Introduction to Superstrings and M-Theory (Graduate Texts in Contemporary Physics)

Graduate Texts in Contemporary Physics R N. Mohapatra: Unification and Supersymmetry: The Frontiers of Quark-Lepton Phys...

Author: Michio Kaku

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Graduate Texts in Contemporary Physics R N. Mohapatra: Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics R E. Prange and S. M. Girvin (eds.): The Quantum Hajj Effect, 2nd ed.

M. Kak:u: Introduction to Superstrings and M-Theory, 2nd ed. J. W. Lynn (ed.): High Temperature Superconductivity H. V. Klapdor (ed.): Neutrinos

J. H. Hinken: Superconductor Electronics: Fundamentals and Microwave Application

Michio Kaku

Introduction to Superstrings and . M-Theory •. Second Edition :: .

:. Wich 45 llIuslrolions •

Springer

Michio Kaku Department of Physics City College of the City University of New York New York, NY 10031, USA

Series Editors Professor Joseph Birman Department of Physics City College of CUNY New York, NY 10031 USA

Professor R. Stephen Berry Department of Chemistry Uni versity of Chicago 5735 South Ellis Avenue USA

Professor Mark Si1vennan Department of Physics Trinity College Hartford, CT 06106 USA

Professor H.E. Stanley Center For Polymer Studies Physics Department Boston University Boston, MA 02215 USA

Professor Jeffrey Lynn Department of Physics University of Maryland College Park, MD 20742 USA Professor Mikhail Vo10shin Theoretical Physics Institute Tate Laboratory of Physics University of Minnesota Minneapolis, MN 55455 USA

Library of Congress Cataloging-in-Publication Data Kaku, Michio. Introduction to superstrings and M-theory I Michio Kaku. -2nded. p. crn.--{Graduate texts in contemporary physics) Includes bibliographical references and index. ISBN 0-387-98589-1 (alk. paper} I. Superstring theories. I. Title. H. Series. QC794.6.s85K35 1998 539.7'258--<1c21 98-26976 Printed on acid-free paper.

to 1999, 1988 Springer-Verlag New York, Inc. Al! rights reserved. This work may not be translated or copied in whale or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the fonner are not especially identified, is not to be take.p as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by A. Orfuntia; manufacturing supervised by Joe Quatela. Photocomposed copy prepared in M-TE'C by The Bartlett Press, Inc., Marietta, GA. Printed and bound by R.R. Donnelly and Sons, Harrisonburg, VA. Printed in the United States of America.

9 8 7 654 3 2 I ISBN 0-387-98589-1 Springer-Verlag New York Berlin Heidelberg

SPIN 10685632

To my parents

Preface

we are all agreed that your theory i.~ crazy. The question which divides IlS

is whether il is crazy enough.

-Niels Bohr

Superstring theory (and its latest fonnulation, M-theory) has emerged as the most promising candidate for a quanrum theory of all known interactions. the past Superstrings apparently solve a problem thai has delled solution

roc

50 years, namely the unification of the two grellt fundamenta l physicallheorics oflhe century, quantum field theory aod general relativlty. Superstring theory introduces an entirely new physical picrure into theoretical physics and a new mathematics that has startled even the mathematicians.

Ironically, although superstring theory is supposed 10 provide a unified field theory of the Universe, the theory itself often seems like a confused j uOlble of folklore, random rules of thumb, and inruilions. This is because the

development of superstring theory has been unlikc thot of any other theory, such as general relativity, which began with a geometry and an action and later evolved into a quannlm theory. Superstring theory, by contrast, has been evolving b..1ckward fOT the past 30 years. 11 has a bizarre history, beginning with the purely accidenUil discovery of the quantum theory in 1968 by G. Veneziano and M. Suzuki. Thumbing through old mathematics books, they stumbled by chance on the Beta fimction, written down in the last century by mathematician Leonhard Euler. To their amanment, they di scovered that the Beta function satisfied

aJmost aU the stringent requirements of the scattering matri.'( describing particle interactions. Never in the history of physics has an important scientific discovery been made in quite this random fashion.

V111

Preface

Because of this accident of history, physicists have ever since been trying to work backward to fathom the physical principles and symmetries that underlie the theory. Unlike Einstein's theory of general relativity, which began with a geometric principle, the equivalence principle, from which the action could be derived, the fundamental physical and geometric principles that lie at the foundation of superstring theory are still unknown. To reduce the amount of hand-waving and confusion this has caused, three themes have been stressed throughout this book.. To provide the student with a solid foundation in superstring theory, we have first stressed the method of Feynman path integrals, which provides by far the most powerful formalism in which to discuss the model. Path integrals have become an indispensable tool for theoretical physicists, especially when quantizing gauge theories. Therefore, we have devoted Chapter I of this book to introducing the student to the methods of path integrals for point particles. The second theme of this book is the method of second quantization. Although traditionally field theory is formulated as a second quantized theory, the bulk of superstring theory is formulated as a first quantized theory, presenting numerous conceptual problems for the beginner. Unlike the method of second quantization, where all the rules can be derived from a single action, the method of first quantization must be supplemented with numerous other rules and conventions. The hope is that the second quantized theory will reveal the underlying geometry on which the entire model is based. . The third theme of this book is duality, which has revolutionized the way we formulate string theory. Duality, which was first discovered in Maxwell's equations, allows us to determine the equivalence of two seemingly different theories. Using duality, we can show that the five different superstring theories in 10 dimensions are actually unified into a single theory in 11 dimensions, a still mysterious theory called M-theory, which reduces to II-dimensional supergravity in the low-energy limit. Since duality allows us to show the equivalence of a weak coupling theory to a strong coupling theory, it has allowed us to probe, for the first time, the nonperturbative region of string theory, where we find a host of new objects, such as membranes, M-branes, and D-branes. Although the complete action of M-theory is still unknown, already it has yielded a flood of new information concerning the strong coupling behavior of string theory. . We now know that strings coexist with membranes of various dimensions; ultimately, the entire theory may be formulated in terms of a master theory in 11 dimensions, perhaps in terms of membranes of some sort. Ironically, although the action for M-theory is not known, physicists believe that the theory exists because of the vast web of consistency checks given by dUality. (However, because M-theory itself is such an obscure theory, we will still refer to the theory by the name superstring theory.) In addition to providing the student with a firm foundation in path integrals and field theory, the other purpose of this book is to introduce students to the latest developments in superstring theory, that is, to acquaint them with the

Preface

ix

...,

fast-paced areas that are currently the most active in theoretical research, such

String field theory; Conformal fi eld theory; Jentitd to understand the miraculous canccUation of d ivergences o f the theory, which separates string theory from al1 other field theories. Because the theory of automorphic functions gets increasingly difficult as one dl.'SC ribcs muit iloop amplitudes. the beginner may skip the disc ussion of higher loops. The serious srudelll, though, will find that multiloop amplitudes form an !l Tea of active research. Part H begins a discussion orthe field theory ofstrings, and Part UI examines phenomenology. The order of these two parts call be interthanged witho ut diffic ulty. Each part v,a<; written to be relatively independent of the other, so

x

Preface

the more phenomenologically inclined student may skip directly to Part ill without suffering any loss. Chapters 6 and 7 in Part II present the evolution of several approaches to string theory. Chapter 6 discusses the original light cone theory and how to quantize multiloop theories based on strings. However, Chapter 7 was written in a relatively self-contained fashion, so the serious student may skip Chapter 6 and delve directly into the covariant theory. In Part III, the beginner may skip Chapter 8. The discussion of anomalies is rather technical and main1y based on point particles, and overlaps the discussions found in other books. Chapter 9 cannot be omitted, as it represents one of the most promising of the various superstring theories. Likewise, Chapter 10 forms an essential part of our understanding of how the superstring theory may eventually make contact with experimental data. In Part IV, we finally have a presentation of duality and M-theory. In Chapter 11, we have a basic presentation ofM-theory, and how the five known superstings can ali be expressed in terms of a single theory. In Chapter 12, we explore the duality relations that exist in D = 8, 6 and 4 dimensions, giving us, for the first time, a look at the nonperturbative region of string theory. In Chapter 13, we explore more advanced topics, such as D-brane physics, matrix models, and applications ofD-branes to black hole physics. The author hopes that this will help both the beginner and the more advanced student to decide how to approach this book. Michio Kaku

Acknowledgments

I would like to thank Dr. L. Alvarez-Gaume for making many extensive and Ydlusble comments throughout the entire book, which have strengthened the presentation and conlen! in several key pll'lccs. Doctors M. Dine, S. Samuel, J. Lykken. O. Lechtenfeld, I. Ichinosc, D.-X. Li, and A. Das also read the manuscript and contributed numerous helpful criticisms that have been incor· paroled into the book. Dr. D. O. Vona carefully read the entire manuscript Ilud made important suggestions thai gready improved the draft. I wou ld especiaJly like to express my sincerest appreciation to Dr. B. Sakila, Dr. J. Birman, and the faculty and students in the Physics Department of the City College oflhe City University of New York for constant encouragement and support thrOUghOllt the writing of this manuscript, without whicb this book would Dot bave been possible. T would also like to acknowledge support from the National Science Foundation and the CUNY- FRAP Program.

Michio Kaku

: Acknowledgments

J First Quantization and Path Integrals

I

I Patb Integrals and Point Particle! 1.1 Why Strings? . .. . . . . . . , . . 1.2 Historical Rcview orOauge Theory . 1.3 Path Integrals and Point Particles 1.4 Relativislic Point Particles . .. 1.5 First and Second Quantization . 1.6 Faddeev- Popov Quantization 1.7 Scwnd Quantization . .. _ . _ 1.8 Harmonic Oscillators _ . . . . 1.9 Currents and Second Quanri7.3tion 1.I 0 Summary . References. . . . . ..

3 3

f'liambu-Goto Slrings 2. \ Bosonic Slri ngs . . . . . . . 2.2 Oupta-Bleuler Quantization . 2.3 Light Cone Quantization . 2.4 BRST Quantization ... . . 2.5 Trees . . . . . . . . . . .. . 2.6 From Path Intcgrals to Operators 2.7 Projecrive lnwriance and Twists

... .. .. ..

. . ..

.

· . . · . ... · ....

. . ..

7

18 25 28 30

..'

34 37

40 44

47 49 49 60 67

.. .. ..

. ..

70 72

78 84

xiv

Contents 87

2.8 2.9

Closed Strings . . . . . . . . . Ghost Elimination . . . . . . .

90

2. 10

Summary..

95

... .

99

References. . . . . . . . . . . . . . .

3

Superstrings

101

3.1

101 104

3.2

3.3 3.4 3.5 3.6 3.7 3.8

3.9 3.10

Supersymmetric Point Particles . . . . . . . . . . . . Two-Dimensional Supersymmetry . . . . . . . . . . Trees . . . . . . .. . . . . . . . . . . . . . . . . . . Local Two-Dimensional Supersymmetry. ... . Quantization.. . . . . . . . . . . GSO Projection . . . . . . . . . . . . .. .. . . . . . . . . Superstrings . . . . . . . . . .. .. .. . Light Cone Quantization of the GS Action ..... Vertices and Trees . . . . . . . . .' . . . . .

Summary..... . . . . .

References . . . . . . . . . . . . .

4

5

. . . . . . . . . . ..

.....

111 117 119 123 126 128 134 136 139

Conformal Field Tbeory and Kac-Moody Algebras

141

4.1 4.2 4.3

Conformal Field Theory ' . . . . . . . . . . Superconformal Field Theory Spin Fields . . . . . . . . . 4.4 Superconformal Ghosts . . 4.5 Fermion Vertex . . . . . . . . . . . . . . . . 4.6 Spinors and Trees . . 4.7 Kao-Moody Algebras . . . 4.8 Supersymmetry......... . . Summary.. ... . . . . . . 4.9 References . . . . . . . . . . . . . . .

141

Multiloops and Teichmiiller Spaces

178 178

5.1 5.2 5.3 5.4 5.5

U nitarity . . . . . . . . . . . . . . . . . . . . . . . Single-Loop Amplitude . . . . . . . . Harmonic Oscillators " . . . . . . . Single-Loop Superstring Amplitudes Closed Loops . . . . . . . . . . . . . 5.6 Multiloop Amplitudes . . . . . . . . . . . 5.7 Riemann Sufi'aces and Teichmillier Spaces .. . 5.8 Conformal Anomaly . . . . . . . . . . . 5.9 Superstrings.................. 5.10 Determinants and Singularities . . . . . . . . 5.11 Moduli Space and Grassmannians. . . . . . . 5.12 Summary.. . ... References. . . . . . . . . . . . . . . . . . . . . . . . .

150 155 158

165 167

170 174 174 177

181

185 192 195

200 210 217 221 224 226 238

242

Contents S~ond

Quantization and the Search for Geometry

247 250 254 261

267 272

.. .

275 280 '"6 290

.. . ·· .. ·· .. ·. ·. ·.

291 29 1

.

7.3

Gauge Fixing . . . . . .. .

7.4

7.5

interactions . . . . . . . . . Winenfs String Field Theory

7.6

Proof of (quivalence. . . . .

7.7

Closed Strings and Superstrings .

1.8

Summary .

References . . . . . . . . . . . . . . . .

245 247

Ught Cone fJ eld Theory : 6.1 Why String Field Theory? . . . _ .. 6.2 Deriving Point Particlc Field Theory Light Cone Field Theory . Interactions . . . . . . . . . . . . . Neumann Function Method . . . .. Equivalence of the Scauering Amplitudes . four-String Intemction . Superstring Field Theory . . Summary . References. . . . . . BRST Field Theory 7.1 Covariant Siring Field Theory . . . . • .. 7.2 eRST Field Theory

xv

.

..

.

297 300 303 308 311 317

328 33 1

.

Phenomclloloi:)' and Model Building

335

Anomlllies lind the Atiya h-Slngn Theorem 8.1 Beyond GUT Phenomenology . . . . . 8.2 Anomalies and Fcynnum Diagrams . . 8.3 Anomalies in the Functional Fonnlliism . It4 Anomalies and Characteristic Classes . 8.5 Dirac index . . . . .. _ . . . . .. 8.6 G ravitational a nd Gauge Anomalies . 8.7 Anomaly Cancellation in Strings 8.H Summary .

337 337 341 346 348 353 357 366

.. ·. ·

.

368

References. . . . . . . . . . . . . . . .

372

Helerotic Strlni:s and Compactifiutioo

373 373

9.1

9.2 9.3 9.4 9.5

Compactificarion .. . The Hcterotic Smog . . . . . . . .

Spectrum . . . . . . . . . . . . . . Covariant and Fennionic Fonnulations Trees . . . . . . . . . . . . . . . . . .

·. .

..

378 383 386

388

xvi

Contents 9.6 Single-Loop Amplitude . . . . . . . . . . . . . . 9.7 E8 and Kac-Moody Algebras . . . . . . . . . . . 9.8 Lorentzian Lattices . . . . . . . . . . . . . . . . . . . . 9.9 Summary . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . "

10 CaJabi-Yau Spaces and Orbifolds 10.1 10.2

Calabi-Yau Spaces . . . . . . . . . . . . . . . . . . . . Review of de Rahm Cohomology . . . . . . . . . . 10.3 Cohomology and Homology. . . . . . . . . . . . . . .. 10,4 Kahler Manifolds .. . . . . . . . . . . . . . . . . . " 10.5 Embedding the Spin Connection . . . . . . . . . . . . . 10.6 Fermion Generations . . . . . . . . . . . . . . . . . . . 10.7 Wilson Lines . . . . . . . . . . . . . . . . . . . . 10.8 O r b i f o l d s . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Four-Dimensional Superstrings . . . . . . . . . . . . .. 10.10 Summary. . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . "

IV M-Theory

391 395 398 400 403

404 404 409 413 419 426 428 432 434 438 449 453

455

11 M-Theory and Duality 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 11.2 Duality in Physics . . . . . . . . . . . . . . . 11.3 Why Five String Theories? . . . . . . . . . . . . . I I .4 T -Duality . . . . . . . . . . . . . . . . . . . . . . .. . .. .. 11.5 S-Duality............ 11.5.1 Type IIA Theory . . . . . . . . . . . . . . 11.5 .2 Type IIB Theory. . . . . . . . . . . . . . . . . . . 11.5.3 M-Theory and Type lIB Theory. . 11.5,4 E8 ® E8 Heterotic String . . . . . . . . . . . 11.5.5 Type I Strings . . . . . . . . . . . . . . . . . 11.6 Summary.......... . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . .

..

" . .. " "

12 Compactifications and BPS States 12.1 BPS States . . . . . . . . . . . . . . . . . . . . . . . .. 12.2 Supersymmetry and P-Branes . . . . . . . . . . . . . .. 12.3 Compactification...................... 12.4 Example: D = 6 . . . . . . . . . . . . . . . . . . . . .. 12.4.1 D = 6, N = (2,2) Theory . . . . . .. . . . " 12.4.2 D=6,N=(1,I)Theories . . . . . . . . . . . 12.4.3 M-Theory in D 7 . . . . . . . . . . . . . . " 12.5 Example: D = 4, N = 2 and D = 6, N = 1 . . . . . ..

=

457 457 458 460 462 465 466 469 471 473 473 476 480 482 482 484 488 490 491 494 496 497

Contents

12.6 Symmetry Enhancement and Tensionless Strings . 12.1 F -Theory .. . . 12.8 F..XliIl1ple: D = 4 . 12.9 Summary . RefcrencC$ . . . . . . . .

Solitons, D-Brane!, and Black Boles 13.1 Solitons . . . . . . . . . 13.2 Supen:Ill:mbranc Actions . 13.3 Five-BnlDe Action 13.4 D-Branes .. .. . . . . . 13.5 D-Bnmc:: Actions . . . . . 13 .6 M(atrix) Models and Membranes 13.1 Black Holes 13.8 Summary .. 13.9 Conclusion . References . . . ..

. . .. ·. ·· .. . . . . .. . ..

. . . ....

:: : Appendh: A. I A Brief Introduction 10 Group Theory . A.2 A Brief lntroduction 10 Genenli Relativity . A.J A Brief fntrodoction 10 the Tbeory of Forms A.4 A Brief lnlrOduction to Supersymmetry . A.S A Brief lntroduction to Supergraviry A.6 Notation Roferences . . . . . . . . . . . . . . . . ..

Indu

xvii

499

501 502 504

510 S1J 511 5J3 516

m 52J

525

m

537 542 544 5<5

545

557 561 566

573

.

. ..

577

579 581

Part

I

First Quantization and Path Integrals

1

Integrals and Particles

Why Strings? of the greatest scientific challenges of our time is the struggle to WIile the fundamental theories of modem physics, quantum field theory and gen-

relativity, into one theoretical framework. Remarkably, Ihcsc two theories

~

~!~~::~:!~;~th::e sum total ofall human knowledge concerningthe most forces of Nature. Quantum field theory, fur example, has had in explaining the physics of the microcosm. down to disless than I O - I~ em. General relativity, on the other hand. is unrivaled in

~

:~~~;~~:t;~~~'~~~~;:2;~ fascinating sueand origin orlhe cosmos, Universeproviding itsel f. Theaastonishing togclhcr thcycan explain the behavior of malleT energy over a staggering 40 orders of magnitude, from the subnuclear to

.ecosrn·ic domain. :: : The great mystery of tbe past five decades, however, has been the total ., I of these two theories. It's as if Nature had two minds, each of the otlier in its own particular domain, operoting of other. Why should Nature, at ils deepcst and most level, require two tota lly distinct frameworks, with two sets of of assumptions, and two sets of physical principles? : Ideally. we would want Ii unified theory to unite these two fundamental

Quantum fie ld theory . . Generol re Iatlvlty

1Unified field theory.

the history of attempts over the past decades to unite these two dismal. They have inevitably been riddled with infinities or

4

1. Path Integrals and Point Particles

have violated some of the cherished principles of physics, such as causality. The powerful techniques of renorrnalization theory developed in quantum field theory over the past decades have failed to eliminate the infinities of quantum gravity. Apparently, a fundamental piece of the jigsaw puzzle is still missing. Although quantum field theory and general relativity seem totally incompatible, the past two decades of intense theoretical research have made it increasingly clear that the secret to this mystery most likely lies in the power of gauge symmetry. One of the most remarkable features of Nature is that its basic laws have great unity and symmetry when expressed in terms of group theory. Unification through gauge symmetry, apparently, is one of the great lessons of physics. In particular, the use of local symmetries in Yang-Mills theories has had enormous success in banishing the infinities of quantum field theory and in unifying the laws of elementary particle physics into an elegant . and comprehensive framework. Nature, it seems, does not simply incorporate symmetry into physical laws for aesthetic reasons. Nature demands symmetry. The problem has been, however, that even the powerful gauge symmetries of Yang-Mills theory and the general covariance of Einstein's equations are insufficient to yield a finite quantum theory of gravity. At present, the most promising hope for a truly unified and finite description of these two fundamental theories is superstring theory and its latest formulation, M-theory. [1-12]. Superstrings possess by far the largest set of gauge symmetries ever found in physics, perhaps even large enough to eliminate all divergences of quantum gravity. Not only does the superstring's symmetry include that of Einstein's theory of general relativity and the Yang-Mills theory, it also includes supergravity and the Grand Unified Theories (GUTs) [13] as subsets. Roughly speaking the way in which superstring theory solves the riddle of, infinities can be visualized as in Fig. 1.1, where we calculate the scattering of two point particles by summing over an infinite set of Feynman diagrams with loops. These diagrams, in general, have similarities that correspond to "pinching" one of the intemallines until the topology of the graph is altered.

=

()Q

FIGURE 1.1. Single-loop Feynman diagram for four-particle scattering. The ultraviolet divergence of this diagmm corresponds to the pinching of one internal leg, i.e., when one internal line shrinks to a point.

I.! WllY Strings?

5

00 1.2. ")\Yo-loop Fcynman diagram for closed string scattering. The diagram finite because it cannOI be pinched as in the point panicle ease. From wguments alone, wecan see that string theory is less divergent than point @le,'h,,,,',I<,fo,,,"'divergences, however, may still exist.

theory that can simultaueously eliminate the infinities of the S-ma\rix "o
6

1. Path Integrals and Point Particles

Unfortunately, the geometry of the superstring and membranes are some of the last features of the model to be developed. In fact, as seen from this . perspe'6:1ve;'uHnn6cld·1Cd"LI~·M-l~~}mltm:lc.vm:d Lhvtlw.S'pl1~;:t1) J")W.x..u, beginning with the accidental discovery of its quantum theory in 1968! By contrast, when Einstein first discovered general relativity, he started with physical principles, such as the equivalence principle, and formulated it in the language of general covariance. Once the geometry was established, he then wrote down the action as the unique solution to the problem. Later, classical solutions to the equations were discovered in terms of curved manifolds, which provided the first successful theoretical models for the large-scale behavior of the Universe. Finally, the last step in the evolution of general relativity is the development of a quantum theory of gravity. The crucial steps in the historical evolution of general relativity can thus be represented as Geometry -+ Action -+ Classical theory -+ Quantum theory. Furthermore, both general relativity and Yang-Mills theory are mature theories: they both can be formulated from first principles, which stresses the geometry and the physical assumptions underlying the theory. Superstring theory and M-theory are just beginning to reach that stage of development. Remarkably, Yang-Mills theory and gravity theory are the unique solution to two simple geometric statements: (1) Global Symmetry The free theory must propagate pure ghost-free spin-l and spin-2 fields transforming as irreducible representations of SU(N) and the Lorentz group. (2) Local Symmetry The theory must be locally SU(N) and generally covariant. What is remarkable is that the coupled Yang-Mills gravity action is the unique solution of these two simple principles l'---;:F11" FloW -2K1 ",-g r-::Rl1"g 11" . L ---4",-g 2

(1 .. 1 1)

(The first principle contains the real physics of the theory. It cannot be included as a subset of the second principle. There is an infinite number of generally covariant and SUe N) symmetric invariants, so we need the first principle to input the physics and select the irreducible representations of the basic fields. By "pure" fields, we mean ghost-free fields that have at most two derivatives, which rule.s out R2 and F4 higher derivatives theories.) The question remains: r¥hat is the counterpart to these two simple principles for superstring theory and M-theory? The plan of this book, of course, must reflect the fact that the theory has been evolving backward. For pedagogical reasons, we will mostly follow the historical development of the theory. Thus, Part I of the book, which introduces the first quantized theory, will at times appear to be a loose collection of

1.2 Historical Review of Gauge Thcory

7

guiding principle. 11131is why we have chosen, in Part "'P""' integral or functional approach to string [beory. Only path integrals do we have a formalism in which we can derive: fonna lisms, such as the hannonic oscillator formalism. AJthough the ji" 'g<'! fonnuJalion ora firs t quantized theory is still woefully inadequate. a genuine second quantized theory, it is the most convenient in which to tie together the loose ends of the first quantized theory. II of the book we will discuss the field theory itself, from which we all the results of the theory from one aClion. However, once again followed hisforicaJ order and pre:"ented llie field theory backward. III we present the "phenomenology" of!;trings. Although it may be 'I'ruo,,,10 do phenomenology starting at 10 19 GeV, it is imponant to the kinds of predictions that the theory makes. in Part IV, we present duality and I I-dimensional M-theory, which w,us" for time, to probe the nonperturbative region of string theory, theory coexists with exotic objects like p-branes and D-branes. ''''''''r. to really appreciate the successes and possible defects of the sutheory, we must first try to understand the historical problems that ,tag"" d physicists for the past five decades. Let us now mcn to a quick of the development of gauge theories in order \0 appreciate the diffi10

".el'm

('f,f~;~~:¥~;:~;:a! fi nite theory of gravity. We will also briefly sketch !he ~i.l

of thc superstring theory.

Historical Review of Gauge Theory 196Os, elementary panicle physics seemed hopelessly mired in confuweak, electromagnetic. strong, and gravitational forces were each separately, largely in isolation of the otbers. Moreover, investigations force had reached a fundamental roadblock:

weak interocrions. Theoretical models of the weak interactions had ~ro!",~"d

cmbarrassingly little beyood the }·erm.i theory first proposed deaJdes earlier in the 19305: (1.2.!)

r" represents various combinations of Dirae matrices. The next step, a theory of W bosoos. was plagued with the probJem of in. Furthermore, no one knew the underlying symmetry among the or whether there was any. strong interactions. In contrast to weak interactions. the Yukawa ')n"',"n theory provided a renonnaliz8ble theory of the strong interactions: ( 1.2.2)

8

1. Path Integrals and Point Particles

However, the Yukawa theory could not explain the avalanche of "ele- . mentary" particles that were being discovered in particle accelerators. J. Robert Oppenheimer even suggested that the Nobel Prize in Physics should go to the physicist who didn't discover a particle that year. Furtbennore, the quark model, which seemed to fit data much better than it had any right to, was plagued with the fact that quarks were never seen experimentally. (3) The gravitational force. Gravity research was totally uncoupled from research in the other interactions. Classical relativists continued to find more and more classical solutions in isolation from particle research. Attempts to canonically quantize the theory were frustrated by the presence of the tremendous redundancy of the theory. There was also the discouraging realization that even if the theory could be successfully quantized, it WOUld ... still be nonrenormalizable. This bleak landscape changed dramatically in the early 1970s with the com- . ing of the gauge revolution. One of the great achievements of the past 25 years has been the development of a fully renormalizable theory of spin-l gauge par- . ticles in which, for the first time, physicists could actually calculate realistic S-matrix elements. Thus, it took over 100 years to advance beyond the original gauge theory first proposed by Maxwell in the 1860s! (See the Appendix for an elementary introduction to gauge theories and group theory.) . Apparently' me Key lcr eminrtann!5'Lru1 inwI'g~r.:a.'<;6'f,r"1bt.:h~~t~l.'1~~mtll.m~. mechanics is to go to larger and more sophisticated gauge groups. Symmetry, instead of being a purely aesthetic feature of a particular model, now becomes its most important feature. For example, Maxwell's equations, which provided the first unification of the electric force with the magnetic force, has a gauge group given by U( 1). The unification of the weak and electromagnetic forces into the electroweak force requires SU(2) ® U(l). The forces that bind the quarks together into the hadrons, or quantum chromo dynamics (QCD). are based on SU(3). All of elementary particle physics, in fact, is compatible with the minimal theory of SU(3) ® SU(2) ® U(l). Although the verdict is still not in on the GUTs, which are supposed to unite the electroweak force with the strong force, once again the unifying theme is gauge symmetry, with such proposals as SU(5), 0(10), etc., symmetry. Although the gauge revolution is perhaps one of the most important devel- .. opments in decades, it is still not enough. There is a growing realization that the Yang-Mills theory by itself cannot push our understanding of the physical universe beyond the present level. Not only do the GUTs fail to explain important physical phenomena, but also there is the crucially important problem of formulating a quantum theory of gravity. Grand Unified Theories, first of all, cannot be the final word on the unification of all forces. There are several features of GUTs that are still unresolved:

1.2 Historical Review of Gauge Theory

9

.' . GUTs cannot rcso! ..'C the problem of why there are three ncarly cxact . ~ .. ~~ : copies or families of elementary particles. We still c~mnot answer Rabi's :. ::. :: "Who ordered the muo n?" still have 20 or so arbitrary parameters. They cannot, for example, ; '",.le" I,," the masses of the quarks, or lhe various Yukawa couplings. A truly unilled theory should have at most one arbitrary parameter. GUTs have diffieuJLy solving the hierarchy problem. Unless we llppeal to supersymmctry, it is hard to keep thc physics of incredib ly massive particlcs from mixing with everyday energies and destroying the hierarchy. The unification of panicle forces occurs around 10-21 em which is very 01,)" to the Planck length of 10-)) cm, where we expect gravitational to become dominan!. Yet GUTs say nothing whatsoever about ",vltation. far, prolon decay has not been conclusively observed, which ai; :;~~~ru~l~e,:',~out minimal SU(5). There is, thcreforc, still no compelling :~. reaso n for introducing the theory. difficult to bel ieve that no new interactions will be found between ~~:;';:':~ energies and the unification scale. The "desert" may very well 1 new jnt craction~ yet unknown. most perplexing and the most challenging of these problems, from

,~;~~~::~~~;~~o~f;:"';·:e'o.V, has been to find a way of quanti ring Einstein's ,~

Although Yang-Mills theoricshllve had spectacular unifying the known laws of particle physics, the laws of gravity ~:~:~:~ different at a fundanlentallevcl. Clearly, Yang- Mills theory and ~ gauge theory arc incapable of dealing with this problem. Thus, arc faced with fonnidab le experimental and theoretical problems when to their limits. (;~::'I relativity is also plagued with simi lar d ifficulties when pushed to it has been established thnt Einstein's equations necessarily .: .:: ,: exhibit poinllike singularities, where we eJ(pect the laws of general rela. ~: :. tivity 10 collapse. Quantum corrections mUSI dominate over [he classical .' in this domain. action is not boundt.'i.i from below, because it is linear in the curvature . lbus, it may nOI be stable quantum mechanically. relativiry is not rcnormali7.able. Computer calculations. for examhave now condusively shown that there is a nonzero countcnerm in nst,I,,', theory at the two-loop level. attempts to quantize Einstein's theory of gravitation have met with failwe. One of tbe firs t to point out that gcncral relativity be incompatible with quantum mechanic:.'l was Heisenberg, who noted

~~:lr::~~"~;~O::f.~'~dimcnSional

coupling constant would ruin the usual

10

1. Path Integrals and Point Particles Ifwe set -

h

2n

= 1,

c = 1,

(1.2.3)

there still remains a dimensional constant even in the Newtonian theory of gravity, the gravitational constant G:

which has dimensions of centimeters squared. When we power expand the metric tensor g(J." around a flat square with the metric 1]1-'" = ( - + + + ), we introduce the coupling constant K, which has dimensions of centimeters: (1.2.5) .• Therefore (1.2.6) In this system of units, where the only unit is the centimeter, this coupling ·· constant K becomes the Planck length, 10-33 em or 10 19 GeV, which is far beyond the reach of experimentation! ..• Renormalization theory, however, is founded on the fundamental premise • • that we can eliminate all divergences with an infinite redefinition of certain constants. Having a negative dimensional coupling constant means that this complicated reshuffling and resuming of graphs is impossible. Negative coupling constants mean that we can always insert the interaction term into .a " Feynman diagram and increase its power of divergence. This means that any , graph can be made arbitrarily divergent by multiple insertions. This means that general relativity cannot be a renonnalizable theory. The amplitude for graviton-graviton scattering, for example, is now a power expansion in a dimensional parameter (see Fig. 1.3):

(1.2.7) where we are no longer able to shuffie graphs in the usual manner to cancel the infinities, which is the heart of renormalization theory. Thus, renormalization theory breaks down. Because general relativity is hopelessly outside the domain of conventional renormalization theory, we must reconsider Dirac's fundamental objection. It was Dirac who said that the success of quantum mechanics was based on ap- . proximation schemes where each correction term was increasingly small. But· renormalization theory is flawed because it maximally violates this principle and manipulates infinite quantities and discards them at the end. One solution might be to construct a theory of gravity that is finite to every order in the coupling constant, with no need for renormalization at all. For a while, one bright hope was supergravity [14, 15], based on the local gauge·

1.2 Historical Review ofey.mge Theory

M..,>-< ..' X +

~)=(

1',I(~ >-0-<

+

,I(' ):c(

+

IJ

+ ...

~)-2-0-<

+ •••

lURE 1.3. Scattering amplirude for graviton-graviton scattering. Because the constant has negative dimensio n any graph can be made arbitrarily divcr,~tll~n'by requiring an infini te number of counlerterms. ThI.L~ theo ries cOnlain ing grdvi ty must be either divergent aT completely finite ordcr-by-order. Pure gravity has been shown on computer to diverge al the two-loop level. Counhave also been found for quantum gravi ty coupled to lower-spin particles. ",_'",>en""'g!h,m,, is the only candidate for a finite theory.

:: ; Osp(N / 4) (see Appendix). which was the first nontrivial extension of .; equations in 60 years. The hope was that this gauge group would of Ward--Takahashi identities 10 cancel a large class The larger the gauge group, the more likely troubl esome cancel (sec Fig. 1.4): Theory Electromagnetism

Electroweak

Strong GUT(?) Gravity(?) S upergravi IY(,!)

Gauge gro up U[I) SU(2) ® U(I) SU(l) SU(5), O( I0) GL(4). 0(3, I)

Osp(N / 4)

• t",iic strategy being pursued was Gaugc symmetry -+ Ward-Takahashi idemitics -+ Cancellation ofgrapbs -+ Renormalizablc theory.

exAInple, even Einstein's theory o f gravity can be shown to be trivthe first loop level. There e"ists a remarkable identity, called the identity, which immedi:1tely shows that all o ne-loop gntphs in relativity (which wou ld take a computer to write down) sum to zero. tbe super-Gauss-Bonnet identities eliminate many of the divergences . but probahly not enough to make the theory tinite. and most promising of the s upcrgravi tics, the 0(8) supergrnvity, divergent. Unfortu nately, it is possible to write down locally supercounterterms at the seventh loop leve l. It is highly unlikely that the of this and probably an infinite number of other counte rterms can ,,..._ without appclliing to an even higher synunetry. This is d iscourag-

12

1. Path Integrals and Point Particles

Electricity V(1) Magnetism Weak Force Strong Force

SV(2)

®

V(1) SV(51, 0(10) ?

Superstrings?

Gravitation

FIGURE 1.4. Chart showing how gauge theories based on Lie groups have united the fundamental forces of Nature. Maxwell's theoI)', based on U(l), unites electricity .•. and magnetism. The Weinberg-Salam model, based on SU(2) 0 U(1), unites the. weak force with the electromagnetic force. GUTs (based on SU(5), 0(10), or larger .... groups) are the best candidate to unite the strong force with the electroweak force. Superstring theory is the only candidate for a gauge theoI)' that can unite gravity . with the rest of the particle forces. ing, because it means that the gauge group of the largest supergravity theory, Osp(8j4), is still too small to eliminate the divergences of general relativity, Furthermore, the 0(8) gauge group is too small to accommodate the minimal SU(3) ® SU(2) ® U(I) of particle physics. Ifwe go to higher groups beyond 0(8), we find that we must incorporate higher and higher spins into the theory. However, an interacting spin-3 theory is probably not consistent, making one·. suspect that 0(8) is the limit to supergravity theories. . In conclusion, supergravity must be ruled out for two fundamental reasons.: (1) It is probably not a finite theory because the gauge group is not large enough to eliminate all possible supersymmetric counterterms. There is a possible counterterm at the seventh loop level. (2) Its gauge group 0(8) is not large enough to accommodate the minimal symmetry of particle physics, namely, SU(3) 0 SU(2) ® U(l); nor can the theory accommodate chiral fermions. Physicists, faced with these and other stumbling blocks over the years, have concluded that perhaps one or more of our cherished assumptions about our Universe must be abandoned. Because general relativity and quantum mechanics can be derived from a small set of postulates, one or more of these .. postulates must be wrong. The key must be to drop one of our commonsense assumptions about Nature on which we have constructed general relativity and quantum mechanics. Over the years, several proposals have been made to drop some of our commonsense notions about the Universe: (1) Continuity This approach assumes that space-time must be granular. The size of these : grains would provide a natural cutoff for the Feynman integrals, allowing .•

1.2 Historical Review of Gauge Theory

13

us 10 have a finite S-matrix. Integra ls like ( 1.2.8)

wou1d ihen diverge as 6- n, but we would never take the limit as f; goes to zero. LAttice gravity theories are of this type. In Regge calculus [161. for example. we latticize Riemannian space with discrete four-simplexes and replace the curvature tensor by the angular deficit calculated when moving in a circle around a simplex:

-~, A 2<

R -T angular deficit.

Hat space, th.ere is no angular deficit when walking around 8 closed :" :: path. and the action collapses.) UsulI.lly, in lattice theories, we take the limit as the lanice length goes 10 zero. Here, however. we keep it filted 81a nwnbcr L17). At present, however, there is no experimental evidence support the ide;! that space-time is granular. Although we can never

out this approach, it seems to run counter to the natural progression particle physics, which has ~e n to postulate larger aDd more elegant

__ upproach allows small violations in causality. Theories that ineor· :" parate the Lee-Wick mechanism (18) are actually renormali7.able, but : :' ::. permit small dcviations from causality. These theories make the Fcynman .' .' . by adding II fic titious Paull-Villars field of mass M that ultraviolet behavior of the propagator. Usually, the Fcynman converges as p-l in the uitnlViolet limit. Howt."Vcr. by adding particle, we can make the propagator converge even [aster, like 1

"p"+'-;-M'" ~

,. p

(1.2.9)

is a ghost becausc of the - 1 that appears . (This means that the theory wiU be riddled with negative I Usually, we lei the mass of the Pauli-Villars field tend to How~e r, here we keep it finite, letting thc pole go out onto the sheet. Investigations of the structure of the resulting Feynman show, however, that causality is violated; tbnt is, you can meet parents befOre you are born.

~~i~~t~rePlaec Einstein's theory, which is based on the cwvarurc tensor, II

conformal theory based on the Weyl tensor:

( 1.2. 10)

14

1. Path Integrals and Point Particles

where the Weyl tensor is defined as ell-up" = RJLup"

+ gll-[" RpJv + gv[pR"JIl- + tRgll-[pg"ju,

0.2.11)

where the brackets represent antisymmetrization. The conformal tensor possesses a larger symmetry group than the curvature tensor, that is, invariance under local confonnal transformations:

0.2.12) The Weyl theory converges because the propagators go as p-4; that is, it is a higher derivative theory. However, there is a "unitary ghost" that also appears with a -1 in the propagator, for the same reasons cited above. The· . most optimistic scenario would be to have these unitary ghosts "confined" ... by a mechanism similar to quark confinement [19,20]. (4) Locality Over the years, there have also been proposals to abandon some of the ... important postulates of quantum mechanics, such as locality. After all, • • there is no guarantee that the laws of quantum mechanics should hold •• down to distances of 10-33 cm. However, there have always been problems whenever physicists tried to deviate from the laws of quantum mechanics, such as causality. At present, there is no successful alternative to quantum mechanics. (5) Point Particles Finally, there is the approach of superstrings, which abandons the concept. of idealized point particles, first introduced 2000 years ago by the Greeks. •. '-'

The superstring theory, because it abandons only the assumption that the fun- •. damental constituents of matter must be point particles, does the least amount of damage to cherished physical principles and continues the tradition of increasing the complexity and sophistication of the gauge group. Superstring theory does not violate any of the laws of quantum mechanics, yet manages to eliminate most, if not aU, of the divergences of the Feynman diagrams .• The symmetry group of the superstring model, the largest ever encountered • in the history of physics, is probably large enough to make the theory finite • to all orders. Once again, it is symmetry, and not the breakdown of quantum· mechanics, that is the fundamental key to rendering a theory finite. . In Fig. 1.5 we see diagrammatically the evolution of various theories 9( gravity. First, there was Newton's theory of action at a distance, where gravi-. tational interactions travel faster than the speed of light. Einstein replaced this •. with the classical interpretation of curved manifolds. Quantum gravity, in tum," makes quantum corrections to Einstein's theory by adding in loops. Finally,. the superstring theory makes further corrections to the point particle . theory by summing over all possible topological configurations ofinU~ra(;tiIllg} strings.

1.2 Historical Review of Gauge Theory Einrtein

15

Ouenlum Grev iry

.. . ..

o Superstrings

1.5. Steps in the evolution of the theory of gntvilation. Each step in this .' builds On Ihe SUCC~!!$ of the previous step. Ne\OI10n Ihought gravity wa>; a . '.. that acted instanlly over a distance. EiMtein proposed Ihat gravitation was . .. . by the CUf\I'dturc of space-time. The naive merger of general relativity and mechanics produces a divergent theory, quantum gravity. which assumes is cuuscd by the exchange of particle-like gravitons. Superstring proposes that gravitation is caused by the exchange of closed !>tring~.

