that it is built out of squares. It consists of big flat patches with sparsely distributed point-like curvature defects introduced ono it in such a way that they close the surface into a manifold of spherical topology. If one considers this object as a sphere punctured at the points where curvature defects occur, one can show by the direct calculationlf, for the simplest configurations, that the conesponding moduli parameters do not fall on the boundaries of the moduli space but are typically in some general position inside the fundamental domain. It would be nice to solve this model (and may be more general cases including the matter fields) by a continuum approach. In the Liouville theory picture the curvature defects correspond to coulomb charges floating in a twodimensional parametrization space. This could be an alternative approach to 2D quantum gravity in general. Let us conclude by noting that our analytical approach to the models of dually weighted graphs can be generalized to a big class of other matrix models which could be of physical, as well as of the mathematical interest. For example, one can formulate a generalized two-matrix model with partition

function: •we thank Paul Zinn-Juatin for clarifying this question for ua

124

Z

=

J

dN 2 XdN 2 y exp Ntr(F(XY)

+ U(X) + V(Y))

(33)

where X, Y are the NxN hermitean matrices and F, U, V are some arbitrary functions. The character expansion allows us again to reduce the number of integration (or summation) variables to "" N thus making it possible to consider it by means of the saddle point methods. Another, more general solvable k-matrix model of this type is: 1:

Z=

j (II dN j=l

1: 2

Xj) exp Ntr(F(II Xj) j=l

1:

+L

Uj(Xj))

(34)

j=l

The expNtrF(fl;=l Xj) factor can be also expended in characters and one can then integrate over the angular degrees of freedom of the matrices by the use of only the simple SU(N)-orthogonality relations. This approach is at the heart of the Itzykson-DiFrancesko formula which we extensively used for our present model of lattice 2D R 2 gravity. A big question left is how to generalize our approach to include of the matter fields on our lattice manifold. The corresponding matrix models containing the R 2 type couplings can be easily formulated but cannot be solved by the character expansion methods presented here. This is not surprising, since models of interacting spins on the 2D regular lattice, even the integrable ones, are very complicated. Nevertheless, we think that our methods could eventually be powerful enough to advance in this direction and provide a missing link between two branches of mathematical physics: integrable statistical mechanical models on regular 2D lattices, on the one band, and on random dynamical lattices, on the other band. In more physical language, it could provide a link between the integrable models of interacting fields with and without the 2D quantum gravity fluctuations. 1. V.A. Kazakov,

2. 3. 4.

5.

M. Staudacher and T. Wynter, Ecole Normale preprint LPTENS-95/56, hep-th/96011069, accepted for publication in Nucl. Phys. B V.A. Kazakov, Mod. Phy•.Lett. A 4, 1691 (1989). M. Staudacher, Nucl. Phy•. B 336, 349 (1990). V.A. Kazakov, M. Staudacher and T. Wynter, Ecole Normale preprint LPTENS-95/9, CERN-TH/95-352, hep-th/9502132, to be published in Comm. Mtdh. Phy•. , (). V.A. Kazakov, M. Staudacher and T. Wynter, Ecole Normale preprint LPTENS-95/24, hep-th/9506174, to be published in Comm. Mtdh. Ph···

n

125

H. Kawai and R. Nakayama, PAy•. Lett. B 306, 224 (1993). M. Douglas and V. Kuakov, PAy•. Lett. B 319, 219 (1993). B. Rusakov, PAy•. Lett. B 303, 95 (1993). V.A. Kazakov and T. Wynter, Nucl. PAy•. B 440, 407 (1995). E. Brezin, C. ltzykson, G. Parisi and J.B. Zuber, Comm. MtdA. PAy•. 59, 35 11. K V. Kazakov, PAy•. Lett. A 119, 140 (1986). 1978. 12. P. Di Francesco and C. Itzykson, Ann. lnst. Henri. Poincare Vol. 59, no. 2 (1993) 117. 6. 7. 8. 9. 10.

126

New Loop Equations in Ising Model Coupled to 2d Gravity and String Field Theory 11 Ryuichi NAKAYAMA and Toshiya SUZUKI Department of Physics, Faculty of Science, Hokkaido University, 060 Sapporo, JAPAN

=t

New loop equations for all genera in c string theory are constructed. Our loop equations include two types of loops with fixed boundary conditions for Ising spins, loops with all Ising spins up and those with all spins down. The loop equations are shown to be equivalent to the w3 constraints derived before in the matrix-model formulation of 2d gravity. By using this result we propose new Hamiltonian for c = string field theory of Ishibashi-Kawai type.

t

1

Introduction

An idea of string field theory (SFTJ is indispensable for nonperturvative study of string theories. Although they considered SFT for critical strings, the notion of SFT is no less important for non-critical strings. Recently new type of SFT was considered for c = 0 noncritical string theory (2d pure gravity theory) in fl. Much effort has been devoted to extending this formalism to other c $ 1 strings. 3 "' 8 • 18 However satisfactory formalism seems to be still elusive. In this talk we will concentrate on c = 1/2 string. In this theory many types of boundary conditions for Ising spins are conceivable. It seems that by restricting the spin configurations on the loops differently we obtain different versions of SFT. Here we will consider only two types of spin configulations on the loops, all Ising spins on the loop are + (up) or - (down). c = 1/2 SFT for such boundary conditions has already been presented in rJ. In their work, however, the connection ofloop equations with W3 constraints 9 • 10 derived in the matrix model formulation 11 "' 13 of 2d gravity is unclear. The purpose of the present work is to derive loop equations which are directly related to W3 constraints and construct SFT from these loop equations. The details of the argument of this work will be presented elsewhere.14

2

Loop Equations for Loops with

+ Spins

A generating function for Green functions of scaling operators On (n :f= 0 (mod 3)) inC= 1/2 string, T(J.') = T(J.l1,J.l2,J.l4,J.l5,J.l7,J.l8, • • ·)satisfies the W3 "Talk presented by R. N. at the Workshop on "Frontiers in Quantum Field Theory", Osaka, Japan, 14-17 December 1995.

127

constraints 0 0

n=-1,0,1,··· n = -2, -1,0, 1, · · ·,

(1)

(2)

where Ln and Wn are differential operators with respect to JJ's, b which generate the W 3 algebra 15 These results for c = 1/2 string were first conjectured in the study of Schwinger-Dyson equations (SDE) for large-N matrix models in and confirmed in 0 ]. These Ln 's and Wn 's can be succinctly expressed in terms of a complex Z3 -twisted scalar field tf>(z)

r1

r

T(z) W(z)

(3) n

where the fields tf>( z),

q,• (z)

have the following mode expansions c

Here g is a string-coupling constant. The relationship between the SDE for matrix models and the loop equations is well known 2 - 5 . d The functional differential operaters appearing in loop equations generate a 'continuum' Virasoro algebra. From the above results we are naturally led to introduce two independent source functions J~1 )(1), J~2 )(1) to construct loop equations for loops with all spins up. Actually it is not difficult to figure out that loop equations take the following forms

1T+(l)Z+[J~1 >, J~2 )] =

0,

(5)

1 2 W+(l)Z+[J~1 >, J~2 )] = 32{66"(1)- tc5(1)}Z+[J~1 >, J~2 )]

(6)

bSee eqs(3 ),(4) below cNonnal ordering:···: is defined by regarding 1.1. as an annihilation operator and 8/81.1. a creation operator. + g ~)IJ¥>) * D~> * v~> + 3g(IJ~3 -r>) * v~>) r=l

(8) Here D~)(l) stands for 6f6J¥>(1). The symbols* and g)(l)

=1 dlf(l' + l)g(l') 00

(10)

In the above formulae J~1 )(1) + J~2 )(1) is the source function for the loop of length I with fixed boundary condition ( + spins). The disk amplitudes

(11) are given 19 in laplace transformed forms" by

Al)(()

=

1oo dle-, J~2 )] = J~ < Ole-DH exp{1 0

00

dl

L r=1,2

Jf)(l)•~)t {1)}10 >

(36)

133

The parameter D may be interpreted as geodesic distance on the world sheet. Here 'l'~)t (I) is a creation operator for a loop of length I with + spins and satisfies with the corresponding annihilation operators 'I'~) (I) the commutation relations (37) The hamiltonian H is a functional of 'I'~), 'l'~)t and related to the loop equations. To determine its form we note that the existence of the limit (36) implies string field SDE

This can be rewitten as a differential equation for Z+ H[J(r) v]z [J(l) J(2)] _ 0 +•+ + + • + - ·

(39)

i;)

where fi is obtained from H by replacing 'I'~) (I) and 'I'~ )t (I) by and D~), respectively, and interchanging the ordering of J's and D's. A simple generalization of the work for c = 0 string in f) to c = 1/2 string will be to take as ii g h

fi = +g

1oo diJ+(I){l W+(l)- p(l)} 2

1 1 00

dlt

00

dl2ltl2(lt

(40)

+ /2){aJ+(lt)J+(l2) + bJ+(lt)h(l2)}T+(lt + l2),

where a, b are some constants and the tadpole term is given by

p(l) = 32(66"(1)- t6(1)).

(41)

J+(l) and J+(l) are defined by J+ = 1/2(J~) + J~2 >), j+ = 1/2(J~1 )- J~2 >). The 'length' dimensions of '1'~), 'I'~ )t are given by L 413 , L - 713 , respectively and that of g is L- 1413 . Hence the dimension of fi is L- 2 / 3 and we get the result [D) ""' L213 • This result is different from those in rJ, rJ,f 3]. This issue needs further investigation. This construction of hamiltonian can also be extended in an obvious way to the theory where loops with - spins are 9 This is not the only possibility for H. hSimilar hamiltonian is constructed by a different method in ~ 2 ]. In their hamiltonian contribution from singular tenns of loop amplitudes is discarded.

134

included. In (40) W+ and T+ have to be replaced by X+ and U+, respectively and terms with + and - interchanged should be added. By construction our Z+ [J~1 ), J~2 )] satisfies loop equations (5), (6) and we may stop at this point. But if our string field theory is to have geometrical meaning, each interaction in the hamiltonian should give proper decomposition of world sheets into propagators, vertices and tadpoles. In other words, the hamiltonian should provide us with Feynman rules for calculation of loop amplitudes with geodesic distances among loops fixed. From this point of view, an additional consistency condition for string field hamiltonian was proposed in rJ. We performed such a consistency check of our hamiltonian (40) but the consistency condition in this sense does not seem to be satisfied. We are now seeking for another form of hamiltonian. The result will be presented elsewhere 14 . Acknowledgments One ofthe authors (R.N.) thanks J. Ambj~rn, N. Ishibashi, H. Kawai, N. Kawamoto and C. Kristjansen for discussions. 1. M.Kaku and K.Kikkawa, Phys. Rev. D10 (1974) 1110, 1823; W.Siegel, Phys. Lett. 8151 (1985) 391, 396; E.Witten, Nucl. Phys. 8268 (1986) 253; Hata, K.ltoh, T.Kugo, H.Kunitomo and K.Ogawa, Phys. Lett. 8172 (1986) 186, 195; A.Neveu and P.West, Phys. Lett. 8168 (1986) 192; M.Kaku, Int. J. Mod. Phys. A9 (1994) 139. 2. N. Ishibashi and H. Kawai, Phys. Lett. 8314 (1993) 190. 3. N. Ishibashi and H. Kawai, Phys. Lett. 8322 (1994) 67. 4. M. Ikehara, N. Ishibashi, H. Kawai, T. Mogami, R. Nakayama and N. Sasakura, Phys. Rev. D50 (1994) 7467; A Note on String Field Theory in the Temporal Gauge, proceedings of the Workshop on Quantum Field Thoery, Integrable Models and Beyond, Yukawa Institute for Theoretical Physics, Kyoto University, 14-18 Feb 1994, Prog. Theor. Phys. Supp. 118 (1995) 241. 5. R. Nakayama and T. Suzuki, Phys. Lett. 8354 (1995) 69. 6. Y.Watabiki, Nucl. Phys. 8441 (1995) 119; Phys. Lett. 346 (1995)46. 7. M. Ikehara, Phys. Lett 8348 (1995) 365; Prog. Theor. Phys. 93 (1995) 1141. 8. A.Jevicki and J .Rodrigues, Nucl. Phys. 8421 (1994) 278. 9. M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A6 (1991) 1385; R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. 8348 (1991) 435.

135

10. E.Gava and K.S.Narain, Phys. Lett. B263 (1991) 213. 11. V.A. Kazakov, I.K. Kostov and A.A. Migdal, Phys. Lett. 66 (1991) 2051; J. Ambj~rn, B. Durhuus and J. Frohlich, Nucl. Phys. B257 [FS14] (1985) 433; F. David, Nucl. Phys. B257 [FS14] (1985) 543. 12. V.A.Kazakov, Phys. Lett. A119 (1986) 140; D.V.Boulatov and V.A.Kazakov, Phys. Lett. Bl86 (1987) 379. 13. E. Brezin and V. Kazakov, Phys. Lett. B236 (1990) 144; M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 635; D. J. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127. 14. R. Nakayama and T. Suzuki, New Loop Equations inc= 1/2 String and String Field Theory, Hokkaido Univ preprint EPHOU-95-006 (1996). 15. A. B. Zamolodchikov, Theor. Math. Phys. 65 (1985) 1205. 16. V. Kazakov, Mod. Phys. Lett. A4 (1989) 2125. 17. For example see I. Kostov, Nucl. Phys. B326 (1989) 583;ibid. B376 (1992) 539; Phys. Lett. B266 (1991) 42; B. Eynard and C.Kristjansen, Nucl. Phys. B455 (1995) 577. 18. F. Sugino and T. Yoneya, Stochastic Hamiltonian for Noncritical String Field Theories, UT-Komaba-95-8, Oct 1995. 19. I. Kostov, Phys. Lett. B266 (1991) 317. 20. D. J. Gross and A. Migdal, the third reference of 13 ; T. Banks, M. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B238 (1990) 279. 21. G. Moore, N. Seiberg and M. Staudacher, Nucl. Phys. B362 (1991) 665. 22. J. Ambj~rn andY. Watabiki, Non-critical string field theory for 2d quantum gravity coupled to (p,q)-conformal fields, NBI preprint (1996). 23. J. Ambj~rn, J. Jurkiewicz and Y. Watabiki, Nucl. Phys. B454 313; S. Catteral, G. Thorleifsson, M. Bowick and V. John, Phys. Lett. B354 (1995) 58.

136

EXACT SOLUTION OF 1-MATRIX MODEL Hiroahi Shirokura" Department of Physics, Osaka Uni11ersity Toyonaka, Osaka 560, JAPAN I review my new method for solving general !•matrix models by expanding in N- 1 without taking a physical continuum limit. Using my method, each coefficient of the free energy in the genus expansion is exactly computable. One can include physical information in a function which is uniquely specified by the action of the modeL My method gives completely the same result with the usual one if the physical fine tuning is done and the leading singular terms are extracted.

In this review article, I want to explain a new method for solving 1-matrix models that I have found recently 1 • Matrix models have been vigorously studied as an exactly solvable toy model for two dimensional gravity. Even though 1-matrix models are the easiest of all matrix models, only the planar limit 2 have been argued mainly until the appearance of the double scaling limit 3 •4 • This limit enables us to convert a recursive equation derived by the orthogonal polynomial method and Iarge-N expansion ( N is the size of matrix) to a non-linear differential equation for the specific heat. The solution of this differential equation contains non-perturbative effects which are absent in the perturbative expansion by the Feynman diagrams. My method gives another way to solve the recursive equation. Instead of taking the double scaling limit, I expand the recursive equation in N- 1 and solve it order by order. What an intuitive way! However, the· equations for the expansion coefficients of r~ (r" will be explained shortly later) derived in this way is hard to solve even when the expansion order is low. The obstacle to advance was the fact that these equations have summations over paths 5 • To overcome this difficulty, I found a beautiful relation between these summations and a function W(r) which is specified uniquely by the action of the model. Using this relation I find the expansion coefficients of free energy until genus three 1 •6 • In principle one can continue this calculation to higher genus. However, the calculation will become rapidly complicated for large genus. I follow the notations of my previous paper 1 • Since in this short review I cannot explain: all of the notations, I will sometimes omit explanation for some undefined signs. Consult the paper 1 for further information. A 1-matrix "JSPS fellow

137

model is defined by an integral over an N x N hermitian matrix ZN({..\})

~,

= Jv~e-Ns(•..H,

S[+,p)) =

tr

[~+'+ t.~••''*']•

(1)

where {..\} denotes the set of coupling parameters. The main purpose of the game is to find the expansion coefficients of the free energy

In (2) I indicated a 2-dependence of e(h). This parameter a 2 is a real solution of W (r) = 1 b , where W (r) is open the function I mentioned before

( ) W r

~

(2p + 2)!

= r [ 1 + ;;r p!{p + 1)!..\pr

pl

{3)

·

Using the orthogonal polynomial and the Euler-Maclaurin formula the contribution from genus h is described as follows e

=

1 1

dx{1- x)r2h(x) + K2h

B

h

-

~(-1) 8 2;)! (

1

{(1- x)r2(h-a)(x)}< 2•- 1)1o

.

(4)

This formula is my starting point. The universal contribution which is survived in the continuum limit lies in the first term on the right hand side. In (4) r2 11 are expansion coefficients of the function r((x) =log r,lz) = 0 e211 r 2.(x), where f = N- 1 • What we need to compute e(h) is to solve a recursive equation for r((x) 5

E:

6 If such a solution does not exist, the matrix integration will be complex and loses its physical meaning.

138

In {5) Lp is a set of all paths with length (2p + 1) discussed in my previous paper and -p ~ s;(l) ~pare p integers assigned to each path l. H the action of the model is suitably fine tuned and the double scaling limit is taken, this recursive equation turns to the string equation. Alternatively I solve (5) by expanding as rE(x) =E. e2•r2a(x). It is convenient to introduce a number Ip(n~, ... , nm) which is specified by a Young's diagram (n~, ... ,nm) whose number ofrows ism. This Young's diagram corresponds to a partition of a positive even integer 2n, i.e. , :En; = 2n. These numbers play an important role in my method, for they appear in the expanded recursive equations derived from (5). H f(x) is an arbitrary infinitely differentiable function, the number is defined by a following Taylor expansion p 00 f(x + Bi(l)e) e2n F2n(p),

L IT

=L

n=O

IEL, i=l

where Y2n is the set of partitions of 2n. The number vanishes when p < m. These numbers depend on p which parameterizes the path length. As the space is limited, I cannot explain how to determine the p-dependence of these numbers. The answer for this problem can be presented in a simple way. To do this it is convenient to define a generating function for these numbers 00

I(ro; (n1, ... , nm))

=L

2(p + 1)..\plp(nl, ... ,nm)rop-m.

(7)

p=m

The p-dependence of lp(n~. ... , nm) is described by the following remarkable relation between the generating function and the function W(r 0 ), n-1

I(ro; (n1, ... , nm)) =

L a;(n~, ... , nm)r~W(i+m+l)(ro),

(8)

i=O

where a;(n1, ... , nm) are positive rational numbers. These n rational numbers are completely fixed by n explicit values of Ip( n 1, ... , nm) for small p. Using a computer these values are computable. One meets the generating functions defined in (7) and their derivatives with respective to ro when the large-N expansion of (5) is performed. For example, one obtains a equation for r2 (rather for r2 = r2 / ro) from the e2 order,

r2 W'(ro)

+ ro { /(ro; (2) )ro( 2) + I(ro; (1, 1) )(ro( 1)) 2} = 0.

(9)

139

The relation {8) is easily verified for the Young's diagrams {2), and {1, 1): /(ro; {2)) = lW< 2>(ro), /{r0 ; {1, 1)) = 112 w< 3 >(ro). The actual computation process of en(a1,2, a2,3 1 • • ·Gn-l,n• Gn,l) is understood to represent 'Dn(a1,2 1 a2,3 1 • • ·Gn-l,n 1 Gn,li G3,1, G4,1, · · ·Gn-1,1)· Putting eq {25) and eq. (34) together, we obtain a formula

Pn(9l,92, · · ·Bn)

_(a'1

2

8A

)

(-1)" 'I

L: 1>,.

(ft sinh(~ i= 2

-lc;- 1)9;) smhm9;

sinh(~ -•1- 1)91 smhm91 (35)

where 'Dn means 'Dn(lc1 - 1, · · ·, lc"- 1). The fact that the different divisions of 'Dn into n - 2 triangles are embodied by this single expression is precisely the statement of the old duality. The object Pn(91,92, · · ·9n) is equipped with 9; and lc; for j = 1, 2, · · ·, n and any 1>3 (lc 1 - 1, lc 2 - 1, lc3 - 1) obeys the rule of the triangle specified above. It is, therefore, natural to visualize this as a vertex which connects n external legs corresponding to n loops. The vertex can be regarded as a dual graph of an n-gon that corresponds 1>n(lc1 - 1, · · ·, lcn- 1). Using the formula (4), we perform the inverse Laplace transform with respect to(, (i = 1,"' n). We obtain

where Cj 1 denotes the inverse Laplace transform with respect to (;. Expressing this by a and t we obtain

X

This is the answer quoted in the introduction.

(37)

152

It is straightforward to look at the small length behavior of eq.(37). This was done in 5 in the case n = 3, using the formula

The agreement with the approach from the generalized Kdv flows 18 (See also 19 .) has been given 5 •

4

Residual Interactions

Our formula in the last section tells how the higher order operators (gravitational descendants) in addition to the dressed primaries included in the form of the loop length are constrained to obey the selection rules of CFT. The twomatrix model realizing the unitary minimal series coupled to gravity as the continuum limit of the (m + 1, m) symmetric critical point knows the fusion rules and the duality symmetry in the form of the loop operators. The term we have dealt with in the last section for general n is, however, supplemented with an increasing number of other terms with n (n;:::: 4). The existence of such terms itself implies that the knowledge we obtain from the two and the three point functions is not sufficient to determine the full amplitude for n 2::: 4. In what follows, we show how to perform the inverse Laplace transformation of the resolvents to get loop amplitudes in terms of loop lengths in the case of n = 4, 5. It is necessary to put

(38) in a manageable form to the inverse Laplace transform. Let us recall that Pn(91, · · · ,9n) can be inverse Laplace transformed immediately. If the function An (61, · · · , 9n) is expressed as a polynomial of P; (61, · · · 1 9n) and their derivatives with respect to A, the inverse Laplace transform can be done immediately. Let us pursue this possibility. We also make use of the fact that when one of the loops shrinks and the loop length goes to zero, the n-loop amplitude must become proportional to the derivative of the (n- 1)-loop amplitude with respect to the cosmological constant. In the limit M ln - O, An(91, · · ·,9n) must satisfy the following relation:

(39) This relation restricts the possible form of .:in (9t, · · · , 9n) .

153

As we want .6.n ((Jt, · · · , 6n) to be expressed as a polynomial of P; and their derivatives with respect to A, we need to introduce a notation

where 'Pn represents the permutations of {1,2, · · · 1 n). It is convenient to represent Pn(61, · · ·, 6n) by ann-vertex which connects n external legs. The n-vertex can be regarded as a dual graph of the n-gon (polygon) which corresponds to 'Dn. In terms of these vertices

. . 1 X

. . .Y

(41)

Now we are concerned with the case of n=4 first. By explicit computation motivated by the loop-shrinking argument given above, we find -1-

2!~4(61, 62, 6s, 64)

a

I= BA P4(61, 62, 6s, 64)- [SP12sP234] (61, 62, 6s, 64)

+ [SP12s4Ps4] (61,82,8s,84)

=(X)

I

154

where the prime represents the differentiation with respect to A. We obtain 1 1)2t -2-~ • •.. ,4 • ) = Afu•ion(• • ) + m2(m+ A 4(ll as dlogu(x)/dlog:r for c = 1/2, l, 2 and 5. WP- have assumed v = 1/4 for c = 1/2 and c = 1, v = 1/3 for c = 2 and v = 1/2 for c = 5. The constants a,b determined this way are quite close to the pure gravity values for c = 1/2 and c = 1. In all cases t.he data include the following discretized volume sizt>S: N = 1K, 2K, 4K, ... , 32K.

165

.

..'

·~---------,.-----~

--t----··---·---· ··-·---·--·--·

t

l

·~----~--------~

.

I

I

••

···--···----·--- ---·---·-·-······· ··-···-··--··········· ·---·-- ···-

Figure 3: The left figure shows 1/v (~ d,.) determined by finite size scaling for c = 0, 1/2, 4/-'5. 1, 2, 3, 4, and 5. The dots denote the best values of 1/v (~ d,. ). The right figure shows the measured string susceptibility '"'f• versus central charge c. We here assume = cons!-. \f1'• -J. This is the reason why the measured '"'f• for c = 1 disagrees with the theort>tical value.

z,,

walk. We dt>not.t>s the discrete version of K\I·(R; T) by kN(r; t). tis the number of stt>p of random walk, which is related with T as

T = (3 te2>.,

(27)

wht>rt> {J is a dimt>nsionless constant parameter. Since kN(r;t) satisfies ')()

L(SN(r))kN(r;t) = 1,

(28)

··=0 the discrt>te Vt>rsion of ( 16) is

k:v(r;t)

=

1 Np(x;y)

r with x = -

N"'

t

y

= N>.'

(29)

p(.r;y) is relatt>d with P(X;Y) as

p(x;y) = P(X;Y), Th~>

;Jy = y.

(30)

rt>turu probability of random walk is _ N>.d./2-t kN(O; t) "' const. td./ 2

for

t"' 0,

(31)

while the avt>rage geodesic distance travel by random walk over time t is {r N ( t))

~

00

L {SN(r)) r kN( r; t)

"' const. Nv->.tr ttr

for

t "' 0.

(32)

··=0

-

4

In fig. 4 we show -2dlogkN(O;t)/dlogt (:::::$ d8 ) and (rN(t)) as a function of time for c = 0 (pure gravity), 1/2, 1, 3, 5. From tbe left in fig. 4 we observe that ds is consistent with 2 for c ~ 1 and that it. decreast>s for c > 1. From tht> right figttrt> we observe that u :::::$ 1/4. We then obtain dh :::::$ 2d8 if the "smooth" fractal condition (20) is satisfied.

166

. '

\..

.'

• ~

. '

..

0

0

eo

60

100

= = = = =

Figure 4: The left figure shows -2dlogkN(O; t)/dlogt versus t for c 0 (top curve), c 1/2, c 1, c 3 and c 5 (bottom curve). The system size is N 16K. The right figure shows (rN(tl) 4 versus t for c 0 (bottom curve), c 1, c 3 and c 5 (top curve). The system size is N 4K. Straight lines (as observed) indicate u 1/4 according to (32)

=

=

=

=

=

=

3

=

Discussion

Wf.' genf.'ralizf.' the definition of the two-point function in (3) as follows: Let I) 9 ( ~, ~0 : M;) be an observable which depends on two coordinates ~ and ~o as wf.'ll as thf.' metric, where M; symbolizes some parameters. Wf.' define the averagf.' of IJ 9 (~.~o; Mi) with a fixed geodesic distancf.' R as '1!v(R;Mi)

def

_1_JrfV(gab]Vt/>6(

{..[9-V)e-Smauer[g,)

(33}

.f "ol(I>if£) .f' d2 ~JY(Od2 ~ov'g(~o) 6(d9 (~.~o)- R) '1! 9 (~.~o; M;).

Cil'(R}

jj

x Note that

I) v( R;

M;) satisfies

1

00

dR (Sv(R)} '1! v(R; M;)

(34}

=

(35)

wherf.'

_1 Jr fV(Yab)V 6( f.;g _ V) e-Smau•• (u., which is the vacuum solution of 20 dilaton gravity (1),

10 =..!... jd3 z.jg(gaPaa;ap; + gaPaaX8pX + 24JR), 411"

(3)

where 4> is the Liouville field which is identified with the space coordinate and X is the eM = 1 matter field which is identified with the (Euclidean) time. A physical state with continuous momentum can only be the massless scalar called "tachyon" in the string terminology. In two dimensions the tachyon mode becomes massless in the linear dilaton vacuum. The tachyon vertex operator with momentum/energy k(> 0) is given by

Tt =

;.

j d'Jze('J-l)•(•,,)±t.lX(.c,,) ,

(4)

where ± denotes the chirality. The selection of k > 0 is called the Seiberg condition n. We will postulate below that the amplitude including the antiSeiberg (A: < 0) states vanishes. There is an infinite number of physical states at the integer momenta called the discrete states 16 , which are constructed from the OPE of the tachyon operators with integer momenta, vn-+l(z, z)V,!+l (w,w) ...... 1.. _1, 12Rn,m(w)Rn,m(11i), where n, mare positive integers c& and vl±(z,.z) is the exponential part of the tachyon operator (4). The states Rn,m are nothing but the remnants of the massive string modes in higher dimensional string theories. The discrete states Rn,m form the chiral Woo algebra 4 • We here normalize the fields such that R,. m(z)R,.• m•(w) ' '

= - 1 -(nm'- n'm)Rn+n'-l,m+m'-l(w). z-w

"We slightly change the notation of the subscript of Rand Bin refs.1,2.

(5}

170

Besides these, at the same momenta, there are the BRST invariant operators with conformal dimension zero, Bn,m 16 , which satisfy the ring structure Bn,m(.z)Bn•,m•(w) = Bn+n'-1,m+m'-1(w). Combining Rn,m(.z) and Bn,m(z), we can construct the Woo symmetry currents 4 Wn,m(.z, .i) = Rn,m(.z)Bn,m(z), which satisfy a,.wn,m(.z, z) = {QsRSTt [b-1, Wn,m(.z, .i)]}

where the algebra

3

I

(6)

a,.= L-1 ={QBRST, b-1} is used.