other

~~~::~:~r~!~:~~h~ow~~Cvtr, is qui te unlike its predcccssors in its historical Unlike physical theories, superstring theory has perhaps

!

in science, with more twislS and turns than a roller

young physicists Veueziauo and Suzuki [21,221. independently ;ov,,,d its qunntum theory when they were thumbing through a mathbook and accidentally noted that the Euler Bela function satisfied postulatcs of the S-matrix for hadronic intcldctions (except un.itarity). Schwarz, and Ramond [23-25] quickly generalized the theory to inspinning particles. To solve the problem ofunitarity, Kikkawa, Saki Is, {26] proposed that the Euler Beta function be treated as the Born to a perturbation scrit.'S. Finally. Kaleu, YlI, Lovelace, and Alessandrini the quantum theory by calculating boson ic multiloop dia· hOlVevcr. was still formulated entirel y in renns of on-shell twO

'v.i.,oco

and Ooto []4, 35] realized that lurki ng behind these scanering was a classical relativistic string. In one sweep, they revolutionthe entire theory by revenling the unifying, classical piCf\lre behind the The relationship between the clossical theory and the quanrum theory y ;~:~;;:G~O~ldstone, Goddard, Rebbi, and Thorn 136] and further b: [3 7]. The Ihcory, however, ""'as still fonnulated as a

16

1. Path Integrals and Point Particles

first quantized theory, so that the measure, the vertices, the counting of graphs, etc., all had to be postulated ad hoc and not deduced from first principles. The action (in particular gauge) was finally written down by Kaku and Kikkawa [38]. At last, the model could be derived from one action strictly in terms of physical variables, although the action did not have any symmetries left. However, when it was discovered that the theory was defined only in 10 and 26 dimensions, the model quickly died. Furthermore, the rapid development of QCD as a theory of hadronic interactions seemingly put the last nail in the coffin of the superstring. For 10 years, the model languished because no one could believe that a 10- or 26-dimensional theory had any relevance to four-dimensional physics. When Scherk and Schwarz [39] made the outrageous (for its time) suggestion that the dual model was acruaUy a theory of all known interactions, no one . took the idea very seriously. The idea fell like a lead balloon. The discovery in 1984 by Green and Schwarz [40] that the superstring theory is anomaly-free and probably finite to all orders in perturbation theory has·· revived the theory. The £8 ® £8 "heterotic string" of Gross, Harvey, Martinec,.· and Rohm [41] was considered to be the best candidate for unifying gravity·· with physically reasonable models of particle interactions. . One of the active areas of research now is to complete the evolution of the < theory, to discover why all the "miracles" occur in the model. There has been . a flurry of activity in the direction of writing down the covariant action using methods discovered in the intervening 10 years, such as BRST. Finally, intensive work on duality in the early 1990s, and especially the work of Witten and Townsend in1995, established M-theory as the latest, most ad~ . vanced formulation of string theory. The five superstring theories could now be unified into a single theory in 11 dimensions, perhaps in terms of membranes. Furthermore, the vast web of dual relationships in lower dimensions has given. us a "road map" by which to probe nonperrurbative solutions to string theory. What is still missing, however, is the geometry on which the theory ultimately lies. Understanding this geometry may allow us to formulate the entire theory into a simple, coherent equation. This could complete the evolution of the •. theory, which has been evolving backward for the past 30 years. .

Quantum theory -+ Classical theory -+ Action -+ Geometry. Let us summarize some of the promising positive features of the superstring • model: (1) The gauge group includes E 8 ® E 8 which is much larger than the minimal group SU(3) ® SU(2) 0 U(l). There is plenty of room for phenomenology in this theory. (2) The theory has no anomalies. These small but important defects in a quantum field theory place enormous restrictions on what kinds of theories are

1.2 Historical Review of Gauge Theory

17

The symmetries of the superstring llieory. by a series of cancel all of its potential anomalies. from the theory ofRiemano surfaces indicate that the is to all orders in perturbation theory (although a rigorous still lacking). is very linle freedom to play with. Superstring models arc nodifficult to tinker wilh without destroyi ng their miraculous Thus, we do not have the problem of 19 arbitrary coupling thcory includes GUTs, super-Yang-Mills, supergravity, and Kaluza-::: Klein thcories as subsets. Thus, many ofthe features of the phenomenology :::: for these theo ries carry over info tbe string theory. ,pe;~.ing

thoary, crudely speaking, unites the various forces and particles same way thai 8 violin string provides a unifying description of the tones. By themselves. the noteS A, B. C, ele., are not fundamcntal. the violin string is fundamental; one physical object can explain ;";i,tios of musical notes and eveD the harmonics we can construct from much the same way, the superstring provides a unifying description of ;""twyparticles and forces. In fact, the "music" created by the superstring forces and particles of Nature.

~~'~~~~~:::~~~:~!

theory, because of its fabu lously large set of symme-

' cancellations of anomalies and divergences, we must

C!

present a balanced picture and point out its shortcomings. To be fair we "~::: list the potential problems of the theory that have been pointed out Cl

of the model:

is impossible experimentally to reach the tremendous cnergies found the Planck sCllle. Therefore. the theory is in !>nme sense unlestable. A th"o,y "hat is untcslable is not an acceptoble physical theory. one shred of experimental evidence has been found fO confirm. the ofsuIP,,,,ym,,,,,'ry, let lone superstrings. . presumptuous to assume that there will be no surprises in the "desert" ::: betwecn 100 and 10 19 GeV. New, totally uncxpected phenomena have alup when we have pushed the energy scalc ofouraccelerators. theory, however, wakes predictions over the next 17 orders of which is unhcan.l of in the history of science. does not ~ Iain why the cosmological constant is zero. Any thaI claims to bc II ''theory of everything" must surely explain puzzle of a vanishing cosmological conSlant, but it is not clear how

""""no<

/;:;;'~;::;'~S:O:II~VC this problem.

)

a

embarrassment of riches. There are apparently millions ways to break down the theolY to low energies. Which is thc correct the superstring theory can produce lhc minimal theory other interucljons that have

18

1. Path Integrals and Point Particles

(6) No one really knows how to break a lO-dimensional theory down to four dimensions. Of these six objections to the model, the most fundamental is the last, the inability to calculate dimensional breaking. The reason for this is simple: to every order in perturbation theory, the dimension of space-time is stable. Thus, in order to have the theory spontaneously curl up into four- and six-dimensional universes, we must appeal to nonperturbative, dynamical effects, which are notoriously difficult to calculate. This is why the search for the geometry under- . lying the theory is so important. The geometric formulation of the model may give us the key insight into the model that will allow us to make nonperturbative calculations and make definite predictions with the theory. Thus, the criticism that the model cannot be tested at the Planck length is actually slightly deceptive. The superstring theory, ifi! could be successfully ...

broken dynamically, should be able to make predictions down to the level of . everyday energies. For example, it should be able to predict the masses of the .... quarks. Therefore, we do not have to wait for several centuries until we have accelerators that can reach the Planck length. Thus, the fundamental problem facing superstrings is not necessarily an ... experimental one. It is mainly theoretical. The outstanding problem of the theory is to calculate dynamical symmetry breaking, so that its predictions

can be compared with experimental data at ordinary energies. A fundamental theory at Planck energies is also a fundamental theory at ordinary energies. Thus, the main stumbling block to the development of the theory is an understanding of its nonperturbative behavior. . In Part I of this book, however, we will follow historical precedent ahd present the first quantized formulation of the model. As we will stress through~ out this book, the first quantized theory seems to be a loose collection of random facts. As a consequence, we have emphasized the path integral formulation (first written down for the Veneziano model by Hsue, Sakita, and Virasoro [42, 43]) as the most powerful method of formulating the first quantized theory. Although the path integral approach cannot reveal the underlying geometric formulation of the model, it provides the most comprehensive formulation of the first quantized theory. We will not tum to the functional formulation [44] of point particle theory,. which can be incorporated almost directly into the string theory.

1.3

Path Integrals and Point Particles

Let us begin our discussion by analyzing the simplest of all possible systems, the classical nonrelativistic point particle. Surprisingly, much of the analysis of • this simple dynamical system carries over directly to the superstring theory. The language we will use is the formalism of path integrals, which is so versatile that

1.3 Path Integrals and Point Particles

19

jt,~~~;~,;,:~~~.~ both first quantized point particJ~s and second qua ntized

&1

equal elise. in classical mechanics. the starting point is the Lagrangian for a point

1m.i:f -

L =

(1.3.1 )

V{x),

p",,;,lo is moving in an external potentia1 . The real physics is con· :d ;n tllO statcmcnt thai the action S must be minimized. The equations of can be derived by minimizing the action:

S= / L(Xi,x;' t)dt.

8S

~

O.

(1.3.2)

i':~:::~I~: equations of motion, let US make a small variation in the path If!. given by

ox,.

(1.3 .3)

r"th;, smaU variation, the action varies as follows:

J/

-ox' oX,

8L 8L dr -Ix· t ' + • • ",Xi

I

I

= 0•

( 1.3.4)

'!lI''''''8 by parts, we arrivc al lbe Euler- Lagrange equmions:

8L d!L - - -~ Oxl tir lix,

~O.

( 1.3.5)

point panicle, the equations of motion become d2Xi

m-

dt 2

~

-

aV(x)

ax,

( 1.3.6)

correspond to the usual classical Newtonian equations of ma ti on. add ition to the Lagrangian formulation of classical mechanics, there is

PI

8L

= 8x, """":"'" '

(1.3.7)

dcfinition of the conjugate variable, we have

H = PiX, - L.

p'

H (PI,Xi) = 2~

+ Vex) .

(1.3.8)

the Poisson brackets between the momenta and the coordinates are (1.3.9)

20

1. Path Integrals and Point Panicles

A celebrated theorem in classical mechanics states that the equations of motion ofNewton and the action principle method can be shown to be identical. Beginning with the action principle, we can derive Newton's laws of motion, and vice versa Equations of motion ++ Action principle.

This equivalence, however, breaks down at the quantum level. Quantum .. mechanically, there is a fundamental difference between the two, with the equations of motion being only an approximation to the actual quantum behavior of matter. Thus, the action principle is the only acceptable framework ... for quantum mechanics. Let us now reformulate the principles of quantum mechanics in terms of ·. Feynmanpath integrals [44]: (1) The probability pea, b) ofa particle moving from point a to point b is the square of the absolute value of a complex number, the transition function K(a, b): Pea, b) = IK(a, b)12.

(1.3.10) ...

(2) The transition function is given by the sum of a certain phase factor, which · . is a ftmction of the action S, taken over all possible paths from a to b: K(a, b) =

I)eiZ1tS/h,

(1.3.11)

patlls

where the constant k can be fixed by K(a, c) =

L K(a, b)K(b, c),

I.-

(1.3.12)

paths

and the intermediate sum is taken over paths that go through all possible intermediate points b. The second principle says that a particle "sniffs out" all possible paths from point a to point b, no matter how complicated the paths may be. We calculate this phase factor for each oftbis infinite number of paths. Then the transition factor for the path between a and b is calculated by summing over all possible · phase factors (see Fig. 1.6). Remarkably, the essence of quantum mechanics is captured in these two principles. All the profoundly important implications of quantum mechanics, which represent a startling departure from classical mechanics, can be derived from these two innocent-sounding principles! In particular, these two princi- . pIes summarize the essence of the quantum interpretation of the double-slit • experiment. which, in turn, summarizes the essence of quantum mechanics · itself. . •• It is apparent at this point that the results of classical mechanics can be reproduced from our two assumptions in a certain approximation. Notice that,

1.3 Path 100cgJals and Point Particles

... ",, I , ,/ ...

Clil$Sical Mechanics

"!

to.., I,

I

I

"

/

(

I

,,-"","

\,

J

",'"

''" I " "\\

"

,

J

.,/

---

I

;' -_ --'I ,

.........

'" '"

I"

21

.-

",""

1.6, The essential difference between classical mechanics and quantum 'iian;e, . Classical mechanics assumes lhat II panicle executes jusl one path be· points based either on the equations of motion or on the minimization of By contrast, quantum mechani~ ~um.<; the conuibUl;ons of probability on 3n action) for all possible paths between two points. Although path is Ibeone most favored, in principle all possible paths contribute to integral. Thus, the action principle is morc fundamental than lhc equations al the quantum level. of S that are large compared to Planck's constant. the pbase facLOr

rapidly. canceling out these contributions: h

oS» -,

(1.3.13)

2n

the only contributions to the path integral that survive are those for which ev;ati,o", in the action from the c lassical path are on the order of Planck's

(1.3.14) that the Euler-Lagrange equations of motion are reproduced only in a classicallimil, that is, when Planck's constant goes to zero. Therefore of Planck's constant ultimately dctennineli the probability lhat 8 parexecute lraje<:tories that are forbidden classically. We see the origin ,is"nl""g'sunccrtainty principle embodied in these two principles. tel us try to reformulate this more precisely in terms of path integrals. "ee~,dpri " '; pl' now reads

K (a, b) =

K(a , c) = /

11> D;ui2d / ",

(1.3.15)

K(a, b)K{b. c)Dx/t

(1.3.16)

22

1. Path Integrals and Point Particles

and

L ~ /DX =

lim /rIIldxi,n,

N ..... oo

paths

(1.3.17)

;=1 n=1

where the index Il labels N intermediate points that divide the interval between the initial and the final coordinate. We will take the limit when N approaches ··. infinity. . •.. It is absolutely essential to understand that the integration Dx is not the ... ordinary integration over x. In fact, it is the product of aU possible integrations • over all intermediate points Xi,nbetween points a and h. This crucial difference . between ordinary integration and functional integration goes to the heart of the path integral formalism. This infinite series of integrations, in turn, is equivalent to summing over all . possible paths between a and h. Thus, we will have to be careful to include normalization factors when performing an integration over an infinite number .• of intermediate points. Ifwe take the simple case where L = all functional integrations can actually be performed exactly. The integral in question is a Gaussian, which is fortunately one of the small number of functional integrals that can actually be · performed. One of the great embarrassments of the method of path integrals • . is that one of the few integrals that can actually be performed is

!mx;,

"f xx d

2n

_, 2.<2 _

e

-

-00

r(n

+ ~)

r 2n+ I

•

(1.3.18) ".

l

We will be using this formula throughout the entire book. Let us now break up the path into an infinite number of intermediate points, xi,n' (Notice that the functional expression integrates over all possible values of the intermediate point Xi ,n. so we cannot expect that Xi,n and XI.n+l are close to each other even for small time separations.) Let us write .

dt

~ E

(1.3.19) . In order to perform the functional integral over an infinite number of ·. intermediate points, we will repeatedly use the following Gaussian integration:

i:

dX2 exp[ -a(xl =

r;;

- x2i - a(x2 - X3)2]

exp [-

~a(x1 -

X3)2] "

(1.3.20) .

One of the crucial points to observe here is that the integration over a Gaussian in one of the intermediate points yields another Gaussian with that interme: diate point removed. This is the fundamental reason why we can perform the functional integration over an infinite number of intermediate points.

1.3 Path Integrals and Point Particles

23

the path integral that we wish to perform is given by K(a , b)

=

!~JJ

... f

dXI

dX2·· ·dXN_I

I

XC~£) -limN exp ~: l/Xtt -X"_1)2 j

(1.3.21)

we have suppressed the vector index i). Using the previous relation the final result is equal to K(a, h)

=

m 21(11> -I.)

'n exp ( 4im(XII -

rb - ta

xoi)

.

(1.3.22)

""";,,ie''0 probability function K has some vel)' interesting properties. For it solves the wave equation: -I a2 a ---2 K(a, b) = i - K(a, b)

2m axo

ata

( 1.3.23)

t.. is greater than I". we will generalize these expressions for the case offrccly propagating ,and we wil l find that these expressions for the Green's functions carry only smail, but important, changes. show the relationship between tbe Hamiltonian :md Lab'nmginn fo rin the path inlegrdl approach, it is helpful to insert a compJete set .~;::~~::: states when we divide up the path from a to b. Let us treat the is as an opt.'TtIlor acti ng on a set of eigenstates:

x

X Ix) = x Ix).

(1.3.24)

: Ix) represents an eigenstate of the position operator, treating i as an whose eigenvalue is I:qua1 to the number .r. Then completeness over ,ils""s Co" ,,";,d;"I" and for momenta can be represented as 1=

1= nonn" li,~

f f

j:c)d.r(xl,

Ip) dp (pI.

(1.3.25)

= 8(x - y), e'Px (pix) = =

(1.3.26)

our states as follows: (xly)

,,2 ~

of the infinite number of normalization constants that constantly the path iotcgral fonnalism, we will often delete them for the sake of in this book. We do not lose any generality, because we can, of course, them into the path integral if we desire.)

24

1. Path Integrals and Point Particles

With these eigenstates, we can now rewrite the expression for the Green's function for going from point Xl to XN.

(1.3.27) In order to derive the previous expression (1.3.22) for transition amplitude, let us insert a complete set of intermediate states at every intennediate point between Xl and XN: (Xl, tllXN tN) = (Xl, tllX2, t2) I

f

" ·IXN-h tN-I)

dX2 (X2, t21

f

f

dX2

dXN-1 (XN-lo tN-t!XN. tN)' (1.3.28)

Now let us examine each infinitesimal propagator in tenns of the Hamiltonian, which we write as a function of the coordinates and derivatives:

H = H(x, ax).

(1.3.29)

Then the transition for an infinitesimal interval is given by (Xl. tllX2. 12) = (XII e-iH(x.az)li1 I X2) = e- iH (x.ai)51 (xllx2)

=

e-iH(x,a.;)~1 (xdp)

= e-iH(x.p)~t

= e-iH (x.pl51

f f

f

dp (plx2)

dp e ip(X2 -x I ) 2:rr dp e ipi8t • 2:rr

(1.3.30) .

It is very important to notice that path integrals have made it possible to . • make the transition from classical to quantum commutators. The Hamilt6nian can be expressed either as a function of derivatives with respect to the position or as a function of the canonical momenta because of the identity:

(1.3.3l) This allows us to make the important identification: H(x. p) ++ H(x. ax), . 0 { p ++ - 1 - .

(1.3.32)

ox

In the functional formalism, the important correspondence between momenta ... and partial derivatives arises because of this identity. . Putting everything together, we can now write the complete transition amplitude as (Xl, tdXN, tN)

= jX'; DpDx exp {i [III [pi' XI

II

- H(p, X)] dt} ,

(1.3.33)

1.4 Rciati,,;slic Point Particles

p' 2m

H = -.!....

+ Vex).

25

(1.3.34)

":,:::;, we have dropped 1111 Ihe intermediate Donnalizations. which are f! of 21r .) Notice that the fu nctional imegrol, which. wall once only of the coordinates, is now B function of both the momenta and the

~~:;;~ retrieve the original Lagrangian, we can perform the p integration because il is 3 simple Gaussian integral, and we arrive at

;:: ., ... I I Ix .... '

IN) =

1.'"

Dx exp

{il IN Until - V (x)] dt } .

(1.3.35)

la~~~,:: made tbe transition between the Lagr,mgian and we Hamiltonian

t8

u<;ing functional m.ethods. We can usc either:

L = ~m..i} - Vex)

+-!o

pi + V(x) . m

H = 2-

(l.J.36)

• the only difference between these two expressions is whether

over the coordinates or a combination of the coordinates and the The transition probability can be represented as

1: 1

K(a, b) =

=

Dxcxp

1" .r,

{i1.-' dt[~mx; -

! f"

DxDpexp i

I.,

V(X )] }

[

II

dl pi; - 2~' p' - V(m)

.(1.3 .37)

Relativistic Point Particles OUI discussion has been limited to nonrelativistic partid cs, where aU of freedom arc physical. However, nontrivial complications occur we generalize our previous discussion to the case of relativistic par. In panicular, lbc (- 1) appearing in the Lorentz metric will, in general, ~:~::fc~;~'~; !>tates to propagate in lbe tbeQry. These nonphysical "ghost" e: have negalive probability, must be eliminated carefully to ensure causal theory free of negative norm statcs. 0'"'" rel"iv,,,k case, lei us 31'.SUme that lbe location of a puint particle is by a four-vector. (1.4.1)

i'~~~::;:~'~~:,,:~ r docs not necessarily refer 10 the time. The action is ~

simple, being proponionalLO the four-dimensional path lenglh : S = - m / ds = -m (length).

( J.4.2)

26

I. Path Integrals and Point Particles

The path length ds can be written in terms of the coordinates:

ds =

J-x Jl2 dr:'

(1.4.3)

where the dot refers to differentiation with respect to the parameter r:. This action, unlike the previous nonrelativistic action, is invariant under reparametrizations of the fictitious parameter r. Let us make a change of coordinates from T to f: (1.4.4) .

T

-* i(r).

d

dr r = di d r:,

Then we find

dx dx df -=--, dr di dr:

1(a::)'j'" ~ 1(a:;)'r dr

(l.4.5)

di

Thus, the action is invariant under an arbitrary reparametrization ofthe variable · . r:.

This can be written infinitesimally as r: -+ r { 8x)i =

+ or:, x)i8r:.

As before, we can now introduce canonical conjugates: 8L

PJl =

oxJl

(l.4.6) l

mXJl

J x;'

=

(1.4.7)

The crucial difference, however, from our previous discussion of the nonrelativistic point particle is that not all the canonical momenta are independent. . In fact, we find a constraint among them: ( l.4.8) Thus, the mass shell condition arises as an exact constraint among the momenta. If we calculate the Hamiltonian associated with this system, we find that

H =

pJlx/J. -

L

= O.

(1.4.9) .•

The Hamiltonian vanishes identically. These unusual features, the vanishing of the Hamiltonian and the constraints .. among the momenta, are typical of systems with redundant gauge degrees of freedom. The invariance under reparametrization, for example, tells us that the path integral that we wrote earlier actually diverges:

/ Dxe

iS

=

00.

(1.4.10)

1.4 Relativistic Point Particles

27

"o.us.e there is a St:par-dle contribution from each particular param. But since /Jx is parametrization invariant, Ihis means that we are ~!~~:,. an infinite number of copies of the same thing. Thus, the integral however, explained how to quantize systems with redundant gauge of freedom. For example, let us introduce canonical momenta P and constraint condition via a Lagrange multiplier as follows:

+ m2 ).

L = Py.x/I. - ~e(p!

(lA.II)

.co,.stmint equation (l.4.8) is imposed here as a classical equation of varying e, we recover the constraint 00 the momenta. Quantum however, this constraint is imposed by functionally integrating the path integral, we have

f

Decxp

[-i f

drte(p2 +m2)] '" o(p2 +ml),

( 1.4.1 2)

we have used the fact that the integral over eik;t (or the Fourier transfonn ,. number I) is equal to o(x). Notice that the new Lagrangian ( 104.11) still the gauge degree of freedom. It is invariant under

exj.l ,

OX~

=

cPr.

= sP/l.' d(se)

oe= dr'

(1.4.13)

that this netion has over the previous one is that all variublcs We do not have to worry about complications caused by the " Ilolld e that we have introduced will become the metrie tensor we generalize this action to the string.)

tet"' now functionally integrate over the p variable. Becausethe integration a Gaussian, we have no problem in performing the p integration:

f

Dpoxp \'

J

d, [pi -le(p' +

'" exp

{jf

rn')ll

dr ~(e- lx2 - em 2) } .

(1.4. 14)

we have now obtained a third version of the point particle action. The lD"'go of this action is that it is linear in the coordinates rnd is invariant

:Xp_ d::;;

[ De _

d,

.

(1.4.15)

i,";::::1~'we have fOWld tluce equivalent ways to express the relativistic [)t The "second~order" Lagrangian (104.14) is expressed in lenns of

28

1. Path Integrals and Point Particles

second-order derivatives in the variable x lL ( 1:) and the field e. The "nonlinear" Lagrangian (1.4.3) is expressed only in terms of xlLC'r). It can be derived from .. the second-order form by functionalIy integrating over the e field. And finally, . the "Hamiltonian" fonn contains both x IL ( r) and the canonical conjugate p( r) (it is first-order in derivatives): first-order (Hamiltonian) fonn: L

= PILi'" -

!e(p~

+ m2),

second-order fonn: L = ~(e-I i~ - em 2), nonlinear fonn: L =

(1.4.16)

-mJ i~.

All three are invariant under reparametrization. Each of them has its own distinct advantages and disadvantages. This exercise in writing the action of ...• the free relativistic particle in three different ways is an important one because it will carry over directly into the string fonnalism. Expressed in terms of path . integrals, the point particle theory and the string theory are remarkably similar. .

1.5

First and Second Quantization

In this section, we will quantize the classical point particle and then show • the relationship to the more conventional second quantized formalism offield .. theory. The first quantization program, as we shall see, is rather clumsy com-, pared to the second quantized formalism that most physicists are familiar with, but historically the string theory evolved as a first quantized theory. The great . advantage of the second quantized fonnalism is that the entire theory can be derived from a single action, whereas the first quantized theory requires many , additional assumptions. The transition from the classical to the quantum system is intimately linked .. with the question of eliminating redundant infinities. As we said before, the . path integral is formal1y ill-defined because we are summing over an infinite . number of copies of the same thing. The trick is to single out just one copy. There are at least three basic ways in which the first quantized point particle may be quantized: the Coulomb gauge, the Gupta--Bleuler formalism, and the BRST formalism.

Coulomb Quantization Here, we choose the gauge Xo = t = r.

(1.5.1)

In other words, we set the time component of the x variable equal to the real time t, which now parametrizes the evolution of the string. In this gauge, the

1.5 First and Se.:ond Quantization

29

.i",du,,,, to

L=-mf JI

I.Il

dr.

(1.5.2)

: limit of velocities smal l compared 10 the velocity of light, we have (1.53)

so that the functional integral is modified to

f

Dx"S(xo -

()e's =

f

DXI

cxp

(i ! lm# dl) .

(1.5.4)

case of the string, this simple c.x.ample will lay the basis for the light

.~:;:~~;;:::~~Th~:.:eadvantage of the Coulomb gauge i~ thai all ghosts have ~

from the theory, so we arc dealing only with physical

ntit;". The other ad .....dI1lagc is that the zeroth component of the position now explicitly defined to be the time variable. The parametrization point panicle is now given in terms of the physical time. disadvantage of the Coulomb fonnaJism. however, is that manifest symmetry is broken and we have to check explicitly that the quan-

~:::;,~~!;~:cJosc correctly. Although this is trivial for the point features will emerge for the quan tum string. fixing the

;

to be 26.

LeuLer QuanlizQlion tries to maintain Lorentz invariance. This means, of course, thai must be taken to prevent the ncg31ive norm states from spoiling properties oftbe S-matrix. The Gupta-Bleuler method keeps the tOluJly relativistic, but imposes the constraint (1.4.8) on state vectors: (1.5.5) . that the above etjuation is a ghost-killing constraint. because we can -. -: it to eliminate Po.) This fonnu lism allows us to keep the commutators fully (1.5.6)

wc choose r]1'-~ = (- + + + ... ). Notice that this gauge constraint "",liy generalizes to the Klein-Gordo n cqualion: (0 - m')~(r) ~ O.

(1.5.7)

Gupta-Bleuler fonnalism is an imponant one because most of the ,o].d,,", in string theory have been carried out in this formalism.

30

1. Path Integrals and Point Particles

ERST Quantization The advantage of the BRST formalism [45,46] is that it is manifestly Lorentzinvariant. But instead of regaining unitarity by applying the gauge constraints .. on the Hilbert space, which may be quite difficult in practice, the BRST formulation uses the Faddeev-Popov ghosts to cancel the negative metric particles. Thus, although the Green's functions are not unitary because of the propagation of negative metric states and ghosts, the final S-matrix is unitary because all the unwanted particles cancel among each other. Thus, the BRST formalism· manages to incorporate the best features of both formalisms, i.e., the mani-· fest Lorentz invariance of the Gupta-Bleuler formalism and the unitarity of .. the Coulomb or light cone formalism. In order to study the BRST formalism, .• however, we must first understand Faddeev-Popov quantization.

1.6 Faddeev-Popov Quantization Before we discuss the BRST method, it is essential to make a digression and ••• review the formalism developed by Faddeev and Popov [47]. As we said earlier, . the path integral measure DxiJ. is ill-defined because it possesses a gauge degree· . of freedom, so we are integrating over an infinite number of copies of the same thing. Naively, we might insert the gauge constraint directly into the path integral. If the constraint is given by some function F of the fields being set to zero: l

F(xiJ.) = 0,

(1.6.1)

then we insert this delta functional directly into the path integral: Z =

f n Dx

8 [F(xlL)]e

iS

•

(1.6.2)

x

However, this naive approach is actually incorrect because the delta functional contributes a nontrivial measure to the functional integral. The key to the Faddeev-Popov method is to insert the number 1 into the functional, which obviously has the correct measure. For our purposes, the most convenient formulation of the number 1 is given by

1=

AFP

f

De8[F(x;)],

(1.6.3)

where e is the parametrization of the gauge symmetry of the coordinate, in (1.4.6), x~ is the variation of the field with respect to this symmetry, and the Faddeev-Popov determinant AFP is defined by the previous equation. Notice that the integral appearing in the previous equation is an integration over an possible parametrizations of the field. Since we are integrating out over all possible parametrizations, then, by construction, the Faddeev-Popov

1.6 Fllddccv-Popov Quantization

Jt

!l"inanl is gauge independent of any particular parametrization: o'FP(X)

= AFP(X').

(\.6.4)

";o.::li~ins~,ert the nwnbcr I inlO Ihe functional integral and make a gauge isl to reabsorb the E dependence in x:

Z=

=

f f

Dx6.}l'(x)

f De~[P(x£)]els

Dxll. }l'{x) /

DetJ( F (x )]eiS ,

(1.6.5)

here the x' was gauge rota led back into the original variable x. Since parts of the fu nctional integral were al ready gauge independent. we

(1.6.6) aow extract o ut the integral over the gauge parameter, which measures inJini,'",ohllln' of the group space: ,;olume =

f

(1.6.7)

De

t~~~~;~, n"w expression for the functional which no longer has this infinite

z=

f

(1.6.8)

DXfifPo[ F(x)]e's.

that a naive quantilBlion of me path inlcgral wou ld simply insert the

[;i~~~W~O~U!I~d~om~it thecome Faddeev-Popov determinant, which is 3. new Lhc measure out correctly.

~

the Faddeev-Popov dctcrminant, which carries all the ho,;ls ,ofthe,lheory. Tbe trick is to change variables because both f1 and F havc the same number of of freedom. Thus, the Jacobian can be calculated:

dec 'can

[~: ] D e =

(1.6.9)

DF.

"'''''(0'' write Il"

~ !f D'Wf = !f DFdet [:~]

a,] ]-' [ I

= del -

of

F= O

['F]

=det -

D£

.

W f (1.6. 10)

F",O

the Faddeev-Popov factor can be expressed as a simple determinant of vari";on of the gauge constraint. It is more convenient to introduce this

32

1. Path Integrals and Point Particles

factor directly into the action by exponentiating it. We use the fonowing LlFP

=

!

D()D8e iSt)1,

(1.6.11)

where the new ghost contribution to the action is given by Sgh

=

!

dr8

[~:t=o

(),

(1.6.12)

where the e variables are anticommuting c-numbers called Grassmann num- . bers. Normally, when performing functional integrations, we expect to find the • detenninant of the inverse of a matrix. With functional integration over Grass- . mann numbers, the determinant occurs in the numerator, not the derlonlimltOlr.. Grassmann numbers have the strange property that

(1.6.13) . In particular, this means (1.6.14) .. '

. ..

Normally, this would mean that e vanishes. However, this is not the case for a •• Grassmann number. Thus, we also have the strange identity ( 1.6.15) This identity makes the integration over exponentials of Grassmann-va1u~ . fields in the functional integral rather easy, because they are simply pol)'I)o- . rnials. More identities on Grassmann numbers are presented in the Appendix, . where we show that

!n

d(); d8/ exp

,= 1

[.f

8iAij()j] = det(Aij).

(1.6.16) • •

I.}=I

This identity verifies that integration over Grassmann variables yields deter- •.• minant factors in the numerator, not the denominator, so that we can express . the Faddeev-Popov determinant in (1.6.11) as a Grassmann integral. Now that we have developed the apparatus of Faddeev-Popov quantization, let us return to the BRST approach, where we wish to impose the gauge condition

e= 1

(1.6. 17) .

(we omit some subtleties with respect to this gauge). In this gauge, we should be able to recover the usual covariant Feynmann propagator. To show this, . . notice that our action (1.4.14) becomes (1.6.18)

1.6 Faddeev-Popov Quantization

33

this Lagrangian, our Green's function for the propagation of a point from one point to another is now given hy .6.F(XI ,Xl)

= (Xtl 0 1m 1-'2) = (x II 2

=

101) dre- «rJ-",I) 1-'2)

l<X>d'l~lDxexp(-~l rd r(i;, - m2)).

(1.6.19)

that this is the usual covariant Fcyrunan propagator rewritten in first ~ti,<ed

integral language. before gauge fixing. our action was invariant under

',= dd(',')·

(1.6.20)

o

Faddecy-Popov detenninafll associated with the gauge choice e = 1 determinant of the derivative. We now use II Gaussian integral over ~"'"""' states to represent the detenninant, using (1.6.10):

.6.~'p =dct loTI = -

f

D9D8CXP(

f dr:iiaTo).

(1.6.21 )

we had used ordinary real fields instead of Grassmann-valued fields. the would bave corne out with the wrong power.} we find that our final action can be represented

(1.6.22)

r:;;;,~~;::fCth" B.RSr

Q.pp,,,",h .~ to notice Ihat Ihis gOllge-fu:ed action has

~X/l

= i£9xIJ.' Jp,. = if;f:J PI<' MJ = j£90,

08 =

i£(}O

(1.6.23)

+ !s(p; + m 2) .

we may wonder why yet anot her symmetry appears after we have the gauge degree of freedom. However, this extra symmetry is and hence does not allow us to impnl'le any eonstmints on the theory. symmetry, therefore. is different from the ones found earlier and cannot used to eliminate gauge fields from the action. summarize the BRST approach by extracting an operator Q that will the symmetry fOWld earlier:

'¢ = L8Q,¢I, Q = 9(0 - m 1 ).

Q2 = O.

( 1.6.24)

34

I. Path Integrals and Point Particles

The physical states satisfY

Q lei» =

o.

Notice that enforcing this constraint recovers the Klein-Gordon equation on-shell particles:

1.7

Second Quantization

So far, we have been analyzing only the first quantized approach to quantum particles. We have quantized only the position and momentum vectors: .

[Pi, X j] = -i 8u.

first quantization:

The limitations of the first quantized approach, however, will soon beC:OIIlle apparent when we introduce interactions. Let us say that we wish to del:;crlibe point particles that can bump into each other and split apart, rather than duce an external potential. We must now modify the generating functional include summing over Feynman graphs:

L

Z=

topologies

f

1ength

Dxe-

•

(Notice that we have Wick rotated the r integration so that the exponential converges. It will be clear from the context when the Wick-rotated theory being used in this book because the exponential becomes real. We will discuss the delicate question of the convergence of path integrals.) In other words, we must, by hand, sum over the various particle tOpIOJ()g1I~S·. where point particles can split and reform. Each topology represents the hi~,t"r'\J ofthe trajectories ofthe various point particles as they interact. The amplllnIO,e,: for N-particle scattering, with momenta given by k], k2, ... , k N , can now represented as A(k 1 , k2 ••••• kN) = .

L

topologies

X

gn

f f

exp (-

DXll.FP

dt L(t)

+i

t

k"xj ) .

Notice that we are taking the Fourier transform of the Green's function, so the amplitude is a function of the external momenta. This formula can be mClrp." conveniently represented as

\ .1 Second Quantization

we associate a fdCtor

e,'h

transfonn terO'i. This path ,00","" because it will configurations and ~_""'tri·x i.s not al all obviolL<;.

35

for each external particle coming from the

fonnu la for the scattering amplitude i fomlalism. is. We must fix the set ofall topologically weights by hand. Furthermore, unitarity of

a second quantized description, however, we introduce a field V(x) t:.:~s~";('" relations between the fields themselves, not between the (1.7.5)

' :"I",.o"'g" of· the second quantized approach is that the inlcracting Hamilcan be written explicitl y, without having to inltoducc sums over 'OS"": Showing that the Hamiltonian is Hennitian is sufficient to fix the of all diagrams and to demonstrate !he unitarity of tbe S-matrL"( . . "unID'''I. the pros and cons of first and second quantizations are as

must be added in by hand. order by order in the coupling of Ihe final S-mlllrix is not obvious. This must be explicitly

order by order. formalism is necessarily a pc:rturbative one, ~ince the expansion in l~;:;:::~s is intimately tied to the expansion in terms of lhc coupling difficult 10 describe the theory otT-sheil. Quanti7.ation interactions arc cxplicit in lhe action itself. ~";"'.rilyiS guaranteed if the J-Iamiltonian is Hcnrulian. theory can be formally written nonperturbatively as well as perturba-

.""'y.thcory is necessarily off-shell. .. . transition from the ~tto the second quanti:tcd.lhcory can also be per... most easily in the path integral forma lism in the Coulomb gauge. we showed that thc Grcen's function tor a propagating free point can be explicitly evaluated:

f,,}

l

[.m(x.-x.)']

' ~ exp /

2{rb

1,,)

.

(1.7.6)

""""·s function can also be written in a second quantized fashion. Let with the Hamihonian:

H

1

,

=-- v . 2m

(1.7.7)

36

1. Path Integrals and Point Particles

The Green's function satisfies

= 8(3)(x -

(iar - H)K(x, t;x' , t')

X')O(t - t').

Solving for this Green's function, we find K(a, b) = [i8r - H]x-1r 'x I ' 6. 11, b, b

where we are treating the inverse Green's function as if it were a discrete trulttrt~i in (x, t) space, and we have dropped trivial normalization factors. This us to write the integral in second quantized language. To demonstrate this, will use the following identities throughout this book:

f0

dx; exp

(i~ -x;Aijxj + n(l/2)N

il: I Jix;

(I"4 i~ Ji(A N

= det [Aijl exp

-1)ijJj

I .

(This integral can easily be derived using our earlier formula for the integral (1.3.18). We simply diagonalize the A matrix by making a change variables in x. Thus, the quadratic term in the integral becomes a function of eigenvalues of the A matrix. Because all the modes have now decoupled, Gaussian integral can be performed exactly by completing the square. we make another similarity transformation to convert the eigenvalues of back into the A matrix itself.) From this, we can also derive the following:

f D il: XnXm

dXi exp (

-xjAijxj +

il: I JiXi

~ [~~ .. J'}] det Ik'I-1 ojn ojm exp {~l'(A-I) 4 J Y 1=0 1

IJ

~ (A -I )nm(det IAu I)-I.

These are some of the most important integrals in this book. Using these equations, we can now write the Green's function totally in terms of . quantized fields: K(a, b) =

f

1/!*(xa, ta)1/!(Xb, tb)D1/!* D1/! exp

[i f

dXdtL(1/!)] ,

where L(1/!) = 1/!*(iar

-

H)1/!,

where we are again treating K(a, b) as if it were a matrix in discretized (x, space.

1.8 Hannunic Oscillators

37

;,ummary. we now have two complementary dcscriplions of che point We can write the theory either in lenns of the particle's coordinates in terms of its fields W(x). free level, both de.<;criplions (ll'e totally equivalent, both in ease of ri~:: and also in mathematics. However. at the interacting level, distinct !'C appear. For example, it is easy 10 write

L, _tj)l,

"'¢.,

(1.7.14)

are guaranteed to get a unitary description of an interacting field. '''''. in the firs t quantized approach, the sum over topologies:

L

( 1.7.IS)

oopolol_

way in which to describe a unitary theory. We must check uniby order in increasingly complicated diagrams. Furthermore, we to adopt a tOIAlly pcrlurbative description for the first quanti7.ed . The sum over topologies in the first quanti7.ed path integraJ is a Feynman diagrams, so the formulation is necessarily from the very beginning. Thai is the fundamental reason why we this book into first quantization and second quantization.

Harmonic Oscillators ~rumpl"h'I will illustrate the relationship between first and scoond quan-

the harmonic oscillator problem. This example. will prove helpful )(\,U~~~h~the harmonic oscillator representation, which we will use ex.h string model. Let us begin with a point particle governed by lowmg Hamiltonian: (1.8.1 ) is the spring constant. Because the momenta and coordinates are we can use the same arguments presented earlier in our discussion ,Ib," "g,.I, 10 SCI [p, xj = -i.

(1.8.2)

now redefi ne our coordinates and momenta in Icnns of hannonic p

= (f mw)I i2(a +at ),

x = j(2mw)-I!2(a _ at),

( 1.8.3)

( 1.8.4)

38

I. Path Integrals and Point Particles

In order to satisfy the canonical commutation relation (1.8.2), we must have . : [a, at ] = l.

If we insert this expression back into the Hamiltonian, we find

H = tw(aa t

+ ata).

By extracting a c-number term, we can write this in normal ordered fashion:

H = w(ata

+ ho),

where ho is the zero point energy. We can now introduce the Hilbert space harmonic oscillators. Let us define the vacuum as

a 10) = O. Then an element of the Pock space of the harmonic oscillator Hamiltonian " given by In) = (att 10)

Jnf such that the states form an orthonormal basis: (nlm) =

onm.

The energy of the system is quantized and given by

En = (n

+ i)w.

So far, the systems has been presented only in a first quantized formal .. We are quantizing only a single point particle at any time. We would now to make the transition to the second quantized wave function by introducing 00

I<%» =

L ¢n In) , n=O

where we power expand in the basis states of the harmonic oscillator. instead of describing a single excited state of a point particle, we are introducing the wave function, which will be a superposition of an
(xlct» = <%>(x). This can be calculated explicitly. Notice that we now have two iI'Idepel1de:nt:::;?4 basis states, the harmonic oscillator basis In) and the position eigenvectors Ix) . We must now calculate how to go back and forth between these two bases. . Let us first analyze the simplest matrix element: Uo(x)

= (xIO).

1.8 HarmonK: Oscillatonl

39

element satisfies the equation

0 = (xl (I 10)

= (xl p~x 10) 2m",

(-

ia - imwx ) (..rIO) ax

= -i(2mW)- I(l

(a~ + mcux) oa(x).

=

(2mllJ)-11l

( 1.8.15)

equation can be solved e.xacuy:

ao(.,r) = (mwljf)I J~e-(If2}f l,

( 1.8.16)

(1.8. 17) 8

straightforward titep to calcu late aUsuch matrix elements. Let

o~(x)

= (xJn )

= (xI (n!)I/lat~ 10)

r

= (n!)- lfl (2mw}-
=

(n!)-ln(2mw)~I/l)n

(-i a~ +

(~IuDion is therefore

a~{x) = iM(2~n!)- If2 (mw/lT)II'

(t - aa~

im(o)x

uo(x).

r

~-(If2)tI.

(l.8. IS)

( 1.8.19)

"::~~thcse are nothing but Hermite polynomials H". In terms of these we Can CJtpress the eigenstaTe Ix ) and III) in lenns of each other:

1)(

~

Ix )

~

= L: In) (nix ) = L: In ) a. (x). ,

In ) = Ix)

!

..

dx (x1n)

=/

dxo~(x) Ix).

( 1.8.20)

>s. '~';', g ,(I. . 12) and (1 .8.20). we have the power expansion of the wave in tcnns of a complete set o f orthogonal polynomials, the Hcnni te

I .

~

lI>(x )

~

= (xl
(1.8.21)

il is not difficult to calculate the Green 's function for the propagahpoi.t particle in II hllrmonic oscillator potential. The Green's function

40

1. Path Integrals and Point Particles

would be the same as if we had started with the second quantized with the action: L = ct>(x)* (iar

+ 2~ V2 -

LV1UH1U"lJ,l:,::;

(1.8.2~~i::m

ikX2) ct>(x).

From this second quantized action, we can therefore derive the equations . motion: .

[-1

V2 + 2m = H¢(x, t).

ia,ct>(x, t) =

!kX 2] ct>(x, t)

From this, we can define the canonical momenta conjugate to ¢(x, t) such the canonical quantization relations are satisfied:

[n(x, t), ct>(x', t)] = -i S(x - Xl).

1.9

Currents and Second Quantization

Let us begin with a discussion ofthe relativistic second quantized theory, WhlCttS;:::: as we have seen, is equivalent perturbatively to the first quantized theory. quantizing the point particle in the Gupta-Bleuler formalism, we were the equations of motion: ( [0 - m2 ]q'> = 0

which can be derived from the second quantized action:

L = Hallq'>i}llq'>

+ m 2q'>2].

One of the most powerful techniques we used to explore the first theory was symmetry. We would now like to study the question ofsyrnrnletri~W within the second quantized formalism. First, let us calculate the equations of motion by making a small in the field and requiring that the action be invariant under this variation:

8S

f

= 0=

dDx (8L 8q'> +

8q'>

~8aflq'» 8i}fl¢

.

Let us now integrate by parts, using 8I1(1.q'> = afl8¢:

8S =

!

dDx

G~

-

a(1.

o~~¢) o¢ +

f

dDxi}1l

(8:~¢ o¢) .

lfwe temporarily ignore the surface term, the action is stationary if we hli;;i@E:l the following equation of motion:

8L oL aJ.L -oaJ.Lq' > oq'>

= 0

(l ....

"'0'1

L . ...

1.9 Current:; lind Second Quantization

41

im"rt the Lagrangian into Ibis equation. we obtain the equations of which reproduce the constraint found earlier in the first quantized noW make n small change in the fields, pammetriz.ed by a small, as ,,,Ui,J, Ilwnber 0":

8¢ = 8¢ as".

'c·

( 1.9.6)

this into the previous equation for the variation of the action, keep term intact, Illld assume that the equations of motion are satisfied,

have the following:

!s~fdDxa" (~'¢)"-' JJ ¢ at"

(1.9.7)

Il

~efin,

the tensor in the parentheses as the current: !ii, 8¢

" J" =

aa ll ¢, oE!"

(1.9.8)

, .",ve the important equation

oS = / dDx0I'J:Ssa.

(1.9.9)

action S is stationary under this variation. we have a conserved (1.9.10)

I

w"' ,hi" ",""ri,," over and over again in the discussion ofslrings when

extrnct the currcnt for supersymmetry and conformal invariancc. note that the integrated eharge Q" associmed with the current is

in time:

f

d D xal' JI"J =

f

d D - 1x8rJJo..

+ surface term,

( 1.9.11 )

(1.9. 12) we wish to construct yet another conserved current associated with Let us make a small variation in the spacHime variable: (1.9.13) charge, the volume clement of the integral changes as

odDx

=dDxallax".

(1.9. 14)

42

I. Path Integra1s and Point Partic1es

Therefore, the variation of the action under this change is

f

dDx[LafLDX fL

+ 8L].

DS

=

DL

= DXfLollL + -8¢' + --8all ¢,.

8L

8L

8¢'

8all¢'

Now, if we assume that the equations of motion are satisfied, we have

SS =

!

D d x all { (

+L8~ - 8!~¢' au¢,) ox

u } •

If we now define the energy-momentum tensor as 8L TJ.l.u = sail¢' au¢,

then we have the equation

SS =

!

- 1)J.l.u L ,

dDxoll(T!'-vS xv ).

So if the action is invariant under this change, then the tensor is conserved: aJ.l.p v = O.

emlrg:y-{nOlnellltu~a~

(1.9.1 (

For example, for the scalar particle action, the energy-momentum becomes TJ.l.v = a/1¢,a,,¢, -

1]/1V

L

which is conserved if the equation of motion holds. Lastly, it is instructive to investigate how the various quantization dures treat the Yang-Mills field (see Appendix). Let us begin with the' invariant action:

where

F;v

= a/1A~ -

avA~ - roc AtA~.

The action is invariant under SA"/1!i = a N l - f abe AbJ.I. A C ' where A a is a gauge parameter. The path integral method begins with functional

Z=

tenlsdj~m

fn

dA/1(x)i fh -(I/4)F; ••

J.I. .x

Now we consider the three methods of quantization.

prclcei~~

Currenl~

1.9

and Second Quantization

43

invariancc permits us to take thc gauge

"ViAr = O.

,"gro,t, 0"" 'h' Ao component because it has no time derivatives, so formulation is explicity ghost-free. (The price we pay for this, of loss of manifest Lorentz invariance, which must be checked by :- :. this gauge, the action becomes

~(F;jl

L = +HaoAn2 -

+ " ',

(1.9.25)

,field, are D'ansverse. This is the canonical form for the Lagrangian.

of the Gupta-BJculcr fonnulation is that we cao keep manifest 'Y'"m,,"y without viol
(1.9.26)

, the propagator for massless vector particles becomes

,. '.'

(1.9.27)

p

"~~;~~'~i'~~,a~:" explicitly eontains a ghost. The timclike excitation i 1 in the propagator. which represents a ghost. However, to !.{uantize in this covariant approach becausc we will impose the "'~""'''01 on the Hilbert space:

:f

(
a" AI''' Iljr) =

(1.9.28)

O.

,on,;", allo,",'5 us to solve for and hence eliminate the ghost modes. the .

will allow ghosts to

the Hilbcrt ghost-free.

approach begins by calculating the Faddeev-Popov determinant , Let us calculate the de terminant of the ma!rix:

M ab (

) _

~(a~A""(.x-)

,sAbey)

x. Y -

= 8 Dj.l SAO(x) "

,sAbey)

= OJ.lDj.l(8\l -

yW b ).

( 1.9.29)

44

1. Path Integrals and Point Particles

As before, we can write the determinant of Mal, by exponentiating it into action using (1.6.10):

L = _1 F a2 4

IlV

+ _l_(a AJ.lQ)2 + ca MObcb 2a Il ,

where the anti commuting Faddeev--Popov ghost fields are represented by and c. This action is invariant under the following BRST transformation:

.sA: = (V BRST:

DCa -

-

-a

IlC)Q

e,

_!.fobdcbCd e

2

'

1 a

DC = -(aJ.l"A"·c<)e.

Once again, it is important to notice that the BRST transformation is nillPoteIit~:W The BRST symmetry is not connected to the conservation of any oO!;enraOl~::::: quantity. From the previous invariance, we can extract out the generator transformation S such that Q2 = O.

The physical states of the theory then satisfy Q Iphy) = O. (

1.10

Summary

The great irony of string theory, which is supposed to provide a framework for all known interactions, is that the theory itself is so sorgatliz(~g.~!:m ! String theory is often frustrating to the beginner because it is full of . conventions, md arbitrary rules of thumb. The fundamental reason for this: that string theory has historically evolved backward as a first quantized th_'nri,,:~1% rather than as a second quantized theory, where the entire theory is <1ellllC~CI:~~ in tenus of a fundamental action. The disadvantages of the first qu:antize:q;~~ approach are that: (1) The interactions of the theory must be introduced by hand. They canlnC!!~m be derived from a single action. (2) Unitary is not obvious in this approach. The counting of graphs must checked tediously. (3) The formulatiop. is perturbative, so that crucial nonperturbative tions, such as dimension breaking, are beyond its scope. (4) The formulation is basically on-shell, rather than off-shell.

By contrast, the advantage of the second quantized approach is that everything can be derived from a single off-shell action, where is manifest and nonperturbative calculations can, in principle, be pelfolrn1!}d;;;

LtO Summary

45

string theory evolved historically a... a first quantized theory. theory has been evolving backward, with the second quantized gestill in its infancy. For pedagogical reasons, we have introduced from a semihistorical point of view, beginning with the first quanand later developing the second quantized theory and M-theory. that future accounts of string theory will reverse this sequence. level of arbitrariness in the first quantized theory as much as in this chapter we have lried to lay the groundwork for slring theory I integrals. This functional fonnalism has Ule great adlbat we can express the first and sccond quantized gauge theories with We fmd. in fuet. that large portions of the path integral formulation particles can be incorporated wholesale into string lheory. integral method postulates two fundamental principles that express c-'--- of quantum mechanics:

i
e p,rn"bllll"y P(a, b) of a parlide going from point a to point b is given absolute value squared ofa transition function K (a, b).

P(a.b) = IK(G.b)I', ~' l"n,lll' on function is g iven by the sum of a phase factor

a to h,

the action. taken over all possible paths from

K(a, b):::

tis. where S

.

L kels. ~

the limit of continuous paths, we have K(a , b) =

Dx:= Io'_oc lim

lb

Dxi s ,

, " TITIdXin. i_ I

~ ", I

'

action S of thc firSI quantized point particle: is given by the length path that the particle swccpS out in space-time. We can represent the ~gl,n for the point particle in three ways:

,urst-{),uer ~ (H a mil tOm3n . )"Jonn: sccond-order fonn: non linear form:

I~ := p",x." - 'if ( PI''+ m, ')

x! - em 2 ),

L = i(t' - r

L=

(LlO.I)

-mJ-x~.

because all three forms of the action are parnmetrizationthe path integral diverges. Thus, the quantization procedure must

~:~~~;~:~i':')'DllTI;' etry and yield the correct measures in the functionaJ. ~

be quantized in three ways, each with its own advantages

46

J. Path Integrals and Point Particles

(1) Coulomb Quantization By explicitly fixing the value of some of the fields, such as

xo=t=r we can eliminate the troublesome negative metric states and the becomes! The Coulomb quantization method is therefore man , Itestl~f.m ghost-free. However, the disadvantage of this method is that it is very ward because manifest Lorentz symmetry is broken and must be chl;:ckfrill at every level. (2) Gupta-Bleuler Quantization The advantage of the Gupta-Bleuler quantization method is that we ha'ie::;:; a manifestly covariant quantization program. Of course, negative mettilllt:;:: ghosts are now allowed to circulate in the theory, but they are eVl~ntual~Y:}':§) eliminated by imposing the gauge constraints directly onto the ,n~J'U!:ij~,!,: space:

mvi.

[p~

+ m2 ] 1<{1) =

O.

Thus, the S-matrix is ultimately ghost-free. The disadvantage of thiS:ru approach, however, is that the imposition of these gauge cOllStrmD:W$.~ especially at the interacting level, is frequently quite difficult (3) BRST Quantization This method of quantization keeps the good features of both apllro;acl:leil;~ The theory is manifestly covariant, but the S-matrix is still uru·ltaI"Yt,ecalli;~J.$ the addition of ghost fields in the theory cancels precisely against , negative metric states. The BRST method imposes the gauge e = I' the first-order form and then inserts the Faddeev-Popov term 6 FP into functional to get the correct measure. We can exponentiate this deternliruili.~1 into the action by using Grassmann variables: AFP

= det lOr I =

f

de dee; f dfoa,9.

The resulting gauge-fixed action has a residual symmetry, called the u~,.~' ;l;.,:,m . symmetry, which is generated by Q, the BRST charge. When we generalize these methods to the interacting case, the path mt~~~ formulation begins with the fundamental formula for the transition functi(ju for N -particle scattering: A(kt. k 2 ,

••• ,

kN)

=

L topologies

X

=

gn

!f

exp i

L topologies

f

Dx 6FP

dtL(t)

+ i tk/kXt }

gn (e l L~~I kpxt').

References

47

quantized descriptiOn for the N -particle scattering amplitude is

Y,~~':'~This ;~~':>:~,~;~I; '~·;It1~~~~:. overcenain topologies, which must be rnenns thlll tulilatily is not obvious in the first quantized

"1:

Later, we wiU sec that this problem in the first quantized point . theory carries over directly into the first quanlized string theory. In the

r~~:~~;:~:ri:l:' :~;hl:~::: all topologies can be derived explicitly action.

~,n

from a first [0 a second quantized description is straightin the path integral formulation. For example, the propagator can be

;,";lh"6"1. o,sec<)Ddqu.nll2.t1I~guag"

an" = =

" 1 ! ~

Dxelt:JII.(f)

Dl/t Dl/r"Vt{xa)ifr'(x,,)t! f

D~LC"'),

(1.10.3)

(xl"'l = ,,(.,). L(t) = ~mi:, L(",) = ",'oa,

(1.10.4)

-

H)",.

equation is the Lagrangian for the SchrOdingcr wave equation., which -derived with the postulates o(path integrals and L:::= ~mu? j also possible (0 extract the second quantized vertices first quantized theory in exactly the same fllshioD. We simply writ!: functional integra] over a world sheet where tbe point partiele splits hcr 00'.'" particles, and then write chis Green's function as a functional over second qualltized fields. . shortly see the advantage o f carefully working out the details of part,;', Ie palh integrals. We will find that almost all of this fonnalism over directly into the string formalism!

reviews of string theory, see J. H. Schwarz, cd., SuperslringlI, Vols. 1
47, :~:~~:~~;f:~z;':~::%~~i,~!~11"";"" New York, 1974. i Rep. ge, 199 ( I

~

Mod. PJrys. PJrys.

1213

Jacob, ed., Dual Theory. North-Holland, Amsterdam, 1974.

48

I. Path Integrals and Point Particles

[10] J. H. Schwarz, Phys. Rep. 8C, 269 (1973). [11] M. B. Green and D. Gross, eds., Unified String Theories, World tic](mt11ic: Singapore, 1986. [12] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Vols. I and .. Cambridge University Press, Cambridge, 1986. [13] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32,438 (1974). [14] D. Z. Freedman, P. Van Nieuwenhuizen, and S. Ferrara, Phys. Rev. D13, (1976). [IS] S. Deser and B. Zumino, Phys. Lett. 62B, 335 (1976). [16] T. Regge, Nuovo Cimento 19 558 (1961). [17] N. H. Christ, R. Friedberg, and T. D. Lee, Nucl. Phys. B202, 89 (1982). [18] G. C. Wick and T. D. Lee, Nue!. Phys. B9, 209 (1969). [19] M. Kaku, Nucl. Phys. B203, 285 (1982). [20] M. Kaku, Phys. Rev. Lett. 50, 1895 (1983); Phys. Rev. 27, 2809, 2819 (\ [21] G. Veneziano, Nuovo Cimento 57A, 190 (1968). [22] M. Suzuki, unpublished. [23] P. Ramond, Phys. Rev. 03,2415 (1971). [24] A. Neveu and J. H. Schwarz, Nucl. Phys. B31, 819 (\ 971). [25] A. Neveu and 1. H. Schwarz, Phys. Rev. D4, 1109 (l97\). [26] K. Kikkawa, B. Sakita, and M. A. Virasoro, Phys. Rev. 184, 1701 (1969). [27] M. Kaku and L. P. Yu, Phys. Lett. 33B, 166 (1970). '1.2.'01' M: ~crK.allM L? rY'! u'i'h'Ys?Iti::1'3...~2.9-1~;0R,7, ,;/}?IllWZ.1,\,. [29] M. Kaku and 1. Sehert, Phys. Rev. D3, 430 (1971). [30) M. Kaku and 1. Schert, Phys. Rev. D3, 2000 (1971). ( [31] V. Alessandrini, Nuovo Cimento 2A, 321 (1971). [32] C. Lovelace, Phys. Lett. 32B, 703 (1970). [33] C. Lovelace, Phys. Lett. 34B, 500 (1971). [34] Y. Nambu, Lectures at the Copenhagen Summer Symposium (1970) . . [35] T. Goto, Progr. Theoret. Phys. 46, 1560 (1971). [36] P. Goddard, J. Goldstone, C. Rebbi, and C. B. Thorn, Nuci. Phys. B56, 1 (1973). [37] S. Mandelstam, Nucl. Phys. 864,205 (1973); B69, 77 (1974). [38) M. Kaku and K. Kikkawa, Phys. Rev. DID, 1110, 1823 (1974). [39) 1. Scherk and J. H. Schwarz, Nucl. Phys. 881, 118 (1974). See also 1 . YOlle~i:::~ Progr. Theoret. Phys. 51, 1907 (1974). [40] M. B. Green and J. H. Schwarz, Phys. Rev. 149B, 117 (1984); 151B, 21 (l __ ,.., .:.~ [41] D. J. Gross, J. A. Harvey, E. Martinec, and R. Rohm, Nucl. Phys. B256, (1986); B267, 75 (1986). [42] C. S. Hsue, B. Sakita, and M. A. VirasOTo, Phys. Rev. D2, 2857 (1970). [43] 1. L. Gervais and B. Sakita, Nucl. Phys. B34, 632 (1971); Phys. Rev. D4, 229 • (1971); Phys Rev. Lett. 30, 716 (1973); see also D. B. Fairlie and H. B. . Nucl. Phys. 820,637 (1970). [44] R. P. Feynman apd A. R. Hibbs, Quantum Mechanics and Path /nt<'O'rll'/.t McGraw-Hili, New York, 1965. [45] C. Becchi, A. Rouet, and R. Stora, Ann. Phys. 98,287 (1976). [46] I. V. Tyupin, Lebedev preprint FIAN No. 39 (1975), unpublished. [47] L. D. Faddeev and V. N. Popov, Phys. Lett. 258,29 (1967).

2

Strings

Bosonic Strings theory. at first glance, seems divorced from the standard techniques 'J~:;! over the past 50 years for second quantized field theories. This !c:' . string theory was first historically d iscovered as a IlI'St qUQlllized This is the reason why string theory al limes ap~ars to be a random

Cri~~~::;;~!~:~,~,;o~nv,;;~en~":·~ons. Although a second quantized field theory

Ie

single action. 3 first quantized theory requires assumptions. In particular, the vertices, the cho ice of interactions, of these perturbation diagrams must be postulated by hand to be uni tary laler. integral formal ism for the first quantized point particle generalized for the string by J. L. Gervaisand R Sakita, which enables write dowo the dynamics of interacting strings with remarkable ease. previous chapter, we laid the crucial mathematical groundwork for a nssk i on of the first quantized point particle theory. Surprisingly, almost all main features of the Nambu-Goto string have some fonn of analogue " f;", quantized point particle theory. Of course, entirely new features are in Ole ~tritlg theory, such as the existence of powerful symmetries on the sheet, but the basic methods of quantization can be carried over directly . '. the point particle case studied in the previous chapter. saw that the usual formulation of second qua nrized field theory can t~,~,~( in first quantized form. Thus, the traditional covariant Feynman ?2 ( 1.6. 19) can be written vin (1 J.28). (1.3.30), ( 1.3.37) as

50

2. Nambu-Goto Strings

where we integrate over all possible trajectories of a particle located at x).t which start at Xl and end at point X2. The interactions, we saw, were introdlucle( by hand into the theory by postulating a particular set of topologies over this particle can roam. The scattering amplitude, for example, is A(k l , k2, ... , kn ) =

L

! n

topologies

;=1

L topologies

Dx 1l.Fpe- f

N

(

LdHi

L:~=! k,·x,

ik,.x' ) ,

where we integrate over topologies that form the familiar Feynman fu¢30r~~~

.• It is important to notice that the resulting Feynman diagram is a graph, not .... :"'n",' manifold. At the interaction point, the local topology is not R1J, so it cannot be manifold. There is no correlation between the intemallines and the I'rlteractlOIl; points. This means that we can introduce arbitrarily high spins atthe int1emcti()t(:W, point of the first quantized relativistic point particle. Thus, the first qU!1p.tizel~:}~ point particle theory has an infinite degree of arbitrariness, corresponding to lU<;!.:.:* different spins and masses we can place at the interaction point. t'1Uthl~mlore; the ultraviolet singularities of each Feynman diagram correspond to the nu:mb,eir::::ili of ways we can "pinch" the diagram by shrinking an intemalline to zero, LlLU",.,.:,:'-": defoIDling the local topology. This picture, however, totally changes with the string. Although the integral fonnalism looks almost identical, there are profoundly important uu-.·.·_-""'" ferences. In particular, the sum over histories becomes a sum over all pO:5silJI~::::::a~

FIGURE 2_1. Vertex functions for point particles and strings. A large number . point particle theories are possible, based on different spins and isospins, be(:ause::;:~ the Feynman diagrams are graphs. Only a few string theories are known, n"'>J",r"lV because the interactions are restricted to be manifolds, not graphs. Conformal syrn';:::::::::::l metry, modular invariance, and supersymmetry place enormous restrictions on th~r:::;:~ manifolds we may use to construct superstring theories that have no counterpart itt:::::IJ point particle theory.

2. 1 Bosonic Strings

51

sheets thai we can uraw bet'ween two different strings (see 'This world sheet, in tum, is a genuine manifold, a Riemann sursci ofintcmctionsconsistem with the propagator is severely limited.

Furthermore. the super~: ~~:~1?:~~~;;:a~3' ~ :~~:~:'~~~1:::::~::'~~~: contrast to the docs not suITer from ultraviolet divergences caused by :r;hrinking lheories we can write.

in

liDes 10 zero. You cannot "pinch" the string world sheet to Thus, string theory is free of ultraviole I diver;: . from strictly arguments. (We must be careful to paim Olll . diagram, can be reinterpreted as an infrared diverof massless. spin-O particles into the vacu um. eliminates these infrared divergences.)

path integral formalism can treat both the first the first quantized string theory with relative ease, profoundly important physical differences between the two !boories from strictly topological arguments. ,~" gi'n our discussion of strings by first introducing the coordinate of vibrating in physical spac(}-{im~. Let the poims along the string be etr~,d by !be variable (J. and then lei thc Siring propagate in lime. Let

(2.1.3)

the spacHUne coordinates Oflhis string (see Fig.2.2) P.'ll"8ffictrizcd variables. When the string moves, it sweeps out a two-dimensional which we call the ··world sheet " We will paramctrize this world sheet variables. u and r . The vC(:tors thai are tangent to the surface arc by the d~rivatives oflhe coordinate; tangent vectors

ax" aXJ< 8a

(2. 1.4)

::::~.

,r

~

~

t~~~T~h~,~t~W~O:~.dimen$iOnal sheet swept out by 8 striTlg. When a string, in space-time, it sweeps out a two-dimensional . by a moves world I

bya and T . The string variable X,,(O,l) is just a vcctortllat to a poin! on tllis two-dimcnsionJI\ manifolcL

52

2. Nambu-Goto Strings

The contraction of two of these tangent vectors yields a metric:

where we have now replaced the two variables (r, a) with the set (a, b), a, b can equal either 0 or 1. The infinitesimal area on this surface can be wriLtt~ij;::::~ simply as

d Area '" Jdet Igabl da dr. In analogy to the point particle case, where the action is the length swept

by the point, we now define our action to be the surface area of this sheet. Our Lagrangian is therefore [1--4]: L = -12rc ex'

wo.rr4:~~

J'X2 X'1L2 - (X. X'IL)2 IL

IL'

where the prime represents a differentiation and the dot represents r dittefl~nt(~::::~ ation. (When we discuss M-theory, we will introduce the theory oflmemt,rali¢~:~m in arbitrary dimensions in the same fashion. But instead of two cO()fdiliaWs::n! on a world sheet, we now have (p + 1)-coordinates on a world volume. Th~@~ action for the p-brane is still the same: the square root of the determinant .• . the metric tensor. The action is just the volume ofthe membrane.) The actIon: is just the Lagrangian integrated over the world sheet, which is the total of the two-dimensional surface:

s=

J

da dr L(a, r).

The Green's function for the propagation of a string from configuration Xa a·!~:*~ "time" ra to configuration Xb at "time" rh, as well as the path integral OV{·:r::;::;~ a surface that expresses the topology of several interacting strings, can b~jm represented as

Z=

L topologies

J

d/lDXe- area ,

where D X = rIll.a. r d X IL (a, r), d ILfepresents the measure of integration ovt~t::§~ the location of the extemallegs, and where we have made a Wick rotation the t variable (r ---* -it) so the integral converges. . The correspondence between the point particle path integral formalism thu~];§l we carefully developed in the previous chapter and the string formalism. quite remarkable. We find that almost the entire point particle formalism c@~m

2.1 Bosonic Strings

53

into the string fom13lism:

xli(r) length

XI'(a, r)

''''

n ,.'

IT dXI'(a, r)

dXI'(r)

1'-.0". t

the path integral for the point particle and the string theory have similarities. The N -point function for the N -string scattering amalso be written as R Fourier transform, similar to point particle path

" 1 .r;

X,

i,.

Dxe-f,;l.I.r"ldt

•

D X e- f.

0

d" f., L{X)d<

(,.,fr ;. ")

:;;:;: ::~~:~in;th~;'~I,~,"=gu:,,~:gc~:Of t

there are remarkable similarities between point particle and

~tring

path i TItegrals.over the cmdal difference the topologles which the objects For the the topologies are graphs. as in Feynman whereas the topologies for string theories arc manifolds: I

Graphs --+ Manifolds. crucial reasons why there are so many point partide actions (and so actions) is the differcncc bctweengraphs and manifolds. Thcnontrivstniotio", pl""d on manifolds severely restrict the number of consistent ... theories. the case oflhe point particle. the choice of parametrization was totally Thus. our actions must· be reparametrization invariant. To see this, ,mak,,,,,,,bitracy oh'"'I!' of variables: (; = (;(0 • • ).

r=

i(a, .).

(2.1.10)

r ti,i,rep"a""'tn,,,,';ion,, the string variable changes as

8X# = X'!'S(Y

+ XlLor.

(2.1.11)

the area of a surface is independent of the parametrization. the action :s'ly ,rep"",m",i,.tioD invariant, which is easily checked. us now write down the canonical conjugates of the theory:

p = OL = I'-

sku.

~ X XI' - (X"X"")X~ 211"'" Jdet la X"3b X.1 f2

Q

54

2. Nambu-Goto Strings

As in the point particle case, these momenta are not all independent. In fac1~/]~ we find two identities that are satisfied by the canonical momenta: . .

I

P~ +

Constraints:

1 (2Jt a /)2

Pp,X'p,

= O.

X~ = 0,

Thus, the canonical momenta are constrained by these two conditions. If ","'.••.,,-"calculate the Hamiltonian of the system, we find that it vanishes identically,'",.,.:-:« in (1.4.9): ..

The vanishing of the Hamiltonian and the presence of constraints among th!'l'" ::;m: various momenta are indications that the system is a gauge system with ar!::;:;m: infinite redundancy. The reparametrization invariance of the system is the on,..:·:,:,:,:, gin of this redundancy. We can therefore write down the close . between the constraints of the point particle theory and the string theory: {p2

+ m2 =

O}

I

1 Xa (2Jt~22 11 =

Pfl.2 +

--+

PI1 X

0,

I

(

= 0.

Before we begin a detailed discussion of the quantization ofthe string, it '.' instructive to investigate the purely classical motions of this string. Let us firs'it8~ classically set the parameter r equal to time, so that

X/1 =

(1, v;),

X~ = (0,

X;>.

Then let us factor out X a from the action (2.1.7): L = __ 1_IX~211/2(l_

2Jta'

where

v~)1/2,

r

r

vis the velocity component perpendicular to the string: _ Vi

=

VkX~

Vi -

I

-a-X;. Xj

The boundary conditions that we derive from this gauge-fixed action UJ\JJUULV.

v~ = 1. This means that the e.nds of the classical string travel at the speed of light. We can also calculate the energy of the classical string. Let us assume the string is in a configuration that maximizes its angular momentum, Le., it , a rigid rod that rotates with angular velocity w around an axis labeled by unit vector r. The string can be parametrized as

X = err,

2.1 Bosonic SlTi llg~

r :::: w x

55

r,

r · w :::: 0,

(2.1.19)

o~1.

>~~~.~;. ::.:the energy and the angular mornenrum of the system. we must ~

the Lorentz generators associated with the ~tring: M il":::: /

da(P~ X/I

- P"' X ") .

(2. 1.20)

that this generates the algcbrd. of the Lorentz group if we impose the brackets: [XII(q), P..(q')l "" 11... ~ 8«(J - a ' ).

(2.1.21)

now calculate the energy and the angulor momeorum from th(: )O,ots of the UlrenlZ generator [5 J:

J = -1- / ' d. ;.,----'...','~-;,c;),cor> [r x (w 2Ha '

_I

(I

w-.....

,•

X

T)]

... w(4a'w·1)-1

""' wE 2a'.

(2.1.22)

tbe angular momentum of me rolating string is proportionai lo tlte square of the system: (2. 1.23) p lot the energy squared on the x-axis and the angular momentum on we obtain a curve called the Regge trajectury. The slope of the is: given by a' and the curve is linear. Thus, we have obtained Regge trajectory for the classical motion of a rigid rotator. We this book. lake the oonnalization a' :::: This is an arbitrary However, we will see laler that the intercept Qo of the leading must be equal to one, which is fixed by confonnal invariaoce once the theory. Thus, we set

i.

,

,

a :::: 2' ao = 1.

(2. 1.24)

we quantize the system, we will find that there is an infin ite number cb IP",IlI,,I Regge trajectories, but with increasingly negative },-interceplS. we bave stressed, there is a crucial difference between the point particle and the string, which is tbat the string system has a larger set of conthat generate the gauge group of teparametrizations. For example, if

56

2. Nambu-Goto Strings

we introduce the canonical Poisson brackets, then we can explicitly show tll~~{~1 the constraints generate an algebra. To calculate this algebra, we decompose the X and P in terms of nOlrni!~~]1 modes for the open string:

+2 L 00

X"(o-) = xl"

pl"(o-) =

n=1

!I

pI"

1 -X~ cos no-,

..jii

+ [; In p~L cos no-

I·

Notice that the X decomposition is given strictly in terms of cosines. This because, when we calculate the equations of motion of the string, we mti$~:j:m integrate by parts and hence obtain unwanted surface terms at 0- = n: aIi~~:~~ 0- = O. In order to eliminate these surface terms, we must impose

X'I" = 0 at the boundary. This boundary condition eliminates all sine modes of tll~::@ string. (These are called Neumann boundary conditions. It was once th~mllnt::W that Dirichlet boundary conditions, where the derivative of the string V
= XI"(-o-),

X~(o-)

=

-X~( -0-).

The same applies for the modes of the canonical conjugate Pw This, in allows us to combine both constraints into one, using the properties of string under reflection from 0- into -0-. If we let the string pruranletJriz:~ti()*:::m length be n: we can define

1 L f = 4n:

l

Jr

-Jr

do-f( 0-)

(

X' .j2a' n: PI" + .j{a l

)2

'

where f(o-) is an arbitrary function defined from -n: to n:. Notice that botl~:@ilil constraints are now combined into one equation because of this ."." .......... vJ....: symmetry. Using (2.1.21), we can show that these generators form a "a.. ,~ . . ... algebra:

2.1 lJosonic Strings

/X8=/8 ' -8/' ·

57

(2.1.30)

to show that this algebra satisfies the Jacobi identities:

[LU' [L &, LhJ l = 0,

(2.1.3 1)

,ot"••:ke,,,ep,·<s,,~t.111 po,;ible cyclic symmetrizations. This algebra Jlirasoro algebra which will turn out to be one of the most tools we have in constructing the string theory. . . I l}. we can elevatc the constraints into the action with Lagrange ""'ta, r) and pCa, r):

X~

L=P j(/'-+JrU'A[p2+ I'

(2lTa')2

/J.

J+PP XiI'. I'

(2.1.32)

integrating out over these Lagrange multipliers, we arrive at set {Jf constraints. Not surprisingly, this new action has its own group parametrized by 'I and €: SXI' = 2Jra'eP,<

,X'

oP,.,. = [ 21f;' (lA

=

lip =

-i;

+ '1X~,

+ TJP/'-

1

+ ).'1/ -1/';" + pie -

-i7 +>..'£ -

(2.1.33)

"

pe',

AS' + P"I- 'l'p.

~

~~7n:~~~O~f~this~'!~fo~nn~,~fO~r'i~theofthe action is thataction. it is firsl-{)rder and docs nol original As in the point particle

there exists yet one more form for the action, expressed of an auxiliary field. To find this third formulation of the action, lei ;od"", a new independent field

(2.1.34)

""P"'.""'"

metric on a two~dimensional surface. UnJike our previous this metric is now totally independent of the string 'lIIinble. Let us i' d,,~ the Polyakov fonn of the action [7] (g = I detgQ"I):

(2.1.35) generalization of the second-order point particle action (1.4.14). No~ Ihe Polyakov aetion resembles an action with scalar fields inleracting external tvJo~dimensionalgravitational field. This action, too, possesses reparametrization invariance:

8XI' = e"8 XP Q

&gQb = eCac8Qb

~

,.;g ~ .,(e',/8).

gQc11"eb _ g',e8"Eu,

(2.1.36)

58

2. Nambu-Goto Strings

The action is also trivially invariant under Weyl rescaling: 8g ub = Agub. The Polyakov action is entirely equivalent at the classical level to the p
Tab

= -4na,1 ..;g 8g8Lab '

~3.'~w.

(2.1..)

Working this out explicitly, we find (2.1.3·~·m~

The moments ofthe energy--momentum tensor will correspond to the generators. Thus, we have another way ofderiving the Virasoro generators tr",,;,;,:.;, this new but equivalent formalism. Notice that the metric field gab is not a propagating field. The metric tenS!..~~; does not have any derivatives acting on it. Thus, we can eliminate it via its owij:~~ equations of motion. This leads us to (

8L = 0 8gab

--+

2BaX/18bXIL gab = gCdBc X v 8d Xv'

Substituting this value of the metric tensor back into the action, we re(lefJlv.~@ the original Nambu-Goto action. Thus, at the classical level, the two actioi1l~~~ are identical. In summary, as in the point particle case, we now have three different ""'''''·irw.:l~ which to write down the action, all of which are equivalent classically. Each hi·:'I$.:a~ its own particular advantages and disadvantages when we make the transiti(jii~ to the quantum system. These string equations are direct generalizations .: .....U h the three point particle Lagrangians found in (1.4.16). As before, we have thii:l:::::! second-order formalism, which is expressed in terms of the string variable .. as well as the metric tensor gab; the nonlinear formalism, which is eXIJressll@~ entirely in terms of X/1; and the Hamiltonian formalism where we have and its canonical conjugate P/1 (or the pair BuXIL and PUlL): first-order (Hamiltonian) form:

second-order form: nonlinear form:

L = P.. X/1 + na' )., [P/12 ~

L

+ (2na X~ )2]· .:.• • '

+ PP/1X'/1 "-' ..;g P: gabPblL + PU/18aX IL ..;g/na,' = ~..;ggUbBuXIL8bX/1, (2.1.
L = _1_(X2 XI2 - (X X'/1i)I/2. 2na' IL v /1

2.\ Bosonic Strings

59

we suspect that thcse actions are totally equivalent, so that we can drop the others. This is apparently not so. fortwo subtle reasons:

>n""'.

dealing with a firSI quamized theory, we have to take the interacting topologies that are swept out by the string. For the string, the precise mlhlJ"e of these topologies is ambiguous be specified hy hand. However. for the Polyakov form of the which contains an independent metric tensor, we can eliminate of this ambiguity by specifying that we sum over all conJonnally .",dula, inequivolent configurallons. (Thesc tcrms wi ll be defined This wiIJ become a powerful constraint once we start to derive nod will determine thc function measure uniquely. The measure and

~

::rl~.~in the Nambu-Goto action, however, are nol well defined.

out, hQwever, that this rule of integrating over inequivalent not automatically satisfY unitarity. This sti ll must be checked

fix ing ofWeyl invariance fo r the Polynkov action, although triva~':;:'1; poses when mak.e the transition to quan twn 18 An anomaly appears when we carefully begin the quaotizaprocess. In fact, this conformal anomaly will disappear only in 26

probk-ms

we

now discuss the quanti2ation of the string aclion. The strategy we will

~~:~i;;j~~'~ h e~fr~~ee~~ili~eo~ry~'~O~obf~ " ~m~l'b~ e ~ph~Y~S~ 1 C ~ '~ I ~H;n~b~ert~~~CeWi l Jbe [

thealgcbra symmetry. (For the string, the symmetry will invariance and the algebra will be the Virasoro algebra..) must apply the constraints onto the Hilbert s pace, which eliminates and creates a unitary theory. It is important to keep thi ~ strategy in begin the quantization of the string.

-. Symmetry -+ CurrCllt -+ Algebra -+ Constraints -+ Unitarity. ~" I)oln< 1",,'lcllcc••:.,~ " ~ begin the quantization program in several arc three fonnalisms in which 10 fix the gauge o f the theory: ~" -B ll eul,"

(eonfonnal gauge). (2) light cone gauge, and (3) BRST The advantages and disadvantages of eacb area as follow~ :

Gupta-Bleuler is perhaps the simplest of the three fonnalisms. We ghosts to appea r in the action, which permits us 10 maintain manifest The price we must pay, however, is that we m ust impose constrai nts on the Hilbert space. Projection operators must in al l propagators. For trees. this is trivial. For higher loops, , this is exceedingly difficult. lIdvulitage of the light cone gauge formalism is that it is expl icitly 'o"-r.",, in the action as well a... the Hilbert space. There are no eo01-

60

2. Nambu-Goto Strings

plications when going to loops. However, the formalism is very awkw~~{l:§I and Lorentz invariance must be checked at each step of the way. (3) The BRST fonnalism combines the best features of the previous two malisms. It is manifestly covariant, like the Gupta-Bleuler formalism, . it is unitary, like the light cone formalism, because the negative mel[nt;= ghosts cancel against the Faddeev-Popov ghosts. Let us now discuss each quantization scheme separately.

2.2

Gupta-Bleuler Quantization

The Gupta-Bleuler formalism will maintain Lorentz invariance by lilljpmim.~r:ill~: the Virasoro constraints on the state vectors of the theory:

{¢I Lf 11fr} = 0, where (¢ 1and 11fr) represent states of the theory. This constraint will ell[l1ln:a~¢} ghosts in the state vectors, allowing us to keep nonphysical negative m~:tiii~~ ghosts intact in the action. ( Classically, the metric tensor has three degrees of freedom that we c~~m~: gauge away, two arising from reparametrization invariance and one from.-v.1Z~lJl:ii invariance. Since the metric tensor has three degrees of freedom, we cang. all of its components away: 8ab

=

Dab

= ( -~

~)

which we call the conformal gauge. (There are complications, as we have in taking the confonnal gauge for the quantum theory and for higher lOOps~rn~ Our action reduces to

r

S = _1_ da !dT(X 2 4Jtct' 10 fl.

_

X,2). fl.

This is exceptionally simple because the action now corresponds to uncoupled free string. This action yields the free equations of motion:

@;~:$

(a~2 - aa;2) Xp.(a, T) = 0 with the boundary condition: X~(O, r) = X~(Jt, T) = 0

which we need to enforce when we integrate by parts and eliminate the surta¢.~::::~ term. The solutions of the equations of motion are arbitrary functions of a and a - r:

XIJ.(a, T)

= Xj(a + T) + Xj(a -

T).

2.2 Gupta-BleulerQuantization

~,,;"'I

61

coo,",u"";O,O relations are now (2.2.7)

, I( + ..f-, 1

6(0 - (f) =:rr

2 ~cosnucosna

') .

(2.2.8)

' However, as

in the case, we can always choose the ;~Of~C~OU:'~S~c"i'~O~~;O~:fi~~ r~u't::,~O;:umb:~":::~;::,:~~~;~~~~, of the path

:~~ Unlike the point particle case, 18~:;~~'~: b we ~;: now ~::'~ have anbecomes infinite osc;II",o<',. one set for each normal mode:

of'

x:: = ii ../2a'(a~ -

a~It),

1 (an, +a,_It, ) PIt" = ..j2a'

(2.2.9)

can satisfy thc canonical commutation relations if we set

(2.2.10) conventional to introduce an equivalent sct of oscillators: m > m

XfJ.(a, t) = x"

0,

> 0,

+ 2a' p"r + j ../20"

."

L ~e-I'"

cos nO" .

(2.2. 11 )

",.0 II

in this basis, tbe Hamiltonian takes on an especially simple f('lnn (see

~

..,

= '"' + O"p'/'-' L natIt/'- aJl. n

(2.2.12)

have made an infinite shift in "the zcro point energy. At this point. of the lowcst-ordcr particle is not well defined because we made ;0;1. 'mill, but we will later show thai this lowest particle is actually n We will show that the intercept of tbe model is fixed at L

'~:~:~:,:~:~,:~~I~;::;~:~l~ is basically uncoupled from the other oscil-

n'

. In fact, the Hamiltonian is diagonal in the Fock space ofbatmonic

excitations. Taking this specific representation oflhe string function

62

2. Nambu-Goto Strings

s_

10> kl1 a?10>lkl1 a;1110>

FIGURE 2.3. Regge trajectories for the open string. The x-axis corresponds to t~~~~11 energy squared and the y-axis to the spin. The particle farthest to the left is t'~~:~~ tachyon, which corresponds to the vacuum of the Fock space. The massless ~ni .. ~,,·~·»!w. particle is the Maxwell or Yang-Mills field, which corresponds to a single creaQ(~~~m operator acting on the vacuum. There is an infinite number of Regge trajectlori(~~J~m.: corresponding to the infinite excitations of a relativistic string or the infinite nUlnbl~~W. of states in the Fock space. from the infinite number of possibilities is a great advantage because lowed eigenstates of our Hamiltonian are now simply the products of the Fo(~~::::=i spaces of all possible harmonic oscillators: ( eigenstates:

rr

[a!.IL} 10} ,

n.1L

where the vacuum is defined as

n > O. The spectrum of the lower lying states can be categorized as (see Fig. tachyon -+ 10) , massless vector -+ a[1L 10) , massless scalar -+ klLai lL 10} ,

(2.2.15:)iW

massive spin-2 -+ ai lL ai 10) , massive vector -+ lL 10} . v

ai

(The fact that the string theory is so simple to quantize can be traced t~~m the fundamental fact that the Hamiltonian is quadratic in the string this means that it decomposes into an infinite series of free particles. A va:i~~:~m collection of vacuum solutions to string theory can be constructed because, i:t.@;§!§ essence, it is a free theory. However, this simplicity breaks down c~:~;~~7ij1~~ when we analyze membranes in M-theory. We will find that the I~ is now quartic, making the quantization intractable. In contrast to the sinllp].ic~:m~ ity of string theory, there is still no satisfactory method for quantizing fte,~@m membranes.) As expected, we recover the leading Regge trajectory that we obtained earli~~:m*l from classical arguments, and also an infinite number of daughter tra,jectori~~:mi

2.2 Gupta-meuler Quantization

63

increasingly negative y-intcrccpl. In this gauge, the propagator or function takes on a simple form: K(a, b):;:: (Xul e- m • IX b )

=

r" Dxex p -jdad,!-I-(X! - X )1 , (2.2.16) lx. 41!'a' '2 /I

DX=

TITI dX'(u) = TI dX:. ,

0

(2.2. 17)

,."

careful to note that the functional integral over DX is an infinite over each point along the string,. or each Fourier mode of ;;'",I.y" this integration can be carried out cxplicitly. Let the particle , :;:: 0 to r = 00. Because the Hamiltonian is diagonal on the harmonic oscillators, we can perfonn the integration over 't: exactly.

(2.2.18) is the zeroth Fourier component of the Virasoro generators (2.1.28). ,"'or,ds, I,ro.pal,ator for the fn."C theory is

0'"

(2.2.1 9)

id"d brlwoo. stales of the string. However, becausc wc have the identity IX)

j

DX (XI = I

(2.2.20)

~:"'m.v<'"" l"lh integrals explicitly at every intermediate point between state. In fact. because of the simplicity of the N -point be able to remove all functional integrations over string states the hamlOnic oscillators. It is important to stress again oscillator fonnalism is nothing but a specific representation int(':l7"L liS simplicity is a consequence of the Hamiltonian being in the Fock space ofharmoni<': oscillator states. the same way, the closed SlI'ing can also be .....'fitten in terms of . oscillators. For the closed string, there is no X' = 0 boundary conwe can use both sine and cosine functions to decompose its normal Thu.<;, we expect twice the nllmb{~r (If (I!':cill~lOr!': for the closed string.

64

2. Nambu-Goto Strings

The oscillator decomposition is given by X/i(a)

I ( = xI' +"2et' )

1/2

~ 1 . rnrr ~ r,z(ane-

.

t .

t

.::

+ ane wa + ana rnrr + an e- l1Ia )/i

n=] V"

(2.2·4.~ J::W

P (a) = PI' I' 2IT

+

1

fJn(-iane-ina-ianeinrf+iateina+iiite'

2IT.J2(;i n= I n n n

The Hamiltonian for the closed string is

H= IT 12rr da (ex P; + 4:fex I

00

l )

-t- ) 2 + nana n + ex P w

( t = " ~ nanan

I

n=1

Again, the Fock space consists of all elements created out of harmonic oscii@ lators, but this "time there is an extra constraint that is not found for the nrii>m string:

(Lo - Lo) I¢} = O. (The interpretation of this constraint is that the closed string must be mdlep1ew. dent of the origin of the 0' -coordinate. For example, the operator f dO't>ia:(:Lo...;;tiiXED can be interpreted in two ways. First, it generates rotations in O'-space, so w.~ED average over a rotation of 2IT in a -space. Second, if we perform the i'Iltel¥~!~WI we have B(Lo - Lo), which is constraint (2.2.23) when applied to the H'.1lbj~m:~ space. We will return to this constraint later.) The Fock space consists of (see Fig. 2.4): tachyon

~

massless spin-2 ~

10>

IO} ,

aitlai" IO} ,

v, ,

k /I k a+/I i+PIO>

FIGURE 2.4. Regge trajectories for the closed string. The Fork space is of two commuting sets of hannonic oscillators. The massless spin-2 particle graviton, which corresponds to the product of both types of operators acting ontMM vacuum.

2.2 Gupta--BJeuler Quantization

6S

massless scalar ....... k"kllar"ar~ 10).

a massless spin-2 particle occurs in the speclruOl of che closod "n th:' "'fig model was first being interpreted as a model ofhadroos. of this gravilonlikc spin·2 particle was

II.

great embarr.lssment.

were made to associate this particle with the Pomeron trajectory . S-matrix theory. It can be shov.'t1 that, when we generalize to trees this OllIssless spin-2 particle has n gauge invariance equivalent to of Einstein's theory. By dropping the earlier interpretation of the th."y as a model ofhadrons, we find a natural place for £his gravilonlikc

graviton itself. ~:~~~~,t~h~:e, Gupla--Blculer formalism in the confonnal gauge appears ir mainJy because we arc allowing ghosts to appear throughtheory reduces to the simplest possible string theory, a free string. we must pay for this simplicity, however, is the imposition of the the Fock space. We ean writo the Virasoro generators as [til:

(2.2.25) 00

1.0 = .

..L,a_~/J.a~ + !a~

.

states of the theory must therefore satisfy Ln 14I} = 0. (L. - I) I ~ ) = O.

II >

0, (2.2.26)

generated by these operators is

[Ln. L .. J = (11

~

D , m)L,,-t .. + 12(n - n)1in._m ,

(2.2.27)

is the dimension of space-time. The fact that there is a c-number appearing in this equation at first sounds surprising, but this can ,1"IOd explicitly by taking the vacuwn expectation product of a simple

(OI(L,.

D

Ld 10) = '2'

(2.2.28)

of this e<:ntral tenn is that we normal ordered the generator 1.0 to matrix elements. This normal ordering was even found in the point in (1.8.7), exeept now the energy shift is infinite. The price we pay . elemenl.t i~· that locality in u is /0.1"/. The normal ordering spoils the generators I. f were originally local functions in 0'. The process

66

2. Nambu-Goto Strings

of quantization necessarily destroys the locality of Viras oro generators variable a, and hence the c-number central term occurs. Thus, the qwmt:Lzal\~QjjI$ scheme and regularization scheme used to extract finite information .. v.LU>'<1IU~ model are actually inconsistent with conformal symmetry. Fo:rtuJnat.el)';t1f1@i inconsistency can be eliminated if we fix the dimension of space-time t(j:ttiim 26. That there are ghosts in the theory is due to the fact that the zeroth cOlnp0Itl~U~1 ofthe harmonic oscillator has negative metric: ghost = {a,;,o} 10) . Thus, the coefficient of the Green's function occurs with a negative V~""~.'c.H,* addition, there are zero norm states and negative nonn states that must be taJ<~ into account. To analyze the spectrum, let us define a spurious state IS) to be oile th~rt"R orthogonal to all physical real states IR). Spurious states can be written

IS) = L_n Ix) ,

n > 0,

{RIS} = 0, for some integer n and some state Ix}. (If we take the matrix elemeqt onfifi~ state with a physical state, the scalar product always vanishes bel::aULse< destroys a physical state.) Now let us construct the spurious state:

It} = [L-2 + aL~tll
SPf:ctrUJjjI~

This fixes the following:

3 - 2a = 0,

!D - 4 - 6a = 0,

which, in tum, fixes D = 26,

3

a= 2'

Thus, this spurious state satisfies (2.2.26) and hence is part of the nh'vsilm: Fock space. At first, this seems disastrous. We want our physical Hilbert to be ghost-free. But notice that in 26 dimensions this state has zero norm: negative norm). Since 1
2.3 Light Cone Quantization

18) = (aO"~J

67

+ bk . 0" _1 + c{k . Il'_J /) 10) .

L,18) = L 2 10) = 0, we find that b = a(D - 1)/5 and c 10. We find that the nOnn of the state is (Big) = i~a2(D - 1)(26 - D) . . dimension of space-time cannot exceed 26 or else negative norm as part ofthc physical states. In general, we find thallhe spectrum if the dimension ofspacc-{ime is Jess than or equal to 26: ghost-free:

I

a ~ I, a:::; 1.

D=26' j

D

~

25.

(2.2.34)

!",.,,;,;o was donc only for a picce of the Fock space.

But call ghosts 10 all orders in Lhc string model? We will return to the difficult ;~::~:o!~e, in Gupr.a-Bleliler formalism at the end oflhis ::, we actually construct the physical Hilbert space and show that negative norm states in 26 dimensions.

m;m,,;ou the

Cone Quantization the light cone gauge, where all unphysical degrees of fn.-cdom are removed from the very beginning, is possible because we have twO and hence two gauge-fixing conditions can be in. One of these gauge-fixing constraints can be the modes from the Hilbert space, as in the Coulom b of ghost states, which is quite involved for the see at the end of this chapter), becomes

light cone gauge. choose the notation

x+=

x-

~[XO+XD- ' J

= _1_ [Xo _ XLH }

-Ii

.

(2.3. 1)

(2.32) 0 11

which version of the action in (2.1.4I) we use, we will have constraints. lfwe Slart with the original Nambu-Got.o action. gauge conditions in tile path integra l arc

DX M t:.FP

n •

S(X+(u) - p+r)S(X!

+ x~ -

2XuX'I')e-s, (2.3.3)

68

2. Nambu-Goto Strings

where M is a measure term that must be added to have a unitary th~~orY";;aff&~ the two delta-functions represent the gauge-fixing constraints. The relltWl~ able feature of the second constraint is that the Nambu-Goto action, WhlCh::~ expressed as a highly nonlinear square root, completely linearizes [12)

/ X! X~ -

(XItX'It)2 "-'

~(X~

X~).

-

(Because the light cone action is no longer a square root, we have a . behaved action that can be canonically quantized.) .. The constraint X+ = p+. means that the a dependence within .. completely disappeared and that the "time" • now beats in synchronism w •. itiW, X+. We can use the second constraint, in turn, to eliminate the X.:. hence all longitudinal modes have completely disappeared. The action can. be expressed totally in terms of transverse ones. :: Next, we will solve the constraints in the first action in (2.1.41). "':"!'~,:_!,,)1 integrate over the Lagrangian multipliers p and ).. in the Hamiltonian tor~~ml the action, and then impose the gauge-fixing constraints:

Z=

f

DXDP

x8

IJ

o(X+(o') - p+.)8 (p+(o') -

(p; + ;;)

:+) (

8(Pp.x'JJ-)e- s.

Because the covariant Hamiltonian (2.1.13) is equal to zero, the only the Lagrangian is PItXp. (the X- term drops out): L =

1](

da PJJ-Xp. =

1](

da(P;Xi - p+ P-(a)).

There are several remarkable features to this formalism First, we can four, not two, constraints onto our Hilbert space, two from gauge fixin~~.~mrn two by integrating over).. and p. Second, because the covariant ~Utll11UUJ.lJ
H = p+ 111" da P-(o'). On the other hand, we can solve the constraint for P-: _

P (a)

=

11:

2p+

(2Pi + Xl) 1<2

•

Plugging the value for P- into the definition of the light cone (2.3.7), we now have

H =

r (p/ + 1<~X2)

"21< Jo

dO'

HamiLt6i~m

2.3 Light COOO Quantization

69

just the Hamiltonian (2.2. 12) defi ned over physical cransvcrse

we can also eliminate the X- modes by solving another constraint: PI'X'1l = 0

(23.1 0)

be solved for X-. yielding X-(u)

~

1" •

du'-"-l P,X;J. p'

(2.3.11)

"" rytJhing together, our functional now becomes Z =

f

D X j DPj ;/dr!d<1CI',X,-H).

(2.).12)

the light cone Hamiltonian density. The great advantage of the light

is that the Vinssoroconsrraints have beeD explicitly solved. so there to impose them on Slates. All + modes have been gauged away from the - modes have been el iminated in Icnns of transverse stales Virnsoro conditions have been solved exactly through (2.3.8) and "",,,d of imposing the Virasoro constraints on the Hilbert space, we

them exactly and eliminate the - modc1:i. the great disadvantage of the fonnaJism is that we must tediously Lo" nl2 inY'Miance at each step of the calculation. Normally, the of the Lorentz group arc given by

MPv =

1"

dO'[X" P'" -

xv P" ) (2.3.13)

[0 eheck from (2.2.7) Ihat this satisfies tbe correct commutation for the Lorentz group:

[M"", M"'''J = ;11"" M'"

+ ....

(2.3.14)

Lorent2. invariancc has to he checked once again in light of the fnc t "~::~:~;,tl; all ghOSt modes. Most ofthe commutators are trivial to el

Ihcyare linear. The troublesome term comes from X -. which and is written as

x- =.c '

~

+ p -. + i L 11

,,0

I

_tr;e- JIIr cos na,

(2.3.15)

/I

(2.3.16)

70

2. Nambu-Goto Strings

and a is the intercept. All commutators are easily calculated, except one involving X-. The calculation, which is lengthy and must be pelrfOl~iU carefully, yields the final answer: [M- i , M-j] = p

}2 f)Q'~nQ'~ - Q'~nQ'~]lln, n=l

where

II = n

~(26 12

D) + ~n [D -1226 + 2 - 2a] .

For this commutator to vanish, we must have D =26,

a=1.

This fixes both the dimension of space-time D and the intercept a model. .

2.4

BRST Quantization

(

As in the point particle case, the BRST quantization method starts Faddeev-Popov quantization prescription and then extracts out a new IJ.Jll.Jl,I~ symmetry operator. The action is invariant Wlder a reparametrization (2.1;$!~1 ogah = gacobovc + o"OV c gel> = 'ilaOVb + 'ilbOva

-

gabOc OVc

(the second line has been written covariantly and contains twl)-djmem>IQi~ Christoffel symbols). This allows us to impose the gauge constraint: gab = Oab. This, in turn, generates a Faddeev-Popov determinant: ..0..FP

= det(Va) =

det('il;J det(Vz}.

This determinant can be calculated by exponentiating it into the ""T,nn do this, we must introduce anticommuting ghost fields b and c (see (l As before, these ghost fields make it possible to place the det.ernninanl:9~iji derivative in the exponential. Let us define z = -r + iu; then:

L =:rr1 (IIazx/J.azx/J.

- ). + baze + bazc

Notice that this action is invariant Wlder

oX/J. = s[cazx/J. DC = s[eazc], DC = S[COiC],

+ COiX/l]'

2.4 BRST Quantization

71

!Jb = t:[ca,b + 2fJocb - -l8, X p oz X lL j, -

-

-

I

ob = elcui: b + 2a:cb o

:'

•

•

2ai X",a~ X"1.

variation end ( 1.9.8), we can extract out the niJpotcnI BRST opHowever, it is also important to nOle that in gcnernl, given any with commutation relations p."" ). . . ! = f! . . )...p. it is possible 10 a nilp'''"' operator Q [10] oul of anti commuting operators e" and 00

Q~

L ,-. lA. -l/!.,-.b,j.

(2.4.6)

ft"'~OO

Ie" , b",l = 0. . .-""

(2.4.7)

.ST· O~"""o,can be wrineo in this form:

I

00

- a)

+ L[c_~L . . + L_"c.. J - '2 ",.1

L ....... 00

:c_ .. C_ ... b"T",:(m - n).

=-«1

(2.4.8)

equals the X -dependent VlI1lsoro genemlOr, and L~' is the ghost

~::;~~i~~::f:t:;~~~~1~:;,~:!~th~c;re~~are two unspecified parameters

and the dimension of spac~ should be zero:

Vll)ue the square o f Q,

(2.4.9) vanish, we [Dust fix the dimension of spaco-timc to be 26 and me to be equal to I. point particle case, we find thai the physical states of the theory

by

Q Iphy) = O.

(2.4. 10)

'CO"P'''''tc ()"' the modes, we find that the lowest !.1 ate satisfies

o. L. I.) ~ o.

( L , - 1)10)

~

(2.4. " )

72

2. Nambu-Goto Strings

2.5

Trees

So far we have quantized only free strings. The interaction of strings can;:~ determined either through functional methods or through harmonic V"'''H''~.'PJ:~ methods. Of the two, functional methods are by far the more powerfuL we can view the hannonic oscillator method as just one specific ref're!;entatJi~ of the functional method. For trees and even the first loop, oscillator me:tnmi: provide a fast and convenient method of calculating the amplitudes, higher loops the oscillator method rapidly becomes impractical. The tuncPO!Ul method, because of its vast versatility, can always be used either to deiive:1 oscillator method for trees and the first loop, or to derive all loops. Historically, the dual model was first discovered accidentally as a "l"""'+~. amplitude for the scattering oftachyons. These amplitudes possessed a rm;n!iii able property called duality, meaning that they could be factorized in tep'!:® s-channel poles or t-channel poles. (This duality between sand t Chfll1Il;¢l$;=Jfi entirely different from the duality that we will find in M-theory.) V.-,'HlH the theory of point particles, the Feynman diagrams sum the s- and t-c.ti@~ poles separately (see Fig. 2.5), but the dual amplitudes already posisessedp'4~ in both channels for the s- and t -channel poles separately. For more QOlnpll~~tij diagrams, we see that the same N -point amplitude could be factorized number of ways. Because the counting of these diagrams was not the . counting of Feynman diagrams, for years it was erroneously believed . true field theory interpretation of the dual model was not possible. Following the analogy with the point particle case, let us begin by detmu the N -point scattering amplitude for tachyon scattering [11-13]:

X 6.FP

exp

[i f L drY dr (D .;g exp(ik'.1l X

- IOP~e.! f dfL(D.;gexP{iki.IlXlli)). Dgab

This expression will be the fundamental path integral from which derive the theory of interacting strings. It will be the single most imlJott:a formula in the firSt quantized formulation. In the expression, momentum k i from the i th tachyon flows into the b01ltiq~ of the surface at a point Zi. The string variable X III is defined at that Zi on the boundary where momentum k i flows into the diagram. The mea&'Pl of integration d fL is an integral over the various Zi, which we will det,e~~ shortly.

,

,

3

H ,

z~

4

=1

.

I

2.5 Tree!;

73

3

4

-, '. '

•

Z1 •

00

- ' - -a Duality of the Veneziano model. The four-point amplitude carl be in tcmlS of either s-chnnncl or t-channel poles. This is in contrast 10 particle field theory, which sums over both .f- and t-channcl poles, i the N-point function. I'or this reason, il was once a field theory of strings was impossible. A field theory of strings would by overcounting, especially at higher orders. ;Xp",,,;ion:silnplifi1;:.'j wl1:siderably

i fw~

lakt: Iht: I.:uuformal gaug::. In

f f 4:':' !(a,xlI.a~X!l)d21. + t X" I L J (fI ,,,,.x) , dp.

DX exp (

dlJ.

1(I!)010fI1..

i

k,I'

(2.5.2)

I~ 1

we show how to simplify the string intemction diagram. By letting interaction length to go to zero for the external tacbyons, wc see that interaction surface can be reduced to an infinite horizontal strip in plane, extending from a _ 0 to a = :rT and 't _ ..too. that the action, althougb it is no longer reparametrization invariwe have fixed gab = 8"b, is still conformal ly invariant. Thus, to of conformRlly equivalent surfaces, we will take t.he set of we rnustsum to be lhe SCI ofall cOlljomwlly inequivalent I complex surfaces.

74

2. Nambu--Goto Strings

tu F

G

I

H

J

u=lI'

1'_

c

D

B

z = e T + io

l

FIGURE 2.6. Conformal surfaces for open string propagation. In the P-~:~:'~;l~1 surface over which a string propagates is a horizontal strip of width rr. Tl)e lines at the bottom corresponds to "zero width" strings or external tachyons. z-plane, the surface becomes the upper half complex plane. The mapping from;l~~m surface to the other is given by the exponential. Let the world sheet of the N -point tree be a horizontal strip of width extends horizontally in the complex plane. The x-axis corresponds to r an4H~[OOr, y-axis corresponds to a. Let us now change coordinates to complex

z=

eT+;cr.

This mapping takes this infinite strip, which describes the world sheet interacting string (with zero width tachyons), into the upper half of the cOlnp!t~ plane. Fortunately, the functional integral is a Gaussian that can be evallua.ted;~i.~m the identities presented in the previous chapter. Let us define Xc\.ssical solution to the classical equations of motion: 2

V X!"cl

= -2i:rra'JI1 •

where N

JI1 (z) =

L k;JLll(z -

z;).

;=1

where Z; are points on the real axis of the complex plane that COlTe!;polnd external zero width tachyons interacting with the string. After a Wick ro~~t@m

2.5 Trees

7S

' variable, this is just Poisson's equ8lion for electrostatics. To solve this, the Green'5 function:

(2.5.6) calculate the Green's func tion, with NeumarUl boundary conditions, ! f!::;~C~,Th~':le~;a~S;iest way to calcuillte this is to bOiTOW n trick '~:~ 0 namely the method of images. Let us place ~t at the point z' in the upper half-plane. CotlSidcr another point the point that is symmetricaUy reflected through thc x-axis; half-plane. If we are silting on a point charge l ill lhe upper then the pol(.:n tial at thnt point is proportional to

z'

zt

0(%. z,) = In II'.: -ll ce~~:;:i !fwe (5

+ In It - tl.

(2.5.7)

arc sitting on the x-axis so that z is real, then the derivative

function nonnal to the x-axis is zero. Thus, these boundary 8re precisely what we want, so (by the uniqueness theorem) this is function for the upper half-plane. now insen this Green 's function back inlo the integral. The classical that solves (2.5.4) is

Xd = -hi ! G(r, z') J(z' )dz:

(2.5.8)

make a shift in the integration variable:

X,. -+ X,. .d + X,..

(2.5.9)

,",.,,,,fo,',, that the functional imegra[s can be performed using (1 .7. 10):

/ DXCXP { - 4;a, j iI:X,.iltX"d2 Z+i ! J"X 1' d1Z }

{~

= cxp

=

IT

!

J,.(z )O(z. l )JI'(r'} dz dl' }

exp (a'k! . k j in IZi

- i:jl j

i'l-J

=

n

Iq - lj]2«'!, ... / ,

(2.5.10)

'
]I'(z) =

L)(z -

zj)k/'.

/ '~"";ng tog,clh "

in (2.5.11). we fi nd (2.5. 11 )

.

76

2. Nambu-Goto Strings

(Notice that we have explicitly removed the "self-energy" term for i which would be divergent. We can truncate the integral and still maintamtll' conformal properties of the theory. We will find that this truncation have to be performed in the harmonic oscillator method.) . Now we must complete the last step, which is to fix the measure d fJ.. • . :: Our first guess is that, when the amplihIde is expressed in terms of <..;;. •: ·n .... measure is simply equal to one. This is the correct choice that is cOlnp:at with conformal invariance. To prove it, recall that earlier we said sum over all conformally inequivalent surfaces. Consider the set OfCOllf(jwR transformations that map the upper half-plane into itself, such that the <""<.""11 is mapped into itself. In general, the points on the real axis that are into each other transform under a subset of the conformal projective or Mobius transformations: I

y =

ay +b

ey+d

for real a, b, c and d such that ad - be = 1. This set offour parameters a real matrix with unit determinant:

In general, the group defined by the set of all real 2 x 2 matrices unit determinant is SL(2, R). Notice that this group of transformations c~~.; generated by making successive transfonnations:

y ---+ y + b, y ---+ ay, 1

Y---+ -.

Y Thus, we wish the amplitude, including the contribution of the me:asure il~~~~ projectively invariant. Let us make a projective transformation on the integrand to see transforms:

I1(ZI - Zj)2a'k,.k j = [1(z; - zjfa'krk j i <}

i <j

fl(a - CZD2. k

We want our measure to cancel out the noninvariant term in the above sion. Let us take our measure to be the number 1 and the integration regtQij: be fixed by Zi > Zi+t. Then there is one last complication. We must " ... t~:.:~~ the gauge" for projective transformation or else we will have oVlercoUJ[ltiI1g;:~~ must integrate once and only once over each projectively distinct COlrIDIM·?M of the Zi variables. If the external momentum flows into the upper hal.f·pl~@gl points given by Zi, then we are allowed to fix three of these points at ra~r~~fi This corresponds to "gauge fixing" the projective invariance, which self:q~:~

2.5 Trees

77

loj,,,,;.voly inequivaiem paramctrizations. Our final resull is d

- 8(41 - 4i+ 1)

IJ.-

dV

~b<'

n" d 4/.

(2.5.16)

i_I

have explicitly removed the contribution ITom the three fixed points (2.5. 17) are Ihrccpoints that can be randomly chosen on the real axis. Notice ~ve fixed three variables so that we only integrate over projectively "" "m configurations. (If we integrated over these three variables. we overcounting of the integration region. We could be integmting ,:~;~~nwnbcr of copies oflhc same thing.) :.5 choices for the three fixed points are Zl

= 1.

1.N ==

(1.5.18)

O.

(o"fig"",rio, . our final result for the N-poinl ampl itude becomes [14-

(2.5.19) region of int~gration is 00

==

XI ::: 41

==

I ::: Z)· • '1.H_1 ~ IN

== O.

(2.5.20)

final result fo r the N -point amplitude, which we derived using .cti,,,oll techniques. summarize what we have done:

~

~:t.i~~~:~~,~;;;~I:,~,~gt:h~S~,~o~:b:~e was zeroaforhorizontal the external so the striptachyons, in the complex has width 1T. Momentum ITom the external tachyons flowed into at designated points along the real axis 1./ (see Fig. 2.6). a conformal transformation from the strip to the upper half explicitly solved for the Neumann function, which we using a trick from electrost(ltics, thc method of images. invariance fixed the measure of integration to be the number I. the external points z, > ZI+I' the gauge" lefl over from projective transformations; i.e .. we three of the N points of the integrand.

1e,;O,,, e'p,,,,,;;,)n for the N -point amplitude (2.5. 19) considered only : with external tachyons entering the world sheet al selected points on

78

2. Nambu-Goto Strings

the boundary. In principle, we can have particles with arbitrary spin boundary. For higher spins, we have the same factor e1k-X, which Ten·Tf'"i>i;i!; part of the Fourier transfonn, multiplied by the polarization tensor of the spins. Thus, the vertex function for the tachyon is not actually elk-X, is universal for all spins because it is part of the Fourier transform. The ... vertex for the tachyon is actually just the number 1. Spin-2 vertices can be represented as

V = ..;gg"bauX"abXve"viki"X~, where e"v is the polarization tensor. For the general case:

V

2.6

1u' __ -gU2."-l U2.,,j)a = '\Ir;:;gg([a 0

121

XI'I

.•• t

a X" 2m e J.iL ..... 1J.2I1J elk~X~.

'122111

From Path Integrals to Operators

We used functional methods to calculate the N -point scattering ampli1tud,e; only nontrivial part of the calculation was the integration over COI[lfq,rml3.nY:~ii equivalent two-dimensional complex surfaces, which determined of integration of the amplitude. The same calculation can also be performed using the harmonic V''''U~J~M formalism, which, as we have stressed, is nothing but a particular rep,re\ilm.i:1 tation of the path integral for which the Hamiltonian is diagonal. For and the first loop, the harmonic oscillator method is quite simple be(~aus~::1~ Hamiltonian is diagonal on the Fock space of the harmonic oscillators .... : ever, for higher loops this is no longer true: the harmonic oscillator mc;~OOn becomes increasingly difficult and impractical. The path integral methc)d:tl:iUll~ provides the only systematic way in which to analyze the higher-loop tudes with relative ease. (The calculation of anomalies, however, is eas,i~~jm the harmonic oscillator formalism, where the integer index n serves as off for the theory. In the path integral formalism, we must use [email protected] techniques and other regulators, as we shall see in Chapter 5.) In the functional fonnalism, we know that the propagator for free stri!j~t. given by

(X" I {'XJ dU- TH IX b} = (Xu I I 1 IX!>} . Loo

J

Similarly, we also know that, by inserting a complete set of intl;:ml~~!* states within the path integral, the vertex function for the i th tachyon

(Xu I elki~xl' IXb} , where

X~

= X,,(a = 0, < =
2.6 From Patb Integrals to Operators

IXI

J

DX(XI

=

79

I

;r~:~~:~:~a:~,;,;~f;:: from the N -point function and make the tTan-

the

~

oscillatOr

formal ism to harmonic formalism. ":~~li~ l e~'l~us with the path integral expression for the N-point ~ and make the substitution to harmonic oscillators:

= =

=

J f !

= /

DX dl.Le- s

" e"t,X(~) n ,-.

dJLe - s { ... i

k / xlf;)

IXi )

f

D Xi (X

j! c itH,X(r, . d ... }

dJ1, {... eltiXlOl jx ,) / DX J (X J! e-(" ~ '- 'IIHe'k,., XIO)._ }

n, dr,

(O, k,f elklX(Ole-i', HelkJ X(O)"' !O,kN)

= (0, kd V(kl}DV(k ))··· V (kN _1 )!O, kN)' points in this derivation must be clarified. First, in the path integral we found that we had [0 omit the i = j contribution to tbe inte(2.5.10). Similarly, we also have to truncate the harmonic oscillator . The expression for the vertex function (2.6.2) is fortllaUy infinite convert to harmonic oscillators. For example, if we naively take the ~_ex!pc"~ '; O~ value of the exponential, we arrive at a divergent sum. ordered. expression is the one we desire. which has finite matrix . Let us exponemiate (2.2. 11): ; exp ik /"X"/ (0" = O.

T/)

:= exp

X

(k.f (X;~ e ,

exp

..

1n

"

1it ~(1.)

(-k .f: :. ,-;'" "-.

I.

(2.6.4)

ordering consists of moving all creation (desttuction) operators to the - so that the resulting operator has fini le matrix elements.} in the transition from path integrals 10 operators we used the fact . '" Hamiltonian is the generator oft" !;hifts. The expression Lo - I acts as . Hamiltonian for horizontal displacements in the complex plane. lind the following formul a useful: yLo f (•• )y -Lo

fu ••";on

f.

Thus, we

= f( •• y - ·)

(2 .6.5)

can define

y 4 Vo(k;)y - LI = V(k y, = ~-"). "

(2.6.6)

80

2. Nambu-Goto Strings

We used this expression in the transition from path integrals to narmofmc<9:~~ dilators when we converted the vertex function at 1:i to the vertex the origin. Third, in our derivation ofthe hannonic oscillator formalism we intJ'Odll¢m the vacuum state 10, k). Notice that the functional formalism started w;:~~I~ world sheet of the interacting strings being an infinite horizontal strip ][ in the complex plane. The external lines were placed along the real ~~~I Therefore the effect of a string coming in from negative (positive) ir"rliTIlity .@;~~ responds to the functional integral over a semi-infinite strip. Thus, the va(:u~ijit.~ state 10) is the functional integral of the string theory over a semi-infinite sfi'iPm~ We can represent the tachyon vacuum with momentum k as:

of

10; k) =

kx -

i

10; 0) ,

where

n > O.

cr:n 10; 0) = 0,

Putting everything together, we can now convert the path integral to naImCIJ;tlj! oscillator form:

An =

f dIL J

DXe-s

D N-I

dZ i exp[ikiXfLi(a = 0;

= (0; kll VO(k 2)D VO(k 3)D ... DVo(k N -

1)

.in

10; kN)

•

It is important to realize that the harmonic oscillator expression is a consequence of the functional formalism. Fortunately, this amplitude is easy to calculate. As an exercise, let us calculate the four-point function explicitly: A4

= (O;kll V(k 2 )DV(k3 ) IO;k4 } =

=

r' dxx

10

10

k3 4 1 • -

{Ol exp (-k2 .

f

cr:n

n=1

n

I

x R exp (k3 .

1

dxx-{l/2)s-2(1 _

X)-O/2)1-2

r( -cr:(s))r( -cr:(t)) r( -cr:(s) - cr:(t)) = B( -cr:(s) , -aCt)),

where a(s) = 1 + ~s,

s = -(kl

+kd,

t = -(k2 u = -(kl

+ k3i, + k3)2,

f

n=1

cr:-

n

n

)10).:: .:

2.6 From Path Integrals to Operators

81

iil:~~~:~l_ ~::'I":::~~~~~~":IlIflplitude R is the number operator: R = L:: Ina!an. ~ :' four-point hn.s an elegant form. In fact, it can ;",.,",,00 as tbe Eu ler fWlction. This Beta fWlclion bas fnscinating properties tha t first excited the imagination of high-energy theorists -' -late 1960s. It was discovered by G. Vcne7.iano and M. Suzuki quite by as they were looking for a way in which to satisfy fmite energy sum the badronic S-matrix.. At lhat time. theorists wanted to construct a ~_:~~::~",~e for hadrons thai had the following ofG. Chew's criteria

· IIUSI";~

theorists sometimes added;

A(s. t) ....., s«(')

(2.6.12)

large s and fixed I, and also ~wllity,

i.e., (2.6.13)

of "axioms"

the hadronic S-matrix "''as so large that physicists

the

ever be satisfied. Then,through when a math Eulerbook, Beta discovered wbile thumbing ;:~:;~;:e~'hl~'~'~\~LCOUld

gSI"ling',

were surprised thRt this analytic function satisfied axioms! For example, Regsc behavior can be s hown by approximation on the r function; r(x):::.:

x'

..fii .,;;e-x.

(2.6.14)

can also be shown by evaluating the poles of the integrand near the of integration x = 0 and x = 1:

) +' ;...:02),-'_"",('"('",)..:+-,,,-n) A(.f,t} ..... -1 -'('_"('",)..:+...:1,,,)(,,'(,,-'" n! a(s) - n

(2.6. 15)

mil" ,ex"","i,>" exists for the r-<:hannel poles. In fact, aU the postulates first, unitarily. can be satisfied. (Unitarily is violated for the simple On":~tthe ana[ytie SlrUcture of the ampJirude is a series of poles in s· f A true Wlitary amplitude. however, should also havc imaginary ,instead ofpoJesand cuts along the real axis, as wcshall sec in Chapter 5). . operator methods, we can now solve for the N -point function.

the

82

2. Nambu-Goto Strings

z,=oo

ZN=O

FIGURE 2.7. The N-point function. Projective invariance allows us to fix thr~ee~1t.: the N variables. The most convenient parametrization is to take the first vaIiabl~;~~ be 00, the second to be 1, the last to be 0, and all the others to be ordered betweJ.~.~H

and O. The N -point amplitude can be written as (see Fig. 2.7):

AN = (O;kll V(k2 )D ... V(kN_1}IO;k N) IN-1dX.

=

joo 0-.' (O;kd V(k ;=3

2,

1)V(k3,Y3)V(f4'Y4) .. ·V(kN-l'e'N-dlV~:~!Nti2

Xl

(2. v..',.•" " _

where

fi 1=3

dXj

=

dyj ,

1=3 YI

XI Yj

n

=

X3 X4"

·Xj.

(Notice that we commuted the factors ofX R to the right, until they amllhl.lat~~i;I:~itm the vacuum.) This expression can most easily be simplified using the COlile,~¢~ state formalism. Let an arbitrary state of the Fock space be represented

L 00

II.) =

n=O

}..n - I (atr

10) = e~aj 10) .

n.

Then we have {J.LI}..} xuta

= ell· ...

I}..} = Ix}..) ,

elLut II.) = I}.. + J.L} • By using these identities in succession, we find

2.6 From Path lnlcgrals 10 Operators

83

find the same expression for the N-point function (2.5. 19) that we using functional methods:

(2.6.21)

. ::

y's are ordered along the real axis as befort. Notice that we have fixed the values nfthree pornES along the real axis to be 0, t, and 00,

. ~,a.

oscillator methods, we can show that the N-point fimction is cydi-

'.r~~~~~i)'W~lhiCh is not obvious when we write the amplilUde as the ~ D V D V. We use the following identity, which expresses

"ppe'" when two vertices arc pushed past each other: Yl) V(kl. )'2)

= V(k2.)'2) V(k" yJ) exp[2ct'1r ik j • k J6(YI - n)l. (2.6.22)

= I if x > 0 IUlde(x) = -I if x < O. have to rewrite the vacuum 10; kN) in a fann where it resembles vertices, so we can calculate the properties ofthc amplitude under a '..,,,,,ng.,menl of the vertices. We wili usc the following identities:

tim VI' . y) 10;0) ~ 10;k).

,..., y Hm (O;OlyV(k . y)~ (k;OI.

(2.6.23)

J~""

in this fashion, the tachyon states at the extreme right and left arc no ...ted unsymmetrically from the others. The vertex 011 the right is now a! O. and the vertex on the left is defined at 00, and all other vertices ,:w,i lhi' n that interval. when we push the Nth vertex completely around the amplitude, we (V(k"

yJl ... V(k N• YN» = ( V(kN, YN)V(k l. y,) . . . V(k .... _h YN-d) x exp [2Cli1fkN.

I:kiE ,., ].

(2.6.24)

. :: : !hal the last factor vanishesNif we have t:onsen-arioa of momentum:

L: k; ~ O. ,-, uscdtheon-shell co ndiliona'k~

(2.6.25)

= I . We sec that only on-

an amplitude that is cyclically symmetric. (Historically, this reason why pbysicists believed that a field theory of strings was not possible. A field theory, by definition, is an off-shell fonnulation, marvelous features of the Veneziano model worked only on-shell .)

84

2. Nambu-Goto Strings

2.7

Proj ective Invariance and Twists

In addition to cyclic symmetry, we saw that the integrand of the N-n.m . 'fd: function was also Mobius invariant; i.e., the function remains the same '. make the following change of variables:

y' =

ay+b , ey+d

where

ad - be = 1. We showed Mobius invariance in the functional formalism. Now let us {l:'~!I a Mobius transformation directly on the hrumoruc oscillator matrix ele'lIie)~~

(O; 01 V(k t , Yt) ... V(kN. YN) 10;0) Yt YN

= (O;OI V(kt;

YD ... V(k Y~) 10;0) N

,'

YN

Yt

Il(a _

ey;}2.

1=1

(

As we saw earlier, this factor, in turn, can be canceled by the trallsti)m18,~~~

in the measure

It turns out that we can actually calculate the generator of these infilnhesQUiji transformations on the vertex function. Not surprisingly, the generator Mobius transformations will be the Vrrasoro generators. Let us define ... :< ::::::m~: T = 1-

6

Lan L n, n

z=

Z'

+6 L

anZ n+ 1.

n

Then TV(k" Zj)T- 1

= (1 - sa'k; ~ nanZR) V(k j , z;},

(Ln. V(kj,z,)] =zn [z:z +nalkl] V(kj,z,). We say that the vertex V has conformal weight equal to a ' k~. (The COl1lIOrm weight will play an important part in the proof that the string model is g' !l~~m free. In Chapter 4, we will explain the origin of the conformal weight, wl:llitfi~ labels irreducible representations of the conformal group generated .. Virasoro Ln. For a vertex function to be properly defined, it must have lon-shell, which is true for the tachyon.)

2.' Projective lnvllriancc and Twists

. the

85

L~

generate conformal transformations when I:Icting on the ver. In tact, we can show tha t a specific representation of the Virasoro

conforma l fields is given by

Ln = ~i'+ 1 0,. "'",nt.ti()O satisfies the definition afthe conformal algebra in (2.2.27) central term) and generates conformal transformations in functions variable :c:. in the subgroup ofthe conformal transformations that maps inlo itself and the real axis into itself. i.e., the projective ': ", This subgroup is generated by only 1..,1> L _h Lo, which generates

(2.7.7)

easily calculate the transformation of a vertex function induced by ea L,

V(y)e-
= V[y(l - ayr- l ].

ebLOV(y)e-t>1.o = V[e"'y ). d

e

. 'V(y)e- cL ,

=

V(y

(2.7.8)

+ c).

element ofSL(2. R), then we also have .

U 10;0) ~ 10;0).

... .

(2.7.9)

" . this might seem surprising, because we know that real states are ",' by L" for positive, nOI negative, integers n . However, 10; 0) is not It is annihilated by L_I because LI IO;O) ...... O:o· Q"_ I IO;O} =0

(2.7.10)

Q"o 10; 0) = O. Thus. 10; 0) corresponds to the true vacuum of the

Igroul', which is not a real Slate of the theory. When we multiply 10; 0) it becomes a real stale (because x has commutation relations with is annihilated by Lit for positive,,). everything together. we have {o; 01 V-I U Vek l . yd ... V(kN. YN) V-I YI YN

~

u 10; 0)

(0;01 VI',';': ) ... VI'",' J'N) 10;0) )"1

"P
YN

fIca - cy:J'

(2.7.11)

1", [

(2 . 7.3). the ViI"asoro gauge group, one other feature ofsiring theory is totally in the point particle case, and that is the "tw:ist" operator. Remember

86

2. Nambu-Goto Strings

that a string sweeps out a two-dimensional world sheet, not just a srnlgl~:lil~Q:il1 Thus, if we were to twist the world sheet, a topologically inequivalent Wl1iilfiit sheet would be swept out. At the first loop level, for example, this is the cniiiif~ topological difference between a disk with a hole and a Mobius strip. A simple expression for the twist operator [20] can be derived observation that a vertex function located on the real axis should cOllVertiJliii: a vertex function located on the top of the strip:

n

Q,V(CT

= n")n- I = V(a = 0).

Notice that the only change in the vertex function under this tra1l1st~orniilitia is that every oscillator at the nth level is multiplied by (-l)u. Thus, the.: . operator must b e < . Q,

= (_l)N+I,

where 00

N=

L a-nlLa~ .

l

n=l

Notice that this satisfies

as it should. This establishes the twist operator up to an overall sign. HOlwe~yjm since N even states are charge conjugation even and N odd states are odd Il·ri.'.;if!iil C, this fixes the value of Q, that we have chosen in the expression. There is an equivalent method of obtaining the form of the twist Opf~rii!~~@ Notice that the action of the twist operator on a tree is to reverse the ori.en1i~tt~~ of the extemallines (see Fig. 2.8) Q, Vo(k J )DVo(k2) ... Vo(k N -,) ]0; kN)

=

VO(kN)D Vo(kN-d ... VO(k2) ]

Notice that the cyclic ordering of the tree has been reversed by the erator. To extract the operator that will perform this reversal, let us first ... 1

2

N-l

N

N

1

2

N-l

NN-l

2

1

FIGURE 2.8. Action ofthe twist operator. The twist operator twists the factorize!l!:r@i~ which is equivalent to flipping the entire diagram upside-down. By duality, ho\~ei7.m:l we can always rewrite the diagram in the original configuration (with ext.ernlaffii§? renumbered).

2.8 Closed Strings

&7

"",,;00 in tenns of y variables: Q

f TI

dYI Vo(k J, Yl)'" Vo(k".. y,..) 10;0),

(2.7.17)

take the limit as Yl --) I and y". -~ O. written in this fashion, we now see that we can reverse the cyclic end interchange YI and YN by a change of variables: y:=I-y/.

wish to write down an opemlor tbat will pcrfonn this change of Previously, we wrote dOWIl the generators of SL(2. R} that will a projective transformation on a vertex function. By examining this - -' of variables, we easily find that the Mist operator mlL"! he given by Q

= (_I}Re- L.,.

(2.7.18)

'O~:l~~'~~,~two forms of the twist operator seem totally different, they arc y on-shell. Since the Veneziano amplirudc is strictly on-shell, the freedom of choosing either form of the twist operator.

.osed Strings our discussion has been specific to open strings. where the external attacb themselves to the endpoints of the conformal SlTip swept out string. Poles cOlerge in the Veneziano model when two points lj and the edge of the strip come close together. Let us now analyze tbe model, which is based on closed rather than open strings I . III a path inlegmllaken on the lull!:: (or sphere) swept out by a The pole structure of this model is much larger than the original because external sl ate..~ can attach themselves anywhere of the tube as it moves in spacIHime [21 J: ::

t u) _

:::: '

'.

- r[ - ~(a(l)

r( - !a(s»)r(-!a(t) f( - !a(u)) 1

2

2

+ a(u»)]r[ -f(n-(u) + a{s»)Jr[ -~(a(s) + aCt))]"

9··1)

,"erio". unlike the earlier Veneziano function, bns poles simuhaneall three channels, rather than j ust two channels. This was quickly aI,,,,d to the N -point fWlction: (2.82)

(2.8.3 )

88

2. Nambu-Goto Strings

As in the case of the Veneziano function, the poles in this amplitude emLe~m;1 when two variables Zi and Zj come close together, except now the poles¢.:~~I occur anywhere in the complex plane, not just on the real axis. Our starting point for the quantization of the closed string is (2.2.21), wn.em we decompose the string X/l(a) and its conjugate into normal modes. canonical commutation relations remain the same as (2.2.7), which leads . Hamiltonian given in (2.2.22). As in the open string case, we can fac;torizc;:».i:~em amplitude into vertices and propagators in the harmonic oscillator tbrma.lis1w.~ but now several important differences emerge: (1) We now have two sets of mutually commuting harmonic oscillators a" . ~ii4~W 0:" to sum over, rather than just one set as in the open string case. (2) The Vrrasoro conditions now consist of two sets of conformal' geller.at~fK:S~ Ln and Ln acting on the physical states:

L" 1¢} =

L" I¢} =

0, (Lo - 1) I¢} = (La - 1) I¢) =

o.

(3) We must integrate over all shifts in a, because the closed s~~~,~:~.~~~~ should be independent of where we chose the origin of the a Cl (4) The amplitude is not just the sequential product of vertices and pn)pagat:drsOO Because external lines can occur anywhere in the complex plane, we sum over all different orderings of the external lines. . (5) Fixing the vertex function to have weight 1 and using the anomaly caniC;~~~ lation argument show that the intercept for the closed string model be 2 and a'k2 = 2, meaning that the theory necessarily contains the nul;~~ less graviton. In fact, the linearized genern1 covariance gauge syrnmleft~ of genern1 relativity simply emerges as the lowest-order Vrrasoro gatlg¢ symmetry (we will expand on this in Chapter 7). Let us begin by discussing the propagator:

D = _1_ ( zLo-2zLo-2d2z 2Jr J1d :5.1 sin Jr(Lo - La)

Jr(Lo -

Lo)

1

Lo

- + La

2

This actually has a simple physical interpretation. Notice that the tactor the inverse L's simply is the usual propagator of a closed string. The HUIIJLlI.I;'""WA involving the sines, however, is equal to zero unless

Lo =

La.

This, in tum, can be represented as

f

dO' exp[i2JrO'(Lo -La)].

2.8 Closed Strings

8.

t,

,

C D

F

,

C

8

A

D

0-

8

2lf

,-

A

C A

2.9. Confonnal surfuee~ for closed stri ng propagation. In the p-plane, the D~,.~~:,r::~~,<;,~(]'n a horizontal s!rip of width 2", such that the upper and lower h i ' identified with each other. whieh is topologically equivalent to . In eontrast to the open string case, the cxlcmallincs attaeh themselves 10 ~:';:;;;;,,;D,;f:the surface, nol the boundllf)'. In the "plane, this $urface IMPS 10 the ~I plane by the exponential map.

are two ways 10 interpret this operator. If we explicitly perform thc oS'' ', we have the operator &(Lo - La), which is a projection operator on the ful l Hilbert space lhat eliminates states I¢) which do not satisfY - La) 14') = O. Second, we notice that this operator is the generator of roo";o" by one fun cycle, so that the propagator simply expresses the fa ct closed string moves, we must integrate over one fu ll cycle. The c losed . is therefore independent ofwhcrc we chose the origin of the . (This constraint w ill !\ave important consequences when discuss the compactification of the closed string and the heterotic s tring in chapters.) Fig. 2.9, we see that the world sheet originally consists of a horizontal in the complex plane that is 211" in width, soch that the top and bottom are identified with each other (creating a long horizontal rube ). External 'h;,;~;,~t;;,n~es entcr this tube intcrna1ly. We see that by exponentiating these !O we can transform this horizontal tube into the entire complex . Extcmallines then couple [0 all pointS within the plane.

can

90

2. Nambu-Goto Strings

As before, the vertex function is once again given by :e ikX :, where the Str~o;~ X is defined at the point in the complex plane where the external m()ml;:nt].li;jjt:~ is entering. Because the two sets ofbarmonic oscillators commute, the veliWii=m function factorizes into the product of two open string vertex functions. final expression for the N -point scattering amplitude of tachyons is given •.: [22,23]; . . AN =

L (0; kJI V(k )D··· V(kN-d 10; kN) . 2

peml

The summation over all possible permutations of the ordering of the exteni.fd:;:: lines guarantees that the variables can roam freely in the complex I./l<1U<'".

'I

2.9

Ghost Elimination

We have developed the harmonic oscillator formalism using quantization. In the conformal gauge, the theory maintains manifest LoreI).~~ij invariance and, in fact, becomes a theory based on free fields. This ac(;ou~~:m for the fact that the theory at the tree level is very easy to write liown. The price we have to pay for this simplicity, however, is that we must II'np()s:t::~ the Virasoro constraints directly onto the Hilbert space to eliminate 51l1J""" •• In general, the proof that ghost states do not couple in tree diagrams is qWlte::;~ simp Ie. (The proofbreaks down in loops, where extreme care must be [email protected]:~ to properly eliminate ghost states.) Let us define a real physical state as otl;~~:ill that satisfies the Gupta-Bleuler constraints:

Ln IR) = 0, [Lo - 1] IR) = O.

n > 0,

Let us define a spurious state as one that does not couple to real states:

(SIR)

= O.

We can conveniently represent such a state as IS) = L_n Ix)

for some state X. We now wish to show that spurious states do not couple ... trees, i.e., (S] Tree) = O.

To show this, we need two more identities:

[Ln - Lo - n + lJVo = Vo[Ln - Lo + 11, 1 1 [L n -Lo+11 1= [Ln- L o-n+l]. L0 Lo +n - 1 (This can easily be proved from (2.7.6), which in tum crucially depends on vertex function having conformal weight 1 with external tachyons ,,"'t-id='lifli~W&

2.9 Ghost E.limination

91

1. Thus, thc restriction on the ronfonnal weight of the vertex is crucial ;mination of all ghosts in the theory.) these two identities, we can now easily show

[t" - Lo - n + I )VoDVQD··· Vo 10) =

o.

(2.9.6)

· is the direct result; it shows tlllll the L 's can be pushed to lhe right until · finally annihilate on the vacuum. • """,mary, we have shown (hat Spurious slates do nOI rouplc to trees:

L" ITree) = 0

~

-+

(SI Tree) = O.

(2.9.7)

:~~'~::i we do 1I0t have to make any speciaJ change in the functional for because of the presence of ghosts. The ghosts stales propagat-

the tree amplitude automatically cancel by themselves. However. we . that the loop functions do indeed have problems with gh~ts propagating · I . This is because

(SI Tree IS)

f

O.

(2.9.8)

we naively trace over the trees to obtain loops, we thus inadvertently ud,,,h,, presence of ghosts, which must be facLOred out explicitly:

th",p =

L (n I Tree In) .

(2.9.9)

•

explicitly contoins ghost slales.

;~;~~~:~~.E~:l~ ~d~":':IU;;:plehappens from trees andooly contribute loop diagrams in the Yang--Mills case. to Faddecv-Popov

~

to the Yang-Mills theory do not contribute to the IrCCS of theory, but they do contribute to the loops. The Faddcev-Popov ghost . . is (see (1 .9.30): (2.9. 10)

that this produces the foUowing interaction of the gauge field A with ghost field c:

L, '" cAe.

(2.9.11)

,i, """piing means that a single ghost cannot couple to a tree diagram congauge fields. They can only contribute to loops, where the e fields circulate internally. Thus, particular attention was required to guarantee lhc Faddeev--Popov ghosts canceled against lhe negative metrie Slates in loops. m(
'~:.' :;,'~:: il:;,~h~~o';w::e:;v:~e'r, an extremely complicated task. In principle, the ability gauge is normally sufficienl to prove that the theory is

i!

. However. we can never be sure that the quantum theory doesn't anomalies thai destroy this fact. Therefore, it becomes important explicitl y that the Fock space is ghost-free in the Gupta-Bleuler

2. Nambu-Goto Strings

92

formalism. There are two independent proofs of this theorem, neither ot1iVhit~li~ is very simple, due to Brower, Thorn, and Goddard [24, 25]. The reader tilliv.ill skip the following discussion. Because of the numerous details involved, let us first sketch our .,tr.,tf'OV the elimination of ghosts. We would like to construct a set of physical opl~ra~t9ijim V~ and V;;; such that: (1) they commute with the Virasoro generators:

[Ln' V~] = 0, [Ln, V;;;] = 0; (2) the V~ and V; generate the physical Hilbert space: [phy}

=

V.:'~, V.:'~2 ... V::::" V:::m ,

'"

V::: mp [0; Po}

such that the states have either positive or zero norm, but never neI5atii¥~~ norm. The original Fock space of D-dimensionaI harmonic oscillators "llU·UH.I. equivalent to the set of states generated by these (D- 2)-dimensionaI operators, the - component of these oscillators, and the original operators:

{V~n'

V::: n, L,,} .

The L-n generate ghost states, so that by only taking the states V~n and v.':,=~u;.;. we will generate the correct ghost-free states. This is the desired gn'OSl"KllllU; result that we now want to prove. Let us begin by first defining

.

A~

1

= 2](

1

21 (

.

V'{nko, r)dr,

0

where

where ko is a null vector: k~ = 0,

{

ko = -1,

k6'

= k~ =

O.

We can view this vertex operator as the vertex for the insertion of a massll~~ vector particle. This particular vertex function was chosen for the .v.""u reason. Notice that, because kQ is a null vector, the Virasoro generators hA}l~ the following commutation relations with this vertex operator:

d . [Lm' VCr)] = -i-(e'mf V(r)). dr

2.9 Ghost Elimination

93

~~;;~~~.~thi~·:S,,~iSS 8 total derivative. When integrated over a circle, this means [L •. A~l =

o.

(2.9.19)

we chose the form of the vertex function such iliat we fulfill the first the A's commute with tbe Virasoro generaum . So far, however, we '. flO constraims on ko other than that it is a null vector. We will need to more constraints so that we fulfill the second condition. us start with a ground state veclOr of momentum Po. such thai Po and k{) Ct "Po::::

1, ko'Po=l.

(2.9.20)

us apply the vertex operator A: on the state vector 10; Po}. The resulting momentum Po + IIkO' We wanl this resulting state to satisfy the mass co"d;irio"118 well. As 3 consequence, we demand (X'M2

= -!(Pll + nkoi = -I -

II .

(2.9.21)

we demand that the momentum vector ko be constrained 10 be a null that the vertex: operator commutes with the Virasoro generators, and I. nmn,d that 4 . Po = 1 in order to generate additional transverse mass states whcn acting on the ground state. we must fuUy implement the last condition, namely that these opin fact, generate the entire physical space of states. There are some

" :,~~~~:~~;:,A::t.:r;,,~'t~,,;wc might suspect that the - modes can be created by ~

. vlllue ofk o. NormaUy, since the conformal spin ofa of operutors their sum, we expect this to be truc. However. the ordering destroys this. In ftlc t, we find thai the - components do not ,.-__ . with the Virasoro generators:

:x-I"x': ] = I"" ( -;

:r + m) :X- e'~X+: + inm1e,,,,,+id' . (2.9.22)

.d,no remedy this difficulty, we must add an cxlra item to tbe definition.

,.",m"',I, definition is now V$«k. T} =

:x$<:I' x + _,' ik,",~(logk. d,

X}e'H.

(2.9.23)

, .es'bo..... thal the above combination bas cooforma1 spin I and reduces, we take the transverse components, 10 the previous expression when we integral. let us take the commutator between these fields:

(2.9.24)

(2.9.25)

94

2. Nambu--Goto Strings

This allows us to write

v,: = om.Q. [V~, V!] = mOijom._n.

[V;. V~] = -nV~+n' [V;, Vn-] = (m - n)V;+n

+ 2m 3 0m ._n.

Notice that the + components are trivial and that the commutation rellltii;lilW1I of the transverse operators are exactly the same as for the usual narmCt~:::m oscillators. Now let us redefine the - modes once again:

Putting everything together, the final commutation relations are

[V~. V!] = moUom._n.

l

V~] = 0. -_ -_ -_ 26 - D 3 [V m • Vn]=(m-n)V m +n + 12 m Om.-n,

[V;;;.

[Ln.

V;;-]

= [Ln. V~] = O.

This is the final set of commutation relations. Notice that new V;;; n",~TR'iilf.i: commute with the original V~ operators and that they both commute wilth;l~ill1 Vrrasoro generators. Thus, we have now created a new Hilbert space independent operators that we can use to replace the original Fock "'1.1<'"' J1

.-

{a_nl""""* {V.: n • V:::: n • Ln}.

The physical Hilbert space is thus generated by the operators {V~n' V,""'j~Jmw which are equivalent to the DDF operators [26] which generate the nm""""" space in the light cone gauge. Furthermore, we know that the following has zero norm:

V::::n1 V:::: n2 ••• V::::nN 10) • This can be checked explicitly by taking the norm of the state and uS~:~':I~ commutation relations of the V;;-. Thus, we have the final statement, set of states generated by

have either positive norm or zero norm, but never negative norm. This cQJi.OOi:S pletes the proof that the Vrrasoro conditions completely remove the gh(f$~~ from the Hilbert space.

2. 10 Summary

95

Summary

~

:E~;%~r:\~~:;E~f~; is remarkably similar to the theory of point [ for the addition of gauge symmetries representing i

invariance of lhe \.\'Orld sheet. A3 in point particle theory .4.1 6), we have three cquivalem aCliOflS for the string:

~t~,.-d"

(Hamiltonian) form:

L = P"X" + "a' ,," [p2+ "

Xil ,]

(21ra ')

+ pP"X'I' ..... .;gPI~g"l>p b" + p aP 8" X/,../i/](cx.' secolld-order form :

- I L = _../ig"bJ"X",abX". 4:rra'

L = ~(Xl X fl

nonlinear form:

21f0"

".

_

(2.10. 1)

(X X'I')2 )1/2 I'

•

us summarize the similarities fOund in the point particle case and the

:th,co,y v,hen expressed in pi:lth integral language: xI'(r)

Dx =

n

--t-

X,, (u, r ),

length -+ area,

d.tp( t )

--t-

DX =

P.T

n

dXI'(a, r),

I',r.a

(fI ""') - (fI ""X') ' I_I

I_I

graph _ manifold,

"

P +m =0-+

(P,,+-X; )' =0. 2Jra'

'~:::;:, for quantizing the Siring action is

\0 write down the symmetry ~I extract the currents from this symmetry. calculate the a lgebra currents, and apply these currents onto the Hilbert space in o ,Itim;,"" ghosts. The Slmtegy we follow throughout the book is _ Symmetry -) Current _ Algebra ~n'~"t; ot"

--t

Constraints

--t

Unitarity.

possess repar~trization symmetry, which generates the

(2. 10.2)

96

2. Nambu-Goto Strings

The algebra generated by these operators is

[L", Lml

= (n -

m)Ln+m

D 3 + 12 (n -

(2.1·~·:_

n)8n,_m'

As in the point particle case, there are three ways in which to qUirntiz¢.~W theory.

Gupta-Bleuler Quantization In the Gupta-Bleuler fonnalism, we fix the gauge: gab

= 8ab

and the resulting action breaks reparametrization invariance but mElml:alrlsJl subgroup of conformal transformations:

L = _1 (X 2 _ X'2). 2][ I' I' (,

This Lagrangian, of course, propagates negative metric ghosts associated . the timelike mode of X. To eliminate them, the Gupta-Bleuler qUirnt'izalti~~ method states that the state vectors of the theory must satisfy

L" I¢} = 0, (Lo - 1) I¢} = o.

n > 0,

This means that the constraints are constructed to vanish on the Hilbert Sp~(~~ Although the action in this formalism is quite elegant, the price we "
Light Cone Quantization In the light cone quantization program, we set X+ = 2a' p+r.

(2. 1"'.'!'r.'m

The advantage of this approach is that we can eliminate all the redun(1@~ ghost modes of the theory from the very start in terms of only the tra'lTerse components. The disadvantage is that the fonnalism is awkward and we ,...;,"~v~ reestablish Lorentz·· symmetry at every step. The surprising feature is the Lorentz generators of the broken theory close only in 26 dj'im~msiIOn!~i;: Specifically, the problem occurs with the commutator:

2.10 Summary

1

n I[D - 26 6."=12(26 - D)+; 12 +2-2a.

97

(2. 10.9)

commutator to vanish, we must MVC D

~

26.

a = J.

(2.10.10)

' RRST method combines the best featuIes of both approaches. We still Lorent7. covariance of the action, but the negative metric states worry us bl..'Causc they cancel against the ghosts circulating in the because of the FaddccY-Popov dctcnnioant. When we exponentiate .d,lce....I'''l'~vd'''nnir''nt,we must introduec two anticommuting ghost . : band c. The resulting gauge-fixed action scUJ has a residual symmetry

:

by the HRST charge: I

00

00

L

'2

co(/o - a) + 2:)cn L .. + L"c..1,,=1

:L",c-"bn+", :(m - n).

11."'.-00

(2.10.11) (2. 10.12)

inlcrccpt 10 be ooe and the dimensioll of space-time to be 26. The states of the theory are defined by Q Iphy) ~ O.

(2.10.13)

in point particle theory, interactiolls are introduced by summing over

topological configurations in the path integral. The fundamenta l func· ~~,:~q~;ii' (>D for the interacting theory. upon whlch this entire chapter rests,

k" .... kN)

J J J '.........

~ E.

X LlFP

~ E IOpOIogics

d"'N

exp

[i /

D.o'

DX

L dO' dr

ID

./i exp ik',Il X ""

I]

JDg"" Jd",(fI .JEexP(ik•.,X"»)' ' .. I

(2. 10.14)

98

2. Nambu-Goto Strings

The amplitude can be computed exactly in the conformal gauge by uSlngtn~ identity

f

DX exp {= exp

=

IT

{~'

f

4~a' aZxfLa~xfL d2z + i

f

f

JfLXI< d

2Z}

JI«z)G(z, z')JI«z') dz dz' }

IZi - Zj 12a 'ki ·k,.

i<j

In the conformal gauge, the sum over topologies is given by the sum oviiW';: aU confonnally inequivalent configurations. If we consider conformal tra'!~(~~ formations that map the upper half-plane into itself and the real axis i~'(W,i itself, then the points along the real axis transform according to a projec'tiWill transformation SL(2, R): ,

y =

ay +b , cy+d

(

where the coefficients are real and satisfy ad - be = 1. This fixes the m~:as':¢~ili§ d fJ." so the formula for the N -point function becomes

where the z's are ordered along the real axis. The transition to the harmonic oscillator formalism is straightforward ..,'",::;"-/h cause, in the conformal gauge, the Hamiltonian is diagonal on the Fock sp~'r¢i'~ of oscillator modes. The propagator from the configuration Xa to Xb is (Xul

(''''0 e-THdr IXb) =

10

(Xul

1

Lo - 1

IXb ).

The vertex is equal to (Xci

eiki"X.i(O.r;)

IXd ).

We can remove aU the string eigenstates because IX}

f

DX{XI = 1.

Thus, the N -point function is equal to AN

= (0; ktl VO(k2 )D VO(k3)' .. VO(kN-d 10; kN) .

In the operator formalism, projective invariance can be reestablished using the fact that the operators L±l and Lo generate the projective gro,liit.=§jl SL(2, R). In fact, under an arbitrary conformal transformation, the V"""""'.~~/...

References

TV(k; , 1./) T - 1 =

(1- 6a.'t: 'tnotn1. .., n )

V(k/.

zj).

99

(2.10.22)

(2.10.23) thai the vertex V h3S conformal weight equal 10 a'kl = I.

ean explicitly conSUUct the operators that will generate only physjcaJ with zero or positive nonn The previous identity shows thai operators ,(",rm=~. ~piD equal to I automatically commute with the Virasoro genThis allows us to create operators, based on the vertex for a massless

, that will generate the physical space. We can coosuuct three commuting operators which together generate Ihe eillirt Fock

(V.'..

ii:•. L

.IIO).

(2. 10.24)

on ly tbuse states buil! out of V~II nnd V:" and dropping the L_ n • obtai, a new Fock. space thai has only Siaies which have posiljve and zero Thus, lhe set of spaces thai satisfy: L.I~) ~

O.

(Lo-I)I~) ~O.

(2. 10.25)

.., ,o f " eg;"i" norm states in 26 dimensions. we nole that string theory is easy to quantize because its Hamiltonian ua~~~; in the string variables. Simple harmonic oscillators ean be used III the theory. However, in M-theory, this simplicity breaks down. the membrane is quartic in the Hl'lmiltonian. At present, there i ~ no '''''Y in which to quantize thc free mambmne.

,r""",y

Y. Nambu, Lectures at the Copenhagen Summer Symposium ( 1970). T. 0 010, Progr. Theoret Phys. 46, 1560 ( 1911 ).

For eartier fonnuJations, sec: also H. 8 . Nielsen, 15th International Co'iferenc~ ofHigh Energy Physics (Kiev), 1970. L Susskind, NuollO Cimento 69A, 451 ( 1970). SeeC. B. Thorn. in Unified String I1leory (ediTed by M. B. Oreen and D. Oross), . World Scientific, Singapore. 1985. M. A. Virasoro, Phys. Rev. 01. 2933 (J910). A. M. Polyakov, " hys. Lell. 1038, 207,2 11 (1981). S. Fubini, D. Gordo n. and O. Veneziano, Phys. Letl. 29B, 679 (1969).

100

2. Narnbu-Goto Strings

[9] P. Goddard, J. Goldstone, C. Rebbi, and C. B. Thorn, Nuc!. Phys. (1973). [10] M. Kato and K. Ogawa, Nuc!. Phys. B212,443 (1983). [11] C. S. Hsue, B. Sakita, and M. A. Virasoro, Phys. Rev. D2, 2857(1970). [12] J. L. Gervais and B. Sakita, Nucl. Phys. B34, 632 (1971); Phys. Rev. D4, (1971); Phys. Rev. Leu. 30, 716 (1973). [13] D. B. Fairlie and H. B. Nielsen, Nucl. Phys. B20, 637 (1970). [14] K. Bardakc;i and H. Ruegg, Phys. Rev. 181, 1884 (1969). [15] M. A. Virasoro, Phys. Rev. Lett. 22, 37 (1969). [16] C. J. Goebel and B. Sakita, Phys. Rev. Lett. 22, 257 (1969). [17] H. M. Chan, Phys. Lett. 28B, 425 (1969). [18] H. M. Chan and S. T. Tsou, Phys. Lett. 28B, 485 (1969). [19] Z. J. Koba and H. B. Nielsen, Nuc!. Phys. B12, 517 (1969); B10, 633 (19~~~ [20] L Caneschi, A. Schwimmer, and G. Veneziano, Phys. Lett. 30B, 351 (I [21] M. A. Virasoro, Phys. Rev. 177,2309 (1969). [22] J. Shapiro, Phys. Lett. 33B, 361 (1970). [23] M. Yoshimura, Phys. Lett. 34B, 79 (1971). [24] P. Goddard and C. B. Thorn, Phys. Let!. 40B, 235 (1972). [25] R. C. Brower and K. A. Friedman, Phys. Rev. D7, 535 (1973). [26] E. Del Giudice, P. Di Vecchia, and S. Fubini, Ann. Physics 70, 378 (1

3

ngs

Supcrsynunetric Point Particles '~~;::'~,,;;'~.,'one of the most elegant of all symmetries. uniting bosons Cl single multiplet: Fermions

No

Bosons.

fields of differing statistics. supersynunetry and supergroups have up an entireiy new area of mathematics. . there is nota single shred of experimental evidence For example. physicists have tried to fit the electron or neutrino into "ym,nOltrio multiplets. but the scalar partners of these leptons have never . In fact, nonc of the presently known particles has a supersyoum:tric Some critics have called supersymmelIy a "solution looking for a

" 1~::',~~~': arc absolut.ely no empirical data to support the notioDof su-

!'l

il is undewable that supcrsymmetry provides a wealth of highly theoretical mechanisms ilial hold tremendous promise. Supersymis more elegant way in which to unite elementary panicles 1 multiplets; it also has definite practical appl ications field theory;

'.~;:':';:~:','Z generales super-Ward-Takahashi identities thai cancel normally divergent Feynma1 grapbs. For example, Feynman loop a"~ttlli· with bosons and fermions circulating internally in the loop differ a facto r of -1. Because of supersymmetry, the boson loop can cancel .'g'",,,,, the fermion loop, leaving us with a much milder divergence, We therefore. thai Ynng- Mills theories with supcrsymmetry have be-tter

\

102

(2)

(3)

(4)

(5)

3. Superstrings renonnalization properties than ordinary gauge theories. In fact, ce.~,tl1.t~ "nonrenormalization theorems" can be proved to all orders in pel1UlrbaQQj~ theory. Supersynunetry may solve the "hierarchy problem" that plaglJes ordi1)~mi Grand Unified-type theories. In GUTs, there are two widely se"flanlte~i;~~ ergy scales, the energy scale of ordinary particle physics in the v~'fit.~i electron volt range and also the GUT energy scale at I 0 I 5 or so t.V~Wl}1m electron volts. In between these two energy scales is a vast :'desert" no new phenomena are found. However, when renonnalization efii~c~!~:lCf.i calculated, the two energy scales inevitably begin to mix. Loop c~~6.'IJ~ ings, for example, to the quark masses can push them up near the' .• scale, which is unacceptable. "Fine-tuning" one's coupling . masses by hand can, in principle, solve the hierarchy problem, ~U".·"" is very contrived and artificial. Fortunately, the Ward-Takahashi la~:.tm ties of supersymmetry are strong enough to enforce theorems" to all orders in perturbation theory. Thus, SUI),er:syrnrnLeti;v,]m! necessary to stabilize these two mass scales in perturbation thp,"ni' prevent mixing. ( ::::::::::W::! Supersymmetry may help shed light on the "cosmological constant" 'P'i(9im lem. Empirically, the cosmological constant term )...,j=g, correction to the Einstein-Hilbert action, is exceedingly small on a#;~i tronomical scale. The problem is to explain the near vanishing OI:)~ cosmological constant without "fine-tuning." Supersymmetry is pn)b~!1.i:JjOO strong enough to force the cosmological constant to be zero to all otiieRm:i in perturbation theory (because this term breaks supersymmetry). 'H""""'" doesn't completely solve the cosmological constant problem., hO'weve~~:~AA cause we must inevitably break supersymmetry to reach ordinary enlel'g~§J (The problem, therefore, is to explain the vanishing of the constant after sypersymmetry breaking has occurred.) Supersymmetry eliminates many undesirable particles. The t~"hv,")n·::·flml appears in the bosonic string model, for example. is elimrijln~~a~:te~:d~~~I~ it violates supersymmetry. By eliminating these particles, SI also reduces the divergence ofthe higher loop graphs. In Chapter 5 show that the potential divergences of the superstring theory are a~~~I~ with the infrared emission of tachyons and dilatons. Thus, by I ing these particles, we simultaneously eliminate the source" of pO"t¢llUi divergences. .:: Lastly, when supersymrnetry is elevated into a local gauge theory, if' rally reduces the divergences of quantum gravity. This is becaus~e::'~1.'~ supersymmetry can be defined only in the presence of gravit()fis Appendix). Local supersymmetry is thus intimately tied up with relativity. In fact, local supersymmetry successfully eliminates the . loop divergences of supergravity. However, the largest of the supergraV~\1 theories, 0(8) supergravity, probably has a divergence at the se\rerrth):95.l§ level, which most likely rules out supergravity as an acceptable qw~~~~~

II

3.1 Supersymmetric Point Particles

103

th,,",y.Only when we combine local supersymmcrry with the coninvariancl'l of the string theory do we have a large enough gauge to eliminate perhaps all the divergences of quantum gravity.

,>","""tc" as an invnriancc of an action, was first discovered in the Gervais and Sakita [1] showed that an extension of the usuaJ possessed a symmetry that converted bosons into fermion:>. it languished for many years because the supcrsymmetry of model was a two-dimensional supcrsymmetry on the world until relatively recently that it was fml:l.!Iy proved that the ~~!:;;~"'''''d borh two-dimensional and IO-dimensional spaoo-tlmc our di.scussion by t:oWiitlcring the siwph::st possible spinning ,e spi~'~i"g point particle. In addition to the \f'.uiable x)J.' which locates of the pOin! particle, let us introduce Dirac spinors 0", where A is In general, a Dimc spinar the Dirac matrix in D dimensions ;n"ioo;b", 2''''''"compiex components. We can writc p. 3J

r" .

'f -'( '

S=i

e

..t)J. -

'£j"r)J. B")'d r.

lv

(HI)

particle action is invariant under

80 A = t: A ,

B8 A = Ox)J.

e,

=ieA r"8 A ,

(3.1.2)

Be = O.

(3.1.3) ~:~~:I, all by itself under this transformation. Thus, any expression

combination will be invariant under this symmetry. The strange action, however. is that half of the components of the fennioruc from the action by itself. "Ying,th'"'' .f.)J.. and e fields. we can derive several equations of motion: IB

n 2 = 0, Ii" =: 0,

r·no =o.

(3. 1.4)

(3.1.5)

aJ!~:~~i,.~~::'J'l~;: :~!"~~:~~~~;~ vanish. Butsincc8 always appears p'e". half they. ncomponents of the Dirac spinor are

:
104

3. Superstrings

actually missing from the action. Thus (J is not an independent splnolf::.t)j satisfies a constraint that reduces its components by half. The reason for this decoupling is that the action is invariant under yet an(lftijijjij local symmetry [2]: '

{

8(JA

= i r • OKA.

8 X IL

= i~ArILS(JA.

Se = 4ee A KA.

Furthelmore, there is also another bosonic invariance of the action: S(JA

= ),J)A.

8X IL = {

ieArIL(j(JA,

lie = O.

When we try to commute two supersymmetry operations given in (3 .1'~I':'"' find that the algebra does not close unless we use the equations of mCIt1o~::::;:: [8 1• 82]eA = (2ir IL Kte Br.

orILK~ +4ir· oKteBK~)-(l ~ 2).

(This is actually typical of supersymmetric actions. Notice that the nU1J1lj:~ of components of x lL and (J A do not necessarily match off-shell. which 1l1~~~ that auxiliary fields are in general necessary to close the algebra.) Lastly, if we calculate the canonical conjugate to the coordinates, we A _

1T:(J -

8L _.

-.- -

(j(JA

I

r IL nile A .

This poses vast complications for the covariant quantization. Because the~t-e:: an explicit dependence on x within the canonical conjugate, the qUJmtiiz.a1.ft relations become nonlinear and hence exceedingly difficult to solve. Wij~1 it turns out that the term r il OIL becomes a projection operator in the th~~~ meaning that we cannot invert the transformation and solve for (JA. In q:~g words, a naive covariant quantization of the spinning point particle dotlS;:~ijffi seem to exist at all! This is a warning that the supersymmetric string is:)~m§ going to be as simple as super-Yang-Mills or supergravity theory. As sequence, we will discuss the simpler two-dimensional world sUI)en;yn:Up:i¢~~ of the N eveu-Schwarz-Ramond (NS--R) model first, and then the more .Cj9~~ plicated lO-dimensional space-time supersymmetry of the Grleenr-SlcMv:im~ model.

3.2

Two-Dimensional Supersymmetry

Given the problems associated with superparticles (such as lack of a cO'vM'iJPll quantization and lack of an algebra that closes off-shell), let us tenlpoltW;!m:: abandon space-time supersymmetry and discuss the two-dimensional

3.2 Two-Dimensional Supersynunl:try

lOS

ym",e~

of the simplest possihle action involving free strings and free . This action will already be gauge fixed in tile conformal gauge, but all the essential features oftwo-dimcnsional supert.ymmctry. In dixod fonnulation, we will impose the gauge COllstrnims on the Fock hand. Lalcr, we will present the complete action, where we will be lhese cOllstraints starring from a locally symmetric action.

gauge, we have [J] 1 L = --2 (a~ X",a~ X" - iy,JJ p "a.. Yt14 ), n

(3.2.1)

I, 2 and labe ls two-dimensional vectors, and J.L is a space-lime is a bizarre object, an anticommuting Majorana spinor in two

,im" ,md a vector in real space-time. Let us define

!/1M =

(~~ ) ,

pO =

(~ ~j ),

, (0 i)

p...

Ip", p'} =

j

0

~jJ ~ iftupo,

'

(3 .2.2)

-2,"".

this out explicitly in components, v.'C have (3.2.3) action is gauge fIXed, it is still inv:lriant under the global

jrmatio",

SX" = e",". Sl/f" = - ;p"a. xl't.

(3.2.4)

abandoning OUf attempt to produce a theory of strings spucHme sp inors, we have achieved a two-dimensional theory that is quite s.imple, involving free bosonic and fie lds. the NS- R theory [4-6J was the first successful attempt at spin inlo the dual model. It was also the first example of a lin. action and was soon followed by fOUI·dimensional actions [7. 8}. wri tten down our two-dimensional supcrsymmetric acn'tn,,, the "lOP"'" '"ed in the previous chapter to solve the system. !.1ep is to extrnct the currents associated with these symmetries, then the algebra that these currents generate, and fmally apply these conon the Hilbert space. TIle sequence we will follow in this section is

"U'

106

3. Superstrings

a straightforward generalization of the steps we developed in the chapter: Action -* Symmetries -)- Current -)- Algebra -* Constraints -)-

nrp',';"

UUHi1I(i~(

Following this strategy, let us now calculate the supersymmetric curretll:m~ sociated with this symmetry. As we saw in the last chapter, the eXllste:n¢I~:; conformal symmetry is sufficient to generate a conserved current. h1 ( we saw that this conserved current could be written as

r=~ 8¢. 8 ap,¢ 8s a

p,

By inserting (3.2.4) into (3.2.6), we find Ja =

! pb p" 1fJ/'/' abx

Jl"

To check that this is conserved, we must first write down·the motion for the system, which is especially easy since it is a~ee action:

[ar + Orr ]1fJt; = 0, [or - arr ]1fJi = o. It is now easy to show

Oa 1"=0. Written out in components, this equals (h

= !(Jo ± J 1»)

L = 11fJt;(o, - orr)X/./" J+ =

!1fJi(ar + aCT)X IL .

U sing the equations of motion, we can show that

(a r

+ arr)L =

(ar

-

arr)J+

0,

= O.

In addition to the supersymmetric current, we also have the momentum tensor, which as we saw earlier in (1.9.17) can be as

8L P' = - 0 A. v SOIL¢ V'Y

-

wr.l~M

LSI'

v'

This can easily be adapted for our purposes. Inserting (3.2.1) into (3.2.1"fj:;::~~ find

Tab = OaX/'/'ObX IL

i -

i -

+ 41fJ IL Pa Ob1fJp, + 41fJ/'/'PbO,,1fJ1L o"T"b = 0,

(Trace),

3.2 Two-Dimensional Supersynunctry

\07

is traceless because we have explicitly subtractcd out the explicitly, this is

1'2

== l(X

+ X ,:z ) -

t rac~.

i" i .. '41Jro (a, - iJ,,)1Jrou - '41Jr1 (a r -+ B")1JrI I,,

== Till

(32.14)

. Ii.. iu = X· X + 40/0 (Of - 8,,)Vto.. - 41P1 {a f

+ (1,,)"11"

i strategy (3.2.5) is to use these currents to restrict the Fock space so ghosts are eliminated:

(T,,/>l =

o.

~

o.

(J.)

(3.2.15)

strcs..~

again, however, that we are simply imposing Ihis constraint the Pock space. Because me action is not locally supcrsymmetric, ,Ier;;vethis constraint from first principles. Later, when we present symmetric action, we will see that these constraints emerge because

""<"

String, we actually have a choice of two distiuct boundthat we can place on the % and 1Jr1 fie lds, either periodic or rio,iio . At C1 = 0, we can always choose "'0 =: 1/1"1' However, at C1 = I(, the choice of two distinct boundary conditions. If the field is periodic p,,,;o,jio), we have the Romond (Neveu-Schwar.£) boundary conditions Ramond: Neveu-Schwarz:

Vro{1r, r) == "'I(re, .), ( 1Jro(n, 1:) = -1JrI(rc, r).

(3.2.16)

two different types of boundary conditions for our SpinOT, we nanually different ways in which to pO'.'l"er expand these two fields in Fourier ipo"",<~ with integer (half-integer) modes [8, 9]:

L 00

1Jr6'("',, ) = rl/2

d~e-i~(r--Gl,

(3.2. 17)

00

1Jri(cr, r) = TI/2 1Jr~(a, T) = riP

L L L

d:e- III!I-'-D 1,

b!'e-'r(r-o),

rEZ+I/2

VrUa,

r)

==

2- 1/2

b~e -ir(t+a).

(3.2.1 S)

rEZ+I/l

",:~~x;~t:~:~~.~:~::;te~~.;(3;;.;2:.5) is to isolate the algebra thai these currents r:;:

we saw that the symmetries yielded the

108

3. Superstrings

energy-momentum tensor, which in tum generated the Virasoro or co'n~81ffl;jil algebra. For the NS-R action, we will find that the superconfonnal alg;l';bi6tJ1 generated. With this decomposition of the fields, the moments Ln of the curren~:~ can now be written as

1'" 2rr

L" = - 1

0

da{e rll " (Too + To l )

,

+ e-,na(Too -

TOI )}.' '

Notice that the definition of the conformal generators now >w..,...,.,,~ contributions from the anti commuting sector. We can repeat this Fourier expansion for the supersymmetric current as,if"eJI: For the R sector, we have '

v21'"

Fn = -

IT

+ e-

da{e rll'" h

,

1ntf

0

L}. •

For the NS sector, we have

v21' " rr

G, = -

,

l

da{e"" J+ + e->r" L}.

0

To extract the algebra from these moments ofthe currents, we need to "U~'~!:!J:Im the canonical conjugates of the fields. Ifwe define .,.,Jl _

_

"A -

-::-

8L _

81/1 A.Jl

,

then we find that the fermionic field is self-conjugate. Thus, we l11l1Dos:ei (1/I~(a, T), l{I;(a', T)}

= rr8 AB 8(a -

a'}r,lLv.

Therefore the oscillators in (3.2.17) and (3.2.18) obey R: NS :

{{d:,d~}=r,JlV8"._m' {b~, b;} = r,JlV 8r.-s .

It is important to notice that the zero component of the R sec:t9tOO proportional to the Dirac gamma matrices:

{db, do} = r,Jlv. Hence, we will see that the R sector corresponds to the fennionic sector, at*~!g~ the NS sector, even though it contains anticommuting operators, still renfAlI2 a bosonic sector. The fact that boundary conditions like (3.2.16) an important role in the development of the fennionic and bosonic of the theory is quite novel and apparently unique to the string ~"""1"\' apparently obscure fact will play an important role when we discuss mUiltr~. amplitudes defined over surfaces with holes. Then these boundary ..v ..,"'~!:!: will determine what are called the "spin structures" of the manifold.)

3.2 Two-Dimensional Supersymmctry

109

,b.,d,w" C"~ construct the algebra generate. As before, the moments of these currents will . :: . closed algebra. The algebra for the NS sector is

D ,

= (m - n)L",+/I + g(m - m)8... _".

(L,.. L II )

[L "" 0.] = (im - r)G", ~"

10,. G,I = 2L r 'i"

(3.2.26)

+ 1D{r2 -

~)o,._,.

we have, in terms of oscillators in the NS sec lor.

I

=

~

2" ~

~

L :CL"a",+,,: + ~ L (r + 4m ) :b_rb",+, :, .. -oo ,-.-00 (3.2.27)

same steps for the R sector, we have

[L"" L,,] = (m - n)L .. +<

D ,

+ Sm

Ol/l._~'

[L"" F,, ) = (im - n)F"'H ' {F.. ,

F4 1=

2L",+"

(3.228)

+ ~Dm2o... _".

explicitly. we have

I

L,. =

00

00

L

2"

:«_,,0'.. +00:

+ 1 L (n + !m) :d.."dm+,, :.

~"-OC>

" .. -DO

(3.2.29) ~ ...d ,,!f1~, H,'mil"mi,,,, ,
H

=

-

rb~,b,,. + er:' p~.

(3.2.30)

L ..,na~"a"" + L_.,md~",d",,. +a' p~.

(3.2.31)

.. "'[

00

H=

-

L na~"a",. + L

,.". 1/2

_

"_. that this Hamiltonian i~ diagonal on the Fock space of bllIOlonic :" . This is cxtremely important, because ihis means thai we can easily . transition from path integrals to the harmonic oscillator formalisDl. ". for the theory becomes a simple integral over the exponential. performed .

110

3. Superstrings

u",IO>

FIGURE 3.1. Regge trajectories for the NS-R model for open strings. The h",","-!« states corespond to the Fock space generated by an possible prQducts of a+ an(I:~~ harmonic oscillators acting on the vacuum. The fermion states correspond ..",,:.U.Mo Fock space generated by all possible products of a+ and d+ oscillators actirigl:;l}1~ the vacuum, which is a space-time spinor, not a scalar. On the left, the mas~l!1i§ spin-l particle corresponds to the Maxwell or Yang-Mills field (all tachyon sWiim including the vacuum, can be eliminated when we impose the G;SO the right, the massless spin-1 fermion is the supersynunetric partner ofthe vector field. ( The states of the theory, as in the bosonic sector, are simply given Fock space of the product of these oscillators: NS eigenstates:

h,,:.1t6,;;s

nn nn

{a!.IlHb;n.o} 10) ,

nip. m.v

R eigenstates:

{a!,1l }{d~,v 10) ua ,

n,ll m,o

where u'" corresponds to an arbitrary spinor, which as yet has no res.tri(~tii'~~ placed on it. Some ofthe lowest lying states in the NS sector are as Fig. 3.1):

NS states:

R states:

2

= 2,

10) ,

k

tachyon

kJ1b~1/2 10) ,

e=l,

massless vector

a~l 10) ,

spin-! fermion

[0)

vacuum

{ {

Ua ,

spin-~ fermion

d~l 10) U a ,

spin-~ fermion

a~l 10} U a •

In summary, we find a remarkable similarity between the NS-R mctdel'!i:ilm the original bosonic string, In each case, we begin with an action, detin¢!::~~:::l symmetries, generate its currents from the symmetries, construct the .' bra from the currents, and then apply the constraints onto the Hilbert spa~~~ The main difference is the addition of the supercurrent, which gel1erates,::tj1ti superconformal algebra (3.2.26) and (3,2.28).

3.3 Trees

111

step in our strategy is then to apply this supcrconfonnal algebra Hilben space of the NS-R model. The ghost-killing conditions IlfC

Fn or Or 14') = 0 nand r . To establish the mass-shell conditiow, however, wc will investigate the tree amplitudes fo r the NS-R model.

. ,our sTarting point for the intemcting string meol}' is the functional Unfortunately, we have no guiding principle fo r the conslJ;Uctioo of ~e,."i' theory except for intuition. As a guess, let us multiply the usual term with a facto r of k/J. Vr~ at the poim at whieh a spio-O particle diab'T3ffi. Then a reasonable asslUuptioD is that the N -point scattering for this scalar particle is a generalization of(2.5.2) and is given by

A(t,2,3 ... ,N)=

L topolOf:;""

- L

lopolDg.ics

D", ~

Jd1).DXDVrfIki",vr~il:,.·X'~ f d,,(fr k"""""".X") , ;",1

IT IT N,(a, ,), II

(3.3.1)

(= I

(3.3.2)

".'

we arc now functionally integrating over an infinite sequence of ,,,,,,",variables. the bosonic functional integral, we can remove the functional integrals intermediate point along the string sheet because the Hamiltonian is on this space. Thu.~, by using

I = IX) ,"'1

f

DX D",

l.pl IXI

(3,),3)

intennediate pornt, we ciln remove all the functional integrals, leaving h,h"mo"i, oscillators. Thus, once again the functional integral permits of the barmonic oscillator formalism, which we vicw as only one representation of the functional integral. '~~i: language, the vertex for the emission of a scalar particle with :tl bc:comes

(33.4)

~;:~;~~~;'I,,:::~:;;';iO,~

::(

for choice

this is the rute thaI vertex functions must weight I, in order to preserve the ghost-killing conditions

•

112

3. Superstrings

(2.9.5). From (3.2.27) and (3.2.28), we can calculate the commutator with 1/11" and we find that it has confonnal weight ~. Since Vo has cOIltOI~~~ weight a' k 2 and 1/1 has weight the weight of V is the sum of the two:

!,

a'k 2 + -21 = 1

f(1 = _1_. 20"

-?

!.

where a' is equal to This vertex function, which corresponds to the ' or absorption of a tachyon, is guaranteed to satisfy the correct ghlost-killiiii§~ conditions. The propagator is easy to calculate. Notice that the Hamiltonian is di;jl~~ onal in the space of harmonic oscillators. Therefore, we will use an eXJ)l~!W representation of the functional based on the normal modes of the harm:Ii'~~W oscillator. Then we have

1

00

D=

e-r(Lo-l} dr

o

=

I . Lo - I

Let us constmct an N -point function of tachyons:

(

{O; kd kl . b l/2 V(k 2 )D ... V (kN-I)k N • b- 1/ 2 10; kN) • where we have placed the tachyon states to the left and right of all the vertJ~m and propagators. In this fonnalism, we can prove cyclic symmetry in the sa1iiliil way it was shown for the bosonic string. We first note: . V(k. y) hm 10; 0) = k . b_ 1/2 10; k) •

y~O

Y

lim (O; 01 y V(k, y) = (0; -kl k . b 1/2 •

y ~oo

This permits us to treat all external tachyons alike. By commuting the ,.I~~~ vertex to the left, we can show, in exact parallel to the bosonic case studle(t~m (2.6.24), that the amplitude is cyclic symmetric. At this point, however, we have a problem. The tachyon state that we Mi(~~ just constmcted corresponds to k/Lb~l!2 10; k), which satisfies

[Lo - l]kl'b~1/210;k} = 0

-?

a'k 2 =

!.

However, this means that the vacuum state 10; k) has even lower mass it satisfies the following conditions:

[Lo -1] IO;k)

=0

-?

a'k2 = 1.

We are faced, therefore, with the unusual problem that the we vacuum theory (3.3.10) does not correspond to the tachyon that we have' t cOlflstrilc:~m (3.3.9), In other words, the Hilbert space seems to be too large; the we is an unnecessary state. We seem to have two lowest lying states. The resolution to this puzzle comes from the fact that the vacuum indeed, is a redundant state that can be removed from the theory. This is accomplished by redefining the Hilbert space of the theory.

3.3 Trees

11 3

that we bave been working with is called the PI formalism clumsy one. The vacuum of the theory is not equal to the tsChYOIl, so

space is acrually larger than necessary. Although cyclic symmetry is :: : to prove in this formalism, we would like to introduce a more streamfonnalism, called F2 [10], which works with the reduced Fock sp.:1ce by

the vacuum state at k2 = ila'. accomplish this, let us rewrite the t!tchyon state as kJlb~ln IO;k) = G_1/ZIO;k},

(3.3. 11)

(0 ;/( 1kp bf/2 = (O;kl G 1f2 .

(3.3. 12)

N-point tachyon scattering ampJirudc (3.3.7) using

(3.3. 13 ) comes the crucial step. We will push the 0)/2 to the right successively

various vertices and propagators. We need the fannulas IC" V) = [L 2r • Vol = [Lo + r - J]Vo - Vo(Lo - 1), I

G I/l

I

Lo

-

1..0

. "son.li. 1 to notice that when we shove of the propagator .,--:-~

i GIn· G!1l

to the right, we change the

1 _ cc-=--,

Lo - l

(3.3. 14)

(3.3. 15)

Lo-!

the othcr tenns involving the L 's vanish as in the bosonic case. Finally. ,,"', p""h"d G 1/2 all the way to the right, where it vanishes on the tachyon (3.3. 16)

",,>Ulling all tenns together, we flnally have AN ::: (o; kl l

=

*1 . blp V(I0)

(0;kdY(k 2 )

1 !

La - '2

I L.

1

... V(kN_dk .... . h- 11 2 10; kN) .

.· ·Y(k N_dIO; k.v} .

(3.3. 17)

we have now completely rewritten the original amplitude sucb. has been shifted. In particular. the old va<.ouum state FI formal is.m at 0'k 2 = , hal'; vanisherl from the Hilbert l';pltce. 11 has ,"p.1,dl coml,letcly in the new fonna iism, whieh we call F2 (I OJ. Instead, we with the tachyon statc being represented by 10; k), which now satisfies >ow cOlodilion [Lo 10;..1:) = 0 with a'k2 = is quite remarkable. The state fO; k}, which used to represent the old "m "," at (1' k 2 = I , has now suddenly transfotmed into the tachyon state

V

t.

114

3. Superstrings

= !.

Thus, in the F2 formalism, the tachyon state and the new va(~u~~m at a' k 2 are the same particle. (y./e should be careful in noting that the same synl11jQru§§ 10; k) can represent either the old vacuum in the FJ formalism or the tachy(j~:§§ in the new F2 formalism.) This shifting of the Hilbert space is a novel feature not found in point l?;~i tide field theories. In fact, when we discuss the conformal field theory l1 next chapter, we will find that this strange "picture changing" phlenc)lru:~9im~ arises whenever we construct irreducible representations of the sU]:lerc:onJtO'J mal group. Hence, it is not a trick, but an essential feature of the group thp;,,,, (We should also mention that when we make the model space-time SU~ler~rvtJj~ metric, we will eliminate the tachyon state as well. Thus, the tachyon de(~Ou~t~~~ from the true superstring Hilbert space, leaving a unitary theory.) . Let us summarize the differences between these two formalisms: vertex = V, propagator = (Lo - 1)-1, tachyon:

kfJ.· b~I/2 10; k) (afe =

vacuum:

IO;k) (a f k2 = 1),

k),

(

vertex = V,

F2 :

I

propagator = (Lo tachyon = Vacuum:

krl, IO;k) (afe = ~).

The advantages and disadvantages of the two formalisms are as follows: (I) In the FI formalism, manifest cyclic symmetry is much easier to However, we must carry along the excess baggage of the vacuum which decouples completely. (2) In the F2 formalism, cyclic symmetry is obscure, but ghost and gauge transformations are easy to perform. The advantage is are now working in a smaller Fock space. Using the F2 formalism, it is not hard to evaluate the four-point furLci%9mm exactly: (0, -kd V(k2)

I I V(k3 ) 10; 1.4) Lo - '2

=

r(1 - a(s))r(l - a(t)) , r(l - a(s) - aCt)) .

(3.3:~91§1

wherea{s) = 1 +ais. There is also a straightforward generalization to the N -point function. ~W N -point function is represented as 1:

3.3 Trees ~.~I,

115

this expression contains many factors that are tedious to work out

method of obtaining the entire resuh is to usc the relation

-V(y) ~

.;;

f

dO exp{ik· X

+ 9k· "".;;).

(3.3.22)

that i f we power expand the exp onential. only the linear teno survives 'j"",,&",u,on, and we wind up with the previous cllpression for tbe vertex. at this point, done nothing. Now, leI us use two identities:

(01 "'"(,,) ,,"(y,) 10) ~

../Yi...!Yi I

V(y,) V(n)

../Yi ../Yi

10) =

f

."" y, - 12

dO, dfh. exp[k 1 . k1ln(y.

-)'2 -

91th)] (3.3.23)

bard to generalize the above relation to lhc N -point tachyon amplitude:

;~:;:~.~ '~:'~~:'~:'~~;~: is that We can now read off the various terms

[:1

the

by successively power expanding and integrating

Grassmann variables. to tachyon-tachyon scattering amplitudes, we can also calculate of massless vector particle!; (which correspond 10 Maxwell and

. The choice of the vertcx function for the massless vector by thl: mclthat it must hnve conformal weight 1 and thc spin. A natural choice for this vertex is

vcclorofthe vector particle amI kl = {·k = O. The of the vertex is equal 10 the sum of the conformal weights of factors. Since '" has conformal weight! and flU has conformal (J'k 2 , this means that the conformal weight of this vcrtex is given by

~+~+O= I , is the desired weight. This vcrtcx is guaranteed to satisfy the G and L conditions. It is not difficult to calculate the N -point scattering trus massless gauge particle us ing thc same fonnalism developed tachyon. For example, the scattcring amplitude for four massless gauge is givcn by

116

3. Superstrings

where the kinematic factor K is given by K =

+ SU/;23;14 + tU/;12;34) . + I s(k I4 k 32l;z4 + k23 k41 /;13 + k 13k 4Z /;Z3 + k24 k 31/;14)

-i(stl;I3;Z4 1

+ !t(k2I k 43 l;31 + k 34k 12 /;24 + k 24k 13 /;34 + k 3I k 42 l;lZ) + ~U(k12k43l;32 + k 34k 21 l;14 + k 14 k23 /;34 + k32 k 41 /;li), where

(The scattering involving fermions can also be calculated; they will also Mj!~ the basic form, except that the kinematic factor K will depend on the ext:eJ·m.«m spinors.) We should add that another advantage of working with the Pi t~~~~(~~~11 that we can explicitly show the invariance of the amplitude under the (

8Yi = eje, 8ej = e. This generates the group Osp(!, 2) (see Appendix), which is the ric generalization of the proj ective group SL(2, R). This group is geller:ate.ct:~~..:3. the algebra formed by the set G±I/2,

L±h

La.

Let Q be an element of the group OspO, 2). Then the proof of invariance can be shown by noting: Q 10; 0)

= 10; 0) .

(Notice that, as in the bosonic case, the vacuum state 10; 0) is not a nn'VSUJa state. Thus n, which does not in general kill physical states, can annihilate vacuum state.) The vertex rotates as nv(y, e)Q-l = V(yl, ( 1 ),

where V(y, e) is the vertex function before we integrate out over 9. Now that we have established the properties of the three-boson vertex tion in the Neveu---Schwarz formalism, the fermion-fermion-boson (with an external boson line) for the Ramond model can also be We choose V(k) =

where

r :e ik .X :,

3.4 Locnl Two-Dimensiona! Supersyrnmecry

117

Yll is the product of the Dirac matrices. In addition to this vertex.

we also need the propagator: D

F. = -FoI = -'-0 ,

(3 .3.3 1)

Fo is given in (3.2.20), We have chosen tbe intercept to be zero (which see is the proper choice in order to guarantee confonnal invariance; lit stage, however, we cannot yet motivate the choice afzero intercept.) the vacuum

Slate

is DOW a spin-~ fermion,. given by

LIO;q), u, .

(3 ,3,32)

o

we sum over the spinar indices a of the spinar 14",. The scattering for a fennien interacting with several bawns is now given by (3,3,33) we have only exhibited amplitudes with two external fermion lines. Out first guesses, based on simple intuition, have all been suethe action in Ule confonnal gauge was thm of a free thcory, model has been exceedingly simple. There is a price to be paid for principle. because lIle suing model can be fdetorized in any channel, hn"ld be possible to factorize the R model in thc meson channel Wld the NS model or to extract out the multifermion vertex functioll. this is quite difficult. In particular, the fennion vertex. function (with :X~::::~~fermion leg coupled 10 internal meson and fermion lines) is an ~ difficult object to work wi th. This, in tum, makes it difficult to multifermion amplitudes in the NS-R fonnulism. Ith'nuJ!b;', ""n be shown that the R model can be factorized in (he various to derive the NS model , the NS-R. formalism is ac ruaU y quite clumsy cak.-ulating multifermion amplilUdes. In the next chapter, we will see ... the techniques of conformal fi cld theory make the covariant calculation of .. amplirudes possible.

Local Two-Dimensional Supersymmetry

no< th'" "C imposed the foHowing conditions by band: (7;"")

(Jo )

= 0, = o.

(3.4.1)

was given. We merely appealed to the fact thallhcrc must elliSI i from which these constraints can be derived ab initio. will now describe the local gencraii7.3lion of the previous theory. The the construction of this locally sypcrsymmetric action is to add more

118

3. Superstrings

fields to the theory. In addition to the supersymmetric pair (X il ,1/ril)

we introduce a two-dimensional vierbein into the theory and supersymmetric partner: (e~,

X,,),

where Greek letters such as Cl and f3 label two-dimensional vectors in c·llii··Y~~ space, Greek letters like f.1, and v continue to label IO-dimensional soatce-4iii:.l1 vectors, Roman letters such as a, b, c label two-dimensional vectors space, where X" is a two-dimensional spinor as well as a tw.)-diffilens:i9~m vector, and where we have suppressed two-dimensional spinor ind_i.~~ce~s~~'::a~~mm both the vierbein and the spinor Xa have four components, as n supersymmetry. Let us now construct the complete action, fir!,t written d9V~ by Brink, Di Vecchia, Howe, Deser, and Zumino [11-13]: L = - - 14I Jg[g"P fJ"XilfJpXil - itil p"V" 1/rJ.l,

net

+ 2X"p{3 i1/rJ.l,ap

+ 4tJ.l,1/rJ.l,x"pP p"Xp]. Notice that the p matrices used above are actually defined in tw()-diffil~ns.iQ~~~ curved space because they are multiplied by the vierbein:

This action is invariant under 8XJ.I, = s1/rJ.l" 81/rJ.l, = -ip"s(a"Xil - tJ.l,X,,).

2DSUSY:

oe p = -2isp"Xp,

8Xa = V"s. It is also invariant under Weyl (scale) transformations: oX il = O.

Weyl:

o1/rJ.l, = -tu1/rl'. ueap = ue ap • ~

ox" = 4a x". It is also invariant, due to certain identities in two dimensions, under 0Xa

= ip"TJ.

oep = 01/r1' = 8X il

= O.

Finally, by construction, it is also manifestly invariant under local .tw.!ffi:: dimensional Lorentz transfonnations and reparametrizations because presence of the vierbein. (The presence of ordinary derivatives like aa utlt.fJ~::::

3.5 Quantization

n;::~,:O::f,:C,~U.r",:e~d~,,:~p;:a::c;~e,~covariant derivatives D" i

:-f.

. .-

119

is duc to the fact that

field vanishes in the action.) arc in a position to derive the constrain[S thaI we imposed on the hu hand. Notice that the variation ofthe action with respect to the

(3.4.8) .nat;''" of the vierbein's supersymmctric partner yields

p"pil8"X1,l/!IJ. - ~(..p.1/f)P"p/lXa = O.

(3.4.9)

the constraints that we promised earl ier, which are now derived as a of the variation of fields.

e"" = 8"If' (3.4.10) that there are enough symmetries on the X" field to eliminate it entirely, ,!o"~~'" of two-dimensional supc rsymmett}' (3.4.5) and the symmetry the constraints reduce to Tap = i:J,.XIlJtJX" .fa

+ ~iJiJp("afJl'Ir -

(trace) = 0,

= ~p~p"atJx!J,l/FjI =0,

(3.4.11) (3.4.12)

the parentheses symbolize taking one-half of the symmetrized sum). or course, are the constraints we originally introduced al the begiruting discussion in (3.2.7) and (3.2.13). Thus, we now have derived these from an invariant action, rathcr than simply imposing them OIl the . of the sySLCm.

Lorentz symmetry;

m'ns;on,d supcrsymmetry; IO-dimensionai space-time supersymmetry.

Quantization 'ti,,,.;m, o:f tl,d~S-R "";;;0' is a straightforward extension of the quantiof the bosonic string. The surprising feature is thai the highly coupled action becomes a free theory after choosing the gauge. . we will usc three methods: (1) Gupta-BlcuJer, (2) light cone, and

120

3. Superstrings

Gupta-B [euler The Gupta-Bleuler formalism is the simplest of the three and is the fonrrui)~~~ we have actually been using all this time. Using the large set ofsYInmletrie$:~~ the model, we can choose the conformal gauge:

Xa = 0, {

b

ers

~b

=

Da'

Notice that in this gauge the action reduces to the one that we have been with the provision that we impose Ln[¢)=O,

{ G, 14» = 0, = 0, {Fm 1\11) [\11) = 0,

NS:

Ln

R:

for positive n, m, and r. Thus, we have been quantizing the NS-R the Gupta-Bleuter formalism.

n ... tlU,I;J" lB,

Light Cone Quantization The essence of the light cone formalism is that we eliminate aU lllll)hy:si~im ghost states by solving the constraints (3.4.11) and (3.4.12) explicitly in ~~m!§ of the transverse or physical modes, rather than applying them onto "toot.. ,,: can impose: 1{f1

')iii~t1~i

= 0,

{ 1{f"'(a,

+ a" )X", =

0.

Most of the generators are only slightly changed. The real difference in the - components of the fields, which contain the transverse of the conformal generators: _

_

Cl n = Cln (X)

~ (

+ 2 1+

~

p '=- 00

1)

i i I aon,o

r - Zn :bn_rb,: -

2'-+-, p ..

where the first term on the right refers to the bosonic generator we found in (2.3 .16), and 1

b; =

+ p

D-2

00

LL

.

,

Cl~_sb~.

1=1 r=-oo

The troublesome Lorentz generator is the one that contains the -

M- I = Mol •

K- /

=

00

D-2

00

"~ " ' " 2-+ /~ ~

P m=-oo

j=l

+ K-1,

'=-00

j I _,b,j - bm_,b,). j i Cl_mCbm

3.5 Quantization

121

(3.5.&)

6n=n (D ;2)+~(2a_

D;2).

(3.5.9)

: Lorentz 10variWlce is achieved only with D = 10 and a = ~.

quantization begins with the calculation of the Faddooy-Popov deassociated with the confonna.! gauge. Because of the large set of of the aClion. we can enforce:

e: = 5:. (3.5.10)

Xa =0. • we must analyze the constraint when we make in (3.4.5) and (3.4.7)

11

variation of the (3.5. 11 )

the Faddccy-Popov determinant (1.6.10) associated with the gauge is given by (3.5.12)

, ::~():~ not contribute to the Faddecv-Popov ghost determinant.) The :'e deteoomanl associated with the Wlticommuting sector is thus (3.5. 13)

this is a bit tricky because of the number of degrees of freedom Xa. whieb is both a rwo-dimeosional spinor and a two-dimensional and thus has four components. b~~:(~ the e:lSiest way to calcu late the detennin,
!

Sll~ = ~

DfiDye- S-,

f

d

2

zfiay + C.c.,

(3.5. 14)

:. now thc ghost fie lds f3 and y are committing variables. (Although this . : looks similar to thc pn;vious ghost action (2.4.4) found for the bosonic . . is essential to point oul thaI these new ghOSt fields {J and y transform

122

3. Superstrings

differently under the conforma1 group than the b, c ghosts because they W~t:~~ derived from variations of a spinor XCI rather than the tensor gab. We will dlSICi.):$:it~ this further in the next chapter.) The complete action, therefore, is now the sum of the original action wim~~~ the sum over the anticommuting ghosts b, c and the commuting ghosts {3, We can easily calculate the energy-momentum tensor and "the SUI]er:syi;tifo:~~ metric current for the superconformal ghosts. Taking their moments, we aml\iti:~ at the super-Virasoro algebra: 00

00

L

L~l =

(!m + n) :{3m"--nYn:,

m.n=-OO

<Xl

Ft = -2

L

(m + n):bm_ncn: +

m,.n=-oo

<Xl

L

b-nYm+n

L

+

Un -

m) LnfJm+n'

m.n=~co

m,n=~oo

Similarly, the combined action, including the ghost contri\mtion, is·: invariant under a nilpotent transformation. The generator of this ... transformation is given by Q, where [14-16] 00

Q=

L

(L-ncn + F-nYn) -

1

L

2"

n=-co

+

00

(m - n):CmLnbm+n:

m.n=-oo

L (3n) ""2 + m LnfJ-mY",+n + L 00

<Xl

m.n=-oo

m,n=-oo

Y-mY-nb",+n(3

where the Land F generators are functions only of the (X and d OSC:lll~ltor'Sa~ we let Q2 = 0, then we arrive at

D = 10,

a={ oj

(NS),

(R).

The physical condition is now

Q Iphy) = O. Solving this, we arrive at the usual physical state conditions:

NS: R:

(Lo -

!) (41) = 0,

Gr

= 0,

{ 141) 141) = { F" 141) (Lo)

r > 0,

0,

= 0,

n > O.

Let us now calculate the anomaly contribution of these oscillators. find that the two contributions add to yield a vanishing anomaly, but 10 dimensions. The ghost contribution yields [F!h, F:h} = 2L;~n - 5m 2 •

3.6 GSO Projection the R sector, the anomaly contribution was 4Dm2

123

+ 2a. Thus, we have

~Dm2+2a-5m2=0

(3,5.22)

D = 10,

a = O.

(3.5.23)

for the NS sector, the anomaly is equal to

IGr . G s } = nr+z

+~-

5r2.

(3.5.24)

• rr,u" cancel against the anomaly coming from the bosonic sector: t DCr 2~ ."". Thus

(3.5.25) vanishes if (3.5.26)

Gsa Projection ie,::~:~e( at

.. the

for Ns-R model was not invariant under IO-dimensional supersymmetry. Although it is not obvious, the NS-R theory is invariant under lO·dimensional spacHime supersytnmctry if we make a truncation of the Fock space. To sec this, leI u.s first analyze tbe space of the ~tring. total number ofbosonie l':tatel': at the nth level is equal to the partition of ";~~~~:;;' p(n}. To see this. let US fu'St take the example of the fourth le number of states that are eigenfunctions of the Hamiltonian at level is five (a_I)4jO) ,

(a_d IO},

a_la_) jO), a-.-~

(a_da'"210),

10) .

(3.6.1)

that this is precisely the total number of ways in which the integer fOUI be broken up as the sum of other integers, i.e., the partition of four. In each state at the Ntb Icvel cm be represented as onc of the various (3.6.2)

R I~I = N I¢), M

N = ' " fI~'

,-,

L

' '

(3.6.3)

124

3. Superstrings

and R is the number operator for this space. In statistical mechanics, we often introduce the partition function that how many states there are at a given energy:

Z

=L

(nl

e- HT In) .

n

We find the partition function useful because it will enable us to number of states of the string:

N\1l1nf,::t'Ii;;m:;

Z = Tr(x R ),

x = e- r • The coefficient of x N in this power expansion is nothing but the partition ,0:',r.'l'y.,~ that is, p ( N). This trace can be performed in a number of ways. The si'mp!l¢~m is to introduce coherent states: ~

1=

f

(

dAdA*e- 1A1'IA) (AI.

Thus, the trace can be represented as Z =

!

dAdA*e- 1A1 ' (AlxR IA).

Performing the integral, we find

n(1 - r 00

Z =

xn

26

,

n=l

As a check, we can power expand this function (in one dimension) and ' that each coefficient of x N is equal to peN), i.e., 00

Z = LP(i)x

i

•

i=1

This function is related to a famous function in mathematics, the ffi4~~ Ramanujan function. In fact, some of the miraculous properties of the model are related to the identities of the Hardy--Ramanujan function. Next, we would like to compute the states of the NS and R sections , if they are supersymmetric. At the very least, we want the number of states to be equal in the two sectors, Notice that the NS-R Fock space actually divides up into two "... spaces, depending on whether a given state has an even or odd oscillators. Because the b oscillators are anti commuting, the total external lines must be even, because b oscillators must be contracted in Thus, we can define a "G-parity" operator that simply counts the b oscillators in a given state. It is -1 for an odd number and + 1 for number: " , U J .,,

3.6 GSO Projection

125

sector of the theory, this elimjllates (he tackyon particle as the lowest lying state, with states.

the R sector can also be reduced. Tfwe demand chat our lowest th(,m it has only 16 surviving modes. goes like this. A Dirac spinor in D dimensions has 2(I/1IV = components. Demanding that it be Majorana (real) reduces this halt: and demanding that it be Weyt reduces this number again by us now calculate the partition function that counts the total number

at each leveL For the NS sec lor, we have ro

iNS

L dim Vn.T~

=

(3.6.12)

R=

00

00

n_ l

r - 112

L na~an + L

rb:br.

(3.6.13)

can be evaluated exactly, leaving us with

m''''',the trace over the R sector becomes (3.6. 15) 00

R ::::

L n(a,;a

n

+ d:d~).

(3.6. 16)

n-" I

n(l 00

I, ~

8

"-,

'T"(I

+ x"J".

(3.6.17)

we might suspect that there is no relationship between these two . However, it was known to Jacobi as early as 1829 that these two ",ion,,,,,< precisely equivalent: iNS

=

JR.

(3.6.18)

of"ou"", i, nol a proof that the reduced NS-R model is supersymmetric. a necessary condition. This projection is called the GSa projection Glio,,"', Seherk. and Olive [17]) and it has a profoWldly important role in the superstring. (In Chapter 5, we will find the GSa projection is ~,," to modular invariance of the closed superstring one-loop amplitude.)

126

3. Superstrings

Thus, one of the great disadvantages of the NS-R approach is that sP~~~:~~1 time supersymmetry is very obscure. A second great disadvantage of the 'o.r:~:« model is that the vertex function for the emission of a fermion is very dltti~1;iat to work with and does not have nice properties under the Virasoro gauulg~~e;ls,d);:~§ a result, it took many years before two-fermion scattering amplitudes CI ' worked out. In the NS-R theory, the vertices for boson and ferniion C01JPl:iUg~; are [18-22]

t

where L is a Virasoro generator and W

= (OIBexp [.

\:;.

Iv

xexp(-~

2

f

(_l)m-r ( ;

m=O.r= 1/2

f

r.5=1/2

r+sC_1)r+s( r

S

r

~

t)d~tbw]

\.

2

-~!)( -4!)b~bs(J.IYllIUJ';;::;;:m 2

S

2

(3 .U:A"tr.:: where this vertex takes us from the bosonic Fock space to the fermionic FQI~w::m: space because we have a bosonic vacuum on the left and a fermionic val;uil~m on the right. UnfOltunately, this vertex has conformal spin ~, making it dlttiC~~fj~ to construct an amplitude that is conformally invariant. ~:;~~i Thus, although the NS-R model is appealing because it is essentially a fi. theory in the conformal gauge, the problems with the fermionic sector fo"x:~::m us to go to higher versions of the theory. Next, we will investigate the Schwarz (GS) action which is equivalent to the GSO projected NS-R the:qij@@ and is manifestly space-time supersymmetric.

3.7

Superstrings

Although the NS-R was simple and elegant, it displayed only n~rD;~~:~~~im~ world sheet supersynll11etry and not genuine lO-dimensiona! try. Let us now present the Green-Schwarz action, where the 10-dlim,em;ioi~~@ supersymmetry is manifest [23, 24]:

where

3.7 Superstrings -

~

127

1,2) afC genuine space-time fermion fi eld.. (rather

:ii~;~::~:' as in the Ns-R model). These 10-dimensional spinors, scalars in two dimensions, not two-component spinors. li ke the supen;.ymmetric point ptlrticle action (3. 1.1), is invariant global supersymmclry: 80" = e", oXIl = ie-A I I'D"'.

(3.7.3)

"""",m the proof, we have to make use of the identity I" ~'p1h l'I'Vtll = O.

(3.7.4)

"",,,his identity, we must make usc of a Fierz Ir3nsformation. It turns out identity holds only under the following conditions: = 3 and the fermions are Majorana;

= 4 and the fermions are Majorana or Wey!; = 6 and the femlions are Weyl; = 10 and the fcrmi ons ale both Majorana lIod Weyl. slaodard Dime representation is complex and exists in any space-time Thus, theyofhave half asmatrices many is one where as ~~~if'l~.:~Th~1e:~~:~;~~rep~,~osc~ntation the Dirac

;

componen~

only in D = 2, 3, 4 mod 8 dimensions. A matrices is onc whl..'I"C we have projected out components by the Wey! projection operfltor: t

;~;~l::~~~~~~~~i~::n~:~::;~~~;~:cven dimensions. And lastly. spinors Ih31 si are defined only in 2 mod 8 dimensions; D = 2 mod 8.

Majornna-Weyl spinor:

I"IP","'01 ""', we will ehoose Majorana-Wey! spinors in 10 dimensions. prove thllt the action (3.7. I) is locally supcrsymmctrie in 10 dimensions, first construct the foll owing projection operators, which project onto and self-dual pieces of two-dimensional vectors:

Pt =

~(g!>.8 ±t"~/.,fi).

projection operalor has the follow ing properties: ·, p" p.' P±.8p~*=*,

P±· , 8.8., p" =F = 0 , Ill!>

=

"

p!>.8/{1

-

K'· -- P·'K' + .8'

(3.7.5)

128

3. Superstrings

where the supersymmetric parameter is given by K Aaa , where A = 1 or:~:::;i;i: represents a two-dimensional vector index, and a is a spinorial mdlex<m::m: dimensions. We will usuaUy suppress the spinorial index. Notice that two equations state that K is self-dual (anti-self-dual) for A = 2 (1). Then the action can be shown to be invariant under '

8e A = 2ir . naKAc<,

ie

8XJL = A r li 8e A , 8(JggafJ) = -16Jg(P~Y KlfJ aye l

+ P~y K2fJ aye 2 ).

The great achievement of the as action is that it explicitly COlrltaing dimensional supersymmetry. Because the fundamental field is now a anticommuting space-time spinor (rather than the anticommu'fing field in the NS-R theory), we can explicitly construct the supersymmetric operator Qa. :' The great defect of the presentation, however, is that naive covariant r.·..;'i;;.~Z;; tization, as in the point particle case, does not exist. Once again, as in for the point particle case, we must face the fact that the quantization are nonlinear because lfA

8L.

= -[)lit A

~ lrlia"Xjte

A

+ ....

As a consequence, the commutation relations between fields are highly #.;&..~~~ linear. Worse, we find that the commutation relations are actual.ly pf()POlrtj()~~ijj~ to the inverse of the constraints; i.e., they probably don't exist [2, "'....-.:J'" As a result, we will be forced to use light cone quantization.

3.8

Light Cone Quantization of the GS Action

The counting of the number of independent degrees of freedom within thii~lli~ spinors is important when we fix the gauge. Normally, Dirac spinors .... n-:,Wfi dimensions CD even) have 2(1/2)D complex components. Thus, the spinors in 10 dimensions have 32 + 32 complex components. However, when. restrict them to being Majorana spinors, we are left with half as many: 32..,..'.'""z real components. Restricting them to being Weyl spinors further reduces the~~2% down by half once again., to 16 + 16 real components. Choosing the light c',:t"~]~ gauge will reduce the number of independent components down further to 8· real components. Lastly, when we go on-shell and impose the Dirac eqllat(Qn::=:::;:~ on these spinors, the number of independent components goes down by hatJ~:~11~ again, to eight. But this is precisely the number of components nec;es:sar;~~:§ to form a supermultiplet with the eight bosonic components of the string Thus, we have precisely the correct number of components necessary to satist:I(;~:~~ supersymmetry on-shell. If N is the number of components in the spinors fj;·t:=:::@:

3.8 light Cone Quantization of the GS Action

N

Dirac;

129

= 32 + 32 complex components,

+ 32 real componeQls. == 16 + 16 real components, = 8 + 8 real components,

N = 32

Majorana: Majorana--Weyl :

N

Light Cone:

N

N = 8 real components.

On-shell :