Scattering Amplitudes of Tachyons

Let us consider the action in the tachyon background

I= Io + J.'sTo ,

(7)

where To = lim,_o T,±. The tachyon is massless so that S-matrix including To vanishes. To ensure the non-decoupling of J.'sTo we must make the bare tachyon background J.'B divergent as follows: J.'B - 7. The S-matrix of tachyons in the tachyon background is defined by

The superscript free denotes the free field representation. The c5-function and J.'s r come from the zero-mode integrals of X and tP respectively 14 • g is the genus, X = 2 - 2g and s is given by s = }:~ 1 ~ +X - N - M. The theory is tranlationally invariant in the time X, while is not in the space coordinate tP· So the factorization property of amplitudes are different from the usual string theory in the zero-dilaton vacuum ~ = 0. Let us introduce the eigenstate of the hamiltonian H = Lo + Lo 6, lh, l; N > with the eigenvalue ih2 + !12 + 2N, where lh, l; N = 0 >= ccexp[(2 + ih)tP(O, 0) + ilX(O, 0)]10 >. The normalization is given by< h', l'; N'lh, l,; N >o= - f{21r) 2 6(h' + h)6(l' + l)6N',N· Note that the on-shell (H = 0) state has purely imaginary h. l must be real to preserve the translational invariance of X. The string "Lo is the zero-mode of the Virasoro generator including the ghost part.

171

:k.

propagator is given by So the factorisation of 20 string am pi tude into two parts is given in the form

< 0 >= -2~

L 110

Jdhf dl 271' 271'


N=O

2

X

ih2

+ ll2 + 2N < h, l; NI02 > + ... '

{9)

where · · · denotes other channels. The aero-mode integral of X ensures the conservation of energy so that l is fixed, while the aero-mode integral of~ does not produce the cS-function in the linear dilaton background. We then obtain the analytic function of h. So h integral is non-trivial even in tree amplitudes. Naively we can deform the h integral to the complex plane. It picks up the on-shell poles on the imaginary axis. 4

Ward Identities of W 110 Symmetry

We introduce the normalized tachyon vertex operator

A{A:) = r{A:) r{-A:)

(10)

and call the amplitude given by replacing Tf in (8) with Tf the S-matrix. Henceforth we consider the Ward identities in the form: J d2 z86 < Wn,m(z, i) 0 >,= 0, where 0 is a product of the normalised tachyon operators. Let us first calculate the operator product expansion (OPE) between the current and the tachyon operators, which is given in refs.4,1,

-+ (0, 0) T~: -+ Wn,m(z, i) T~: 1 3

= ; n!

• • •

-+

T~:,.

(Ji A:,) T~+···+A:,.-n+m(O, 0)

1

(11)

•=1

where Tt(z, i) is defined by replacing the integral in {4) with c(i)c(z). It was computed step by step from the n = 1 formula to the general n. This is analogous to the calculation of the OPE coefficients in CFT, where ft, (i = 2, · · ·, n) just play a role ofacreening charges. Note that the OPE with the zeromomentum tachyon T0 vanishes, but the OPE with the tachyon background J4BTo becomes finite due to the renormalization of J4B· The OPE with the tachyon f,- is easily calculated by changing the chirality. It is carried out by changing the sign of the field X such that f+ -+ 1'and Wn,m-+ -Wm,n (the roles ofn and mare interchanged).

172

The OPE singularity gives the linear term of the Ward identity. In addition we get the BRST-trivial correlator < : J d;az{QsRST 1 [L1, Wn,m(z, i)]} 0 >1 . Usually such a correlator would vanish. In this case, however, it gives the anomalous contributions from the boundary of moduli space. The boundary is described by using the string propagator in the form

(12)

r-oo,

where the second term of r.h.s. is the boundary. Let us first calculate the n = 1 anomalous contribution. We then have to evaluate the following boundary contribution:

. ->. ~ hm L,., ., ..... oo 2 N=O

JJ1 -dl

211'

-dh

211' e-~:SI•I:S1

-2 "HI xQsRST He-

-

d :a z

< 01 [-b-1 W1,rn(z,i))

h, -l; N

1

>
. (13 )

We consider only the N = 0 mode. As a result, the N :f. 0 contributions vanish exponentially as e-:aN.,. The z-dependence of the integrand is given in the form 1 H [b-1, w1,rn(z,i)]QsRST He_., 1- h,-l

>=

(14)

/(h,l)lzl{(rn-1)(-iii-IH)-:arn}e-"("3+13)/:11- h + i(m- 1), -l + m- 1 >

'

where we use QsRST = ~c0 H + · · ·. f(h,l) is the calculable coefficient. Changing the variable to l.zl = e_.,.,, where 0 ~ z ~ 1, we get the following rdependence: 211'T exp[-r{i(h:a + z:a) + z(m- 1)( -ih -l)}). Thus the integrand is highly peaked in the limit T - co. So we can exactly evaluate the integral of h at the saddle point h,.,. = i( m - 1)z. We then get the expression

{2; >.ry-:;:

J 1 dl 211'

1

0

T ;a] /(h=l(m-1)z,l) . dzexp [ -2{(m-1)z-l}

x < 0 1 li(m- 1)(1- z), m- 1 - l >< i(m- 1)z,liO:a > . (15) The z integral is also evaluated at the saddle point z,.,. = rn~ 1 and produces the coefficient rn~ 1 /( h = il, l) = A( m - 1 - l)A( l). We then get the boundary contribution

(16)

173

where the A-factors are abscubed in the T,!_ 1 _ 1 and 71+. The l integral is restricted within the interval 0 :5 l :5 m-1 because the saddle point z •.,.. should be located within the interval 0 $ z •.,.. $ 1 to give the finite contribution. Assuming that the boundary structure does not change in higher genus, we then get the expression

(17)

The second term is a variant of the first term, which comes from the configuration that two surfaces are connected by a handle. The factor in the first term is to correct the overcounting of the summation and that in the second term is to correct the double counting coming from the interchange of 1',!_ 1 _ 1 A+ and 71 . We next consider the n = 2 case. We then have to evaluate the following quantity:

i

where 1: = J d 2 zV1,+(z, z). The primes on 01,2 denote the exclusion of the After carrying out the integration of w, we evaluate the z and h operator integral using the saddle point method. At T -+ oo we obtain the contribution

1:.

(19) where 9 = 91 + 92 • There also is a variant of this contribution coming from the configuration where two surfaces are connected by a handle. As an another variant of (19) we furthermore obtain the boundary contribution with the triple product of amplitudes,

174

where 9 = 91 +92+93· Noting the factorization property discu1111ed before that the intermediate state becomes on-shell after integrating over the intermediate momentum, it is calculated by replacing the vertex operator in (18} with the factrization formula < i;t 02 >,3, where ~. = J ~!nh2 + 12 )- 1 = fr and only theN= 0 mode is considered. The general n formula is

v,+

-i ;:., v_,,

il

A4 -

1

n

n! Cfi" k,) J X

< T,A+ 01I 1

dl,9(l,)

>g 1

/j

(t

I < n:.+ .LI~ 02 >g

l, -

2 • • •

"~" k, + n- m)

A+ I < Tz. 0

4

>g.

(21}

1

where 2::= 1 9i = 9 and a = 1, · · ·, n + 1. 9 is the step function. The a = 1 formula is nothing but the contribution of the OPE (11). In addition, as discussed in the cases of n = 1, there are many variants of this expression coming from the boundary configurations that some of the surfaces are connected by handles. The formulas with the vertex i',- are given by changing the chirality; i'+ -+ i'- and Wn,m -+ - Wm,n. Summarizing the boundary contributions, we can write out the Ward identities. For example, we get

o = ; =

j a
2, further boundary contributions are necessary in order that the solution exists. It is easily imagined that there are the contributions shown in fig.l. On the basis of this figure we can speculate a generalisation of the formula (21) as follows:

..xa-l+h

f rr

dl,tJ(l,) v;:-(-lt ..... -la, kt. .... kn+t-a-2h;n,m)

i:::;l

(23)

=

=

where E::::;l Yi g- h and a 1, · · · , n + 1 - 2h. h stands for the genus of the surface E in fig.l. -1, represents the conjugate mode of T1~. The E-part just gives the connectivity matrix D;i(h) at T -+ oo. The h = 0 formula is nothing but (21). The h #: 0 contributions exist for g ~hand n ~ 2h, where note that the h = 1 formula would contibute in the W2, 1 identities, but it vanishes due to the Seiberg condition. The direct calculation of the connectivity matrix v;= .

........ 00

In Fig.5 we show the distribution of triangles whose position z is mapped within a circle by the transformation az z~---z+a-1

l i'f-.

with a= + There exists a theoretical prediction of the distribution [6], but it has not been presented in a form ready for the direct comparison to the numerical experiments.

198

-o-~o"=.s---o:".o:----~o~.s----'1.o_ _ _-..J1.5

=

Figure 5: A plot of the obtained coordinates of all vertices for c 0 with five independent configurations superimposed. Ae a matter of convenience all pointe are mapped into a circle with the center at ( ~, and radiue

1-.

'f-)

5

Complex structure of the T 2 surface

As far as two dimensional orientable manifolds of 5 2 topology are concerned they all have the same complex structure, i.e. any manifold can be transformed to the other by a combination of a general coordinate transformation(Diffeo) and the Weyl transformation{ WeyQ. On the other hand for the surface with T 2 topology the moduli space

{ga&}/Weyl ® Diffeo is spanned by a complex plane of moduli integrals of an Abelian differential T=

T.

It is defined by the ratio of the

J6 i,.dz"' + i J6 ],.dziA £. i,.dziA + i £. -j,.dziA '

where integration contours (we call them as the a.-cycle and the b-cycle) are chosen to be two closed paths on the manifold which cross each other only once. Here, j,_.dziA is a harmonic 1-form and ],.dziA is its dual. If we regard i,. as a current density on the torus, r J4 j,_.dz1A and J~~.],.dziA correspond to the voltage drop along the a-cycle and the total current crossing the a.-cycle, respectively. In practice, we select two closed paths made up by connecting edges of triangles, which intersect only once. Then we cut the surface along one of

199 b

Figure 6:

the path for which we choose the b-cycle, and apply constant voltages(lV} in between neighbouring triangles dual to the cut(Fig.6). Solving the Kirchhoff equation we can determine all the currents jp. and Jp. along links between neighbouring triangles in the dual graph. By construction of the network the following two integrals are trivial;

J:1 jp.dzP. = !,r ~1 jp.dzP. = 0. Thus

T

is obtained by measuring total currents crossing two cycles; ii&

-r- !+'I

,.

'.

where I,. and I& represent the total currents crossing the a-cycle and the b-cycle, respectively. Instead of measuring the resistivity of this surface we employ the value determined previously for the surface with S 2 topology and corresponding central charge. Fig. 7 shows the distribution of T for configurations of 4,000 and 8,000 triangles. Each value ofT is transformed appropriately by the SL(2, Z) transformation a-r+b Tl-+ c-r+d (ad- be= 1) so that it is in the fundamental domain. By summing over real -r's with a fixed imaginary part we get the distribution for surfaces with two, four and eight

200

Pure Gravity (4000 triangles)

4.0

2.0

0.0 ,__~......_~__._~--'--~-'--~--'-~--' -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Re-.: Pure Gravity (8000 triangles)

4.0

2.0

0.0 ....__~_.__~__.-~_,__~__._~_...._~...... -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Re-.: Figure 7: Plot of the moduli{T) on the complex-plane mapped to the fundamental domain with a total number of triangles of 4, 000 and 8, 000. Each dot corresponcla to the moduliT for a sample of DT-surface.

201

Pure Gravity

•2K~;qles

•4K~;qles

.. 01 in such a way that '5>.. 01 -£. • 0 Then the left hand s1de of eq. (12) and eq. (13) become homogeneous quadrat1c * S0 • equations which we write symbolically as r * r and In our derivation of the WT identities, we have assumed the invariance of the bare action under the gauge transformation (7). However we start with the tree level action which possesses only the volume preserving diffeomorphism invariance. The crucial question is whether we can choose the counter terms of the theory in such a way to satisfy these identities by starting with such a tree action. We answer affirmatively to this question in this paper.

= -£ =

J

~

so

3

Analysis of the Bare Action

In this section, we solve eq. (13) to determine S0 • S0 will be simply denoted by S in this section. Let us examine the general structure of the bare action. By power counting, it has to be a local functional of fields and sources with the dimension D. We also have the ghost number conservation rule and its ghost number has to be zero. By these dimension and ghost number considerations,

211

it is easy to see that K and L appear only linearly in S:

{14) where 68 denotes most general BRS like transformations with the correct dimension and ghost number. It is also easy to see that there are no A and hence no (; dependence in 68. Since A has dimension 1, S can be at most quadratic in .X:

(15) where EofJ and F0 are general local functions of A, C and C with dimension zero and one respectively. G 0 is the bare gravitational coupling constant and it is the only quantity with dimension - f . We denote below by Oi the set of all anticommuting fields Ki, C 0 , (;o and x; all commuting fields A;, L 0 , M 0 • The fundamental equation for the action S takes then the form: {16) The equation {16) is invariant under the following canonical transformations. Let us make the change of the variables (9, x) -+ ( 9', x'): X;

Ot.p (

=

I

)

oO; x,9'

(x',9). • = :~ X;

9~

{17)

We can verify that we recover the equation {16) in the new variables. It has been shown that the set of the canonical transformations ( 17) is the most general set of the transformations which leaves the equation {16) invariant!• 11 • Let us write these transformations in the infinitesimal form:

X ~a = (0;)'

X;-

o

quired counter term of J R type at order G1- 1 apart from the change of the definition of a up to order G1- 1 • Now the circle is complete and we have proven the renormalizability of quantum gravity near two dimensions. The major assumption we have made in this proof is that the solutions we have found in this section exhaust the solutions of eq. (25). Under this very plausible assumption, we have established that this model is renormalizable to all orders with the following bare action:

a, a,_

IL;ff jlR(Za+at/J+ebt/12)- ~a~-~t~;8,t/J?"1+ jr~a~-~rpi8,rpiJJIA" -£bRrp~).

(32)

where Za = 1 - A1 fe and 2(D- 1)a2 = e- A1 • Here we have omitted the BRS exact part of the action. Of course we also need counter terms which correspond to the wave function renormalization of the action. Let us consider the physical significance of the coefficient a 2 • We note that it. measures the conformal anomaly of the theory. At the critical point where a vanishes, the conformal mode becomes indistinguishable from the scalar fields which couple to the gravity in the conformally invariant way. The Z2 invariance under t/J - -t/J is restored at the critical point since the odd parity sector of the effective action vanishes. We have suggested that this Z2 invariance may distinguish the different phases of quantum gravitf. Vle would like to interpret this phenomenon as the signature of the conformal invariance. a 2 can be expanded in Gas follows: (33) This quantity measures the conformal anomaly of the theory as a function of G. For the particular value of G, it vanishes and the theory becomes conformally invariant in the sense we have just explained. As it is well known the {J function is related to the conformal anomaly. Therefore it is reasonable to adopt (33) as the {J function of G by resorting to this connection. We also recall that G is the gravitational coupling constant. It measures the strength of the coupling of h.IA, field at the momentum scale p..

218

In the conventional definition of the {J function, it is defined through the bare gravitational coupling constant: 1

JJ(

ao = Gz~.

{34)

The {J function of G is obtained by demanding that the bare quantity is independent of the renormalization scale p.. However there is an ambiguity in this procedure since the bare gravitational coupling constant changes if we rescale the conformal mode'. The relation (33) is free from such an ambiguity since this ambiguity does not alter the classical relation. Therefore we adopt the right hand side of eq. {33) as the {J function of G. We expect that this {J function is also obtained by the conventional procedure since Z~ = Z Za in our scheme. Through the conventional procedure we find that P.op.G = £GZ~/(1- G/a)Z~. If Z = (1- G.Jh)Z~, we find the same {J function with eq. (33) by the conventional method. In our renormalization procedure, we have classified the nontrivial solutions of eq. (25) into those with the conformal anomaly and those with the vanishing conformal anomaly. The latter is associated with the higher poles in £ in general while the former is associated with the simple pole. This classification must be generic in field theories. We expect that such a classification of divergences underlies the pole identities which ensure the finiteness of the conformal anomaly.

5

Conclusions and Discussions

In this paper we have further studied the renormalizability of quantum gravity near two dimensions. We thereby put the 2 + £ dimensional expansion of quantum gravity on a solid foundation. We have proven that all necessary counter terms can be supplied by the bare action which is invariant under the full diffeomorphism. However the tree level action itself is not invariant under the general coordinate transformation. Only after adding the counter terms and thereby considering the bare action, we can recover the action which is invariant under the full diffeomorphism. We have chosen the tree level action to possess the volume preserving diffeomorphism in variance. In order to recover the full diffeomorphism in variance, we need to require that the theory is independent of the background metric. This requirement has led us to search a theory which is conformally invariant with respect to the background metric. Obviously the Einstein action is such a theory and we conjecture that the requirement of the background independence leads us uniquely to the Einstein action as the bare action.

219

In our perturbative expansion, we need to introduce not only the gravitational coupling constant G but also another coupling constant a. G controls the dynamics of h,. 11 field while a controls the dynamics of the conformal mode. a 2 is related to G since it is nothing but the f3 function of G modulo a factor of G. However a itself cannot be expanded in G since such an expansion is singular in f.. We have constructed a proof of the renormalizability of the theory to all orders in the perturbative expansion of G. In this expansion, a 2 is also perturbatively determined in terms of G. Since G is O(e) at the short distance fixed point of the renormalization group, G may be regarded as a small expansion parameter as long as we regard e to be small. Our proof is base on the plausible assumption concerning the solution of eq. (25). This assumption is in accord with the investigations of the gauge theorie/• 11 . We hope that it can also be proven in the near future.

1. S. Weinberg, in General Relativity, an Einstein Centenary Survey, eds. S.W. Hawking and W. Israel (Cambridge University Press, 1979}. R. Gastmans, R. Kallosh and C. Truffin, Nucl. Phys. B133 (1978) 417. S.M. Christensen and M.J. Duff, Phys. Lett. B79 (1978) 213. 2. H. Kawai and M. Ninomiya, Nucl. Phys. B336 (1990}115. 3. H. Kawai, Y. Kitazawa and M. Ninomiya, Nucl. Phys. B393(1993) 280. 4. H. Kawai, Y. Kitazawa and M. Ninomiya, Nucl. Phys. B404 (1993) 684. 5. T. Aida, Y. Kitazawa, H. Kawai and M. Ninomiya, Nucl. Phys. B427 (1994) 158. 6. Y. Kitazawa, Nucl. Phys. B453 (1995) 477. 7. For a review see J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford Univ. Press, 1989) sect. 21. 8. A.M. Polyakov, Mod. Phys. Lett. A2 (1987) 893. V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819. 9. F. David, Mod. Phys. Lett. A3 (1988) 1651. J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 504. 10. I.R. Klebanov, 1.1. Kogan and A.M. Polyakov, Phys. Rev. Lett. 71 (1993) 3243. 11. G. Barnich and M. Henneaux, Phys. Rev. Lett. 72 (1994) 1588; G. Barnich, F. Brandt and M. Henneaux,Local BRST cohomology in Einstein- Yang-Mills theory, preprint KUL-TF-95/16,ULB-TH-95/07, hep-th/9505173.

220

Woo GAUGE THEORY B. SAKITA Department of Phy1ic1, City Collge of the City Univer1ity of New York, New York, NY 10031, USA We pretent a general method of constructing W ao and tuao gauge theories in terms of d + 2 dimensional local fields. In this formulation the W ao gauge theory Lagrangiana involve non-local interactions, but the w 00 theories are entirely local. We discuss the 1o-called clu1ical contraction procedure by which we derive the Lagrangian of w00' gauge theory from that of the corresponding W ao gauge theory.

It u a plea.sure to dedicate tkil article to th.e 60th. birth.day celebration of Keiji Kilclcawa, a clo&e dear friend and colleague for over th.irty yeara. 1

Introduction

W 00 algebra and its so-called classically contracted 'Woo algebra appeared recently in various problems in physics, in particular in the study of c = 1 string theory 1•2 •3 •4 •5 •6 •7 and quantum Hall system 8 •9 • The gauge theory based on these algebras also appeared in these studies 7 •10 12 •13 • In fact the same algebras and the gauge theories based on them had been proposed previously as the theories 14•15 •16 •17•18 •19 •20•21 relevant to the large N limit of SU(N) gauge theories 22 • In this article we elaborate the description of the theory systematically by using the technique developed in the study of quantum Hall system 8 •11 •12 • The W 00 algebra is a commutator algebra of Hermitian operators of one harmonic oscillator 8 •9 •23• It is an infinite-dimensional Lie algebra. If we choose a set of linearly independent real function of z and z as the parameters of W 00 group, the structure constants of the algebra are expressed in terms of Moyal bracket 24 • Replacing the Moyal bracket by a Poisson bracket we define the 'Woo algebra. It is an algebra of area-preserving diffeomorphism. As an introduction we discuss this issue in section 2 together with the so-called classical contraction procedure by which the W DO algebra is transformed to the W 00 algebra. W DO gauge theory is a gauge field theory of W DO as an internal symmetry algebra. The W00 gauge potential is a space-time dependent Hermitian operator of harmonic oscillator. In the coherent state representation it is a function of z, z, which we call the color space coordinates, and z"" (I'= 1,2, ... d), the space time coordinates. Thus, we can express the W 00 gauge theories in terms

221

of d + 2 dimensional local fields. The interactions of the fields are necessarily non-local in the color space in W 00 theories, but they are local in w 00 theories. In section 3 we define W 00 theories as d + 2 dimensional field theories with non-local interactions and w 00 gauge theories as d + 2 dimensional local field theories. Since the W00 algebra is closely related to the w 00 algebra, the gauge theories based on these algebras may also be closely related. In order to see the relationship at classical Lagrangian level, we introduce the l -+ 0 contraction procedure by which we derive the W 00 gauge theories from the corresponding W 00 gauge theories. The procedure consists of the introduction of a length scale l in the color space, an appropriate scale transformation of the fields, and the l -+ 0 limit. In this section we also introduce matter fields analogous to the quark fields and the Higgs fields. 2 2.1

W 00 and

W

00

Algebra

Woo Algebra

We define the W 00 algebra as a commutator algebra of all Hermitian operators {(a, at) in the Hilbert space of a harmonic oscillator 8 • A convenient parametrization for these operators is achieved by using a real function {(z, z) as

(1) where at and aare standard creation and annihilation operators and t t stands for the anti-normal-order symbol, i.e. all the creation operators stand to the right of the annihilation operators. The last expression 1 is in the coherent state representation. Obviously the product of two fs is not anti-normally ordered and we bring it to the anti-normal-order form by using the commutation relation [a, at) = 1. We obtain the following commutation relation:

(2) where {{1 , 6} is a Moyal bracket

24

defined by

The commutation relation 2 is that of the Woo Lie algebra in the fundamental representation. The W00 is an infinite-dimensional Lie group with parameters being a set of linearly independent real functions {(z, z). The generators of

222

W 00 are the linear functionals of {{.z, z). Thus we write for arbitrary representation:

[p[{t], p[{2]] = ip[{{t,6}],

(4)

where p is the generator of W 00 group. 2.2

W 00

Algebra and Contraction

The Lie algebra of W 00 , the area-preserving diffeomorphism&, is defined by the commutation relation

(5) where { , } is the Poisson bracket symbol. It is well known 8 that one can obtain the w 00 algebra from the W 00 by a contraction. For this purpose let us introduce a length scale l in the z, z space, which we call the color space, and set

{6) The Poisson bracket is the leading surviving term of the Moyal bracket in the l -+ 0 limit. In this paper we call this procedure as the l -+ 0 contraction.

3 9.1

W00 and w 00 Gauge Invariant Lagrangians Pure Yang-MilLs Theory

The W 00 gauge theory is a gauge field theory of W 00 as an internal symmetry algebra. Let us discuss first the pure Yang-Mills theory. We introduce a gauge potential .A,. which is a Hermitian operator in the harmonic oscillator Hilbert space as well as a function of space time:

The action is given by SyM

where

F,.v

= -~ Jddz tr(F,.vF"v), 4g

{8)

is the field strength defined by

F,.v = a,..Av -OvA,.- i[A,.,Av]·

(9)

223

We then rewrite these by using the coherent state representation:

S -

Y M - - _1 2

4g

jjd"

~(-)nan-r :z: d2 :ZL...J I ,_.rJ.&II ( z,:z,:z-)an-rJ.&II( ~.r z,:z,:z-) I n=O

n.

(10)

where :F:J.&II(:z:, :z, z)

= 8J.&AII(z, :z, z)- 811AJ.&(z, :z, .z) + {AJ.&, Av}(:z:, :z, .z).

(11)

This action is invariant under the W 00 gauge transformation: (12)

In the coherent sate representation the gauge fields are formally d + 2 dimensional local fields, d for space time and 2 for color space. However, the interactions are non-local since the action involves derivatives of infinite order. In the action 10 we no longer have damping factor e-lr-1 2 because it is a trace expression. Therefore we have to restrict the field configurations so that we can integrate by parts in the color space. In this paper we define our W00 gauge theories as d + 2 dimensional field theories such that all the fields and their derivatives vanish at :z = oo.

3.2 Matter Field8 Next, let us introduce a fermion field, which is a fundamental representation of W00 , namely a field which transforms as a bra or ket vector in the Hilbert space of harmonic oscillator: 11/J(:z:)) =

j

l:z)d2 :ze-l'-l 2 (:zi,P(:z:))

= j l:z) d 2:ze-l'-l ,P(:z:, z). 2

(13)

We write the action as

SF=

j j ~zd2:ze-l.zl\,b(z,:z)"Y~-'(i8J.&-

AJ.&(z,:z,z)},P(:z:,z),

(14)

which is invariant under the W00 gauge transformation 12 and 61/l(:z:,.z)

= -iU(8~,z)f,P(z,z),

(15)

where f f indicates that the derivatives are placed on the left of .f. As a last example· of W 00 gauge theory let us consider next a scalar field which is in an adjoint representation. In this paper we call this field simply a Higgs field.

224

The action is given by: SH

=

I [I d11 z

d2 zit,

(~~ 8:'(8,.M(z,z,z)- {A

1"

M}(z,z,z)) x

x8;'(8"M(z,z,z)- {A",M}(z,z,z}))- trv(M}, {17}

v(M)

= L9nMn,

dim(gn)

= d (i -1)- n.

n

Here again we require that the fields and their z, z derivatives should fall off to zero at z = oo. We can check that this action is invariant under the W 00 gauge transformation 12 and

6M(z, z, z} =

{~,

M}{z, z, z).

{18}

Notice that here again the interactions are non-local in the color space. The quantization of the theory is done by the standard canonical quantization. Although the interactions are non-local in the color space, they are local in the ordinary space. Accordingly there is no problem for the quantization. 9.9 w00 Gauge Theories and Contraction

Let us discuss next the contraction procedure which will allow us to obtain the w 00 gauge invariant actions from W 00 ones. For the fields in the adjoint representation such as A,. and M this procedure is straightforward. So we shall discuss here only these cases. As we mentioned earlier, by introducing two real coordinates u, and u11 as in 6 and by taking the l--+ 0 limit we reduce Moyal bracket to Poisson bracket. If we simultaneously rescale the fields and the coupling constants as

A,.(z, z, z} =

z- 2 A,.(z, u)

M(z,z,z} = {2?r}ttM{z,u) 93

6

=g 3 l-,

_

1. 2-n

9n=9n{2?r} 2 l

{19}

,

we obtain from 10 and 17 the following Woo gauge invariant d + 2 dimensional local field theory: SyM

=- 4~ 2

I

d11 zd2 uF,_ 11 (z,u)F""(z,a)

{20}

225

SH

=

j d zd c1 [i(a,M(z,U)- E•;a,A,(z,c1)8;M(z,c1)) x 4

2

x (a" M(z, c1)- £'; 81A"(z, c1)8;M(z, c1)) - ii(M)] (21} ii(M)

=I: inM"(z, cr}. "

Here again we require that the fields vanish at c1 = oo. 2{(z, cr} we obtain the Woo gauge transformation: Setting {(z, z, .z)

= z-

cSA~'(z, cr}

= 8~'{(z, U)- £ 0; a,{(z, c1)8;A~'(z, cr} cSM(z, cr} = £'; a,{(z, c1)8; M(z, cr}.

(22} (23}

Even though the transformations 22 and 23 are obtained from 12 and 18 by the l -+ 0 limit, we can independently check that the actions 21 is really invariant by the w 00 gauge transformations 22 and 23. We remark that 23 can be written as

cSM(z, cr} = M(z, cr+cScr(z, cr)}- M(z, cr}, which is a local area-preserving coordinate transformation. We should mention that the YM lagrangian 10 had been written down in the literature. 14•15 •16 •?• 18•19•20 References

1. J. Avan and A. Jevicki, Phy1. Lett. B266 (1991} 35; 2. A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Nucl. Phy1. B357 (1991} 565; 3. D. Minic, J. Polchinsky and Z. Yang, Nucl. Phy1. B362 (1991} 125; 4. G. Moore and N. Seiberg, Int. J. Mod. Phy1. AS {1992} 2601; 5. I. Klebanov and A.M. Polyakov,Mod. Phy1. Lett. A6 (1991} 3273; 6. E. Witten, Nucl. Phy1. B373 (1992} 187; 7. S.R. Das, A. Dhar, G. Mandai and S.R. Wadia, Int. J. Mod. Phys. A7 (1992} 5165; 8. S. lso, D. Karabali and B. Sakita , Phy1. Lett. B296 (1992} 143 9. A. Cappelli, C. Trugenberger and G. Zemba, Nucl. Phy1. B396 (1993) 465 10. A. Dhar, G. Mandai and S.R. Wadia, Mod. Phy1. Lett. A7 (1992} 937 11. Rashmi Ray and B. Sakita, Ann. Phy1. 230 (1994} 131 12. B. Sakita, Phy1. Lett. B315 (1993} 124 13. K. Shizuya, Phy1. Rev. B52 (1995} 2747

226 14. E. G. Floratos and J. Iliopoulos, Ph.ys. Lett. B201 (1988) 237; 15. E. G. Floratos, J. Iliopoulos and G. Tiktopoulos, Ph.ys. Lett. B217 (1989) 285; 16. E. G. Floratos, Ph.ys. Lett. B228 (1989) 335 17. D. B. Fairlie, P. Fletcher and C. K. Zachos, Ph.ys. Lett. B218 (1989) 203; 18. D. B. Fairlie and C. K. Zachos, Ph.ys. Lett. B224 (1989) 101; 19. D. B. Fairlie, P. Fletcher and C. K. Zachos, J. Math.. Ph.ys. 31 (1990) 1088; 20. C. K. Zachos Hamiltonian Flows, SU(oo), SO(oo), USp(oo), and Strings in Differential Geometric Methods inTh.eoretical Physics; Physics and Geometry, NATO ASI Series, L.-L. Chau and W. Nahm (eds.), Plenum, New York, p.423, 1990 21. I. Bars, Ph.ys. Lett. B245 (1990) 35 22. G. 't Hooft, Nucl. Ph.ys. B72 (1974) 461; Nucl. Ph.ys. B75 (1974) 461 23. For earlier work see J. Hoppe, MIT Ph.D. Thesis (1982); Hoppe and P. Schaller, Ph.ys. Lett.B237 (1990) 407; C.N. Pope, L.J. Romans and X. Shen "A Brief History of W 00 " in Strings 90, ed. R. Arnowitt et al (World Scientific 1991) and references therein. 24. J. E. Moyal, Proc. Camb. Ph.yl. Soc. 45 (1949) 99 25. A. Jevicki and B. Sakita, Nucl. Ph.ys. B165 (1980) 511. 26. B. Sakita, "Quantum Theory of Many- Variable Systems and Fields", World Scientific, Singapore (1985)

227

MULTI-PLAQUETTE SOLUTIONS FOR A DISCRETIZED ASHTEKAR GRAVITY K. EZAWA Department of Physic1, Osaka Uniuersity, Toyonaka, Osaka 560, Japan In this taUt I will make a survey of developments and issues on the Wilson-loop solutions which were found in Ashtekar's formulation for canonical quantum gravity. After explaining the issues on the Wilson-loop solutions in the continuum theory, I will investigate a discretised version of Ashtekar's formalism. Then I construct the simplest non-trivial solutions, namely 'multi-plaquette solutions'. I will also discuu what lessons are extracted from the multi-plaquette solutions.

I would like to congratulate my supervuor Prof. Keiji Kilclcawa for ku siztietk birthday.

1

Introduction

Constructing the quantum theory of gravitational interaction is one of the most challenging problems in modern theoretical physics. In the mid seventies perturbative quantization of general relativity (GR) turned out to be non-renormalizable. Most of the particle physicists consider this fact to be an indication that general relativity is the low energy effective theory of a more fundamental theory, which should be quantized when we construct the theory of quantum gravity. However, the non-renormalizability of GR does not contradict the possibility that G R is quantized in a nonperturbative manner. It is therefore worth examining whether the nonperturbative quantization of GR exists or not. Canonical quantization of GR has been investigated vigorously as one of the candidates for nonperturbative qauntum gravity. In the traditional canonical formalism called the ADM formalism,1 we take as canonial variables the induced metric ha.& on the spatial hypersurface :E and its conjugate momentum ;rat. While this formalism is suitable for extracting geometrical picture, it has a serious drawback that the Wheeler-Dewitt (WD) equation~ namely the dynamical equation in its Dirac quantization, is too complicated: 7i'l[hab] r;-:-;-1 1 6 6 =- { vdeth (h .. ehtt~- 2h..thct~) oh..t ohct~

= 0.

~3)

+ v'deth·

}

R(h) 'l[hab]

(1)

228

In consequence, not any solutions to this equation have been found so far in the full theory. Naturally a question arises as to whether there are any canonical variables which simplify the WD equation. The answer is yes. Ashtekar's new variables 3 are the desired ones. Ashtekar's formulation for canonical general relativity is derived either by prescribing a complex canonical transformation to the first-order ADM formalism 3 or by performing a (3+1)-decomposition of the complex-chiral action 4 which is classically equivalent to the Einstein-Hilbert action. Here we skip the detail of these derivations and refer the reader who is interested in it to refs. 3,4 and 5. The essential thing is that the first class constraints are written by polynomials in the new variables, namely the self-dual connection A~ and the densitized triad E"'. In particular the scalar constraint S takes the following form s EiilcF!bEi"Eicb, (2)

=

=

where F! 6 84 Ai- 8&A~ + Eiilc A{ A~ is the curvature of the connection A~. Because this scalar constraint is almost equal to the volume element times the Hamiltonian constraint 1{. in the ADM formalism S

~ Vdet(E"') x 11.,

the WD equation in Ashtekar's formalism is considerably simple: ~ . S'lt[A~]

1 .. lc . 6 6 . 3 F! 6 - . - . 'lt[A~] = 0. = -t' 4 6A~ 6A 6

(3)

We therefore expect that we can find solutions explicitly. Actually, several types of solutions have been found in Ashtekar's formalism." They are roughly classified into two types, topological solutions 7 and Wilsonloop solutions!' 910 Topological solutions are solutions to the SL(2, C) BF theory, namely the topological field theory which is closely related to Ashtekar's formalism. These topological solutions are considered to be 'vacuum states' of quantum GR.11 Wilson-loop solutions are essentially particular linear combinations of products of Wilson loops which are composed of paralell propagators

ha[O,l]

=Pexp

1dsa"(s)A~(a(s)), 1

(4)

where P denotes the path ordered product, a : [0, 1] - E is a curve, and J, is the SL(2, C) generator which is subject to the commutation relation

[i,, Jj]

= EijlcJic.

'"For an extemive review, see e.g. ref. 6.

229

Recently there have been remarkable developments in the application of spin netwoek states to quantum Ashtekar's formalism.12 Associated to this, progress in search for Wilson-loop solutions is expected to be made in the near future. I will henceforth concentrate on the Wilson-loop solutions.

2

Wilson-loop solutions

Let us now look into how an appropriate linear combination of Wilson loops is annihilated by the scalar constraint. For this purpose we first calculate the functional derivative of a parallel propagator:

=

=

l,From this we see that an expression like F!6 (z)a 4 (6)a 6 (t) (a(6) a(t) z) appears whenever the scalar constraint S acts on the parallel propagator ha[O, 1]. As a corollary, products of Wilson loops evaluated along smooth loops without any intersection become solutions to the WD equation (3):

S· (

II ae{a}

a]) = 0,

{6)

W[a, 'II"

where {a} is a set of smooth loops which have no intersection and W[a, ?r] denotes a Wilson loop in the representation ?r which is evaluated along the loop a (7) We will call these solutions 'non-intersecting loop solutions'.13 The above result indicates that non-vanishing contributions to the action of the acalar constraint arise only from kinks or intersections. If we consider only parallel propagators ha[O, l]A B in the spinor representation 1r = 1/2, the action of S on a kink and on an intersection is calculated as 6

=

=

"We auume that the curves a and fJ intersect at a point a:o a(•o) fJ(to). a1 and fJ2 respectively denote the curve from a(O) to a(•o) and that from fJ(to) to fJ(l). h 41 .1'2 [0,1] denotes ha{O, •o]hj!J(to,l].

230

where au( a, z) stands for the area derivative 14 acting on the point z E a. t,From these results we find the following solution:

S · (W[a, 1/2]W[,8, 1/2]- W[a · {3, 1/2]) = 0.

(10)

This is the 'double-intersecting-loop solution' found in ref.8. This double-intersecting-loop solution has been extended to the intersections at which three or more loops intersect.9 It can also be extended to the case where 'Ira and 'lrp are arbitrary finite dimensional representations of SL(2, C).10 Thus we have found a considerably large space of solutions to the WD equation which are composed of parallel propagators. However, these Wilson-loop solutions are physically of little interest because of the following reasons. First, these solutions correspond to degenerate metrics. To see this explicitly we rewrite the scalar constraint as the product form (11)

where iJo.i = lEo.bc is the magnetic field and ~: _ lEijl:Eo.&cE 6j Eel: is the 'densitized co-triad'. Because the action of S on a Wilson-loop solution 'lliwL [A~] algebraically cansels out, it follows that the densitized co-triad annihilates the Wilson-loop solution:

F:c

(12) On the other hand, the squared-volume element operator is also expressed by A0 means of e4

(13) In consequence the Wilson-loop solution has zero-volume (at least naively)

(14) Moreover, there is an argument that these solutions are 'spurious solutions' which do not enjoy (quantum version of) spacetime diffeomorphism invariance.15

231

These undesirable features of the Wilson-loop solutions suggest that physically intriguing solutions 'i'phv• [A] must satisfy the following condition

(15) We easily realize that, in serch for such solutions, the area derivative will play an essential role. However, we do not know as yet how to express the area derivative in the continuum theory without any ambiguity. So, from now on, we will investigate as a heuristic model a discretized version of Ashtekar's formalism. 3

A discretized Ashtekar's formalism

In this talk we will adopt a discretized version of Ashtekar's formalism which was proposed in ref.l6 and which was revived in ref.17. This model is defined by analogy with the Hamiltonian lattice gauge theory.18 The arena is therefore an N x N x N cubic lattice whose site will be denoted by n. We will use a to stand for three positive directions of links. The connection A~ is replaced by the link variables V (n, a) which take their values in SL(2, C) and the lattice version of the conjugate momentum ,., i4

E

1s

(16) where a) and translation operatorsf p 1(n,

p 1(n,

p 1(n,

-a) are respectively the left translation and right

' -a)· V(n- a, a) = -V(n- a, a)~ 2i.

{17)

Let us now consider a discretized version of the constraints. Among the three constraints in Ashtekar's formalism~ Gauss' law constraint is solved by considering only gauge-invariant functionals of link variables. The diffeomprhism constraint is formally solved by regarding our lattice to be a purely topological object.17 Thus we are left only with the scalar constraint (3). As a candidate for the discretized scalar constraint I propose the following operator:l 9 .

SD(n) =

2: 2: 4.

=

E k~ + const. =-{3 E k~ + const. i=l

(6)

i=l

The Calogero-Sutherland system is "almost" free in this sense. The nontriviality comes from the fact that the momentum of two particles should be at least separated by {3, which is a manifestation of fractional statistics. It is also interesting that there are alternative description by "holes" where the minimum separation is separated by 1/{3. This duality is one of the interesting features of CSM. 4

Collective Coordinate Representation

To find a correspondence between CSM and Virasoro symmetry, we introduce a collective field that describe the system in the large N limit.

239

We introduce a standard free boson, tP(z)=q+aologz- E;anz-n,

tP-(z) =

(an, am]

= nc5n+m,o,

L

;a-nzn,

n>O

n~O

(ao, q]

= l.

(7)

The boson Fock space :Fa is generated over oscillators of negative mode by the state Ia) such that (8)

(al is similarly defined, with the normalization (ala') = c5a,a'· We consider the following map from a state I/) into :Fa to a symmetric function /(z), I/)

1--+

/(z) = (alGI/}, C

= exp( J.BE ;anPn),

Pn

n>O

n j'; i;,. ,

= Ezi,

(9)

where is a parameter. Under this correspondence, 0-n and On are interpreted as the creation and annihilation operator of power-sum, v'PPn and respectively, because (aiCa-n = ..fPPn (aiC and (a ICon = a::(aiC.

j';

The correspondence between the Hilbert spaces is one to one in the large N limit. Therefore we can translate any operator which act on one Hilbert space to another which is acomplished by aplying the rule, O(aiC = (aiCO. In particular, the Hamiltonian is translated as, (10)

where

Hp =

L:

J.B(a-n-mOnam+O-nO-mOn+m)+ L:a-nan((1-,8)n+No,8).

n,m>O

n>O

(11) This is a typical tP3 Hamiltonian which appears in some string theories. To find the eigenstates of CSM now reduces to the diagonalization of this Hamiltonian. This is of course nontrivial task but the knowledge of CFT gives some crucial hints as we shal see below.

5

Virasoro Singular Vectors

A critical hint to find the Jack polynomials in the collective field language is to express the Hamiltonian (10) in terms of the Virasoro generators. Let us

240

define them as,

T(z) = ,LLnz-n- 2 = n c

~: 84>(z)84>(z): +ao82 4>(z),

= 1-12a~,

2ao =a+ +a_,

a+=VP

a_= -1/.fi.

(12)

We note that these generators satisfy the Virasoro algebra with central extension c = 1- S(l;P>2 • If we further put {J = pfq, it is identical with the central charge of minimal model. With this definition, the Hamiltonian can be rewritten as,

Hp = v'2P,L a_nLn + ,L a_nan(NofJ + {J- 1- v'2Pao). n>O

(13)

n>O

Our main idea to diagonalize the Hamiltonian is following. The cubic term which was the main obstrucle to diagonalize the Hamiltonian is now replaced by the Virasoro charges. If we apply this operator to the singular vectors of the Virasoro algebra, this term vanishes and the remaining bilinear part is already diagonalized. The explicit form of the null state is well-known. We introduce ar,• as

(14) The singular vector indexed by {r,s} has two representations, which we write,

lxt,,) =

Applying the Hamiltonian to these states, we find,

Hplx;,,) = rs ((No- r){J + s) lx;,.)

(17)

By comparing the eigenvalues, we find that the singular vector lxt,) (resp. lx;:,)) may be identified with the Jack polynomial whose Young diagram is given by a rectangle sr.

241

By using the dictionary we introduced, we can get the integral representation of the Jack polynomial in two different ways,

6

Generalization

The identification of the Jack polynomial with singular vectors of the Virasoro Algebra is simple and exact. However, what is the interpretation of eigenstate with more general Young diagram? The answer was that they were described by the null states of W algebr&Sl. This completes the correspondence between the eigenstates of Calogero-Sutherland model with the quantum algebras obtained by the Hamiltonian reduction of KP type equations. Finally let us mention that the Calogero-Sutherland model with spin degree of freedom is recently studied in the similar fashion'. References 1. H. Awata, Y.Matsuo, S.Odake, J.Shiraishi, Phys. Lett. B347 (1995) 49-55. 2. H. Awata, Y.Matsuo, S.Odake, J.Shiraishi, Nucl. Phys. B449 (1995) 347-374. 3. H. Awata, Y.Matsuo, S.Odake, J.Shiraishi, hep-th/950321. 4. H.Awata, Y. Matsuo, T.Yamamoto, hep-th/9512065.

242 BEING FASCINATED BY STRJNGS AND MEMBRANES: IS KIKKAWA-TYPE PHYSICS POSSIBLE AT OCHANOMIZU A. SUGAMOTO Department of Phr11ia, Ochanomizu Univer1it,, !-1-1, Otluka, Bunkf10-ku, Tokr~o, 11! Japan On the occasion of the 60th birthday of Professor Keiji Kikkawa, Kikkawa-type physics performed at Ochanomizu is personally reviewed, and the generation of the metric is discussed with the condensation of the string fields.

1

Personal Memories with Professor Keiji Kikkawa and the KikkawaType Physics at Ochanomizu

It is my great pleasure to contribute to the proceedings of the workshop held at Osaka to celebrate the 60th birthday of Professor Keiji Kikkawa. I am very much influenced by his physics, especially by his papers on 1) the light-cone field theory of string (Its Japanese version included in Soryushiron Kenkyu is my favorite.), 2) his lecture notes on superstrings given just before the string fever started (I think everybody should begin with this lecture note when he or she wants to do something in strings.), 3) hadronic strings with quarks at the ends, and 4) the path integral formulation of the Nambu-Jona-Lasinio model. Professor Kikkawa cited my paper on the dual transformation in gauge theories at the Tokyo conference in 1978, without which I could not have survived in our particle physics community and would definitely be engaged in another job now. Therefore I am greatly indebted to him for his guidance in physics. At Kikkawa-san's 60th birthday Conference, everybody was talking about "p-branes and duality transformations". I really thought we were timeslipping to 15-20 years ago. At that time "dual transformations, membranes and n-dimensionally extended objects (now called p-branes)" were my favorite themes. 1 2 If my paper on membranes (which was the theory of n-branes) had a little influence on the famous membrane paper by Kikkawa-san and Yamasaki-san 3 , I would be very happy. After moving to Ochanomizu University in 1987 from KEK, I have been working with my students mainly on the phenomenological problems ofnon-Kikkawa type physics, including beyond the standard model effects in the e+e- -+ w+w- process, the effect of the top condensation in B-physics, neutrino physics, CP violating models and the baryogenesis of our universe. Postdocs Yasuhiko Katsuki, Kiusau Teshima, Hirofumi Ymada, Isamu Watanabe, Mohammad Ahmady and Noriyuki Oshimo

243

did their own physics on beyond the standard model, multiple production in perturbative QCD, non-perturbative QCD, linear collider physics and the two photon process, rare decays and the heavy quark symmetry in B-physics and CP violation in SUSY and SUSY breaking, respectively, with the help of then students, Miho Marui, Kumiko Kimura, Atsuko Nitta, Azusa Yamaguchi, Fumiko Kanakubo, Tomomi Saito, Tomoko Uesugi, Tomoko Kadoyoshi, Minako Kitahara and Rika Endo. I have, however, sometimes come back to the Kikkawa-type physics of the string, membrane and gravity theories with my postdocs and my students: For example, (1) Orbifold models were firstly studied with Ikuo Senda. 6 (2) Using the light-cone gauge field theory of strings invented by Kaku and Kikkawa, we, along with Miho Marui and Ichiro Oda, have derived the AltarelliParisi like evolution equation, since the decay function of strings works naturally in this light-cone frame as has happened similarly in QCD. 7 (3) Knotting of the membrane was studied. 10 (4) With Ichiro Oda, Akika Nakamichi and Fujie Nagamori, we studied four dimensional topological gravities, mainly their quantization. 8 (5) Concerning their topological nature at high energies, the estimation ofthe membrane scattering amplitudes was performed with Sachiko Kokubo, giving an indication of the structural phase transition among the intermediate shapes of the membranes when the scattering angle is changed. 9 Other Kikkawa-type physics performed by our postdocs at Ochanomizu were; (6) Kiyoshi Shiraishi studied some 5 years ago the BPS soliton and Born-Infeld theories as well as finite temperature field theories, (7) The dilatonic gravity and black holes were investigated by Ichiro Oda and Shin'ichi Nojiri, and (8) Hybrid model of continuous and discrete theories are examined by Toshiyuki Kuruma. Pertaining to Kikkawa-type physics, recently I have become interested in (9) the generation of Einstein gravity from the topological theory 4 , (10) the swimming of microorganisms viewed from string and membrane theories, 5 and (11) phase transition dynamics viewed from field theoretical membrane theories. Issue (9) is being investigated with Miyuki Katsuki, Hiroto Kubotani and Shin'ichi Nojiri, issue (10) with Masako Kawamura and Shin'ichi Nojiri with the aid of our Barcelona friends, Sergei Odintsov and Emil Elizalde, but the last issue ( 11) is still at the stage of promoting a vague idea. In the next section I will mainly explain issue (9), and will comment on

244

my vague idea of issue (11). 2

Generation of the Einstein Gravity from the Topological 2-Form Gravity

The topological 2-form gravity is given by the following chiral action for the self-dual part:

S=

J~f~~"~P (B:1 (x)R~p(x) + «P06 (x)B;1 .(x)B~p(x))

,

(1)

conotraant term

B:

where 11 (x) is the anti-symmetric tensor field or the Kalb-Ramond field and R~p(x) is the SU(2) field strength for the SU(2) spin connection The constraint condition expressed by the Lagrange multiplier field «Pa6(x) can be solved naturally by introducing the vierbein and the t' Hooft symbol as 11 = !'11Jce! e~. Then we have the Einstein action. In the process of solving the constraint the extra Kalb-Ramond symmetry possessed by the topological ""BF" theory is broken in an ad hoc way. Instead we wish to start with the Kalb-Ramond invariant action and derive the constraint spontaneously. The Kalb-Ramond symmetry, A! A~, was originally the 11 --+ 11 + gauge symmetry of strings. Therefore, by introducing the string field, we write down the following Kalb-Ramond invariant action:

w:.

B:

B:

S =

B:

v:"

v:"

Jd4x~f~~"~P B;11 (x)R~p(x) +L: L: L: E""~P[Csc"~(x)+roB;vlc;x,xo])tifC;xo]r · C

x [

+L c

~o(EC) ~(EC)

Csc~~(x) +rB~p(C;x,xo]) ti[C;xo]]

L V[ti(C; xo]tti(C; x

(2)

0 ]].

~0

B:

In this expression we need to modify the Kalb-Ramond field 11 and its transformation to the non-local ones, reflecting the difficulty of their non-Abelian versions. Now the condensation of the string fields

(3)

245

plays the role of the Lagrange multiplier. If the condensation becomes large for the symmetric (isospin 2) part of (a, b), then its coefficient gives the constraint, leading to the Einstein gravity. For the details refer to Ref. 4. 3

Phase Transition Dynamics and Field Theory of Membranes

During the temporal development of the 1st order phase transition, like the cooling down of vapor (unbroken phase), liquid droplets of water (bubbles of the broken phase) are nucleated, they fuse with themselves, and finally the whole vessel (the whole space) is filled up with water (broken phase). It is really amazing to know that for such a difficult problem there exists a solvable theory called the Kolmogorov-Avrami theory 11 , if the critical radius of the bubble vanishes and the wall velocity is constant. "Solvable" means that we can exactly know the probability of the arbitrarily chosen N spacetime points to belong to the broken or the unbroken phase. This may suggest the existence of a solvable non-relativistic membranic (interfacial) field theory. It is another Kikkawa-type physics to pursue. References 1. A. Sugmoto, Phys. Rev. Dl9 (1979) 1820 ; K. Seo, M. Okawa, and A. Sugamoto, Phys. Rev. Dl9 (1979) 3744; K. Seo and M. Okawa, Phys. Rev. D21 (1980) 1614 ; K. Seo and A. Sugamoto, Phys. Rev. D24 (1981) 1630 . 2. A. Sugamoto, Nucl. Phys. 8215 (1983) 381. 3. K. Kikkawa and M. Yamasaki, Prog.Theor. Phys. 76 (1986) 1379. 4. M. Katsuki, H. Kubotani, A. Sugamoto and S. Nojiri, Mod. Phys. Lett. A29 (1995) 2143; M. Katsuki, S. Nojiri, and A. Sugamoto, OCHA-PP61, NDA-PP-20 (1995), to be published in Int. J. Mod. Phys. A. 5. M. Kawamura, A. Sugamoto and S. Nojiri, Mod. Phys. Lett. A9 (1994) 1159 ; S. Nojiri, M. Kawamura and A. Sugamoto, Phys. Lett. 8343 (1995) 181 ; S. Nojiri, M. Kawamura and A. Sugamoto, preprint, NDAFP-21, OCHA-PP-65 (1995), to be published in Mod. Phys. Lett.A ; E.Elizalde, S. D. Odintsov, S. Nojiri, M. Kawamura and A. Sugamoto, preprint, UB-ECM-PF 95/9-13, NDA-FP-22, OCHA-PP-66 (1995) hepth/9511167; M. Kawamura, preprint, OCHA-PP-71, hep-th/9601156. 6. I. Senda and A. Sugamoto, Nucl. Phys. 8302 (1988) 291 ; Phys. Lett. 2098 (1988) 221 ; Phys. Lett. 2118 (1988) 308 . 7. M. Marui, I. Oda and A. Sugamoto, Int. J. Mod. Phys. A5 (1990) 4257.

246

8. I. Oda and A. Sugamoto, Phys. Lett. B266, (1991) 280; A. Nakamichi, I. Oda and A. Sugamoto, Phys. Rev. D44 (1991) 3835; F. Nagamori, A. Sugamoto and I. Oda, Prog. Theor. Phys. 88 (1992) 797. 9. S. Kokubo and A. Sugamoto, Computer Phys. Comm. 75 (1993) 311 . 10. A. Sugamoto, in the "Proc. of the Trieste Conference on Supennembranes and Physics in t+1 Dimensions, M. J. Duff, C. N. Pope, and E. Sezgin (Eds.), World Scientific, pp16-28, (1990). 11. A. N. Kolmogorov, Bull. Acd. Sci. U.S.S.R. , Phys. Ser. 3 (1937) 335; M. J. Avrami, Chern. Phys. 7 (1939) 1103 ; S. Ohta, T. Ohta, and K. Kawasaki, Physica A140 (1987) 478 .

247

THE RUBAKOV-CALLAN EFFECT AND BLACK HOLES

Tab.hiro KUBOTA Department of Phy•ic•, O•alca Univer•ity, Toyonalca, O•alca, 560, Japan The Rubakov-Callan effect ia reexamined by conaidering the gravitational effect. cauaed by the heavy monopole mau. Aaauming that the Higp vacuum expectation value ia u large u (or larger than) the Planck mau, we ahow that the calculational acheme of Rubakov and Callan may be extended in the preaence of curved background field. It ia argued that the denaity of the fermion condenaate around a magnetically charged black hole ia modified in an intricate way.

1

Introduction

It was early in spring of 1983 when I moved to Osaka University that I began

to have the good fortune to work with Professor Keiji Kikkawa as one of his research associates. No sooner than I arrived at Osaka, Kikkawa suggested to me to work with him on the finite temperature effect on the monopole catalysis of the baryon number violation, then a brand-new phenomenon predicted by Rubakov 1 and Callan 3 • We started to work on this fresh idea together with H.S. Song and Ref. 3 is an outcome of our joint efforts. The Rubakov-Callan effect attracted a lot of broad attention because of its phenomenological and astrophysical importance. In my opinion, however, it is even more attractive because of the novelty and boldness shown at the cutting edge of the development of non-perturbative field theories. Here I resume the Rubakov-Callan effect, but as a theoretical laboratory with a hope of gaining an insight into gravitational effects in quantum field theories. Rubakov and Callan have shown that the fermion condensate around a magnetic monopole is long-ranged, i.e., falling off by an inverse power of the distance from the monopole. Kikkawa et al. 3 on the other hand have claimed that the condensate at finite temperature becomes short-ranged, and that the typical length scale is set by the inverse of temperature. If we consider the gravitational effect caused by the heavy monopole mass, we have another length scale, namely, the Planck length. It looks more than likely that the spatial dependence of the fermion condensate should be at variance with RubakovCallan's at this length scale. I will argue in this short note that this is in fact the case, and will give necessary calculational formulae.

248

2

Magnetically Charged Black Holes

To make things as clear as possible, we consider the conventional Einstein gravity coupled with SU(2) gauge theory which spontaneously breaks down to U(l) group by an SU(2)-triplet Higgs scalar ~~~. We are concerned with the behavior of left-handed fermion 'IIi' around a magnetic monopole in the presence of the gravitational background. The action consists of three parts

S=

- 16 ~G j

f14z..;=9R + Sboson + Sfermion 1

(1}

where the bosonic and fermionic actions are given, respectively, by Sboson =

j tl'z..;=9 { -~(F;"f' + ~ID,.~"I 2 - ~ (1~"1 2 J "' .~.i. (a 4w,. i i ,.r -

v 2) 2 },

S fermion='·

,. t a-zy-gyet'Y

I'-

(2)

mn O'mn- 2e A" ") 1 - -'11'. 'Y5 (3) 2

Here the gauge coupling is denoted by e. We will consider the static and spherically symmetric metric ds 2 B(r)dt2 - A(r)dr 2 - r 2 (d1P + sin2 6dt;2 ) (4) as a solution to the Einstein equation and and w;:'n in Eq. (3} are the vierbein and spin connections respectively associated with the above metric. The 't Hooft-Polyakov monopole in the gravitational background (4} has been considered in literatures 4 5 6 • Numerical solutions to the Einstein-YangMills-Higgs equations are also available. There are two mass scales in this system: one is the Higgs vacuum expectation value v, and the other is the Planck mass Mp. If v is as large as Mp and the Schwarzschild radius is comparable to the monopole size, then the monopole should become a black hole. Lee, Nair and Weinberg 6 classified the classical soultions, drawing a "phase diagram" according to the value of v and the monopole mass. They also discussed the relation of their solutions to the Reissner-Nordstrom solution. We are now interested in solving the monopole-fermion dynamics, considering (strong) quantum fluctuations of Yang-Mills field in the presence of the gravitational classical background (4). Following Rubakov and Callan, we will consider a particular type of quantum fluctuations around the 't HooftPolyakov monopole configuration, ~"r" = vh(r)r 1 , (5}

=

,. A"r"e" 0

e:

--

2 (r, t) r 1 , -ao e

2 (r, t) r 1 , A",. r " e"1 -- -a1 e

u(r) u(r) cotO 1 A"r"e"2 = - - r 2 , A"r"e"3 = - - r3 + - - r . (6} " er 1' er er Here ao(r,t) and a 1 (r,t} are the quantum fluctuations and u(r) and h(r} are the classical part.

249

To put our system into a solvable form, we have to restrict ourselves to a limiting case, in which the radius of the monopole is vanishingly small. More specifically we will assume that vis much larger than Mp. To take this limiting 1 and u( r) 0 outside the horizon. The Higgs and case implies h( r) gauge fields take the same configurations as those at large distance even at the horizon. The bosonic action (2} then becomes

=

Sbo•on

= ::

=

J

dtdrVA(r}B(r} r 2

8a1(r, t) _ _1_ 8ao(r, t) } 2 ( 7) at ~ ar up to an additive constant. This is an action analogous to the two-dimensional QED in the gravitational background and ao(r, t) and a 1(r, t) are the vector potentials. X {

3

1

JEW

The Fermion Dynamics

A fermion moving around the monopole comes close to the core of the monopole if it is in S-wave. From here on we will consider only the S-wave fermions which is defined, if we use the chiral basis for the gamma matrices, by

Vr sin6fiW "t'T = (O,x), 2

Xat(r, t} = cSatXl(r, t)- i(T 1}atX2(r, t). (8} Here a and l are spin and isospin indices, respectively. The S-wave part of the fermionic action is reminiscent of the two-dimensional fermion with two components if we set XT = (X1 1 X2), Srermion = -411'

J

dtdrv'A'W xf DJ=O x,

(9}

where the Dirac operator in this case becomes DJ=O

= -i (

k!