~~~~':!::~.~Ihi:::·s. reduction down to the light cone gauge. We will choose r+61.l = 0, r:l = rl/l(rO

±

r 9).

(J.8. 1)

these gamma matrices satisfy

(r+)' ~ (r-)' ~ 0

(3.8.2)

'C:;~, ::~I exactly half of the original 16 + 16 components of the spinors J:r . The fiDal light cone action becomes remarkably simple in this

s=

24ifQ"

J

dd dr(OaX i

au Xi -

i spi>as),

(3.8.3)

we mnke the substitution

..;p+fJ _

S.

(3.8.4)

,. .,."nl',1 poi nt is to observe that all the Complicated oonlinear lenns in that prevented a simple covariant quantization of the superstring have ~~::Pb~~a;:;~;., (Notice that a curious phenomenon has also occurred. i n the ,'11 action, l and 0 2 in two dimensions were scalars, independem of other. Now. in the light conc gauge, thesc two independent scalars have into a single two-dimensional spinor.) .,,",,",''" is exceptionally simple because the system reduces to that particles (while the covariant theory was intractibly coupled). The mti,oos of motion are those of free strings:

•

e

(aT

+a,,)sld =

0,

(8, - 8.)S'" ~ O.

(3.8.5)

",,,,,muta"'on relations are (3.8.6) '''''''' . we can have, as beforc, sevcral boundary condition!; on these fields. an open s tring (Twe J). wc bave the following boundary conditions:

I

SI,,{O, r)

= Sht(O. r).

'''' S"(1r,r)=S (If .!).

(3.8.7)

130

3. Superstrings

Notice that these two string fields have the same SO(8) chirality. because only these relations are compatible with a global sUI)efi>ynam~m~ transformation that equates the two supersymmetry parameters . This that the open Type I superstring has only N = I supersymmetry. the signs in the previous equation, yielding opposite chiralities, would supersymmetry totally impossible.) The normal mode expansion thus

L 00

SIII(a,

T)

=

2- 1/ 2

S~e-ill(T-a\

n=-OO

L 00

S2a(u,

r)

=

2- 1/ 2

S;e- ill (T+al,

It=-oo

where

IS:, S!} =

(jlIb(jn._m.

For the closed string (Type II), however, we actually have two options; the fields can either be chiral or not. Closed strings are, by periodic in sigma, which yields the following normal mode expansion: 00

L L

S11lI e-2in(r-a) '

n=-co 00

S2a(a, .) =

S~e-2i,,(<+a).

n=-oo

If these two fields have different chiralities, then they are called Type they have the same chiralities as in Type I strings, they are called Type. I.e.,

Type I = { Type ITA = {

open and closed string, same chiralities, closed string, opposite chiralities,

closed string, Type lIB = { same chiralities. The spectrum of the light cone superstring is especially appealing be(:al,i·~~m the theory is on-shell supersymmetric, meaning that we should iI'nmled:iat~~l~~ see all particle states arranged such that the helicity states of the bm;ti~;~::::::~ match the number of fermions. The mass of each particle is deterrnin.ed: Jj~~w.' the Hamiltonian: 00

N

= a:'(massi =

L(a:~IIa:~ n=]

+ nS~"S~).

].8 Light CODe QlJlI.ntization of lhe GS Action

131

I. the ground slate of the theory consists o f the massless vector and ils spinar partner. Let Ii) represent the eight physical transverse ~:;::~ of the maMiless vector field. Then the spinor partner la) can be

" .,

(3.8.13) normalize our states as (iU}=Oij.

{alb) = Hhy+)d, ; Ii) = s:(SOYi+)" In) ,

(3.8.14)

is the Weyl projection opemtor. (We will use the notatioD !.hal is equal 10 the product of gamma m3trices summed over ull IU",tiO'" in the indices. The normalization is such that y n = y 1y1.) were to quantize the lO-dimensional super Maxwell theory in the light '8,ug", then the supcrsymmctric pair (All' '/fa) would correspond to

I

supersymmelric multiplet

Ai ~ Ii) ,

1/ta

~

In}.

(3.8.15)

the light cone theory reproduces the super-Ma;(well theory at the lowest the next level, we have 128 boson and 128 fermion states:

128 bosons; 128 fennions;

I (

a ~ 1 U) -+ 64 states, ~J

Ib}

-+

64 states.

a~1 Ib) -> 64 states. ~lli) -+ 64 states.

(3.8. 16) (3.8. 17)

N = 2 level, we have 11 52 bosons lind nn equal number offemllons:

a~I Q:~1 Ik)

-+ 288 states,

~I~~I Ii} -+ 224 states,

1152 bosons:

a~llj) -+ 64 stales. a ~lS~ 1 Ib)

-+

~2Ib) -+

64 slales,

Q: ~1~!

-+

U)

~2Ij) -+-

1152 fermions:

a~IO:~J la)

512 stales. 512slales.

64 states, -+-

288 stales,

a~2Ia) -+ 64 states,

s.'. I S~ 1 Ie) -+- 224 Slates,

(3.8. 18)

132

3. Superstrings

This procedure can be repeated at the next level, where we have 1 states. Not surprisingly, we can also regroup these massive multiplets ac(;¢.t1tm ing to 0(9). This is because massless D-dimensional supergravity, w~iW:::: compactified, yields D - 1 massive states. Thus, the N = 1 sector, .,,'dm~ 128 bosons and 128 fennions, can be regrouped as 44 + 84 bosons reducible 0(9) representations, and a single spin-~ 128 multiplet fennions. At the N = 2 level, we have the bosons being regrouped 9 + 36 + 126 + 156 + 23 I + 594 representations of 0(9), while the ferrnl~imi can be reassembled in 16 + 128 + 432 + 576. For the Type II closed string, the spectrum becomes even more ~:!~~:l~t1!IiOO because we obtain supergravity at the massless level, with 128 b9sons and"';;;;.'-1~ fennions: ~

SUSY multlplet:

128 Bosons Ii} Ij} ; la} Ib} , { 128 Fennions Ii) la) ; la} Ii} ,

where the first state refers to the S oscillators and the second to the S If the fermions have the opposite handedness (Type IIA), then this the N = 2, D = IO-dimensional reduction of ordinary N = 1, D supergravity. However, if the fennions have the same handedness, then there are c9'!)~::::: plications. This theory contains a fourth-rank antisymmetric tensor dimensions, with 35 independent components in eight dimensions. It has shown that no covariant action for such a particle exists! Thus, we unusual situation where a supersymmetric action does not exist for this tor. The theory, of course, is still well defined. The light cone theory S-matrix elements are explicitly calcUlable. However, the S-matrix, speaking, does not seem to arise from a covariant action [33]. Finally, if we take the restriction to symmetrized states of the Type I mel9fYi we obtain N = I, D = 10 supergravity. Let us summarize the zero slope of these theories:

= 10 super-Yang-Mills,

Type I ---+ N = I,

D

Type 1---+ N = I,

D = 10 supergravity,

Type IIA ---+ N = 2,

D = 10 supergravity,

Type lIB ---+ nonexistent. At higher and higher levels, it becomes prohibitive to continue this of the spectrum to show supersymmetry. We can prove, however, that the spectrum, at arbitrarily high orders, is supersymmetric simply by that supersymmetry generators exist with commutation relations that with the Hamiltonian. We will now show this by explicitly supersyrnmetry generators to all orders.

18 Light Cone Quantizatioll of the GS Action

133

light cone gauge, the two supersymmetry transformations (3.7.6) and become quite simple: 8S' = (2p+/,2'1 a ,

18X

i

=

0,

&5" =

- j

1

Jjj+pt:<8aXiY~Eci.

SXi = (p+rl !2Y~ili ';S".

(3.8.21)

, ~:!:~f~;,n;:;,;. ~~~o,~~a representation of the y matrices. There arc three 8 ~ The first is a vector representarion 8v which we denote

~

~iD~d~':~X:i~;.~T~h~':O~!h~'~'~:IW;~o~~~~~~~i~ boththespinorial. the fortile other.are Thus. y matrix We bas use indices

: ric,.) We can find an explicit fonn for the two supersymmetry

,"C .. in terms of the fie lds that reproduces the transformations given in

QIi =-

(p+)-'l2 v,,~. L

ro

'"' <'" I ~ "_nan"

(3.& 22)

3 straightfor.vard t85k 10 calculate the anticommutation relations

these generators (3.8.23)

H =

~ I f:[a~~a~+n~nS:J + !P:) . P

(3.8.24)

,,=J

'pc<",f,h" the light cone theory is Lorentz invariant is now expanded to tbe the superstring is super-Poinca.rC invariant. In addition to the usual mulal'''', we must show that (3.8 .25) re,~~~~~~~,~~::spinor indices in 10 dimensions. Again, the only difficult the - component of the oscillators:

in

(3.8.26)

till;,ul! commutator has an extra piece:

(3.8.27)

132

3. Superstrings

This procedure can be repeated at the next level, where we states. Not surprisingly, we can also regroup these massive multiplets . ing to 0(9). This is because massless D-dimensional compactified, yields D - 1 massive states. Thus, the N = 1 secton 128 bosons and 128 fermions, can be regrouped as 44 + 84 om,onR:' reducible 0(9) representations, and a single spin-~ 128 multiplet fermions. At the N = 2 level, we have the bosons being rpo'rl"\111'\p,i"I 9 + 36 + 126 + 156 + 231 + 594 representations of 0(9), while the can be reassembled in 16 + 128 + 432 + 576. For the Type II closed string, the spectrum becomes even more' because we obtain supergravity at the massless level, with 128 bosons fermions: SUSY multiplet:

128 Bosons Ii) Ij) ; la) Ib) , { 128 Fermions Ii) la); la) Ii),

where the first state refers to the S oscillators and the second to the Sosc:Hll If the fermions have the opposite handedness (Type IIA), then this rp1'\"'" the N = 2, D = lO-dimensional reduction of ordinary N = 1, D supergravity. However, if the fermions have the same handedness, then there p1ications. This theory contains a fourth-rank antisymmetric dimensions, with 35 independent components in eight dimensions. It shown that no covariant action for such a particle exists! Thus, we unusual situation where a supersymmetric action does not exist for tor. The theory, of course, is still well defined. The light cone theory S-matrix elements are explicitly calculable. However, the S-matrix, speaking, does not seem to arise from a covariant action [33]. Finally, if we take the restriction to symmetrized states of the Type I we obtain N = 1, D = 10 supergravity. Let us summarize the zero slope of these theories: Type I --+ N = 1,

D = 10 super-Yang-Mills,

Type 1--+ N = 1,

D = 10 supergravity,

Type IIA --+ N = 2,

D = 10 supergravity,

Type lIB --+ nonexistent. At higher and higher levels, it becomes prohibitive to continue this of the spectrum to show supersymmetry. We can prove, however, that the spectrum, at arbitrarily high orders, is supersymmetric simply by that supersymmetry generators exist with commutation relations that with the Hamiltonian. We will now show this by explicitly "al"UJ,aUILlE supersymmetry generators to all orders.

3.8 Light Cone Quantization of the OS Action

133

cone gauge, the two supersymmetrytransformations (3.7.6) and be<~rr'e
l

I

oS' = (2p+F2'1'"

oX' = o. 8S' = -i,fii+paiJaX IY~citci,

oX' =

(p+)-1/2y~"EaSa.

(3.8.21)

for a representation of the y matrices. There are three 8

ofSO(8). The first is a vector representation 8v which we denote i. The other two representations are both spinorial. We use the

for onc representation and il tor the other. Thus, the y matrix has indices "foon ,,,: yj".) We can find an explicit form for the two supersymmetry in tenns of the fields that reproduces the transfonnations given in

00

Q~

= (p+) - t/2 v,~U L '" ~

<'
;

" - ~Q'n'

(3.8.22)

a straightforward task to calculate the anlicommutation relations

en I~e" generators (Q", Q'J ~ 2p+,",

I

lQ", QUI = .J2y~~pl,

(3.8.23)

!Q~, Q6) = 2Hf/,iJ,

H =

~ 1 :t[a~nQ'~ + nS~nS:l + !pf j . P

(3.8.24)

n., 1

,"of,t", 'lh, light cone theory is Lorenlz invariant is now expanded to the the superstring is super-Poincare invariant. In addition to the usual we must show that

""to,;,

(3.8.25)

f3 represent spinor indices in 10 dimensions. Again, the only difficult io~'",involves the - component of the oscillators:

LY; =

21+ p

f

"'--00

(O'~_'",a~ + (m -

tn)S;:_mS:)'

(3.8.26)

IIk'olt commutator has an extra piece: (3.8.27)

134

3. Superstrings

Because the theory is already uniquely defined to be in lO dimensions, [M- i , M-j] = O.

Furthermore, we also find the commutator for the super-Poincare CTr{"\'''''''': [ M-i , Qa]

-i i Qil = -./2 Yail .

This complete the proof that the spectrum of the GS model is lU-d.lmceu$; supersymmetric to allievels~

3.9

Vertices and Trees

We now turn to the question of the interactions. We will have to rely on educated guesses as to the nature ofthe vertex functions, but demanding vertices transform under supersymmetry places stringent conditions final form. Having established space-time supersymmetry for the sUj:'erS let us now show that this restriction is sufficient to allow us to ('{"\r, "tri.;";; trees. We will demand that the supersymmetry operators take us from vertices to boson ones and vice versa. This will impose an enormous . . .. constraints on our theory, which will, in effect, uniquely determine the" itself. We demand that, under a supersymmetric transformation generated Q's, the vertices VF and VB transform into each other [ltQa, VF(u,k)] [ltQa, VB(S',k)] [SilQil, VF(u,k)] [suQu, VB(~,k)]

rv

VB(~,k),

rv

VF(ii,k),

rv rv

VB(~,k), VF(u,k),

where S stands for the polarization tensor of the massless vector and U stands for its Majorana-Weyl spinor partner. Notice that are complicated by the fact that a supersymmetric transformation generates rotations in space-time, so that the sand u' inust also properly into ~, ~, and ii, u as we make this supersymmetric A natural assumption is that these vertices can be expressed as VB(S, k) =

s . B,eik-X,

[ VF(u, k) = uFe'k-X,

for some vector Band spinor F. Remarkably, when all the details are out in the light cone gauge, we can satisfy all these conditions for '''Ull-'1\'':~ of Band F. We begin by first calculating the rotated spinors and polarization To calculate how the polarization and the spinor transform, we take

3.9 Venices and Trees

13 5

,mllO,,,n! of the supersymmetric generators and see how they transform At the zeroth level, we find Q~

_ (2p+)If2 ~. (p+)~I f2 (y{ pl )....

Q" -;.

s:.

(3 .9.3)

zeroth components transform the polarization tensor imo Majorana-

pinors. Le: us define . ' Q'l u ) ~

Ii), . ' Q' ln ~ I' ), ," Q"ln ~ luI. e~ QIl

(3.9.4)

lu) = Ii).

(3.9.3) inlO (3.9.4) and solve for the transfonned spinors and vec-

is now a simple matter to show that the rotated spinors and polarization

f

= (yl)aal)"U", ii~ = ,tk+(yi{ i)",j. U.! = ~f

I

~

~(EYijik',j, -

1

...Ii

.,

(t"v -u) r ""

(3 .9.5)

.fi . . +s"u"k', k+

we insen 0.9.5) into (3.9. 1), we find the solution for the 8 and the F B+ = p+, 8 1 = Xi _ Rll k },

r

r

= (2p·· yl/2 [(y. XS>" + Hy iSi Rij :ki], = (p+ / 2)112 S".

(3.9.6)

(3.9.7)

'"" n',,!,." .""""IC>V"."'rt;'" and propagators to obt..'lin trees. The great of this approach is tbat we can calculate multifennion amplitudes the same ease with which we calculate multiboson amplitudes. """""e, the propagator and the amplitude for the scattering of fermions are given by

D

~

I--,f: a~"a~

+ nS'!.."s,; + ~p2 ) -'

AN = (0; kd V(k 2)D· . . V(kN _I) 10; kH) ,

(3.9.8) (3.9.9)

128

3. Superstrings

where the supersymmetric parameter is given by K Aaa , where A = 1 or:~:::;i;i: represents a two-dimensional vector index, and a is a spinorial mdlex<m::m: dimensions. We will usuaUy suppress the spinorial index. Notice that two equations state that K is self-dual (anti-self-dual) for A = 2 (1). Then the action can be shown to be invariant under '

8e A = 2ir . naKAc<,

ie

8XJL = A r li 8e A , 8(JggafJ) = -16Jg(P~Y KlfJ aye l

+ P~y K2fJ aye 2 ).

The great achievement of the as action is that it explicitly COlrltaing dimensional supersymmetry. Because the fundamental field is now a anticommuting space-time spinor (rather than the anticommu'fing field in the NS-R theory), we can explicitly construct the supersymmetric operator Qa. :' The great defect of the presentation, however, is that naive covariant r.·..;'i;;.~Z;; tization, as in the point particle case, does not exist. Once again, as in for the point particle case, we must face the fact that the quantization are nonlinear because lfA

8L.

= -[)lit A

~ lrlia"Xjte

A

+ ....

As a consequence, the commutation relations between fields are highly #.;&..~~~ linear. Worse, we find that the commutation relations are actual.ly pf()POlrtj()~~ijj~ to the inverse of the constraints; i.e., they probably don't exist [2, "'....-.:J'" As a result, we will be forced to use light cone quantization.

3.8

Light Cone Quantization of the GS Action

The counting of the number of independent degrees of freedom within thii~lli~ spinors is important when we fix the gauge. Normally, Dirac spinors .... n-:,Wfi dimensions CD even) have 2(1/2)D complex components. Thus, the spinors in 10 dimensions have 32 + 32 complex components. However, when. restrict them to being Majorana spinors, we are left with half as many: 32..,..'.'""z real components. Restricting them to being Weyl spinors further reduces the~~2% down by half once again., to 16 + 16 real components. Choosing the light c',:t"~]~ gauge will reduce the number of independent components down further to 8· real components. Lastly, when we go on-shell and impose the Dirac eqllat(Qn::=:::;:~ on these spinors, the number of independent components goes down by hatJ~:~11~ again, to eight. But this is precisely the number of components nec;es:sar;~~:§ to form a supermultiplet with the eight bosonic components of the string Thus, we have precisely the correct number of components necessary to satist:I(;~:~~ supersymmetry on-shell. If N is the number of components in the spinors fj;·t:=:::@:

3.8 light Cone Quantization of the GS Action

N

Dirac;

129

= 32 + 32 complex components,

+ 32 real componeQls. == 16 + 16 real components, = 8 + 8 real components,

N = 32

Majorana: Majorana--Weyl :

N

Light Cone:

N

N = 8 real components.

On-shell :

~~~~':!::~.~Ihi:::·s. reduction down to the light cone gauge. We will choose r+61.l = 0, r:l = rl/l(rO

±

r 9).

(J.8. 1)

these gamma matrices satisfy

(r+)' ~ (r-)' ~ 0

(3.8.2)

'C:;~, ::~I exactly half of the original 16 + 16 components of the spinors J:r . The fiDal light cone action becomes remarkably simple in this

s=

24ifQ"

J

dd dr(OaX i

au Xi -

i spi>as),

(3.8.3)

we mnke the substitution

..;p+fJ _

S.

(3.8.4)

,. .,."nl',1 poi nt is to observe that all the Complicated oonlinear lenns in that prevented a simple covariant quantization of the superstring have ~~::Pb~~a;:;~;., (Notice that a curious phenomenon has also occurred. i n the ,'11 action, l and 0 2 in two dimensions were scalars, independem of other. Now. in the light conc gauge, thesc two independent scalars have into a single two-dimensional spinor.) .,,",,",''" is exceptionally simple because the system reduces to that particles (while the covariant theory was intractibly coupled). The mti,oos of motion are those of free strings:

•

e

(aT

+a,,)sld =

0,

(8, - 8.)S'" ~ O.

(3.8.5)

",,,,,muta"'on relations are (3.8.6) '''''''' . we can have, as beforc, sevcral boundary condition!; on these fields. an open s tring (Twe J). wc bave the following boundary conditions:

I

SI,,{O, r)

= Sht(O. r).

'''' S"(1r,r)=S (If .!).

(3.8.7)

130

3. Superstrings

Notice that these two string fields have the same SO(8) chirality. because only these relations are compatible with a global sUI)efi>ynam~m~ transformation that equates the two supersymmetry parameters . This that the open Type I superstring has only N = I supersymmetry. the signs in the previous equation, yielding opposite chiralities, would supersymmetry totally impossible.) The normal mode expansion thus

L 00

SIII(a,

T)

=

2- 1/ 2

S~e-ill(T-a\

n=-OO

L 00

S2a(u,

r)

=

2- 1/ 2

S;e- ill (T+al,

It=-oo

where

IS:, S!} =

(jlIb(jn._m.

For the closed string (Type II), however, we actually have two options; the fields can either be chiral or not. Closed strings are, by periodic in sigma, which yields the following normal mode expansion: 00

L L

S11lI e-2in(r-a) '

n=-co 00

S2a(a, .) =

S~e-2i,,(<+a).

n=-oo

If these two fields have different chiralities, then they are called Type they have the same chiralities as in Type I strings, they are called Type. I.e.,

Type I = { Type ITA = {

open and closed string, same chiralities, closed string, opposite chiralities,

closed string, Type lIB = { same chiralities. The spectrum of the light cone superstring is especially appealing be(:al,i·~~m the theory is on-shell supersymmetric, meaning that we should iI'nmled:iat~~l~~ see all particle states arranged such that the helicity states of the bm;ti~;~::::::~ match the number of fermions. The mass of each particle is deterrnin.ed: Jj~~w.' the Hamiltonian: 00

N

= a:'(massi =

L(a:~IIa:~ n=]

+ nS~"S~).

].8 Light CODe QlJlI.ntization of lhe GS Action

131

I. the ground slate of the theory consists o f the massless vector and ils spinar partner. Let Ii) represent the eight physical transverse ~:;::~ of the maMiless vector field. Then the spinor partner la) can be

" .,

(3.8.13) normalize our states as (iU}=Oij.

{alb) = Hhy+)d, ; Ii) = s:(SOYi+)" In) ,

(3.8.14)

is the Weyl projection opemtor. (We will use the notatioD !.hal is equal 10 the product of gamma m3trices summed over ull IU",tiO'" in the indices. The normalization is such that y n = y 1y1.) were to quantize the lO-dimensional super Maxwell theory in the light '8,ug", then the supcrsymmctric pair (All' '/fa) would correspond to

I

supersymmelric multiplet

Ai ~ Ii) ,

1/ta

~

In}.

(3.8.15)

the light cone theory reproduces the super-Ma;(well theory at the lowest the next level, we have 128 boson and 128 fermion states:

128 bosons; 128 fennions;

I (

a ~ 1 U) -+ 64 states, ~J

Ib}

-+

64 states.

a~1 Ib) -> 64 states. ~lli) -+ 64 states.

(3.8. 16) (3.8. 17)

N = 2 level, we have 11 52 bosons lind nn equal number offemllons:

a~I Q:~1 Ik)

-+ 288 states,

~I~~I Ii} -+ 224 states,

1152 bosons:

a~llj) -+ 64 stales. a ~lS~ 1 Ib)

-+

~2Ib) -+

64 slales,

Q: ~1~!

-+

U)

~2Ij) -+-

1152 fermions:

a~IO:~J la)

512 stales. 512slales.

64 states, -+-

288 stales,

a~2Ia) -+ 64 states,

s.'. I S~ 1 Ie) -+- 224 Slates,

(3.8. 18)

132

3. Superstrings

This procedure can be repeated at the next level, where we have 1 states. Not surprisingly, we can also regroup these massive multiplets ac(;¢.t1tm ing to 0(9). This is because massless D-dimensional supergravity, w~iW:::: compactified, yields D - 1 massive states. Thus, the N = 1 sector, .,,'dm~ 128 bosons and 128 fennions, can be regrouped as 44 + 84 bosons reducible 0(9) representations, and a single spin-~ 128 multiplet fennions. At the N = 2 level, we have the bosons being regrouped 9 + 36 + 126 + 156 + 23 I + 594 representations of 0(9), while the ferrnl~imi can be reassembled in 16 + 128 + 432 + 576. For the Type II closed string, the spectrum becomes even more ~:!~~:l~t1!IiOO because we obtain supergravity at the massless level, with 128 b9sons and"';;;;.'-1~ fennions: ~

SUSY multlplet:

128 Bosons Ii} Ij} ; la} Ib} , { 128 Fennions Ii) la) ; la} Ii} ,

where the first state refers to the S oscillators and the second to the S If the fermions have the opposite handedness (Type IIA), then this the N = 2, D = IO-dimensional reduction of ordinary N = 1, D supergravity. However, if the fennions have the same handedness, then there are c9'!)~::::: plications. This theory contains a fourth-rank antisymmetric tensor dimensions, with 35 independent components in eight dimensions. It has shown that no covariant action for such a particle exists! Thus, we unusual situation where a supersymmetric action does not exist for this tor. The theory, of course, is still well defined. The light cone theory S-matrix elements are explicitly calcUlable. However, the S-matrix, speaking, does not seem to arise from a covariant action [33]. Finally, if we take the restriction to symmetrized states of the Type I mel9fYi we obtain N = I, D = 10 supergravity. Let us summarize the zero slope of these theories:

= 10 super-Yang-Mills,

Type I ---+ N = I,

D

Type 1---+ N = I,

D = 10 supergravity,

Type IIA ---+ N = 2,

D = 10 supergravity,

Type lIB ---+ nonexistent. At higher and higher levels, it becomes prohibitive to continue this of the spectrum to show supersymmetry. We can prove, however, that the spectrum, at arbitrarily high orders, is supersymmetric simply by that supersymmetry generators exist with commutation relations that with the Hamiltonian. We will now show this by explicitly supersyrnmetry generators to all orders.

18 Light Cone Quantizatioll of the GS Action

133

light cone gauge, the two supersymmetry transformations (3.7.6) and become quite simple: 8S' = (2p+/,2'1 a ,

18X

i

=

0,

&5" =

- j

1

Jjj+pt:<8aXiY~Eci.

SXi = (p+rl !2Y~ili ';S".

(3.8.21)

, ~:!:~f~;,n;:;,;. ~~~o,~~a representation of the y matrices. There arc three 8 ~ The first is a vector representarion 8v which we denote

~

~iD~d~':~X:i~;.~T~h~':O~!h~'~'~:IW;~o~~~~~~~i~ boththespinorial. the fortile other.are Thus. y matrix We bas use indices

: ric,.) We can find an explicit fonn for the two supersymmetry

,"C .. in terms of the fie lds that reproduces the transformations given in

QIi =-

(p+)-'l2 v,,~. L

ro

'"' <'" I ~ "_nan"

(3.& 22)

3 straightfor.vard t85k 10 calculate the anticommutation relations

these generators (3.8.23)

H =

~ I f:[a~~a~+n~nS:J + !P:) . P

(3.8.24)

,,=J

'pc<",f,h" the light cone theory is Lorentz invariant is now expanded to tbe the superstring is super-Poinca.rC invariant. In addition to the usual mulal'''', we must show that (3.8 .25) re,~~~~~~~,~~::spinor indices in 10 dimensions. Again, the only difficult the - component of the oscillators:

in

(3.8.26)

till;,ul! commutator has an extra piece:

(3.8.27)

136

3. Superstrings

where the various vertices can be either fermionic or bosonic. The amplitude, for example, is equal to A4

=

r(-!s)r(-!t) K

I

I'

r(l - zS - zt)

where K is a matrix that gives us the appropriate spin structure for tude. If we let u represent a spinor and ~ the polarization of 1l1
+ k4)v~4)' + !s(UtrllrVr).U3)~41l(k2 + k3)vb, -!S(U2rllu3)(Utr IlU4) + !t(UtrIlU2)(U4r IlU3)

K (UI' ~2, U3, ~4) = - !t(Ut rllrVr).U3)~21l(k3 K(ut, U2, U3, U4) =

As we mentioned before, the basic form (3.9.10) for the massless vector scattering amplitudes is the same. The only difference COlnel~; factor K.

3.10

Summary

In summary, there are two equivalent versions of the superstring relative advantages and disadvantages of the N~R and the GS actlOnSi follows: (l) The N~R action is linear and easily quantized, both p(),mrj"nth) in the light cone gauge. Trees and loops for bosons are ,,<>c.h';,', structed. It is manifestly invariant under two-dimensional sUI,en;ynlt However, these great advantages must be balanced against the fact dimensional space-time supersymmetry is very obscur~, to say Because it is based on anticommuting vector fields 1/1Il' there are with spin statistics, which requires a GSO projection. FwthfmIll()t~ vertex for fermion emission is prohibitively difficult to use, thus multifermion amplitude calculations beyond practical reach. (2) The GS action is manifestly supersymmetric in 10 dimensions at levels. It is based on genuine space-time spinors, not' . fields. However, the action is highly nonlinear, and a covariant 1l11
-1 L = - 4,y'g[gufJauxllafJxll -

net

+ Fir," 1/I 1l xupfJ pU XfJ]'

-

i1/l ll pUV u1/l," + 2 Xup fJ pU 1/1'" afJ

3.10 Swnmary

137

is invariant under

oXiJ. :::: s1/l", 2D SUSY:

81/1" = -p"e(8"X" -lPl'x,,),

81 =

(3. 10.2)

-2isp o Xr;,

ox,,:::: V"t, as Weyl, reparametrization, and local two-dimensional Lorentz invarioftbis formalism is that the energy-momentum tensor and current generate an algebra, the super-Virasoro algebra, to explicitly eliminate all ghosts in the theory. We do not have to ~~,:;:.!.~a.uge constraints by hand, as in the case of the supcrconformally strategy for quanlization will be the same one used for the bosonic We first isolate . action, extract the algebra from to eliminate the ghost states and obtain '-+ Symmetries ·-+ Current -+ Algebra -+ Constraints

--+ Unitarity.

~;~~~: :th~':;v>,~'~erbein and the X field, we get the NS-R tbeory in the (3 . 10.3)

of the invariancc of the original action, which is locally super, we can construct the following currents and set them equal to

10 = ~/Pd1/l1'&bXI" Toh :::: 8aX I' &hX"

(3. 10.4)

i _

i -

+ 41/1" p"8h1/l,, + 41/11'Ph8a1/l1'

- (Trace). (3.10.5)

calculate the momenls of these consttainls, they fonn the algebra: [L",. L it ]:::: (m - n)L", ..."

NS:

D

+ S(m

,

- m)o",._n,

[L m • G, J:::: (-~m - r)G",+"

{G" C s }

::::

2LrH

+ ~ D(r 2 -

[L .. , Ln] = (m - n)L",+n

D

~)O,._S,

+ 8m

,

[L"" Fn]:::: {~m - n)F",+n.

0",._,., (3.10.6)

{F",. f~l = 2L,.+n + ~Dm2o,"._",

the bosonic case, it is straightforward to make the transition from path to the hannonic oscillator formalism. The Hamiltonian is diagonal

138

3. Superstrings

in the Fock space of harmonic oscillators, so we can remove all i'Iltelrtnc functional integrations. We find that the tachyon vertex function with weight I can be \XT1'itt"n kJl1/fJ.i Vo.

However, we find that there are two distinct ways or "pictures" in can write down the N -point amplitude, called the FJ and Fz tbrmall1slm rules for forming the N -point functions are ' vertex = V, I propagator = - - Lo - I tachyon = kJl . b- 1/2 10; k) ,a'k 2 =

4,

2

a'k = 1,

vacuum = 10;k), vertex = V, propagator =

I J '

Lo - 2 tachyon = vacuum = 10; k) ,

2 "/k ... -

The advantages and disadvantages of the two formalisms are as

!2' lUlJlUW

(I) In the Fl formalism, manifest cyclic symmetry is much easier to However, we must carry along the excess baggage of the which decouples completely in the theory. (2) In the F2 formalism, cyclic symmetry is obscure, but ghost is easy to perform. The advantage of the F2 picture is that we working in a smaller Fock space.

,",UJ""""J

The N -point function is represented as

An = jd fL N (0;01 V(k1,Yl), ... , V(kN,YN) Yl

YN

=1 0 ;0).

Although the NS-R model with GSa projection can be shown to same number of fermion and boson states, the proof that the theory is space-time supersymmetric is prohibitively difficult. As a result, we space-time supersymmetry by postulating the new GS action, where genuine space-time spinors. The GS action is S=

4a'~

j

daar{Jgg"fJ IT" . ITfJ

- 2s"fJ81rJ.io"eI82rJ.iofJe2},

where

+ 2is"fJ o"xJ.i(8 1r J.iOfJe 1 -

2 8 r J.i .

(3. '

References

139

'action is invarianl under

S9 A = e A ,

sx" = 8(J'" = 2; r

ii A r"9'\

(3.10.13)

. n...K"" •

ax" = iO"r

ll

8l/ .",

8(Ji8 ..~) = - 16Jg(p:ri'fJ oy 8'

+ p:rI(2fJ(jy02),

(3.10. 14)

~~,:~;~n~~,:~~~~:S~~S, formulation is that it is spare-time supersymmetric

.u levels.

the price we pay for this is that covariant quantization

. theory is nOl0riously difficult, because it is a coupled system even al the Therefore, as we will go to the light cone gauge, where the theory a linear theory:

S=

2411"0"

J

du dr(J"XiaO Xi - i Sp"abS).

(3. 10. 15)

",,,ion is also spaco-timc supersymmetric, with generators given by Q~

= (2p+),n SO. 00

i Q'" = (p+)-I f1".,. r , " ""... L ... _~a:".

(3 . 10.16)

Intiico,,,"wt,"i,," relations between these generators are given by

I

IQ"Q'I~2P+""

IQ", QIi ) =

../2y:iJp i,

(3. 10. 17)

IQ', Q'I ~ 2H"'.

(3. 10.18) ,we will investigate the conformal field theory, which combi nes the best of both the NS-R and formalisms.

as

J, L. Gervai s and B. Sakila, Nucl. Phys. 834, 632 (1971).

W. Siegel, PhYi. Lett. 1288,397 (1983); Nue!. Phys. 8263, 93 (\985); Classical ua.,;m Oral'if)' 2, L95 (1985). Brink and J. H. Schwarz, Phys. Leu. 1008, 3 10 ( 1981). P. Ramond, Phys. Rev. D3, 2415 (197 1).

140 [5] [6] [7] [8] [9] [10] [II] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

3. Superstrings

A. Neveu and 1. H. Schwarz, Nucl. Phys. B31, 86 ( 1 9 7 1 ) . . > / A. Neveu, J. H. Schwarz, and C. B. Thorn, Nucl. Phys. B31, 529 (1971······· 1. Wess and B. Zumino, Nucl. Phys. B70, 39 (1974). See also Y. A. Gol'fand and E. P. Likhtman, JETP Lett. 13,323 (197 Volkov and V. P. Akulov, JETP Lett. 16, 621 (1972) (English, p. 438). M. A. Virasoro, unpublished. Y. Aharonov, A. Casher, and L. Susskind, Phys. Rev. D5, 988 (1972). L. Brink, P. Di Vecchia, and P. Howe, Phys. Lett. 65B, 471 (1976). S. Deser and B. Zumino, Phys. Lett. 65B, 369 (1976). For an earlier attempt, see Y. Iwasaki and K. Kikkawa, Phys. Rev. (1973). N. Ohta, Phys. Rev. D33, 1681 (1986). M. Ito, T. Morozumi, S. Nojiri, and S. Uehara, Progr. Theoret. Phys. (1986). 1. Schwarz, Suppl. Progr. Theoret. Phys. 86,70 (1986). F. Gliozzi, J. Scherk, and D. Olive, Nucl. Phys. B122, 253 (1977). C. B. Thorn, Phys. Rev. D4, 1112 (1971). L. Brink, D. Olive, C. Rebbi, and 1. Scherk, Phys. Lett. 45B, 379 (1 S. Mandelstam, Phys. Lett. 46B, 447 (1973). ,. J. H. Schwarz and C. C. Wu, Phys. Lett. 47B, 453 (1973). D. Bruce, E. Corrigan, and D. Olive, Nuc!. Phys. B95, 427 (1975). M. Green and J. H. Schwarz, Phys. Lett. 136B, 367 (1984). M. Green and J. H. Schwarz, Nucl. Phys. B198, 252, 441 (1982). T. Hori and K. Kamimura, Progr. Theoret. Phys. 73,476 (1985). M. Kaku and J. Lykken, in Symposium on Anomalies, Geometry, and (edited by W. A. Bardeen and A. R. White), World Scientific, Singap10re See also 1. Bengtsson and M. Cederwall, Goteborg preprint 84-21. T. 1. Allen, Cal Tech preprint CALT-68--I373. T. Hori, K. Kamimura, and M. Tatewaki, Phys. Lett. 185B, 367 (I C. Crnkovic, Phys. Lett. 173B, 429 (1986). R. E. Kallosh, Pis 'ma JETPh 45, 365 (1987); Phys. Lett. 195B, 369 (1 1. A. Batalin, R. E. Kallosh, and A. Van Proeyen, KUL-TF-87117. N. Marcus and J. H. Schwarz, Phys. Lett. 115B, 111 (1982).

4

nformal Field Theory Kac-Moody Algebras

Confonnal Field Theory the mysteries of the superstring is the existence of cwo ways of forthe theory: the first is the NS-R model (after the Gsa projection)

~

:~;~~~I:~(~~'V~'::'~'O;":~', the second is town he GSdistinct modeladvantages with genuine . Eacb ~"'~~d formulation has its and discuss theconfonnal field theory, which to sec the dynamic link between the two formalisms. The confonnal

. theory of Friedan and Shenker [1] combines the best feature s of both . . Conformal field theory allows us to:

""xl""

covariant anticommuting spinor fields based entirely on free

The GS formalism, on the other hand, is based on complicated

in",,,,,;','g fields that make covariant quantization exceedingly difficult.

Construct explicit covariant tree gmphs for multi fermion scattering. 10 NS-R formalism. however. this is prohibitively difficult because it is ":~~~:~,to introduce complicated projection operators that extract out tI . The confonna! field theory replaces these awkward projection operators with free Faddccv- Popov ghosts, which are easy to manipulate. and NS-R formulations and see their relation, ~::~'7.I~~: between the ~,~ . . It provides the bridge by which we can interpret the resulls of one formulati on in tenns of the other:

as

conformal field theory

++- {

as model. NS-Rmodel.

142

4. Conformal Field Theory and Kac-Moody Algebras

(4) Construct the covariant supersymmetry generators. This is ImpOi,Slljli the NS-R formalism, and is possible in the GS formalism only in cone gauge. (5) Describe both the NS and R sectors of the theory with the same . . . . .. rather than using the clumsy formalism of two distinct Hilbert on the vacua IO)NS and IO)R U a . This is accomplished by a process bosonization, i.e., constructing fermions out ofbosons in two' ..

There is a small price we must pay, however, for confolmal field Ghosts and antighosts proliferate quite rapidly in this formalism, p"r"'ii" for superstrings, and a strange phenomenon called "picture changing" introduced. Fortunately, these ghosts and antighosts are free fields, easy to manipulate. Furthermore, conformal field theory does not Setlm derived from any action, as in the GS and NS-R formalisms. Coni()mll:ll theory stresses the group-theoretic behavior of the fields, rather than nrC"'''t from an action. Therefore, we suspect that there is a higher, yet uU~'1i'''V' first quantized action that exists beyond the GS and NS-R actions. (We shoul4 stress that conformal field theory is not a field theory in of second quantization, i.e., where we start with the formalism of :SC1IWl Tomonaga, and Feynman. The second quantized field theory of strings discussed in Part II of this book.) The essence of conformal field theory is that it stresses the use of invariance alone to calculate correlation functions between different 6]. It is quite remarkable that conformal invariance by itself is determine almost completely the structure of N -point scattering amlPU For example, in (2.6.9) we have encountered operators that have the matrix elements:

where the function f can be a power or a logarithm. For example, the element of tree amplitudes between two bosonic strings is (X(w)X(z))

~

log(w - z),

while the matrix element of two normal ordered vertices is (elkX(w)e-lkX(z)) ~

(w _

zrk2.

So far, we have used explicit representations of the fields in order to the matrix elements. However, the process can also be reversed. If all about a field's group properties, we should be able to calculate elements and even reconstruct the field. The essential idea behind conformal field theory is to use the properties of the field ¢ to completely determine all of its matrix elelnel1 even reconstruct the field. In conformal field theory, this is acc:oITlpllst

4.1 Confonna1 Field Theory

14]

the short-distance behavior ofleft-moving or right-moving fields (4.1.2)

all possible matrix elemenLS can be calculated from the oflhefields themselves. field theory, we also construct a spinfield Sa that transfonns as space-time Lorentz transfonnations and has eonfannal

. However, coming from the Faddeev-Popov ghost sector, we find yet j,,,-fie.ld 'with ,on 'o=a""ight ~ . It is the product of these two fields, one field and the other being a ghost field, that allows us to construct ~~r!~e~~ fermion vertex with the proper weight one. This vertex., although ~\

exponentials of fields, is defined totall y in tenns of free fields and solves the problem of constructing a simple fermion vertex function.

which tbis spin field is actually introduced is through the process i " [7,81. i.e., creating a femti OD out of a boson. II is easy, to create a boson out ofrwo fennions. However (in two dimensions) the option of being able to create fermions out ofbosons, which was thought to be impossible. Ac(utllly. the hint of how to "bosonize" was given in the previous chapter, when we introduced the venex function (4.1.3) (2.,6.22) we saw that (4.1.4) ti~'':;;~:~bY choosing various values of the momenta in the exponential, we

create operators that satisfY the relation

Ii

(4.1.5) words, we can create a field with fennionic commutation relations ,fl,,,oniici,,,me.ni, oscillators. The key to this construction is the nonnal of oscillator modes in two dimensions. This feature does not carry .• -,~ . fie ld theories. (In hindsight. it is possible to see wby and bosons are so closely linked in two dimensions but nOI in higher For the Lorentz group O( I, I), which has only one generator, tile called "spin" in one spatial dimension does not have much meaning.) method of bosonization via nonnal ordering of fields will also be the ~ ,onst""cting the confonnal field theory. We will construct the fermion Sa via normal ordering of exponentials of bosonic fields. From this, ;11 ,w"truct the covariant $upersymmeuy operator. The advanlage of this we can now discuss spaco--time spinors using one common vacuum bosons and fennions and that the entire construction is based on free

' ' ' m".

we discuss the superconformal case, let us begin our discussion by conformal field theory. Let us make the most general conformal

144

4. Confonnal Field Theory and Kac-Moody Algebras

transfonnation on the world sheet variable z:

z --+ z(z). Under this conformal transfonnation, we say that a primary analytic transfonns with conformal weight h if it transfonns as .

¢(z) =

¢(Z) (dZ)h dz

(Secondary fields transfonn as derivatives of ¢(z).) This conformal the same concept we introduced back in (2.7.6) when we were dlsCU~fSj conformal properties of vertices. We see now that mathematically mal weight is an index to label irreducible representations of the group generated by the Virasoro algebra. We can now construct objects that are invariant under a transfonnation: ¢(z) dz h = ¢(Z) dz h. Once we have defined how fields transfonn under a conformal trilJ1St()lti:l we mustthen check that we have closure of the group under two such operations. Let us say that we make two successive confonnal traJlstl:)11l:

z --+ ZI(Z)

--+ Z2[ZI(Z)].

Then the field transforms as V1¢(z)V]- 1 = ¢(z]) (dZI)h dZ ]

2

V2V]¢(z)Vi Vi = ¢[Z2(Z](Z))]

'

(dZ 2 dZ])h dz] dz

Thus, the confonnal transfonnations fonn a group with the CO]TIPoslti, V3 = V2V], Z3

= Z2(Z] (z)).

The closure of the algebra can most easily be seen if we take an imal conformal transfonnation. Consider an infinitesimal vru'i/'lticm coordinate: 8z = s(z). Under the transformation (4.1.7), we have the infinitesimal traJlstlJrII 8¢e = [sa

(We will abbreviate az by arrive at

a.)

+ h(as)]¢.

If we commute two such variations,

4.1 Confonnal Field Theory

[El. e2l = £13£2 -

£2a£l.

145

(4.1.1 6)

this relation with (2.1.30).) shown once again that the group closes with arbitrary weight h. understanding of the meaning afthe conformal weight h, let liS conformal weight a f the string. Under the infinitesimal variation we use the chain rule 10 show that the string transforms as

(4.1.17) the string field was weight O. Likewise, il is easy to show thaI the of the string field has weight 1:

S3X .. (z) = rea

+ (3e)J3X,,(z).

(4.1.18)

swnmarize the weights of some common string fie lds: Field

Weight

x ax ax· ax

o

,

(4.1.19)

2

this means thai the energy-momcntum tensor or the Virasoro have 2. . . The product of two fields of weights h) and h2 produces + h2 at the same point: ¢(hd(Z)¢(/!l\z:) = ¢(h l +~lJ (z:).

(4.1.20)

the important fac t, which w ill be used throughout thi s book, that the of an objcct of weight I is an invariant:

8

f

l,b(ll(z)dz = fr vi.l¢ + (i.lv)4>Jdz

=

f al'¢ld,

=0.

(4.1.2 1)

use this fac t in constructing vertex operators and also tbe action of the q"',"';;'od theory. us investigate in detail how the energy-momentum tensor acts on

,Ic,~:~~~: of the theory.

e

is written in tenns of X,,(z. z),

s=

_, 2rr

f

d 2zaX 1'i.lX" ,

(4.1.22)

'''''~!)'-ID''m''nn"," tensor in (1.9 .17) associated with this action is

Ts = - ~ax"axl"

(4.1.23)

146

4. Confonnal Field Theory and Kao--Moody Algebras

The transformation of fields under conformal transformations is by the integrated energy-momentum tensor parametrized by a small

s:

1,

Te = _1_.

ko

21T I

dzs(z)T(z),

where we take the line integral that encircles the origin in z-space. portance of the energy-momentum tensor T is that it is the
[2~i

dzs(z)T(z), ¢(Z2)]

1,

= _1_.

k

21T I

1 = -2.

dzPs(z)T(z)¢(Z2) O.2 -Co

1,

k2

1T I

=

to

Ps(z)T(Z)¢(Z2)

~ 1, 21T1I

k2 dzs(z)T(z)¢(Z2).

Notice that the curve CO.2 encircles both the origin and the point Z2, the curve C2 encircles only the point Z2, because

C2 = CO•2 - Co (see Fig. 4.1). Notice also that we have adopted radial ordering operators in the complex plane, where the operators are ordered accord their distance from the origin:

Izl

<

IzIl·

In the last step, notice that we have dropped radial ordering, because ordered products are analytic. From before, we also know that 8¢ = [sa

+ h(as)]¢.

By comparing the two expressions (4.1.25) and (4.1.28) for 8¢, we read off the small-distance behavior of the field ¢: _ h _ 1 T(z)¢(w, w) rv ( )2 ¢(w, w) + aw¢ + .... .

z-w

,.-.

z-w

Since the energy-momentum tensor itself has weight 2, we can into the short-distance equation. Thus, we derive the short-distance of the generators themselves: 1

c

T(z)T(w) = -2 (z-w )4

2

1

~

LU .... " .....

+ (z-w )2 T(w)+ (z-w ) awT(w)+···. _

4.1 Confonnal Field Theory

147

4.1 . Contours for confonnal field theory integrations. In the first diagram, lies between two concentric circles. In the second diagram, the direction contour has been reversed and merged with the outer contour, forming path. The poin! z lies within the closed path, but the origin does not. eo,'ndte,m '~n the right-hand side oflhe equation shows that the generators transfonn as weight 2 confonnal fields . ., us now introduce nonnal modes and make the link with the oscillator We can always decompose any field 1/r of weight h as follows: 00

L

y,(z) ~

,-'-'y,•.

(4.1.31)

. -~

,Ik,w, us to decompose the energy-momentum into normal modes . We

J

L n= r

dz ~+l T() z,

21fiz

00

T( z) ~

L

z-·-'L •.

(4.1.32)

be instructive to actually derive, stcp by step, the Virasoro algebra from '~~,~;,:'::x~p:,:e'SSions to see the equivalence of commutators and operator

c!

' . Using (4.1.30) and (4.1.32), we find

-f -

dwwn+l 21fi

f -z

dz ~+I 2Jri

148

4. Conformal Field Theory and Kac-Moody Algebras .

X

=

!c 2

[ (z - W)4

f :~

+

n 1 w + \ (n

2

(z -

T(w) W)2

+

1 ·

(z - w)

+ l)w n2T(w) + wn+1awT

+ !c ;, wn- 2(n + l)n(n = ,[ dW.\(2n j 21C1

" ]

aw T

I)

j

+ 2)wn+m+iT(w) _ (m + n + 2)w n+m+1

+ ~n(n + l)(n 12

I)Wm+n-1j .

This finally reduces down to the usual Virasoro algebra:

[Lm' Lnl = (m - n)Lm+n

c 3 + 12(m -

m)om,-n.

Inserting the expression for Ln into the line integral, we also obtain

[Ln, ¢(z)l = (zn+la

+ h(n + l)zn}¢(z) . .

(This can also be derived from (4.1.28).) We have shown that this description is equivalent to the usual one in commutators that we studied in previous chapters. In fact, the previous . is nothing but (2.7.6), which we used to calculate the conformal vertex function. .."",.". Not only can we express the usual conformal commutators in this ,.•..••' . we can also express the Faddeev-Popov ghost contribution in the . gauge in this equivalent language. The Faddeev-Popov ghosts will different representation of the Virasoro algebra based on band c rather than the string variable X. If we write (2.4.1), we have

ogzz = \lzosz, 8g zz = \lzoSi' These constraints, in turn, allow us to calculate the change in the measure. A simple calculation of the Jacobian of the transformation

Dg"Dg zz = (det \lz)(det \lz)DszDs z. To exponentiate the determinant into the action, let us introduce fields, b and c. Then the ghost action (2.4.4) can be written as Sgh

=

~

f

2

d z(b"azc'

+ bzzazcZ).

(Note that we have written the ghost action such that the tensorial the fields is manifest under the conformal group. In this way, the in""r,.,~r /

4.1 Confonnal Field Theory

149

under conformal transformations is transparent. Thus, [he field b"

~~~.i;::~~:::' while c' is a first-rank tensor.)

Ie

.

action, we can construct the energy-momentum tensor associated Tgt. = ca,b

+ 2(a,c)b.

(4.1.38)

fields have the following matrix element: (b,,(z)c'(w)) ~ ---'-z-w carefully calculate the !ihort-rli!itance hehllvinr nfrhese fields, we

-13 2 I T (z_w)4+{z_wF gt.(Z)+ z_waTgh{z)+·· ·. (4.1.39) factor of -13 in the anomaly calculation for the ghost energytensor. (fwe take the sum of these two energy-momentum tensors, and (4.1.38), we find that the combined tensor has a vanishing anomaly T(Z) ~ Tx(z)

+ T",(,).

I D - 26 2 (z W)4

T (z)T(w) - -

+ ....

(4.1.40)

that the sum of the two energy-momentum tensors has zero central in 26 dimensions. This, in fact, fixes the dimension of space-time To summ.arize, the bosonic representation of the energy-momentum of ",·io. fields X yields D for the anomaly term, while the

~:~::~,~~~nt~:!o yields -26. The

'-

sum

of these two tensors is the true tensor, which has vanishing central term only if 0 = 26. we have a Lie group, the first question to ask is: What are its Let us now say a few words about the representations of the

or SU(3), for example, we know that the representations can be by using the familiar "ladder operators" to construct triplets, octets, representations. For a general Lie algebra, we can construct repby taking all possible products of "raising operators" acting on weight" vacuum. The set of all such states created by the raising is called the "universal enveloping algebra." The same process can for the Virasoro algebra, treating L_n as ladder or raising operators . • ill de'no the highest weight veclOr lh) as follows:

Lo Ih)

~

h Ihl.

Ln Jh) = 0,

where n > O.

(4.1.41)

150

4. Conformal Field Theory and Kac-Moody Algebras

Notice that any physical state 1R) with h = 1 is a highest weignt-.,p("t"" it satisfies (Lo -1) 1R) = O. Notice that the vacuum state 10; k) is also a ) weight vector, but that 10; 0) is not. Now define the set of states generated by all "raising" operators L_ n i:,:: on the highest weight vector:

Iw) =

L),.1 L),.2 ••• -nl -n2

eN

-nN

Ih).

The set of all such states is called a Verma module. If 1h) is considered a state, then the Verma module associated with it is the set of all that we can generate from that real state. Notice that, under an arbitrary conformal transformation, thill state transforms into another member of the Verma module: Q Iw) = Iw'),

where

This is because a Virasoro generator that hits an element of a Verma simply creates another state within the same Verma module. The Verma therefore forms a representation of the group. In addition to the highest weight vector, let us also define the invariant vacuum, which we introduced earlier: L±l 10; 0) = 0,

Lo 10; 0) =

o.

The relationship between these two different types of vacua is Ih) = ¢(O) 10; 0) , where ¢(O) is a primary field of weight h. This is easy to check if we sides of the equation with the Virasoro generators.

4.2

Superconformal Field Theory

Now let us discuss the more complex question of writing the su]:)er'COI1J field theory. Instead of writing down the transformation of a field as a of a complex variable, we write down the most arbitrary trallls:Domllati pair of variables: ,

z = z(z, 0), where 0 is a Grassmalllll variable. The most arbitrary transformation pair is given by i(z) = (z(z, 0), e(z, 0)).

4.2 Superconformal Field Theory

15 1

disappointing that this transformation is much too general. In fact, it close properly if we take the product of two such transformations. to impose a canslraint on the system in order to make the group of properly. define the supersymmetric derivative as

,fo,"""ti<,., 01,,,,,

D=

a a ae +Baz'

(4.2.3)

(4.2.4) lei us calculate how this supersymmetric derivative transfonns under a (4.2.5)

oorn1",,',',m law is also disappointing because it is highly nonlinear. We thc conformal case, a transformation of the derivalive to be linear. "'0''', we simply eliminate the nonlinear terms by imposing the constraints D ~ (DO)O, Dz -800= 0.

(4.2.6)

i linearize the transformation of the reparametrizstion, the constraints we want. By imposing them, we can show that s uperconformal transformations close properly:

- -

z_z--+z.

(4.2.7)

: summary, we say a field transformation is superconfonnaJ if it satisfies (4.2.8) to check that if we eliminate the (j components, this expression to the conformal equation written earlier in (4.1.7). the previous section, we can write this down as an infinitesimal Oil the fields:

o,¢ = [Ea

+ h De)D + hCfJe)]¢,

(4.2.9)

parametrizes the superconformal transformalions. Notice that this is exactly identical to the bosonic transfonnation (4.1.28) except term thai has two D's in it. The closure of the algebra can now be

E

0[0,.',1 = [0",3,,],

(4.2. 10)

152

4. Confonnal Field Theory and Kac-Moody Algebras

where

[.0], .02]

= S]aS 2 - S2aS] + !(Dsd(Ds2).

Notice that we reduce back to the usual Virasoro case (4.1.16) when Ds] and DS2 equal to zero. We find that the Grassmann parametrization can be written in tenus parameters, az = Sa and ae = Sb:

s(z) = 8z + e8e = Sa + esb, 8z = .0 - e8e = Sa + !esb, 8e

= !Ds = !(Sb + e8sa).

The previous equation is a specific solution of (4.2.6), so we know group closes properly when we make successive transfonnations on 'U~'"", This solution will be the basis on which we will construct the sUI'~n;olllf
S = _1 jd2zdedBnXJ1.DXw

2n

The equations of motion are

DDXJ1. = 0 whose solution is

Thus we choose

XJ1.(z, e) = XJ1.(z)

+ e1/lJ1.(z)

so that the action can be written as

S = _1 j d 2z(aXJ1. aXJ1. -1/Ia1/l -liralir).

2n

Notice that this is nothing but the original NS-R (3.2.1) action written in mal language. From this action, we can read off the sUI,er·"enerf~y--m()mleqw. tensor with a slightly different nonnalization

T = h

+ eTB =

-!DXJ1.aXw

Written out explicitly, this is

h

= -!1/IJ1. ax J1.,

TB = -!axJ1.aXJ1. - !a1/lJ1.1/Iw This is nothing but the currents (3.2.7) and (3.2.13) written in corlfmma guage. Note that the supercurrent fa is now written as part ofthe same tpl1,oit\j

4.2 SuperconfonnaJ Field Theory

153

y-;morrrenrumlen,,,, To, which means that they are supersymmetric each other. in (4.1.25), the variation of a superfield is given by

1. dz s(zl)¢(Z2). I 1:1

O¢(Z2) = _'-. 2Jl"

(4.2.20)

l

field ¢ has weight Ii and the energy-momentum tensor has weight ~, and T transform under a superconfonnal transformation as

8 12 T(zd¢(Z2) '" hT¢('ld + -

! D2¢ + 0-a '2 2¢ +"',

:::,2

l12

C I

T(ZI)T(Z2) ....... - - 3

ll2

38]2

+ - 2 T (22)

4 :::12 2Z12 I I 812 + - - D2 T(22) + -3zT(Z2) +"', 2 ZI2 ll2

(4.2.21)

19 12 = 81 - (h.

=

ZJ2

(4.2.22)

ZI - ZI - fllfh .

ctuall·v. these relations are a deceptively compact way of expressing a large ofe~".,!,ons . To see this, let us expand oul the previous expression. omg ,·, ..•.. 18) into (4.2.2 1), we find

/.",,)18(2:2)"'( )TF(zz)-(

3,,/4 ZI:::2

ZI

).+(

(ZI

<:1 - Z2

, ')ZTF(zz) + <:2

c/4

)Tp (Z2) '"

2

Z2)~

+

T )28(:::2)+ I

Zl

! 2

ZI - Z2

<:2

Zl

~h+""

T B(Z2)

l2

82T8 + ··· . (4.2.23)

+ ... .

"",Ialjo,•• when expanded out, yield the superconformal algebra thai we out explicitly in (3.2.26) and (3.2.2&). Thus, these equations contain a ,id,,,,bl, amount of information. field transformation in (4.2.21) can be wrineD out in detail if ¢ ()¢1.

, TF (zd4>tJ(zz) ""' ~''-:-¢I(Z2) ZI

Z2

+ ... ,

(4.2.24)

154

4. Confonnal Field Theory and Kac-Moody Algebras

Ifwe power expand as

n

n

A, ( ) _ "~z -n-h-I/2A, 'l'1.n.

'1'1 Z -

n

then we obtain the transformation of q>(Z. e) under a transformation:

[L m•
+ h(m + l)zm
[sCm.
+ ~)hZm-I /2
[L m•
= [(h -

~)m - n]
[sCm.
c = -2s(6A(A - 1) + I). where s equals + 1 (-1) for Fermi (Bose) statistics. Summarizing this, Field

Anomaly

X",

D !D 2

1/11'

b,c {J.Y

-26 11

4.3 Spin Fields

155

og ,t he", together, we find that the total anomaly contribution is

D+ tD - 26+ 11 ""

hD -

(4.2.30)

10)

==- lOin order to cancel this term.

Spin Fields almost nothing is new. We have only redecived old results that could obtained with the hannonil; oscillator fonnalism of the previous . We have simply chosen to rewrite the generators ofthe NS-R algebra that stresses the confonnal weight and the ?--piane singularity rather than expanding them out in tenns of their

moments. What is new, however, will be the introduction of a new that transforms as a genuine spinor under the Lorentz group. This will us to rewrite the NS-R model with only one vacuum, rather than two,

'P"""", called bosonization. us write down the generators

PJ ~

of the lO-dimensional Lorentz group

terms afthe anticonunuting vector field 1/1 found in the NS--R theory: (4.3.1)

de
anticommutallon relations of the 1/1 field given in (3.2.23), we can

J" "(z)j'''{w) '""

,- w

+

TI'''' j"T(w)(1 -

I (z - w)2

J,t

(IJu.r Il"u - JJ,

H

..-+

\!)( J v)

0' ++

+ ....

r)

(4.3.2)

that the generators of this algebra are functions of <: and hence form a I than that of SOC I O). In fact, this algebra is called a Kac-Moody and we will use such algebras extensively in this book. h:::,~~~,::'~~ the group theory behind Lorentz symmetry, we know thai the Sl i can be grouped into tensors and spinors. The trans-

properties of these fields can be uniquely determined from group

~

E~~" Similarly, we will define tensor and spinor representations oftbe algebra associated with SO( 10), whose transformation properties detennined by group theory alone. These transformation properturns out, arc so powerful thai we can determine the matrix elements

156

4. Conformal Field Theory and Kac-Moody Algebras

In particular, a vector, by definition, transforms under the SOO Moody algebra as follows: j"'"(z)l{!a(w)"'"

I

z-

w

(r!').al{!" - TJ"al{!/J.)(w)

+ ....

A spin field Sa, because it transforms as a spinor under SO(lO), must by definition, the following transformation property: j"'"(Z)Sa(w)"'"

~

I (y[/J.yvl)f3Sf3(w) 4z - w a

+ ....

The remarkable claim of conformal field theory is that the two identities, which show how vectors and spinors transform under SO( sufficient to determine practically all of the correlation functions of the The energy--momentum tensor, in turn, can be written as the normal square of the Lorentz generators: '" 1

T'" B

-4 ·/J.V· () = D _ 1) J/J." Z •

Explicitly calculating the commutators of the previous equation verifies is the Virasoro generator. , ;;; ; Unfortunately, the spin field Sa has conformal weight ~, which we will later in this chapter when we construct an explicit representation ofthese . . . . We can see this intuitively, however, for the following reason. We saw in (4.1.46) that the highest weight vector 1h) of a Verma module can be as

Ih) = ¢(O) 10;0), where ¢ is a field of weight h and 10; 0) is the SL(2, R) vacuum. superconformal case, we actually have two highest weight vacuum

Ih)

= S(O) 10; 0) , Go Ih).

We wish to eliminate the second highest weight vacuum vector in preserve two-dimensional supersymmetry. In the superstring, we we only have one spinor in our theory at the lowest level, not two. In eliminate the second state, use the identity A

2

Go = Lo

C

-16'

(We are taking a slightly different form for the c-number anomaly in mond algebra in (3.2.28). This choice also satisfies the Jacobi Hle:ntltle we hit the state Go Ih) with another Go, then we want the state to V(1lU"ll' means

GoGo Ih)

= (h -

lC ) Ih) 6

= O.

4.3 Spin Fields

157

the weight h of the spin field must be :~ = ~,as expected. key to calculating the matrix elements of any confonnal field is to ,01,." its short-distance beh.avior with the superconforma! group and olher Therefore let us now calculate the short-distance behavior of the spin li:';:~'::~":~~:::'I~oe,,~ its short-distance

i

alone to dctcnnine

structure

of all, the previous identity (4.3.4) tells us that the operator product of S's must contain at least a 1/1 field: S.,(z)SP( w) = (z

lW)"~ (y/l~>:1/I$«z)1/r"'{ w) + other tenns.

(4 .3.1 0)

calculate the operator product of '" and S. we consider the three-point

(4.3.11) the vacuum is the NS vacuum. In the limit of z\ -4 00 and Z3 -4 0, field changes the NS vacuum into a vacuum with spinor quantum

mb'''', i.e., the R vacuum jOk 5.(0) 10)Ns = 101. u. ;

(0 1",5.(00) =

U.

(4.3.1 2)

(0 1..

that the spin operator allows us to go from the NS vacuum to the R ;,""m" which was not possible in tbe earlier NS-R theory.) This means that J J) can be written as

(4.3.13)

'" c~ly the zero modes of the '" field survive this vacuumexpcctalion product, we are left with the matrix element of a Dirac matrix: I 1 1J.r1' (z )S.,(w) ""'" -Ji (z _ w)1/2(yJol)eS.6(z) + ....

(4.3.14)

in tum, means that 1/1 occurs in the operator product of tv.'o S's:

S.(z)S'(w) -

I .(y )',,"(z)+ .. ·. -Ji(z _ w)l/4 JoI tI

(4.3.15)

, we need to know the short.distance behavior between two spin fields. saw that the confonnal weight of the spin fi eld is ~. Thus,

S,,(z)S.6(w) "" - ~:(z

-

W) - 5/4 +"',

(4.3.16)

~ is twice the dimension of the spin field. Finally, we can bring (4.3. 10), (4.3.14), and (4.3.16) together, which yields short-distance behavior of two spin fields. In summary, invariance arguhave established the short-distance behavior of the spin field, which can represented as p

-1

.6

S,,(z:)S (w) ...... (z: _ w)5/4~"

I

+ -Ji(z _

.6

W)l/4 (y" ),,, '"

I'

(z:)

158

4. Conformal Field Theory and Kac-Moody Algebras

+ .J2(z ~ W)1/4 (y !-,V)"p Vr!-'(z) Vrv(z) + ....

i.

We saw that the spin field has dimension What we want is a vertex of dimension 1, suitable for use in a multifermion amplitude. the missing factor of ~, let us now tum to the ghost sector of the the
4.4

Superconformal Ghosts

Using conformal language, let us now rewrite the Faddeev-Popov del:errn arising from the superconformal gauge fixing. Using (3.4.5), we find Dg zz = V'zD;z,

[

DXz = V'zDS, and their complex conjugates. Thus, the Faddeev-Popov determinant detg(V'z) detx(V'z), and their complex conjugates. At first, this looks very much like the del:errn we found in (2.4.3) for the Nambu-Goto string, i.e., the determinant of az • There are, however, several important differences. The Hilbert which these operators act has now changed. If we expand the detl~rm~i in terms of basis states, we find that the conformal weights have cIlaI~g~ the tensor transformations under the conformal group have changed, statistics of the fields is reversed. When we exponentiate these determinants into the action (see (1.6.1 arrive at the ghost action: Sgh

= Jr1

f

d 2 z de

deBDC

+ c.c.,

where B(z) = f3(z) + eb(z), { C(z) = c(z) + ey(z), where f3 and y are commuting operators. Performing the integration we have .• Sgh

=

~

f

d z(bac + f3ay) 2

+ c.c.,

DB =0, DC =0,

where(b, c) are the ghost fields arising from gauge fixing the metric, "U'"V~~" f3, y are ghosts arising from fixing the Xa field (either in the NS or the R sec·tOrI

4.4 Superconfonnal Ghosts

159

"p',mon:fo,ma] ghosts (which have commutation relations among them) encountered in (3.5.14) when we quantized the NS-R model. The now is that we wish to stress the confonnal properties of these fields, weights. summarize the weights of these fields, we find

,n"

Field

Weight

,b

2 - i

p

1

y

,

Statistics Penni Fenni Bose Bose

,,

(4.4.7)

the action (4.4.5), we can extract out the energy---momentum tensor, is given by the sum of two pieces:

I

Tx = -! DXi
h

+ e T8 ,

" Tgh = -Ca B + 'i DCDB - '2aC B .

(4.4.8)

out explicitly, we have Tf' (z) = -cafJ - iacfJ

+ ~yb,

Tt(z) = cab + 2acb - !y8fJ - ~ay.8.

(4.4.9)

;".wl,w,ud always having to write down band c and.8 and y, especially the equations for the bosonic and fermionic fields are so similar. We will, adopt a system where we can describe all ghosts fields at once, with weights. Let us write down the generic ghost action, using boldface to represent either the commuting or anticommuting ghosts: S = 7ri

f

- + c.c., d 2zb8c

8b = 8c = 0,

(4.4.iO)

(4.4.11)

we define the field b to have arbitrary weight A and the field c to have I - A. Remember that the anticommuting b, c ghost system has A = 2 the commuting {3, y ghost system has weight A = ~. This action is a 'P'
(4.4.i2)

b{z) =

L n.. ~ - ~+ Z

z-~ -"' bn.

160

4. Conformal Field Theory and Kac--Moody Algebras

L

e(z) -:-

Z-n-IH en ,

neHA+Z

where 8 equals 0 for the NS sector and ~ for the R sector. representation, the Virasoro generators are

L~c = L(k - (1 - )...)n)bn-kCk. k

We can check that these generate the usual commutatio'n arbitrary }.. In addition to the energy-momentum tensor, we can construct two .• ••. currents from the action, the BRST current and the ghost number Following (1.9.12), the BRST current is due to the fact that the gauge action, plus its Faddeev-Popov ghosts, has a residual gauge that is nilpotent (and hence cannot be used to eliminate any more U""'''''J gauge symmetry has a current associated with it, so the BRST current derived directly from the action: JBRST(Z)"':' DC(CDB - ~DCB).

Following (1.9.12), the BRST charge is the superintegral of the BRST = _1_. 2rrl

QBRST

rJ

dzdeJBRST'

This charge can be divided into three pieces: QBRST

=

Q(O)

+ Q(l) + Q(2),

where

rJ J 2rri r

Q

1 - 2rri

dz(cTn(X,

Q(l)

= _1_

dz1Y1/l aXil

(0) _

Q(2) = _1_.

2

1/1, {j,

JL

y) - cocb),

.

'

J dz1y2b.

2mr

4

By carefully analyzing each piece, we can show that the sum is nilpotenl$

Q~RST

= O.

In addition to the BRST current, there is also the U(1) current, "ghost number current." Naively, because the ghosts always occur in action, we expect that they must conserve a certain quantum number, analogous to the situation with baryon number. However, surprisingly we find that there is a correction term to the ghost current. Let us write the ghost number current, which is just bilinear in the ghost fields: j(z) = -be = I:>-·-ljn, n

4.4 Superconfonnal Ghosts

in =

L £cn-! bb

161

(4.4.21)

,

+1 (Fermi) or e = ~1 (Bose). We have all the identities necessary the short-distance behavior between the ghost current and the

T(z)j (w) "- (z

Q w)J

+ (z

it,)

W)2

+ .. ..

(4.4.22)

= e( I - 2.\.). (Notice that the presence of this Q factor is anomalous.) ,impl ;es (4.4.23)

,g1><"( number current assigns a ghost number to each of the ghost fields. 0 " __ '

number current has unusual properties unde r complex conjugation:

j ~ = - j _", - Qo".,o.

Q ~ ,( I - 2),). ,\ = ~(l - sQ). e = + I (Femti statistics);

(4.4.24)

(4.4.25) - 1 (Bose statistics).

Qw"" missing in (4.4.24), it would have normal complex conjugation .",es.) The quanrum numbers of the ghost fields are

h,c:

E:=

{J, y:

s = -I ,

I,

A=2,

Q=-3,

c= - 26,

A =~,

Q = 2;

c = II ,

(4.4.26)

anomaly contribution of the ghost is c = - 2e(6)'(,l. - I) + I). O ne unusual features of this ghost structure is the existence of an infinite v"eUG. which arises because of (4.4.23) and (4.4.24). Let us define

'b".(

b. lq ) ~ O. ('~

Iq } = 0,

n >eq - ).., n ::: - eq + )...

(4 .4.27)

zeroth component of the ghost number current and the Lo have the action on these SUItes:

i, lq) ~ qlq ). L~ Iq ) = !eq(Q

+ q) Iq) .

(4.4.28)

'rt;,,,h,,.the last identities sbow that the only nonvanishing matrix elements (-q - Q lq )

~

1.

(4.4.29)

"',;,,' way'o show this is take the matrix elements ofthe currents between

162

4. Conformal Field Theory and Kac-Moody Algebras

different vacua, where each vacuum is labeled by the number q:

(q'ILo Iq) •

(q'ljo Iq) .

All the matrix elements are zero, except when q' = -q - Q. This is highly unusual. In the usual Veneziano model, there was unique vacuum. Now, there appears to be an infinite number of vacua by q for the ghost sector of the NS and R models! The existence ofan

number of Fermi and Bose vacua is one of the unusual features of . . u,Hu, field theory. This means that there is an anomaly in the ghost current. The prO'Olelll in (4.4.22) and (4.4.23), i.e., the fact that the U(l) ghost current-has anol~ commutation relations with the energy-momentum tensor. The an()Dli term corresponds to the violation of the conservation of the ghost In fact, the divergence of the current is given by 8zj z = ~ Q ..fiR(2) , two-dimensional curvature density. The ultimate origin of all these .. actually lies back in (4.1.36), when we calculated the Faddeev-Popov . minant over \/z and \/z. A careful analysis of the eigenvalues of these ooe:ratf shows that we have to remove the zero modes, or else the determinants sense. When we exponentiate these determinants by expressing them in ofFaddeev-Popov ghosts, these zero modes, in turn, correspond to solutions to the equations 8zcz = 8zbzz = O. We cannot give. a full UU)\.",,; of these zero modes, unfortunately, because it requires new inionna1tioll will not be discussed until Chapter S, when we analyze anomalies in (Briefly, this anomaly can be shown to be related to the topology of mann surface swept out by the string. By integrating the divergence for the U( 1) ghost current, we can use the Gauss-Bonnet theorem to d 2z,JgR(2) = -Sn(g - 1), where g is the number of holes or 11
J

4.4 Superconformal Ghosts

163

",itj" frequency parts of the oscillators) are

1-11. / HI· NS:

{I-II.

(4.4.31)

are normalized as follows :

H 1-11 ~ I, ( - II-I I~1.

(4.4.32)

h"" 'P'"'''''''' of' '" infinite number of inequivalent vacua appears at

a disasler, we will show latcr on that this yiclds a perfectly acceptable

In particular, we will show that, on-sbell, it really makes no difference of tbe various vacua we choose. All on-shell matrix elements for the physical process, with arbitrary choices of vacua, will yield the same . In fact, we will he able to create "picture changing" operators that take us from one vacuum to another. The situation is exactly analogous one found earlier with the FI and F2 pictures discussed in (3.3.18) and worked out the structure of the ghost sector of the NS-R model, our islo fin" thdie'd that gives us conformal weight ~. the missing piece fermion venex function. us now bosonize the ghost current by introducing a new field rp into the

j (,)

,a,,(,).

~

(4.4.33)

object we wish to investigate is :e'l<1:.

(4.4.34)

,hort-
e'l~II))

w r'k·,~wl ~ [~tq(q

+ Q)(z -

W) -2

+ (z -

+ ... ,

w)""la",jeq¢{w)

e'l¢{O) 10) = Iq) .

weight:

!eq(q

+ Q).

, m""" that multiplying by the bosonized fie ld eq ,

(4.4.35)

+ ....

(4.4.36)

(4.4.37) (4.4.38)

allows us to go from vacuum to another. Notice that the NS ghost vacua are integral, while K t~:~~: vacua are fractional. Because q can be fractional , this allows us o and forth between the various NS and R vacua by multiplying with

164

4. Confonnal Field Theory and Kac-Moody Algebras

Let us use this bosonization technique to write the anticommuting fields in terms of a new scalar boson field a: b(z) = e(z) =

e-"(z), e"(z).

(We can check that the fields have the correct conformal weight. From a field eq " has conformal weight ~q(q - 3). Then for q = -;-1, the has weight 2, while for q = 1 the field eO' has weight - 1. They thp.lrp.tr.ri the correct weight.) Notice that both the left and right sides are anl:}C(llllltni:ii fields, even though a itself is a commuting field. It is now easy to ",hr.,,;'>: (a(z)a(w)) = log(z - w),

z-w

For the NS-R ghost sector, however, this is more subtle"Th:ese already commuting, so bosonization does not seem possible. We can, use a trick:

fJ

= e-4>a;,

y = e4>T),

where the left-hand sides are commuting fields, and the right-hand product of two anticommuting fields; i.e., ; and T) are also anticolmrt fields (which, in turn, can be bosonized). Thus, we have written COlnrt fields in terms of anti commuting fields. We can also reverse this pnlced T) = aye-4>, a; . afJe4>.

Notice that the; and T) fields are themselves anticommuting, so bosonize them. Let us express these two anticommuting fields in tennsl boson field x: ; = eX, T) = e- x .

Although conformal field theory has the great advantage that all fields fields, one small price to pay is that we must keep track of all these free fields. Because it is crucial to keep all these boson fields clearly let us tabulate their quantum numbers: Boson

Charge

Anomaly

Weight

¢ X

Q=2 Q =-1 Q =-3

c = 13 c= -2 c = -26

wt(expq¢) = -~q(q + 2) wt(expqX) = ~q(q - 1) wt(expqa) = ~q(q - 3)

a

There is a crucial subtlety in the definition of the Bose NS-R glllJM~·.T"! that fJ is defined in terms of the derivative of ;, so that it is indlep(~nd¢)

4.5 Fermion Vertex

165

s

mode of the field. Thus, the usual Fock space is independent of the mode of the field. Thus, we have two possible Foek spaees. The "small" kSI>aoe is. missing the zero mode of ~. The " large" Fock space contains the mode of S and is reducible. Because

s

(4.4.44)

:""".,"',,tthe vacuum of the s system is degenerate. 1),

purpose oftbis construction is that we can now write down the missing of the fermion vertex operator.

Fermion Vertex us calculate the confonnal weights of the bosonized fields:

+ Q) = i, = !sq(q + Q) = -~.

wt(t-(·f2k11) = !sq(q wt(eon)f»

(4.5.1)

ow have found the missing piece. Since the bosonized fiel d e-('/2~ has ~, we call now construct the true fermion vertex a/the theory: v

-

-1/2 -

u"e-(I/2k/1S eii:.X

(4.5.2)

",

i+i+a'e.

(4.5.3)

'pI,,, the external fermion line on-shell k 2 = 0,

y/l.k"u" = 0,

(4.5 .4)

we have a vertex of conformal weight J. This vertex bas the correct D,"iesun,d" the BRST charge: (4.5.5) te1TI1S that vanish on-shell. This is one of the great accomplishments of fi eld theory, the creation of a genuine fennion vertex function with I weigbt I based emireiy on free fields. The key observation was to sector to suppl y the missing conformal weight of we have now anained our goal, we are not yet out of the woods. in the last section that there is a problem with respect to Bose sea. It turns out that if we insert this fermion vertex within a element, we get 0:

i.

(... V_ I/2''' V- In •... ) = 0.

(4.5.6)

needed, of course, is a venex with ghost charge +~ to cancel the -t .gfcom !!h' fermion vertex. This new vertex V'/2 must anticornmute with

166

4. Conformal Field Theory and Kac-Moody Algebras

the BRST charge up to terms that vanish on-shell. It is easy to show vertex

v=

[QBRST,

cI>]

for arbitrary cI> has a vanishing anti commutation relation with the because Q is nilpotent. However, all these vertices are spurious. are null and do not couple to real states IR), which satisfy the QBRST

IR) = 0

so they cannot be used as vertex functions. They simply produce elements with the physical sector of the theory. However,there is a function for which this reasoning does not apply: VI / 2 =

2[QBRST,

;-V- I / 2 ].

Normally, we would expect that such a vertex is also spurIous and couple to real states of the theory. However, ;- VI/2, as we said v~. . .v. part of the irreducible Fock space of the theory, and thus we cannot that this commutator vanishes on contraction with real states. This vertex not necessarily vanish because

After working out the details, we find that this vertex is equal to VI/2

= ua(k )e ik . X[e O/2)
This, it turns out, is the correct fermion vertex that allows us to COlltnlc V-1/2. However, at this point we realize a rather disturbing problem. We too many possible vertices! For example, we could also have written

V3/2 =

[QBRST, ;- VI/2],

VS/2 = [QBRST, ;-V3/ 2 ], V7/ 2 = ....

In fact, there is an infinite number of such vertices, each one linking alent Bose sea vacua. This is certainly an embarrassment of riches. it is possible to show that we need only use VI / 2 and V- I /2, and that vertices do not yield any new matrix elements. We wish to show the identity:

(... V_ I/ 2 (Uj, kl , ZI),

... ,

V1/2(U2, k2, Z2)"')

= (",VI/2(UI,k l ,ZI), ... , V- 1/2(U2,k 2,Z2)"')'

The proof that we can simply switch the various ~ and - ~ ghost "'U"'¥" the vertices at will involves some rather subtle arguments that take us the irreducible small Fock space (which does not include the zero moo.¢! and the reducible large Fock space (which does include the zero mode

4.6 Spinors and Trees

167

/{eo.!,mby rewri!ing the vertex V'/l in the equivalent fonn

V1/ 2 (Z2)::::

f

2~j

dWjBRST(W)S(Zl)V_ L/2(Z2).

(4 .5. 14)

that the vertex is now written in the large Fock space, so we must

.,i ,.",th,,, ,(z) to redefine the vacua so thai we have a nonvanishing matrix However, because the position of Hz) is irrelevant (because we want zero mode), we can now rewrite the matrix clemen! as

(- .. oHzd V-1/2 ...

2~ i

f

dwjBRST(W)i(Z2) V-L{l(Z2} .. -) .

(4.5.15)

'{arwe have done nothing. We have simply gone from the small Fock space, the zero mode of never appears, to the larger Fock space, where it

s

. However, the value of the matrix clemen! remains precisely the same. we make the following observation. The contour integral that encircles

changed at will. so lei us enlarge it until it goes around the back of the surface (a sphere) and finally comes around and encircles the point ,the enlarging of the contour cuts across other bosonic vertices, that jaRST commute w ith all of them on-shell, so that we can move the current jSRST at win unti l they encircle ZI. point. notice that the contour integral is now around ~ V- 1/2. which ; ",=hewrinen as V+ 1!2' Thus, we have totally reversed the position of the integral, and we can eliminate all the and go back to Ihe sman Fock Symbolically, the steps can be represented as follows:

s

.. V_ 1/2'"

V /2"' ) -+ '

C.(~

V_1/2)'"

[1 dW~jBRSTV-In] '·.)

~ (--. [f dWV"'T V_'n j ... (~V_",) .. -)

..,. ( ... V ... V_In' .. ). 'f2

(4.5. 16)

' . '"",'" of this exercise was to show that we can successfully reverse the of the and - in the matrix element This means that, although Ihere infinite number of fennion vertices, they will al l yield the same on-shell . dement.

!

i

Spinors and Trees to calculate tree am plitudes, it is necessary to construct an explicit of the spin fields in terms of NS-R operators. Although these were highly coupled operators in lhe OS formalism, the great advantage confonnal field theory is the fact that the interacting spin fi eld can of free fie lds, making lhe calculation of correlation function s

:s,',",,;on ,orr,oo'''"

lJU!jsiblt:.

168

4. Conformal Field Theory and Kac-Moody Algebras

We will construct the spin field out of the generators of the SO(lO) which in tum are composed of the 1fr(z). The SO( 10) algebra is a Lie algebra (see Appendix). This means that, out of the 10(10 - 1)/2 generators of the SO(1O) algebra, five mutually commute among the:msl!~h forming the Cartan subalgebra. The commutation relations among theSi;t' commuting generators and 40 noncommuting elements are [Hi, H j

]

= 0,

[Hi, Ea] = aiEa, [Ea, E,6] = 8(a, (3). E a+,6

The last identity is true if E a +,6 is a raising or lowering operator. Each, root vector of the group S0(10). The 8 terms are the structure constants group, which obey various symmetry conditions and associativity. Let us now introduce five mutually commuting fields (Pj and p.Yr\rp';t;; 45 generators ofS0(10) in terms of them. We can represent the five mu.too. commuting members of the SO(IO) algebra as a¢j = H j .

To represent the remaining 40 generators, write

E a -- .·eifJ~
fJa = (±1, ±1, 0, 0, 0). We also include its permutations. Notice that there are four possible ments ofthe + and - signs and that there are 10 possible ways in .....vu ,~, signs can be placed into five slots. Thus, this matrix has 40 elements in slot corresponds to one of the five ¢ j. The purpose of the factor Ca in (4.6.3) is to make the commutation come out correctly. If we insert these expressions back into the defini'til the algebra, we find that we must set ca c,6

= 8(a, (3)ca+,6'

If we demand associativity of the commutators, then we also have 8(a, (3)8(a

+ (3, y) =

8(a, (3

+ y)8«(3, y) .

.,;

One of many possible representations of these two cocycles is 8( a, (3) = (_1)0(a.,6), where a(a, a) = !(a, a),

4.6 Spinors and Trees

a(a , (3) + a(j3, a) = (a . (3) mod 2

169

(4.6.9)

[9, 10J for other conventions.)

can also represent the NS anticommuting vector fi eld in terms of this

'.

~"iza,'io;" . Let us write

.,'PO J.c

,/,p. 'I'

Po

• . • Po'

(4.6. 10)

(±, 0, 0, 0, 0)

(4.6. 11 )

-

~

h,~~~~~~~:::~, There are 2 x 5 elements within p. which is the number of III

1/1" . us now represent the spin field in terms of this bosonized picture. Let us the matrix

)..'" =

!<±, ±, ±, ±, ±).

(4.6. 12)

that there are 2$ = 32 terms in this matrix. Let us define

S", --

,

.

.·,/~ ·" 'C 0'

(4.6. 13)

",field Sa has the correct number of components for a spinor in 10 dimen. Furthennore, it has weight This is because each individunl fuctor has ~, and there are five of them, for a total of ~. This confirms the ear· e'~~:::~~t~':7 made in (4.3.9), which was based purely on group-theoretic [8 that the spin field Sa has weight field e --{I!214> has weight ~. and e(lf214> has weight Thus, our vertex 1 weigl'tl is given by

i.

i·

v _ I/l.a --

- i.

Sa ,-( l{2I4>.,/~' x. . ..

(4.6.14)

':'e~::,:~~bave an explicit representation of the spin field, we can calculate

tri

for fermion- fermion scattering. us calculate the four·fennion scattering amplitude, represented by the ofthrce independent factors involving X. Sa, and e- f1/ 2\f1:

(4.6.15) ZI

=

00,Z2

=

I ,Z)

=Z , l. = 0.

us calculate each factor separately. Using the formula derived earlier in we find tbal the X -dependent ractors are equal to

"p'ee :',

( V(k l .Z I ) V (k 2• Z2)'" V(k ...,. ZN ») =

D (li -

Zj)u

J

•

(4.6.16)

llej

V (k. z)

the sum or all k, equals zero.

= ;eiU{lI:

(4.6. 17)

170

4. Confonnal Field Theory and Kac-Moody Algebras

Let us now specialize to the case of the ghost contribution, which (e-(l/2)rf>(OO)e-(l/2)rf>(l)e-(l/2)rf>(z)e-(l/2)rf>(O))

= [z(1-

z)r l / 4 .

Lastly, the spin field contribution is

(S,,( 00 )S{3( 1)Sy(Z)S8(0)) =

[zO - z)r3/ 4 {0 - z)(y/i),,{3(YI'J,,8 - Z(y/i),,8(Y/i){3y} ,

Putting all three pieces together, we arrive at

A4 =

g211

x

dzl

J •k2 -

0-

1

Z)kJ .... -1

{(1- z)(y/i)"{3(Y/i)y8 -

Z(y/i),,8(Y/i){3y},

Rewriting this amplitude, we finally arrive at

A4 = g2

{BO -

'-'

~t, -~s)(y/i)"{3(Y/i)Y8

- B(-~t, 1 - ~s)(y/i)"8(Y/i){3y}. This calculation and others involving N -point fermion scattering amp}f are not that difficult with conformal field theory [9, 10], but would extremely difficult in the older covariant NS-R and GS formalisms.

4.7

Kac-Moody Algebras

Although the results of conformal field theory are powerful, to the the various rules and conventions may seem a bit arbitrary and appears, at first glance, that conformal field theory depends on and accidents rather than anything fundamental. In reality, the underlying consistency and elegance of conformal are due to new infinite-dimensional Lie algebras, called Kac-Moody [11-22], which are powerful extensions ofthe usual . gebras. They were discovered by mathematicians V. G. Kac and R. in 1967, although one form of them was already known to nh'VSl,dst rnid-1960s as current algebras. Together with the superconformal two dimensions, the SO(IO) Kac-Moody algebra provides the U"'.•u.,. framework for conformal field theory. Many of the matrix elements formal field theory, in fact, can be viewed as Clebsch-Gordan cO()mlCl Kac-Moody algebras. We define a Kac-Moody algebra to be a generalization of an algebra such that its generators obey I i Tj] - 'fijlTm+n [Tm' n - I

+ k rno~ih0m,-n·

The algebras looks very much like an ordinary Lie algebra, exc:epI infinite integer index rn on each generator and the COl1stant k,

4.7 Kao--Moody Algebras

171

i The zeroth component of the T 's is nothing but the algebra of a finite . We will often find it convenient to rewrite the generators of the ~~~;:~; algebra as the Fourier components of a single functi on defined (4.7.2)

• also break down the generators of the Kac-Moody algebra into the subaigebra and ils eigenvectors:

£.(9).

(4.7.3)

a Kao-Moody algebra looks like an ordinary Lie algebra smeared over us now construct what is called the "basic representation" of the K.acalgebra using vertex operators. This representation holds only for laced groups (i.e., groups with roots of equal length, which are A, E Lie groups) y..ith level one (k = 1). Let us first define the string

(4.7.4) introduce the basis vectors for the lattice of our Lie algebra such that (4.7.5)

,,110'" us 10 write the string variable and the vertex function as vectors lallice:
L',."

q/CO) =

:e'\tI(B):C;.

(4.7.6)

the CI are the familiar cocycles introduced in (4.6.5) in order to gel ':::~:~~:~~ relations with the correct sign. There are many possible Ifl one of which is (4.7.7) this allows us to introduce the generators of the Kao-moody algebra: H,(9) =

Eo

•

;q, (O)q,(O); =

d¢/ -de'

= :qj(/:J)qk(B): = ±icjCt:ei04J(6):,

(4.7.8)

and we use the + sign fork > j and the - sign fork < j. the ilh and jth eiemenlS commute, the H 's are generalizations of the subalgebra, i.e., the set of mutually conunuting elements of the Lie

= Ct -

e)

172

4. Conformal Field Theory and Kac-Moody Algebras

Let us now calculate the commutator between these generators. In same way as before, we find that the product of two vertex functions

V(a, 8) = :eimp(O):, V(a, 8)V(,8, 8') = !':!..(8 - 8,)-a·{3:e ia q,(O)+i{3q,(O'):, where !':!..(8-8') = e- i(l/2)(O-O')(l_(l_s)e- i(O-O'))-1 ~ 8 -~,

+ is +....

We now have all the identities necessary to compute the commutator all the generators. We find, by direct computation,

[Hi(8), Hj (8')] = 0, [Hi(8), Ea(e')] = -2n8(8 - 8')aiEa(8). Notice that the H's are mutually commuting, as they are in the ornin
[Ea(8), E_ a (8')] = 2n8(8 - 8')

L aiHi(8) + 2ni8'(8 --:- 8')

and

[Ea(8), E (8')

={

{3

2n8(8 - 8?E a+{3(8) 0 otherwlse

if a +,8

E

r,

(r is the root lattice.) Thus, we see that the commutators of a n ..a'.-n algebra are very similar to the ordinary ones for a Lie algebra, except generators are smeared over a circle. It is also crucial to notice that the semidirect product between the algebra and the Kac-Moody algebra is possible. The commutators

[Lm' T~] = -nT,~+n' [Ln' Lm] = (n - m)Ln+m

c 2 + 12n(n -

l)8 n.-:m.

Depending on the representation ofthe algebra that we choose, we correlation between the level k ofthe Kac-Moody algebra and the C ofthe Virasoro algebra. One remarkable property ofthe Kac-Moody is that we can construct a representation ofthe Virasoro algebra entirely of the Kac-Moody algebra. Let us write the Virasoro generators "U".a.l.' a circle:

L(8) = !N

= !N

(~:Hi(8)2: + ~ :Ea(8)E_a(8))

(t :T;(8i:) . 1=1

This is called the Sugawara form.

4.7 Kac- Moody Algebras

173

• ~''''' an the necessary identities to commute these past each other, and [L (e). L(e')] ~ 2rr;8'(e - e')(L(e) + L(e' )), C

- I 2 2rr ;(!'''(e-8')+!'(8- 8'»,

(4.7.16)

(4.7.17) d is the dimensionality of the group and C2(G) is the vaJue of the Casimir operator for the adjoint representation. commutator yields thesemidirect product between the Kac--Mood y Virasoro algebras:

[1.(0), E.(O~1 ~ 1'(0 - O')E.(O).

(4.7.18)

Dmiput, "" value of the central cbarge for a few groups: Group

c • - I

,

n +!

• •

n(2n

+ I)

• +2

SU(O) SO(2n + I) SO(2n)

E. Sp(n)

the perspective of Kac-Moody algebras, let us now reanalyze conforMid th".ry, "iin!<'"""ri·,. the results of conformal field theory from group . We see that the spin given in (4 .6.13), under the SO( lO) Kac-Lie algebra, transfonn as a 32-componenc spinor. Likewise, the NS vector fields given in (4.6.10), under the SO( I 0) Kac- Moody as 10-component vectors. We can also view the b, c, (J, y fonning representations of the superconforwith different values of the centro] term given in (4 .4.43). Finally, view the correlation functions, which. are the heart of conformal , as the Clebsch--Gordon coefficien ts found in the different tensor representations of the SO( 10) Kac-Moody algebra coupled group.

174

4. Conformal Field Theory and Kac-Moody Algebras

4.8

Supersymmetry

Finally, we can now construct the supersymmetry operator in this allows us to go back and forth between the fermion and the boson the theory. Because V±I/2 has conformal weight 1, this means that is an invariant under the conformal group. Thus, if we take k = 0, supersymmetry operator Q±I/2.a =

Depending on whether we take the

2~i

f

+ or -

dZV±I/2.a'

fermion vertex, we find

Q-I/2.a = Sa e-(1/2)¢, QI/2.a = e(I/2)¢ SfJ(yf.L)fJaBzXw

These operators indeed have the property of converting the fermion a boson sector, and vice versa:

[Qa, VF(u, k, z)]+ = VB(l;f.L = ufJ(yf.L)fJ a, k,z), [Qa, VB(l;, k, z)] = VF(u fJ = ikf.Lt;V(Yf.Lv),k, z). Once again, however, we are faced with the problem of having an ber of such supersymmetry operators, which link the various inp,nlll""j' sea states together. However, because of arguments given earlier, have to worry about this. Although there is an infinite number of Bose vacua, the matrix elements between them yield the same

4.9

Summary

The great advantage of the superconformal field theory is that we,} best features of the NS-R and OS models. The superconformal ... , has the advantage that it is manifestly covariant, based ehtirely on possesses a covariant supersymmetric generator, has spin fields and fermion vertices of weight 1, and is easy to manipulate. There are two drawbacks, however, to the conformal field theory must carefully keep track of the many bosonized ghosts. Fortunlat(}l ghosts are free fields and operate on different Hilbert spaces. SelcOIld, an infinite number of ''pictures.'' Fortunately, the final S -matrix is mclem of whatever choice of "pictures" we take. The essence of conformal field theory is that we can calculate the functions of various fields simply by knowing their transformation and their short-distance behavior. For example, the correlation furlctj.~ (eikX(W)e-ikX(z)) ~ (w _ Z)-k 2

can be calculated once we know the short-distance behavior of two

4.9 Summary

175

of the achievements of the conformal field theory is the construction ,~~~;:~ver1ex function with conformal weight I . This vertex function is td a ~pin field S'h whicb must transform as a genuine spinor unde r 0) Lorentz group:

(4.9.1) :co05id".ti"ns, such as the field having dimension

y,"(z)5o (w) -

1

(y"tos,(z)

.J2(z -

P S.,.(z)S (w) '" (

hallows us to fix the

+"',

(4.9.2)

W)2

-\ p )S/40" z-w 1

+ Ji

2(z -

I ,8 jJ )"4(Y"r,,1/1 (z) v2(z-w

+:;:;

W)l/~

(y"')!y,"(z)y,"(z ) + .. . .

(4.9.3)

;'ekn<,w.ll ,thesh,m·dis,,,,,cebol>aviocofth. spin field, we can construct funct ion that has weight I. We need an extra piece that has weight comes from the ghost sector:

1 Sgh=;r

J'

--

d z dO dOBDC+c.c.,

(4.9.4)

B(z) ~ P(z)+ Bh(z), ( C(z) ~ c(z)

",o:,tsoc"" can be bosonized according

f3 =

(4.9.5)

+ By(z). to

e-4>&~.

(4.9.6)

y = e<>T/.

in turn. finally allows us to write the vertex function with conformal l on-shell: V_ 1/2 = u"e--{I /2}1; S«tI~·x

(4.9.7)

has conforma l weight (4.9.8)

t

order to write amplitudes that can contract w ith the - vertex operator, a corresponding vertex function with +~, which is given by

= u"(k)e r.X [e(I ,l2lt; (aX"

+ iik ·l/il/I")(y"kpSP + ie3,,/2T/bS«].(4.9.9)

these two vertices, we can now construct multifennion scattering litud".

176

4. Conformal Field Theory and Kac-Moody Algeb!as

One drawback of this formalism, however, is that we have an infinite of vacua and hence an infinite number of vertex functions. These vacua· constructed as eq
weight:

10) = Iq) ,

~eq(q

+ Q).

This, in tum, allows us to construct an infinite number of functions. For example, V3/2 = [QBRST, ; V 1/ 2 ],

VS/ 2

= [QBRST, ;

V3/ 2 ],

V7/2 = "',

are all acceptable vertex functions with weight 1. This is an emlbalrrai~s of riches. However, it can be shown that, at least on-shell, all these are equivalent to each other. Thus, we can take anyone of these at construct the fermion-fermion scattering amplitudes. , One of the great advantages of this formalism is that we can now the supersymmetry generator, which is nothing but the vertex grated over z. In fact, we actually now have an infinite number of supersymmetry operators, all of them being equivalent on-shell: Q-I/2,,,

=

S"e-(1/2)
Q 1/2,,, -- e(l/2)
a xiL' et c. Z

Finally, it is possible to obtain a coherent overall picture OfCorlfOIll theory if we view it as one way of calculating matrix elements for .. dimensional Kac--Moody algebras and the superconformal '. Moody algebra has commutation relations I - 'fijlTm+n i Tj] [Tm' n - 1

+ k modj 0m,-n, 0

where the !'s are the structure constants of a finite-dimensional For level-one simply laced algebras, we can construct the basic repJr~$:€ of the Kac-Moody algebra. The basic representation is given in tenns~ operators first found in string theory. In this forinalism, the spin fields S" and the NS-R 1/1 fields can as spinorial and vector representations of an S0(10) 1'>..111"-1'V1UIJU~ and the various (b, c) and ({3, y) ghosts can be viewed as tonmng representations of the superconformal algebra. Correlation lUtLCtlOnft.1 be viewed as Clebsch-Gordon coefficients for the combined U~'\·.~c' Moody and superconformal algebra.

Refcrences

177

Sec D. Friedan, in Unified String Theon'u (edited by M. B. Green and D. Gross), World Scientific, Singapore, 1986. v. G. Knizhnik and A. B. Zamolodchikov, Nue/. PIrys. 8247, 8] ( 1984). A. A. Belavin, A . M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys. 8241 , 333 (I984). D. Fricdan. E. Maninec, and S. Shenker. Nuci. Ph~. 8271 , 93 ( 1986). O. Friedan, E. Martinec, and S. Shenker, Phys. Lett. 1608,55 ( 1985). V. G. Knizhnik, Plrys. Left. 1608,403 (1985). S. MandclsllIm, Phys. Rn>. 011 , 3026 (1975). S. Coleman, Phys. RI!:'o!. 011 ,2088 (1975). J. Cohn, D. Friedan, Z. Qiu. and S. Shenker, Nucl. Phys. B278, 577 (1986). V. A. Kosrelecky, O. Lech!enfeld, W. Lerchc, and S. Samuel, Nuc/. PIr),s. B288, 173 (1987). v. Kac, Functional Anal. Appl. 1.328 ( 1967). R. v. MoodY,J. Algebra 10. 2 11 (1968). P. Goddard and O. Olivc, Internot . J. Mod. Phys. At , 303 (1986). P. Goddard, W. Nahm, and D. Olive, Phys. Lelf. 160B, III (1985). P. Goddard, A. Kcnt, and O. Olive, Comm. Moth. Phys. t413, 105 (1986). J. I..epowsky and S. Mandelslam, in Vertex Operaton in Marhemarics and Physics, Springer. Verlag, New York, 1985. K. Bardak~i and M. B. Halpern, Phys. Rev. 03,2493 ( 197 1). M. B. Balpem. Phys. RI!:'o!. 012, 1684 (19'75). P. Goddard., W. Nahm, and D. Olivc, Phys. Lelf. 160B, II I (1985). P. Goddard, A. Kent, and D. Olive, Comm. Math. PhYJ. 103, 105 (1986). v. Kac, Infinile Dimensional Lie Algebras, Birkh!user, Boston, 1983. R. V. Moody, Bull. Arner. Math. Soc. 73, 217 (1974).

CHAPTER

5

Multiloops and TeichmOlier Spaces

5.1

Unitarity

One of the most exciting aspects of string theory is the possibility of gravity that is totally finite and therefore independent ofcOIlventijDll~l1j malization theory. String theory may provide a framework in .' first time, a finite theory of quantum gravity may emerge. What is larly fascinating is the mechanism by which the cancellation of all " divergences takes place, namely the use of topological arguments to certain divergences. Once again, we see the enormous power ofsYlDIIl<;?IT. is built into the string model. We will show, for example, tltat the that are potentially divergent are topologically equivalent to the vUJL>,,,,>,, effective dilaton. Therefore, by eliminating the dilaton from the theory tain a theory without any apparent divergences. Thus, mechanisms before appeared in point particle quantum field theory are 'n ~spomnbl,~ elimination of potentially harmful graphs. So far, we have developed only the first quantized theory of . strings without loops. This, of course, cannot yield a unitary theory. ler Beta function, we saw earlier, has poles on the real s-plane axis, imaginary parts or cuts, and therefore the theory describes only tree Early attempts were made to modify the original Beta function by imaginary part to the mass of the resonances: A(s,t)~

"

L..

(X,

2'

, S-M,+lr,

but then the marvelous properties of the Beta function were destroyed.

5.1 Unitarity

179

correct proposal to unitarize the model was finally made by Kikkawa, and Virasoro (KSV) [I] in 1969 by adding loops, treating the Beta the Born tenn in a perturbslive approach to defining the S-matrix. multiloop amplitudes were calculated by Kaku, Yu, Lovelace, and i [2--8]. (We should point oul. however, lhat the KSV urutariuuion discuss M-theory. For membranes, it is not clear even the free theory, Jel alone the theory of interacting memD-branes bave proved useful in understanding the excitations found in "o'Cj,ou, a complete understanding of me imcraclions in M.theory is still o

)w.~~::~;,,,:,~, 'c

how the perturbation series is set up, let us beygin with the operator U, which transforms an initial state at t = -00 into state at t = 00. The S-matrix is the matrix element of U:

s,'

~

(il U( -00. (0) Ifi .

(5.1.2)

operator is unitary, the S-matrix ilSelf is also

SiS

=:

sst =: I.

(5.1.3)

form. this reads

L {il Sin) {nl Sl u) = 8

1},

(5. 1.4)

•

the n's are a complete set of intermediate states. If we separate out the corresponds to no scattering, we get the T matrix: S=I-iT.

(5. 1.5)

(5.1.6)

'" ,"" matrix elements for the scattering of the multiparticie initial stale to the multiparticle final state U), then we have

1m T I) =

I

-2: ~ iii T ill) (n l rt u)

(5.1.7)

5.1). If we represent the four-string scattering amplitude as (i I T Ik). ~"""'y we must combine various four-point functions to obtain the next the perturbation series. Because strings can "twist" as they sweep out the set of Feyrunan diagrams for the loops is larger diagrams. In facl, as Fig. 5.2 shows, there are three types thaI we can construct using the optical theorem. string, the interactions sweep out a world surface that is topoequivalent to a disk with holes. In addition, we can also have "twists" (A twist is created when we cut a line between two holes and then

180

5. Multiloops and Teichmiiller Spaces

1m

=

~

• ,

FIGURE 5.1. Unitarity of the S-matrix. The imaginary part of the sca.ttelrin~d tude is proportional to the square of its abso lute value. In this fashion, we can higher loops from lower-order tree diagrams. This was the original unital"i scheme advocated by Kikkawa, Sakita, and Virasoro.

=

-J r

.~

FIGURE 5.2. Planar, nonplanar, and nonorientable single-loop open string Unitarity forces us to sew together three types of graphs. (The x on ./ corresponds to twisting the leg.) The nonorientable diagram corresponds to a ... strip. The nonplanar diagram corresponds to placing some external tachyon the boundary of the interior loop.

rejoin the cut by reversing the orientation of the points along the cut.) Th'dOl'" types of open string diagrams are: (1) Planar diagrams, which are topologically equivalent to a disk with punched in the interior with the external lines located on the . (2) Nonplanar orientable diagrams, where the external1ines can be lU"''''~'' some of the internal holes as well as the external edge, or where overlap.

5.2 Single-Loop Amplitude

181

IO~~~'~:"t~~ diagrams, where we have an odd number of twists in the

of the disk. The Mobius strip is an eXllmple of II nonorientable

UI

closed string, the tree diagram swept out by the interacting string >o1"S;"" equivalent to a sphere. There are two types of loop diagrams create out of the closed string. Let us cut 2N holes in the sphere and mark the orientation of points along the circular edge of each hole. i;';~':::~rs oftlOles to obtain a sphere with N handles. The two types

PI,n., d,i,s",ns, where the orientation of the circular edge of each pair of is preserved when we resew the diagram. A doughnut, for example, II planar single-loop diagram. ~o"ori.' nl"bl, diagrams, where the pairs of points along the circular edges the holes are joined by reversing tbe orientation of the holes. A Klein , for example, is a nonorientable diagram (in Fig. 5.3, we see how bottles can be assembled from fiat two-dimensional surfaces that "vt"h,'~ boundaries identified). (We note that only Type I strings, whieh no orientation or direction, have Klein bottles in their perturbation Type n strings carry an intrinsic orientation and cannot p roduce bottles.)

",i".

r.

~th~'~fu?~nfc~ti~o~na~I~"~o~n ~:aliJ.~sm[~,~w~c~sa~W~"'~H~Cf'~th~'~t :"'~I~"~C~'~1~'~~

spbere[9-II] . . method of calculating this Newmann function was to conformaJly were conor sphere to the upper half-plane or the entire complex plane. We a method developed in electrostatics, the method of images, to the Neumann funclion: G(z, z) = In Iz - z'l

+ In Iz - tl ·

(5.1.8)

we will generaJjze this discussion 10 Riemann surfaces with holes. Formathematicians long ago wrole down the Neumann function for the sphere with N holes. In fact, Burnside [12] solved this problem in ! The solutions 10 this classical problem are given in terms of automorphic wruch we will now analyze.

Single-Loop Amplitude let us set up the functional integral for the single-loop diagram:

A=

1

DX dtte lS

s

fI

elk, x, ,

(5.2.1)

1=1

we functionally integrate over a horizontal strip in the complex plane

182

5. Multiloops and Teichmiiller Spaces A

B

C

0

C

·0

A

0-

C

•

B

,

A

B

0

C

E

0

o

FIGURE 5.3. The Klein bottle. By identifying the opposite sides ofa rp("hn can obtain either a torus or (ifthe orientation ofthe sides is reversed) a shown here. The Klein bottle is a two-dimensional closed surface with only .,

(see Fig. 5.4) that has finite length and then identifY the left and right this way, we construct a surface topologically equivalent to a disk The functional integral, of course, can be calculated explicitly, I P,,'VIn the factor exp

[L.:kiN(Zi, zj)kjJ,

where N is the Neumann function. Now, let us topologically deform the horizontal strip into the surface. Consider an annulus, defined as the region in the upper that has an outer radius of rb and an inner radius ofra and a ratio w Now impose the fact that the outer perimeter is to be identified with perimeter. This means that a point z on the surface of the outer pe.ri·m~)t¢J

,0'''' '.' . Confonnal surface of the single-loop open string diagram. In Ihe p Ihe surface corresponds to a rectangle of width :r and of arbitrary length, we identify the ends. Rectangles of constant width with varying lengths confonnally inequivalent, so we must integrate over all lenglhs in the path The wavy lines correspond [0 "zero width" strings or tachyons, which may to the surface on cilher the upper or lower boundary. In lhe ~.plane, the

corresponds 10 a narrow tube that is bent into a half-circle. with external

!' ,m,,";';. from the ends. id,,.,;fied with the point on the inner perimeter with the same polar angle:

z -+

WZ.

(5.2.3)

identification creates a semicircular tube in the upper half-plane. The

,fo,,,,,,1 '''''J'pi'g from the horizontal strip 10 this tube isjust Ihe exponential. tube, in can be mapped inlO a disk with a ho le by stretching one of the tube until it becomes a large circle and then shrinking the other When we construct Neumann functi ons in this annulus, we obviously are 'rest,d in functions that have the property ~(z) ~ ~(wz).

(5.2.4)

take arbitrary powers of w, this idemificalion acrua!ly divides the upper . in fi nite number of concentric circles. Each concentric circle radius w~. By identitying the outer perimeter of one annulus with its inner

184

5. Multiloops and Teichmilller Spaces

perimeter, we can create an infinite succession of tubes. Thus, the entfi't!tl\ilii

half-plane can be decomposed into an infinite sequence of these £uu'eS{ QUl are interested in only one of them. Thus, we have divided up the upper half-plane by making an 1·(lentlti,c~ concentric circles generated by multiplication by the number w. This thus parametrized the disk with a hole. This is called the Teichmiiller for the single-loop diagram. These parameters have a: natural geller:ali:l~~' surfaces with N holes. . In general, we can also divide up the upper half-plane by using an projective transformation, which, as we saw earlier,maps the real the real axis. (In general, circles are mapped into cirfles under a transformation.) We define anautomorphicjUnction as one that has the 1/I(Z) = 1/I(z'),

.0

where we make a projective or SL(2, R) transformation:

,

Z =

SL(2, R):

az + b , cz +d

_

0

{ ad - bc = 1.,

for real a, b, C, and d. (Phases can sometimes enter into the erties of these functions.) Fortunately, mathematicians have function. The Neumann function is N(z, z')

"al"Ula.,

= In 11/I(z' Iz, w)1 + In 11/1(2' Iz, w)l,

where we demand, up to a phase, that the function be periodic: 1/I(z, w) = 1/I(wz, w).

Explicitly, this periodic function is In 1/1 (x, w)

= ln(l

- x) -

! In x +

ln2 x 2 In w

00

+ ~)ln(l -

wnx)

+ In(l -

w nIx) - 21n(l - w n)

n=1

Notice that this function explicitly has the required periodicity mentioned earlier. Thus, when we exponentiate the Neumann turlctlq momentum factor in the integrand contains factors like

nn(Zi N

WnZj)kiok j •

n i<j

This function can also be rewritten in terms of Jacobi theta turlCtJ01l!~;: .

.

2

1/I(x, w) = -21fl exp(lJ!'~ Ir)

8,(~lr)

, (I ' 8, 0 r)

5.3 Harmonic Oscillators

185

Inx

~:::::2Jri' In w r=2Irj'

(5.2. 12)

are four Jacobi theta functions, given by

~

__ oo

..n, ~

= 2f(ql)q 1/ 4 sin Jr LI

(1 - 2q 'bt cos 2lT V + q"'), (5.2.13)

L ~

0 J (v I r ) =

q~ ' e21"~" ~

.., +

= f(q2) n (l

2q 2,, - 1cos 2:rr v + q411 - 1) •

~

f(q') = TI(I - q"'),

q =~If I,

(5.2.14)

'~::':'~~;:d~function. is thus

;:

confonnal im'atiance, we can also determine the factors a and dlL . ~~~:~ we will find it convenient simply to swnmanze the results of the ; i oscillator approach, which also arrives at the correct integmtion

Harmonic Oscillators convenient language in which to discuss the planar, nooplanar, and

~[~:i;;Single-l00P diagrams is the operator (anguage, which. ofcoursc. representation of lhe functional integral. Lei us begin with a strip

~

I

plane r + ia t the sets orall points from a = 0 to If. Now place with exlemallines along the x-axis. As before, if we take this

186

5. Multiloops and Teichmilller Spaces

to be our surface for the functional integral, then we can insert a COlmp of intermediate states everywhere: IX)

f

DX{.t (XI

=

1,

and we obtain for the functional integral

... f

DX I (XII VO IX 2)

f

DX2 (X21

1

Lo - 2

....

As before, we note that the above expression for the functional possible because the Hamiltonian is diagonal in the harmonic v~,,,ucnVJ By eliminating the functional integration at all intermediate points ....... strip, we obtain the first planar loop amplitude, in operator language:

Ap =

f

,)

d D p Tr[VoDVoD .. : VoD].

Carefully choosing the gauge for the multiloop calculation is v. u.v.u •••• tree amplitudes, we saw in (2.9.5) that ghost states do not couple to they can essentially be ignored. However, ghost stp.tes do couple to trees because of (2.9.9), so they can propagate internally in a loop are carefully eliminated. An identical problem occurs when qmmtizing Mills theory (which is not surprising, since both Yang-Mills and gauge theories.) There are three standard ways of eliminating the ghosts in loop First, we can insert projection operators which explicitly remove from the Hilbert space. However, this technique is rather cumbersome hibitively difficult for fermion loops. Second, we can allow ghost states to propagate and cancel the ghost state,s: Using BRST ~...~'" mal field theory, even higher fermion loop amplitliQes can be vu.'v ....,,,.v'" third, we can use the light cone gauge, which we will use in this ch~LPt~ The advantage of choosing the light cone condition is that it vastly the calculation. Ghost states, for the most part, can simply be H111I{"\r,'11 In addition to the light cone gauge, we will also choose a specific frame for the external legs of a multiparticle amplitude:

k+ =

o.

(We should caution the reader that certain complications exist WlLHU to the kinematics of enforcing this for an arbitrary number of external with arbitrary spin. For example, we may have to perform analytic tions of the external components of momenta in order to preserve A careful analysis shows that we can always choose this frame for 26 external boson legs and less than 10 supersymmetric legs. inherent problems, however, with the light cone approach because ways perform a Lorentz rotation on the vertex functions to a frame wbl~t~ij + components are nonvanishing. The light cone gauge is compatible

5.3 Harmonic Oscillators

187

...,/ ""Iw,,,,f uh, + components of the momenta. As a result, we will omit discussion arlhis delicate point in our discussiOlL) i~I1W""'ly. Ihis trace is easily evaluated using the coherent state formalism we used for the tree diagram;

I Ii,., ! dx}

dO P T rrVo(k .. x, )Vo{k1 , XIXl )' .. VolkN. X,XI •.. .IN }w"'-l ).

(S.3.S) (5.3.6)

trace can be explicitl y calculated using cohercIlI Slate methods. We usc

Tr(M)

=

~

f

d 2At-llf (AI M I'>").