+iT2ao(r,t))

1 a - \T. 2 a1 ( r 1 t )) . (10) IAT::\yA(r) 8 r The fermionic action (9} and the bosonic one (7} are saying that our dynamical system looks similar to the Schwinger model. There is, however, an important difference, that is, the boundary condition to be imposed on the fermion field. In the case of Rubakov and Callan, the boundary condition was imposed at r = 0, so that the approximation of vanishing monopole size is justified. In our case we have assumed v > Mp and the Schwarzschild radius • 2 +\T

(

250 is greater than the radius of the monopole core. It is the most reasonable

choice of the boundary condition that the flow of the probability should cease to go inside at the horizon (located at r = rH) so that the hermiticity of the Hamiltonian is preserved. This is achieved by setting (r 1 + ir 2 )x(rH, t) = 0. (11) Now we will take the gauge fixing condition Ag 0 and will discard ao(r, t). The technique used by Rubakov and Callan to solve the monopolefermion dynamics was the bosonization method, which is also vital in our case. Let us denote the bosonized field by fP(r, t). The fermionic kinetic term is expressed by the bosonic one, and furthermore IP(r, t) is related to the gauge field via O!p

( r, t )

=

1

aa1(r, t)

yB(r)

at

JDT::\

(12)

,

where 0 is the two dimensional Laplacian associated with ds 2

= B( r )dt 2

-

A(r)dr2 • A close look at Eqs. (7) and (12) shows that the kinetic term of the gauge field may be easily expressed by IP(r, t) and we arrive at the effective action Seff

=Sbo•on + Srermion = ~

J

dtdrJA(r)B(r)1p(r, t)L,.tiP(r, t),

(13)

where 81r

L,.t = 2 J e

1

A(r)B(r)

NJ

oy'A(r)B(r)r2 o + - 0 .

(14)

1f

The number of fermion flavors is denoted by N1. Thus in the curved background as well, the two kinds of dynamical degrees of freedom due to fermionic and bosonic parts are described by a single scalar field fP(r, t). The boundary condition (11) is transcribed into 8,.1p 0 at r rH.

=

4

=

Gravitational Effects on the Fermion Condensate

We are now in a position to see how the effect of the curved background appears in particular on the fermion condensate around a magnetic monopole. We will consider two flavor fermion doublets (NJ = 2). The fermion condensate that we are concerned with is < f(r, t) >, where ) (:1)( ) (1)( ) (:1)( ) f ( r, t ) = x(1)( 1 r, t x1 r, t + x2 r, t x2 r, t .

(15)

The superscript denotes the flavor index. The dynamics described by Eq. (13) can be solved if the full Green's function (L,.t)- 1 is known exactly. The operator (14) is very much simplified if we restrict our considerations to the Reissner-Nordstrom case I i.e., A(r)B(r) = 1. We will denote this Green's function by P(r, t; r 1, t') for this particular case and 0 at the horizon r rH. According to the the boundary condition is a,. p

=

=

251

analysis by Rubakov and Callan, the density of the fermion condensate is given by < f(r, t) >- exp{2'P(r, t; r, t)}, the coincidence limit of this Green's function. It should be noticed that the Green's function P(r, t; r', t') is expressed as a combination of two other types of Green's functions defined by

DV(r, t; r', t') = 6(r- r')6(t- t'), (0

+ ~) 'R,.(r, t; r', t') = 6(r- r')6(t- t').

(16)

=

Here we used the notation " N 1 e2 /87r 2 • It is rather straightforward to obtain V(r, t; r', t') by using the method of DeWitt and Schwinger 7 • Applying their method to our case, one finds that the Green's function is expressed by the Hankel function H~2 ) (JJyC'2U) and its derivatives. Here u = u(r, t; r', t') is the two-dimensional geodesic interval between (r, t) and (r', t') and J.' is an infrared cutoff. The leading term in the DeWitt-Schwinger expansion turns out to be

!

log(-2JJ2 u(r, t; r', t')], (17) where A is the hi-scalar function defined by DeWitt. The boundary condition is made satisfied by putting mirror images at 2r H - r and 2r H - r'. The coincidence limit of the Green's function V(r, t; r, t) contributes partly to P(r, t; r, t), and it entails the Coulomb interaction between (r, t) and its mirror image. The density of the fermion condensate is described thus by the hi-scalar functions A and u. As to the other Green's function 'R,.(r, t; r', t') , the calculational method of DeWitt 7 does not apply in its original form. It is, however, easy to see that the term ~t/r 2 in (16) is playing the role of infrared regulator (or the positiondependent mass), and that there must exist a term like logr in the coincidence limit 'R,.(r, t; r, t), contributing to the fermion condensate. 5

Summary

In the present paper we have seen that the mathematical framework ofRubakov and Callan may be generalised in an analogous way to the case of the curved background. The density of the fermion condensation is deformed not simply by the local deformation of the space-time but also intricately by the mirror image. Throughout the present note the gravity has been treated classically and quantum aspects of gravity are all neglected. It would be of particular interest if we could include strong gravitational fluctuations in this framework. Gregory and Harvey 8 studied the zero-energy soultion of the Dirac equation in the presence of the magnetic monopole and gravitational background. They argued the possibility of baryon number violating scattering processes

252

of£ the magnetically charged black hole. The relation between their approach and the present one will be discussed elsewhere. Acknowledgements I would like to express my sincere thanks to Professor Keiji Kikkawa for his continuous guidance and encouragement. The present essay is dedicated to him on the occasion of his sixtieth birthday. This work is supported in part by the Grant in Aid for Scientific Research from the Ministry of Education, Science and Culture (Grant Number: 06640396). References 1. V.A. Rubakov, Nucl. Ph.y11. B 203, 311 {1982). 2. C.G. Callan, Ph.y11. Rev. D 25, 2141 {1982); Ph.y11. Rev. D 26, 2058 (1982). 3. K. Kikkawa, T. Kubota and H.S. Song, Prog. Th.eor. Ph.y11. 11, 1346 {1984). 4. F.A. Bais and R.J. Russel, Ph.y11. Rev. D 11, 2692 (1975); Y.M. Cho and P.G.O. Freund, Ph.y11. Rev. D 12, 1588 (1975). 5. P. van Nieuwenhuizen, D. Wilkinson and M.J. Perry, Ph.y11. Rev. D 13, 778 {1976). 6. K. Lee, V.P. Nair and E.J. Weinberg, Ph.y11. Rev. D 45, 2751 {1992); Ph.y11. Rev. Lett. 68, 1100 (1992). 7. B.S. DeWitt, Dynamical Theory of Group11 and Field11 (Gordon and Breach, New York, 1965); J. Schwinger, Ph.y11. Rev. 82, 664 (1951). 8. R. Gregory and J.A. Harvey, Ph.y11. Rev. D 46, 3302 (1992).

Note added: After submitting the present paper to the editor of the Proceedings, I was informed of the related work by Piljin Yi [Phys. Rev. D 49, 5295 {1994)]. He also worked out the generalization of the Rubakov-Callan 's bosonized effective action around general static spherically symmetric black holes. I would like to thank Dr. Piljin Yi for calling my attention to his paper.

253

t;3 GRAPHS ON A RANDOM SURFACE NOBUYUKI ISHIBASHI KEK Theory Group, T1ukuba, lbaralci 905, Japan We consider a di•cretised 2D quantum gravity model in which the matter degree• of freedom are expreued by a gu of networb in the •hape of the ~~ Feynmann graph8. We expect that tbi. model hu •ome feature8 in common with c > 1 noncritical•trinp. It i• .bown that 'Yot"i"IJ = ~ for thi• model, indicating that it i• a •y•tem of branched polymer..

Noncritical string theory for c > 1 has been a big mystery ever since the KPZ formula 1 was derived. The critical exponents become complex, if one naively substitutes c > 1 into the KPZ formula. This fact implies that it is impossible to take the continuum limit of the dynamically triangulated worldsheet of c > 1 noncritical strings. Many people conjecture that such triangulated worldsheets yield a system of branched polymers. Recently results of numerical simulations indicating such a behavior of c > 1 worldsheets are reported 2 • In this short note, we would like to present an analytically solvable model which captures some features of c > 1 worldsheets. In this model, the matter degrees of freedom are represented by a gas of networks in the shape of the t;3 Feynmann graphs. Technically what we will solve is equivalent to the matrix model which corresponds to 2D quantum gravity coupled to the Yang-Lee edge singularity. We will show that it yields a system of branched polymers as expected. We will start from the following observation. One of the most convenient ways to realize the c $ 1 noncritical string theory is to use the O(n) model 3 4 • The matter degrees of freedom on the worldsheet are expressed by the configuration of a gas of non-intersecting loops. Assigning the Boltzmann weight n (2 ~ n ~ -2) to each loop, one can realize all the c $ 1 conformal matter. The O(n) model (on a fixed lattice) is related to the continuum Coulomb gas picture as follows. The model can be transformed into an SOS model, so that the loops in the former is identified with the domain walls of the local heights in the latter. In the continuum limit, the local height of the SOS model turns into the scalar field in the Coulomb gas picture. Therefore, roughly speaking, the loops of the O(n) model correspond to the domain walls of the scalar field. Generically a domain wall of a scalar on a two dimensional surface is a loop. The value of the scalar field can be used as time coordinate at least locally on

254

the worldsheet 5 • Then a domain wall is considered as a component of a time slice, which is generically a loop. Hence we expect that the O(n) model can realize matter described by one scalar field in the continuum limit. What if we consider a theory involving, say, two scalar fields? The value of the two scalar fields can be considered locally as coordinate on the surface and a point corresponds to each value. If one discretizes the value of the scalars as above, a domain will be a cell-like region in the two dimensional space and the domain walls will form networks rather than disconnected set of loops. Generally we expect that the domain walls intersect each other when c exceeds 1. Therefore we do not expect the 0( n) model to express c > 1 matter. Our expectation is consistent with the fact that one can obtain only c ~ 1 matter by changing n. If one wants to obtain a generalization of the 0( n) model for c > 11 a natural guess would be a model where the matter degrees of freedom is expressed by a gas of networks. In this note we will consider a simple example of such a model. We are not sure if our model really corresponds to c > 1 matter but we believe that at least it captures some features of c > 1 noncritical string theory. A gas of networks on a random surface can be realized by a simple modification of the 0( n) model on a random surface 4 • The matrix model action corresponding to the O(n) model on a random surface is 1 2 - -A 91 a + "'(1 It =NT" [-A L- -B,2 - 92AB12)] • 2

3

.

2

•

(1)

Here A and B, (i = 1 · · · n) are N x N hermitian matrices. A integration generates ,pa graphs, dual of which are identified with the dynamical triangulations of surfaces. In the following, we take the large N limit so that we restrict ourselves to the surfaces with topology of sphere. B, integrations generate the gas of loops on the surface. Because there are n matrices B,, the factor n#

of loops

(2)

I

comes into the Boltzmann weight. In order to generate networks instead of loops, all one should do is to add interaction terms of B, 's to the action. Here we will consider the simplest case n = 1 and add only a three point interaction term: 1 2 - -9tAa + -1B2 - 92 AB2 - -9aBa] · (3) I2 = NT" [-A 2

3

2

3

In this case, !J integration generates networks in the shape of the ;a Feynmann graphs on the dynamically triangulated surface. We would like to look for the

255 critical value of the coupling constant 93 at which the critical behavior of the surface changes. For n = 1, the matrix model in eq.(l) is equivalent to the two matrix model describing the Ising model on a random surface 6 • Indeed by the change of variables (4) M1 A+ B, M2 A- B,

=

=

!

one can recover the usual form of the two matrix chain model. Near the c = critical point, Tr B 3 corresponds to the dressed spin operator in the continuum limit 4. Therefore turning on a small 93 is equivalent to turning on the magnetic field. One can show this fact directly by changing the variables as

(5)

M1 =aA+{3B, M2 =-yA+6B,

so that the three matrix interaction terms in I2 be of the form 9eH Mf + M?. Here Hex 93 for small 93· Therefore, obviously there is a critical value of 93 corresponding to the Yang-Lee edge singularity, where the critical behavior of the surface changes. What happens to the networks generated by B integration, at this imaginary value of 93 ? As we will show shortly, the partition function becomes nonanalytic with respect to 9 3 at this point. Hence this critical point is the point where the networks become very fine in the surface. If one considers the networks as domain walls of scalar fields, this behavior is exactly what we want in the continuum limit. Now let us derive the partition function of the model. The string equation corresponding to the Ising model in the magnetic field is 1 9e-H

1f.2

t=13-r·

(6)

Here t is the cosmological constant and I is the second derivative of the partition function by t. 1i denotes the continuum magnetic field and H const. x 1ia5 16 , where a is the lattice spacing. Near the critical value 1i~ = - 3~!;~3 t 5 13 , I behaves as

=

(7) Therefore the partition function becomes nonanalytic with respect to 93 at the critical point. The first term on the right hand side of eq.(7) is the contribution of the surfaces on which the network is not fine. If one discards this term as a "nonuniversal term" as we do usually in the matrix model, one can see from the second term that 'Y•tring = This is exactly the 'Y•tring for a system of branched polymers.

!·

256

We have shown that the gas of networks on a random surface becomes a system of branched polymers. Although we restrict ourselves to the simplest case of n = 1, it will be possible to solve more general cases. The scaling dimension of the operator corresponding to E Tr Bf' was obtained for general n 3 • Therefore it may be possible to identify such an operator with a known operator in 2D quantum gravity and obtain the partition function when the system is perturbed by using such an operator. Then we may be able to explore the examples of similar critical behavior. Acknowledgments It is my pleasure to thank Prof. Kikkawa for his great influence on me as a physicist. He has been one of the role models for the physicists in my generation. We learned many things from his lectures, his talks and especially questions asked by him.

References 1. V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phy•. Lett. A3{1988)819; F. David, Mod. Phy•. Lett. A3(1988)1651; J. Distler and H. Kawai, Nucl. Phy•. B321(1989)509. 2. H. Kawai, N. Tsuda and T. Yukawa, Phy•. Lett. B351(1995)162. H. Kawai, N. Tsuda and T. Yukawa, in this volume. 3. B. Nienhuis, J. Sta.t. Phy•. 34(1984)731; B. Duplantier, J. Phy•. A19(1986)L1009; B. Duplantier and H. Saleur, Nucl. Phy•. B290[FS20]{1987)291; Phy•. Rev. Lett. 57(1986)3179; H. Saleur, Phy•. Rev. B35(1987)3657. 4. B. Duplantier and I.K. Kostov, Nucl. Phy•. B340(1990)491. 5. N. Ishibashi and H. Kawai, Phy•. Lett. B322(1994)67. 6. V. Kazakov, Phy•. Lett. A119(1986)140; D. Boulatov and V. Kazakov, Phy•. Lett. B186(1987)379. 7. E. Brezin, M. Douglas, V. Kazakov and S. Shenker, Phy•. Lett. B237 (1990)43; C. Crnkovic, P. Ginsparg and G. Moore, Phy•. Lett. B237(1990)196; D. Gross and A.A. Migdal, Phy•. Rev. Lett. 64(1990)717.

257

Wave Packet in Quantum Cosmology and Semiclassical Time Y.OHKUWA Department of Mathematics, Miyazaki Medical College, Kiyotake, Miyazaki 889-16, Japan T. KITAZOE Faculty of Engineering, Miyazaki University, Miyazaki 889-21, Japan We consider a quantum cosmology with a massless background scalar field 4Ju and adopt a wave packet as the wave function. This wave packet is a summation of the WKB form wave functions, each of which has a definite momentum of the scalar field. In this model we calculate the time variable TE explicitly, which gives rise to the Ehrenfest equation, and compare it to another time variable Tw , which is derived by the WKB approximation. It is shown that TE is a smooth extension of Tw and they become identical in the narrow limit of the wave packet.

1

Introduction

The notion of time is one of the most serious problems in quantum cosmology.PI Though this is still controversial, many attempts have been done recently. One of them is to utilize the semiclassical approximation. Danks and others assumed that the solution to the Wheeler-DeWitt equation has the form of the WKB approximation, namely 'llwKB = 41et 8 o , where So is the Hamilton's principal function, and they introduced a time variable Tw , using So .!21-11 31 They showed that , when a quantum matter field (/>Q is coupled to the system, its wave function satisfies the Schrooinger equation with respect to T w in the region where the semiclassical approximation is well justified. However, this formalism crucially depends on the assumption that the solution has the WKB form. Grcensite and Padmanabhan extended this notion and introduced a time variable TE , using the phase of an arbitrary solution to the Wheeler-DeWitt equation .! 1411151 They obtained TE by requiring that the Ehrenfest principle holds with respect to this time variable. It has not yet been Clarified how the classical universe emerged from the quantum universe. However, it seems probable that the wave function of the universe forms a narrow wave packet in the classical region. In this note we will consider a quantum cosmology with a massless background S 8 • In this model the time variable Tf; will be calculated explicitly and compared to T w .

258

2 Wave Packet and Time We consider the following minisuperspace model in (n+1)-dimcnsional spacetime .1161 Though n = 3 in reality, we calculate in the more general case. The metric is assumed to be ds 2 = -N2(t)dt2 + a2 (t)d0~ , where dO~ is the flat metric. We take a massless background scalar field 8 (t) . The WheelerDeWitt equation for a wave function llf(a, 8 ) reads

'H.s

= =

Cn

=

'H.s l{f

(1)

0'

rt2

2Vnan-2

~

02 ) ~ + U ( a )

( - - aa 1 P o oa_ - -

aPoo

1611'G

2n(n -1) '

aa

aa

U(a)

a2

1

= Vn 1 !~0 an .

Here Poo is a parameter of operator ordering, A is a cosmological constant, Vn is the spatial volume, and we assume that Vn is some properly fixed finite constant. The WKB approximation to this model was investigated in Ref. [16], and the results are as follows.

So

= =

Is

=

4>o

= =

lJiwKB

(2) (3)

('y = 0) ,

(4)

(5) l[x]

(11: = 0)'

(6) (7)

lwt

259

where fa= ±1,eu = 4v~A/16?rG, rw(4> 8 ) is any function of 4> 8 and K.,-y,e,c,p are arbitrary constants. We can identify K. as· the momentum of 4> 8 • Now let us adopt the following Gaussian wave packet as the wave function: 'II A(~t)

=

Jd~tA(~t)WwKB(K),

=

..fi;u exp -

1

[

(8)

(~t- ~to) 2 ] 2u2

'

where K.o and u are arbitrary constants. Note that, in the limit u -+ 0 , 'II becomes identical to WwKB . We choose Poo = 1 , 'Y = 0 and define cp = -~ln(eua2n + ~t 2 ) . If we assume that A(~t) has a narrow peak at K.o , namely u is small, then So and cp can be expanded around ~to . We neglect the higher terms than (~t- ~t0 ) 2 and integrate with respect to K. , and we obtain

_I

1

1

(

u2 12

'II~ C,pv 1 - u 2,,exp I+ 2(1 - u 2/") where f = s 8Q>s

(14)

260

As proved in Refs. [14],[15], the Ehrenfest principle holds with this time variable TE:

where 0 is an arbitrary observable, g is the determinant of the Wh 0 . We find from Eqs. (6),(7) and (16),(17) that, when 11: in Eqs. (6),(7) is replaced by Ko, ~ = ( , Tw = TE and a -> 0, TE becomes equal to T w , which may be expected from Eqs. (8) .

-a

3

Ja

Summary

We considered a wave packet in quantum cosmology that is a summation of the WKB wave functions, each of which has a definite momentum of the back ground scalar field. We calculated the time variable TE explicitly, which gives rise to the Ehrenfest equation, and compared it to another time variable T w , which is derived by the WKB approximation. It has been shown that Tf: is a smooth extension of T w and they become identical in the narrow limit of the wave packet.

261

Acknowledgments The authors would like to congratulate the 60th anniversary of Prof. K. Kikkawa. One of the authors (Y.O.) would like to thank his group for hospitality and would also like to thank Nishina Memorial Foundation and Inoue Foundations for Science for financial supports.

Note added Very recently after finishing this work we received Ref. [17]. According to it, a solution to Eq. (14) is not always an Ehrcnfest time. Our Tf; can be understood as a candidate for an Ehrenfest time in the sense that Te is a smooth extension of T w.

References 1. See, e.g., C.J. Isham, in Integrable Systems, Quantum Groups, and Quantum Field Theories, eds. L.A. Ibort and M.A. Rodriguez (Kluwcr, London, 1993); K.V. Kuchar, in Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, eds. G. Kunstattcr, D.E. Vincent and J.G. Williams (World Scientific, Singapore, 1992). 2. T. Banks, Nucl. Phys. B 249, 332 (1985). 3. J.J. Halliwell, Phys. Rev. D 36, 3626 (1987); J.J. Halliwell, in Quantum Cosmology and Baby Universes, eds. S. Coleman, J.B. Hartle, T. Piran and S. Weinberg (World Scientific, Singapore, 1991). 4. J.J. Halliwell and S.W. Hawking, Phys. Rev. D 31, 1777 (1985). 5. S. Wada, Nucl. Phys. B 276, 729 (1986) ; erratum, ibid. B 284, 747 (1987). 6. J.B. Hartle, in Gravitation in Astrophysics, eds. B. Carter and J.D. Hartle (Plenum, New York, 1986). 7. A. Vilenkin, Phys. Rev. D 39, 1116 (1989). 8. T.P. Singh and T. Padmanabhan, Ann. Phys. (N.Y.) 196, 296 (1989). 9. T. Padmanabhan, Class. Quantum Gravity 6, 533 (1989). 10. C. Kiefer and T.P. Singh, Phys. Rev. D 44, 1067 (1991); C. Kiefer, in Canonical Gravity -from Classical to Quantum, eds. J. Ehlers and H. Friedrich (Springer, Berlin, 1994). 11. Y. Ohkuwa, in Evolution of the Universe and its Ob.~ervational Quest, ed. by K. Sato (Universal Academy Press, Tokyo, 1994); Nuovo Cimento UOB, 53 (1995). 12. Y. Ohkuwa, Int. J. Mod. Phys. A 10, 1905 (1995).

262

13. Y. Ohkuwa, T. Kitazoe and Y. Mizumoto, Int. J. Mod. Phys. A 10, 2317 (1995). 14. J. Grecnsite, Nucl. Phys. B 351, 749 (1991). 15. T. Padmanabhan, Promana Jour. Phys. 35, 1199 (1990). 16. Y. Ohkuwa, preprint MMC-M-7 (1995). 17. T. Brotz and C. Kiefer, preprint Freiburg THEP-96/1 (1996).

Part III Supersymmetry

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EXPLORING GAUGE THEORIES WITH SUPERSYMMETRY KENNETH INTRILIGATOR ln1titute for Advancecl Study, Princeton, N J 08540, USA We review 110me of the recent work on the d:yD&mica of four dimenaicmal, •uper•ymmetric gause theorie..

1

Introduction

Recently, it has become clear that certain aspects of four dimensional supersymmetric field theories can be analyzed exactly, providing a laboratory for the analysis of the dynamics of gauge theories {for a more complete presentation on which this review is largely baaed and a list of references, see Ref.l). For example, the phases of gauge theories and the mechanisms for phase transitions can be explored in this context. The dynamical mechanisms explored are standard to gauge theories and thus, at least at a qualitative level, the insights obtained are expected to also be applicable for non-supersymmetric theories. We summarise some of the recent ideas. The basic approach will be to consider the low energy effective action for the light fields, integrating out degrees of freedom above some scale. Aasuming that we are working above the scale of possible supersymmetry breaking, the effective action will have a linearly realized supersymmetry which can be made manifest by working in terms of superfields. The light matter fields can be combined into chiral superfields xp = ~p + 9a'f/J: + ..., where the ~p are scalars and the are Weyl fermions. In addition, there are the conjugate anti-chiral superfields X) = ~t + Bc.'f/Jta + .... Similarly, light gauge fields combine into aupermultiplets involving a gauge boson A" and gauginoa ~a and ~l. We will focus on two particular contributions to the effective Lagrangian - the superpotential term

,p:

f

d28 Wetr(Xp,g,,A),

(1)

and, when there is an Abelian Coulomb phase, the gauge kinetic term

f

d29lm[Tetr(Xp,gJ,A)W!J.

{2)

Upon doing the 9 integral, the superpotential yields a potential for the scalars and a Yukawa type interaction with the scalars and the fermion&. W! gives

266

the auperaymmetric completion of F 2

+ iFF so

9etr

Tetf""

41ri

+g!tr 211"

(3)

is the effective gauge coupling constant. The auperpotential and the effective gauge coupling constant are functions of the light chiral auperfielda Xr, various coupling constants g1, and A, the dimensional transmutation scale associated with the gauge dynamics, - ,'!{;) - log A/I'· We will think of all the coupling constants 91 in the tree level auperpotential Wtree and the scale A as background fields 2 • Then, the quantum, effective auperpotential, Wetr(Xr,gi,A) is constrained by: 1. Symmetries and selection rules: By assigning transformations laws both to

the fields and to the coupling constants, the theory has a large symmetry. The effective Lagrangian should be invariant under it. 2. Holomorphy: Wetr is independent of gJ2. This is the key property. Just as the auperpotential is holomorphic in the fields, it is also holomorphic in the coupling constants (the background fields). This is unlike the effective Lagrangian in non-auperaymmetric theories, which is not subject to any holomorphy restrictions. This use of holomorphy extends considerations of3 •4•5• It is similar in spirit to the proofs ofnon-renormalization in sigma model perturbation theory 6 and in semi-classical perturbation theory 7 in string theory. 3. Various limits: Weff can be analyzed approximately at weak coupling. The singularities have physical meaning and can be controlled. Often these conditions completely determine Wetr. The point is that a holomorphic function (more precisely, a section) is determined by ita asymptotic behavior and singularities. When there is an Abelian Coulomb phase, TeJ J is similarly constrained by the same considerations and can often be exactly determined. The results can be highly non-trivial, revealing interesting non-perturbative dynamics. Different examples exhibit a variety of phenomena, for example: • Smooth "moduli spaces" of physically inequivalent vacua. This is in contrast to the lore that accidental classical degeneracies, not protected by any symmetry, would be lifted by quantum effects. This phenomenon is perhaps special to auperaymmetric theories.

267

• Mtu1le11 mesons and baryons. This is in contrast to the lore that baryons are large and heavy bound states which do not enter in the low energy dynamics.

• Confinement without chiral symmetry breaking. This is in contrast to the lore that confinement leads to chiral symmetry breaking because a chiral symmetry is like an arrow which must be reflected when put in a bag. This phenomenon is perhaps special to theories with charged scalars. eMtu1le11 magnetic monopoles and dyons. This is in contrast to the lore that these are large and heavy collective excitations, not entering in the low energy dynamics. •Mtu1le11 quarks and gluons in an interacting non-Abelian Coulomb ph.tue. This is in contrast to the lore that non-Abelian gauge theories get a mass gap via confinement or the Higgs mechanism and that there are no interacting, four dimensional, fixed points of the renormalisation group. •Compo1ite "magnetic" gauge invariance with compo1ite gauge field. and quaru." This is in contrast to the lore that gauge invariance is fundamental.

By examining these theories we can thus teat and revise the lore about quantum field theory. In many cases dualitr governs the dynamics. Here "duality" simply means that strong coupling phenomena can often be described in terms of a dual set of (possibly weakly coupled) variables. This is common in condensed matter: for example, the Kramers Wannier duality of the Ising model, bosonisation, and various other contexts in which strongly interacting electrons lead to a set of collective excitations which are the more sensible variables for describing the system. As we now review, there are several notions of duality in gauge theory, associated with the various different phases of gauge theories. The phases of gauge theories can be characterized by the potential V (R) between electric test charges separated by a large distance R. We can also consider the potential V(R) between magnetic test charges (or the ends of test solenoids) separated by a large distance R. One phase is the Coulomb phase, with m_!88leas gauge fields leading to longrange potentials V(R) e2 I R- 11Rand V(R) g2 I R - 11 R. The electric and magnetic coupling constants satisfy the Dirac condition eg = 211'h. NonAbelian as well as Abelian theories can have a Coulomb phase. Because there are massless, interacting, non-Abelian gluons, this requires an interacting, four

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dimensional, scale invariant theory. It was pointed out in Ref.[9] that N = 0 (i.e. non-supersymmetric} SU(Nc} QCD with Nt flavors of Nc + N c should have a non-Abelian Coulomb phase fixed point for Nt = ~1 Nc -1. The theory is just barely asymptotically free, with the one loop beta function negative and the two loop beta function positive. In the large Nc limit, fJ has a zero, which is an infrared fixed point, for coupling e~ ,... 1/CNc, where Cis some large constant. This fixed point coupling appears to be sufficiently small so that we can trust perturbation theory and the argument for the fixed point is consistent. The theory with e = e. is a scale invariant theory of massless quarks and gluons with Coulomb interactions and is an attractive fixed point of the infrared renormalization group flow. Note that with fewer numbers of massless flavors the theory is more asymptotically free, the fixed point coupling e. is larger, so we can not be as confident about if there is a fixed point and how it should behave. There are N = 1 analogs of the fixed points described above - for example N = 1 SUSY SU(Nc) QCD with N1 just below 3Nc flavors. Although, as mentioned above, we can not be so confident in the non-supersymmetric case about what happens for fewer flavors, the infrared behavior of SUSY QCD was described for all Nc and Nt in Re£.[10]. We will discuBB other examples below. There are also SUSY theories which have a non-Abelian Coulomb phase for arbitrary coupling e. Examples are N = 4 SUSY Yang-Mills and N = 2 SUSY QCD with Nt = 2Nc. Another phase of gauge theories is the free electric phase, where there are massless, electrically charged fields which renormalize the electric charge to zero at long distances as e- 2 (R) ,... log(RA), leading to a potential V(R) ,... 1/Rlog(RA). Because of the Landau pole for R :$ A- 1 , this phase only makes sense as a low energy effective theory; it needs to be regulated by a bigger theory in the UV. Again, the free electric phase can be non-Abelian as well as Abelian. Examples are non-Abelian gauge theories which are not asymptotically free and are thus free in the infrared. The potential between magnetic test charges is V(R) = g2 (R)/ R where, by the Dirac condition, e(R)g(R) ,... 1. In the free electric phase the magnetic coupling g(R) is thus renormalized as g2 (R) -log(RA) to be infinitely large at large distances. Giving the electrically charged fields a small mass, m, leads to free electric behavior for distances smaller than m- 1 and Coulomb behavior with e 2 ,... 1/log{A/m) and g2 -log(A/m) for distances large compared with m- 1 • This is, of course, the situation for ordinary electrodynamics, which is in the Coulomb phase for R > 1/malactron· Another phase is the free magnetic phase, where massless magnetically charged fields renormalize the magnetic charge to zero at long distances as

269

g2 (R) ""' 1/ log(RA). The Dirac condition then implies that electric charge is renormalised to infinity at large distances as e2 (R) -log(RA). Another phase in the Biggs phase, where the condensate of an electrically charged field gives a mass gap to the gauge fields by the Anderson-Biggs mechanism and screens electric charges, leading to a potential which, up to an additive non-universal constant, has an exponential Yukawa decay to sero at long distances. As in a superconductor, magnetic flux is confined to a thin tube, leading to a linear potential for magnetic teat charges, V(R)""' pR, where p is the MeiBBner effect string tension. Another pOBBible phase is the confining phase, where there is a mass gap with electric flux confined to a thin tube, leading to a potential between electric test charges of V(R) ""' uR, where u is the string tension. The potential between magnetic teat charges could behave as V(R) ""'conn., which would be consistent with a monopole condensate. The above behavior is modified when there are dynamical matter fields in the fundamental representation of the gauge group because virtual pairs can be popped from the vacuum and completely screen the sources. Indeed, in this situation there is no invariant distinction between the Biggs and the confining phases 11 • In particular, there is no phase with a potential behaving as the "confining" potential V ""' u R at large distances - the flux tube can break. For large expectation values of the fields, a Higgs description is most natural while, for small expectation values, it is more natural to interpret the theory as "confining." Because there is really no distinction, it is poBBible to smoothly interpolate from one interpretation to the other. Generalizing the Higgs and confinement phases are the (p, q) oblique confining phases introduced by 't Hooft. Consider dyonic teat charges n± with magnetic and electric teat charges ±(nm, n.). In the (p, q) oblique confinement phase, the flux between these teat charges is confined unle. (nm, n.) ""' (p, q) where, as discuBBed above, the similarity is up to the part of the gauge group which can not be screened by the dynamical fields. So V(n_,n.) ""' U(n-,n.)R for (nm,n.) -f.- (p,q). The potential between dyons with (nm,n.) - (p,q) could behave as V(p,g) ""' conri, which would be consistent with (p, q) dyon (0, 1) is the Higgs phase and (p, q) (1, 0) condensation. The case (p, q) is the confinement phase. There could be other, physically distinct, oblique confinement phases with different order parameters. A concrete realisation of this will be discussed below. Electric-magnetic duality, which exchanges electrically charged fields with magnetically charged fields, exchanges the behavior in the free electric phase with that of the free magnetic phase. Mandelstam and 'tBooft suggested that, similarly, the Biggs and confining phases are exchanged by duality. Confine-

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ment can then be understood as the dual MeiBBner effect associated with a condensate of monopoles. As we will review, in supersymmetric theories it is pOBBible to show that this picture is indeed correct. Dualizing a theory in the Coulomb phase, we remain in the same phase (the behavior of the potential is unchanged). For an Abelian Coulomb phase with free maBSleBB photons, this follows from a standard duality transformation. What is not obvious is that this is also the case in a non-Abelian Coulomb phase. This was first suggested by Montonen and Olive 12 • The simplest version of their proposal is true only in N = 4 supersymmetric field theories 13 and in finite N = 2 supersymmetric theories 14•15•16 • In the N = 4 case, the statement is that an "electric" theory with gauge group G and couplinJ e is pArrice&lly mtlutiaguuAable from a "magnetic" theory with gauge group G and coupling 1/e. Here G is the "Langland's dual" ofG, which was discuBSed some time agcl- 2 • For example SU(N) +-+ SU(N)/ZN and S0(2r + 1) +-+ SP(r). So the G theory at strong coupling behaves as a weakly coupled G theory, with the G degrees of freedom coming from collective excitations (-solitons) of the G theory. TheN 2 theory with NJ 2Ne is thought to be dual to itself, withe +-+1/e. Seiberg extended these ideas to asymptotically free N = 1 theories 10 • Here there are two cases which should be distinguished. In one case, two (or more) different theories have the same, interacting, infrared fixed point. This is quantum equivalence, a phenomenon which is familiar in two dimensional quantum field theories. The non-Abelian Coulomb phase fixed point is a four dimensional, interacting, N = 1 superconformal field theory which can be described in terms of any of the dual theories. Effects which can be seen at weak coupling in one description will often arise as strong coupling effects in the dual description(s). As in the N = 4 and finite N = 2 cases discuBBed above, the duality is an exact statement about the theories at the fixed point values of their couplings. Starting away from the fixed point, for example at small coupling, it is a statement about the infrared behavior of the different dual theories. The other case of Seiberg's duality is that an asymptotically free "electric" ultraviolet theory can be free in the infrared with a low energy spectrum consisting of collective excitations (""' solitons) behaving as the gauge fields and matter of a different, "magnetic", non-Abelian gauge theory which is not asympt~tically free. The low energy theory is an a aoa-Abeliaa free magaetic pAue. This is "effective duality:" the magnetic theory is the infrared solution of the electric theory. The gauge invariance of the magnetic theory is comporite. A rich daBS of examples are N = 1 SUSY SO(Ne) gauge theories with NJ

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271

matter fields Qi E Ne 1 i = 1 ... Nt. For various values of Ne and Nt these theories exhibit 10•17 • Genuine confinement. There ia genuine confinement for teat charges in the

apinor representation of the gauge group (we do not distinguish between SO(Ne) and Spin(Ne)), which can not be screened by the dynamical fields. • Oblique confinement. This ia the first realisation of oblique confinement as

a new phase, physically distinct from confinement. •Confinement without chiralaymmetry breaking. eMasslesa composites such as glueballa, exotica, monopoles, and dyona. •Non-Abelian Coulomb and free magnetic phases with dual descriptions. At the classical level, these theories have a "moduli apace," Melt of degenerate vacua with different (Q). In the bulk of this apace, (Q) breaks SO(Ne) to SO(Ne - Nt) for Nt ~ Ne - 2 and it breaks SO(Ne) completely for N1 ~ Ne- 2. The mass of the Higgaed gauge fields are m - (Q), so the different vacua in Mel are physically inequivalent. The degeneracy implies that there are masslesa "moduli" matter fields, which can be taken to be the "mesons" Mi; = Qi · q;. The expectation values of the Mi; (subject to the classical relation rank(M) ~ Ne) provide gauge invariant coordinates for the apace Mel· There are singular aubspaces of the moduli apace with enhanced unbroken gauge symmetry. In the quantum theory, asymptotic freedom implies that the gauge coupling has strength g2 ((M))- 1/log(M/A2 ) and thus the gauge dynamics are weakly coupled for vacua with large (M) and strongly coupled for vacua near the origin of the moduli space. Therefore, quantum effects could lift the classical degeneracy and/or change the shape and the singularities near the origin. Consider first Nt = Ne- 2. In the bulk of Mel the gauge group ia broken to 50(2) ~ U(l), so there is a masslesa photon supermultiplet. In the quantum theory it follows from holomorphy, symmetries, and various limits that Wecact(M) = 0- i.e. there ia a quantum moduli apace of physically inequivalent vacua. These same considerations imply that the Abelian Coulomb phase has an effective gauge coupling T'etr(det M, A) which, much as in Ref. 18, ia exactly given 'l;ly the curve 17

(4) For example, at weak coupling (large det(M)), (4) properly reproduces the one loop beta function of SO(Ne) with Nt fields Q.

272

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=

The curve (4) is singular for det M 0 and det M 16A3Nc- 4 , Classically, the submanifold det M 0 has a singularity associated with a non-Abelian Coulomb phase with some of the SO(Nc) gluons becoming massless. However (4) implies that, near det M = 0,.,. behaves as -1/-r s=::: ; ; logdet M, or e!JJ ,... -log det M, which reveals the presence of magnetic monopoles, with mass matrix M'i. So the submanifold det M 0 is actually in an Abelian free magnetic phase associated with massless monopoles. At the origin there are N1 massless monopoles qf. Away from the origin, they obtain a mass matrix proportional to (M) via (5) W ,... M ;.; q,+q;.

=

=

=

Similarly, the behavior of "•II (M, A) near the singularity at det M 16A3Nc- 4 reveals that this submanifold is in a free dyonic phase, associated with a single dyonically charged field, E, which is massless at det M = 16A3Hc- 4 ; near the singularity the dyon gets a mass via

(6)

=

Having described the theory with N1 Nc - 2, it is possible to obtain results for N1 < Nc- 2 by adding mass terms for some Q flavors and flowing to the low energy theory with fewer flavors. For example, consider giving one flavor, QNJ, a mass m and flowing to NJ Nc- 3. The equations of motion imply that there are two inequivalent branches with a dual Higgs mechanism: One branch is at det M 0, where the monopoles are massless, and a monopole condensate (qj,1 q"N1 ) ,... m forms, leading to confinement by a dual Meissner effect. The condensate also breaks the chiral SU(NJ) flavor symmetry to SU(N,-1). On this branch there is a moduli space of degenerate vacua with additional massless fields q, at the origin: W ,... M'i q,q;. These fields arise from components of the monopoles qf of the high energy and are baryonic exotics of the low energy theory. Having baryons arise as monopole is reminiscent of the idea of having baryons arise as solitons in the pion Lagrangian 19 • There is confinement without chiralsymmetry breaking at (M1i) 0. The other branch of vacua occur at det M 16A3 Nc- 3 in the high energy theory, where the dyon is massless, and a dyon condensate forms, (D+ n-) ,... m, leading to oblique confinement. On this branch a superpotential is generated, lifting the moduli space of vacua. Because this branch is physically distinct from the previous branch, we have a realisation of confinement and oblique confinement as two distinct phases. Note that the transition from the phases of NJ Nc- 2 massless flavors, namely Coulomb, free magnetic, and free dyonic, to those of N1 Nc - 3 massless flavors, namely confining and oblique confining, is a second order

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phase transition: (q+q-) provides an order parameter for confinement and (D+ v-) is an order parameter for oblique confinement. For NJ > Nc- 2 the dynamics is governed by duality. The infra-red behavior of these theories has a dual, magnetic description 10•11 in terms of an SO(NJ- Nc + 4) gauge theory with N1 flavors of dual quarks 9i and an additional gauge singlet field Mi; with a superpotential

(6.1) (an additional term is required for NJ = Nc - 1). For ~(Nc - 2) < NJ < 3(Nc- 2), both the electric and magnetic theories are asymptotically free and flow to the same non-Abelian Coulomb phase fixed point in the infrared. At the fixed point, the two theories are indistinguishable. For example, the composite operator Mij = Q' · qi of the electric theory and the elementary field Mi; of the dual, though very different at weak coupling, become the same operator at the fixed point. For Nc- 2 < NJ ~ ~(Nc- 2), the magnetic theory is not asymptotically free. For this range of N,, the electric theory thus flows to a free magnetic phase in the infrared with a dynamically generated, composite SO( NJ - Nc +4) gauge invariance. The magnetic dual is the infrared solution of the original electric theory, with the magnetic gauge fields and matter arising as some sort of collective excitations(- solitons) ofthe original SO(Nc) theory. A special case of the duality is for NJ Nc - 2, where the dual is a "magnetic" 50(2) gauge theory with NJ charged fields qf with W - Mi;qtqj. As discUBBed above, for NJ = Nc- 2 there is a moduli space of vacua with SO(Nc) everywhere broken to 50(2). The "electric" 50(2) discuBSed earlier and the "magnetic" s0{2) are related by the standard, Abelian, electricmagnetic duality, justif~g the name "magnetic" for the dual theory. Indeed, the elementary fields qi of the magnetic theory are the NJ 't Hooft-Polyakov magnetic monopoles of the electric theory, which we found earlier from the exact expression for the photon gauge coupling. This duality satisfies a number of other non-trivial checks. For example both the electric and magnetic theories have the same global SU(NJ) x U(1)R symmetries and their 't Hooft anomalies match. To get a feeling for the nontrivial nature of this matching, consider, for example, the U(1)~ anomaly. In the electric theory there are the NJNc quarks '1/JQ with U(1)R charge -(Nc2)/NJ and !Nc(Nc -1) gauginos with U(1)R charge 1. In the magnetic theory there are NtNc quarks '1/11 with U(1)R charge -(Nc- 2)/N/t ·!NJ(NJ + 1) fermions '1/JM with charge 1- 2(Nc- 2)/NJ, and !Nc(Nc- 1) gauginos with U(1)R charge 1, where He E N1 - Nc + 4. These numbers are completely

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determined by the condition that U(1)R is anomaly free and respected by the dual superpotential. There is already a non-trivial check, which is that the charge of M'i in the magnetic theory, as determined above, agrees with that determined by the map M'i = Q' · Qi. The 't Hooft anomaly condition from U(1)~ is

which, with some work, is indeed found to be true! Another check is that the duality is preserved under the renormalillation group flow associated with various deformations. For example, adding a mass term on the electric side takes the theory to SO(Nc) with NJ - 1 flavors in the infrar~. The infrared theory is at stronger coupling. On the magnetic side, SO(Nc) is Higgsed to SO(Nc -1) with one flavor eaten. The low energy magnetic theory is at weaker coupling and is indeed the dual of the low energy electric theory. Duality is preserved under the RG flow with strong coupling effects exchanged with weak coupling effects. Starting from NJ just below 3(Nc- 2), which is the limit for asymptotic freedom, and adding mass terms to decouple flavors one by one, the theory flows to stronger and stronger coupling, undergoing several phase transitions. These strong coupling phase transitions of the electric theory are seen at weak coupling in the magnetic dual, where they can all be understood as the Higgs mechanism. The electric transition from non-Abelian Coulomb phase to free magnetic phase simply occurs because the dual theory is Higgsed to a low energy theory which is no longer asymptotically free. Similarly, the transition from free magnetic to confining simply corresponds to the dual gauge group becoming completely broken by the Higgs mechanism. Conversely, some obvious properties of the electric theory arise from strong coupling effects in the dual. For example, moving away from (M} = 0 Higgses the electric theory, yielding an infrared theory which is at weaker coupling. In the dual theory, (M} gives a mass to some of the dual quarks, giving a theory which is indeed dual to the low energy electric theory and at stronger coupling. Again, duality is preserved under the RG flow, with strong coupling effects exchanged with weak coupling effects. Taking strong coupling effects Mmagnetic in the dual theory into account, it is seen that M.rectric {(M'i)lrank(M} ~ Nc}· While an obvious classical consequence of M'i = Q' · Qi in the electric description, the constraint rank(M} ~ Nc arises only as

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275 a consequence of non-perturbative effects in the dual. (There is an additional Z2 modulus for rank( (M}) = Ne which, on the electric side, is seen claasically as a baryonic sign. In the magnetic description, this Z2 is reproduced from the sign of a gaugino condensate in the dual group.) There is a growing list of dual examples but there is presently no known general understanding of duality. To conclude, gauge theories exhibit a variety of interesting phenomena which can be explored by using supersymmetry. Often strong coupling effects have dual, weakly coupled descriptions. A hope is that these ideas will lead to a general, deeper understanding about the dynamics of gauge theories. In addition, they could be useful for realistic model building, with supersymmetry dynamically broken. Finally, duality in string theory could give new perspectives about duality in field theory and visa-versa.

Acknowledgments I would like to thank N. Seiberg for the collaboration on which this talk is based and and for many illuminating discussions. I would like to thank the organizers for arranging such an interesting conference in honor of Professor Kikkawa's birthday and for the opportunity to participate in this happy occasion. This work was supported in part by NSF grant PHY-9513835 and the W.M. Keck Foundation. 1. K. Intriligator and N. Seiberg, hep-th/9509066, to appear in the Proc. of Trieste '95 spring school, TASI '95, Trieste '95 summer school, and Cargese '95 summer school. 2. N. Seiberg, hep-ph/9309335, Ph71•· Lett. 318B (1993) 469. 3. D. Amati, K. Konishi, Y. Meurice, G.C. Rossi and G. Veneziano, Ph711. Rep. 162 (1988) 169 and references therein. 4. M.A. Shifman and A. I. Vainshtein, Nucl. Ph711. B277 (1986) 456; Nucl. Ph71•· B359 (1991) 571. 5. J. Polchinski and N. Seiberg, (1988) unpublished. 6. E. Witten, Nucl. Ph711. B268 (1986) 79. 7. M. Dine and N. Seiberg, Ph11•· Rev. Lett. 57 (1986) 2625. 8. M.T. Grisaru, W. Siegel and M. Rocek, Nucl. Ph711. B159 (1979) 429. 9. T. Banks and A. Zaks, Nucl. Ph711. B196 (1982) 189. 10. N. Seiberg, hep-th/9411149, Nucl. Ph711. B435 (1995) 129. 11. L. Susskind, unpublished; T. Banks, E. Rabinovici, Nucl. Ph711. B160 (1979) 349; E. Fradkin and S. Shenker, Phy1. Rev. D19 (1979) 3682. 12. C. Montonen and D. Olive, Ph711. Lett. 72B (1977) 117; P. Goddard, J. Nuyts and D. Olive, Nucl. Ph711. B125 (1977) 1

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13. H. Osborn, Ph11•· Lett. 83B (1979) 321; A. Sen, hep-th/9402032, Ph11•· Lett. 329B (1994) 217; C. Vafa and E. Witten, hep-th/9408074, Nucl. Ph71•· B432 {1994) 3. 14. N. Seiberg and E. Witten, hep-th/9408099, Nucl. PA11•· B431 {1994) 484. 15. A. Hanany andY. Os, TAUP-2248-95, WIS-95/19, hep-th/9505075. 16. P.C. Argyres, M.R. Plesser and A. Shapere, IASSNS-HEP-95/32, UKHEP /95-06, hep-th/9505100. 17. K. Intriligatorand N. Seiberg, hep-th/9503179,Nucl. Ph11•· B444 (1995) 125 18. N. Seiberg and E. Witten, hep-th/9407087, Nucl. PA11•· B426 {1994) 19. 19. T.H.R. Skyrme, Proc.Roy.Soc. A260 {1961) 127; E. Witten, Nucl. Ph11•· B160 {1979) 57; Nucl. Ph11•· B223 (1983) 422; Nucl. Ph11•· B223 {1983) 433.

N

=2 SUPERSY'MMETRIC GAUGE THEORIES and SOLITON EQUATIONS

TOHRU EGUCHI

lJeptJrtrner&t of Phy1ic1, Urait1er1ity of Tokyo Tokyo 119, Japan Using recently proposed soliton equations we deriw a basic identitY for the scaling 2 supersymmetric gauge theories «i8F/8a.;- 2F 811'ibt u. violation of N Here F is the prepotential, o.;'s are the expectation v:..tues of the scalar fields in the wctor multiplet, u = 1/2Tr(iP2 ) an~ bt is the coefficient of the one-loop beta function.

=

E·

=

1

During the last two years there has been a major progress in our understanding of the non-perturbative strong coupling behavior of 4 dimensional supersymmetric gauge theories a.nd superstring theories in various dimensions based on the idea of S (strong-weak) duality 1 - 13. In the case of N = 2 supersymmetric gauge fields for various gauge groups, many of their low energy effective Lagrangians have been determined exactly with or without the presence of matter fields 14- 22 • There now exists a considerable amount of data for the algebraic curves and differential forms which are used to compute the prepotentials of theN= 2 effective Lagrangian. Recently an idea has been proposed which may possibly organize thel!ie data. within the framework of some known integrable systems 23- 26 • In this talk we will adopt a. machinery from the soliton theory and derive a.n important relation obeyed by the prepotentials F of general N = 2 supersymmetric gauge th~ories coupled to massless matter fields

oF I: a{}a, ; - - 2F = 8rib1u. r

(1)

i=l

Here b1 is the coefficient of the one-loop beta function and a; ( i = 1, · · · , r) are the expectation values of the scalar field in the vector multiplet. r is the rank ofthe gauge group G and u = 1/2Tr (t,6 2 ). Eq.(1) holds in the Coulomb branch of the theory. The right-hand-side of the above equation shows precisely the amount of the scaling violation dictated by the beta function. The simplest version of the formula {1) was first obtained by Matone for the case G = SU(2),NJ = 0 making use of the Picard-Fuchs '(P-F) equation 27. In the case of the SU(2) theory it is easy to generalize the analysis to the

278

case of matter fields Nt :f. 0. The method ofP-F equations, however, becomes complicated when we go to higher rank groups and we need an alternative method of derivation. It turns out that the machinery of soliton equation is particularly suited to our purpose and it is easy to derive the formula (1). My discussions will follow our article 28 • Eq.{l) has also been derived in ref.29 independently. Let us first recall the SU(2) case without matter fields. Accorrding to the analysis of Seiberg and Witten the expectation values of the scalar field a and its dual an are given by integrals of a certain meromorphic differential ~ over homology cycles of an elliptic curve E. E and .\ are given by y 2 = (:e 2 - A4 )(:e- u) and.\= {!- (36) tan {3 = H .




We also have the additional relation that expresses the Jl parameter in terms of the gravitino mass (37) Using the invariances of the superpotential, we can express two of the anomalies in terms of the integer powers Ccolor = -(N + nd)x ;

c,rav = (1 - 3nd - ne - 2N)x

I

(38)

so that the Kac-Moody level is given by _ kcolor- 12 3nd

N+nd

+ ne + 2N

_ 1

(39)

For consistency it must be a positive integer. Using Eq. (38), the non-vanishing of the color anomaly implies N ~ 0, or nd ~ 0, or both. Also, it tells us that, when Nand nd are positive integers, C1 rav/x is negative and the DSW mechanism preserves supersymmetry. X -symmetry is broken, producing the expansion parameter (40)

296

In terms of anomalies, the mass ratio becomes 2 mb

=

).(Cy+Cweak-8/3Ccolor)/2z-N .

(41)

mT

We can express this equation in more physical terms by appealing to the GreenSchwarz mechanism, according to which the anomalies must satisfy Eq. (14) (42)

In string compactifications, we have Cweak

kweak

= kcolor

1

so that (43)

= Ccolor •

The Ibanez relation (44)

is then used to fix the Weinberg angle. We then find that 2 •16 mb

= ...J:_).(C.,..k/2z)(tan

mT

2

9.,-5/3)

(45)

m3/2

This equation is consistent with the data. The left-hand side is of order one. Numerical simulations of the MSSM with soft supersymmetry breaking suggest that the J.L parameter is also of the order of the gravitino mass. In this case, this equation implies that tan 2 Ow = 5/3, which agrees very well with the extrapolated value at unification! This also happens to be the value of compactifications that go through 80(10). The agreement of this formula with experiment lends credence to our scenario of mass hierarchies. We can proceed, using Eq. (43), to express all the charges and anomalies in terms of the positive integers N, ne, nd, n 0 ,

X

X

d = -(n -N) 2 0

'

s= -N. (47) 2

In particular it follows that

(48)

297

If we fix the Weinberg angle at its SU(5) value, we can generate one more relation between the integers

(49) Hence ne ~ nd, and from 38 ne ::/: 0, resulting in a hierarchy between lepton and quark masses. The Kac-Moody level number is now given by ne (50) kcolor = 12 3 ne +nd- 1 There is no solution with kcolor = 1, 2, 3. The lowest interesting value, kcolor = 4, requires nd = 1. In general, however we expect to have a hidden sector with non-Abelian gauge groups. The Green-Schwarz mechanism requires that the mixed anomalies with the hidden sector gauge group not vanish, but be proportional to Ccolor· This implies that there are massless fermions in the hidden sector with X charges. These fermions will in turn contribute to the mixed gravitational anomaly. In the above we have not taken into account the contribution of these fermions. We should write instead

(51) so that kcolor

= 12 3n

N+nd

d

+ n + 2N _ e

1 _ Chid •

(52)

where Chidx is the contribution from the hidden sector massless fermions. This shows that this contribution must be present in order to have a low value for the Kac-Moody. integer kcolor· Alternatively, if we require kcolor = 1, as in models that do not have grand unified groups, we find that the contribution to the mixed gravitational anomaly from the massless particles in the hidden sector is fixed to be (53) If the hidden sector contains a non-Abelian symmetry, its Kac-Moody level will also be equal to one. Thus we know from the Green-Schwarz cancellation that its mixed anomaly coefficient is just -neX· Thus these equations give us a glimpse of the physics of the hidden sector as well. This instructive example serves as an illustration of the power of the Green-Schwarz restrictions. It yields the correct sign of the mixed gravitational anomaly. If we fix the Weinberg angle, we find this model to be very restrictive. First the expansion parameter is found to be

rg;::;;

v

>. = 192;2

0

(54)

298

Since mb m.,.

to get mb ,..,. m.,. Masiero. Then

15 ,

= >.N = _!!:__ ,

we deduce N

(55)

m3/2

= 0,

the original proposal of Giudice and (56)

Since ne is a positive integer, we see that the bottom mass is suppressed, suggesting that tan {3 need not be that large. Finally we note that all the predictions of this model are, once the Weinberg angle is fixed, in terms of one positive integer ne.

5.2

Le Moins Petit Modele

We can .improve the petit Modele in several ways. One is to increase the number of chiral families to the observed three. Less obvious is the addition of pairs of fermions which are vector-like with respect to the standard model, but chiral with respect to symmetries beyond. These fermions appear as chiral fermions in the effective theory and contribute to anomalies that do not involve electroweak and color quantum numbers. Below the scale at which the nonstandard symmetries break, these fermions acquire masses. A second way to improve on the model is to increase the gauge symmetry by adding to the X symmetry non-Abelian or Abelian non-anomalous symmetries. Thus consider a standard model with one chiral family and two Abelian symmetries, one of which is anomalous. Assume two electroweak singlet fields, 80 , a= 1, 2. Its superpotential is

W=QiiHu

(57)

where n~o), n~o), and n~o) are positive integers. through the Kahler potential

The

J.1.

term is generated

(58) The scales at which the extra symmetries are broken are determined by the DSW mechanism 2

xtl8tl 2 + X2l82l 2 +

l~1r 2 M~tanckCgrav = 0 ·

(59)

299

The D term for the non-anomalous symmetry must vanish so as not to break supersymmetry (60) Thus unless the charges are very large, we expect both X and Y1 to break at similar scales. We note that it is possible to generate a much lower breaking scale for Y1 by taking into account soft supersymmetry breaking. For instance, the treelevel hypercharge D term does not vanish (unless {3 = 1r /4), because the soft supersymmetry breaking terms induces a negative mass squared. With only one chiral family, the most general non-anomalous Y1 , consistent with the tree-level superpotential, is of the form

(61) where the first two symmetries are contained within SO(lO). Since this symmetry is vector-like with respect to the (} fields, the vanishing of its D-term requires (62) Hence both X and Y1 are broken at the same scale. Invariance under Y1 yields (1)- (2) nd,e - nd,e = nd,e '

(63)

and there is only one expansion parameter. This case is not very different from the petit Modele. We find that (64)

which has the proper sign to preserve supersymmetry. Here N = N1 = N2. however we are left with much arbitrariness. In special cases, this model has some distinctive features. For instance if Y1 is chosen so that y = 0, it can be broken at a very different scale, say when the right-handed sneutrino gets a vacuum value, perhaps around 10 11 GeV. Another choice is when z = 0, in which case, the X symmetry loses its anomaly, contradicting our assumptions. Finally, when w = 0, we find that nd = 0 indicating no suppression of the bottom to top ratio. Thus unless the non-anomalous symmetry is determined in a special way, we do not have enough data to narrow down the anomaly structure. With several families, this situation may be different. We leave to a future publication 12 the study of such models.

300

Acknowledgments I wish to thank the organizers, H. ltoyama, M. Kaku, H. Kunimoto, M. Ninomiya, and H. Shirokura for their marvelous hospitality. This work is supported in part by the United States Department of Energy under grant DEFG05-86-ER40272. 1. C. Froggatt and H. B. Nielsen Nucl. Phys. B147 (1979) 277. 2. P. Binetruy and P. Ramond, Phys. Lett. B350 (1995) 49. 3. P. Binetruy, S. Lavignac and P. Ramond, preprint LPTHE-ORSAY 95/54, UFTIFT-HEP-96-1, hep-ph/9601243. 4. M. Green and J. Schwarz, Phys. Lett. B149 (1984) 117. 5. L. Ibanez, Phys. Lett. B303 (1993) 55. 6. L. Ibanez and G. G. Ross, Phys. Lett. B332 (1994) 100. 7. CDF collaboration, F. Abe et al., Phys. Rev. Lett. 74 (1995) 2626; DO collaboration, S. Abachi et al., Phys. Rev. Lett. 74 (1995) 2632. 8. P. Ramond, R.G. Roberts and G.G. Ross, Nucl. Phys. B406 (1993) 19. 9. L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. 10. A.E. Faraggi, Phys. Lett. B274 (1992) 47, Phys. Rev. D47 (1993) 5021; A.E. Faraggi and E. Halyo, Phys. Lett. B307 (1993) 305, Nucl. Phys. B416 (1994) 63. 11. M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B289 (1987) 317; J. Atick, L. Dixon and A. Sen, Nucl. Phys. B292 (1987) 109. 12. N. lrges and P. Ramond, in preparation. 13. V. Jain and R. Shrock, Phys. Lett. B352 (1995) 83; preprint ITP-SB95-22 (hep-ph/9507238). 14. G.F. Giudice and A. Masiero, Phys. Lett. B206 (1988) 480. 15. H. Arason, D. J. Castano, B. Keszthelyi, S. Mikaelian, E. J. Piard, P. Ramond, and B. D. Wright, Phys. Rev. Lett. 67 (1991) 2933; A. Giveon, L. J. Hall, and U. Sarid, Phys. Lett. 271B (1991) 138. 16. Y. Nir, Phys. Lett. B354 (1995) 107.

301

INTEGRABILITY and SElBERG-WITTEN THEORY H. !TOYAMA and

Departement of Phy1ic1, Graduate School of Science, 01aka Unit1er1ity, Toyonaka, 01aka 560, Japan. A. MOROZOV 117!59, ITEP, Mo1cow, Ruuia.

A summary of results is presented, which provide exact description of the lowenergy 4d N 2 and N 4 SUSY gauge theories in terms of ld integrable systems.

=

=

This conference is devoted to the impact that the ideas of Prof. K.Kikkawa - especially those concerning duality and string field theory - have had in modern theoretical physics as well as to their interplay with the other concepts and methods. The subject of this contribution is related to duality through the remarkable achievement of N.Seiberg and E.Witten 1 •2 1 who used the duality to obtain the explicit answers for the low-energy effective actions of certain four-dimensional gauge theories. Below is a brief presentation of the results of papers 3 •4 •5 , which- as a minimum- allow to represent the answers of 1 and 2 in a simple and compact form, and - as a maximum- should attract attention to the pertinent role that integrability (and thus abstract group theory) plays in the description of exact effective actions, namely, the role which becomes especially pronounced in the low-energy (Bogolubov-Whitham or topological) limit of quantum field theory. See refs.6 } 4 for related developments. All the relevant references can be found in 4 •5 • The presentations of K.Intriligator and T.Eguchi at this conference allow us not to repeat all the discussion of 4d physics and the methods used in 1 • 2 • We can directly proce.ed to our main subject.