(5.3.7)

the different fac tors, we have AN =

I Ii,., d DP

T - f(w)

I 1 dx/x: /2Ir.-

r,

(5.3 .8)

'-DO O~ 1(1- w"'-Ic}/)( l w-)2

(I

I<J ,. ... 1

ul"Cftl» ) i, '~J

(5.3.9)

00

few) =

0(1 - w"), ".,

(SJ.l0)

ell = Pi/Pi.

also explicitly perform the p integmtion:

Id 0• aP

I_ I

( 1m,; -

XI

-

-2Jt) n,. (" -

In w

0

1~I<Js.N

[e

-1 (1

JI

['

exp In Cjl 2lnw

JJ'" (5.3. 11 )

""')1 = I/I(Cj/, w), VI

= tnA / lnw,

(5.3. 13)

188

5. Multiloops and Teichmuller Spaces

and the f) function here simply orders the various Vi factors along This is the final result for the planar single-loop amplitude. l~011ce features about the result: (1) As predicted by the path integral method, the integrand is an au.~,~\, function. (2) The measure is easily evaluated in the harmonic oscillator aPl:)roz calculation is a bit harder in the path integral approach.) (3) The integral diverges at q = 0, which corresponds to the inner".v •.,,,:'~ ing to zero radius. The divergence is a mild one and can be eli·rnin:~~ we add superpartners to the strings. .

For future reference, we will define the planar (P), nonplanar nonorientable (NO) integrands at the same time: 2

_ 1- x ( In x ) Vrp(x, w) - .jX exp 2ln w (In2 x) _ I +x VrNP ( x, w) .jX exp 2In w

VrNO

(x,

W

)

2

( In x ) .jX exp 2ln w

_ 1- x -

CXl n n=1

CXl n n=1

CXl n n=!

{(1-

n n w x)(l - w /X)} (1 -;- ivn)2

{o{(1 -

n w x)(l +wn/X)} < (1 _ wn)2 ..... (_w)nx)(1 -

0 _ (_

These functions, in turn, can be reexpressed in a form in which Jacobi theta functions is more apparent: - 2rr. Vrp(x, w) = lnq smrrv

IJ {(I - (1(1-

2q2n cos 2rrv q2n)2

CXl

n_1

+ q4n) }

.

'

-4rr. 1 rrCXl {(1-2(-Jij)nCosrrv+qn) VrNO(X, w) = lnq sm2: rrv n=1 (_Jij)n)2 VrNP(X,

_ -rr -1/4 CXl w) - In q q n=!

IT

{(I - 2q)2n-1(1 _cos 2n)2 2ftv + q4n-2) } . q

Written explicitly in terms of theta functions, we have

Vrp(x,

vi) =

-2rri exp X

VrNP(X,

(;~::)

In x Ell ( - . 2m

In w ) I 1 ( - . El2rrz !

In2 X ) w) = 2rr exp ( 2ln w

° lnw) 2rr i '

5.3 Hannonic Oscillators

x E)' (_In_X _In_W_) 9,-1 (0 Inw) -

2;rri

"",(x" w) = -2rri exp

x9 1

2;rri

2:rr i

I

(In'

Inx

( 2n,

-.

189 (5.3.21)

'

x )

21n w

lnw

-2rr-j

[)"'_1(

+2

\,"JI

0

In WI) -. + -2 . (5.3.22) 2Jrl

that we have calculated the planar single-loop graph, let us calculate . single-loop function. The trace we want to evaluate is

ANO =

JdDpTr(~VoDVo'"

VoD),

(5.3.23)

n is the twist operator in (2.7.13). Notice that the twist can he placed along the chain, and nothing changes. The trace can be evaluated same coherent state techniques, and the major change is that w turns The final result is

(5.3.24)

that the imegralion region is twice the usual one from 0 to I. This is for the external lines to go completely around the Mobius strip, the go around the edge twice (see Fig. 5.5). This fact will eventually crucial role in the cancellation of anomalies in Chapter 8. A

8 A

-

L-,.-----'

8

8 A

L -_ _- '

A

-

-

A

8 A

8

L-----r---'

A

8

A

L -_ _--'

8

5.5. Integrating around a Mobius strip. For the Mobius strip, an external travel twice the length of the strip in order to make a complete path around Thus, the nonorientable strip has an integration region that has hvice

190

5. Mu1tiloops and Teichmill1er Spaces

The nonplanar diagram can also be evaluated. Notice that placing ...... .. number of twists on the loop separates the external lines into two ....la"".," that revolve around the outer edge and those that revolve around the final answer depends on which lines are on the inner and outer ""'E,"" ..• .-:; place the twists so that the first to the Lth lines are split apart from the ANP

f

= .

D

.

d pTr(QDVJD··· VLQDVL+ J ··· DVN ).

The integral can again be perfonned, with the answer [15] ANP =

1 1 -t N-J IT dVi

R i=J

J

0

d

q

(

2

-In 7r q

2)N (f(q2)r24 IT(VrNP,ij ,~ or i<j

where we use VrNP if the ij lines are on opposite sides of the disk they lie on the same side. The integration region for the lines reflects that there are two disjoint regions of the disk. The external lines are mt,egt sequentially, except that there are now two disjoint regions. For convenience, let us now put all three amplitudes in the same expressions: ,

where J = P, NO, NP,

Rp = {O < VI < ... :::: VN = I}, RNO = {O :::: VI < ... < VN = 2}, fp(q2) = fNP(q2), fNO(q2) = f( _q2), q

4 = qp = qNP = qNO'

and where the nonplanar region of integration reflects the "fact that two disjoint regions of integration. Now that we have written explicit representations for the various ( string amplitudes, let us draw some rather remarkable conclusions . amplitudes.

(1) Closed Strings from Open Strings One of the strange features of the nonplanar single-loop diagram is that more poles than those found by factorizing on the usual open string [ 16]. By examining the factor

5.3 Hannonic Oscillators 2

19 1

3

• 4

1 2

3

1

4

5.6. Emergence or the closed string theory rrom open strings. A curious the open string theory is that, at the first-loop level. it already contains string theory as a "bound state." The nooplanar diagram ror open strings stretched unlil it becomes a cylinder, which in tum can be ractorized into two cylinders. Thus, the intermediate state must be a closed string. the ~s factor comes from the momentum-dependent integrand) we find there are extra poles at s/4:::: -2, 0,2,4,6, ... , which are precisely Ihe of the poles of the closed string sector. Tbus, the open SIring sector I the closed string sector. This can most easily be shown dual diagrams by thinking of the nonplanar diagram being a cylinder, with sets of extcrnal lincs that can rotate around the top and bottom edges of However, by factorization, we can slice the cylinder horizontally, the. intermediate state is a closed loop. Thus. the closed SIring emerges "bound state " o/the open string sector (see Fig. 5.6). The closed string by itself, is an entirely unitary theory. However, the open string sector, is not. We see that the presence of open strings demands the existence strings as intennediate states. or else the theory is not unitary. In fact, by Lovelace [171. this unwanted singularity in the complex plane oonplanar diagram is 8ctu81ly a cut (which would be disastrous), but )"". 'pole only in 26 dimensions. In fac t. this was the first indication that model was consistent only in 26 dimensions.

:h'h"

Renormalization

"'" th," the divergence of the planar diagram arises from

l

o

divergence at

Q --Jo

ldq , QJ

l idQ. 0

(5.3.30)

Q

0 corresponds to the hole in the disk shrinking to

"~i);~'~f~~;~;~;~:,:"~iSCS from the fact that we arc summing over an infinite ~

stales propagating in the interior or the loop.

192

5. Multiloops and Teichmiiller Spaces

This is not, however, the ultraviolet divergence that we usually with Feynman diagrams. For point particles, the divergences of amplitudes arise when we deform the local topology of a particular .. such that a propagator shrinks to a point. Thus. divergences are associated with deformations of the local topology of the graphs. } In string theory, however, because of conformal invariance, we '.' a propagator to a point. Thus. conformal invariance on the world out ultraviolet divergences. However, we still have the infrared diverli:et the interior points shrinking to zero. Again, conformal invariance tells us that we can always map the ing hold into a dilaton or tachyon vanishing into the vacuum. We . "pinch" this shrinking hole and extract a closed string resonance with quantum numbers vanishing into the vacuum. Thus, conformalm'lVaJ~Ul1Jl()1 us an entirely new interpretation of the divergences of string theory. interpretation associates with each divergence a closed string state "~,i"" off from the hole, with zero momentum, corresponding to tachyons divergence 'or dilatons for the q - I divergence vanishing into the va(:uUti! Fig. 5.7). When we pinch the shrinking hole and extract a closed string vanishing momentum, we notice that the remaining diagram looks tree without the hole. Thus, by extracting out the divergent pole C0I1ID what we have left is a finite tree. This, in tum, allows us to treat the as a redefinition of the free parameter, the Regge slope a'. This prc)ce!,s "slope renormalization," works fine for the dilaton q-I pole, but it is how to handle the tachyon pole contribution q -3 . Thus, the bOSOnIC might not be totally free of divergences.

(3) Finite Superstrings Experience with supersymrnetric loop calculations for point particle has shown that the internal bosonic line cancels against the internal line, yielding amplitudes that are much less divergent than expected. thing occurs for superstrings. Let us now tum to the superstring graph:s.? we will be able to perform this "slope renormalization" [18, 19] ~hIJU',H~j will find that the q-3 divergence in the open string Type I theory "''''TIN''. itself (this is also expected because the theory has no tachyons). only the q-I pole, so slope renormalization is possible. The truly renml1' thing, however, is that the Type II theory is actually finite by itself, slope renormalization!

5.4

Single-Loop Superstring Amplitudes

The single-loop open superstring can be calculated in either the GS NS-R formalism. In the GS formalism, we might use the light cone forl[jj'g

5.4 Single-Loop Superstring Amplitudes

193

o

5.7. Emission of the dilaton. By confonnal invariance, we can defonn the string diagram and pinch the interior loop. The resulting diagram 10 the emission of a tachyon or dilu ton into the vacuum, which results divergence. Super.;tring theory is constmcted so that these poles do so the theory is fonnally finite.

,,~~::~,:,~; satisfactory covariant onc does not exist. The advantage arthis ~

however, is that the graphs are manifestly space-time supersym. In the NS-R fonnalism, we must either use a projection operator to

bu,,, out the ghosts or use the SRST techniques that allow the ghost to prop-

and cancel the negative metric ghosts. Unfonunately, supersymmetry is I",mifes, until we add bosonic and fermionic loop contributions separately insen the GSQ projection operator into each loop. us use OS fonna lism developed in Section 3.9, where supersymmetry As before, the massless vector boson vertex is

(5.4.1)

Bi(r) = p i(r)

+ k J R1j(r),

Rij (r) = ~S(r)yij- S(r), 00

S'(r) =

L "", - co

S:e-'''·.

(5.4.2)

194

5. Multiloops and Teichmiiller Spaces

where we have also set t + = 0 for the polarization vector of a ma.ssl'es.$ The fermion vertex is

where Fa(r)

= i(p+)-1/2(S(r)y.

P(r) - ~:Rij(r)k;S(r)yj:t.

Let us consider only the trace over external bosons. There are a greati simplifications. For example, the trace over the So operators eight of these operators. Thus, amplitudes with two and three f>yt,>rri, vanish all by themselves. As a result, there are no self-energy corrections at all in the theory. The first nonzero amplitude occurs for four external legs. Even amplitude actually vanishes, except for the contribution from the

t; k j R~ :e ik .X :. In fact, the only diverging contribution to the trace is '. Tr(wDI/2)n.LuY-Su) = f(W)8. ··0:::.>:.

As expected, this cancels against the other contribution coming from .•••> loop. Thus, the final single-loop open superstring amplitude is .

Aloop

=

K

t Jo

n

8(vl+I - vI)dvI

1=1

t Jo

dq TI(1/I/J)k l .kJ , q 1<1

where K is the same kinematic factor (3.9.11) found in the tree external massless boson legs. As expected, the graph diverges because there are no tachyons in the theory to give us a q-3 diverjl~enld:e Let us now extract out the infinite piece of this graph. By carefully . •.•.. the finite piece near q = 0, we find that the 1/1 function reduces to an sine function, so that the finite piece A becomes ...

To show that we can perform slope renormalization on this graph, let perform the V2 and V3 integrations. If we define sin Jr (V2 - VI) sin Jr V3 x = sin Jr(V3 - vl)sinJrv2' then, after the integrations, we find A\oop'"

K

{I (InX + In(1 -

Jo

1-

x

x

X)) x-a's(l- x)-a'r dx.

5.S Closed Loops

195

checked that this, in rum, is precisely the derivative afthe Born term r.,,~C110 the slope. Thus A,

's

I,

a

00p

I -----A (a,)2 oCt' 1=-

(5.4.ll)

we can absorb this divergence into a renormalization of the slope. s~;,~;::;:~ result, thal a redefinition ofthe Regge slope can render the

theory renormalizable al the first loop.

us consider the case of the closed string amplitude, where even sw~ri"s

take place.

Closed Loops a straightforward process to calculate the single-loop amplitude of bosonic string. Let us stress the differences: world sbeet of the string, which was topologically equivalent to the

>P" h,lfCpllane for the open string case, now becomes the entire complex

;'~";n "n;n',g...;on 0'''" "othat it is independent :ti~:'::~~~:':;:~:~~, of space. 0'

must sum over different orderings of the vertex functions. extemallines, which were once attached to the boundary oftbe strip, not attached to the interior of the complex surface .

= 0 to a = 21l' , such top and bottom edges are identified with each other. For the single we must now also identify the left and right edges. By the exponential we then map the finite horizontal strip to the entire complex plane. the usual coherent state methods, we find [20J (taking the slope ex' == ~)

. 5.8, we see the horizontal strip defined from

Cf

A= jd Dp Tr (V! DV2 ... VND)

(5.5.1)

Vj

= {21l'ir! InZ I l. 2 '··l.j.

VJi

==

v} -

T

=

V,w

w= eji

==

V;.

== (2ni)-lln w,

ZlZ2 "'ZN,

Zi+1Z/+2" 'Zj,

(5.5.2)

196

5. Multiloops and Teichmiiller Spaces

t a

A ,...--...........-----/---+---, A D 'D

C

C

8

8

a = 27r

A

A

8

z=e

T +

ia

FIGURE 5.8. Conformal surface for the single-loop closed string dmgra.n:i.' plane, the surface is a rectangle of width 2n and arbitrary length, with edges identified. In the z-plane, the surface corresponds to a doughnut. can attach themselves to any point within the surface. and 2

(

)TI OO

m

)_ (ln lzl) -1/2(1{(l,-W Z)(l-W X z, w - exp 21n Iwl Z Z m=\ .,. (1 _ wm)2 Xij

= X (Cj; , w) = 2n exp

(

-n(lm V.;)2) 8\(v·; I r) J J.. 1m r 8;(0Ir)

Let us rewrite this as

where

Notice that the integrand is doubly periodic:

xCv + 1, r) = xCv + r, r) = xCv,

r).

m ;

5.5 Closed Loops

197

easy to see because the theta function has the following properties: 8 1(1.1+ II f)=8 1(v I f), 9 1(1.1 + fir) = _e- l lI'(2w+f)8 1(v

I f).

(5.5.6)

' is important, because it shows that a naive integration over the v varidrastically overcounl the proper region of integration. Consider the ~1~log~~formed by the origin and the points 0, I, f, and I Y. When are identified, this becomes topologically equivalent to a torus )Ughm... Notice that the double periodicity divides up the complex p lane infinite number of these parallelograms. Thus, we want to integrate only one parallelogram. or else we will have infinite overcounting. Thus. we must restrict the integration over the v variables or else we will over an infinite number of copies of the same thing. We w ill I

+

0 :5 1m

Vj ::::

1m Y,

-!
(5.5.7)

';"'gly. in addition to this truncation in v space, we must perform yet truncation in r space. The integrand of the closed-loop diagram is invariant under yet another ttansformation. given by

,

ay

+b

, -- c::--,--; cr +d'

(5.5.8)

C, d are al l integers and ad - be = I. This generates what is cal led ::::~~:;';: .SL(2. Z). Let us show that the integral is invariant unde r d1y _

le r

+ d l- 4 d 2y,

Imr ~·.IC't+dl -2 1mY.

(5.5.9)

1",,10""",. following are actualty invarianl under this modular transformad'f d 2r Tm2 r = 1m2 T .

(5.5. 10)

lei us calculate how the other lenns transform under a modular (5.5. 11 )

(5.5.12)

X(

b) cr+d

"a, + .

cr+d

=Ier+d l-'x(v.y}.

(5.5.13)

198

5. Multiloops and Teichmiiller Spaces

Thus, F(i) = F(i).

The integrand is therefore invariant under a modular transionnaltiof was first pointed out by Shapiro [20]. But what is the intuitive me:anll~: symmetry? The Polyakov action tells us that we must integrate over all cOll,tbti inequivalent surfaces. We first observe that two parallelograms values of i are confonnally inequivalent when we identify .... Thus, naively we expect that the integration over i automatically·· over all confonnally inequivalent surfaces. However, this is not the . .• There are actually two kinds of reparametrizations of a surface carefully distinguished. The first are the reparametrizations that be defonned back to the identity, i.e., the set of smooth rep'aram~1 that contains the identity map. The second is the set oflreparame:tri,~al cannot be smoothly defonned back to the identity. Global dltJteolmolrp of this category. For example, take the parallelogram and identify of opposite sides. This creates a tube. Nonnally, we would bring of the tube together to make a torus. However, now twist one of the of the tube by 2rr and then resew the ends together. By examing surface, we find that this has caused a genuine reparametrization but that the identity map cannot be represented in this way. This a "Dehn twist" and it generates a discrete group. For the torus, it that the group generated by Dehn twists is the modular group Sp(2 Concretely, the integral is invariant under, i -+ ;-1 Ii and i -+ successive applications of these two transfonnations, we can generate the entire modular group. But carefully examining the two transfonnations shows that they simply interchange the b01Ullcia parallelogram, generating Dehn twists. In summary, this second symmetry, called modular invariance, the fact that we must gauge-fix not only reparametrizations that reach the identity map but also global diffeomorphisms that are not to the identity map. Thus, we must divide up the complex i-plane only integrate over one surface invariant under the.transfonnations and i -+ -liT. Thus, the complex i-plane is divided up into an of redundant copies. To eliminate this infinite overcounting, we following fundamental region of integration:

fundamental region

(see Fig. 5.9).

~

I

I -2I < Re i < 2' 1m i > 0,

Iii 2: 1,

5.5 Closed Loops

199

I -j

j

5.9. Fundamental region for the s ingle-loop closed string amplitude. Modol"h~ amplitude divides (he complex plane into an infinite number of regions. Thus, ..... e must choose only one such regioll, oreise the amplitude and with ITI . The most convenient region lies between ReT = gr<~"''';h." one.

"",ri,"'"

-i

+!

will shortly see that modular invariance is perhaps one of the most tools we have in checking for the self-consislcncy of new sIring said, !.he effect of a modular transformation is to perform a Dehn ; ;",., !,hume the boundary conditions on the parallelogram. For example, have a Siring defined on the parallelogram labeled by X(UI ' 0"1), then a transformation changes the boundary conditions by (5.5.16)

];",Ily', w " , , " ,eh'"kth''' tl" ,,,,,.f,,",,,,,;,o", r --+ t the boundary conditions in the following fashion: T -+

r

+1

r -+ - l /r

--+

+ I and r

( X(CTI,U2)

-+

X(O'I +(12 , (12),

X(O'"I ' 0'(2)

-+

X(O'"l' - 0'1)'

-,I>

-

LIT

(5.5. 17)

let us analyze the divergence structure of the closed string amplitude.

"' """ thaI x(v, r) -+ brlll i

(5.5.18)

O. Thus, poles occur, as expected, in the amplitude when external lines The poles occur at s = -8, 0, 8, 16, etc. This divergence can be as coming from the self-energy diagram on the external leg, which, I is on mass shell. this calculation to the Type II superstring, which call using the same techniques. TIle vertices aU have the form W = VV,

\' ~~: ~

.~;,: ~;~~;~~~~;~~t:o OpeD string vertices

(5.5.19)

(with half the momentum) . Once again, the two and three point loops van ish

200

5. Multiloops and Teichmiiller Spaces

because of the trace over the So modes. The trace can be evaluated in the open string case, except we now have double the number of~~~'uI< The answer for the boson scattering amplitude is Aloop

= K

where Fs(r)

= (1m r)-3

i

2

d r(lm r)-2 Fs(r),

fn

d2VI O(X/J)(1/2)k 1okJ.

1=1

I<J

Notice that the factor C( r) is missing and that the power of 1m r has The remarkable thing is that this amplitude is totally finite! This UUIL".~' due to several factors: ~

..

(1) The absence of two and three point amplitudes makes it im))osst place tadpole or self-energy insertions on the external lines. not have the poles found earlier for the closed bosoIlic string. (2) We can reduce the divergence ofthe graph by taking a fuIlldalmellltalld free of divergences. (3) There are no contributions from tachyons vanishing into the there are no tachyons. (4) Fermion internal lines cancel with boson internal lines to divergence of the graph.

5.6

Multiloop Amplitudes

The multiloop function can also be written explicitly in tenus of path over Riemann surfaces with holes. The main problem with """ct."'l',r"""", amplitudes is the choice of parametrization of the Riemann surface. ofparametrizations have been developed for the multiloop i:lllJVllLUU.C",

(1) Schottky groups. Multiloop amplitudes, which were originally in this formalism [2-8], are discussed in this section. There advantages to this parametrization of a Riemann surface. First, it is There is no guesswork in the choice of integration variables, known exactly. The integration variables, in fact, are intuitively the topological structure of the Riemann surface. Second, the factorize (because this representation was originally calculated ...... . together multiresonance vertex functions). Thus, unitarity can be . . The disadvantage of this formalism, like other formalisms, is that invariance is not obvious at all. The integration region must be by hand. (2) Constant curvature metrics. The formalism, which will be the next section, is based on Riemann surfaces with constant

5.6 Multiloop Amplitudes

201

The advantage of this fonnalism is that it arises naturally when the

action. The other advantage is that a considerliterature exists for Riemann surfaces with CXInstant The disadvantage of this approach, as in the Schottky repreis that modular invariance of the higher loop amplitudes is still . The integration region must be truncated by hand. Unlike the representation, however, explicit representations of the 6N - 6 parameters for arbitrary surfaces with constant metrics are rare. because this formalism is not derived by sewing together vertex functions, factorization and hence unitarity are not functions. This is perhaps the most natural fonnalism. because

od"I" invariance is built-in from the very start. We will discuss it in 5.11. This method is based on generalizing the theta functions '",do-ood in (5.2.11) for the single-loop amplitude to include functions are quasi-periodic in several variables. The natural integration vanis the period matrix fl lj itself, which can be defined for any Riemann This fonnalism can also be easily extended to include theta funcdefined on various spin structures. Although this formalism holds promise and is the subject of much research, there are also severe awbach. For example, beyond three loops, the period matrix becomes extremely awkward method of parametrizing moduli space. (This dif. moduli by the period matrix beyond three :' Only recently have mathematicians this problem. Unfortunately, the solution is highly nonlinear, and more work has to be done to develop the fonnalism beyond three As in the previous formalism. factorization (and hence unitarity) obscure. Higher loop theta functions are constructed by making eduguesses and appealing to the uniqueness of the final result, not by together vertex functions.

g:,:::;~~[;~~~:::::; Because the light cone method is based strictly on variables, without any ghosts, this fonnalism is manifestly unitary hence we intuitively expects thaI it automatically provides one cover ,f.::~:~;:;:1:;:,ThiS conjecture, which was never considered before by :u in their smdy of moduli space, has recently been proven orders. We will only briefly discuss the light cone formalism at the of this chapter because this fonnalism will be further developed in next chapt~r in the context of developing the field theory of strings. advantage of the light cone fonnalism is that it is manifestly unitary, . I . to a genuine second it is obviously gauge-fixed. l)i

four fonnalisms, of course, must evenmally yield equivalent results. first discuss the Schottky groups and rewrite the single-loop amplitude that can most easily be generated to the multiloop case. (We will

202

5. Multiloops and Teichmiiller Spaces

discuss the theta function method in Section 5.11). Let uS define a transformation (2.7.1) that forms the group SL(2, R), in away that its geometric properties. Let us define the invariant points XI and projective transformations to be the points that are left invariant transformation: . P(XI) = XI,. { P(X2) = X2 ..

invariant points:

Then we can rewrite an arbitrary projection transformation, which arbitrary parameters, in terms of the two invariant points and the P~)=

Z(X2 - XXI) - xlx2(1 - X) . z(1 - X) + X2 X - XI

Another convenient form for the projective transformation is P(z) - X2

----'--'---- = X

P(z) - XI

z - X2 . z - XI

The advantage of writing the projective transformation in terms of plier is that products of projective transformations have simple UH'H'I! (X P

r=

X(P")'

X PQ = X QP , (Xpr l = X(P)-I.

Under a projective transformation, any P can be brought to the form'

Notice that the trace of P can be written as 1

Tr P(z) =

.JX + v'x'

This, in tum, allows us to define "conjugacy classes." Two pf()1ec:t1v,etr mations PI and P2 belong to the same conjugacy class if they have multiplier. Depending on the multiplier, we can define several types of transformations: (1) (2) (3) (4)

P is hyperbolic is X is real, positive, and not equal to one. P is parabolic if X is real and equals one.

P is elliptic if IX I equals one and X does not equal one. P is loxodromic if X is complex and none of the above.

For a real projective transformation, the multiplier can be gre;ater~J less than one, depending on whether XI < X2 or vice versa. We lUU." ....., freedom to choose all our projective transformations to be hV1Der·bO.l1C~:

5.6 Multiloop Amplitudes

203

jp;,:,::';,:~;o~;: one. For the single loop, we now recognize the projective ;Ii to be

P(z):: wz

./W P(z) ~

0

(

l ).

(5.6.7)

./W '

tbe multiplier of the single-loop transfonnation is simply w itself. that we have 3N parameters associat.ed with N projective transfor-

However, we noted earlier thai three points along the real axis can be fixed because of projective invariance. Thus. the open string N -loop will have 3N - 3 parameters that describe the surface. These are

Teichmiiller parameters and are the minimum number of parameters to characterize inequivalent Riemann surfaces with a boundary. For ,[rn"d string, we have spheres and complex projective operators, so we - 6 parameters. In summary:

I

Teichmuller parameters:

6N - 6,

closed string:

open string:

3N - 3.

(5.6.8)

"j""II:y, we may say that it takes two parameters to locate the position of the hole and one parameter to label its rodius. Thus, 3 N pammeters are

"!,:;~;:~~:~~:~~:;~~~,'~~ N holes. We must subtract three to eliminate ~I

equivalent ways of putting the exlemal lines on the real axis. This

a total of3N ~ 3 parameters.)

us now rewrite the integrand of the single-loop function in terms of ''''y ;~vari'ao' funclions. From (5.6.4), we note that the partition function rewritten as

nt' - w") ~ n(J -(X,),) ~ n(J -X,. ). ro

".,

00

00

n=1

(5.6.9)

"=1

the partition function is now written in terms of the sel of projective po. we can rewrite the momentum integrand in terms of projective Before, we showed that the combination dlL TI(Zf - Zd ,·kl !l

(5.6.10)

j <j

fonned nicely under a projective transfonnation because of momentum '''';atj,m and the fact that the extemallines are on-shell . Now, we want to that the single loop can be written as an invariant integrand:

n

(z/ ~ w"Z; )";·ll 6. "" dtL

n

n.. U<j

(z; ~ P"z j l ,·kl !l.'

(5.6.ll)

204

5. Multiloops and Teichmiiller Spaces

FIGURE 5.10. Action of a projective transformation. Under successive of the same projective transformation, circles surrounding one' invariant into circles surrounding the other invariant point. After an infinite """""."" jective transformations, a point on one of these circles comes arbitrarily invariant point. Thus, the single-loop integrand can be rewritten as

f D(Zi dJL

pn Zj l;·kj (1-

,

Xp,,)-24~.

i<j

It is crucial that we identify the region of the complex plane integrating over. In Fig. 5.10 we see how the upper half of the cOlnple is divided into equivalent sectors by the action of the operator P.

a projective transformation maps circles into circles. Thus, a point invariant point, after being repeatedly hit with P, gradually ffil·Jgra1teSi the other invariant point. In the process, the upper half-plane is ;:'11\.,"",,:, disjoint regions. We can take any of these disjoint regions for our the figure, we have taken the region that is midway between the points. The arrows show how a point migrates under P, i.e., we see tification of the points along the surface. When we add in external lie only on the x-axis and can only move up to the edge of these This procedure easily generates to the multiloop amplitude. We (1) Projective invariance under SL(2, R) transformations. There Sfi()U~1 projective operators for the N -loop amplitude. (2) Modular invariance for the closed string, so we integrate fundamental domain in parameter space.

From these invariance arguments alone, we c,an almost write multiloop amplitude. First, let us write the multiloop divergence. Let

represent N real projective transformations, such that their invariant ....... arranged sequentially on the real axis. Let {P} represent the set of distinct products of the various P transformations, raised to any

5.6 Multi]oop Amplitudes

205

B

A X~I

5.11. Schottky representation for the double-loop open string diaglllm. are two sets of invariant points lying on the real axis, corresponding to each twO projective transformalions. The circles surrounding the invariant points

identified with each olher. i~~:0"~ (In Fig. 5. 11 we see thai the invariant points are all aligned on 8: The set {P) fonns a group. Our region of integration will be the

of the real axis that exists between the limit points of the elements of the timit points of all the elements of {P} fills up the entire complex then this is uninteresting. Then there is no region of integration for our interest lies in groups (P} whose limit points fonn a discrete set on the , so there is a finite region of integration. Groups with this property

Scholt/g.' groups. [P} represent all distinct products of these transfonnations modulo permutations. Then the dive rgent term for the N -loop amplitude is a over the set (PJ [4, 5J:

n( l

~

xjP}~ )-"

(5~6.l4)

IP,

momentum-dependent term is given by a product over the set I PI:

Tl

(z/ - {P}Zj)UI .

(5.6.15)

i<j.{P)

integrand for the N -loop amplitude is therefore surprisingly easy to The only difficulty is determining the region of integration such that p"o,joo'ivo invariance. The final formula for the N -loop amplitude external tachyon legs is [2--8]

",,,ve

AN

~

f fI

a= l

d"k. d"

nn(l {PI

(PI

X",)-"

206

5. Multiloops and Teichmiiller Spaces

where

n

M

dll

I"""

=

N

ndZ' dV- 1 dx(l)dx(2)(X(I) l abc a a a i=1 ,,=1

X(2))-2 a '

where the Roman letters i and j represent the external tachyon uU"". letters represent only the loop variables. Each projective two invariant points and a multiplier: . X(2)

x(l)

"

'

"

X".

'

The product over {P} is taken over all inequivalent products of operators P". (We exclude those products where there is ov(~rC'oUllur member of {P} is a product of transformations that ends with the P", then it cannot be allowed to operate on the invariant point of we will overcount.) The essential aspect of this integrand is the region of integration, count each conformally inequivalent configuration once and only On the real line, we can always make a projective tra:nstorrnation all the external lines are bunched together on one side and all points are bunched together on the other side. (If a point z lies of the invariant points of a projective transformation P", then P z across its invariant points to the other side. Thus, by SUl;ce:SSl've :p) transformations we can shove all external lines to one side of and all the invariant points to the other side.) We must be careful, exclude the possibility of a projective transformation that whips a way around the diagranl. The transformation

has the property that it moves points all the way around the real the invariant points. Therefore, we must be careful to extract the . ..... that enters in by circling the entire real axis an al'bitrary number .. '. Let x(1) and ~(2) represent the invariant points of this product. the limits of integration: x(1)

<

P(ZI)

<

ZM

<

ZM-I

< ... <

ZI

<

X\I)

< X~2) < x~1) <

with the restriction that all multipliers for P", including the prodU(;t less than or equal to one. Of course, there are other equivalent chIQIc:e,l

5.6 Multiloop Amplitudes

207

i.,nn"!:,,,,;on because we have taken a specific truncation afme region. we could have extemal lines placed berween the invariant points projective operators. Because the set I P} was a Schonky group, (;,':~~:'~~ that there is 8 finite region of integration for our variables. out that the calculation of the multiloop amplitudes was done using projection operators that eliminated the ghosts on the leading With BRST ghost fields. the calculation can be redone to eliminate ;bl, ghos~ . By the uniqueness of the Neumann function over a surface, same answer.) we have discussed only open string multiloop amplitudes. Now, let these previous statements by actually writing down the Neumann for a sphere with N handles or holes for the closed string sector [2 1]. first consider the complex plane with 2N holes cut out. Call this .· ~"c~;: I~i:es exterior to 2N holes, the region S. Let us call these pairs of .( . and ail where; = I to N. Projective transformation s, we have circles into circles, so let us define N projective transfonnations PI us from one a -cycle into their partner (see Fig. 5.12)

II;

;rnI;"

(5.6.21) transformations Pi, of course, can in tum be parametrized by points 4~11 and Zl2) and by the multiplier Xi. We will Ctnd thaI I points lie within each of the a-<:ycics, so they do not appear in plane with holes cut out. In general. a point that lies within the region exterior to aU a-cycles) is mapped into the interior of one ~ by the action of a projective transformation P,. to the a-cycles, we also have b-cycles, corresponding to cutting "'"w."n a, and The b-cycles are thus circu lar lines that encircle when analyzing a sphere with N holes. Notice that the projective z --t wz in the single-loop diagram took us from the interior radii of the annulus. Thus. the projective transfonnation moved the b-cycle. In genera l, the projective transformations P, will map one a-cycle into its partncr, so that we arc moving along a b-cycle. now define VI 10 be the product of all distinct products of the various that i mns from I to N, while I runs over an infinite number of Now let us define the function", :

0·')'''."

a,.

,',(

')

.,.. z. z

~ In( _ ') + " ' In (z - VdXi - V'Z) z z ~ (Z V)( V/Z ,)' I /Z t! -

(5.6.22)

prime in the summation means that we include either VI or V,- s but Normally, we would expect to define the Newnann function as N(" ") ~ In 1"'("

, ')1.

(5.6.23)

this function actually does nOI have the correct propenies. We wanl automorphic functions that change only up to a constant wben we go

208

5. Multiloops and Teichmiiller Spaces

a, 8,

( '. a3

8, a2

I

..

a,

a2

0

0

I

I

I

l b,

I b2

a3

0

\

I

6

0

8,

112

\ b3

b ·1l3

FIGURE 5.12. Homology cycles of an arbitrary closed Riemann surface. ··.•• .. shrunk to a point. By cutting the surface along the a-cydes, we can flatten '.' into a plane with 2g holes cut out. Notice that the b-cycles are now lines . pairs of holes, which are defined by the a-cycles. ' b cycles are 2g closed lines on the surface of genus g that cannot be

around the various loops in the z-plane. The above expression does this periodicity property, which was the crucial criterion for the function. Thus, this function cannot be an automorphic function. The single-loop function, by contrast, did have this property. NOUClili:' function In z, for example, changes by 21l'i when moving around In(ze21ri )

= Inz + 21l'i

and by a factor ofln W when going across a b-cycie:

In wz

= In w + In z.

These factors are called periods of the function In z. The ratio of periods is

--=. Inw

21l'i

5.6 Multiloop Amplitudes

209

is called the period matrix. Thus, r is (he period matrix for a single way, we can show that the entire single-loop Neumann function :esonly up to a constant when traveling around the a- or b-cycles. that the previous N -loop candidate for the Neumann function ;";,n".h,,. traveling around the rth a-cycJe; 8"ft(?. t') = O.

(5.6.27)

we can show that this function. when we travel around a b-cycle by ng" "'arufmm'''ion ? -+ P,( l), is not invariant. It is not periodic because (5.6.28)

_ ,", (rl l

( )- L lirl

I

l-

n

V

(I)

Il,

(5.6.29)

(21

,-Vll,

r~~~!~J:SY'nb<>1 (r) simply means that, to avoid double counting in

~

we must delete any P, that might be acting on the . I and c and d correspond to the faclors found in the original of the projective trnnsfonnation. When we take the function v,{z) , around the s th Q- or b-cycles, we find

.5.... I),(?) = 21rili". (5.6.30)

Dlnli,(,) = 2niT".

I TfS

L (,sl l

= 2rri

I

V,,(I)X,(l) - V, ,(l») S n (d ) _ V'Z~1)X?;2) - V' Z~I»' ' (Z ( I ) S

f

f

(5.6.1 I)

Tfl> which we by taking lI, around the s th b-cycJe, is again 11", p,,,j,,,hna,·"x. which is a natural generalization of the period matrix earlier for the single-loop diagram in (5.5.2) and (5.6.25). the generalized definition of the period matrix, we can write the com~"'m'mn function that has the correct periodicity properties. We modity

. In *(z. z') = In

t(z. z') - 2rr ~ 'L" Re(lIf(z) "

1J,(z'»((Im tTI)n (5.6.32)

Neumann function becomes N(z. z') = In I"'(z, z')I.

(5.6.33)

can put everything together and write the N -loop amplitude for the string. The changes that occur from the single-loop amplitude are;

210

5. Multiloops and Teichmuller Spaces

(1) The single-loop partition function now becomes

where we take the multiplier of the product over all inequivalent classes of VI. (2) The factor (In Iwl)-l, which was related to the period !HaLHA,. replaced by the determinant of the N -loop period matrix: det I 1m Tlrs. (3) Another factor of

n

Iz~l) - z~2)1-4

r

occurs in the amplitude.

where we integrate over the fundamental region F of the surface with

5.7

Riemann Surfaces and Teichmuller Spaces

Although the previous calculation gave us an explicit formula for the amplitude in terms of Schottky groups, we also had to be careful in the region of integration so that we avoid overcounting. This an inherent problem with the Nambu--Goto formalism, where the integral does not uniquely fix the region of integration. The Polyakov formalism, however, provides a way in which we inate the overcounting from the very beginning, using powerful theot¢ Riemann surfaces. One of the greatest advantages of this IVl.li1<111"!l~, we can bring to bear the full force of the last century's worth of cal research 011 Riemann surfaces. Of special importance will be the the determinant found earlier, which contains the singularity stnlcttlte:(tlU N -loop diagram, can be expressed in terms of the Selberg zeta fiulctl(:il Recall that the Polyakov action is given by (2.1.35) L = -1 2rr

f

d 2z.Jig ab aa XlLabx/L'

5.7 Riemann Surfaces and Teichmuller Spaces

2 1I

generating functional is

Z

='

L !. lOpOla&ies

.

Dg""

,neme

Jr .

s DX"'e- .

(5.7.2)

cmbtdd,nr,s

"re we must be careful to divide oul Riemann surfaces that are equivalent to by a confonnal transformation. We wi ll follow the derivation of Alvarez

for closed S[fings. functional integration over the string variable X is a Gaussian and hence to perform (see ( 1.7. 10»:

(5.7.3)

Vl =

- I -a "'ggm'a~, ..Ii '"'Ill>

(5 .7.4)

' : ~:,';,~~:e~:~. the delenrunant will always mean deleting the zero mode. ill

function (5.2.14), which contained the divergence of the single comes out of this detenninant.

ow,ov"., the functional integration over the metric is considerably more because of the presence of gauge parameters and also the presence . As we saw in (2.4.1), the measure is invariant under 2D general

iari.m,..md rescaling: (5.7.5)

·we""",•• h" C \ri,,,oifel symbols, we can rewrite this expression covariantly: (5.7.6)

parametrizes a Weyl rescaling, and OVa parametrizes a reparametrizaoor",,, two-dimensional surface. In general, this means that the measure of over gab is actually infinite. Notice that gab has three independent and that 5va and du also have three components, so that naively we can set all the components oftbe metric to the delta function:

(5.7.7) choosing the conformal gauge is equivalent to factoring out the inteinfinite volume due to Weyl rescalings and to two-dimensional over the surface. Gauge fixing thus means replacing the integration over metrics with

Dg"h -

Dgab(QOitr) -l(flweyt)-l,

(5.7.8)

the infinite volume of the space can be represented as (sec (1.6.7» QD iff

=

f

OVa,

212

5. Multiloops and Teichmiiller Spaces QWeyl

= / DO'.

For spheres and disks without holes, this is actually· true. surfaces with larger numbers ofloops, or handles, this isno longer true . .•••.. • of complications due to the parametrization of the loops. . In general, a sphere with N holes or handles requires a number eters to describe the location and size of each hole. Basically, for a holes we need one parameter to label the radius of each hole and parameters to give us the coordinates of the center of each hole. 3N parameters to describe N holes. (For a sphere witli!. handles, we .. . complex parameters to describe N pairs of holes.) As we saw parameters can be fixed overall (and set equal, for example, to 0, 1, . . so a disk with N holes can be described by 3N -3

real parameters, or double that number if the surface is a sphere with N •.. •• Thus, gah cannot be set equal to flab for surfaces with higher . •••... However, any two-dimensional metric is conformally equivalent to /> of constant curvature. By a conformal transformation, we can ... . curvature to a constant. Thus, the space of metrics over which integrate is the space of constant curvature metrics divided out by morphisms on the surface M. Moduli space is the space of constant · metrics when we eliminate the overcounting arising from rp"'
r

Mconsl d mo u 1 space = Diff( M) .

However, as we saw in the single-loop discussion in Section 5.5, actually two kinds of reparametrizations, those that can be corme1cte( identity and those that cannot. In the previous identity, we divided set of all diffeomorphisrns of the surface. Thus, we also have the dividing out by the diffeomorphisrns Diffo(M) that are connected identity. The resulting space is called Teichmiiller space '

..

Tetchmuller space =

Mconsl

Diffo(M)

.

The relationship between moduli space and Teichmiiller space, of necessarily very close. In fact, they are actually equivalent up to a discrete group, called the mapping class group:

r

modu t space =

Teichmiiller space MCG

Thus MCG = Diff(M) . Diffo(M)

5.7 Riemann Surfaces and Teichmilller Spaces

21 J

,, I

5.13. Action of a Dehn twist. The torus has been cut along its a-cycle. One been twisted one full tum and then rejoined. Notice that the original h-cycle become the sum of an Q- and a b-cycle. This mapping ofllle torus into itself

~: ~:::~~~~i :\~:~;::; back to the identity map. The set of al l Dehn twists

,I(

mapping class group.

words, the only differences between Teichmuller space and moduli global diffeomorprusms thai cannot be connected to the identity. For , think of cutting a torus as in Fig. 5.13, twisting one of the sliced ,b)' 2". Imd then reconnecting the two edges. This is called a Oehn twist. this transformation generates a diffeomorphism that cannot reach ':;~; 11~;,:;;,:,,:glObaJ diffeomorphism. Thus, Teichmiiller space is "larger"

n

because you have to divide out the global diffeomorphisms. twists. to arrive at moduli space. The dimensions ofbotb these spaces by

n T,;chm,iilJ·" = dim moduli :::

O. 2, ( 6N - 6,

if N = 0, if N == I, if N

~

(5 .7.15)

2.

course, is precisely the number of parameters necessary 10 describe with N holes. Thus, this description gives us the necessary paramparametrize the N -loop diagram. (We note thai the modular group

class group are identical, and we will use these two tenns

volume factor due IQ the mapping class group can be factored so that, for our purposes, we may treat Teichmiiller space and

. space as essentially the same. > the

discussion has been general. To actually extract out these extra that describe the holes of a Riemann surface, we need to use the spaces.

of the manipulations are a bit involved, so we must always clearly goal in mind, which is to rewrite the functional measure DEal! in of '~'P"''"''''''' OVa D(1 and the 3N - 3 Teichmiiller parameters Dc/_

214

5. Multiloops and Teichmiiller Spaces

Thus, our goal is to establish

Dg ab = P,DvaDa Dti.

objective:

Then, by simply dividing out by DVa and Da, we will have eliminated the infinite redundancy introduced by reparametrization mvarlance. Let us now rewrite the variation of the metric tensor (5.7.6) in a venient form, revealing the fact that gab is also a function of ti, the parameters. Using the chain rule, we can formally represent the OelDeti of the metric on the Teichrnillierparameters via 8tilJ/8ti:

8g ab = [Va8Vb

+ Vb8va -

(V c8VC)gabJ

+ C'Vc8V'C)gab + 28agab +

where

. agab T' = - . - (trace), at' where we have explicitly taken out the variation ofthe metric as a the 3N - 3 parameters (Teichrniiller parameters) that we label t i with the N loops. Notice that we have subtracted out the trace in and that, at this point, we do not have to specify precisely how depends on the various ti. This variation can be rewritten simply as 8gab = PI (8V)ab + 28agab + 8t iai gab, where

PI (8v)ab = Va8Vb

+ Vb8va -

gab V c8v c.

The operator PI plays a crucial role in the theory ofTeichrnuller SP3LCe$ that it is an elliptic operator that maps vectors into .traceless syInrrLetri¢ Let ker PI represent the kernel of the operator, i.e., the set of vp."tn·r~ mapped to zero by the operator. Ker PI is called the set of "ccmf()mLIi vectors." For surfaces with genus N, the dimension of the kernel nt1th" is given by dim ker PI = 6 dim ker PI = 2 dim ker PI = 0

for genus 0, for genus 1, for higher genus.

We also wish to define the adjoint of PI, which we will call PIt. of course, we first have to define how to take the inner product.

118ga bl1 2 = 118va bll 2

=

f f

d2zyggacgbd8gab8gcd, d2zyggab8va8vb.

5.7 Riemann Surfaces and Teichmiillcr Spaces

2! 5

have defined a scalar product, this allows us to define the adjoint definition (a IPb) = (pta lb}:

Pt

(5.7.23)

pt,

i.e., the space of vectors that are annihi· now analyze the space ker ]fwe rew rite everything in terms of l and ~. then the elements of satisfies

p/.

(5 .7.24)

I of Pt is spanned by what are called quadratic diffe rentials. Forthe dime nsions of the spaces of quadratic differentials for Riemann are known:

dim ker

pt = 0

for genus 0,

= 2 for genus I, dim ker Pit = 6N - 6 for genus N.

dimkcr

Pit

(5.7.25)

nwnberofparamcters within the kernel of Pit isequallo the numberof ,tin,Olle"",,,.,".,,,, needed to describe N loops or handles. Symbolically,

summarize how 10 divide me measure of integration into its constiruent (5.7.26) has a rather simple meaning. It says that tbe components within up into three parts, a dilation part, a traceless part, and also Teichmuller parameten;. 11 also means that we are very close to (5.7.16), but there are some subtle complications. make a change of variables and calculate the Jacobian ofthe . Lei us first assume that there are no Teichmiiller parameters Then we make a change of variables from hub (which is the pan of the metric) and r (which is the trace of 8gb) to oVa and 0' ;

r

h)]

ae"~ Dgub = del a(O', v) Do-DI)G'

IX] PI

= det [ 0

= detP I = [detPt PIt 11/2 ,

(5 .7.27)

(5 .7.28)

th,,,,,lue of X is arbitrary, since it drops out of the determinant. that the square root of the delerminant of PI Pit is just the Faddee\'detenninant for the confonnal gauge, which in turn can be written in BRST ghosts. Thus, we can write the Faddeev--Popov determinant in (2.4.3) as (5.7.29)

216

5. Multiloops and Teichmiiller Spaces

Next, we wish to calculate the Jacobian factor due to the fact thaltijf~ sure actually depends on the Teichrniiller parameters t,. The the Teichmiiller parameters are not orthogonal to Pl 8va • so the Jac:ohili contain cross terms that have to be factored out. Let us begin our discussion by introducing a set of 3N - 3 cornpI!i.ti~ 1fta that will be an orthogonal basis ofker PIt. Let us now del;oIlo.pC)Se:tb;e: [22,23J:

into the following identity: Ti =

(1-

PI

+ PIt) PI PI

Ti

+ PI-;-plT i • PI PI

It is important to note that we have done nothing. We have only acld.~li subtracted the same term. However, the term that we have mtrocluced:¢qj~. the operator PI acting on another state. Thus, the above identity 1~:::~ because it allows us to extract the piece of Ti that lies along the rurl~~~ PI:

where

. 1 t· v' - - - P T'

- Pt PI

(1ft", Ti) =

I

'

f d2z..;glcgdeT~dl/r~e·

As expected, there is a piece of Ti that lies along the direction PI accounted for when we write the Jacobian. Let us now insert this back into the measure for 8gab :

118gab l1 2 = 118all + (T i , l/r°)(l/ra, l/rb)-I(l/rb, Tj)8ti 8tj + IIP I DVa Il 2 • This, finally, gives us the Jacobian that includes the contribution .. Teichrniiller parameters. We have also explicitly taken into account .. : •• : that Ti originally contained a piece that lay in the direction of PI : Dgob = Du DVaDti det

1{2

t

(PI

Pd

det{l/ra, Tb) 1{2

•

det (l/ra, 1ftb) This, in turn, allows us to put the entire functional integration I" the following factor:

f

DXe-

s

DgabQDilffnw~YI =

f DVanDilffn~eyl 1{2

t

du

x Dt, det (PI PI)

dete1ft°, Ti) 1{2 . det (l/r0, l/rb) .

5J:i Confonnal Anomaly ,

X (

J cf2:..;g der'( -

217

) - (1/2)0

(5.7.37)

"11)

finally attained our goal , (5.7.16), up to the question of anomalies ," mo·il scale invariancc. this discussion migllt appear long and difficult, the end result is

~

i~~;~;;Th~ie;:fi~n::a;!I~:an~~"~v~e~r ;Sh~O~:ws~~that thebemeasure integration, ,can written of totally in lennsincluding of dcterPI and . There are three parts to the measure: is the FaddL'Cv-POpov tenn, which is written as the square root of the

run", of P1 pt; (2) the second is the term involving T

i,

which arises the moduli parameters ti are not pe:fCndicular to the basis vectors fonn an orthogonal basis for kcr PI ; and (3) the third factor is the of the Laplacian, which we will also show can be VoTitten in terms

P,. , it is not yet possible to remove the integration over the scalar Cf in (5.7.37). Unfortunately, the various measure terms contain thc ~,~~:~~,;~. We will find, in fact, that in general the scale factor CUIUlot IT from the mcasure terms at all, breaking conformal invariance. section, we will show that there is an obstruction, an anomaly, to ;~:'..:;~;' called the confonnal anomaly, which can be removed only if !iI of space-time is 26. This fixes the dimension of space-time. process, we will show how to write al l detenninants totally in tenns P1 operator.

ConfonnaI Anomaly that we can cancel the integration over the volume of the ,mclt;;"" 'ion group, because

nDi~

!

DVa

= 1.

(5.8.1)

we wish to eliminate the Weyl rescaling term:

nW~l

f

DCf

= 1.

(5.8.2)

"~;:';:~:,,:in~:,:egration over the Cf rescaling term is considerably more comCf terms exist in the other factors of the measure as well. we must very carefully extract all a-
!tI

218

5. Multiloops and TeichmuUer Spaces

First, we note that the term (T i , 1{IU) is already Weyl invariant. if we rescale according to ~

(J

gab

yi

= e gab, = erfTi,

we notice that by the very definition of (Ti, 1{Ia), this teim lo;;UIlliU:ll>iXIl'II (The 1{1 variable remains unchanged under a Weyl rescaling.) Thus, the only term we have to worry about is

)}-(1/Z

det'ptPl lI/Z{detl (-VZ

)D

f d 2z,,[K

[ det( 1{Ia, 1{Ih)

(The prime still means that we have extracted out the zero determinant. ) Fortunately, the extraction of the Weyl rescaling parameter can be:~1i simultaneously on both terms if we use a few facts about Ri~:Iru:riUi· The trick is to rewrite PI, PIt, and V Z in a way such that all three be expressed in terms of the same differential operator.. Let Tijlkmn ... abcdef···

represent an arbitrary tensor in the two-dimensional Riemann ·sw.;W~~ course, we can always rewrite this tensor in terms of complex';';'",K;.-A z and Z. In these coordinates, we denote as K n the set of all teIllsQj~l transforms as

(e.g., see (2.7.6) and (4.1.7)). Clearly, the operator ViT = (g
maps a tensor T into another tensor. More precisely, it maps Kq (Vi: Kq -+ Kq-I).

It is also possible to define an operator that acts in the opposite operator

maps Kq into K,.q+ I:

v;: Kq -+ Now let us define the operators

VZ Pq = (

oq

Kq+l.

,"fli·:·1(;

5.8 Confonnal Anomaly

pt = ,

(-"". ;;

219

(5.8.12)

0

notation, we can show

,

- PI PI

det

=

(V:Vf

(5.8.13)

o

pt PI = d ct(V~VDdel(V:2V_. . ') = del.6i del.6. ::: I .

(5.8. 14)

: :"+ Laplacian is the complex conjugate of the l2 del / e ""n

Laplacian, we have

Pt PI = del t::.r·

(5.8.15)

""ID define .6t

!S

VfV~.

(5.8.16)

these definitions, we have now expressed the determinant in the

"","",

11l [dct'(_V2) ] -(1/2JD ]

f d'a.ji

-

-

det.6.+ I

dot'''(,,·. ,,')

(del.6. i r(l12l0

•

.

(5.8.17) both pieces of the Jacobian, which at first appear dissimilar. by the same operator for q = 1 and for q = O. Now, calculate the variation of the previous expression when we do a :~~~~,:: <"","nlon' to write the equation in terms of the heat kernel

6.;,

!ogdeIH = -

l,

.

~d'

_ Tr'e- rH ,

(5.8.18)

a small number. Using this equation, we can now express the variadeterminant due to a Wcyl rescaling. The actual calculation is quite

we will only quote the final result 1 22~24 J : det 6.: 1+6q( I +Q)j' dct1/2{1/!a, Vh ) = 1211" d z..JiR80',

(5.8.19)

"'the contracted curvature h.:nsor in two dimensions. Thus, we want this fRctor when Q = 1 and q = O. The final resuil is / !In [ det l12 Pt PI (det ( ~ V2» ) g del l / 2 Cv". Vb) J d 2 "l..j8

-(/2)D]

220

5. Multiloops and Teichmi1l1er Spaces

=

13 -!D 2 12n

f

d 2 z,.j8 Rlia.

Notice that for D = 26, the conformal anomaly disappears. Thus,~ . !~~tii shown

QW~YI

D = 26:

f

Da

=

1.

..

Our final result for the functional is thus

Z =

f

Dr det(ljra . T f) , det I/2 (ljra, Ijrb)

det~/2 pt P gil

2))-13

(det'(_V d2zJg

J

where all terms are evaluated with the metric gab where the a term ' completely factored out. Thus, our goal of attaining (5.7.16) has .·· accomplished, at least in 26 dimensions. Now that we have developed this powerful formalism, it is reduce this case to the problem of the single-loop amplitude and earlier result (5.3.12) in terms of our new perspective 011 Ri(~m;mIll < ~~ffi1 [25]. . .. For a single-loop graph, the region of interest is a doughnut. Our .. ' : . ... metric gab corresponds to a lattice in C: W1Z+W2Z,

where Z corresponds to an integer. Thus, the region of interest is determined by the ratio r=W2/ W I .

~:·~~II

And the fact that we normalize the domain to have unit area. Then, the .• class group is simply SL(2, Z), and the fundamental domain, as befor~Ylml taken to be (7: = 7:1 + i'f2)

The measure (det' 1/2 pt P ) ( 1 1

J

2 ) n det' 6.+ d2zJg 0

- 13

can be further reduced. For example, the factor (5.8.26) equals

"

!(det' 6.)-12(2nr2- l

r I3 ,

where det' 6. =

e 1rr2l3

ri

n(l -

4

00

n=1

e21finr)

5.9 Superstrings

221

"",1m" mc.asure can be written, as before, as dr, = d 2T/rf.

(5 .8.29)

everything together, we arrive at

(5.8.30)

the generalization of multiloop amplitudes to superstrings is except for the complications due to defining spinotS on of genus g. consider the torus in two dimensions 0"1, 0"2, constructed on a parnllel-

' ~~~I ~;'~~~;\:OiPPOsite ~

sides. A string X defined on the parallelogram

propenies:

X(O'I, 0"1)

=

X(O" I

+ 211', , 0"1) =

X(O"Io 0"2

+ 2rr).

(5.9. 1)

me addition of spinor.,; to this surface increases the number ofpossi:. A can be either periodic (+) satisfying R boundary conditions satisfying NS boundary conditio ns in either "I or "2. (So considered different boundary conditions in the 0 direetion the T.) Thus, the total number of different combinations ,""druy ,:ol,d",'o,, , is four. The possible choices we can take to dcfine a torus correspond to (±. ±). We say that these four possible choices define its ~p;n l·tructure. we know tha t modular transfonnations will mix up the bound he nce the spin slTUcrures. so all four of them can contribute For example, the transformation (0"10 0"2) -+ (0"1

+ 0"1, 0"2)

(5.9.2)

(- , -) into (+, -). Likewise. the lr.lnSformation (UI, (2) -+ (0"2, -01)

(5.9.3)

-) and (-, +). Thus modular invariance, which interchanges conditions, forces us to have (+ . - ). (-, +), and (- . -) in the is invariant by itself. to the one-loop amplirude, the lrace of the Hamiltonian over all four possibilities. The for the NS and the R sector are ~

HNS

=

L

r_I/2

r1/l~,'1/I: -

is·

(5.9.4)

222

5. Multiloops and Teichmuller Spaces

(2~ comes from zeta function regularization. See Chapter 10.) .. tonians are either NS or R depending on their periodicity propertiesIij'[:ii~ direction. However, when we take the trace over these .. inserting a complete set of intermediate states in the r direction. integral in the • only selects out the antiperiodic boundary '"'v•.u.u •.•..,.J~; is incomplete. We must modify the sum to include all possible sp1)r(~ji tures, which is made possible by the insertion of a factor of ( -1) F, \y:ij~ is the fermion number. The final amplitude is the sum of amplitudes,:'):"!. \,~ defined for all four traces, multiplied by some coefficient C: ..

A(r)

=

L C(±, ±)A(±,±). ±

Explicitly, each trace can be written as follows: A(-, -) = Tre2lTirHNS = 183 (01 r)/1/(.)t,

A(+, -) = Tre 2rrirHR = 182(0 1.)/11(r)]4, A(-, +) = Tr(clTirHNS(_I() = 184{0 1.)/1](.)]4, A(+, +) = Tr(e2rrirHR{_I() = 0, where 1/ is the Dedekind eta function:

n 00

11(.) = elTir/12

(1- e2lTirn).

n=1

Notice that the last trace vanishes by itself. (The first three theta fun1~il 8 2 . 3,4 are all even under z -+ -Z, while 8] is odd. Thus, only the" structures survive in the trace.) The significance of this is as follows. For the NS sector, for "A,UUjJ""f elude both the ( -. + ) and ( -, -) sectors, so we have to add their cOIltcimm to the trace: . Tr(l + (_l()e 2rrirH • But this is precisely the GSa projection [26], which projects out sta~~~::m under (-It! We therefore have a new, physical interpretation of projection operator, which was introduced in (3.6.12) to elilmiJlatle<'~~~~~1 supersymmetric sectors of the NS--R theory. It's surprising invariance, supe~symmetry, and the GSa projection are so i'IItirnal:el)dij~. When we calculate the vacuum amplitude, which has no extemal-t.i~g~!(il which appears in the calculation of the cosmological constant), we :.~'~~~Jm!1 all four contributions to the spin structure. But then we have the result ofJacobi:

8i(0

1 .)

+ 8:(0

I

r) - e~(O

1 .)

= O.

5.9 Superstrings

223

that the vacuum energy of the superstring, i.e., the onc-loop 10 the cosmological constant, vanishes exactly! ""'"I'" we bave proved that modular invariance and the GSO projection are essentially the same for the closed superstring. Modular invarimixes up the boundary conditions, demands that we add all four ruCO"'" to the final amplitude, which in tum corresponds to adding preGSO illsertions of the operator (_I)F into the trace. Originally, the

roj,,,,i,ODW'" imposed on the NS-R superstring in order to have superWe now know that the GSa projection enforces modular invanance

",'C .

• ,~;:;':~i;we find that the one-loop contnbution to the cosmological equal to zero. (However. this docs not mean that superstrings :..~ vanishing cosmological constant problem. After supersymmetry is we nO longer expect that the vacuum contri bution is zero, and hence theory still does not explain why the cosmological constant is zero up""""m""" b"aking.) stateme1l1S reinforce our conviction that the internal consistency of "'IT"'g theory is quite remarkable. lot us generalize our commenlli to construct multiloop amplitudes with lines. the appararus that was constructed for Rielllllrm surfaces carries to the superstring case if we use the NS-R formulation of (5.7.2), the correlation functions can be computed from [28 J:

I: I: J[ . 0,., J(

10pt>1"~",,

'!>in

m.tncs

.

mo.bcJJ'O);i

OX'

f f D.p'

Dx.,-s.

SlnlOlllre.

(5.9.11) addition here is the intcgration over the two anticommuting vector and also the sum over all spin structures. sh"w,'
(5.9.12) ~OW.

in analogy with the bosonic string. coostruct two operators Pi n such that

(5.9.13)

10 the integration over moduli. we now have to integrate over mo,dul;·, which are defined as the space of traceless Xa that cannot be away by a local supersymmctry LTansformation. Thus, they satisfy

(5.9.14)

224

5. Multiloops and Teichmiiller Spaces

For a surface of genus N, the dimension of the supennoduli space '·:. t the dimension ofker P1/ 2 : ...•, ·•.·c·.c.·c.:.,

o o

if N = 0, if N = 1 (periodic-periodic), if N = 1 (otherwise),

4N -4

if N :::: 2,

2

(where periodic--periodic stands for boundary conditions on the l".::lC':",m;,::n We wish to to construct the Jacobian for

DgabDXa' The Jacobian for the fermion field is calculated as before: 4N-4

DXa = (detPi/2Pl/2)-1/2e-(11/2)S(0')D~DAn dai, ;=1

where ai are the supermoduli, the counterpart of the Teichmiiller p~1~i and where the integration over the fermionic A and ( replresents'11~~1ii1 gration over the redundancy introduced by supersymmetry and SU~I~!I transformations. We define S=

4~Jt

f d2z~(gabaaaaba + /L2 (e"

- tiAya baA

- 1) +

r 3/ 2 /LAAe(2 j 2)a + ! Rga ),

where R is the curvature tensor. Putting everything together, we <1m,"':';'::'>+:!: following: A

DgabDXa

A

A = (det 1/ 2 PIt Pl)(det_lj2At PI/2Plj2)

n n

4N-4 X

e-15SLDaDvaDADl;

6N-6

Daj

i=l

Dt

i",l

By dividing out by Da, Dv a, DA., and Dl;, we extract out the ~~ dundancy introduced by the symmetry of the superstring action. Uo!':Ii".'~~ however, the parametrization of the moduli, especially beyond the is quite difficult. The nature of the supermoduli, unfortunately, isj~~~ understood. These are some of the main stumbling blocks to a def&.'\jt~Y.I derstanding of the multiamplitude, i.e., the choice of coordinates ' Riemann surface consistent with modular and supermodular invarii~fl¢:f~ ;'

5.10

Determinants and Singularities

The advantage of the Riemann surface method is that, at least tormaUY;i:W$ obtain results for the general singularity structure of the N -loop """"",":.7.<

5.10 Delenninants and Singularities

225

out thai tbe determinant that contains the singulari ty of the N -loop ud,,·" the Selberg zeta function. earlier that projective transformations naturally fall into conjugacy Two projective transformatiof':;; arc said to be within the same conclass if they have the same multiplier. Thus, any two members of the class can, by a projective transformation , be brought 10 the (5.10. 1) l is sometimes called the "length" of a closed geodesic. Let z and ""..nt tv.'o points in the complex plane. Theil we define the "distance" these two points as );:-;:'1 2

,

d(z.z)= 1 +21

(5.102)

mz 1mz ,.

,~;::al.~ :~t~::~:::;~;~ afthe "distance" between two points is that it

::1

all clements of the same conjugacy class . . - . saw, from the example of the single loop, that the multiplication by I w moved 8l.TOSS the h-eyele. Topologically. this simply means a closed geodesic on the Riemann swface. Thus, l is the length closed geodesics on the surface. We say that a transformation is if it is not a power (greater than or equal to 2) of any other element set of projective transformations. Thus, we arc interested in the set of geodesics. We dcfme the Selberg zeta function liS l23] ~

Z,(,) ~

0 0 [1 - ,(y),-('+"I.

(5.10.3)

I primitive p...o

we takc products over the lengths of dosed primitive Seodcscis y on """. and ~(y) == 1 for bosons and v(y) = ± 1 for feIlllions, depending structure.

m~'~~::~'~":'~'O~can write the various determinants found earlier in the

Ie

in terms afthc Selberg zeta function [28): del' == Z'{I)e- ruIM ).

66'

det 6i = Z'(2)e- CIXlM) ,

(5.10.4)

"mion detenninants can also be expressed as follows: (det j~:/2PI/2)~12 == e-<'(!!lJX(M)Z,,(t), (del' y" Day BDp )~{2 -

e '·Ojl/t>x(M)

(2p)!

Z~2p)

(i),

(5.10.5)

is the Euler munber (p is the number of zero mode.<.: afyo Do). The constant c is given by

L (l:o"'
(2Inl-2m -1)log{2Inl-m)

226

5. Multiloops and TeichmUller Spaces

- (1111

+ ~i + (11 -

[n]i

+ (n + ~)log2J'f + 2(-1).

Thus, by analyzing the structure of one function, we obtain the structure of the N -loop superstring! Mathematically, it is known that the Selberg zeta function is unless the Riemann surface degenerates topologically. For eX~lillJ)le.; 'Wfi:m a divergence as the length of the primitive geodesic approaches i:'~jmi1 corresponds to the "neck" of one of the handles of the sphere ~tr.·t:ril". infinity. By carefully analyzing the Selberg zeta function as one ofthi:Jrui of the primitive geodesics goes to zero, we find detl/2 pt P (det I

1

l

(-Ll.t»)-13 ~

J .J8d2z

r2e4Jf2/1

'

which reveals the pole corresponding to the tachyon vanishing into the:ya:iiii Being able to isolate the singularities of the multiloQP amplitude int¢ii!.lii known mathematical function, the Selberg zeta function, is a great solving the main problem facing string perturbation theory, i.e.,' :Q~~ti. proving finiteness to all orders. Formally, it can be shown thatthe div'er~[~ the multi loop amplitude may occur when the integration over mc.dlil~:¢} the topology of the Riemann surface, e.g., when two holes separate ~W~;qjl a long goose-like neck. However, there are many delicate points rel:at~i~:fi~ question of the cancellation of these divergences that have not beeu:t(~ solved. Although preliminary results are encouraging, the l'~rOf()Us: the cancellation of divergences is still an outstanding problem.

5_11

Moduli Space and Grassmannians

Although enormous progress has been made in elUCidating the 111i1.u!J;<~ structure of multi loop amplitudes, in some sense the tangible lC~,WL:>.Uj~¥J< disappointing. Back in 1970, it was already known that the dr,,'er~~enc~:~ multiloop amplitude, by explicit calculation, corresponded to the deip#.ffi ofthe topology of a Riemann surface [4,5]. So far, t1ie elaborate maltll~:lffii techniques we have introduced still haven't answered the key we rigorously show the theory is finite to all orders? If so, how the perturbation series? How do we extract nonperturbative the theory? Knowing that the divergences of the theory can be terms of Selberg zeta functions [23, 29-32], although an iml)Ol"tan·t~e;:~@M hasn't solved these mysteries. In summary, the progress has been of mathematics, rather than physics. At this point, we can take at least two diverging attitudes. We c!f:::~m on the perturbation series and proceed directly to the field theory which nonperturbative information might be extracted. This is the' approach taken in ordinary point particle theories and is the subject 0";nJjl~ series of chapters. Or second, we might imagine some kind ofsytnri!t¢ft.Y~1!m

5. 1 I Moduli Space and Gll:lSsmannians

227

inwriance, by which we might be able to manipulate the entire Riemann surfaces of arbitrary genus. point particle Feynman diagrams, the Gecond strategy i!> probaThe symmetries arc 100 smal l. and the Feynman diagmms arc manifolds. However, the Feynman series for string theory are sums ....,ifi',,"'. Riemann surfaces, where modular invanallce pl!lY a crucial it is conceivable thaI the entire perturbation series might be roa· mathematically. This is the approach o f Friedan and Shenker [33], >rol""~ exploring the propenies of the ·\miversal moduli space" of all surfaces, including i.nfinite genus.

li:,~~.~~.t1ri:::~~~:;:::' was too ambitious and complex to extract mean-

("

. Ilowever, two dcvelopments have given some impetus to this

Bclavin and Knizhnik l34] have shown that the measure for the I bosonic amplirude is simply the absolute value of a certain 38 - 3 fonn. This is cnlled "holomorphie fa ctorization ." it may D1lIke possible wriring the multiloop meas ure by inIn practice, however, there arc eertain problems in spectfYing of the period matrix beyond three loops. This is called " HOlomorphic factorizat ion has made the two- and measure almost nivinl, but the Schottky p roblem prevents the measure from being easily written. 1984 mathematicians finally solved the Schottky problcm. Thus, a key " "",iion LO utilizing holomorphic factori7.ation seems to be removed. the Sehottky problem, moreover, provides us with an even more w,,;fullooi. It allows us to describe an infinite-dimensional space, called ··Grassmannian" (Gr), in which all Ricmunn surfaces of genus g are as single points [35-37). Thus, by studying the properties of the in the Grassmaonian, it may be possible to manipulate the set of all irt,,,b,a'ion diagrams at once.

~~~i"~'~ Grassmannian finally provides us with the conceprual frameto treal the entire penurbation Geries as a whole. we sliU do how to sum the entire series. For example, phase space reduces the . all po~sihle positions and momenta of a panicle to a series of points. . space, in principle, conlains all possible motions of all possible in the Universe. However, this says everything and it says noth.i.ng. have to impose equations of mol ion IUId boundary conditions to exmeaningful informa tion from phase space. The same applies to the

in

ambitious program requires fu lly exploiting the modular transfonna,[BU"n"," surfuces of gellus g. Of particular importance is the mapping l'O"P :MC;G •. wl,ii,h reduces to the modular group SL(2, Z) for the tolUS. want is a way of studying the mapping class group for arbitrary will be the constructiOD. of theta funct ions defined on Riemann

'''''Y

228

5. Multiloops and Teichmilller Spaces

surfaces of arbitrary genus. This is in contrast to the Schottky metl10;fi: studied in Section 5.6, where modular invariance was not so In Fig. 5.12, we have written the a- and b-cycles for an artlltriitM~ Riemann surface, which we call the "canonical homology basis.'· :"",,,;I,l" ~::IHI tisymmetric symbol (a, b) represent whether or not two cycles int,'(i' 'fii~~ equal to 0 if they do not intersect and equal to ± I if they do. ror:ex--Jl consider the torus, where we have only the a- and h~cycles. NotlCc~tI1tlWl is equal to I, because the a- and b-cyc1es intersect, but (a, a) Then the four possible combinations for the torus create a ma,trDe: (a, a) = (b, b) = 0, { (a, b) = -(b , a) = 1. The elements of the mapping class group do not change this j'I1ltersectim1ii However, we know, by definition, that the group that leaves this is Sp(2, Z). (See the Appendix.) To describe the elements a Dehn twist Da created by slicing the surface along the a-cycle; cut by 2n, and then resplicing the surface back again. Under ll~'crl~l= the b-cycle converts into the sum ofthe a-cycle and the b-cycle (s Thus,

Da(a) = a, Now represent this in terms of matrix language. Let the Dehn the column vector [a, b]. Then the Dehn twist can be represented as

Similarly, we can describe the mapping class group for the (W(H9(QRJH by taking Dehn twists along ai, a2, hi, ~ as well as the cycle aj"I/.f':"l'~ ::)ly.J;MJ! a circular line that encircles the two holes. Using same reasOIl~'~~~.~~ , show that the Dehn twists, operating on the column vector [ai, be represented as [38]

the

1 0 0 0

o .'

100

101 010

1 0 I 000

0 1

001 000

I

0 0

0

o o o

100

1 0 0 010

0 1 0 101

001 000

",

5. J I Moduli Srace and Gmssmannians I D II ,· ' • •

=

0

0 -I

0

0

0 0 0 -I

0

let us Ireut Ihe closed Rieman surface of arbilrary genus. The matrix can be represented as (aj,aj ) =(b/.bJ)=O,

!

(ai, b) j =-(b1 ,Qj)=8/j

(5.11.5) ,

in,>th,ing bUi a block-diagonal matrix. with equation (5.1 1.1) in each group that preserves this matrix is Sp(2g, Z). We might suspect, , that the mapping class group is j ust Sp(2g. Z). This is not quite right. 'ie~:::;~D~e.hu twists D~ around cycles that arc homologically trivial and ::(I the a- or b-cycJes. Thus. they are represented as the unit lhe basis that we have been using. Ncvenhelcss, these Dehn twists tim.to g lobal diffeomorphisms that must be included in the mapping This subgroup is called the Torelli group T, and hence we finally , d",;",d result

MeG

----,,--

~

Sp(2g. Z).

(5.11.6)

the effect of the Torelli group on spin structures is trivi81, so we funho< discussion of it) we wish to describe the period matrix for the Riemann surface and "::~~):~ under Sp(2g, Z). In addition to the 28 cycles we caD write on ~ surface, we can also write 2g independent harmonic one·forms on the surface (see Appendix). Because we have equal numbers of one· form s (due to the Hndgo-de Rahm thoorem), we can always ze "h. integration over a-<:ycles such that (5. 11.7)

the b·cyc!c integration will produce a g x g square matrix: (5.11.8)

period matrix , which generalizes the variable T introduced for the I~:~:l~~':~~ in (5.5 .2) and equals the period matrix introduced for the ~ in (5.6.31). It can be shown ralher easily that the period symmetric and has a positive definite imaginary part In general, + 1) element... of a symmetric g x g mimix (with positive definite

'i:::,:~,~~~'~~:' a space called the Siegel upper half-plane. a(

ofinlIooucing the period matrix is that no two inequivalent surfaces can ha . . .e the swne nij. whieh is called Torelli's theorem.

230

5. Multiloops and Teichmilller Spaces

Thus, up to Sp(2g, Z) transformations, the period matrix gives us a way in which to characterize different Riemann surfaces. Under an " transformation, the period matrix transforms as ' Q' =

(AQ

+ B)(CQ + D)-I:

where the A, B, C, D are each symplectic g x g matrices. Thus, TW(~'" UI period matrices may describe the same Riemann surface if they an Sp(2g, Z) transformation. N ow that we have a mathematical description of the mapping , in terms of Sp(2g, Z) defined on Dehn twists, the next task is to wt:lltt:£1 functions defined over a Riemann surface with g handles. We saw the single-loop amplitude is described in terms of the qmiSl·,aO'UI)I,'t:il,e~ theta function e. Now, we wish to write down geneci1ized theta furi,,:e,iti6~~ have certain periodicity properties defined for the surface with g ," Ifwe take the single-loop theta function and then force it to odicity properties over the Riemann surface with g handles, we natuti1; an additional set of infinite terms to the summation. Eventually, approaches [39,40] 8(z

I Q) =

L

ei1tn.n.n+21tin.Z,

n€Z'

where the sum over the vector n is taken over a g-dimensional 1~!~~ theta function is defined not over the complex variable z, but over" ' defined by

z = (Z W,

JpD

where Po is an arbitrary point on the surface. Since w has g cOlnp@l~ then z is also a vector with g components Zi. It also possible to construct theta functions with spin torus, we saw that a spinor parallel transported around a up a phase of + 1 (-1), depending on whether the spinor was periodic). Likewise, the genus g Riemann surface be polygon with 4g sides (Fig. 5.14). We now have to define 22g pmlses;W::!?J describe the different ways in which we can parallel transport a spiMt:;~ft this polygon. The set of characters [a] = (a, b) defines the spin s~Pi(~1 the manifold. We can now define the generalized theta function with spin strulcmf.~

can

e[(~J(z

I Q) =

L

e i1t (n+a)·n.(u+a)+2Jri(n+aHz+bl

nEZ'

=

eirra.!:!-a+2rria.(z+b)8(z

+ Qa + b I Q).

Under a shift in the lattice, we find that the generalized theta up to a phase:

8[a](z + Q . n + m I Q) = e2".ia.m-irrn.n.n-21fin.(z+ bl 8[a](z

I

5.1 L Moduli Space and Orusmannians

231

" .-',

., 5.14. Canonical homology cycles of the Riemann surface of genus 2. This 'onnod by drdwing the a and b homology cycles of /I sphere with two holes, cUlling along the lines that form the cycles, and then uDmveling The advantage of this basis is thaI the homology cycles for II surface or " "'" CIIn be repTeSet\ted as a polygon.

change Z

-7

-z the theta function transforms as

0[.J( - z I 0) = ( -I)Q·'''[.J(z I 0).

(5.11.14)

h w;<,,,,/;o, ,,,, spin structures, as we did for the single-loop function according to whether or Dot they are even or odd under 1 ~ -1. there are 2 8 -'(2' - 1) odd aDd 21'- t{2 X + I) evcn spin structures

~~~~S"~'~fu~C~C,:U~f:,g~'n~'~>s~;.~~~~,~~~~:

these generalized tbeta funcunder a mapping class group transfommtioo (5.11.9). Under we find

el.d/,' I n') =

Be

;".6 del(Cn

t =

[ :.] =

(.:'8

+ D)I!2Q(a](z I Q).