302

1

The problem: from 4d to group theory

Renormalization group {RG) flow for the 4d N schematically shown in Fig.l.

1/e2

2 SUSY YM theory is

.....

r IJ.. "

I

IR

I

"

Aaco

"I I I I

{hj}

uv m

ll

ll') ,. .

Perturbative expression e 2 f3w log Xreceives, of course, nonperturbative corrections, which do not change the shape of the curve too much, except at the infrared (IR). We assume that theory is 110/tly regularized in the ultraviolet (UV) by being embedded into some UV-finite theory, say, N = 4 SUSY YM. (An alternative possibility could be the NJ = 2N., model). Such regularization affects the theory for Jl. > m. From the UV perspective m introduces the mass scale in conformally invariant theory at Jl. > m.a. In the IR the model is •oftly regularized by the condensates of the scalar fields, h" = f(tr t!J"), which break the gauge symmetry down to the pure abelian one. Then all the minimal couplings of fields (which are in adjoint of the gauge group) disappear. In the UV the N 4 SUSY YM model is completely characterized by the gauge group G and the bare coupling constant r = ~i + :.C (a single complex number). The abelian model in theIR contains ra = rank G abelian gauge fields, and the corresponding complex charges

=

"Actually, m can be identified with the maaa of the extra N in the acljoint of the gauge group, which is added to the pure N convert it into the N

= 4 SUSY one in the UV.

= 2 SUSY aupermultiplet = 2 SUSY YM in order to

303

form a matrix 11;, s(IR)

J =f =

i,; = 1, ... , ra:

d':z:Im To;(G'Gi J4

2

a ~:z:d 8Im

=j

.

+ iG'CJi) +... .

To; W'W1

s(UV)

+ ...

J f

=

=

d':z: Im Ttr(G 2 + GG) + ...

d 4 :z:d2 8 JmTtr W 2 + ...

=j

d':z:d 4 8 lm F(cJ')

d':z:d 48 lm F(cJ)

{1.1) The three lines here are written in the N = O, N = 1 and N = 2 notation respectively. Only kinetic terms for the gauge (super)fields are presented explicitly and the rest is denoted by dots. The N 2 (and N 4) superfields in the UV are non-abelian and thus contain infinitely many auxiliary fields. At the same time the abelian superfields in the IR are equaly simple for N = 1 and N = 2 SUSY:

=

=

(1.2)

As a result, the abelian charge matrix is just

11;(a) =

a2F

-a'·aa .. a1

(1.3)

Duality properties become transparent when one introduces a "dual scalar": {1.4)

so that

=

It is a distinguished property of N 2 gauge models that the Wilsonian background fields - of which a' and af are examples - and not just the moduli ~. r, m - are physical observables. This is due to the important result of 15 , which says that the mass spectrum (of BPS saturated states) - a physical

observable- is exactly given by M "" 1~::>7~a;

+ n'af)l.

;

The problem of defining the low-energy effective action in this framework can be formulated as follows: INPUT: G (the gauge group), T (the UV bare coupling constant}, m (mass scale) and ht (symmetry breaking v.e.v.'s). OUTPUT: a1(h) (the background fields) and .F(a1) (the prepotential) - and thus also af = a.r;aa• and 11;(a) = a2 .Ffaa•aai. In other words, all what one seeks for in the setting of Seiberg-Witten theory is the RG map:

{G, r; m, ~}

-+

{a1(h); .F(a')}

(1.5)

304

One can see here an analogy with the c-theorem of A.Zamolodchikov in d = 2 physics: the detailed description of RG flow from one fixed point (conformal model) to another is somewhat sophisticated, but the net result is simple: when we get from the UV to the IR it is enough to say that the central charge has jumped to its adjacent value (in the corresponding series of conformal models, specified by the symmetries preserved during the RG flow). 2

Intermediate data: Riemann surfaces

The discovery ofN.Seiberg and E. Witten was that the mapping (1.5} is actually decomposed into two steps: Riemann surface C;

{G,

Tj

{ meromorphic one - differential dSmin on C m, 14} __.. "hh w1t t e property 8dSmin = hl o omorph"1c

814

{C, dSmin}

--t

{a'( h); F(a)}

(2.1}

The second step is simple: given C one can define the set of conjugate A and B cycles on it, and given dSmin one can write a 1(h} = ,[ dSmin 1

JA,

af(h) = ,[. dSmin, F(a} =

Js•

!L 2

,[

1 JAr

dSmin {

Jsr

dSmin .

(2.2} with 14 = 14(a; ). In the last formula (the one for the prepotential), the set of A and B contours should be enlarged to include those wrapping around and connecting the singularities of dSmin: see 5 and section 7 below. Also, this formula, when dSmin{h} is substituted into, gives an answer as a function of 14. One should further express 14 by a' with the help of the first formula and substitute it into the last one in order to obtain the prepotential F{a'). The genus of Riemann surface C need not coincide with ra = rank G: it is often larger. Then eqs.(2.2) seem senseless since the number of A and B contours on the r.h.s. can exceed that of the a' and af on the l.h.s. However, in such cases - given appropriate dSmin (as defined in eq.(5.2} below) - all the extra integrals at the r.h.s. of (2.2} automatically vanish - and there are exactly ra non-vanishing a' - as necessary. This is of course also important to make the dependence a'( h) invertible. Thus the second mapping in (2.1) is completely described and very simple. The real issue is the first mapping in

305

(2.1}.

3

What is C? emergence of integrable systems

This mapping

{G,

Tj

m, h.&} -

{C; dSmin}

(3.1}

contains no reference to four dimensions, Yang-Mills theory or anything like that: it is clearly something much simpler and general. With no surprise, it can be described in a language far more primitive than that of 4d gauge theories: one should look for this mapping at the first place where the group theory (the input in (3.1}} meets with the algebraic geometry (the output of (3.1)}. A natural place of such kind is integrability theory 16 • Namely, the map (3.1} possesses description in terms of 1d integrable models. In other words, the particular question (the UV I R map) in 4d gauge theory appears to be equivalent to some (actually, almost the same) question in the 1d integrability theory. In this language theory (3.1} can be described as follows: Given a simple Lie group G one can construct an associated 1d integrable model. Parameters T and m naturally appear in this construction. The only thing that we need on the model emerged is its La.x operator L(z}, which is a G*-valued function (matrix} on the phase space of the system and depends also on the "spectral parameter" .z. Thus! (3.2} G ~ L(z). Then the first ingredient of the map (3.1} is C:

det (t- L(.z)) = 0.

(3.3}

If the spectral parameter z in (3.2) belongs to some complex "bare spectral

surface" E, this equation defines the spectral curve C of L(.z} as a ramified covering over E. (For every z E E there are several points on C, differing by the choice of the eigenvalue t of L(z). The sheets are glued together at the points where some eigenvalues coincide.) For the given L(.z) eq.(3.3} depends •The map (3.2) is actually a canonical one in the framework of geometrical quantization (Kirillov-Kostant method): it can be nicely described in terms of coadjoint orbits of G, Gq and

G, momentum maps, Hitchin varieties etc.

What we need to know here about this map

is much simpler: that it exi•ta, is naturally defined entirely in terms of group theory, and mo•t of explicit formulas (in convenient coordinate•) are well-lmown since 1970's.

306

on the integrals of motion of integrable system. These are identified with the moduli "' at the l.h.~. of (1.5). 4 -4.1

Examples ld •ine-Gordon model {G

= SL(2)}

In this case

L(z) = (

p

ell +z

-p

)

(4.1)

.

The Lax operator depends on the coordinate q and the momentum p, but the spectral curve (3.3) depends only on their particular combination h2 = p 2 +(ell+ e-ll)= p 2 + 2coshq:

(4.2) or

(4.3) This h 2 is nothing but the integral of motion (the second Hamiltonian) ofthe sine-Gordon system. Thus we see that what remains in (3.3) from the phasespace dependence of the Lax operator is just that on the constants of motion (this is one of the central facts in integrability theory). For us this means, that once "' are identified with the integrals of motion of integrable system, we indeed get a map {G, r, m, "'}-+C. -4.2 SL(2) ld

Cr~logero

•y•tem

This time the Lax operator is expressed through elliptic functions. Elliptic functions live on elliptic br~re spectral curve E( T ). Its modulus T is exactly the one which is identified with the bare coupling constant in 4d theory. One can choose a coordinate on E(T) in two essentially different ways: the one is the flat coordinate and elliptic functions are (quasi) doubly periodic in and the other is the elliptic parametrization

e

e

'307

and 17

e1 (r)- e2 (r) = 8~0 (r), etc.

The Lax operator of Calogero-Moser model is

(4.5) where F{qle) = z,...,. p(e)

.:Ce>! q)

and Weierstrass function

=-Be2 logu(e) =

e:~1 + "'' L..J ( (e + m1+ nr)2 m,n

1

- {m + nr)2

) .

{4.6) The full spectral curve is now

c:

det (t - L(e)) =

o,

(t- p)(t + p) = g2 F(qle>F< -qle> = 9 2 (p(e)- p(q)),

{4.7)

or simply

(4.8) where this time h2 = p 2 + g 2 p( q) is the second Hamiltonian of Calogero-Moser system. Calogero coupling constant g is to be identified with the parameter m in 4d considerations:

{4.9) Thus, in the framework of Calogero-Moser models we devised a mapping (3.2) parametrized by two variables, T and m ,...,. g, which is exactly what necessary for our purposes. 5

What is dSmin?

The last thing which is necessary to formulate our description of SeibergWitten theory is an explicit expression for dSmin, which enters the formulas (2.2) for the background fields ai and af. Now we know that the appropriate bare spectral surface is the elliptic curve E( T) and the full spectral curve C is a ramified covering over E(r) defined by the equation det(t- L(e)) = 0. We are ready to give an explicit expression for dSmin· On E{r) there is a distinguished canonical holomorphic !-differential

308

Its periods on E( T) are 1 and T. dSmin is just twice the product of the Laxoperator eigenvalue t and this dw 0 f dSmin

6

= 2tdwo.

(5.2)

Examples

6.1

SL(2) Calogero and it11 limiting ccue11

Since the curve C in this case is given by eq. (6.1) eq.(5.2) says that dSmin,....

Vh + gllp(e) de

c"'_e(e)

V:~rz dz.

(6.2)

Despite the full spectral curve C is of genus 2 {C is obtained by gluing two g-llh and copies of E(T) along one cut, which connects two points with z

=

±yfll!=

two different values ofy = 1 ( :2 - e4 )). However, the differential (6.2) can be essentially considered as living on some other - genus one curve

c:

{6.3)

which is different from E(T) (except for theN= 4 SUSY limit of h = oo), but of which C is also a double covering. In fact, this is another way of saying that dSmin has only two - rather than four (= twice the genus of C) - non-vanishing periods: this is an example of how the extra periods are automatically eliminated by the choice of peculiar 1-differential (5.2). In terms of Cwe have dSmin ,....

h-gllz y( z) dz,

{6.4)

and we remind once again that g ,.... m = the mass of adjoint matter multiplet. eAgain, as in the case of (3.2), there are dift'erent interpretations of this formula: their origin. range from the theory of prepotential ( quasiclassical T-functiona) and BogolubovWhitham theory to Hamiltonian structures of integrable theories and symplectic geometry of Hitchin varieties. Again, the only thing that is essential for us is that all these interpretations are essentially the same and that the explicit formula (5.2) is true.

309

There are two interesting limits of this formula that deserve attention. The first one is the N = 4 SUSY limit, when the adjoint multiplet gets massless: m 2 ,..., g2 = 0. Then obviously C-+ E(-r},

(6.5} dSmin -+ 2Vhdwo,

and the periods (background fields) are a=

t

dSmin -+ 2../h,

aD

=

t

dSmin -+ 2-r../h.

(6.6)

Of somewhat more interest is the opposite limit, when the matter multiplet decouples, m -+ oo. Of physical interest is, however, the situation when the mass scale survives, i.e. the case that the dimensional transmutation takes place. This is achieved in the double scaling limit, when simultaneously m2,..., g2-+ oo, T

-+ ioo,

or

q :::

e,. . ,. -+ 0,

(6.7)

A2 = m 2 q

(6.8)

but ANc = mNcq,

i.e.

remains finite. In this limit the bare spectral curve E(-r} (parametrized degenerates into a double punctured sphere with coordinate z, and 1

z

2?n

q

~-+-.log-,

i

Since g2 ""' and~+ n-r""' which survive in the sum

2;,

and

dwo=~-+

1 dz -.-. 2n z

by~)

(6.9)

log(zq 2n- 1 }, it is clear that the only two terms

(6.10} in the double scaling limit are those with n = 0 and n = 1, so that

(6.11)

Here C( ) .,.

= _1 8loga(-r) 3'In·

a.,.

DO

'

a(-r}

= q2 II (1 _ q2n>24. n:l

(6.12}

310

Accordingly, the scaling rule for the h parameter is (6.13) and it is u that remains finite in the double sacling limit (while h -g 2 C( r = oo) ..., - 311 ). As a result,

-+

oo as

Ju- A 2 cos 1P dv>, (6.14) where :i -!A2 (.z + .z- 1 ) -A2 cos IP· The r.h.s. of (6.14) is exactly the original Seiberg-Witten differential of 1 , which describes the N = 2 SUSY pure gauge SL{2) model. It can be of course immediately reproduced from the sine-Gordon description of our section 4.1. See 6 for more details.

=

6.2

=

Toda chain for any SL(Nc) and beyond

In this case the bare spectral surface is a double-punctured sphere obtained by degeneration of elliptic E(r). Other formulas from s.4.1 are generalized as follows:

L(.z)

=iii+

(6.15)

In the fundamental representation of GL(Nc) the roots are represented as matrices E,; with non-vanishing entries at the crossing of i-th row and j-th column. For positive roots i < j (upper triangular matrices), for negative roots i > j. Diagonal matrices represent Cartan elements. The simple positive/negative roots belong to the first upper/lower subdiagonal, the affine roots ±ti0 are located at the left lower/ right upper corner respectively. Thus !e91-9Nc a

P3

0 0 0

0 0

PN.-1

1

efNc-fNc-1

PN.

P2

0 1

0

e'"-'2

0

0 0

P1

L(.z) =

1

ef2-fl

z

0 0

311

The full spectral curve is given by the equation 0 = det (t- L(z)) = 1

N.

=(t- Pl)(t- .P2) ... (t- PN.) + ~ ef>+l-9> + ... - z - :; = i=l

(6.16)

N.

= ~ S1(h1)tN·-I = 2PN.(tlh). 1=0

In this formula one should take into account the periodicity condition qN.+l = q1 . As usual all the p, q-dependent terms gather into the Toda Hamiltonians h&, h1 = EiPi• h2 Ei (iv1 + e"+ 1 - 9'), ••• (h1 = 0 for G SL(Nc)). Finally, S1(h) are Schur polynomials. One can easily see how they appear by omitting all the interaction terms (with q-exponents) and substituting the free Hamiltonians h~o) f Ei pf for h1. In order to introduce the interaction back it is enough to substitute back h~o) -+ h 1 : all interaction effects enter only through h1. Thus we obtain the spectral curve in the form:

=

=

=

C:

(6.17)

=

It can be brought into a more familiar form Y 2 PJ,. (t)- 1 by a substitution z - .z- 1 . If Nt N 2 SUSY matter superfields in the fundamental 2Y representation of the gauge group are added to the model in 4d, the curve (6.17) is replaced by

=

=

(6.18)

with the polynomial QN1 (t) depending on the masses of the new fields, and the same PN.(tlh) as in (6.17). Finally, the !-differential in all these cases (when the bare spectral curve is double-punctured sphere) is

dSmin

tdz = -:--. \'71'

z

(6.19)

312

6.3

SL(Nc} Calogero-MotJer model

This time the bare spectral curve is elliptic E{r} Pt

L{{) =pH+ g

L F(qai{)Ea =

gFu

a gFN.l

gFN.2

gFN.3

PN.

(6.20} In order to handle the elliptic functions routinely in this case one needs more relations than given in the section 4.2. Namely, a few identities for symmetric combinations ofF-functions are required 4 :

S2 (F}

S3 (F}

=

i

= F(qj{}F( -qj{) = p({)- p(q},

(F(qt2!{}F(q231{)F(q3tle) +permutations of q1, q2, q3)

= -8ep(e},

(6.21} With the help of these identities the full spectral curve can be represented as

C:

o = det(t- L({)}

N.

=L

s,(h)TN.-l(tl{).

(6.22}

1=0

This time, in varience with the Toda-chain case, eq.(6.16}, the n-th order tpolynomial Tn(tj{) - while still h-independent - depends nontrivially on{: cnTn(tl{) = 1 +

(-i)L.

m.r.

II (a(·-'Jp({))

m•

• (6.23}

313

Converting this expression from the flat coordinate { to the elliptic ones z and y

= Jn!=l(z- e,.), one obtains To= 1, 'li =t, 72 = t 2 -

z,

Ts

=t 3 -

3zt + 2y,

3

(6.24)

14 = t 4 - 6zt2 + syt - 3z 2 + E e!, ... G=l

- some linear combinations of Donagi-Witten polynomials introduced in 10 • An advantage of (6.23) as compared to 10 is not only the simple derivation (and thus the possibility to obtain a general explicit formula (6.23)), but the full separation of h and { variables achieved in eq.(6.22). See 4 for details and discussion. The 1-di:fferential dSmin is, as usual, just dSmin = 2t({)d{.

(6.25)

6.4 SL(Nc) Ruijsenaars model To finish with our examples, we present a few formulas for the further generalization of Calogero-Moser system- its relativistic (from one point of view) or quantum group (from another perspective) generalization: the 1d Ruijsenaars system. The bare spectral curve is still elliptic E(T), the Lax operator is composed of the already familiar elliptic functions, but it is given by a different formula 18 :

L,;({)

= ep• ;~q~~~e~ II n(p)y'p(p)- p(q,,). q,, J' l;ti

(6.26)

Calogero-Moser model is the p -+ 0 limit of this one with the coupling constant g ,..., m emerging from the scaling rule for P,-varibles. If one takes the normalization function to be n(p) u(~-&) ,..., ~-&+o(~-& 2 ), rescales as P1 i"Pi, and takes into account that p({) ~-&- 2 + o(1), F(qi; !~-&) ~-&- 1 (1- 6~;) + 6,; + o(~-&), it is easy to see that

=

=

=

=

(6.27) and the order-~-& term is exactly the Lax matrix (6.20). For Nc spectral curve can be written as 0

= det (t61; - L,;({)) = t 2 -

t · trL + det L,

= 2 the full (6.28)

314

and

1 H =. -(-)trL 2n J'

1 = -(ep + e-Ph/P(J')- p(q), 2

det L =

n 2 (14)

(6.29)

(P(J') - p{e)),

so that (6.28) gives

t= n(J')dS!!in

H

± yH2 - p(J.') + p({) and n(J')

,

(6.30)

= 2n(J.£)tdwo = 2H(J')dwo + ds~r~lhfg2=H2(1')-P(I') ·

We remind that

JF-d d v~(t)..lt 92 + Col

smin,....

1

p~~;, ..... ,....

y(z)

(6.31)

z.

Theory of prepotential

There are several presentations at this conference devoted to the prepotential in Seiberg-Witten theory. Instead of repeating the same things we rather outline here a general theory - not only applicable to the Seiberg-Witten case ( which is associated with Riemann surfaces and corresponds to d 1 and 0 dS below). This general theory can be given different names: that of quasiclassical r-functions, of prepotentials, the Whitham theory, special geometry etc. see 19•20•21 •7 •9 for various presentations. Applications to the Seiberg-Witten case are straightforward see 5 and references therein. The real meaning of the prepotential theory - and the very fact that a more fundamental object (prepotential) than the action exists in a rather general setting in classical mechanics - remains obscure. It should be somehow related to the fundamental role that quuiperiodic (rather than periodic) trajectories- which exhibit some ergodicity-like properties - play in the transition from classical to quantum mechanics. Why is the theory of quasiperiodic trajectories expressible in terms of deformations of Hodge structures - and how general this statement can be should be a subject offurther investigation: we do not touch these fundamental problems in what follows.

=

7.1

=

Notation and definitions

Consider a family M(h) of complex manifolds M of complex dimension d (in the previous sections d 1 and M (h) are some families ofspectral curves, M

=

=

315

=

= = =

C ). The family is parametrized by some moduli h,, 1c 1, ... , K dime M. Let us fix some canonical system of d-cycles on M: {A,, B, }, i 1, ... , p !dimH4 (M) with the intersection matrix A,#B; 6,;, A,#A; B;.#B; 0. Finaly, pick up some holomorphic (d, 0)-form 0 on every M .4 Its periods

=

a.(h)

=J'A,1 o,

=

=

(7.1)

are functions of moduli. Consider now a variation 60 of 0 with the change of parameters (moduli). 60 is also a (d, 0)-form, not necessarily holomorphic. Still, always 0 1\60 = 0 (just because 0 is a maximal-rank form), and integraion of this relation over entire M gives

up to the contributions from singularities. Imagine that the last item at the r.h.s. - the contribution from singularities of 0 and 60 is absent. Then we obtain from (7.2):

,E a.6af = ,E af 6a..

(7.3)

i

This implies that the

prepotentit~l,

:F = !

2

defined as

E. a.af = !2 E. hA· 1 o 1 o, hs· '

'

.

(7.4)

..

possesses the following property: 1

6:F = - E (a.6af + af 6a.) 2

.

'

=E. af 6a.. '

If the freedom of variations is big enough, e.g. if #K

as #p = !dim H 4 (M), we conclude from this that D

(7.5)

= dimcM is the same

8.1"

a·=-

'

(7.6)

8a. dJt is clearly a restriction on M that such n exists: examples of suitable M are provided by K3 (d = 2) and Calabi-Yau (d = 3) manifolda. In our di1c:uuion below we ahall see that this restriction can sometime be weekend, by admitting O's with simple singularities. Additional requirements for 0-dependence on moduli will be specified later.

316

and 1"

D

a:r-

1

:F=-L..Ja;a, =-Ea;-. 2 . 2 . 8a;

•

(7.7)

•

In other words, we can consider a; as independent variables, and introduce the prepotential :F(a) by the rule (7.4}- and it will always be a homogeneous function of degree 2- as follows from (7.7). The two requirements built into this simple construction are (i} the absence of singularity contributions at the r.h.s. of (7.2};

(ii) the matching between the quantities of moduli and A-cycles, K =: dimcM = p =:

7.2

~dimH 4 (M)

Comments on requirement (i)

The problem with this restriction is that variation of holomorphic object w.r.to moduli usually makes it singular - by the very definition of moduli of complex structure. Thus, even if 0 is free of singularities one should expect them to appear in 60. The only way out would be to get the newly emerging poles cancelled by zeroes of n - but often the space of holomorphic O's is too small to allow for adequate adjustement. Fortunately, requirement (i) can be made less restrictive. One can allow to consider n which is not holomorphic, but possesses simple singularities at isolated divisors. As a pay for this it is enough to enlarge the set of A-cycles, by adding the ones wrapping around the singularity divisors, and also add all independent B-chains, connecting these divisors (such that 8B div 1 -div2 ).e At the same time residues at the simple singularities should be added to the set of moduli {h}, thus preserving the status of the second requirement (ii). This prescription is still not complete, because the integrals over newly-added B-chains are divergent (because these end at the singularities of n). However,

=

=1), n can be a meromorphic {1, 0 )-differential with •imple (order one) poles at some punctures e.. , a = o, 1, ... , r. Then one should add r circles around the point. e1 .... ,e.. to the set of A-cycles, and r lines (cut.) connecting eo with e e.. to the set of B-contours in eq.(7.2). Then the last term at the r.h.s. can be '"For example, if M is a complex curve ( d

1 , ••• ,

omitted in exchange for enlarging the sum in the first term.

317

the structure of divergence is very simple: if a cut-off is introduced, the cutofF-dependent piece in :F is exactly quadratic in the new moduli - and does not depend on the old ones. If one agrees to define the prepotential - which is generic homogeneous function of order two - modulo quadnJtic functions of moduli, the problem is resolved. Thus the real meaning of constraint (i) is that 60 should not introduce new singularities as compared to 0 - so that we do not need to introduce new cycles, thus new moduli, derivatives over which would provide new singularities. Since now the freedom to choose 0 is big enough, such special adjustement is usually available. The non-simple singularities (higher-order poles at divisors) should be resolved - i.e. considered as a limit of several simple ones when the corresponding divisors tend to coincide. The corresponding B-chains shrink to zero in the limit, but integrals ofO over them do not vanish, ifO is indeed singular enough. This procedure of course depends on a particular way to resolve the non-simple singularity. Essentially, if we want to allow the one of an arbitrary type on the given divisor, it is necessary to introduce coordinate system in the vicinity of the divisor and consider all the negative terms of Laurent expansion of 0 as moduli, and "weighted" integrals around the divisor as A-cycles. In the case of d = 1, when the divisors are just points (punctures) one can easily recognize in this picture the definition of KP /Toda-induced Whitham prepotential with one-parameter set of "time"-variables (Laurent expansion coefficients or moduli of coordinate systems) for every puncture as additional moduli (see, for example, 7•5 and references therein). As often happens, it is most natural from the point of view of string theory (integrability theory in this case) to put all the moduli in a single point (or two), but from the point of view of algebraic geometry it is better to redistribute them as simple singularities at infinitely many divisors. Finally, singularities of 0 on subspace& of codimension higher than one do not contribute to eq.(7.2) at all- and often variation w.r.to moduli produces only singularities of such type as d > 1.

1.3

Requirement {ii}

Thus, what essentially remains is the other requirement (ii) - the matching condition between the number of moduli and A-cycles. Since the procedures involved in resolution of (i) do not change this matching (they always add as many new moduli as new A-cycles), this requirement can be analyzed at the very beginning- before even introducing 0.

318

8

Picard-Fuchs equations

Dependence a'(h), af(h), described by eq.(7.1) can be also formulated in terms of differential Picard-Fuchs equations for the cohomology classes of 0. They are often convenient for comparison of Whitham universality classes of different models: Whitham-equivalent models should have equivalent Picard-Fuchs equations. We refer to 6 on details of how this idea can be elaborated on. Here we just list some important examples (all for G SL(2)).

=

8.1

Pure ga.uge N = 2 SUSY model in 4d: 1ine-Gordon model in 1d

In this case

~

{8.1)

dSmin-y~d:C

and Picard-Fuchs equation is 22 {8.2)

8.!

=

The flow from pure ga.uge N 4 SUSY model to the N in 4d: Ca.logero-Mo1er model in 1d

= 2 SUSY one

Now

h-z

dSmin....,

(8.3)

{z- el)(z- e2){z- ea) dz.

ea.

The branching points of the elliptic ba.re spectral curve E( T) are functions ofT, and the simplest Picard-Fuchs equation looks like 1 a 2wi a-r

f

dSmin

=

(y (h) ah.a + 2h - 2hC(-r)2

2

A

2

[1

A

2

1

A

1 ] 12 g2(-r)

a) . ah.

f

dSmin

{8.4) Here C{-r) is just the same quasimodular form ( the logarithmic derivative of the Dedekind function) that appeared in (6.12) above, while the modular ~ [9~0 {-r) + 9~ 1 {-r) + 9~0 {-r)). The differential operator eq.(8.4) form g2(-r) is presumably convertible (by conjugation and change of variables) to the Schroedinger form:

=

a - ( y(h)--.. a ) 2 - h= -1 .a- - a-2 - p(x), -1. A

2w' a-r

A

ah

2w' a-r

ax 2

(8.5)

319

which (if true) would reflect the interpretation of Whitham theory as that of quantization: (we consider essentially classical Calogero-Moser model, but the Picard-Fuchs equation on the moduli space is the Schroedinger equation for this model). See 6 for more details. One can derive an infinite set of Picard-Fuchs equations, with difFerent powers of r-derivative, eq.(8.4) being the simplest one in the series (first Tderivative ). Only two of them are algebraically independent, because dSmin depends only on two variables: h and T. Still, the entire infinite series, once derived, can exhibit some new nice structure- as it usually happens (compare with the Virasoro etc constraints in matrix models).

J

8.3 Ruij•enaar• model in ld dSRu - H( ) minI"

J( z -

dz e1 z - e2 z - ea A

)(

A

)(

A

)

+ dSCtt.ll min h=H(~&)2-p(~&) ·

(8.6)

Here dS~f~ is given by (8.3). Picard-Fuchs equation is not drastically difFerent from (8.4), most important, the lowest equation seems to be still of the first order in 8J8r- what does not allow to identify it with the Picard-Fuchs equation for Calabi-Yau model, where all the derivatives are of the second order (see 6 and below). Again, there are only two independent Picard-Fuchs equations.