(Cn +

C -A ) [ : ]

Dr' . z

=

H~~ ~;~:l

(5.11.15)

(5.11. 16)

(5 .11.1 7)

ph"sef"t., ;s given by = aO T 83 + b eT Ab _ 2aB TCb _ (aD T - bCi)(A BT)" ,

I ~::;::;~:::~ the transpose and cJ represents taking the d iagonal clement it

rcpreseOis the eighth root of unity (with a cenain restriction that worry us here). we have defined theta functions with tbe proper periodicity on surface, our next task is to actually calculate the measure of amplitude. Here, we arc greatly aided by a remarkabl~ result

232

5. Multiloops and Teichmuller Spaces

of Belavin and Knizhnik [34], which simply states that the measU!i;it.:~ multiloop amplitude is the square of a certain holomorphic furlctiloij::{U zero modes):

z-

holomorphic factorization:

r.

71/\ij

- JF (detlmQ}13'

where F is the fundamental region of the surface, and 1'/ is a 3 g - 3

n

3g- 3

71 =

dYiF(y} ,

where YI is a parametrization of the Teichmiiller parameters, and morphic, has no zeros, and has second-order poles at the bOlllll(~~:~ the surface degenerates. It is easy to check, for example, that the function (5.5.4) possesses this property (up to zero mode tennl~s~)'~itiV~i.~1 The statement of hoI omorphic factorization, on one level, is ir ous. It simply says that the closed loop is a product of the left- and 19t1't~Bi1iii modes of the string, except for the zero modes. The proof of this pci'Wej~ suIt, however, is rather jnvolved, but can be summarized as techniques derived from the previous sections, we know how to forrrui;~~m the measure of the multiloop amplitude in terms of Teichmiiller P$~~ and complicated determinants. Then, it can be shown that

a8log W = (D -

26) ...•

where W is the measure term up to the factor raised to the Notice that the right-hand side is the conformal anomaly term, whicb:;J.iIi~ in 26 dimensions. Thus, in 26 dimensions we have aa log W = v . :·.....·:'.,.-, case it can be written as the absolute value of a holomorphic tilltc:tt~ When we are in any other dimension, there is an obstruction to hol[~~~f1j factorization. Then we say that there is an analytic anomaly. . .. What is remarkable is that the conditions on F are probably strjingeq1~l#ii to uniquely fix the measure for all multiloops [41-61]. This is an en~!~§M powerful result, which may eventually allow us to write the mltltillooltHttml by inspection. For example, consider the two-loop function. Let us choose the Te1pf!!~ parameters as the period matrix itself, since they both have three in(j;~m complex variables. Then the following combination is modular inv'ati.:~~ij dQ

'caHmrH)-' We wish to find an expression for IF12 (detlm Q)-lO. Using the theory of Riemann surfaces, we find that the unique answer is two-loop:

F =

In R.b

8[a](O

I Q)j-2 •

5.11 Moduli Space and Gra5$mannians

233

product is taken over the 10 even characters such that 48 . b = using similar arguments, about holomorphic functions, we can the F function for the thrce-Ioop amplitude as

three-loop,

F

=

(!]

6[al(0 I el)

rn

(5.11.23)

again the Teicbmuller variables dy, arc take n 10 be the clements matrix dn//. ':, i,",pl. however, stops abruptly at the four·loop level. In + I) elements in a symmetric matrix defined in the while lhe number ofTeichmiillcr parollTleters is 3g - 3. the same number of clements only for g = 2. 3, which makes of the period matrix of higher loops exceedingly difficult. origin of the Schottky prnblem, the fact tbat the set o f elemenls I upper half-plane agrees with the elements in the period matrix = 2.3. The question is: What conditions do we have 10 place on "tit)~,. defi ned in the Siegel upper half-plane such thai they become of the pcriod matlix? mathematicians have recently solved tile Schottky problem. possible \0 ~i m pl y eblU"aelerizc period matrices for arbitra ry g and I"', it. holomorphic functions defined over Riemann manifolds of Y,;::i;l "~'fact, this allows us to envision an infinitc-dimensional space, ~: where Riemann surfaces o f arbitrary genus arc represented

:w,,",,,"ti, a genemlizatioo oran operator formalismfor a con/ormal defilled em u Riemalln manifold of arbitrary genus. We want

10

operatol"l'i acting on fI generating function [35-37} that will allow us all correlation functions defi ned 00 surfaces of genus g. Let us begin fennion operators

w(z) =

L W~z -..

- If!,

(5.11.24)

" usual antieommu13tion relations

(5. 11.25)

;bilooal current l(,. w) =

,~(,)~·(w),.

(5.11.26)

operator is the diagonal element

J(z, z) =

L nEl

}"z. - It-I.

(5.1 1.27)

234

5. Multiloops and Teichmiiller Spaces

Now define the generating function rex) = (0] eH(X)g ]0) ,

where 00

H(x) =

L

Xn

In,

,,=0

g is an element of the Clifford group given by

g = exp

{t=oo dz t=oo dwj(z, w)l(z, W)} ,

where the integral is taken around infinity. It is important to Im:lW' • mil~ a function of the period matrix Q as well as the spin'structure dei¥i~fi the surface, which is parametrized by the characters a and h. information concerning the nature of the Riemann surface is en(;odl¢iJ~iif the function g. In ordinary point particle path integrals, we know that the actiQti~:ijJ operator 8/8 1 on the generator of the Green's function creates of a field into the functional integral. The analog of this is given operator "oc "1 0 +""" -I 0.... v(z) = e"'--n=OxlI'z e ogZQ(J ,L../1'>=l n z -0

where xn , am have the same commutation relations as the oscillators. By construction, we have the identity (Oj1/l(Z}l/1*(W)g [0) -1( ) ( ) *( ) ( ) j (O[ g [0) = r x v z v w r x x=O·

Thus, the action of the vertex operator is to create the cOITelati()rI::::~ for the conformal field theory, similar to the action of 8/8 1 on the~.~~mll function in ordinary field theory. Now let us impose the conditions, called the Hirota equations,

i=oo dzv(z)r(x)v*(z)r(y) = O. The Hirota equations can be shown to be intimately linked to the .. the KP (Kadometsev--Petviasvili) hierarchy. .. We now have all the tools to WTite an explicit expression for r(xY=

eE".",~oxoQ"'",t'"e[a] (- f

A"x" [

Q),

,,=1

where the An appear in the power e~ansion of the period mat.IU~;; jm~mli Q(z) =

L Anz-nn=O

I

dz

5. 11 Moduli Spacc and

_I

= 4l(m - I)! lI(n - I)!] a,'"

G~smannian.~

E(t,y) a; log ;;,'''"''; (, - y)

235

(5 .11.36)

, y) is called the "prin\c form" and is a straightforward generalization """tion ,- ,. found in the conformal field thcory of the spbere. ln fact, W, it behaves like z - w even on a surface of genus g. Explicitly. it

E(" w)

~ 8[.] (I.' wi Q) (h.(z)h.(wn-"', h.(,) ~

L a,8(a ](0 I Qlw,«)·

(5.11.37)

(5.11.38)

,o,n"o the main result afthis discussion. We find that lhe tau function Hirota equations of the KP rucl"'dfchy if and only if Q is, in fuct. .

g Riemann surface. This is the constraint that we dimension ~g(g + I), can now be restricted to - 3 and is equal to the period matrix of a Riemann surface of , in principle, solves the Schottky problem (although the result : '.' nonlinear). i .dded bonus, because r(z) can be 'v;cwed as the generating function . field theory defined on an Riemann surface of genus g, we can : . arbitrary correlation functions. The matrix clemcnt of two fermions ",umu surface is given by

_ 8[a J(/;,., I Q) ~ P - 81al(0 I Q)E(" w) .Il, wJ.

(5. 11.39)

know this as the Szego kemel, which is the unique meroin l and w with II single pole with residue 1 for know this simply as the two-poin t function. The N -point is equal to

~

rI E(l" I
l,)

rI E(w., w,)(r' [aj(O I Q) . <1

236

5, Multiloops and Teichmiiller Spaces

There is a way in which to check the consistency of¢.ese eqlll.ationls.lWe:~ that there are two ways in which to calculate the N -point function, '-'-"'~~Hfl either fermionic oscillators or, through bosonization, bosonic OS(;lll~tQ~ the form ei
n N

(01

1{I(Zi)

i=l

N

IT 1{r*(wi)g 10} = deti} Pa(Zf, Wi)' j=l

Equation (5,11.40) can be obtained using the bosonic rep1resentatll:n@@ (5.1 1.41) can be obtained using the fermionic representation. Forttitiatel equivalence of these two different expressions was proved by Fay [3~'~~jll called the "trisecant" addition theorem for theta functions. The matl1~:tt equivalence of these expressions is another check on our .. In summary, what we have developed is an operator formalism fO)rr.~·,t~11 field theory defined on an arbitrary Riemann surface. The ket ~ represents a specific Riemann surface, and applying various on it corresponds to taking matrix elements over that Riemann ~'tJ\~~ important to note that each element r (x), if it satisfies the Hirota coiri
...J~

.,

5.11 Mo:iuli Space and Grassmannians

237

,

,,,,'"

I,

--+..;

- .....

\

'---_ ..... ;

"

" >,

>,

'.

,,

"-

'.

'. >,

'.

~~.~.~I':n~Ih;:e~l~ast diagram, we see the angles, lengths, and "timcs" that

~

swface. These parametern make up the Tl:ichmOlier space,

required to paramctr1U each internal loop. (These par.amctern

"I'yyield one cover of the fundamental modular region.)

a field theory of strings. We thus intuitively expeet that the conof ligbt cone Feynman diagrams by assembling vertex functions in natd'x yields a specific "triangulation" of moduli space. This result, proven in [63, 64], was never previously considered by the malh (Driefty. the proof is based on constructing certain integrals on a surface of genus g and calculating lhcir periods as we move around p n" han,dle. Then il can be shown that the periods arc purely imaginary, a modular lnV'.mant description of the surface.) '~:;~:' . in Fig. 5.15, we sec bow a multiloop surface is parametrized Itl parameters, with T, repre3coting the various interaction times string splits or joins, and 8i representing "twiliting" each string by ,.,,'Oluli.,n. To complete 6g - 6 parameters for a surface of genus g, ta integmtc over the circumference of each cylinder. Thus, all I parameters have a nantral interpretation in terms of the physofa Feynman diagram. The light cone fonnalism has several the previous fommlisms. First, there is no need to truncate thc which is necessary in the Schottky and the constant cut. modular invarianec. Second, it easily generalizes to an -. number ofloops (whil::b cannot be said for the theta function method.) - is unitary and based on physical variables. And fourth, it is naturally a field tbeory. of discussing this fonna lism, we will discuss the light cone method of the field theory of strings. We will devote the next series of ' Iothi, impOrtant topic.

238

5.12

5. Multiloops and Teichmuller Spaces

Summary

In summary, we have shown that the unitarization scheme for the ; .:: be implemented by using the unitarity equation, using the N -point ;: ' as the Born term: ', I . 1m T,'j :=: '2 (il Tin) (nl Tt Ij).

L n

The function integral is now modified to

A=

i

DX dJ.Le iS

s

fI

, eik1i •

;=1

where we sum over all conformally inequivalent surfaces S that aI"e: :ill~ii spheres with N holes. In particular, the open string graphs come in planar, nonplanar, and nonorientable (as in a Mobius strip). The graphs come in only planar and nonorientable (as in the Klein bo1tle.:';;:::~~ Fortunately, mathematicians have already calculated the Ne:untanl)tl~ on these surfaces, which are particular examples of automorphic tut:l,(:~tR For the first loop, the Neumann function is given in terms of In 1fr(x, w) = In( 1 - x) +

!

00

+ L[ln(l -

ln2 x In x + 2 In w wnx) + In(l - wnIx) - 2In(1 -

n=1

We found that the string graphs never have ultraviolet diverg,~~l~~ infinite sum over the intermediate Regge trajectories renders the the ultraviolet region. However, as in Feynman graph theory, the , a graph is given by the local change in topology of the surface. The ,'" . open string graph diverges when the interior hold vanishes. This c~t~'6!il by conformal invariance, to extracting a closed loop with zero ni~!J:ij§!QI Thus, the divet;gences correspond to a tachyon or a dilaton . vacuum. The divergence structure of these graphs are as follows: '"

(1) The Nambu-Goto open string diverges as q-3 and q-l. The gence, associated with a dilaton, probably can be absorbed inta::!"l~DI renormalization to all orders. The tachyon divergence is more

S. 12 Summary

~

2~i~~~&<~apl:ts. when analyzed in the

239

complex

$- and r-plane, flCtu io them, which reduces to poles in 26 dimensions. These to closed strings. Thus, the open string theory, by itself, Closed strings emergc ru; "bound states" at the first-loop

closed string amplitude divergc.'i because of tadpole or self-energy '~~:~~~ extemallcgs, which are on-shell. ;:.; I superstring has only q _l poles, and bwee we can have slope ""","""IO(IO to absorb this infinity. The q- J divergence never occurs ~;r;;;.;the boson and fermioo intemal legs cancel against each other.

":~~~i~E:i;:~l~':~~:~~~~; :Th~Iere are no two- and onthree-point ;'; external results generalize nicely to the N -loop amplitude using the theory functious. There arc several pammetri:aHions we can usc. The group method, which has the advantage thnt its ehoice of and was dcrived in a manifestly faetorizable fash ion by vertices together. However, modular invariance is not is (hI; theta fu nction method, in which modular invariance .' in from the very stan. It also easily generalizes to spin S[I'Uetures. : of this method is that the choice of varinbles is far from factori7..ation and hence unitarity are obscure. These ampli rudes postulated by hand to satisfy tbe boundary conditions, rather than by sewing vertices together. Ultimately, these methods are identical. slart with the complex plane with 2N arbitrary boles CUI out. Lei us hole!> imo N pairs, labeled ill and a/ . These are called a-cycles. If line between any pair of a-cycles, we obtain b-<:yc\es. Lei us define operators P, iliat map one a -cycle 0/ into its partner a/. Then these operators can be parametrized by two invariant points 41 and Z2 and X, sueh that

lO""u",,d

P(Z) - ZI = X Z - Zj •

P CZ)

<:2

Z

Z2

(5. <2.5)

open string, thc centers of the a-cyeles lie on the real axis, as well ;U>Un,n< point of the projective transformations , The final open string '. '. amplitude is given by

240

5. Multiloops and Teichmiiller Spaces

(2»)"'.k

X

IT

(I)

;=1

x where

I

{PIx).

Zi -

{P) XI..(I)Xfj(2)

(I) -

Xfj

{PI x),(2)

- ,

) (1/2)kp ,kl

x12) - {p}Xi1)X~2) - {p}xi2)

M

dll = f""'I

f

{P}x),

M ( Zi -

N

IT dz' dV- IT dx(l) 1 abc

r

;= 1

a

, dX(2)(X. (I) a ..a

X(2») - 2 a ,

a=1

where the Roman letters refer to the extemallines and the Greek le~~~i resent the loops. Notice that the region of integration lies between points of the set {P}. We must have these limit points forming a discrete~~ on the real axis. Thus, we must have Schottky groups. The N -loop amplitude can also be directly reformulated in the lanl~ Riemann surfaces if we use the Polyakov action instead of the NlamiD:~~1 action. There are several new contributions to the integral: (I) A factor ~FP coming from the Faddeev-Popov determinant, wl111C~l:~it rewritten as

det l/2

Pt PI.

(2) A factor from the integration over the X variable:

f

DX exp ( -

= (

f

d2Z.JggabOaXILabX,.,) -(l/2)D

21L

I

2

Jd2z../i det(-V ))

,

where '12 = -1

../i

am .Jgg"'" on .

(3) The Teichmiiller parameters t; must be added to the integral. To ,." this last factor, let us write ',

8gab

= [Va8Vb + Vb8va

- (VcovC}gabl + (V c8V )gab + 28ugab + 8t; Tl, C

where ti are,the Teichmiiller parameters and

, agab T' = ---. - (trace).

at'

We can rewrite this as

8g ab = PI (8V)ab

+ 2augab + l)ri OJ gab.

5.12 Summary

PI(8v)..,,:::: V"ou"

+ Vb 8Ua -

gfl"Vc~vr.

24 1

(5.12.14)

change variables to

r l:::: 1/f1l(lJrll, r l )+ Prvl ,

(5 .12. 15)

~ptTI

(5.12.16)

./ =

P it PI

(l/t", Til =

I

'

f d2Z..;gg""gt!~TJ.Vt:f_

(5.12.17)

",-fin.l result for the measure is therefore

n-'

n-'

- 1 ("')-1 D VaaDifT D q ~ ~WtyJ D g"b .... ~IODI1J~ ·Wcyl =

x Dti de ll llC p t p ) dct(l/r". Ti) I I det 1!2(!/I., I/Ih) X

(I ~7f,./8

dct·<_ V'Z»-(I!2IO

(5. 12.18)

d

. great advantage of this approach 10 the multiloop amplitude is that use the power of Riemann surfaces to analyze the divergences of the . In panicular. we can analyze the singularities aT the Selberg zeta

n n[l ~

Z(s) =

, (y )e-" +"].

(5.12. 19)

/ primitNe. pzO

of the Selberg flJllctions confinns Ihe facI that the measure has a where the topology of the genus g s urf.lce degenerates. "",oat.ely, the cOnstructiOll oflhe measure on multiloop surfaces is greatly by the holomorphic factorization theorem, which states thai the equal to

,we"

z -• -: 1]

1 F

1/Aij

(det [at Q )13 '

(5.12.20)

is a holomorpltic 38 - 3 form. The formu la is intuitively obvious if

". the equal contribution o f Id't- and right-movers to the amplitude '. for zero modes). The only complication to the proof is an anomaly the analytic anomaly) that vanishes in 26 dimensions. Using this result, basically guess the answer for lhe integrand of the multiloop 9mpl.irude, function with the COl"l"CCt periodicity and holomorphic singularity must be unique. method, based on theta fu nctions, exploits the modular properties of perio< functions from the very beginning. For surfaces of genus g, wc

242

5. Multiloops and Teichmiiller Spaces

have 22g spin structures, corresponding to all possible periodic (antip!m~ boundary conditions on the sillface. Oill task is therefore to construct 1"\''I¢Uliiai functions on this surface with spin structille that have the correct oeiiM properties and singularities. The answer is given in terms of two turlCtiOOS:;; generalized theta functions and the prime form. The theta function is. 8[a](z I Q) =

Le

i J\"(n+a).O.(n+a)+2ll"i(n+a Hz+b),

nEZ'

where a represents the spin structure and the prime form ,, holomorphic generalization of z - w found for the sphere) is given 0''y::::::~~~ E(z, w) = 8[a]

(it I Q) W

(h",(z)h",(w))-lj2,

where h", =

L, 8,8[a](01 Q)Wi(Z).

Given the theta function and the prime form, we can calculate Green's functions for bosonc and fermionic operators on Ri~~rnaru($~ Thus, we can construct a new conformal field theory on Riemann just spheres. For example, the two-point function of two ferrnions the Szego kernel: 8[a](j~w 1 Q) 8[a](0 1 Q)E(z, w)'

One difficulty in solving the multiloop problem is the question ofCh()Q~ilP.j correct coordinates. One ideal choice would be to use the period tmimJ~~ the coordinate. The problem, however, is that there are 4g(g + 1) dit~~iI to a square matrix like the period matrix, but only 3g - 3 complex n~l~:~ parameters. For two and three loops, they coincide, but for higher do not. This is the Schottky problem, which only recently has The solution to this problem makes possible the use acterize the entire perturbation series. Each multiloop amplitude :W,I:,_: structure represents a single point in the Grassmannian, so we may(';iUii principle) have a way of manipulating the entire perturbation . mann surfaces all at once. This may, in turn, eventually yield information about the superstring theory.

References' [1] [2] [3] [4]

K. Kikkawa, B. Sakita, and M. B. Virasoro, Phys. Rev. 184, 1701 ' M. Kaku and L. P. Yu, Phys. Lett. 33B, 166 (1970). ' M. Kaku and L. P. Yu, Phys. Rev. D3, 2992, 3007, 3020 (1971). M. Kaku andJ. Scherk, Phys. Rev. D3, 430 (1971).

References

243

· Kalcu and J. Scherl:, Phys. Rev. D3, 2000 (1971). Alessandrini, Nuuvo Cimento 2A, 32 1 (197 1). . LovellicD, Phys. Lett. 328, 703 (1970). Lovelace, Phys. I.eu. 328, 203 (1971). B. Sakita,and M. A. Virasa ro, Phys. Rev. 02,2857 (1970). . S. . L. B. Sakita, ' >hys. Rev. 04,229 1 (197 1);Nuel. Phys. 834,632 ' ~;;;;;:', Rev. Lett. 3U 716 ( 1973). 8. F i and H. 8. Nielsen, Nile/. Phy~. 820,637 (1970). · Burnside, Proc. London Math. Sci.n, 49 (1891 ). Amllli, C. ]3ouchiat, and J-L. Gervais, Nuovo Cimento Left. 2, 399 ( 1969). Bardakci, M. B. Halpem, and J. A. Shapi ro, PhY$. Rev. 185, 19 10(1969). · Kaku and C. R. Thorn, Phys. Rev. 1D, 2860 (1970). c"m,""and J. Schcrk, Nlld . Phys. 850, 222 (1972). '. Lovelace, Phys. Letl. 34 8 , 500 (1971). · Neveu and J. Scherk. Phys. ReY. DI , 2355 ( 1970). H. Schwarz, Phys. Rep. 13e, 259 ( 1974). '. :.' . '. :.

~ .•~~~;;:~~P~~'h;YS'

Rev. OS, 1947 ( 1972). Unified Siring Theori/ts (edited by M. 8. Green and D. Oros~) :. Scientific, Singapore. 1986. :' . Alvarez, Nud. Phys. 8216, 125 (1983). : D. H. POOng, Nuel. Phys. 8269,205 ( 1986) . .: . M. Polyakov, Phys. Lell. 1038,207,2 11 (1981). .' . Polchinski, Comlll. Mat". Phys. 104,37 ( 1986). , 0Ii022i, J. Scherk, and D. Olive, Nuc!. Phys. B122, 25 3 {1977}. '. . Seiberg and E. \Vinen, Nud. Phys. B276, 272 (1986) . . D'Hoker lind D. H. Phong, Nw:I. PlTys. 8278, 255 (1986); Comm . Malh . ,I

104,537 ( 1986). I Nue/. Ph),s. 8 277, 102 {1986}. A. Namazic and S. Rajeev, Nuc/. Phys. 0277,332 (1986). ;. .. Steiner, PhY.f. Lell. 188B, 447 ( 1987). ~: . ~Iberg, J. Indian Malh. Soc. 20, 47 (1956). ;~ ~';',iriied9-iaiin and S. Shenker, Pllys. utI. 81 75,287 (1986); Nuc!. Ph),s. 8281,

A. Belavi n and V. G. Knizhnik, Phys. Lim. 1680,201 (1986). L ~1~~:'::~;Y~.~M~~atsuo , and H. Ooguri, Mod. Phy.s. Lell. Al, 119 (1987). :. . C. Gomez, and C. Reina, Phys. Lell. 558, 55 (1987} . . Lett. 19IJO, 4 7{ 1987). ,~ I ,.~,G'~m',O. Moore, and C. Yara, Comm. Math. Phys. 106, I (1986). Theta Functions on Riemann Suifaces. Lecture NOles in Mathematics, ~"!'. S~ri ,~g"-Ver1ag, Berlin, 1973 .

Lectllres on ThetQ, Birkhauser, B
244

5. Multiloops and Teichmul1er Spaces

[48] A. Belavin, V. Knishnik, A. Morozov, and A. Perelomov, Phys . ..,"''',,·.·.'·'e.IJ (1986). [49] G. Moore, Phys. Lett. 176B, 369 (1986). [50] A. Kato, Y. Matso, and S. Odake, Phys. Lett. 179B, 241 (1986). [51] H. Sonoda, Phys. Lett. 178B, 390 (l986). [52] O. Lechtenfeld, CCNY preprint (1987). . [53] E. Verl1nde and H. Verlinde, Phys. Lett. 192B, 95 (1987). [54] A. Morozov, Phys. Lett. 184B, 171, 177 (1987). [55] F. Gliozzi, Phys. Lett. 194B, 30 1987. [56] M Bonini and R. lengo, Phys. Lett. 191H, 56 (1987). [57] A. Restuccia and J. G. Taylor, Phys. Lett. .187B, 267, 273 (1987). . [58] F. Steiner, Phys. Lett. 188B, 447 (1987). . . [59] H. Sonoda, Phys. Lett. 184B, 336 (1987). [60] A. Parkes, Phys. Lett. 184B, 19 (1987). [61] A. Morozov and A. Perelmomov, Phys. Lett. 183B, 296 (1987). [62] J. J. Atick, G. Moore, and A. Sen, NucZ. Phys. B308, 1 (1988). [63] S. Giddings and S. Wolpert, Comm. Math. Phys. 109, 177 (1 [64] E. D'Hoker and S. B. Giddings, NucZ. Phys. B291, 90 (1987).

Part

II

Second Quantization and the Search for Geometry

6

Cone Field Theory

String Field Theory? we saw that the development of the first quantized theory often disjoint and seemingly random. appealing to ru l e~ of thumb and often seemed quite arbitrary. The choice of vertex functions, the of integration, the counting of diagmms, etc., all were insertccl by summary, the first quantized theory suffers from several important 'i~~;::,'jOilS

'nn,' ',e introduced ad hoc. without any rigorous, overall

cannot easily be shown in the first quanti7..ed approach. There enni,i,n Hamiltonian from which we can derive the interacting '~::;;,:i:s~~necessarily perturbative and on-shell, which makes difficult :c of nonperturbative ellccis.

:~~::~th~:e~Iin;::,::q:~n~"'~breaklng. ' iz~';cd~:'~tn~"n~g!~\th~';,'U~'Y~: is unsuitable for aincalUnfortunately, to all orders the

~

to be stable. This means, thaI there is little hope withill the first quantized approach of calcu""mical breaking of 26 or 10 dimellsions down to four dimensions. doub tful tbat rigorous phenomenology can be done within the first framework. The first quantized theory, although it can yield virtually

~:':::~~,~r ~~:,::','~ solutions, cannot select out the true vacuum from ~

vacua that the model can exhibit. Part II we now tum to ll1e.field theory of strings [1-5J, which has "'" ulryid,lin, ,. nonperturbalivt: formali~m in which llit: true vacuum

248

6. Light Cone Field Theory

may be found. (yVe should point out, however, thai"second qmmtitiitf(l:ii:li: been developed only for string theory, not M-theory. In fact, it is H:>!~::~tl>ll how to formulate M-theory in a satisfactory first quantized forlllili.lisltll/f~~~iiI a second quantized one. Thus, our remarks about second qu:mtiza"tio;~j!itIl confined strictly to superstrings.) At first, the second quantized approach seems to be totally redlundlii)ij~~~ usual first quantized approach. Perturbatively, we simply reproduce diagrams as the first quantized approach. However, there are several it',~"P".i advantages to the second quantized field theory: " " ::~:::;:~

>

(1) Interactions are introduced through a new gauge group whose close on interacting strings. There is now a group-theoretical for introducing the interactions of string theory. (2) The theory is manifestly unitary because the Hamiltonian is H¢~~ All weights of perturbative diagrams are fixed at the very l)el~l1llll.in: (3) Most important, we have, in principle, a method for calculating dYiiJiil l effects in the theory.

Historically, it was once thought that a field theory of strings was u"n"]":w:&.i because it would violate a host of fundamental and cherished n';'rl;;~ quantum mechanics. Specifically: (1) A field theory of strings would be a nonlocal theory riddled problems, such as the violation of causality. ::: (2) A field theory of strings would necessarily be off-shell, yet ""tJ"i:~:;jifj properties of the Veneziano model, such as cyclic symmetry;." on-shell. Thus, a field theory of strings would not replro(luclm::~~~1 model. (3) A field theory of strings would not be both Lorentz invariant a.n;~~~ at the same time. This is because a quantization program for not exist that is both Lorentz invariant and unitary off-shell. quantized theorem. this is not important because the theory is the field theory of strings is necessarily off-shell and hence either Lorentz invariance or unitarity. (4) Most, important, a field theory of strings would be plagued of unitarity because of severe problems with overcounting of I)~~I diagrams. Field theories add the sum of s- and t-channel poles'~¢i~WJ!tl which violates duality.

Fortunately, a field theory of strings that answers each oeQ.(~~ objections is, 4,ldeed, possible. First, string field theory does not violate causality because the "iJltl~ such as the breaking of a string, take place instantaneously. information concerning the change in the topology of the string tra~r#~ the string at or less than the speed of light. In other words, smng::w:~ multilocal. Thus, string field theory is the only known nonlocalfi:":"~"~~~UI consistent with the principles of quantum mechanics.

6.1 Why String Field Theory?

249

IOnd,smin, fi"ld theory generates Green's functions that necessarily viimportant properties of the Veneziano model. But this is m; s~;;:.": these Green's functions correctly reproduce the Veneziano 01 and only onoShell matrix. clements can be measured. BRST method explicitly breaks unitarity off-shell with Faddccv, . This makes DO difference. however, because the Faddeev-Popov -: cancel with the unitary ghosts on-shell. Similarly, the light cone method , unitarity but breaks Lorentz invariance off-sh.ell. The point is tbat the theory is both Lorentz invariant and unitary, so the theory is still

"'.th'

string field theory actually breaks duality off-sheU, but this makes on-shell . For example, in light cone string field theory we sum s- and t-channel poles in the Fcynman series:

"A,

A = L.,. f

"A,

l+L

s - M,

J

(6.1.1)

2+ · .. ·

f-M J

theory of Slrings solves (he problem of double counting by breaking amplitude into its separate t- and s-chatlllel parts, consistent with a in~::,::;::~:~ (see Fig. 6.1). Thus, sln'ngfield theory breaks manifest I, it at lhe level of the S-matr1x. Although string field theory solves the problem of un itarity, it does so at the price of breaking duality, which is recovered only at the end when we sum over all cio,sed strings, we also find that the individual Feyrunan diagrams break modular invariaucc. Onlythc sum is modular invarianl. Thus, the light

, 2 4

,

,

.

2

.

,

4

by the light cone field theory. As in any field theory, .I i duality. Only the sum is dua1. Thus, light eooc theory of strings solves the duality is explicitly broken for each diagram. Only the . duol.

250

6. Light Cone Field Theory

cone field theory is manifestly unitary (because 'the Hamiltonian is . Hermitian) but the price we pay is that we break manifest modular ~V~~4 First quantization (modular invariance) ---* second quantization (u.~~~~ We will begin a discussion ofthe second quantized theory string staJ'®it~ the light cone formalism [I], because this retraces the historical CI·4~vellt1fH; of the theory and because it is the most fully developed tbr:malllSD1; because the light cone gauge is'a gauge-fixed formalism, in which aH~q'@~ gauge degrees of freedom have been explicitly removed, there is no::trfI:il~ the elegant group-theoretical formalism from which string field the()l1~1iaiii derived. Thus, for pedagogical reasons we begin our discussion .un-",:·" cone field theory. We begin by once again discussing the field theory ofpointpartlCI.es;::;Vl trace how Feynman derived the Schr6dinger equation from the '~U'.H"\<;l theory of point particles.

6.2

Deriving Point Particle Field Theory

In Chapter 1, we started with the path integral for a point particle moviffi a point Xi to point Xj' Associated with each path connecting is was a phase factor e • The fundamental postulate of quantum ~~w~11 that the probability amplitude of a particle moving between these is the sum over all possible phases associated with each path. rotation, we have l!.jj

=

l

;Ij

Dxe- s •

X;

where

1 1j

s=

dt (!mv; - Vex)) _

The evolution of a quantum-mechanical wave, by assumption, nh,'v
foo b.(Xj, tj;X;, t/)'l//(Xi. ti)dxj. -00

Now we would 'like to calculate the variation of this wave fuIllCb9.W!tlJ~ small displacement s in time. Earlier, we found that the pf()pagat.Q:(.:'WJI:I~ to

6.2 Deriving Point Particle Field Theory

e=

• .. nnw intervals

tj

fi.

-

we find, therefore,

> /OO! ;m(x ~ :: 1/I(x, r + e) = A-' exp 26

y)' )

_QO

.

::

251

l/!(y, t)dy ,

(6.2.5)

is a normalization constant. Remember that the time interval e is very the scpardtion between x and Y Deed not be smaiL to keep only terms of order e. If we let y = x + '1. where 11 is not a small number, the integral becomes

l/'(x,t+e)=

00""'"

i:

A- ' e i "'Ql{1t1{l(x+'1, t )d'l'

(6.2.6)

to notice that there is no limit, in principle. to the size of 1/.

in the functional integral, keeping terms of order e will necessarily

to terms second order in

1}.

Now let us power expand the left-hand

ier"" of e and the rigbt-hand side in tcnru; of 11: 1/!(x,t)

+ e&1/I Dt

=/0:> A-lei",~If2f _00

"a''' )dll·

!

x l/I(X,t) + '1 aw ax + '211 axl

(6.2.7)

can be cxplicilly performed. First, the integration constant can be to be A~

(-21riE)'" m

(6.2.8)

the terms in the right-hand side, we notice thai the only terms that the Gaussian integration arc those for which 11 appears with an even the integrand. Finally, we are left w ith .81{1 j

at :::: -

I a1 1{1 2m &x2 .

(6.2.9)

have now derived the SchrOdinger equation, slarting only from the that L :::: and the basic assumptions of quantwn mechanics. the effects of a potential term and extend the expression to all directions, the derivation is basically unchanged, and we arrive at

!mx7, i

a1/l = __ , I V 2 1/1 + V(x)y,. "

(6.2.10)

m

'1:::;,:';;;' with e;oo;ternnl potentials entirely and introduce 1/13_ or1{l4directly into the action. This is the second quanti7M analog

:r;

over y ~shaped or X~shapcd topologies in the first quantized point

:th,ory. Feynman's original derivation of the Schrodinger equa~

'" ''" ,,,lou',,,;n, the time evolution of the wave. However, there is yet

252

6. Light Cone Field Theory

another way in which we cart make the transition from the first to mef:iijj;1ffil quantized formalism in which the derivation is more direct. Let us nQ'i\'-~ to the path integral action and show that" we can make the tral[lsili6bd& second quantized formalism starting from the Green's functions theittiS~ffi rather than appealing to the equations of motion. In Chapter I, when we made the transition from the Hami.ltOiliiti(~~ Lagrangian formalism in the first quantized point particle theory, ...."··:u·,, an infinite set of intermediate states, the eigenvectors of the x-coordii:iiU l=lx],t;)

f

DXj(xj,t;!

into the expression AU = (Xj, tilxj,tj)

at each intermediate point between Xi and x.j. This allowed us to' transition between the Hamiltonian and the Lagrangian approaches .. Now we wish to replace this with the integration over a cOlnpl¢i(!=:$Jj second quantized fields: 1 = IVi)

f

D 2 te-(tlt) (Vii·

Following (1.8.21), we define t(x) = (xl1fr), Vi*(x) = (tlx), 2

D t =

TI dVi(x)dt*(x). x

Thus, at each intermediate point between the initial and final states H~::!"::1!'!1'I'l particle, we are now going to introduce an infinite set of intermediate fi"~. states IVi) rather than x-eigenstates Ix). The easiest way to check ttt~'lJJj~ of (6.2.13) is to take the following matrix element and insert a cOInpJ'~m intermediate string states: 8(x - y)

= (xly) = (xilly)

• . (xly)

=

f

f

2

2

D Viexp {-

f f f (1frlz)

D Vit*(x)t(y) exp {-

DZ(Zlt)} (.,.,:t.\l:t.::,,7.'!

DzVi*(z)t(z)

1

(where as usual we drop ovemll normalization factors in the path ';t"¢Igm treat 1jF(x) as an element of a column vector labeled by discrete· .' in'

6.2 Deriving Point Particle Field Theory

253

Jizingthe expressioo.. we find

;" - Jr;Id";dV,"';"',ex ~ ";,,, 1 p[-

(6.2. 16)

relation is just ( 1.7.11). Thus, we have now shown thai we can move

'"~::~:~i from firs t quantized basis elements Ix) to second quantized i~ I"'). such thut 1j!(x ) = (xlw). ' " u, beboln our derivation of the Green's fun ction for the SchrOdinger entirely in lenns of sccond quantized field functionals, without rethe equations of motion or Huygens' principle. We will joserr the identity at every intermediate point along the path :

l1Jrl}

f D2"'I D21/tlexp{-1jI~("'2

-1/1)) -

1Jr;l/rd ("'21 .

(6.2.17)

"'t,

,p,ove this identity by functionally integrating the expression over reduces to the completeness expression written in lenns of the tfl

the above expression between two infinitesimally close position in (6.2. 12):

= (xlle- iH"!x2) = (XdX2) - i (xd Hot IX2) + ... = ! LY1fr12{Xd wl}{¥rzlx2 ) e;o;p[ -l/rj(1/I2 - 1/Id - i

f

2

D ""m4 (x d 1/l1)

",; 1/111

H~t I"'J) (,y4Ix.)

x e;o;p[ -wi(Y,2 - 1/11) - 1/Ii1/l1 - 1/tj(1/I4 - y,) - \11; 1/1)]

+"' ,

(6.2.18)

subscript 12 or 1234 simply means the product of two or four of IOc,iocall differentials. We now take the limit of small time separations:

i):

1/ti(l/i;+1 - l/I;)""'"

/,'~ ,y";Ydt.

(6 .2.19)

-: taking the limit, we find =

, ou,

f

2

D 1/t1/t(XI) 1fr' (XN)CXP

[i J

dt .1jt"(iiJ, -

H)tl

(6.2.20)

""w Lligrangian is L = ",.

[i:, - HI"',

(6.2.21)

hc~~;::~:;;:"~O~';':h~e:;S:~C~hr6dinger wave equation. Therefore we bave te given the posndates of quantum mechanics

254

6. Light Cone Field Theory

and the classical first quantized formalism for a point particle, WlthOllttl:ta use of the equations of motion or Huygen's principle. The main reason why we went through this analysis for the point ~" .."".~~ that we will now repeat the same steps for the light cone and the theories. Surprisingly enough, we will find that this entire functional _ carries over directly into the string field theory for the light cone and ,actions.

6.3

Light Cone Field Theory

As in the point particle case, we now want to make the transition rrom _l~~~ quantized to the second quantized string fornialism, using (1.8.21): .

We must be careful to state that the field functional is not a local X (a) at a specific point a on a string. By contrast, it is actually a functional defined at all points along the string. lfwe discretize the sttijlj a series of points:

then the field functional becomes

and we take the limit N -+ 00. Thus the stringfunctional is sin:!ultan~~~ function of every point along the string [1]. Let us begin by defining the Hilbert space of string excitations. -irretrt.l.O power-expru'u1'1n€rMlo1'~j·iil·"t:er11!Sf,r1imrm.l)m'C'Ui~~llit&ffi;~i:i'i@ case we have

leI» = ¢(x) IO}

. j + Aia~1- IO} + hija~la_1 10) + ....

We see immediately the difference between the first and second fOImalism even at the free leveL In the first quantized tbrmallsrJl; sic object is X f.I. ' which represents just one possible cOIrlig;tmlltiQ.~:::~tt.~ string. In the second quantized formalism we are dealing functional eI>, which is simultaneously a composite of all pOl~sib!l;~J~ configurations. To make things concrete, let us now introduce a specific reI're1;en~~~'ij1~ IX) eigenstates in terms of harmonic oscillators. We want the X to act on the string eigenvector IX) such that we reproduce tL. ••'.)!J';:::::::~

6.3 Light Cone Field Theory

255

w oment, lei us make the assumption that the eigenvector IX) can be

IX} = (l IO) - k

ncxp(aX~~ + bXI,"a,~ + ca:,,,a,~,,) ..

(0) .

(6.3.6)

L.

C, and k are arbitnuy constants.

X,

these constants a, b, and

c by acting on this slate vector

XI ,,. IX ) = 00- 1 ~i ("i... - ai~n)n 10)

= n
(6.3.7)

b = -2i,

e == ~.

(6.3.8)

we wanl the operator ax to have the correct comm\llation relations ", ;:. Ifwe operate on the state veClor, we find ;

I I X } = ('aX i ." XI, ..

;

8X -

'."

I X) =

the coefficient to be a IX l = k

n'.

exp ( -

. - 1(01,11

t

+ b.,.") IX}.

(6.3.9)

.

+ a;.,,) IX)

(6.3. 10)

= -1. OUf final result for the state vector is

xln - 2i X/

Jl

aJ.1I + tai'.,.al.. ) (0)

I

(6.3. II }

is H norm.1 lizalion constant. This expansion, in turn, allows us to ".the field function as 8 power expansion in Hennite polynomials. For we eM calculate (01 X ) = k

ne-x,'.. .

(6.3.12) j.",

tw11 , allows us to rewrite the original field functional (6.3.4) in lenns

polynom ials, using (6.3J I):

(X I"') = "'(x)

n

H,(X i . )

+ A, (.t)H1(XI,l)

n .", i,

Ho(XI.~) + _..

(6.3 .13)

256

6. Light Cone Field Theory

(where we have taken a different normalization of the Hermite pol[~~m. than usual). .. Let us now quantize this field functional, following closely the •...• in ordinary point particle field theory in (L8.9) to (1.8.12). In ... power expand the field function in an arbitrary series of orthogonal •.• also Let us choose an arbitrary element of the Fock space, which is .a•• tlmlil of creation opemtors raised to some power. Let the set of integers<~..:·g~Me the number of creation oscillators in a particular state with Lorentz 11l~!~~ level number t. Thus, an arbitrary state of the Hilbert space can be .v•..•!Jij.,tttlfe.

I{n}} = c n(ai._,)n; IO}, i ,1

where this state is a product of an arbitrary sequence of creation ()..:~.,Uil and we choose the nonnalization constant to be c = ({{n}[{n]})~1/2

such that ((n}l{m}) = b{nl.lm)'

Thus, these states can be normalized so that they form an In fact, the basis is complete

ortlt:lOllLOn~:~~

L l{n}){{n}l.

I =

[nl

The matrix element of I{n}} in (6.3.14) with the string (6.3 .11) is just a product of Hermite polynomials: (XHn})

= H{n)(X)e- L;.• xl.•.

Let us now explore some useful properties of I{n}}. If we power field functional in terms of this orthonormal basis, we can

eitf~~1

let>} = L¢(n)l{n}) {nl

to the following

(XI ct» =

L ¢(n1 H(n1(X)e- L ;JI Xl. •• {nl

The inner product between two such fields functionals is easily "al"'H~~ (\III ct» =

L(nl t/!{:)¢(n)'

We can also show B[nl.(m)

=

f

¢[nl¢(ml exp (-

L ¢f

n ))

(n)

d¢,

6.3 Light Cone Field Theory

257

,,,n","u« of integration for this space is given by

D2¢ =

n,.,

(6.3.22)

d¢l"i dq,(nj-

tdentititls, we can now show that lhe number I can bl: wrillcn in a oJO!,OU, 10 (6.2.13),

1

~

f

1<1>1 n' ("I""pl-("'I"'I).

(6.3.23)

we

equation can he proved by power expanding each of variouS oCliooo11 , via (6.3. 19) anu explicitly performing the inlcgrdtion overthc

. Thus,

1~

[I: f.,., 110 n] nd'.'m' II: .i"IIPII ) ,-I.'.' 1m)

(n)

L ,prnll{n}) ({plitJIjr) / L Ifn)) ({p}16

Ipl

d2¢lmle-l:!f,~·,I:

(,,).(p).)nrl

=

(6.3.24)

1"j.lpl·

IIIJ.(pl

are justified in taking this expansion of !.he number I given by no" i, ci
" n ,(X,(o) - Y,(o)) y,IXIYI ~ n ::: '

; .. 1 0<0'
; ::

~

- (XI"'I D'IIYJ'XP( -I"'IZI

f

f DZ(ZI .. I) (6.325)

we need to generalize (6.2.17) before deriving the Schrodinger strings (6.326) this identity is most easily proved by fitst functionally integrating we have all the identities in place, let us first consider the matrix b'lw.,,, two string states tha t arc infinitesimally close

(XII e-iH1T IX 2 ).

(6.3.27)

r~:,~::e:, this matrix e lement either in terms of the first quantized pic-

le

insert a complete set ofmomenrumcigenstates. or in terms of the

258

6. Light Cone Field Theory

. second quantized picture, where we insert a complete set offield Thus,

first quantization:

1 = IP)

second quantization:

1 = I ct>)

f f

DP

(PI,

D 2 ct>e- (I1 '

Retracing our steps in Section 1.3, we inserted the momentum and then derived the infinitesimal Green's function for the . formalism: (Xtie- iHdr IX2)

= (XilP)

~i:g;~

1

dP(Ple-iHdrIX2)

f DPe-i~(x.p)dreIJd"p(Xl-X2) . =f =

DPe(fdnPXdIe-iH(X. p)dr,

where we used the fact that (P IX) '" ei J du p" X", Then we in~,,,Ttp'A· :i>;';~:{' number of these first quantized intermediate states between every ill.~·~t~~ interval between the initial and final points:

where L = P X - H, Thus, we could go back and forth between Hamiltonian and ~~jR formalisms in the first quantized string theory, N ow let us repeat all our steps, inserting a complete set of seconcttU:imm field functionals into our action. Let us start with an II'mn1te1;unatJ~ amplitude 1l.12

= (XII e- iHdr IX2)

= (X,IX2 ) =

f

i (XII H dT IX2)

+,.,

1;fct>d X tlct>I)(ct>1IX2 } exp{-(ct>dct>2 - ct>1) - (

- i

f

x exp

1

D ct>1234(X 1 lct>l) {ct>21 H dr 1ct>3} (ctJ 4IX 2 )

1- i~

(;JCPj+, - ctJ i )

-

(ctJHdctJ;)

I+, ".

6.3 Light Cone Field Theory we now take the limit as

L I;I ,+, - <1>,)

,

~

f

259

d' I"'I
inscn this complete set of intennediate field states between all in· bctweelllhe initial string configuration and the fina l string Then tht: matrix element becomes

f

¢&(X.)¢(X N )D 2 cbexp

[i f dr«(¢ !ia. -

H Icb})].

(6.3.32)

""""in ,,,,,,it for thls section. Therefore, our second quantized action I. = ¢'ua, - H )¢.

(6.3.33)

we nave now s hown the equivalence of th~ first and second quan· for the free light cone smog theory. We have ~hown that we usi ng (6.3.28), the Green 's function in either the fI rst or second language -All

=

l .XIDx exp[i [" drL (r )]

=

f

2 D
[if

d.

l' dC1{XJ''2 1..' (p?+ X")

X~l.

t(.) = - I 2. 0

H =

L~' L(¢l )DX

dCl

]('2

•

l

(6.3 .34)

(6.3.35)

(6.3.36)

this second quanti7.ed field theory ac tion from the requirement lbat the Green's function for the propagation of a string. This s hows pw~lIl el between finit and second quantization for the case of free parnllcl, however, will be dras tically broken when we discuss the .: . now quantize our action. The equation of motion derived from our

(ia, -

H )"'(X) = O.

(6.3.37)

,,,bat corresponds 10 II partic ular b
Ilnl>

=

E,",llnl>. l,",

ill

EI~I = -----:;: ~ln,

2p

1./

+ap"

(6.3.38)

260

6, Light Cone Field Theory

We will find it convenient to take the Fouriertransfonn of these c·.·p:«ft~1 with respect to x -: ¢[nj(X+, X-, Xi) ~ ¢p+,[nj(T, Xj).

Thus, the coefficients must satisfy the equations of motion + {

,f.

'Y P •

'(I' , X-) = / r

/l r

dp,ei(P·J:.-Er"f ) A + . { , I p' . P" n r'

Notice that we have introduced a new operator A; which is the annihilation operator associated with the state {n}, It is crucial to noti¢¢:: is not the same thing as introduced in Chapter I. creates a sl'lllgle~':ibi mode ofthe string, while A creates or destroys an element of the ~U.'U."'H spanned by all possible products of A creates or destroys states an infinite-component field theory, It satisfies the commutation relaWlltl$

a!

a!

a!.

[Ap+,p,,{n). A;+.qj.{mj] = 8(p+ - q+)8(Pi -

Qi)8[nj,{mj'

Combining everything, we now have an expansion of the field furl~~:~~Wii terms of plane waves: 1<1»

=L {II I

/ T1 dPiAp+,Pi,lnjei(P'X-EW) I{n}). i

We can also express this power expansion in the X basis (see (1.8.2

(XI <1» = ct>p+(X)

=L [n)

/ T1 dPi Ap+,p,,(n)H(n)(X)e- L.,... Xf."ei(P·X-E"f i

T )

I{n}) .• ·.t·.!;!.~s:r,"",~

>.

We are now in a position to derive the canonical commutation the second quantized string field theory. Because we have power field in terms of orthonormal polynomials, it is easy to show that [1 [ct>p+(X, T), ct>t+(Y, T)] = 8(p+ - Q+)

T1 T18(X;(a) -

Yi(a)).

rr

i

Let us denote the vacuum ofthe A oscillators by 10)). Notice that ml:SV~ll'>l state is the product of the vacua of all the higher spin fields cOIltaine(J'W, <1>. This state is the vacuum of the infinite-component field theory ",,-,',1;·»:1 has nothing to do with 10). Using the previous identities, find an eX]~(¢j~ for the Green's function in terms of the field functionals: ll.12

= ((01 ct>p+(X[, T[)ct>;+(X 2 • T2) 10)) = 8(p+ - q+) / DXeifLdrrdr

IJ

x T18(X(cr', T2) - X 2 (cr)). rr'

..

8 (X(cr, T[) - X[(cr))·

6.4 Interactions

26 1

initial and final states are labeled by X(a , rd and X(a', <2). It is to notice thnt we have now expressed the Green's function for the of a free string in both the first and second quantized language.

",h,,f',,,, level, we can go hack ~md fortb between these two forWe now have the tools with which we can write an explicit expression :- Greeo's function for the free string. which gencralizc..<; the expression It I,m'"

point partides:

D 00 [

- q+) x

1r

D k jl/2I - 2) sinh(kT /0:)

a) IIUD-» cxp I- 4T(X a , -X, ) (4JrT 1

x eXP{kSinh-

1

2)

k: [COSh~ (Xi +Xi)-2XI ,Xl] },

(6.3.46)

is the time intervaL Notice that now we have a derivation of the function aij entirely in leons of second quantized field functionals. run"""'Y, we have derived the free light cone string fi eld aCl ion by cli-

: inserting a complete set of intennedlate string slates at every string '.

between the initial and [mal string configurations.

';""~ '''rr~'~''D'O

'"

d" question of interactions. Our derivation of the from the finit quantized theory can now be generalized to calculate ""'Clio[>s oflhe second quantized field theory. Again, we will insert the identity (6.4.1) integral in order to extract the vertex function. "o,;, ~lly, many oflhe early pioneers in qUlIntum physics, sueh as HeisenYukawa, looked into the question of non local field theories and found theorics violated causality, i.e., interactions could propagate faster !..peed of light. The nonlocal interactions, which can in,'olve two di.sXI and X2 , cOllld trnnsmit informAtion fasler thRo the speed oflighl, is fo rbidden.

.~:g~Jfield theory solves this problem. The solution is simple field theory does not violate causality and the laws of quan!!~:;:'~:~i~'~trm~ it is not a nonlocal theory, it is actually a multilocal I

The interactions of the string, in which·strings can break or refonn, are

<1",,,,,,",., break or reform instantaneously and then the vibrations travel the string at or below light speed. Thus. we do not have a violation of

262

6. Light Cone Field Theory

)( -x c\-o

)-) -

0 0

0

-

C

FIGURE 6.2. The five interactions of the ligbt"cone field theory. ,,--_ . strings can break, fission, and pinch. Notice that each interaction takes pJI!I~MlI along the string. This is the solution to the problem of causality, which t&J)WJ~ all nonlocal point particle theories. Thus, string field theory is the only 1¢.~~ theory based on extended objects that preserves causality.

Because all the symmetries have been extracted out in the light co~te::ji there is no overall guiding principle for the theory. We will, there'£o¢i:mli postulate the following principle:

The only interacting string configurations that are allowed action are those that instantaneously change the local . strings. Although this principle is defined only in the light cone gauge, UIP,:.;v., that it is sufficient to determine all the possible interactions of the fi~l~~ij.1l There are only five such local interactions (see Fig. 6.2) that are COllSi"l'~~ this new definition oflocality and also the conservation choose the parametrization length of the light cone string to be pto'P-Q!• •• to p+, that is, IX = 2p+, and hence momentum conservation u'LlPJ1~j?:;.~ sum over string lengths is conserved.) Thus, we postulate

where each term corresponds to a specific interaction for an UI1<;U:.' 1wm and a closed'string field W. These five interactions, sYlnbl::llt(~ally:':~P.~ can be represented as L/ = <1>3

+ <1>4 + w3 + <1>2\11 + cl>1JI.

We will write explicitly what some of these interactions are, cOI]t;ii!~ the conditions of locality. Once we have written specific rep'resent~~~

6.4 Interactions

263

interactions, we must eheck that they reproduce thc known results first quantized theory. i;~~;,:::,i~:ntcrdction involves the breaking ofa string into two smaJler ti' the condition o flocality, the string can break only at a single along the string. The disturbances from this rupture shou ld laler ,k,ng the string at velocities less than or cqual to the speed of light. arc led to pos tulate that the points along the ~tri ng are continuous Ie l)o,..d'>«:1 o,ftb""""",tic''''. The uniquc fonn of the vertcx function, with momentum conservation and locality, is a series of Dirac della ,,,hal continuity of the three s(("ings. Our vertex is [lJ

",u"e

dp~8

(t J p+F)

DX m ¢J t (X ) ¢I'( X])¢I(X 2)om

+ ac.,

(6.4.4)

0, = DX 1 DX 2DX 3•

(6.4.5)

. 6.3). We will usc the notation

o S :s: JT(al + 0:1). (1

al =

a

for

a2 = a - ;ra] O'J

0 .:: a ::: JTO:]. ;rO: l::: a ::: ;r(a ] + a 2).

for

= ;rr(Q'] + Q'2) -

(J

for

0.:::

(f :::

;r(al

(6.4.6)

+ az),

i length of ellch string is given by ;ra; and whcre the onl y thc transverse modes of the s tring . . clillllcrually perform the D X integration in the above vertcx it is a simple Gaussian. Lct us defi ne

" ;(X) = (XI";).

(6.4.7)

" " 0-0

"

PaTlimelrization of thc Ihree-string vertex. The parametrization length ill given by Ira,. The lIum of all three lTtX( is equal 10 zero.

264

6. Light Cone Field Theory

Then we can rewrite (6.4.4)

S3

f dp+r (t p+r)

=

8

{
where we have introduced the vertex function

1V123) =

f

DXl23lXd IX2 ) IX) 8\23.

Since the eigenvector IX) in (6.3.11) is a simple Gaussian in X • . plicitly calculate the integration over DX'23 and obtain an exactt,>tmu the vertex function in terms of harmonic oscillators. The actual calculation will be more conveniently carried out mltl,l~::ml resentation. By taking Fourier transforms, we can easily r.orlvp,rr eigenstates in (6.3.11) to P eigenstates: .

IP) =

kIT exp (-~p?n + Pi.na: n- ta:na:n) 10). i.n

We easily check that this expression reproduces the correct el~i~.1 equation (2.2.9):

= (ai,n + aln) IP} .

Pi,n IP)

Let us write explicit Fourier decompositions for all three strings·t~~~~ vertex, in both the x and P representations:

Pi(r) (r)

Xi

=-

1

n lar I

=

(rPi + ~ r.: r na ~ VI!Pi.n cos -

r )

(r + ~ Xi

ar

n=1

1

~

n='

r

eXi,n

Or.

naT) 8"

cos -

ar

vI!

where

8\ = 8(na, - a), 82 = 8(a - nO',),

I

= 0, + 82 =

~

1.

The integral that we would like to perform is given by 1V123)

f =f =

DP123

a

DP 123 8

IPi ) 8123

(~(}r(ar)P?)(a»)

IT exp(-! 3

X

4

r=1

p .(r f

t.n

+ p.(r)a!r)t _ r,n t,n

!a!r)! a(r)t) 2 I.n I,ll

6.4 Interactions

265

is simple and stnlightforward because it is a Gaussian. The only we will face is the explicit form of the delta fimctionsl h123 in in terms ofhannonic oscillators. Let liS now extract the Fourier uf the delta functional om in order to perform the functional . Let us take the cosine transform of the constraint equation 8J.Hong

P's. Ordinarily, CQsine and sine lransfonns yield delta functions. "." "''' the three strings all have differenl lengths, we find that the cosine -;. : transforms yield matrix equations, nol delta functions. In particular,

tb" (6.4.15) 0 In"" when we take the cosine transform. 00

. + "(A (I) p(1l + .1(2) pi") + 8 (1) pm + 8(2}p(l) = W ",n 11,/ "'" n.t '" I m /

0 ,

(6.4. 16)

,,:1

A's and B's are the various overlap integrals between the cosine . " . Instead of enforcing this identity as an exact equation, this operator only be satisfied when acting on the vertex function. We want . function \ Vln ) to vanish when acted on by the 8m . This means [I,

(6.4.18) Fourier coefficients are given by laking all Fourier transforms of ",;';'~~ modes. Because the three strings all have different lcngths, '! >ave ~.olltri"i" Fourier coefficienls. By explicit construction, following Fo urier coefficients:

1 o:::2(II/m)lf 2(-1)"'iTal

i"a,

11(1 mu cos-cos-da

(}

= -(2/rr)(mn) I!2( -I )"'.Mfl .

at

:in(m:fJ/. m f3

n

00:)

(6.4. 19)

266

6. Light Cone Field Theory

~

A (3) mn

-

B~)

= 2(m)-1/2(_l)m_ I

Dmn,

1H¥ I

1""" cos-dO' mO' a3 0

= 2(m)-1/2(_l)m_

1

Jra2

"al

= 2a3(Jral)-I( _1)m(m)-3/2 sin(mJrfJ), B~)

1

,,(al+a 2)

Ina

cos -dO' a3

r

= -2a3(rra2)- 1(-l)m(m 3f2 sin(mrrfJ), where iJP. = al/a3 , B(1) m = -a2BIth and B(2) m = al B m· Although the expressions may seem complicated, the actual caJ,ql1J~~~!j#J the integral is simple because it is only a" Gaussian integral. By pe~;1 the integral, we find an explicit form for the vertex function. We .'" the integral over p(3), which is trivial because the delta function " be a combination of the momenta for the other two strings. The integration over p(1.2) is also easy, because it is also a combine terms, we find the compact result for the vertex function harmonic oscillators [1, 6]:

where

and N mr

-

-

K =

_(m)-1/2(A(r)Tr- 1 B) flit

-!Br-l B, •

t

P = alP~ - a2PI'

r =

3

L

A(r) A(r)T.

r=l

Let us now summarize what we have done. We have postulated thllt:fjj1OjJjl interactions be~een strings are those that are local; i.e., only ins~~ local deformations of the topology of strings are allowed. In avoided problems with the violation of causality, which . , attempts over the decades to create nonlocal field theories. What is<,': , is that this principle alone is sufficient to determine all five string . ,.'" in the light cone gauge.

6.5 Neumann Function Method

267

that the Neumann matrices N~~ arc an determined uniquely from

""'poOl,dl.,lo", (6.4.15). Locality and the conservation ofmomcntum sufficient to determine the precise oscillator representation of the wt: wish to l;:heek thai this WprodUl;:C5 the usual Veneziano formula. two things that have to be checked:

,must show that we reproduce the Neumann function for the Vencz'iano

" :~~~' ~,:~;~;'~;i~~:~l~~b;~o~~:~: traosformation of coordinates from e r to the usual upper balf z-planc is the correct now show that we rep roduce the usuaJ string amplitudes, dcmonof the firsl and sccond quantized interacting theories the

Neumann Function Method 2 we saw that we could perform the functional integral over a disk

(6.5. 1)

j) is the NeuffilUlIl functioo between the points i and j on the rim or upper half-plane with handles, and M N . L includes all measure ",Iwn',g,.

0" this approach, however, is that the fWlctional intcllral

or thc upper balf-plane. The string interprctation of on thc 'nf""",,' disk is ohscure. To make the connection between the string and the conformal disk. lei us firsl make the conformal transforp :::: In l. This conformal map takes the upper half z-plane wd into a horizontal strip in thc p-plane that has width JT. Tbis horiin tum, ean be interpreted as the surface swept oul by a single . What we want is a generali7..ation of rhis map to the N-pcint

folio., t.iIWld,lstam [8J and make the following conformal transformaupper half-plane: p ::::

L

0'; log(z -

z;) ,

L

0'/

=

0,

(6.5.2)

paramctri:Ullion length of each ~tring is Jto; . This conformal map the upper half-plane into long horizontal strips, which correspond to of splitting strings.

268

6. Light Cone Field Theory

Analytically, we can see qualitatively that this transformation maps'i~ half-plane into the proper light cone diagram. Let us start our vat~~t positive infinity on the real axis and then slowly move to the l¢fjt;:::i~ approach one of the singularities on the real axis, p moves rapidly infinity on the real axis. But we hop over one of the singularities the logarithm picks up an imaginary part: p ~ a, 10g(i1f z)

= bra; + ai log z.

Thus, in p-plane, we jump na, vertically upward. This c01TesiponruaO:tt from the bottom of a string oflength na, to the top of the string at ~~tj we keep on moving in the z-plane to the left, t4e point in the p-I"liW,@li to move to the right (displaced a distance nOli upward). As we continue to move to the next singularity at Z;-l on m~:'''''''~.2 something strange happens. At a certain point, the p variable can s.l. qW~~\li stop, reverse direction, and begin to move left in the complex plame infinity. This turning point in the p-plane can be calculated by ~i derivative Yl.

turning point:

dp =0

dz

and solving for z. As we hit Zi-I in the z-plane, the point in the P-I)~a,!~'Jtifdi to negative infinity. As we hop over the Zi-I point, we move afl()tll~~~:~~ vertically upward in the p-plane. In this fashion, by carefully mO~.!.t9)H real axis in the z-plane, we sweep out the light cone cOI1fi!5UTati~)n .&~J~ Fig. 6.4. Fortunately, we know from (2.5.7) that the Neumann function iIi . halfz-plane is the sum of two logarithms. Next, we want to power e'j~'ti~iMl} same Neumann function in terms of variables defined on the f'-"H
FIGURE 6.4. Parametrization of the N -point function. The incoming . legs have positive (negative) parametrization lengths. 8y changing the lenl~Wt~ interior horizontal lines, we create the various field theory graphs that s'\W(~1 dual amplitUde.

6.5 Neumann Function Method 3

269

straightforward process to power expand the Neumann function in and rr

N(z, z') "'" In Ie! - ef t + In le{ - efl

= -

't ~i!-~If-fl ~_ I

n

COSflTj cosnrl + 2

max(~, f).

(6.5.5)

nfth" this expansion reproduces the Neumann function is rJiher simwe notice that the function COSnl} has :tero (1 derivative at the which is consistent with the fact that X' ;:: 0 at the endpoint of

Second. by acting on the power expansion hy the operator \]2. we can it vanishes except at the point Z = z', Then we can show that the lenn which selects the maximum of the two, is the correct expression ~ . . . . . ±oo. ay uniqueness, this expression must therefore be for the Neumann function for the interacting string is done in the First, we make an educated guess that the Neumann function is

f (~)• cosml, cosn,,; exp [-nl~, - {;!] ..,\"..N(p, p' ) = -!J", ...1 •

-." (6.5.6)

prime means to delete the n = 0, In == 0 lenn. and where we can coordinates directly on the rib string: p

=

a,(~,

+ i1),) + const.