The W P/1 2, 2,6 -induced Ctllabi- Yau model

8.-4

The manifold is a factor of the one, defined by the equation

The 3-form 0, which is used in the construction of the prepotential on the lines of s. 7, is a restriction of dz1 A dz2 A dza A dz4 A dz5 (one should take into account the quasihomogeneity of (8.7)- this allows to eliminate two variables to get a 3-form). Thus its periods are proportional to

f

0 .....,

j 1JA(z) j dz1 ... dz5eil(a )p(a) ....., j dz1 ... dz (J dAeilp(a)) ...... 6

""'j dzt ... dz5e'(a) (8.8)

320

(the quasihomogeneity of p{.z) is used to eliminate ~) and satisfy the set of Picard-Fuchs equations (which are nothing but Ward identities for the integral (8.8)). The simplest one is

[

(:~) (~ :~+~~a~+ ~r] f n = 0. l-

(8.9)

Again, since there are two moduli, only two equations from the whole set will be algebraically independent. In the particular ("conifold") double scaling limit, when ~. ~ ; 2

oo with

= ~ - i~ 6 fixed, this equation (8.9) reduces exactly to the sine-Gordon-

case one, eq.(8.2). This reflects the fact that in the target space this limit cortresponds to the a'- 0 limit, when the d = 10 Calabi-Yau model reduces to the d = 4 one - and the sector described by the periods of {l is exactly the gauge sector described by the Seiberg-Witten theory. See l 3 for details and references. There is no doubt that the equation (8.9) itself, not only its conifold limit, can be represented in terms of some simple ld system. However, at the moment we do not know what this system is. Neither Calogero nor Ruijsenaars models seems to suit. Technically, the lowest Picard-Fuchs equations for these models contain only first derivative w.r.t. one of the variables (r), while in (8.9) both derivatives are of the second order. Physically, the relevant models should not be assocaited with particular groups (only the rank of the group should be fixed): this is because the variation of moduli of Calabi-Yau model can change one group for another (in one point of the moduli space one can have SL{3) symmetry, while in another one it would be SL(2) x SL(2) -and neither one is a subgroup of another). This phenomenon is not directly relevant for the rank-one example of eq.(8.9)- still it explains why Calogero model itself should not be enough - and shows the direction for the search of the relevant models. 9

Instead of conclusion

Following refs:l• 4 •5 we presented some evidence that the results like those of l,l can be nicely systematized in the a priori different language - that of the ld integrable systems. We do not find it very surprising, because the question that was addressed in l,l is very special: the one about the low-energy effective actions, and the adequate terms in which the conformally invariant theories in the deep UV are related to the topological ones in the deep IR are necessarily rather simple.

321

In fact, the general scheme that one can keep in mind is as follows 2": exact Wilsonian effective actions, defined by the functional integrals like es.tt(t(+)

=

J.v;

es.(;)

(9.1)

t,..

naturally depend on two kinds of variables: the coupling constants in the bare action St(t;),..., L{l.} t{,..}tr Tia t;!• and the background fields~ (examples of the latter ones are our ai and af above). Such exact effective actions (the generating functionals for all the correlators in the given field theory) - as one knows well from the example of matrix models- are "infinitely symmetric" because of the freedom to change integration variables. This symmetry is often enough to identify them with pure algebraic objects: generalized T-functions, defined as generating functions of all the matrix elements of a universal group element g, T(tlg) = E,..,i. (A:clglk.,)t{l.,1.}· In general case (non-vanishing normalization point) both effective action and the T-function (for quantum group) are operator-valued; the IR stable point of RG flow should then correspond to the classical limit in the group theory language. Such considerations are, of course, very general and can seem almost senseless: still they imply something both in the general framework (for example, so defined T-functions always satisfy some bilinear Hirota-like equations), and in concrete examples. The most famous example is the one of matrix models. Another - newly emerging example- is that of the low-energy theories: restricting consideration to the IR stable points of renormalization group flow, one drastically diminishes the number of degrees of freedom (moduli)- what in the group-theory language corresponds to consideration of small enough groups (not necessarily the 3-loop group, as at generic normalization point for the 4d field theory)/ To put it differently, various theories flow to the same universality class in the IR limit - thus these classes can be (and are) rather simple. What I A nice particular example of the relation between RG flows end integrability theory is

by now famous identity

fJw (tr tP2) "' 2:Fred -

L a --.-, B:Fred Ba' i

i

- a member of the anomaly family (together with fJw(tr G2 )

""

T,.,. end axial anomaly),

where the l.h.1. ia clearly of RG nature and the r.h.1. repre.entl the breakdown of homogeneity of the prepotential :F which

occur~

by fixing one of ita argument. (the scale A).

322 the general identification of effective actions with the tau-functions (i.e. with group theory) teaches us is that these classes should be also representable by some r-functions. However, these cannot be just conventional r-functions - defined in the Lie-group terms - because some parameter of the effective action (the normalization point) is fixed. But in order to understand whtd are these relevant objects one can consider just the RG flow within some simple enough integrable system - and then discover that the relevant objects are quuiclu•ical r-functions (or prepotentials). This can provide a kind of an explanation of why it was natural to try to identify the results of 1 •2 with those of integrability theory and where exactly (the Whitham theory) one had to look for this identification. This also explains why there should be no big surprise once such correspondence is established. What needs to be understood, however, is the general description of how group theory (represented by generalized r-functions) always flows to that of Hodge deformations (represented by prepotentials). The main message of this presentation can be that such phenomenon exists, and one should think of what could be the reasons behind this and what is the adequate technical approach to a more generic situation. Once found, the answers can shed new light on the implications of symmetries (group theory) for the low-energy dynamics and algebraic geometry (of moduli spaces) - and this would be of definite use for the future developement of quantum field and string theory. (9.1)

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=

=

=

=

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21.

22.

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=

=

23.

24.

=

325

INTEGRABLE SYSTEM AND N = 2 SUPERSYMMETRIC YANG-MILLS THEORY TOSHIO NAKATSU Department of MathematiCI and PhrJ1ic1, Ribumeilcan Uniuer1ity, Ku1at6u, Shiga 5!5-77, Japan

KANEHISA TAKASAKI Department of Fundamental Science1, Faculty of Integrated Human Studie1, Kr~oto Uniuer1ity, Yo1hida-Nihonmabu-cho, Sa/cyo-ku, Kyoto 606, Japan

=

We study the exact solution o( N 2 supersymmetric SU(N) Yang-Mills theory in the framework of the Whitham-Toda hierlll"chy. We show that it is in (act obtainable by modulating the solution of the (generalized) Toda lattice associated with moduli ol curves. The relation between the holomorpbic pre-potential of the low energy effective action and the T-(unction ol the (gener-alized) Toda lattice is also clarified.

1

Recently Seiberg and Witten 1 obtained exact expressions for the metric on moduli space and the dyon spectrum of N = 2 supersymmetric SU(2) YangMills theory by using a version of the Montonen-Olive duality 2 and holomorphy 3 of 4d supersymmetric theories. Their approach has been generalized to the case of other Lie group 4 5 • Especially surprising in these results is unexpected emergence of elliptic (or hyperelliptic) curves and their periods. Although these objects appear in the course of determining the holomorphic pre-potential :F of the exact low energy effective actions, physical siginificance of the curves themselves is unclear yet. It will be important to clarify their physical role. An interesting step in this direction has been taken 6 from the view of integrable systems, in which the correspondence between the Seiberg-Witten solution 1 and a Gurevich-Pitaevsky solution 7 to the elliptic Whitham equations 8 is pointed out. The exact solution (Seiberg-Witten type) of N=2 supersymmetric SU(N) Yang-Mills theory is described by the following data 1 4 :

326

Data 1 "family of the hyperelliptic

cun~es" N-2

P(z) =

zN

+

L

(1)

UN-II:ZI:,

11:=0

where u = (u2, · • • , UN) are the order parameters and A is the this theory. Data 2 "the merom orphic differential on C"

~-parameter

of

zdP(:~:)

dS= __g_dz

(2)

y

Using these two data the holomorphic pre-potential :F=:F(a) is prescribed from the relation, ai

where

Oti,Pi

=

£,

dS

(1

~ i ~ N- 1)

(3)

are the standard homology cycles of the curve C.

In this talk we discuss the exact solution of N = 2 SUSY SU(N) YangMills theory in the framework of the Whitham equations 8 9 and clarify the relation with Toda lattice. Especially we would like to explain (i) the integrable structure of SU(N) Seiberg-Witten solution has its origin in (generalized) Nperiodic Toda lattice (or chain), and (ii) the holomorphic pre-potential :F(a) is obtainable from the r-function of this Toda lattice (or chain) by modulation (Whitham's averaging method 8 ). 2

e.t Let us begin by dscribing the integrable structure which already appears implicitly in these data. For the convenience we introduce the meromorphic function h by h=

y+ P(z),

and consider the partial-derivation of dS by the moduli parameter fixing this function h. It satisfies

(4) ai

with

(5)

327

where dzi is the normalized holomorphic differential, fore these holomorphic differentials satisfy {)

-dz·

fJai

1

=

{)

(1

-dzi

oa;

~ i,j ~

i.e

Ja. dzi = Di,J. '

There-

N- 1),

(6)

where the derivations by the moduli parameters a= (at,···,aN-1) are performed by fixing h. These equations are the compatibility conditions for the system of holomorphic differentials under their evolutions by the moduli parameters a. So these compatibility conditions define a integrable system. dS gives a solution for this system.

Nextly let us explain the moduli curve C in the data can be understood 10 11 as the spectral curve 12 of (N-periodic) Toda chain. (N-periodic) Toda chain is a 1d integrable system defined by the equations,

8lK(r)'Y.,,) -ai 21'pl/J2K(r)'Y.,,}, (20)

where we have used an abbreviation such as Kpv'(t) a± and l:t: are functions of cl defined by

=

= Kpv(z, z'; t) and N±,

=

N± da± 2 +2a± + 1, 2a:t: l:t: -1, l:t: =a'± /(a'), f(a') = vf(1- a') 2 +4a'/d.

(21)

By using (20) together with heat equations for the kernels, cyclicity of the trace, and identities 4 such as 1'" Kpv•(t) K(t)'Y.,, and D~' Kpv'(t) -D~K(t), we can show that the left hand side of (18) is equal to

=

2 lim """" l t-o i:+,L...J

=

+;id'Tt-ys(Dpe~' +'eP D11 ) (K(t)- K(li 2t)} = 0. ·

(22)

I

Thus, the FTY identity holds for any value of a'. Note that, in contrast to the a 1 = 0 case, the identity holds only after the limit oft -+ 0. Let us proceed to the a'-independence of the Einstein anomalieS. Since we have now the FTY identity, we do not have to investigate both anomalies corresponding to 6.ym and 6Aw. Here we examine the latter anomaly, whose a'-dependent terms are written as 1 ' ' lnJAw[e)j 01 ,_dep 2 !~Tt')'s (Dpe 11 +(lA Dp + 2F)L(t), (23)

=

where L(t) = e-t'Z>(OI') 2 - e- 1.¥' 2 , a'-dependent part of the kernel Kp 11 (z, z'; t). Using again (20) together with heat equations, cyclicity of the trace, etc., we

353

can conclude the gauge invariance of the anomaly: In JAW [e)

Ia'-dep

=

_! jdz ~ 2

X

{

.

•=+.-

tr'}'s(e'1; 11 J.l

lnli 2 Ni

+ ev;p p) bd/2-l;v(z, z')

-ePw J.l.,tr1s bd/2-l(z,z') }L=J:'

(24}

= 0, where bd/ 2_ 1 (z, z') is the (d/2- 1}-th Schwinger-De Witt expansion coefficient 11 of the spin-1/2 heat kernel. In the last equality, we have used tr'}'sbd/ 2_ 1 (z,z) = tr'}'sbd/ 2-l;p(z,z')IJ:=•' = 0, which follows from the fact that these quantities include at most d- 2 '}'-matrices. Acknowledgments

One of the authors (R.E.) would like to thank Peter van Nieuwenhuizen for his interest in this subject. References 1. L. Alvarez-Gaume and E. Witten, Nucl. Phys. B 234, 269 {1984). 2. W. A. Bardeen and B. Zumino, Nucl. Phys. B 244, 421 (1984). 3. K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979); Phys. Rev. D 21, 2848 (1980}. 4. R. Endo, Class. Quantum Grav. 12, 1157 (1995). 5. N. K. Nielsen, M. T. Grisaru, H. ROmer and P. van Nieuwenhuizen, Nucl. Phys. B 140, 477 (1978). 6. R. Endo and M. Takao, Prog. Theor. Phys. 73, 803 (1985); Phys. Lett. B 161, 155 (1985}. 7. K. Fujikawa, Nucl. Phys. B 226, 437 (1983). 8. R. Endo and M. Takao, Prog. Theor. Phys. 78, 440 (1987). 9. K. Fujikawa, M. Tomiya and 0. Yasuda, Z. Phys. C 28, 289 (1985). 10. J. de Boer, B. Peeters, K. Skenderis and P. van Nieuwenhuizen, hepth/9509158. 11. B. S. De Witt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965).

354

PHASE OPERATOR PROBLEM AND AN INDEX THEOREM FOR Q-DEFORMED OSCILLATOR Kazuo Fujikawa Department of Physics, University of Tokyo Bv.nkyo-ku, Tokyo 113, Japan The notion of index is applied to analyze the phase operator problem associated with the photon. We clarify the absence of the hermitian phase operator on the basis of an index consideration. We point out an interesting analogy between the phase operator problem and the chiral anomaly in gauge theory, and an appearance of a new class of quantum anomaly is noted. The notion of index, which is invariant under a wide class of continuous deformation, is also shown to be useful to characterize the representations of Q-deformed oscillator algebra.

1

Phase operator

The notion of the phase operator was introduced by Dirac in his study of the second quantization of the photon field[l]. The free classical Maxwell field is equivalent to an infinite set of oscillators

H

= ~ Lw(k)[af>.ak>. +ak>.af>.]

(1)

k,>. and the essence of the phase operator is understood by considering the simplest one-dimensional harmonic oscillator h

= (2)

=

where a and at stand for the annihilation and creation operators satisfying the standard commutator

[a,at]

=1

(3)

The vacuum state IO) is annihilated by a

aiO) = 0

(4)

which ensures the absence of states with negative norm. The number operator defined by

(5)

355

then has non-negative integers as eigenValues, and the annihilation operator a is represented by

a = J0)(1J + 11}(21¥'2 + l2}(3lv'3 + ....

(6)

in terms of the eigenstates Jk} of the number operator

Nlk}

= kJk}

(7)

with k = 0, 1, 2, .... The creation operator at is given by the hermitian conjugate ~fa in (6). . Indez and Phase Operator

In the representation of a and at specified above we have the index condition

dim ker at a- dim ker aat

=1

(8)

where dim ker at a ,for example, stands for the number of norma.Iizable basis vectors u,.. which satisfy afaun = 0 ;dim ker at a thus agreeswith the number of zero eigenValues of the hermitian operator a.t a. In· the conventional notation of index theory, the relation (8) is written by using the trace of well-defined operators as

(9} with M 2 standing for a positive constant. The relation(9) is confirmed for the standard representation(6) as

(10) n=l

independently of the Value of W. If one should suppose the existence of a well defined hermitian phase operator ¢, one would have a polar decomposition

(11) as was originally suggested by Dirac(1]. Here U and H stand for unitary and hermitian operators, respectively. If (11) should be valid, one has

(12)

356

which is unitary equi'Va.lent to at a= H 2 ; ata and aat thus have an identical number of zero eigen'Va.lues. In this ca.se, we have in the same notation a.s (9)

=

Tr(e-at4/M2)- Tr(e-4at/M2) Tr(e-H2/M2)- Tr(e-ua,ut;M2)

=0

(13)

This relation when combined with (9) constitutes a proof of the absence of a hermitian pha.se operator in the framework of index theory(2). In the cla.ssica.llevel, one may define

a

= ei?JN = ~(ip+q}

(14)

or p

=

q

=

VNsin¢ VNcos¢

(15)

This transformation is confirmed to be canonical, and thus one ha.s a Poisson bracket (16) {N,t/J}PB 1

=

which naively lea.ds to a commutator

[N,¢1 = i

(17)

and the well*known uncertainty relation

(18) The absence of the hermitian pha.se operator ¢ shows that this uncertainty relation ha.s no mathematical basis[3]. The ba.sic utility of the notion of index or an index theorem lies in the fact that the index a.s such is an integer and remains invariant under a wide cla.ss of continuous deformation. For example, the unitary time development of a and at dictated by the Heisenberg equation of motion, which includes a fundamental phenomenon such a.s squeezing, does not alter the index relation. Another consequence is that one cannot generally relate the representation spaces of annihilation operators with different indices by a unitary transformation.

357

2

Various phase operators and their hnplications

Although the hermitian pha.se operator , as was introduced by Dirac, does not exist, there exist various interesting proposals of alternative operators which may replace the pha.se operator in practical applications. We here commnet on two ·representative ones from the ·view point of index relation. !.1 Phase Opemtor of Susskind·Glogower The pha.se operator of Susskind and Glogower[3} is defined by .

1

e"" =

=

-/N+1

a

10){11 + 11}(21 + 12}(31 + ....

(19)

in terms of the eigenstates jk} of the number operator in (7}. This pha.se operator is related to the operator a in (6) by a= tf'{1 N 1 12 • The above operator satisfies the relations ei~P(e'I(J)f

=

1

(eil(1)teil(1

=

1 -10){01

#

1

(10)

and apparently cp is not hermitian. The analogues of cosine and sine functions, which are hermitian and thus observable, are defined by

C(cp} S(cp)

(21)

These cosine and sine operators satisfy an anomalous commutator 1

[C(cp),S(cp)} = 2iiO}(OI

(22)

and an anomalous identity

(23) The modified trigonometric operators also satisfy the commutation relations with the number operator N,

[N,C(cp)} = -iS(cp) [N,S(cp)} = iC(cp)

(24)

358

For a representation with non~negative N in (19),one obtains the index relation

dim ker ei"'- dim ker (ei"')t:::::: dim ker a...., dim ker at= l

(25)

namely, the operator ef·'P carries a unit index. The index relation written in the form

(26) which is in agreement with (20), is directly related to the above anomalous commutator (22) and the anomalous identity(23).

!.! Phase Operator of Pegg and Barnett One may consider an s + 1 dimensional truncation of annihilation and creation operators

a, =

al

=

10}{11 + 11}(21¥2 + .... + 11}{01 + f2}(1fv'2 + ..... +

Is -

1}{slv'S ls){s- 1fy'$

{27)

and then lets -+large later. In this case, one has the index condition TrHl(e-a!a./M~)- Trs+l(e-a.a!/M 2 )

=0

(28)

in terms of s+1 dimensional trace Tr,+ 1 ; ker at a= {10)} and ker aa.t ={Is}}. In fact one can confirm that any finite dimensional square matrix carries a vanishing index. The operators a! and a8 thus satisfy a necessary condition for the existence of a hermitian phase operator. Pegg and Barnett proposed[4] to define a hermitian phase operator by eit/1 = 10}(11 + f1}{2l + ... +Is- 1)\sl + ei(s+l)~fs){Of

where ~ is an arbitrary constant

+ [N,sin]

=

(s + l)[ei(•+l)~ols){OI- e-i(•+l)~oiO){sl] 2 icos

i (s + l) (ei(•+l)~o ls)(OI + e-i(•+l)~o IO){sl] 2

(32)

and [cos¢, sin¢] cos2 + sin 2 t/>

=

0

= 1

{33)

The last expression shows that these operators satisfy normal algebraic relations. !.3 Physical Implications of Various Phase Operators The state Is} is responsible for the vanishing index in (28) and thus for the existence of the hermitian phase operator in {29). In other words, the hermiticity critically depends on the state Is} and the phase operator is cut-off sensitive. The state Is) does not decouple even for large cut-off parameter s from the hermitian phase operator ¢. It has been shown[2] that the algebraic consistency, i.e., the presence or absence of minimum uncertainty states is a good test of the effects of Is} and the index idea. In fact, one can show the uncertainty relations for a physical state IP)

ll.Nt:uint/>

~ ll.Nll.S(v>) ~ ~l(piC(v>)IP)I = ~l(plcosIP}I

{34)

~ ll.C(tp)ll.S(v>) ~ ~l{piO}(Oip} ~ 0

(35)

ll.cosll.sin

These relations show that the uncertainty relations are always better satisfied for the operators C(v>) and S(v>) of Susskind and Glogower which exhibits "anomalous " relations {22) and {23). In particular,for the states with small average photon numbers, the relation in {34), for example, becomes an inequality[2] ll.Nll.sin¢

> ll.Nll.S(v>)

~ ~l(piC{v>)IP}I = ~l(plcosjp}l

(36)

360

due to the effect of the state Is} even for arbitrarily large s, In other words, one cannot generally judge if the measurement is at the quantum limit or not if one uses the phase operator 4> of Pegg and Barnett, since one cannot achieve the minimum uncertainty in terms off/>. The operator tp, which satisfies anomalous relations in (22) and (23) , is in fact more consistent quantum mechanically; one can judge the quantum limit of measurements by looking at the uncertainty relation (36).

3

Analogy with chiral anomaly and a new class of quantum anomaly

It is known(5) that the chiral anomaly is characterized by the Atiyah-Singer index theorem(6)

(37) where the Pontryagin index 11 is expressed as an integral of FF, if one defines a by the chiral Dirac operator

(38) A failure of a unitary transformation to an interaction picture specified by a vanishing index

(39) with a0 = -y~'8~'(1 + -rs)/2 leads to the anomalous behavior of perturbation theory. Also, the failure of decoupling of the cut-off parameter such as the Pauli-Villars regularization mass is well known: If the regulator mass is kept finite one obtains a normal relation, but in the limit of infinite regulator mass, the mass term gives rise to the chiral anomaly. The failure of the decoupling of cut-off also takes place in the phase operator . One may rewrtie the index relation (9) as

(40) where Tr. stands for a trace over the first s dimensional subspace, and the right hand side of this equation comes from the contribution of the state Is}. By taking s -+ oo in this expression, one recovers the index condition (9). The effect of the cut-off parameter Is} in the limit s -+ oo gives rise to the non-vanishing index; this is the same phenomenon as the chiral anomaly in the Pauli-Villars regularization. It has been proposed to regard the non-vanishing index (9) and the associated anomalous behavior in (22) and (23) as a new class of quantum

361

anomaly[2), which arises from an infinite number of degrees of freedom associated with the Bose statictics; one can put an arbitrary number of photons in a specified quantum state. In this view point, the absence of the hermitian phase operator is an inevitable quantum effect, not an artifact of our insufficient definition of the phase operator. We have shown in (36) that the apparently anomalous behavior exhibited by the operator tp in (22) and (23) is in fact more consistent with the principle of quantum mechanics.

4

Index as an invariant characterization of Q-deformed oscillator algebra

The index such as

dim ker a- dim ker at= 1

(41)

is expected to be invariant under a wide class of continuous deformation of the operator a. The index may thus provide an invariant characterization of representations of Q-deformed oscillator algebra[7). We analyze this problem by considering the Q-deformed algebra

[a, at] = [N+1]-[N] [N,at] = at [N,a] = -a where

[N]

= qN

{42)

-N

-q q- q-l

(43)

with q standing for a deformation parameter. The above algebra satisfies the Hopf structure, and the Casimir operator for the algebra (42) is given by[8]

C

= ata- [N]

{44)

and thus [N] and ata are independent in general. Practically the above Qdeformed oscillator is useful to study a Q-deformed SU(2) in the Schwinger construction[9]. To deal with a general situation, we start with a matrix representation of the above algebra defined by 00

a

=

'Ev'fkllk-1}(kl k=l 00

at

=

L: v'fkllk}(k- 11 k=l

362 00

N

=

Lklk}(kl A:=O

c =

(45}

ata-[N]=O

where the state vectors stand for column vectors such as

IO) =

0), u), I!)=

(46)

We examine several cases specified by a different deformatiom parameter q.

l.q>O In this case, (k] ::/: 0 for k ~ 1, and one has an index condition

dim ker a- dim ker at= 1

(47}

since ker a = {IO}} and ker at = empty. We thus have no hermitian phase operator, and the operator defined by 1

-y:;;:::[N::=+=.1] a =

10}{11 + 11}(21 + ...

{48}

becomes identical to that of Susskind and Glogower. Namely, not only the index but also the phase operator itself are invariant under Q-deformation[10]. The deformation with lql = 1 is also known to be allowed. We thus examine

11-1, q = e2" 11 ,9 =irrational In this case, [k]

= qk- q-k = sin 211'k9 q - q- 1

sin 211'9

::/: 0

(49}

fork ~ 1, and thus the index as well as the phase operator are the same as for q > 0.

11-2, q = e2 "i', 9 = rational

363

In particular, we concentrate on the case where 8 is a primitive root of unity[ll], 8 = _1_ (50) s+l with s standing for a natural number. In this case, we have [s+ 1] =

q•+I - q- 1 has

= 1 + _!_[22] + q2 + q4 [24] + (q)2 .

(q)4

been conjecturedf and proved~ 6

qs + q6 + q1 + qg[26} + •... {q)e ·

(5)

In each term, the factor affected by level k is only. the kink factor. The reason of this fact is explained in the following manner. Even in higher spin (S k/2) XXX models, the particles are still of spinS= 1/2. They, however, interpolate k + 1 different ground states (domains), say a = 0, 1, • · ·, k, as a-+ a± 1 (0 . X= x- R, with [X, Y) = -i, [Px.Py) = i. Here, we have used quantities with breve because they are bosonized-electron variables. Note that the same formulas as we give below follow also for electron variables without breve. It is convenient to make two independent sets of harmonic oscillators, (11)

satisfying [a, at)= [b, btl= 1 and [a, b)= [a, btl= o. The LLL projection is to suppress the relative motion of the electron, which is achieved by imposing the condition (12)

on the state in the Hilbert space. 5 W"" xSU(2) Algebra

In various places we deal with operators of the form Q.\ =

Jd x.\(x)p(x), 2

(13)

where .\(x) a c-number function. The state Q.\ I'I') does not belong to the lowest Landau level even if 1'1') does, because (14)

It is necessary to project .\(x) to the lowest Landau level, whose result we denote by A(x). We do this in a systematic way as follows. We make a Fourier transformation of .\(x),

)-J

~( "x -

d2q

(2rr)2 e

ixq~

"lf·

(15)

The above LLL projection is reduced to that of the plane wave eixq. We separate the coordinate x into the guiding center X and the relative coordinate R, or equivalently into the two sets of operators (a, at) and (b, bt),

388

with pAq

= EiiPiqj.

We make normal ordering as

with (18) The operator a acts on the state cp(x)l'l'} in (14) and vanishes because of the LLL condition (12). We eliminate the operator by inserting the completeness condition 1 =I l'l'i}{'l'd of the LLL states l'l'i). As a consequence, we get aCJia(x)QAI'I'} = aA(x)CJ!a(X)I'I'} = A(x)aCJ!or(X)I'I') = 0, (19)

at

in place of (14), where

(20)

We have also defined (21)

which we call the LLL projected density operator. Similarly, we can make the LLL projection of the SU(2) generator sa, which we denote by Sq in the momentum space. As we have stated, the dynamics arises solely from the commutation relation [X, Y] = -i, which leads to the Woo algebra. With the pseudospin incorporated, the WooxSU(2) algebra, [pp, p4 ) = 2ifip+4 e01 2>P4 sin [

[Sp, fi4 1= 2iSp+4 eOt 2>P4 sin (

p;q ],

";q ].

(22) (23)

= iEabc §cP+4 eU/2)P4 COS [ pAq] + !:_Oabp~ ell/2)P4 sin [ pA q] (24) [SapI §b) 4 2 2 P+4 2 I governs the dynamics of the bilayer system.

389

6 Ground State The Coulomb term in the Hamiltonian(!) has the formof(l3), and needs the LLL projection. After the LLL projection the Hamiltonian becomes A

He=

I

rr) 2 V+(tl)P-qPq + 2 21 (2d2q A

A

I

d 2q (2 rr) 2 V_(q)S_ 4 S4 , A3

A3

(25)

in the momentum space, where V±(q) is just the Fourier transformation of the potential V±(x- y), with V± = !z + PA(o ai + __o uz 2 k 2 k 2C '

(32)

where lui « 1 but 0 is not small in general. This is the effective Hamiltonian governing the dynamics of the Goldstone mode. The fields O(x) and u(x) in the above effective Hamiltonian are classical fields. However, the canonical commutation relation naturally follows,

ih Po

[u(x), O(y)) = -6(x - y).

(33)

391

To derive it we evaluate the equation of motion

We then take its expectation value by the state lc$) =ei0 1go). For a smooth pseudospin configuration, we use the expression (28) to obtain 2Po,... -da=PAn2,..._e 17 v v v dt Po C '

d PE 2a -U=--V' 17. dt

Po

(35)

It is trivial to check that this set of equations agree precisely with the Heisen-

berg equations of motion derived from the Hamiltonian (32) together with the commutation relation (33). The Goldstone mode is a superfluid mode. Diagonalizing the Hamiltonian by the Bogoliubov transformation we obtain the excitation mode E(k)=

(36)

where M* =Pol JPAPE and 1J = PEM* /4C. Thus, the excitation mode has a linear dispersion relation as k - 0, which is the superfluid mode. 10 Electric Currents We analyze the current in each layer in the presence of the external It is convenient to define electric field

1r

Er.

] ± _ _

i -

_!_('''tp., + ,,tp .• , 2) _ !(Jl + ]2) 2M ., 1 I.,1, 1 ., 2 1., - 2 i - i •

(37)

The current is defined as the Nother current, but it may equivalently be defined in association with a local phase transformation: Jj is associated with the in-phase transformation generated by p, fJJa - eif!x)cpa, while Jiis associated with the out-of-phase transformation generated by S 3 , fJJ1 eif 0. The exponential growth implies rapid excitation of high harmonic oscillator levels. With the short time average it can be interpreted that particle production takes place with (Nw) ex: e>..mEt. After coarse graining of the time average one can discuss the decay law of the initially prepared ground state: it follows the exponential form, 00

rn =

me 2V

""' L...J ken-band

(6)

r is the total decay rate per unit volume and per unit time. Computation and interpretation of the decay rate r n of the n-th band is our main task in the rest of discussion. 2

Small Amplitude Analysis Revisited It is customary to recast the classcial equation into a dimensionless form,

d2 u dz 2

z = met/2'

+ [h- 294 cos(4z)- 29y sin(2z) ]u = 0, f2 + 2 2c2 2c2 h=4 2m + 294 ..20 ' (J4 = 94~0 ' me

me

me

(7) 9y

= 29Yeo ,

(8)

me

with m the mass of the bose quantum field 1(). Following the general theorem for solutions of differential equation with periodic coefficients, one expands the solution in the form 6 , 00

u(z) =

L

Ck e(>..+in+2ik)z'

(9)

k=-oo

with n = 1 , 2, 3 , · · · . The condition of the existence of non-trivial solution is then equivalent to the condition of non-vanishing matrix determinant of

397

infinite dimensions, with the diagonal entry of 'Yi = h +(..\+in+ 2ik) 2 , and the next off-diagonal entry of ±i9y and the next-to-next off-diagonal entry of - 84 and all others = 0. In the small amplitude limit of leol ~ 1 the n-th instability band starts at h n 2 • The boundary curve h h(n>(eo/m() dividing the stable and the unstable bands corresponds to the eigen-modes of the form, cos(nz) and sin(nz), in the eo - 0 limit. With the assumption that the coefficients, Co and C-n, are dominant and the rest of c", ..\, and h- n 2 are all small, one may simplify the structure of this matrix such that 'Yi = 'Y-n-i = - 4k(n+k) (k :1 0, -n) and 'Yo= h- n 2 + 2in..\, 'Y-n = h- n 2 - 2in..\. Dividing the matrix into a few parts, one first solves the top (k ~ 1) and the down (k ~ -n- 1) infinite dimensional parts in favor of c1 , c2 and C-n-1, C-n-2,

=

=

=

=

with e1 -iOy and e2 04 and the matrix inverse v- 1 defined as a limit of big matrix in the left-upper and the right-down (identical) corners. One next solves the central block ( -1 ~ k ~ -n + 1) for c-1 , c-2 , C-n+2 , Ln+l in terms of co , C-n. Here one needs to invert, ignoring subleading terms, 'Y-1

-

{1

-{2

0

- £i - £*2

'Y-2

- {1

-(2

0 0 0 0

0 0 0

0

0

- c; - ci 0

'Y-n+2

- c; - ci

0

=

- (1

0 f2C-n

'Y-n+1

f1C-n

- {2

0

cico

c;co

(11)

One finally solves the deter min ental condition for the equation co , c_n, which reads to the leading order as

h - n 2 + 2in..\ - A det ( -C*

- C ) h-n2-2in..\-A =0,

(12)

= 1Ed 2(Di"/ + E!/), C = E1(e1E!,~- 1 + e2E!,~- 2 ) + t:2(t:1Ei.~- 1 + t:2Ei.~- 2 ), {13) where E is the matrix in the left hand side of eq.ll.

398

=

=

=

We regard e1 O[e] and E2 O[e 2] since g4 fgy 0[1], and work out matirix elements to leading orders of E. The net result may be summarized as the formula for the growth rate A,

1

An=2n

A 2 _ (h _ n2 _ u.n

- i8y -1-1

Cn = det

0 0 0 0

82 2(n 2

84 -i8y

Y -

1)

0 84

)2

'

dn

ICnl = 22(n-l) [(n -1)!]2' (14) 0

0

0 0 0

0

-1-n+2

-i8y

0

-1-n+l

,(15)

with 1-Jt = 4k(n- k). This result generalizes the previous one in ref 4 in which 84 = 0 was assumed, hence Cn = (- i8y )n. The decay rate of the n-th band is computed by summing modes within the band in the narrow width approximation, (16) The case of n = 1 is an exception to this general formula that must be treated separately, resulting in

3

Physical Interpretation in terms of Familiar Perturbation Theory

We shall now offer interpretation of the decay rate r n of n-th band in terms of the ordinary perturbation theory. Ordinary perturbation uses the particle picture and for that purpose it is useful to recast the rate formula in terms of reaction rates of indivisual particles by dividing some powers of the particle number density n{ = ~ m{{~. Let us first note the dependence of the rate r n on the oscillation amplitude, ex: e~n' and the momentum of final I(J particles in the zero momentum limit of E{ - m{ for the process, n { - I(J I(J; 4;n 2, • This factor appears in the rate formula, eq.16. pV' = nm 2' n m,

Jt -

399

The simplest case is the decay rate of the first band ft. the rate per unit volume and per unit time. Since the unit volume contains ne particles, the one particle rate having the dimension of the inverse time is 4 rl_g}me 3271"

ne -

v~ . - mr .

(18)

This exactly coincides with the decay rate of one { particle computed in the conventional way using the Yukawa coupling to rp, ~ gyme {rp2 . The next thing to be checked is the decay rate from the 2nd instability band f2 ex {~. This time one divides the quantity f2 by since two { particles are involved,

n1

(19) In ordinary perturbation theory the amplitude for the 2-body process, {{ -+ rprp, consists of 3 distinct Feynman amplitudes, the contact term with a single quartic coupling, the t-channel and the u-channel exchange diagrams with two vertices of the Yukawa coupling, adding to -

2 • 2 2( 1 21.94zgyme - - - 2 +

t-m

1 ) 2 · u-m

(20)

In the zero momentum limit of Ee --+ me, t- m 2 = u- m 2 -+ - m1, giving the total amplitude 2i(g}- g~). Working out the phase space factor, one finds out that the invariant rate VreJ (f given by flux x cross section is identical to f2/n~ given above. To proceed to the general n-th order case, one notes that the propagator in the zero momentum limit is given by

0:::::=

1

i Pi- q1 )2- m2

--+

- i k(n- k) m1'

(21)

where Pi is an initial { momentum and q1 is one of the final rp momenta. Hence the n-th order invariant amplitude containing the Yukawa couplings alone is

i (gyme)" nrr-l 1 ig; 1 k(n- k) = mn-2 [(n- 1)1]2 . m2(n-l) (

k=l

(

(22)

•

It is not difficult to check that this leads to the invariant rate precisely equal to the corresponding decay rate of the n-th band, r n/n{. What is left to be

400

shown is then the relative weight of the Yukawa and the quartic contribution. Subdiagrams of cpcp for the whole ne - cpcp process contribute with a • 2 factor, -,./:,"!::,.) for the Yukawa coupling case, considering the propagator above, and with -ig~ for the contact quartic coupling case. The ratio of these two terms is exactly equal to the ratio in

ee -

det (

2i

.k

- YYmt -4k(n- k)

2~

94 mt

-2igy~

)

=-4k(n-k)[

gy -g~](~) 2 , (23) k(n- k) me 2

&

that appears in the decomposition of the submatrix of Cn, eq. 15 for the formula r n. This proves our assertion that the decay rate of the n-th band r n/n'l is equal to the zero momentum limit of the invariant rate computed in the ordinary perturbation theory. Needless to say, this interpretation is valid only in the small amplitude limit. In applications to realistic problems analytic formula in the large amplitude regime is indispensable. In cosmological application the parameter region 294 , or more precisely h - 294 ( 4 E2 *T 2 ) < 94 with the large along h me amplitude of 94 ~ 1 is most important, whose case has been worked out in refs. In the present note we clarified the physical meaning of the n-th band 0 limit. In view of our result here decay formula when it is taken to the the large amplitude formula in refs may be considered non-perturbative effect which is directly joined to the present perturbative result.

=

=

eo -

1. L. Kofman, A. Linde, and A.A. Starobinsky, Phys. Rev. Lett. 73,

3195(1994), and references therein. 2. I. Affleck and M. Dine, Nucl. Phys. B249, 361(1985). 3. G.D. Coughlan, W. Fischler, E.W. Kolb, S. Raby, and G.G. Ross, Phys. Lett. 131B, 59(1983). 4. M. Yoshimura, Prog. Theor. Phys. 94, 873(1995). 5. H. Fujisaki, K. Kumekawa, M. Yamaguchi, and M. Yoshimura, Particle Production and Dissipative Cosmic Field, TU /95/488 and hepph/9508378. 6. For instance, E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, (MacGraw-Hill, New York, 1955).

401

Wilson-type Renormalization Group Study of Fermions Interacting with Dissipative Gauge Field at Finite Temperature H.Takano Joet.u Uniuer1ity of Education ,Joet6u,Niigata,9,49 Japan M.Onoda and I.Ichinose /nditute of Phy1i1c1, Uniuer1ity of Tolcyo,Komaba, Tokyo,159 Japan T.Matsui Department of PhyliCI,Kinlci Uniuerlity,Higa•hi-O,alca 577,Japan We extended the Wilson-type renonnalization group study of the system discussed by Onoda et ml 1 at T 0 to the cue of finite temperature. We study a twodimensional non-relativistic electron ayatem with diaaipative gauge theory. Renormalization constants are calculated in the one-loop level at finite temperature and the renormalization group equations of the parameters are obtained. We ahow that the renonnalized coupling constant which has the nontrivial infrared(IR) fixed point at zero temperature is growing up as the scale parameter increases at finite temperature.

=

1

Introduction

In high-Tc superconductors or in the fractional quantum Hall effect, the systems of two-dimensional non-relativistic fermions interacting with a gauge field are of current interest. Recently, it was suggested that the renormalizationgroup(RG) study is very efficient in investigating low-energy behaviors of nonrelativistic fermion system 2 • Onoda et al applied this method to the twodimensional non-relativistic fermions interacting with the dissipative gauge field 1 • This model is motivated by the work of Halperin et al. 3 A dissipative term of gauge field is induced by the quantum correction of non-relativistic fermion. In Ref 1 , it is shown that the non-trivial IR fixed point of coupling constants is generated in some parameter region by solving the renormalization group equations. In this article, the extension of Wilson-type RG study of our model to the finite temperature is proposed and the finite temperature corrections to these RG equations are calculated .

2

Model

We shall consider a model given by Onoda et al. 1 at finite temperature. This model is two-dimensional system of non-relativistic spinless fermion .,P(z,T) interacting of a dissipative gauge field Ao(z,T) (i = 1, 2).

402

In the standard imaginary time path integral formalism, the partition function is given by Z=

j [d~][d,P][dA]ezp( -s.,

The free part of fermion action

J

s., = ~ L

dp

n

(1)

-SA - Sint.a).

s., is

L ,P(a;p,wn)(-iwn + VFea · p)t/J(a;p,wn)

1

(2)

a

where _

/3-(kBT)

_1

_

,Wn=

(2n+ 1)11" {3

_ d2p ,dp=( 211")2"

(3)

When we shall consider excitations around the Fermi surface and perform the Wilsonian-type renormalization group approach, it is useful to introduce a momentum cutoff as shown in Figs.! and 2 of Ref 1 • In this case , fermion field is divided to N components labeled by a. The momentum p is measured from Fermi surface kF,a 1 and unit vector en points towards the center of the a-th part. The momentum cutoff A is introduced as IPI = ikF,a- kl . 1 FAf3eFAfJ'II ->. dylsinr/J(y)j (efJFA'II + 1)2 '

V(/3, FA, A)

a= 2~~ 7r:I1IB,

o

Y

=11Fea · q = 11Fqcosr/J, FA= T~A

(11)

;.From these equations, we can obtain the RG equations of a(t), 11F(t) and A(t), such as

~~t)

=

{1 - b)a(t)- H:a(f3(t), FA(t), A(t))a:l(t),

(12)

405

dvp(t) dt d.\(t) dt

=

= (2- b- H2({3(t), FA(t), .\(t))).\(t),

(13)

where {3(t) ::= {3e-t, FA(t) = v~~!JA. These equations are coupled differential equations of three parameters. In the zero temperature limit {3 --+ oo, the function H 2 takes the constant value, H 2 = 1, then for the parameter region 0 < b < 1, the coupling constant a(t) has the non-trivial infrared fixed points, a• = 1 - b. At finite temperature, the point a•(t) in which the l.h.s. of Eq.(12) ({3 function ofthe running coupling a(t)) takes the zero value depends on scaling parameter t such as

(14) By the definition of FA(t) and from the Eq.(13), the scaling of FA(t) has the form such as e< 6- 2 )t. Then from the definition of Eq.(ll), the function H2(f3(t), FA(t), .\(t)) becomes to zero as the scale parameter t becomes larger. We find that the renormalized coupling constant a•(t) is running to the strong coupling region at which perturbative calculation breaks down.

4

Conclusion and Discussions

We have extended the Wilson-type RG study for 2-dimensional fermion system with dissipative gauge field to the finite temperature. We obtain the renormalization group equations of the parameters, a(t), vp(t) and .\(t). We find that for a parameter region 0 < b < 1 the coupling parameter a(t) which has the IR fixed points at zero temperature goes to the strong coupling region in the infrared limit at finite temperature. It suggests that this fermion system has a scale t•, such as a*(t*) ""'0(1) and this scale may be related to some phase structure. The same thing is occurred in the model of anisotropic C P 1 model in (3+1) dimensions at finite temperature. In that case, the the scale t• is related to the correlation length and it plays the role of Neel order parameter of spin fluctuation, see appendix A in Ref 4 • On the other hand , at the point of a*(t*)""' 0(1) the perturbation calculation breaks down, so it may be happen that higher order loop correction suppress the growth of the coupling. In this article, we have discussed only about the coupling flow for special region of value b. We can find the low-energy behavior of fermion propagator and discuss about the picture of quasiparticle of the system and what kind of liquid does the system becomes for example Luttinger liquid ,marginal Fermi liquid

406

or ordinary Fermi liquid. The more detailed calculation and discussion is in the Ref. 1 and a forthcoming paper 6 • References 1. M.Onoda,I.Ichinose and T.Matsui Nucl. Ph:g11. B 446, 353 {1995) 2. R.Shanker,Phy•icll A 177{1991)530; Rev.Motl.Phyl.66{1994)129. 3. B.I.Halperin,P.A.Lee and N.Read, Phy11. Rev. B 47, 7321 {1993). 4. H.Yamamoto,G.Tatara,I.Ichinose and T.Matsui Phy11. Rev. B 44, 7654 {1991). 5. I.Ichinose and T.Matsui Nucl. Phy11. B 441, 483 {1995). 6. H.Takano, M.Onoda,I.Ichinose and T.Matsui in preparation.

407

FIELD THEORY ON VON NEUMANN LATTICE KENZO ISHIKAWA, NOBUKI MAEDA Department of Phv•ic11, Hokkoido Univer•itu, Sopporo 060, Jopon

Professor Kikkawa has been working on many topics of quantum field theory. Field theory is a key ingredient in understanding many body systems. Soliton quantization is one of them 1 , which I had an overlapp in research with Professor Kikkawa in 1970's. Sakita and his collaborators 2 introduced collective coordinates based on path integral method and Kikkawa and his collaborator 1 gave its counterpart in canonical method. I extended them into manifestly Lorentz covariant collective coordinates. 3 One of the key ideas there was to find good coordinates and a good gauge fixing. Physics becomes easy, transparent, and tractable with such variables. One of the most exciting physics ofrecent years which has deep connections with field theory is the quantum Hall effects, integer Hall effect 4 and fractional Hall effect. 5 I have been working on them for some time 6 •8•9 •10 and have had nice discussions with him occasionaly. So I gave a short summary of my works on the quantum Hall effects and dedicate it to Professor Kikkawa on his sixtyth birthday. In quantum Hall effect, physics becomes tractable also if such good variables are found. I believe that a description using von Neumann lattice may correspond to it. Ward-Takahashi identity and other rigorous field theoretical relations become transparent forms, in fact, in this method. von Neumann lattice is a minimum complete subset of coherent states. 7 Whole set of eigenfunctions of an annihilation operator of a harmonic oscillator is complete but is overcomplete. A subset of eigenfunctions whose eigenvalues are defined on lattice sites on complex plane is also complete if the lattice spacing is smaller than or equal to a critical value. The lattice with the critical spacing is called von Neumann lattice. They have interesting and usefull properties and are used by us for studing the quantum Hall effect. Based on von Neumann lattice representation of two dimensional electrons in the magnetic field we have succeeded 8 •9 in giving the exact low energy theorem about the Hall conductance, ul/: 11 , in quantum Hall regime, where there is no two dimensionally extended states around Fermi energy. Namely u ICII agrees to a topological invariant of mapping defined by the propagator and is given by exactly quantized integer multiple of e2 /h in the quantum Hall regime. It is amazing to see that many body interactions and disorders give no correction to the quantized Hall conductance. For its proof, the current conservation and equal time commuta-

408

tion relations between field operators and charge density play important roles. They are combined with multi-pole expansions and lead Ward-Takahashi identity between the vertex part and the propagator. Hall conductance G'zy is given by the slope of the current correlation functions at the origin of the momentum and satisfies the exact low energy theorem owing to Ward-Takahashi identity. We see now how von Neumann lattice appears in quantum Hall system. 9 One-body Hamiltonian of non-relativistic two-dimensional electrons in the strong magnetic field is given by, (1)

and is expressed with two commuting sets of non-commuting variables,

1 e= eB(p

11

1

+ eA11 ), '1 =- eB(Pz + eAz),

(2)

ih

[e' TJ) = - eB '

x

= x-

e. Y = Y- TJ,

(3)

[X, Y) = ih

eB

as (4)

Coherent states on von Neumann lattice in guiding center coordinates (X,Y) is the following :

(5) AIO) = 0, A=

~(X+ iY),

[A,At] = 1.

They are complete set. The detailed properties are given in Ref.9. Electron field is expanded with base functions that are obtained from the products between the eigenstates of one-body Hamiltonian Eq.(l) and the coherent states on von Neumann lattice and expansion coeflicents, a,(m, n), are regarded as field operators of Landau level, l, and lattice spatial coordinates, (m, n). The theory is described by a parity breaking lattice field theory of an internal freedom.

409

Propagators,vertex functions ,and other Green's functions have dependence upon Landau level index in addition to momenta. Spatial components of the momentum are defined on compact torus. In this representation we have shown the following theorems. 9 •10 Theorem 1 The Uz'fl is given by a topological invariant of the mapping defined by the propagator in the momentum space, (T Z'fl

=

e2

_1_

h 24

71'

2

jdPf

IJV

Tr[os-t(p)S( )os-t(p)S( )os-t(p)S( )] p $:1 p $:1 p $:1 p I vp/J vPv vpp

(6)

and is quantized exactly as (e 2 /h)N in the quantum Hall regime. Theorem 2 In systems of random short range impurities one particle states are localized if their energy E satisfy,

IE- Ed> 6,

(7)

where E, is the Landau level energy and 6 is a small constant determined by impurities. In finite system , one dimensionally extended states exist around edges and have energies which bridge from one Landau level to next Landau levels. If the system has a suitable size, the quantium Hall regime does exist, neverthless. Theorem 3 A finite size correction disappears in the sigma xy at the quantum Hall regime. Theorem 4 The quantum Hall regime disappears if the current exceeds a critical value and breakdown of quantum Hall effect occurs. The critical Hall electric field is proportional to 8 3 12 and the proportional constant has no dependence on Landau levels. We are developing a mean field theory 11 of the fractional Hall effect based on von Neumann lattice representation. Since Landau level electrons are represented on the lattice which has spacing fixed by the external magnetic field.

410

Continuum electron system are mapped to lattice electron systems. They give a clear picture about fractional Hall effect from former works on twodimensional electrons in the magnetic field of tight binding model, such as Hofstadter's work. Readers who are interested in more details should consult works of the present authors and his collaborators. Acknowledgments

One of he present authors (K.I) thanks Professor Keiji Kikkawa for his warm encouradgements for many years. This work is supported by Grand-in-Aid for general scientific research, (07640522), the Ministry of Education, Science and Culture, Japan. References

1. A. Hosoya, and K. Kikkawa, Nucl. Phys. B 101, 271 (1975). 2. J.L. Gervais, and B. Sakita, Phys. Rev. D 11, 2943 {1973). 3. K. Ishikawa, Nucl. Phys. B 107, 238,253 (1976); Prog. Theor. Phys. 58, 1283 (1977). 4. K.V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980); S. Kawaji, and J. Wakabayashi, in Physics in High Magnetic Fields, edited by S. Chikazumi and N. Miura(Springer-Verlag, Berlin, 1981). 5. D.C. Tsui, H.C. Stormer, and A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1992). 6. K. Ishikawa, Phys. Rev. Lett. 53, 1615 (1984), Phys. Rev. D 31, 1432 (1985); 0. Abe, and K. Ishikawa, in Rationale of Being : Festschrift in honor of Gyo Takeda, edited by K. Ishikawa et al. (World Scientific, Singapore, 1986)p.137. 7. A.M. Perelomov, Theor. Mat. Fit. 6, 213(1971); V. Bargmann et al., Rep. Math. Phys. 2, 221(1971). 8. K. Ishikawa, and T. Matsuyama, Z. Phys. C33, 41(1986); Nucl. Phys. B 280, 523 (1987). 9. N. Imai, K. Ishikawa, T. Matsuyama, and I. Tanaka, Phys. Rev. B 42, 10610 (1990); K. Ishikawa, Prog. Theor. Phys. Suppl. 107, 167(1992). 10. K. Ishikawa, N. Maeda, and K. Tadaki, Phys. Rev. B 51, 5048 (1995); Phys. Lett. A 210, 321 (1996) 11. K. Ishikawa, N. Maeda, H. Suzuki, and P. Wiegmann, in preparation.

Pro!! ram : Chairman· A SuganlClto _ ----,,c,;oo;;;-'C"'hru..,rrn=an~'--T'"_-"In,anu,_·-------jl : Chainnan: K. Shizuya 9 00 9 00

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Opening Address

Comecting the Superstring wJth the Standard Model

y_ Nambu

P. Ramond

Phase operator problem and an index theorem for Q-ddormcd oscillator

10:00

J H. Schwarz f-10-:5_0______________ Du

=~-Chairman: M. Fukuma

11:00

II :l 5 Schwinger-Dyson Equation and String Field Theory

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A. Tseytlin

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extreme black_h_o_l_es

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Exploring gauge theories with supersymmeLry

K. Intriligator

~--------+1-2-:00 _ _ _ _ _ _ _ _ _ _ __

New Loop Equations for Ising

Prepotentials of N=2 Supersymrnetric Gauge Theories and Soliton Equations

Matter Coupled 10 2D Gravity and Non-Critical String Field Theory

H. Ooguri

T. Eguchi

R. Nakayama

50

12:4.5

12:45

Lunch

Photo

Lunch Lunch

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14:15 Critical behaviour of two~dimensional quantum R2-gravity near the flat metri' Integrability and SW theory

Enjoying String ·String Duality

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S. Kachru 1505

A. Morozov

I Ward ldenl!Ues of theW..., Symmetry

St~e~~d ~lobal ~~of S-Duallty -

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Rey

K Hamada

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Massless Hyperrnultiplet

Klto

Fusion Rules and lnteractions for Macroscopic Loop Amplitudes

Qlaimlan: H. Kunitomo

H_ Itoyama

16:20

I

16:00 Integrable System and N=2 Super· syrnmetnc Yang-Mills Theory

--two

T Nakatsu

16:30

2+E Dimensional Quantum Gravity

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M. Ninomiya

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l?:OOChainnan: N. Kawamoto ~ _ ,Qlaiffilan:_'(. Y~

17 00

16:50 Renorrnalizability of Quantum Gravity near Two Dimensions

Strings in Computer

Y. Kitazawa

GKZ-Hypergeometric System and Applications to Mirror Symmetry

T Yukawa

S. Hosono

-- -----------+-------------!

17:20 Multi*plaqueue solutions for a discretized Ashtekar gravity

i 17:45

! 1

K. Ezawa

Fractal Structure of Space·Time in Two--dimensiOnal Quantum Gravity

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1--------- ---19:00

Prepotentials in N=2 SU(2) Supersynunetric Yang-Mills Theory w1th

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and Higher Genus Amplitudes m 2D Stnng Theory

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ll50

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V. Kazakov

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TopologiCal Prut1U0n Function of BPS In

~- Chairman: N. Ishibashi

14:00

--- ""11 18:30

l7:45 Lowest Landau level projection and W·infinity algebra in bilayer Quantum Hall system

Y____w_a_ta_b_iki_-_ _ _. L_ _ _ z__F___E_z_a_w_a_________ 18:30

T. Yoneya 10 JO

Chairman: M. Kato ~ ------ --·-- II :00

H. Kawai

Five-branes and singularities of K3

b

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L Bars

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f--wo-·---'

I 9:45 I String Theory and Matnx. Models

Fundamental strings and

U-duality and non-pcnurbative tnulliplets of the superstring

Superstrings from Eleven Dimensions

W _ Gauge Theory

_ _ __:B::c·cc:Sakita

9 =45

Chairman: Y. Kazama

9,50

I

K Fujikawa

Closing address

K Kikkawa

Excursion

Frontiers • 1n Quantum Field Theory Toyonaka, Osaka, December 14-17, 1995

Topics: dualities in string theory, matrix models and noncritical strings, lower dimensional quantum gravity, topological and superconformal field theory and ... Speakers: I. Bars T. Eguchi K. Ezawa Z.F. Ezawa K.Fujikawa K.Hamada S. Hosono K. Intriligator K.Ito H. Itoyama

S. Kachru H. Kawai V. Kazakov K. Kikkawa Y. Kitazawa A. Morozov T. Nakatsu R.Nakayama M.Ninomiya H. Ooguri

P.Ramond S. -J. Rey B. Sakita J. H. Schwarz A. Sen A. Tseytlin Y. Watabiki T. Yoneya T. Yukawa

Supported in part by Nishina Memorial Foundation and Inoue Foundation for Science.

Organizers: H. Itoyama M.Kaku H.Kunitomo M.Ninomiya H. Shirokura

Phone : +81 6 850 5343 Fax : +81 6 850 5379 e-mail : inquiry9funpth.phya.sci.osaka-u.ac.jp

LIST OF PARTICIPANTS AT THE CONFERENCE

I. Bars A. Filippov K. Intriligator S. Kachru M. Kaku V. Kazakov A. Morozov Y. Nambu

H. Ooguri P. Ramond s. -J. Rey B. Sakita J. H. Schwarz A. T. A. K.

Sen Tada Tseytlin Ishikawa

N. Kawamoto R. Nakayama H. Suzuki M. Yoshimura Z. F. Ezawa

Univ. of Southern California JINR Institute for Advanced Study Harvard Univ. CCNY Ecole Normale Superieure ITEP E. Fermi Institute, Univ. of Chicago Berkeley Universtiy of Florida Seoul National Univ. Physics, CCNY, New York, NY10031 California Institute of Technology Mehta Research Institute U.C. Santa Barbara Imperial College Hokkaido Univ. Dept. of Physics Hokkaido Univ. Dept. of Physics Hokkaido Univ. Dept. of Physics Hokkaido Univ. Dept. of Physics Tohoku Univ. Dept. of Physics Dept. Tohoku Univ. of Physics

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] morozov@desyvax. bitnet nambu@chaos. uchicago. edu [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] .res .in [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] p [email protected]

414

S. Watamura R. Endo T. Eguchi K. Fujikawa S. Iso T. Yanagida C. Xiong K. Hori Y. Sun T. Kuroki T. Yoneya Y. Kazama M. Kato Y. Sato A. Kato T. Tani T. Asatani

Tohoku Univ. Dept. Physics Yamagata Univ. Dept. Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo, Institute Physics Univ. of Tokyo, Institute Physics Univ. of Tokyo, Institute Physics Univ. of Tokyo, Institute Physics Univ. of Tokyo, Institute Physics Univ. of Tokyo Dept. Physics Univ. of Tokyo, Institute Physics Univ. of Tokyo, Institute Physics

of

[email protected]

of

siDOS@kdw .kj .yamagata-u.ac.jp

of

[email protected] [email protected] iso@danjuro. phys.s. u-tokyo.ac.jp

of of

of

yanagida@danjuro. phys.s. utokyo.ac.jp xiong@danjuro. phys.s. utokyo.ac.jp hori@@danjuro. phys.s. utokyo.ac.jp [email protected] kuroki@hep l.c. u-tokyo.ac.jp

of

[email protected]

of

[email protected]

of

[email protected]

of

[email protected]

of

akishi@ms. u-tokyo.ac.jp

of

[email protected]

of

[email protected]

of of of of

415

F. Sugino S. Hirano Y. Sekino S. Hikami S. Higuchi K. Ino S. Tamura T. Sato Y. Kitazawa Y. Watabiki M. Natsuume M.Abe M. Asano A. Sugamoto Y. Nishimori A. Nakamula C.ltoi K. Aouda H. Suzuki

Univ. of Tokyo, Institute of [email protected] Physics Univ. of Tokyo, Institute of [email protected] Physics Univ. of Tokyo, Institute of [email protected] Physics Univ. of Tokyo, Dept. of [email protected] Pure and Applied Sciences Univ. of Tokyo, Dept. of [email protected] Pure and Applied Sciences Univ. of Tokyo, Inst. of Solid [email protected] State Physics Institute for Cosmic Ray Re- [email protected] search Institute for Cosmic Ray Re- [email protected] search Tokyo lnst. Tech. Dept. of [email protected] Physics Tokyo Inst. Tech. Dept. of [email protected] Physics Tokyo Inst. Tech. Dept. of [email protected] Physics Tokyo Inst. Tech. Dept. of [email protected] Physics Tokyo Inst. Tech. Dept. of [email protected] Physics Ochanomizu Univ. Dept. of [email protected] Physics Tokyo Metropolitan Univ. [email protected] Dept of Physics Kitasato Univ. Inst. of [email protected] Physics Nihon Univ. Dept. ofPhysics [email protected] Nihon Univ. Dept. ofPhysics [email protected] lbaraki Univ. Dept. of [email protected] Physics

416

K. Haga H. Igarashi N. Nagao T. Sugimoto

N.Ano H. Fujisaki

K. T. T. H.

Nittou Suga Murakami Kawai

N. Ishibashi T. Yukawa

' K. Hamada H. Aoki N. Tsuda N. Sasakura A. Tsuchiya S. K. Yang

Dept. of Ibaraki Univ. Physics lbaraki Univ. Dept. of Physics Dept. lbaraki Univ. of Physics 2-5-12-414 Nishiwaseda, Shinjyuku-ku Tokyo 169,Japan Rikkyo Univ. Dept. of Physics Rikkyo Univ. Dept. of Physics Chiba Univ. Dept. of Physics Chiba Univ. Dept. of Physics Chiba Univ. Dept. of Physics National Lab for High Energy Physics(KEK) National Lab for High Energy Physics(KEK) National Lab for High Energy Physics(KEK) National Lab for High Energy Physics(KEK) National Lab for High Energy Physics(KEK) National Lab for High Energy Physics(KEK) National Lab for High Energy Physics(KEK) National Lab for High Energy Physics(KEK) Univ. of Tsukuba Inst. of Physics

[email protected] .ac.jp [email protected] [email protected] g3k050@cfi. waseda.ac.jp

[email protected]

[email protected]_.ac.jp [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

417

K.Ito T. Soda S. Odake S. Hosono H. Terao Y. Igarashi K. Itoh

S. lchinose M. Hayashi T. Matsuoka T. Nakanishi S. Yahikozawa Y. Kikukawa T .Takahashi E. Nakai N. Maekawa T. Inami

Univ. of Tsukuba Inst. of [email protected] Physics Univ. of Tsukuba Inst. of [email protected] Physics of [email protected]