(6.5 .7)

1,2, ), and where the constant can be choscn so thal lhe coordinates nearest ruming point The proof that this expansion is correct is out in the same fashion. Notice that the first two terms on the side of the equation arc simply the tenns coming from the free wh;ch guarantees that the equation V2N = br o2(Z - r) is satisficd. ... ,we nOle that the rest of the function is nothing but a power expansion s of"""" which solves Laplace's equations. Thus, sincc the Nel101llnn is unique and the expression solves Poisson's equations, it must be

~~~f~~t~o'~~no~t~;c~,:th~a:it~w:c~~h:~afvc:~dOOC nothingexpansion new at this point. We as a Fourier in the various

~

Thc crucial factor in the above expression is N~'~, which directly in the three-string vertex. These coefficients, in rum, can calculated because we know the form of the Neumann function ''::' :':~:;~;I~:~Which is the sum of two logarithms. Thus, we can tak.e )l transforms of this known Neumann function, defined ovct

270

6. Light Cone Field Theory

the upper half-plane, and extract out the Fourier coefficients Nnm • vv¢'i~ the previous equation and now solve for the Neumann cO()m'Clelllts: of the Neumann function. If we take the Fourier transfonn of ....".>I<..:\,':W! expansion (6.5.5), we arrive at [6]

N's = ~ nO n 's -

Ix 2ni dz 1 zx,

_

e-n~(z)

Iii

N mn -

mn (2)2 n

=

=

x,

(n > 0),

Xs

dZ r

x,

'e-m~,(;:,)-n',(z,) )2' ( Z, - Zs

dz s

(Xl = I, X2 0, X3 00). If we insert the Neumann turlctlorf::a over the upper half-plane into the previous ·equation and make ..,..~"...,..,. change of variables, then we have an explicit formula for the· coefficients. Now that we have an explicit form of the Fourier moments V4,>Wl2nl mann function defined over the light cone diagram, let us COJlsici¢t::Ui of three strings. The transformation of the upper half-plane to the configuration is simply .

p = a1ln(z - 1) + a2 lnz.

The turning point for this mapping can be calculated by taking

dp = 0 dz

--+

Zo

\.1J"'.· '1'1;IU.'

= _ a2 • a3

Thus, the time at which the string splits is equal to 3 1'0

= Re p(z

= zo) = La, In la,l· ,=1

What is remarkable is that the mapping (6.5.9) can actually be . for z in terms of p. This means that we can explicitly calculate functions, rather than appealing to abstract expressions such as

We begin our discussion of calculating the Neumann c~:~~~li!:jll fully placing coordinates on the various three strings. For example, : the coordinate ~3 on the third string;

S3

=~ + in = ~3 + i1]3. a3

By dividing out by a3, we guarantee that the '1 parameter ranges to n. In termS of these new variables, the mapping (6.5.9) can hp·... ,itmltr.

):~:rn!ll

- In z -

S3 + in = -

al

a3

In (1 -

= _ al In(l a3

~) z

+ eCJe-Cl+irr-lnz).

6.S Neumann Function Method

271

int,rod'.", .,"" variables: )' 3!!! - ~)

x

lOi

+ i 1f -

In 4.

(6.5. 14) (6.5.15)

e ll,

a,

r---, a,

(6.5. 16)

,,,d.,,, to

+ .xe)").

Y = Y l.o{l

(6.5. 17)

,.",Iy., !hii, equation can be solved expJicilly by a power expansion of the ~

..,

y = LyO,(y)..-' .

(6.5. 18)

(6.5.18) into (6.5 .17), we fin d an explicit expression for 0,, : (/~

I

= ;;!(ny - I)· · ·(ny - n + 1). ~3

back our expressions for z and

(6.5. 19)

into (6.5 .13), we now find (6.5.20)

omp~·.lh.is equation 10 (6.5.6), letting the variable z' go to zero. In that ~

Inlzl = -$J + "L,N;J cosn'13 .

--,

(6.5.21)

Oruogth.,,~st two expressions, we see that N:~ is proportional 10 alt. analysis can be carried oullO the other strings. and eventuaUy we ,.Iate. the Neumann coefficients in (enns ofa" . The final result is (8) _,.

mil

N ,; .. = . mal

_,_.

+ lIa,

Cl ]U2Q ]N

...

N~.

it: = a;ll,,(- a r+l/a,)ex.p(mro/a,). =

- a,+t/a,

(6.5.22)

and

f,(y,) = (" y,)- '(":'). K = -ro!(2ala2ClJ). )

TO=

LCl, ID la,I,

,,'

(6.5.23)

272

6. Light Cone Field Theory

At first, we see a vast difference between the functions delive:dJr.~ second quantized action in (6.4.24) and the Neumann functions derived from conformal maps. The first version of the N matrix is d¢~.1 terms of products of inverses of overlap integrals. The second vel:S1(!t(ijfi ••N mamcek is Itrrem,1S+.]tTi'\\maml.fi'ru:1.1.nu.<;.J.)TLa Riemann surfac,e;; Miraculously, however, we can show that they are the same [6, T$ N- rS Nnm = nm' Nm' = N-'m·

Thus, although the two derivations seemed at first totally U"""u..,uw~;:~?~ up with an exact equivalence. (The actual proof that the Neumann c~~fflii found in the field theory (6.4.24) and the coefficients found in the ti,.c'h'ii.:;;. theory (6.5.22) are the same can proceed in two ways. First, we known theorem that the solution to Poisson's equation with Km)WIH'~~n conditions is unique. We can show that both Neumann functions are S'j,Q'j:~ the Poisson's equation and are continuous across the -r 0 COlrfi~~l~t~~~~ the strings split. Thus, although the two expressions appear V,""thii,::(.f.:m they are actually the same. There is a second proof, however, wruLch:: brute force to show the equivalence ofthese two expressions. The that we manipulate complicated expressions through the identity Details of this calculation, however, are very tedious and will not 0" ielilltftl The next step is to prove that the Veneziano model is reproduced cone theory.

=

6.6

Equivalence of the Scattering Amplitudes

Let us begin our discussion by writing the N -point amplitude in tne~':mm formalism. The amplitude is a straightforward generalization ofthe,lUlilt found in the light cone point particle field theory [8]: AN =

f ~ f ndP/~\II(')(Pj~~)W, d-rj

1=2

r.n.r

where

where \11(') represents an incoming or outgoing state vector of string, -rj represents the interaction times, P- represents the Fouril~~jMI component necessary to convert the expression to the on-shell arLlp,H

6.6 Equivalence of the Scattering Amplitudes

273

rel,,,,,,,,,momenrum distribution smeared out over the string (which an arbitrary collection of higher resonances). that there are two immediate problems with this amplitude: Neumann function occurs only with the transverse momentwn exci· We must show that the complete expression is Lorentz invariant, the longitudinal momentum factors. interaction lime variables Tj found in the string field theory must '" , rransfonncd into the usual Koba-Nielsen variables l,. This requires II ;ob,iru" whicb in general is quite compl icated,

ft""

~ ",,,,,solv"lhc prob,lern, th,,;e.mi"gly ",oMd" iiv·i sri'c form of the amwith the transverse and Jongiructinal factors occurring in qualitatively fu.~ hion, However, this is an illusion, There is a trick that converts ..,;sic,n into a Lorentz-invariant integral. Notice that the T variable is of Laplnce's equation, TII\IS, because it has the correct boundary wc can write it 8S a line integral ovcr each extemallcg at infinity

"j du'N(u, r;"., '1',).

r = - I L...

(6.6.3)

2rr ,

3 rather strange idemity. which expresses r in lenns of expressions . . To prove Ihis equation, simply multiply both sides of (2.5.6) perfoml integrals over the twCHfimcnsionaJ space. By carefully elim. . terms by integration by parts, we find the above equation.) Thus, we . the T variable with its Neumann function lind write

LP,- Tr =

2~ L

r

r.f

P,-

f

du' N{a,T,;u',T,)

-1"p, -pi j d jd 'N( L

= 1

11'

(1

'oS

(T

(1,

,

T,;a, <,).

(6.6.4)

a,Ols

this longitudinal contribution to the Neumann function containing the transverse contribution containing PIP}, WI! wind up with a runction. Thus, the terms of the rorm p - y CQvariantizc the terms PI Pj. the Lorentzinvarianceofthe integrand consequence of the theorem in electrostatics. The converting the interaction times
are the usual Veneziano variab les on the rcal axis.

(6.6.5)

274

6. Light Cone Field Theory

= 0, XN-l = 1, and XN = 00. Now cOInpl~

Let us choose the frame Xl Jacobian with the factor

where Ci

=

P=

a2 p OZ2 IZ=Xi'

L

Oli

In(x; ~ z).

Notice that the Jacobian and the factor involving the del:iVl!ltiv~lS}~ transformation have the same analytic structure. They both singularities when the various Xi touch and have the same bOlmdai1i:::~ tions. Thus, they must be the same, up to numerical factors! To calC'91~"ii overall constant, we are free to take the Xi spaced widely apart, so tlI~*:~~~ P(Zi) =

lnXi

L

Ols.

s~i

so that

(In addition, there are also complicated terms which include the det'~iIli of the Laplacian defined over the four-point configuration and the :z'~~r@'.~ components of the Neumann functions. These terms cancel among e~~!i!:$] if we carefully look at their analytic structure and their ""·'U5'U"'ULJ"").:::~:>1Sl: demonstration of their explicit cancellation was performed in [6] t{\Y::fh;~' point function. The cancellation for the arbitrary case, including l,c"ip'IP'~~ performed in [12].) Thus, we have reduced the Jacobian to a trivial fa~~~1 12], the product of the various Koba-Nielsen variables. Putting ~ together, we have AN =

f fl 'lI(P?;) f dJL I.n

x

n

exp

r<J

(~L fda' da" pt)(a')N(a', i,;a", is) r.S

If we take external tachyons instead of arbitrary external find

as before. Thus, the N -point amplitude is recovered.

6.7

Four~Slri ng

Interaction

275

Four-Siring Interaction i led us to postulate tne eltistencc ot'a four-string interacwnere two strings can combine at their interiors and instantaneously ~eth,,;'r local topology. AI first. it doesn 'I appear as if this diagram should the Neumann function method. since the upper half-planc was al'.:::~.~ into a configuration thaI was planar. HowevCT. the fact that the ~i (and all the other) interaction tenns are presenl in the Veneziano can be secn if we rigorously examinc the region of integnllion of variables. begin with a four-string mapping and sel XI = t, .f2 = 00, X3 = 0, a nd . Then the four-string mapping becomes

:oba-N;"ls,'o

p=a: 1 In(z-I)+a3 In z+a:.In(z-x).

(6.7. 1)

where the mapping is singular, let us set

dp =

dz

o.

(6.7.2)

og ,';s equation yields Ihe turning points of the transformation:

(6.7.3)

(6.7.4)

":::~~;cthC two solutions of the turning point equation, given by the

:::

±

rool, show thaI there are (wo turning poinls on the Riemann have the freedom of moving past each other. In the s- and t-channel the world sheets of thc strings smoothly transform into each other. a n interesting thing happens when the imaginary parts of the two point'> in the p-pJanc coincide.

~;~~~2;~lei:t~u~s~ls~:t:UdY the Feynman graphs for the f - and II -channel . (sec Fig. 6.5). We sec Ihat these two graphs ealUlot

~

into cach other. The sirings meet either near the top or near the : of the diag:ram.. so there is 00 way of continuously defonniog these 'g"'ms iOlo each other. But this is impossible. The conformal map, by was a smooth one Ihat aUowed us 10 go continuously from the t to ita,,,,,,,1and viee versa. In oth~r 1II0nl.t. a piece of the integration region To see this, let us calculate precisely when the 1- and u-channcl

276

6. Light Cone Field Theory

----------------------~~~ 1 _________ ~ 2

1

4

---------- ---::;-, L_____________ __________ ~

~

2

FIGURE 6.5. The four-string interaction. It is impossible to the t- and u-channel four-string scattering diagrams into each other three-string vertices. Since the confonnal mapping is continuous, there is a missing piece of the integration region, which can be SUI)pli:e:~:~ postulating a new four-string interaction.

diagrams meet We set /). equal to zero and then solve for x:

Notice that there are two solutions that give us the t- and u-(:ha~~~t~HI points. The region (x+ < x < (0) gives us one diagram, aW",,:LI1<>:"" (x_ > X > -(0) gives us the other diagram, but what about between? This is the missing piece. It represents a continuous de:torm2Iticlp::!~19i channel graph into the other graph, such that the local 'VfJ'UllJ'''J,,'V'''::J~ graphs is only instantaneously deformed into the other (see Fig. ~,,~,~~::«::::;

FIGURE 6.6. Topology of string interactions. The equipotential lines on a disk with external charges are isomorphic to the interactions of we have like charges located on opposite sides of the disk, then the lines collide in the very center and rearrange their topology. This is interaction. Likewise, all five allowed interactions can easily be fushion.

.".:S:tl

ft'

6.7 FQur.String lnlcrac.cion

277

278

6. Light Cone Field TheoI)'

In this intermediate region, the local topology of the four such that the strings reconnect in a different sequence. graphical proof that the four-string interaction that we lier is, indeed, part of the Veneziano formula (for the t- and ,"E',,,·...= graphs). Without this missing piece, the string field theory is actu:af.~ complete and violates conformal invariance and other n1"rln"rt;":~:: S-matrix. The four-string interaction, therefore, requ4"es an extra int~~gr:ati(I.lt: This is like a "zipper" that allows us to locally change the topolo&i;=:6) four strings. At first, we might suspect that this interaction takes pl!i~~~1 than the speed of light. After all, the four-string interaction instantaneously in time because the da integration happens lstalntlyt:)Va to be violating our postulate of locality, which we originally ImlPOsedJ,i: a causal theory. The resolution ofthls puzzle is that the four-string interaction to the Coulomb interaction term that arises in Yang-Mills theories ··f.ii*-fi~ tizing in the Coulomb or light cone gauge. In the Coulomb gauge, appears quadratically in Ao'112 Ao (without any time derivatives) . in the coupling to ferrnions Ao y °1/t. By functionally integrating otdi:~i! we find . m'

'i.\I"

t

which is the four-fermion Coulomb term. Notice that the o~:~~~lil an instantaneous operator (Le., has no time dependence). This ently violates relativity, but the S-matrix is actually strictly in the presence of this term. It is an artifact of the gauge any apparent violation of causality disappears when we _..';";4';'."" S-matrix. The apparent violation of special relativity is only an lllUISl0rt; integration dIY over the zipper in the four-string interaction is COll~f~~~.1 special relativity. Let us write down the four-string interaction term for the tion. We stress that the four-string interaction term can be postulating locality but is also consistent, as we have seen, with amplitude. The interaction term is [1]

where J.L is a measure term and

6.7 Four-String Intcl1\ction

279

,., > 0.

a)

Q"L <:

<12 = 1r laL

I-

0,

O"J

(12

=

0""2

= JrIQ"zl - u.

lfa) -

Y= x

(I]

+ 7r(a-
((4

> 0.

a-2 < 0. if y < 0"4 < .1TQ"4 . ifD <: <13 < x.

ifx < 0"] <

iro < U4

(6.7.7)

lfet].

< y.

la- d).

0, = 8(a. - y), 9, ~ 9(y - •• ),

iii

=8(al-X),

01 =

9(x -

IT).

appears : ~~~:~:~;:;~ithi~"'it!:o: ith;c~comp1cte amplirude. Thcrc appear . I mat mustN-point tediously bc checked. However, the four-sIring interaction can be explicitly calculated, it

~

that wi ll reduce this problem cnomlOusly. we notice that jf external e10ctric charges ql are placed on a circular the equipolcntiallines in the disk are easily drawn . The key o bscr'O''''~'" is that the cOllformaJ map takes these equifXJlcnlial lines and into vertical lines on the light cone diagram. Thus, the topology

. lines mu.st repmtluce the precise IOpology afinleroctingopen . This relDllrkab lc observation reduces a seemingly hopeless relatively simple problem, th,lt of drawing equipotential lines for a external charges. autl,II", we know that T :::: Re P(l) = Q"i In Iz - ld. But WI;: also electrostatic potential in two dimensions at a point r produced ~'" 01' dha"g" qi is b>iven by

L,

p"jnt

potential

=-

L, q/lu Ir -

rll,

we can reintcllI ret T as the electrostatic potential created by point Therefore, lines of equal potential arc precisely lines of equal "t o a~c'n~ the Mandelslam strip are just ve rtical lines, which t . Thus, rhcequiporenn·allines on any cunf(lnnal ct.)

correspond

/0

rhe phy.~jcal evolution of an

280

6. Light Cone Field Theory

From Fig. 6.6 it is obvious that a four-string vertex must also · light cone formalism. In fact, by analyzing equipotential .. show that exactly five distinct interaction terms must be added inW(tM~ integral. Notice that closed string interactions emerge from the
6.8

Superstring Field Theory

It is possible to express both the NS--R and the OS action as a light . quantized field theory [13]. New features arise when we do this: (1) The field functional 1oj, point at which the strings break. Without this extra in,,,·rt;r.": theory is neither supersyrnmetric nor Lorentz invariant.

Let us discuss the OS action in light cone language, because explicitly supersymmetric. Our first quantized light cone ""L.'V" (3.8.3) Sic

=

4n~'

J

2

d z(8u X

i

arzxi -

(2i ct')p+OUyCl o",oa),

where e1,2 are spinors with eight components in SO(8) space aiia!::SJJiIIl neously spinors with two components in two dimensions. RecalH~[ml dimensions a Dirac spinor has 32 complex components, a MajoraLIll1,~P.l.:. 32 real components, a Majorana--Weyl spinor has 16 real cOlnp,ori~:*~~ the light cone gauge the spinor has eight real components traJlsfi)tii:i#lJ@I SO(8). The problem with quantizing this action is that the fermion fiel~~i conjugate. We have the equation

Thus: {o'la(a), elb(a')} = {e 2u (a), e 2h (a')}

~ 8ab 8(a -

{Ola(a), 02b(a')} = O.

Notice that this is not in canonical form. These fields are self-clon:iu~ fermionic fields form a Clifford algebra, while we would prefer to ·ml.;'!fK~ mann states without the delta function on the right-hand side of

6.8 Superstring Field Theory

281

way out of th is problem is to divide up the eight states in the spinor 4 staleS, with one sct of four states being the conjugate of the other One way to do this is to use the SU(4) subgroup of SO(8}: SO(8) :0 SO(6) ® 0(2) = SU(4) ® U(I).

(6.8.4)

decomposition, the 8 of 50(8) decomposes intO

8 = 4 ED 4. ,w d~""op<,sethe

(6.8.5)

eight components ofthc spinor inlo 810 = (81A, "IB).

elil = (e2.4 , ,\u).

(6.8.6)

and B range from I to 4. We use the notation

4:

0;'=6",

4:

)." = ).;,.

(6.8.7)

definitions, we have the new independent variables 9~, which anticommuting witho ut any delta function as in (6.8,3). The relations with the canonical conjugates are 19";'(u), )JB(a')I = .5(a - U')8 AB c,U.

(6.8.8)

there arc so mnny stages in thc reduction of a Dirac spinor w ith 32 component, let us summariles bow we reached onl y four independent Dirac = 32 complex. components. Majorana = 32 real components, Majorana-Weyl = 16 real componcnts. light Cone = 8 real components. canonical = 4 real components. we have decomposed the spinors according to the SU( 4) subgroup nm to decom pose the vectors under SU(4) as well. The Vt..'Ctors . :.' 1,2, ... ,8, will be broken up as follows:

. w,

(6.8.9)

A" = 2- 1!l(X 7 +iX 8), A L = 2-1t2(X7 _

j

XS) .

(6.8. 10)

,..I ' c'"oo for the free theory is given by

s=

f

D I6 z[8+* ':L lJI+Tr{8+¢iL ¢I}].

(6.8 .1 1)

282

6. Light Cone Field Theory

where the independent set of integration variables is given by

n l6 z = n 8 x l n4elAn4e2A, and where ct> are open string fields and \II are closed string fields . . OUf basic field functional is now given by (where we have plalce:l[t~ labels on the field):

ab[X(O'), 0 1(0'), e2 (0')] = -ba[X{rra - 0'), e2 (rra - a), el where the 0' variable is purely symbolic but was added to show hm~: tl1'D transforms under a twist Following (6.3.44) fOf the bosonic case, we can construct quantization relations:

1

.

P

.

[ct>ab(l), q,cd(2)] = 2 +o(al +(2){8aCobd~16[Zl(0') -Z2(0'

- oadC"c~16[ZI(0') - z2(rr!a21- a)]}, where

z = (Xl, elA , e2A ). Now that we have established the free theory of superstrings, let:Us,:t'I! the difficult task of constructing the interacting vertices for sUf,e;'~f~J will find several complications:

(1) There will be two sets of oscillators, not one, and sef,antte: R:9.I~~~ conditions arising from the overlap delta function. sets of oscillators commute with each other and do not mix. (2) There will be extra terms defined at the joining point of the tnr~(~:$J In general, extra fields cannot be introduced along the string b~·.~~~j will violate Lorentz invariance and conformal invariance. HCIWIl~\i~i!:~ can be placed at the precise point where the string breaks. We~:m1i:ili careful, however, because of singularities that exist at that po;trit~::}~~mil (3) The greatest restriction is supersymmetry, which will cOlnpll¢(~:ij::~ mine the nature of these insertions at the breaking point. We begin our discussion of the vertex function by postulating ~#>:t~:iH it will take. Based on an analogy with the bosonic case, we poi5tur~t~~~lml superstring vertex must look like

IV} = Zi

expJ~o + ~s] IO} o( ~ar};( LP:}S( LaAA

where the Zi fields are insertions at the breaking point and ~o bosonic term found in (6.4.23): 1300

~ = _ '\' ~ air) Frs a(s) o 2 ~ ~ -m mn -n r.•=1 m.n=1

300

+~ ~ Fl' air ) p ~~ m -m r=1 m=1

_ ~ p2 2ot'

6.8 Superstring Field Theory

283

the bosonic theory, n~;:n~' ;:~:'~:: ~~~::';,f~~!~:~:~~l:~~o:with operators. Let us adoptthat the decomposition:

e1';: =

_ ,_

8 2';: =

~

AlA

L

RA "~I,,alk..l ,

h. "

L RA e-j""/~1 I L

.J2a.

=

n

n

RAeina/lrrl

Jml.l" ,1.211

,

=

"

'

L RAe-i-/I,,1

1

~

./2:rrlal"

.

{R~. RB "I = as,,,+,,.08 I1 B• ",,3

lis =

_003

L L

R~'~A U~~R~~A

+L

m.n .. I,...=1

__

L

V; R~!~eA.

(6.S. iR)

",_ 1 r=1

U and V matrices are totally unkno ....'Yl and I· . A e A- = _(£110 1 _ R l ).

.,

is no justification for this fonn (6.S. 18) other than that it satisfies boundary conditions thaI we will now apply. We will enforce the

(6.8.19)

is + 1(-1) if the string state is incoming (outgoing) and the tilde ',"uh,,''',md oscillator. Tllesc conditions, when written oul in Fourier resemble the conditions found for the conservation of momentum.. I we gcnerdlize (6.4.17)

tf../ii A~!(R~);' ,=1 ,,=1

t f: ~A~!(R~)'

J 1,. ".1

/1;1

"II"

a,

+ R~;') -

-

R~!A)IV} =0.

I

h.BmEl' IV}

~ 0,

(6.S.20)

enforce the continuity conditions

LA:(O) ~ 0,

" -,

~J..,(n)=O .

(6.8.21)

284

6. Light Cone Field Theory

These continuity conditions, in tum, require the following CO]rrdltlOWfiil Fourier modes: 3

00

• - : : :.

LL -In A~~(R~)A - R~~A) IV) = 0.• •••• a, :.. r=l n=]

3001

" " _A(r) (R(')A (

L L -In

mn

r=1 11=1

11

a) IV\ - 0 ..•.•

1 + R(r)A) + -8 -_ -n ..fi mae A

1-

.•... , ..

We now have enough conditions with which to solve for the matrices. The calculation is arduous, but we finally find

Next, we wish to construct the supersymmetry operators in the theci@~iOOl ing on our experience with the generators of the free sUJ)er:syrnmletritilrliill~ Let the first quantized supersymmetric generators in (3.8.22) be then the second quantized supersymmetric generators Q are related free level by

1 f 00

Q2

=

ada

16

D Z[Tr

~-",q¢a + *-",q*a]'

Notice that the canonical quantization conditions (6.8.14) gmrrarlte~ q's form a supersymmetric algebra, then the Q's must also. In n'n-N;;~fil~m first quantized q's obey {q-A, q-B} = 2h8AB, {q-A, q-B}

= {q-A, q-B} =

O.

oftheise.t#l~11r

Now we wish to construct the second quantized version Specifically, the interacting part of the second quantized generator .... :~;.I:}'~ Q-A

_

= ... + A

x

~16

f Dda,D

I6

Zr8

(t

(ts;Z;) (~tI (~21

a)

(
If we now substitute the expression for the second quantized the commutation relations, we have a series of tenns that must "ULl-<,.~~ To the zeroth order in the coupling constant, this is guaranteed be4;al.iIi!~:~1 satisfy the relations of supersymmetry. However, the tenns that the coupling constant contain mixing between the free q's and the

6.8 Superstring Field Theory

term". Specifically, we find

,

285

,

Lq,-'I Q-') + Lq;'IQ-') = 21H) 8", ,:.1

,

,=1

Lq;' IQ-') +(A

-

0) = 0,

(6.8.27) C-" ' -lhese equations, we postu{Qre that the Q's have the general form

IQ-')

=

r'IV), (6.8.28)

now assembled a formidable appararus, mostly constructed by gue:ssanalogies froJU the bosonic C
Zl = 12ar ,j2

(pi _a L m N:rt~;:), t,m at

y..t = l~rtl l/2 (e..i + ~ L ~N;;'R~~..i).

,,2 t.'" rtt

(6.8.29)

let us tum the crank. We must generate IH) by the conunulation '"' of' ,up",,..,,,,,,"y. By brute force , we now find that =

(r1/2ZL _ Z(P~R y.o:yti + ';;ZHeABCDYAyiiycyb) IV).

(6.8.30) we have a complete expression for the venex functions that satisfy i)'m""'''Y, let us try to rewrite the Z aod Y functions. We noted earlier were chosen such that they commuted or anticommuted with rue conditions. We suspect, therefore that they actually vanish except ~;~~~.'; poinl of three strings. Because functions can easily diverge at ~ point, we must carefully take limits as we approach that point. analysis, we can rewri te Z and Y as yA

=- lim(~E)I!2(O"I(Jral -.c:) +8 Aj (JraL - e)).

,-0

(6.8.31)

286

6. Light Cone Field Theory

The crucial thing to note here is that the Y's and Z's are both string; i.e., they occur only at the point at which the strings join. spared any nonlocalities that would spoil Lorentz invariance.

6.9

Id¢i~hii

Summary

Historically, it was conjectured that a field theory of extended Ohlle~j~Ii;'Ui possible for several basic reasons: (1) Attempts by Yukawa, Heisenberg, and others found that •• ~'''''''''' deviated from locality in quantum mechanics, we violated vibrations at one point of a blob would propagate faster than light throughout the blob. (2) The theory would not reproduce the Veneziano model, be(:&'\i::~~ symmetry and many of the properties of the Beta function onlY:'Jl.'l shell, while an action by definition is off-shell. (3) The theory could not be both Lorentz invariant and unitary OI:I.~:SU cause a quantization program for strings does not exist WhJql.!:~:~ Lorentz invariant and unitary off-shell. (4) Likewise, in string theory, a field theory was thought to because of overcounting introduced through duality. The uU~l,I;-:U=lim:l already have sums over s- and t-channel poles, and hence ad;iMl!fll diagrams with poles in each of the various channels would overcounting, especially at the higher loops. Fortunately, the field theory of strings evades all these problems ... first problem of nonlocality by introducing a multilacal theory. in the string topology only take place locally; i.e., strings can eitller:!m:j reform only at a single point in the interior or at the endpoints. Th~y@jgl from the break travel at velocities equal to or less than the (neglecting Coulomb effects in a light cone theory.) String field theory also solves the second problem because it some of the properties ofthe Veneziano model. This is still ac(:eptabl~:i~~ the theory reproduces the on-shell Veneziano model. Third, the BRST quantization program is Lorentz invariant, but:6ri:t\l price of adding ghosts. This breaks unitarity off-shell, but the Ull;,":·.t~~'"' unitary, so no physical principles are violated. Lastly, string field theory solves the problem of duality by eXI,li¢W,1.. ing duality. Owy the sum over all Feynman graphs yields the¢.·~.~#~~ diagrams. Thus, at any intermediate stage of the calculation, dUlilit:~j~.:1!i nonexistent. We have stressed throughout this book that the second qu~mtizect allows us to derive the entire model from a single action. In pal;(f~~ light cone formalism gives us an interpretation of the Feynman seI;jf~~~ the string picture is clear. In the light cone gauge, we use the sin1pl,~rl~$.'~!l.I

6.9 Summary

287

:' . interactions must be local; i.e., strings can either break or reform points along the string or the endpoints. This uniquely fixes terms in the action. In particular, we get lenns that involve by deriving the theory in the same way that feynmall derived .o.iill.g ",cquation frOID classical mechanics. We use the fundamental l!.U = (XI. rdX2, t 2}

= {Xd e- 1HT

IX 2 )

(6.9.1)

L(t ) ==

~ Jdrr (X; 2.

L(") ~

"'(ia, - H)".

H= ;

X?) ,

f du (P/ + :1:).

prove aU these relations by simply inserting different sets of states between the initial and final states:

IX)! DX (XI = 1,

first quantization:

I ~) ! D2 ~e- (,zI¢l1 (¢I I =

second quantization:

I.

(6.9.2)

field functiona l ¢ is not a function of 0-. 11 is a functional of a siring X that is defined over a region of 0': ""''' ~

IXI"I ~ "'[Xla,). XI",). X(a') •.... X(a.)].

(6.9.3)

.",io" way to decompose this function is in a basis state of all possible afthe Fock space:

1"'1 ~

I>,., ,. , 11'111.

(6.9.4)

11> satisfies the string SchrOdingcr equations, we can power expand

functional in eigenfunctions of solutions to the string SchrOdingcr

1-

• (r,Xi o/p+Jnl

J

d PiC.J(p.• ··EjolflA p*.",.ln)·

(6.9.5)

288

6. Light Cone Field Thea!),

Notice that A is the creation lannihilation operator for creating or q~1~~ possible excitations of a string. Thus, it cOll'esponds to an u· UinJte.'11O;ii1 field theory. We impose the standard canonical commutation rell'ltio~ force us to choose

[Ap~ ' Pi.lnl'

A!.... q"tmd =

8(p+ - q+)8(Pi - Qi)8{nl.(ml·

Now we can power expand the field function c.I> in terms of these elgl~lD

(XIp+(X)

L fIT dp;Ap+.p,.(n)H(n)(X)e-L."Xt"

=

Inl

j

We can reproduce all the identities found in field theory. In )JI~Ul1:;:: show that the Green's function is the matrix element of two "<;;'UJ>".

D.12 = ((01 c.I>;.,.(X t , q+(X2' <2) 10}}

= 8(p+ x

q+)

f

DXe i f

IT 8(X(a. <2) -

l-dadr

IJ

8(X(u,
X2(U)).

Interactions are uniquely determined by our rule that the local t('!~ ' ~~~ only change locally. Thus, our three-striitg vertex function is ·y<·,"UI.:" ,..,..... function: "

S3 =

f

dp+r8

(t f p+')

DX123(X2)8m-+.·':Il'.'<1:;:;:

Fortunately, it is possible to perform the integration over the strlng~i;)~!:~i simple Gaussians. After doing the integration, we find

Iv123 ) =

1 exp

3

00

-2 "" "" a(r) ~ ~ -m N's mn a(sl -n

1

r.s=1 m.n=1

3

+ ~~N~a~~P+Kp2 , 00

where

and N~

= _(m)-1/2(A(r)Tr- 1B)m.

K = -~Br-IB,

r

3

= LA(r)ACr)T.

,=1

}

6.9 Summary

289

we must show that this vertex function can yjcld the usual modeL The simplest way to show this is to calculate the Neumann fo"h, three-string configuration in the fir!!t quanti7.ed fonnalisrn and the results. OUf starting point is Mandelstam's mapping: p=a 1 ln(z - !)+a:2 In z.

know the Green's fum:tion in the upper half-plane, we just take orr<po,",n~ of the Neumann function. The Fourier coefficients look

N;'~ = - (2~)212:tr dT/,d7lse-m{,-nv 1n(<::({r) 1 = -

[

mn(2;rr)

[

2 r:r

dZ r

z(';s»

e-m(,(t,)-~t,(=, J

s,

dZ J

\2

(Zr - ZsJ ·

(6.9. 13)

'~::I~insert the Neumann function over the upper half-plane into

. The calculation is straightforward, and we get

: ,I

T.t

_

N nm ,

mn

-

mal

·,· s

+ IICt,

Ot]Ct20t)N m N n •

N~ = a;1 fmC -Otr+!/a:,)exp(mTo/a,),

InCY,) =

(IIYr)-1

C:').

K = -T~/(2ala20t3},

(6.9. 15)

,

'l

=

(6.9.14)

Larln la:rl. f =: 1

P

i

i

:::: alp:! - a2PI'

analyze the question of superstrings in the GS light CODe formalhave two sets of oscillators. fennionic and bosonic, Our basic

i.~,;;'~~;;~.:'",::a~~n~"~ttz for the vertex function

and

force it to obey the Based Oil an analogy with the bosonic theory, we

0.

003

003

llI ,n = J T,.=1

",=J ,,,,J

L

LR~~AU~~R~~j,+ LLv~R~~AeA,

(6.9.17)

290

6. Light Cone Field Theory

In Chapter 3 we showed that the first quantized supersymmetric satisfied (q-A, q-B} = 2h8AB, (q-A, q-B} = (q-A, q-B} = O.

Now we must show that the second quantized versions of these generators also satisfy these relations. We take the ansatz

IQ - A} =

yA

IV}.

IQ-A} = keABCDyByCyb IV}. Remarkably, we can satisfy all conditions placed on the vertex turlciji)i the choice

It is important to notice about this vertex that the condition of still enforced. The additional insertions with Yand Z take place .• breaking point of the string, and hence locality is preserved. Thus,. Ii· ~iw.1i~ treatment of the light cone theory is consistent with our original l~ij oflocality. as ·

References [I] M. Kaku and K. Kikkawa, Phys. Rev. mo, 1110, 1823 (1974). [2] M. Kaku, Introduction to the Field Theory of Strings, Lewes Workshop, World Scientific, Singapore, 1985. [3J M. Kaku, String field theory, Internat. J. Mod. Phys. A2, 1 (I [4] P. West, Gauge-Covariant String Field Theory, CERN- H~~~~;~~:~ll~~ [5] T. Banks, Gauge Tnvariant Actions for String Models, ...LL-I'lv-r [6] E. Cremmer and 1. L. Gervais, Nuc!. Phys. B76, 209 (1974); 410 (1975); see also M. Ademollo, E. Del Guidice, P. Di Vecchia, an:ql:$.=lm Nuovo Cimento 19A, 181 (1974). [7] J. F. L. Hopkinson, R. W. Tucker, and P. A. Collins, Phys.

(1975). [8] S. Mandelstam, Nucl. Phys. B64, 205 (1973); B69, 77 (1974). [9] M. Green, J. H. Schwarz, and L. Brink, Nucl. Phys. B219, 437 (19'8:~}t:I 475 (1984). [10] O. Alvarez'; Nucl. Phys. B216, 125 (1983). [II] H. P. McKean, Jr., and I. M. Singer, J. Differential GeometlY 1, [12] S. Mandelstam, in Unified String Theories (edited by M. B. Gfj:~ji Gross), World Scientific, Singapore, 1986. [13] M. B. Green and 1. H. Schwarz Phys. Lett. 149B, 117 (1984); 43 (1983); Nucl. Phys. B243, 475 (1984).

7

Field Theory

CO'lari.ant String Field Theory advantage of the light cone field theory, as we saw, was thai it wns ro;.:;~:'~~. gbost-froc, and could reproduce the string amplitudes from 3' was 00 need to appeal to intuition in order to conslruct S-matrix. cone theory, however, was still a broken lhoory. We would like a i d""';p';on in which aU the guuges oftbe string arc in operntion. The in the development of !'Itr1ng field theory j~ the use of BRST techIi> ,.,.;'" a covnrianl description afstring fi elds. The power ofllie BRST i~

that we can reformulate string fie ld theory in a fully covariant

the introduction of Paddeev-Popov ghosts. extract the BRST neld theory in the same way that Feyrunan exSchrOdingcr equation from the classical first quantized theory. We with the first quantized theory quantized in the BRST formali sm extract the field functiona1s. It must be stressed that the ERST field the light cone formalism. is still a gauge-fixed field theory. Bewill c,.;tract the BRST theory from the gauge-fixed first quantized will find mther biza.rre remnants of the first quanti7.ed theory inIn~:t~:~second quantized field theory, such as Faddeev-Popov ghosts, ~1 numbers, parameu-ization midpoints. and parametrization also some irony here. OriginaUy. the light cone field thcory was provide a coherent, comprebens ive formalism in which to express , tl,,"ry. Unfortunately, the attempts to covariantize the model have not one but two competing covariant BRST !>tring field theories! Q EIR'IT 1~"Ori,,, '""ba;.do~,"I""ly different string topologies, and

292

7. BRST Field Theory

there does not appear to be any link between them other than the ~~r:tlji3i both can successfully reproduce the Veneziano model. Previously, we saw that the Gupta--Bleuler formalism, instead . the constraints (as in the light cone formalism), applies the corlstraitir~:~ onto the states

n > O. Normally, we would expect that all 26 components of the theory the action. However, this is not true if we are careful to include the constraints being applied to the state vectors. The effect of the ;~ni~ equation is to kill off the ghost states from the spectrum. There is yet another way to eliminate the ghost states of the rh"" ..,', is to follow the analogy of gauge theory. Instead of applying these c: .~·m··$fj on the Hilbert space, we will now demand that the action be invari~·ii,iq.1 transformation of the field cP when it rotates into a ghost

n

Notice that the state L_n Ii\) is a ghost state, as we saw in (2.9.3). this choice because of an analogy with electromagnetism, where \V.i~Ji;~~ gauge symmetry

BAI' = (V'" The action for the Maxwell field is invariant when we rotate the ghost field al'A. Let us now prove that the string variation actually gauge variation of both the Maxwell field and the linearized gravit~lti9~m As we saw from (6.3.19), the field I
+ Al'a~1

IA) = AeX) 10)

+ ... ,

L-l

= k]

. a_I

10)

+ ....

+ ... .

Inserting this power expansion into our gauge transformation ofthe~.t.ij).\flt~ (7.1.2), we find .. IlAl'a~1 ]0)

+ ... =

alkAa~1 IO}

+ ....

Equating terms in the power expansion, we obtain (7.1.3). derived the gauge transfonnation of the Maxwell field by power eXj:j®.i:Jm field function [1-3). Similarly, we can also prove that the closed string sector linearized gravitational field. We now demand n

7.1 Covariant Stri ng Field Theory

293

bar designates the second, independent Hilben space. Power this expression. we arrive at

10.11) = ¢(x) IO} + hJ!~a~ta ~ t 10) IA ) = )."a~t 10) + .. "

+ ....

IA) ~ A,O~, 10)+·· . ,

(7. 1.6)

+ .. " L t = k$la~ ! + .. '.

L_ t = k"a~t

everything together, we obtain ih",O:"o:" ,t 10)

+ ... =

a"A ~ a~la:t 10)

+ (Il

++

II)

+ ....

(7.1.7)

ogeo,dll"ie"ts, we find (7.1.8) recover the original variation of the linearized graviton field. we wish to find an action that is invariant under this gauge ,imatioo (7.1.2). now repeat the arguments made in the previous chapter concerning d"i',"the string field Iheory from the firstquantized fonnalism. The key inserting a complete sci ofintennediate states at every intennediate ~:~:.:~tb~C twu string states. The insertion of the intennediatc states is 1)] by me fact that we must preserve the gauge conslrnints at of the calculation. Generalizing (6.3.29), we find that the new set of odi.""srates is therefore

1~

PIX)

f

DX

IXI

P,

(7. 1.9)

a proj ection operator which guarantees that ghost states arc ' co "ea,," intenncdiate step of the calculation. It satisfies [4]

p2 = P. b~i~~~ ::~; steps used in the previous chapter to extract the Lagrangian ~I

integral, we now find

tha~

the aClion must be

L = cl>P[Lo - I]P4>.

(7.1.10)

fact is that this action possesses the local gauge syrrunetry (7. 1.2).

;u,~~::~:,:~~ ~i:'i;,es:~p::o~nsible for eLiminating the ghosts thai appeer in

p:

would propagate all 26 modes if the projection

~

~~:;::'::~:: 10 power expand this action and then compare the resulls and gravitational actions. To lowest order, we can write the in the actions as (7 . 1.11)

294

7. BRST Field Theory

Let us now power expand this in tenns ofthe lowest-lying states. now becomes

A IL (l1/LVD - alLaV)Av which, of course, is proportional to the Maxwell action. Thus, to I()~(~~ the action (7.1.1 0) contains Maxwell's theory. Similarly, we of our steps and show that the closed string action reproduces graviton action. At the next level, however, there are cOlnp.lic~ltioltlS'i:A~ By brute force, we can power expand our projection on,p.r"tn" levels and we find

P(L o - l}P

= Lo -

1 - kL-ILI

+ L~l(4Lo +!D - 9)l1L~

+ L: 1(6L o + 6)l1L2 + h.c. - L-2(4Lo + 2}(2Lo + 2)l1L2

+ .....

where 11 = 2(16L~

+ (2D -

10)Lo

+ Drl.

The operator 11 is non local because it contains the inverse of a p2. For example, we know that p-2 is nonlocal because it equals propagator in x-space:

1

l1(x - y) = (xl Diy) . Since x and yare two distinct points, the expression p-2 in X-i)"',,,,,,,,, Fortunately, there is a way to remove any nonlocality nomials in p-2 and that is to introduce more auxiliary fields. consider the Lagrangian VrAVr

+ VrB¢,

~ttle('ilill

where A and B are operators. By eliminating the Vr field (e lating its equation of motion and reinserting it back into the ..... "f)!:·~~ by functionally integrating over Vr in the path integral), we lilnj>!;\~'9..· Lagrangian - ~¢BA-lB¢.

Notice that if A is a polynomial in p2, then A -1 is a nonlocal o":p~~~; point here is that we can trade off nonlocal actions for . these auxiliary fields decouple from the final action (because they do not affect any of the physics.

7. 1 Covariant String Field Theory

295

wish to construct an explicit form for the full projection operator P 1",,<,1,. We can l:onstruct P iu sev~ral ways, such as a power cxpan!iion operators, or a projector onto each state of level N. We will explore

,

us define the ~'ma module (sec (4. 1.42» as the set of all possible ::op<""lmsL~" acting on the vacuum 10) (a, < Cli+d:

(7.1.18) symbolically represents a vast collection of indices. Then let us i:q,,,,,d the projection operator in terms of L 1aj . We assume that the operator P has the following form: P = 1

+L

L-aFa.fI(Lo)L(I,

(7.1.19)

0.'

arc arbitrary functions of Lo, and we must remember thai we use Q' and fJ 10 represent large collections of indices. If we demand op,rnIO! P vanish when multiplied by the Vrrasoro generators, then F's exactly. Let P' represent the ghost projection operator. Then

P=l_p s ,

L "N, L 00

p'=-

,.·",1

L_aF~(Lo)L~.

(7.1.20)

a.J!=1 lal=IJ!I=N

explicitly written in the sum over different levels N. By demandvanish when multiplied by L~, this gives us a rccuT"5ion by the polynomial F . Let us define the F matrix as

,

F~{LQ):= L).~lI"~

-

A~p- I(Lc)J(S- I );y(Lo).

(7.1.21)

,nJOn·i"" S nnd A have yet to be determined. By forcing the operator correct properties, this gives us a rceurSion relation where the A be determined iteratively: P{M)

L

L_F::(Lo)L~Ly Ili'l

E

L6Af/(Lo)lR).

(7.1.22)

matrix S is defined as ::'".r(J IR) ;:; LaLfJ IR),

(7.1.23)

is a real state and In-I is equal to L,,,, I i)../. this expansion is iterative. The Nth level is detemlined in terms

""llo,w,,, stal
296

7. BRST Field Theory

This projection operator is nonlocal in p2. This is easily renlle(Jll~'BiS:; calculate the F's exactly because we know the precise location the determinant. Fortunately, the determinant of the matrix elements of Verma::~I~ known exactly. We know that, at the Nth level, the determinant ot·1:1ie:~ element between (01 La and L_p ]0) is given by det

II (01 LaL_pIO) II

=

f]cLo -

hp.q)p(N-pql,

p.q

h p •q

=

[(mq - (;

c = 1-

m(m

+ l)p)2 -

+ 1)

1]/4m(m .: :·:~~~~~I

,

where p, q are positive integers and their product is less than "'....,;,,,••:.: This is a remarkable formula, first written down by Kac [5], wbl¢~t:~ ramifications. For example, if the Kac determinant is nonzero, tbl:s.:w~ the matrix S introduced in (7.1.23) is invertible. This means tl1~'HjHl module forms an irreducible representation of the conformal gr()uti!~:: Kac determinant is essential in determining the.irreducible reI',res;enltil the string model. The principal use of this result is to locate all zeros of the de1:eriJ~ hence the location of all the poles of the projection operator. lm;erl~ffli(a; into our action, we find 00

p(Ml

L L

cJ>* [ Lo - 1 -

M=l

u,P=l

lu I=IPI=M

where F is related to F by a multiplication by Lo. The adv:ap,t~g expression is that we know the location of all the singularities '~igll Now let us repeat the analysis of (7 .1.16) and (7.1.17), where" auxiliary fields in order to soak up the nonlocal terms in our acti9ij~1 decompose the poles within F:

F:p(L o} =

L A~r[L -

0 - a~Jrm

+ B~.·

I.m

We can now introduce auxiliary fields p to soak up the extra [6] contained within F: piN)

L L 00

cJ>* [ Lo - 1 -

N=l

]

L-uB~LfJ cJ>

a.Jl=l

lal=lfJl=N 00

p(N)

+L L N=I

a.Jl=1 lal=IJlI=N

7.2 BRST Field Theory

+ p;NI"'(A'jm}lf2 Lf!4> + ·[L-(f(A:J"')l/2]p~I"'.

197

(7.1.27)

afthis was to show thai the action can be written in a form thai is ,10<.1 in p 1. . The price we paid for locality, however, was introduction sct of auxiliary fields p:/m. Om next task is to find a simple way . to compactly represent this formula. to re!lmlllge the lcnns in the above action in an elegant fasnion,

to use the power of symmetry. First, we will introduce the '~~~;: in which these auxiliary fields can be rein terpreted as being

¥

ghost degrees of freedom." We will see that lhc auxiliary can be represented in terms of states defined on anticommuting fact that tbese auxiliary fields form a Verma module is the first there is an underlying group-theoretic origin (0 the ghost fields. fields. by themselves, are strange objects in a second quantized but the presence of Vemm modules shows that these ghost fields

.,

Field Theory uses the Faddeev- Popov ghosts to express these auxiliary introduced in (7.1.27). These Faddeev-Popov ghosts will kill 2 of _ uS the required 24 physicaJ modes. --review lhe first quantized BRST string theory. In the Lagrangian choose the conformal gauge rewritten in the form :.. = I; p = O. the Faddeev- Popov determinant for the confonnal gauge 85

1l1'1'

J

dEdI'/5[).{e. lJ) - IJ.5[p(e. I'/}]

= I.

turn, explicitly calculate this determinant by introducing the .pOllOV ghOS[ field in () as in (1.6.22)

(7.2. 1)

(7 .22) this expression allows us to rewrite the original action of our

.dud,,,,,,, Faddccv-Popov ghosts: .

\

2

L(rY. r) = PII.X~ - 2[P~

Pauli spin matrix rYl .

+ XII.,2 + P9po,, 01.

(7.2.3)

298

7. BRST Field Theory

As before, this Lagrangian has a local gauge invariance that on the fields by the application ofanilpotent operator Q [7]. to see this is to make a two-dimensional Wick rotation and revvrij;e.;~fW;ii in a manifestly conformally invariant fashion in terms of the COllDp·~~;n:~ z and anticommuting fields b and c. Rewriting (7.2.2), we find, "". .uil·.'. L

= ..!. BazXIlOzX/l + bOiC + boze} . 'f(

As before, we see that this is invariant under 8XIl = s[COZXIl

+ cOzX/l]'

lie = s[cozcl, oe = s[eoze], ob = s[cozb + 204 cb - ~OZX/lo;;Xi'l, -

-

8b = s[eozb

+ 20iCb -

1

ZOlX1,OzXIl].

From this variation, we can extract out the nilpotent However, it is also important to note that, in general, given with commutation relations P"m, An] = f:nAp, it is possible nilpotent operator Q out of anti commuting operators Cn and

{O;·:~::om

00

L L,,[A

Q=

n -

~f!",Lmbp].

-00

For our case, we have two sets ofFaddeev-Popov ghosts with · {em. bm} =

o".-m.

Thus, our nilpotent BRST operator can be written as 00

Q

=

:L :cn(Ln + ~L!C) 1£=-00

00

= coCLo - 1) + :LrcnLn

+ L-ncnl

n=1

1 2

:L 00

:C-mLnbn+m:(m - n),

n,m=co

where 00

L~c =

L (m + n):bn-mc",:, m=oo

Q2 = 0,

where the last identity fixes the dimension of space-time to . . intercept to be equal to 1.

7.2 BRST Field Theory

299

the analogy with the point particle case, we can make tbe fran· the second quantized formalism by taking time slices. This time • .', , the basis states IX } are replaced by the complete set of lntcmtediatc

because afme presence of the "'addeev-Popov ghosts. This means we insert an infin ite number of intermed iate slates iOio our path inW~'d~:~.~~;:;:"' ~~': rt):"l,complete sct of intennediatc states corresponding ~i Because the e variable occurs in the path integral .: derivative, we no choice bUI to include it in the complete set of states \\'ben we factorize the functional integral. Thus, OUI field ¢

bocomes a function ofthcse ghosts fields. Equations (6.3.29) and

replaced by 1 ~IB)IX)

f

DXDB(BI( XI .

(01 (XI"')

~

"' (X. 0).

(7.2.9)

(7.2.10) field functional is now a function ofllIe Faddeev- Popov ghost fields . the tr'".msition from IX) 10), we find

L

~

I"')[L, - Ill"').

(7.2.11)

e

is now a function of both the X and vnriables. This action was down by Siegel (9-14]. ' fwe power expllnd the BRST field 1¢t) ; of;~ ghos t modes, we find I"'(X» ~

L

~,.II,, (X){b_. llc_. I I-).

(.. 11,,1

explicitly propagales not only the 26 degrees of freedom of the

;:~~i:.tw~.O::g~hO~s::'~m~:od~;'::S~'.~~:~.:',~'h~:'~ .Orynumbers "
ml,,,)>o"«

;I "' )~QIA).

(72.12)

7. BRST Field Theory

300

The state (AI Q does not couple to physical states \phy), so this ~. (~w.ijj conjecture. Because Q is already nilpotent, our choice for the .. ",,;.:,:.;.:~.

[15-24] L

=

(IJI[ Q I\II) ,

where we take only the "ghost number -

ktruncation" ofthe ori~~t.'

[\II} = P-1/21 \II} ,

where P is the projection operator that extracts only ghost nUllllbier:~~§~ that the new field I\II) is a subset or truncation of the original If we decompose this action, we find that the lowest exc:i'1tatilonl~:n~ precisely (7. L 10), so this is the desired generalization of our actJlOll~ auxiliary fields that we found there are now regrouped oo,,{'nrnin expansion found in 1\l1}, In summary, we have now compactly (7.1.27) as (7.2,13). The equation of motion corresponding to this action is Q IIJI} = 0,

which reduces to the usual constraints Ln I¢} = 0 for the !',lUUll\-!';l>l
7.3

Gauge Fixing

It can also be shown that the previous action (7.2.13) can be g.ii.P.~ij obtain either the action (7.2,11) or the light cone action (6.3 . gauge, it is sometimes helpful to decompose the operator Q acc:Ilf~~ zero modes

Q = coK -2boR +d +8, where 00

K = Lo - 1 + I)ncnCn + nb_nbn), n=1 00

R = - LC-nCn, n=1

+ !!m.nb-pCm + !cmf:nb-p), -p 1 p .: : ' cn(L_n + !m .-nc-mbp + 2b-p!-m.-ncm}, :: :

d = cn(Ln 8=

where the f's are the structure constants of the cause Q is nilpotent, these operators must satisfy a large identities:

d 2 = 82 = 0, [K, d] = [K, 8] = [K, R] = 0,

nUlq1f)~~1

7.3 Gauge Fixing IR , d]~[R , ;] ~O,

[d ,;]

~

301 (7.3.3)

-2RK.

of reducing this aellon down to the usual one begins by writing ghost vacuum of the theory. 4 we introduced the "ghost counting number" operator (4.4.20): ~

l)c_~b"

+ b_.c + Hco, hal

(7.3 .4)

lI )

.,,'or counts the number of c modes minus the number of b modes. usc it to label eigenstates. Because both of the b, c ghosts have zcro

r

quite

from

:~;:~;:,:0~f~th~e:!Fa~1ddeev-Popov ghost fields is different . The ghost vacuum (as we saw in Chapler 4) is can

ghost number either! or

col +I ~O,

-i. We can define

b,1-I~O,

1 +1~b, I -).

(7.3. 5)

thus two ghost vacua. Let us fix the ghost number of IP to be - ~ omp~" it in telll1$ of the two vacua: 1 "' )~v'I - I+~I+ ) .

(7.3.6)

'eq,uotion for action (7.2.13) becomes (using (7.3.1):

lC"'IKy,) + C"'ld~1 + C8~1"') + C~ I R~I.

(7.3.7)

IA)~"I-I+wl+),

(7.3.8)

",",,,,i,,n of the field becomes

8", ~ (d + 8)" - 2Ro>, o¢= - K)'+(d +.5)w.

(7.3.9) (7.3.10)

the relalio01l that we have written above can be reexprcssed in language. We can always decompose a field func tional into its 00

L

c_~ IC_~l'" c_~,. b_""b_"" ... b_IOII ~"J .••,."""" ... .... .. 10) .

(n ],(",1

(7.3.11) the indices are antisymmctric with each olher becausc of the nu.ati,)n"h"k'ns between the gh09lS. This allows us 10 write

1"'1 ~

L (N) "'t:. M

M ./I

(7.3. 12)

302

7. BRST Field Theory

where (Z) represents the product of N antisymmetrized and metrized c and b fields. Notice that this allows us to mbrod'uce defined on anticommuting variables in the same way that diffenmtj'iitai a l are introduced via anticommuting dx il [10]. We now have the apparatus necessary to perform gauge fixing. L~l;::tJSEIi choose the covariant gauge [9]:

bo I\II) = 0, which eliminates half of the fields in 1\11), that is, tjJ = 0 in (7~p::;<;J"J~8 calculate the Faddeev-Popov detenninant, which arises from the V'l~riti;m bo I\II} = O. We find

L FP

= {Alba Q IA} =

(AIK 1M,

where (A 1 is a two-form. The strange feature of this ghost action '.0,':I:....., turn, possesses its own gauge invariance: .

Thus, the ghost action itself requires yet another Faddeev-Popov "!'il'l"~~ (or else the functional integration over the ghost fields is it·lfinite;k ghost term coming from the Faddeev-Popov determinant of the gh~~~~:~ LFP

= (A2IK lAd.

~~i1i

But this action, in turn, also contains is own gauge symmetry, requires its own Faddeev-Popov gauge term, and so on. It is clear th~Hwiil infinite tower of "ghosts of ghosts" [10, 12]. This is unavoidable, bell~ Faddeev-Popov determinant is necessary to eliminate the infi.nitieS~!~1 by the previous set of ghost fields. Fortunately, it is possible to sum the series. If we start wiltI!;::.. i~}i:JlI (7.2.13), which has only ghost number and then add t-o-'bidt(:~'Ji.~ml tower of ghost actions, we simply retrieve the action (7.2.11), Vi " over all possible ghost numbers. Thus, (7.2.11) is the gal[ge,-nx.eQi:~ (7.2.13). In other words, a string functional with ghost number 7,:,;··j,JIg transformed, with the addition of the ghosts of ghosts, into a '" of arbitrary ghost number. Similarly, it is also possible to use the gauge degrees of fre·¢~~ (7.2.13) to reach the light cone gauge [25]. The problem, h01~e\te! the equations of motion must be used explicitly for certain fiellds:jl1;~ others. Thus, only on-shell can the BRST fonnalism reach the li~~~1 which is a limitation ofthe formalism. In the geometric formaliisslIli have enough gauge symmetry to eliminate the redundant 10ngi~\i4~l1.tlR in the action in order to obtain the light cone field theory ou-snel~i;:::::~W The transition to the light cone gauge is made by observing that states, which must be gauged away, can be arranged as the

-!,

7.4 Interactions

{a~"a:",b_"c-fIIO).

]03

(7.3.17)

number of such a stale is given by the sum of the level numbers of

(7.3. 18)

be shown that these unwanted states can be eliminated through a of mechanisms. For example. for the lowest slates n+ = " _ = 0 , I we , .., "'''" with n( = n,,; n_ ~ 11 ... can be eliminated through $$ = QA. with fir = 0, I1b = 1; 11_ ~ "+ arc Lagrange multipliers. oat iocluded in (a) with n t = I. nb :::: 1; tJ ::: n.L are eliminated by L'Il""8' multipliers. s !:~:;:::;,~~ ~~~'~ unwanted stales. In general, all these unw.mted Slates ~ by using the gauge invarianc~ of the theory or by using multipliers. Only the physical transverse DDF states arc left after

, we hav~ shown that the action (7.2.13) can be gauge-fixed to original BRST action (7.2.11) or the light coile action of the chapter. We can also show (by eliminating out all higber excitations that the action reduces back 10 (7. 1.10).

tn.,,, '" tho Ugb, cone field theory, where strings simply split in their inleinteracting BRST fonnalism of Witten [16] relies 00 the configuration in Fig. 7. 1. (At first, this configuration seems to violate momentum However, only in the light cooe gauge is the momentum of a length. In the covariant approach, the parametrization length 10 the momentum. so this diagram is allowed.) The length of all generalize the delta functions appearing to include this new configwation: S) =

=

J J

DX m ¢l(Xd41(Xl)41(X 1)om

DX m (<1>. 1(<1>2 1(¢llIXd I X 2) IX .l )8 12).

(7.4. 1)

,

'm ~ n n' I X,(u) 05<1' 5,,12 r=l

X,_,(n - u)].

(7.4.2)

304

7. BRST Field Theory

3

2

~

FIGURE 7.1. Symmetric interaction of BRST string field theory. Th(~sl1iifli out using this vertex is not flat, as in light cone field theory. There IS ~~~~~ four-string interaction.

As in (6.5.2), we must find the conformal map that takes us half-plane to the configuration being studied in order to COIlstrucf;tti~~j oscillator form [26-32], Unfortunately, a conformal map oHhrele;·etifib the upper half-plane (without cuts) does not exist for the BRST v¢tt~f:1 the sum of the changes is usually taken to be zero, while here tIl~.Jl.m1~ three charges for the symmetric configuration must add up to is to write electrostatics with six charges adding up to zero cb~u;~tf~ identify boundaries to simulate the presence of three chlrrgl~s. : cutting and pasting together several regions of the complex plan~~ ~.I

FIGURE 7.2. Conformal surface ofBRST string field theory. This c6tlfo.jjl can be represented either by taking six cbarges and resewing the a surface with three charges or by placing a Riemann cut on the

7.4 Interactions

"

+J

3

•

J

•

P= L.,0/ln(Z-Zi)=ln ' -, -;1£/2. 1__ ) (Z + I)

0_1

=

0_2 =0]

J1t 16

_

e-,

l3

=

Citf/l,

Z_I

<'. - 2

=

-l2,

Z_3

.. 1 -

7

(7.4.3)

1,

01 = 0:2=0_3=

7

3o,

= -J , _

<-2 -

= =

JSJrj6

t:

(7.4.4)

,

-Zh

-<'.3·

so we can

z

";::;~~:,:':Pe~~;;::~:~~~~;~' solve for in tcnns of,. which to explicitl y construct thc Neumann functions. Invening the

tI

_

<'.-t/l

p

(J+ie(.),/3 1-

= {" =

.J

I C'0'

~

+ilJlI

(7.4.5)

(7.4.6)

1,2,3 and P = -({" - ;1£)

(7.4 .7)

-I, -2. -3. The Fourier coefficients will occur in the eombina·

(7.4.8) 'ev':ry1h;'8 together, we find that thc Newnann function appearing in

(7.4.9)

306

7. BRST Field Theol)'

where

Cnm = (U + U)nm = (U _

2( - I t Ind n.m.

-2(-lt [AnBm

+ BnAm + AnBm -

n+m

Bn

n-m

U)nm = -2i [AnBm - BoAm + AnBm + BnAm] n+m

tJ-m

Written explicitly in terms of operators, this equals 3.

IV1231\ X = exp [ "21a r_n

N r, . .

nmCCm

+

p' r N rs

+

S l A T '""' p2 oma-m "2"00 L.... Or

0

ji: .r:a1:m

1'>1

r=1

We can, by using the continuity conditions for the ghosts, ghost contribution to the vertex function. We simply replace the band c ghosts. The ghost vertex is IVI23}gh = exp

(~~b':...nX~:"c:.m) 1+ + +)123.

where the vacuum 1+ + +) is the product of three vacua ltrodUl~~¢!:m~ For s = r or s = r + 2, the X matrix is given by 1J1'

rs = -m(N rs X nm 11m

_

r•s + 3) N nm

andfors=r+l rS X" nm = m(Nnm

_

r •s +3 ) N nm

4

The final vertex is the product of these two vertices, which are d~t:O!'"!9 entirely different spaces 1V123} =

IVl23h 1V123}gh .

By a lengthy calculation, it is possible to show that this vp'ITp'1ir:~;K~~ BRST invariance [29]:

There are several different ways in which to write this vertex. from conformal field theory that we can bosonize the antico'IDllnutm:g system in terms of a scalar field, which we will call ¢. Then "'.~. ~.':.~,!'! vertex can be"written as .

fn 3

V =

DX r D¢r ei (3/2)¢,(11/2)

r=1

X

IT

0::;<7::;(1/2)"

8(X~(a) - X~+I(n - a))8(¢r(a) - ¢r+l(n

....,..:,,1-:1:1);

7.4 (nteractio ns

307

somewhat surprising about this bosonized vertex is the presence of insertion. This term does not spoil locality in q because it apat 11C • Furthermore, il provides the correct ghost number ~ for lipl;,,"" '" symbol *. Because the gauge parameter A has ghost numsymbol must have ghost number ~ so that the product of two ""meters 1\ 1.2 yields a third gauge parameter 1\3 with the same ghost

*

is yet another way in wruch to ctl1culate lbe symmetric vertex, which using confonnal maps with Riemann cuts (rather thnn splicing regions of the complex plane together). Consider the conformal map

(.'_~+~I)~(I~-~2)~(I~-~I~)+~(I~'__'~+~I~)'_n

=klog-

(7.4.17)

z(l - z)

bo.s the usual singularities at <. = 0, I, and 00 (which correspond to

;~~!~:;Iand ingoing strings at ±oo). However, what is new is that map bas an explicit Riemann cut thai creates a nrultisheeted p cut is precisely what is necessary to Create the surface in thc pdescribes the symmetric collision of.three strings. The cut extends yfrom in the lowerhalf-planc to + in Iheupper halfcan understand the multisheeted p-plane by tracing our movement real z-axis in fig. 7.3. begin al 1. = +00 on che positive real z-axis and move leO, we can th" co",tM" k such that we lrace out a similar line that moves along the p-sxis (0 negative real infinity. When we reach lbc point z = I (at we reach negative infinity on the rClli axis. By hopping over z = I. fa

l- t../3i

t t./3i

E

H G

I

Foe B

H

c

I

o

,

:, ,, , ,

A

,, E , I

tg __ ___________ J F

B

A

FIGURE 7.3.

308

7. BRST Field Theory

we suddenly jump 7r units in the vertical direction in the p-plane. MnV,'N, Z = 1 to z = ~, we move horizontally right in the p-plane from C we hit the y-aXis at p = i7r. If we now move vertically up the ~,",'...lll
!7r

7.5

!.

Witten's String Field Theory

If the parametrization midpoint of the string is singled out as a .. then it becomes possible to define a closed algebra among string ... tionals. A string represented by IA) can be joined with a string I their midpoints exactly coincide. By contracting over their are left with yet another string of equal length, so we have detineClii,t.jJi similar to multiplication [16]. For example, the abstract notation V""·ULCI.LV<.

means concretely

where we have contracted over the first and second harmonic osc:i1Ial(p~~~ left with a string defined over the third harmonic oscillators. Bel=:l:ij~~~ aU have equal parametrization lengths, the product rule closes IHlle·.tlili zation midpoint is singled out as a special point; i.e., the product of length one is equal to another string of length one. This, in mn'L:·a to define a gauge transformation of the field functionals as .5A = QA

+A *A-

A * A.

Ifwe define a "curvature" as

F

= QA+A *A,

then we find that

8F = F * A - A

* F.

7.S Winen's String Field Theory

309

jfwe can now define an operation called "integration" that preservcs

!

A*B=(_1)"6

J

B*A.

(7.5.6)

we use the anticorrUUl,lIatioll sign for Grassmann odd forms), then we invarianl:i. a surface lenn. and the action itself,

f

F. F = surface tenn. L = A.QA+~A*A*A.

(7.5.7)

J"",u'kalble nature of this approach is that we can reduce the essential of the theory to five "axioms" as in the case of gauge thcory:

~"co"e of a rulpotent derivation

QI =0.

(7 .5.8)

;'""i,,,ivlty ofthe. product [A

* Bl. C =

* C].

A • [8

QIA.B ] = QA.B+(- I)'A .QB.

Ii P''''''''' rule

J

A.B=(_l)"B

J

B.A .

J

(7.5.9)

(7.5.10)

(7.5.11)

(7.5. 12)

QA = O.

,.",(- 1)" is -1 if A is Grassmann odd and + 1 ir il is Grassmann even. eO,,, "', intcgration operation as follows :

J'" J n =

DX

S(X(u) - X(rr - u))"'(X)

C!!cr!!OIl)or

= (II"') .

(7.5.13)

m'.on I is a rather strange object. It means that we take a functiona l fmd its midpoint, and then integrate such thai we fold the string around its midpoint.. Thus, a functional of a string maps into a ~ "nd,,, this idcntity operation. also give an explicit representation oflhis idenlity operator I : I(X) = (XI I ) =

n

S(x.).

" .. U.L..

(7.5. 14)

310

7. BRST Field Theory

In addition, the vertex function is invariant under a subgroup of1:1'):~i~ transformations, i.e., those that do not affect the location of the ~ii"il us define 1 . (

Kn = Ln - (-ItL-n' This operator acts like a derivative on products of forms

Kn(A

* B) = K"A * B + A * KnB .

The integration satisfies the following relation:

f

Kn A = (AI Kn II) = 0

while the vertex function satisfies 3

L K~r) IV} = o. r=1

The advantage of this cohomological approach is that it cohomological formulation of gauge theory in terms of forms. equations, we have

l'Ol:·M~~

d = dxIL0I" A = dxJl.Aoro. IL

We can write this in compact form:

F=dA+A*A, J.

,

oA • al~ T A 'f ~~

8F

=

F

*A -

A

-

H

·

* F.

From these equations, we can write invariants

f

F*F=

f d[A*dA+~A*A*A).

The strength of the BRST approach is that we can write and the definitions of curvature fonns such that the action is be gauge invariant. These elegant axioms sum up a tremer information, capsuling the string model in five statements. The disadvantage of the BRST approach, however, is that t1 axioms is obscure. These five axioms can, in principle, appl: any cohomol9gical system. We have no understanding, the these five axioms came from, For example, general relativi derived, as we have seen, from two principles. These principII reformulated into the five axioms of cohomology. The five a are not fundamental, but only a compact and convenient metl tensor calculus. We seek, therefore, the underlying geometr will allow us to derive these five axioms from first principles

7.6 Proof of Equivalence

3 11

;"'''''' ,.I<,uh,,) ,h,oul!d arise naturally from a new infin itc-
1roc,f of Equiva lence

:h Ihl' not

RRST vertex]

when the

par-

is obvious. this Vm ), c:xtemw , . placed on-shell, is equivalent to any other on-shell vertex. fu nction

reached by a conformal transfoIllliltion generoled by L". For ex-

', .;

I

define another vertex functio n ii Ill) that can be expressed as a transformalion of the original vertex:

(7.6.1)

this equation with a physical state {pllyJ that satisfies (phyl L_II = (phy[ Vm ) = (phYI V12l)'

(7.6.2)

twO verrex functions have Ihe same on-shell matrix clement... In has been shown that the ven ex func tion shown here is equivalent to the covariant oscillator veISion of the Cancschi-Schwimrner-

34J vertex found in the early days of the dUlIl model. (This

:I'~Ei'~v,,~a~'~tu::a~I:';Y:~:d~";iv~"':b~Y:"::ki~·"~,g~!N~-~PO~;I"~il:i",,~:'~'~"':~d~l~h:enfunction.) factoriz-

~

tha t we must obtllin the same S·matrix I by using either proves that the three-lltring vertex fo und in the BRST fonnalism =ii",ly".. same on-shell mntrix clements as the old CSV venex [29, though Ihe geometry of the BRST vertex is completely different. can use this fact to show that all th ree vertices (the old light cone old covariant vertex, and the new BRST vertex) all yield the same ciemenls. that the vertex func tion gives rise [0 the correct on-shell vertex : afthe Veneziano model is quite simple (30- 32. 35, 36]. We will take '. of both vertex functions with coherent stales that are functions

1

z and

then show thai the resu lting expressions arc related 10

by a conformal transformation. We then show that the conformal ""lion can be expressed as (7.6. 1). CSV vertex is

(7.6.3)

312

7. BRST Field Theory

where the C matrix satisfies the following property: (1 - z)n - 1

zm

00

~Cnm Jm=

~

(As a check., we can always hit one of the Reggeon legs with ~~;:iill convert a three-oscillator Hilbert space into one oscillator, andr~lf.w;lU usual vertex function for the tachyon. Thus, (011 Vcsv) is equaltO)t}i($; Veneziano vertex function.) . We will show that this vertex function, up to a transformation ~t(~ffi the L_n , is equaJ to the symmetric vertex. To show this, let us action of a conformal transformation on a coherent state (which in (2.6.18)):

Iz, k) = e L " k"a~.z· IO} . By a direct calculation, we can show that the Virasoro operator a.c.;·u. ti~:t state yields Ln Iz, k) = [en + I)k 2 zn + z"+'dz] Iz, k). This is the transformation of a coherent state when one Ln is a.l!~I~~I?!f!":;I:~ wish to apply an arbitrary number of the Ln's to the coherent Sral[e:·::,:·:·:;~

~ ~ anLn Iz. k) =

[2k f (z) + fez) dzdJ Iz, k) , I

where we define function fez) as

Thus, a generaJ conformal transformation generates the ~Vll'U\l\lll'I;:IJl,'~ tion on a coherent state: _

eL~oa.L" Iz, k) = (~;)

k2

Ii, k} •

where

and dg = f-'(z).

dz Equation (7.6.9) should be compared with (4.1.8), which giv'es : I;liJ.'f.~ mation of a field of weight h under the conformal group. Armed with these results, we can now show the equiVall~e:n~c;e:_~~~~1 on-shell vertex functions. We begin by contracting the tl

7.6 Proof of Equivalence

3 \3

on three arbitrary coherent states [32J:

(7.6. 12)

be calculated explicitly ITom the definition of tbc C matrices s tep is to C()ntract the symmetric BRST vertex (7.4 .11) with three coberem states and compare it with the expression given above. We the following formulas:

'" £..... N~!z7z; =

I

log Z(Zl) -

'.-

."~ Z(Z2) Z(Zl) - I Z(Zl)

'.-

d,

+ log IZll + log -d, , "v' " '" -I 1 L I ~",Z l ZI - og (,,)-H,;1_1 , og n.m

ZI - Z,

' " Nil ~~zn = log

W

It.....

~m"l

1

d-

_z _ d ZI

(7.6. 13)

-

d;

d,

dz

log _

d

Z :..0

d""i.ies allow us to construct an explicit fonn for the matrix element

n:.,

. : the coherent state (k;. zd and the symmetric BRST vertex funcwe compare Ihis expression to (7.6. 12), we find the relations hip ..' . the old CSV vertex and the ncwe r symmetric BRST vertex:

(7.6.14) expression that we want. II shows that the matrix e1emC1lt of the and the matrix clement of the symmetric BRST vertex are rea factor raised to the k 1 power. Now comparc this wi th (7.6.9), tha t this fuctor can be reexpressed in terms of the L" 'so In we now have an explicit solution for (7.6.1 ) (at least 00 coherent ",~~~~ de monstrates that the CSV vertex and the BRST vcrtcx are re'. a trans rormation and hence have the same on·shell matrix

i;un:,:;.

another way to show the equivalence of the CSV vertex with ~~;':,:;wbich is based OD a power expansion of (7 .4.17), rather the conformal properties of complex functions. We know that

314

7. BRST Field Theory

Ln acts like zn+ 1dz on a complex function. Let us define -8(z

+ I)(z -

2){z -

z(I - z)

W(z) =

= I

1) - ez 2 - z + 1)312

+ i1 Z + 169 Z2 + ....

It is always possible to find a series of coefficients an such that W(z) = eL :

1 a.L. Z.

In practice, solving for these coefficients might be difficult, but '-",'11;1.·.. these coefficients are known to any degree of accuracy. For eX!lmlpl¢(aif~ and a2 = 156' Then it can be shown that the symmetric vertex ~,""""!IlJo(.,,, vertex are related by the following conformal transformation: (V [ e- 1n(3./3)/4 L~~I(L~-l) = (Vcsvl e- 2::~:1 2:::1 a.L~.

J1111

J

Although we have now established the link between the syrnmettl:[;~ of Witten and the CSV vertex, we are still facing the problem ~,.,: ...",.actually two BRST string field theories that do not seem to have any 1 v~i":1;!.tl~ to each other. The other BRST formulation of the interacting the;ori;H:~:~ on the old light cone vertex, where strings simply split or rejoin at'!!j:.'jJutili point. We can also postulate that the three-string vertex function :' the covariant Dirac delta function [18--24]:

1V123) =

f

DX123IXI) [X2) IX 3}

IJ (t 8

8r (ar)Xr(ar}).

It is important to realize that this new vertex function is precisely tl1¢:5ifi

the old light cone vertex function (6.4.5), except that the harmoniC'~'1111 are now fully covariant, rather than just tranl)v.erse, and that multiply it with the ghost vertex. When we multiply these two, "covariantized light cone" vertex. We can check that the covariantized light cone vertex satisfii~~) invariance:

(t

Q(r)

jlV) = o.

It is also possible to prove that this covariantized version of vertex is equivalet;lt to the on-shell CSV three-Reggeon vertex fuljlcnl will use a different proof, however, from the one used before. , will show that the continuity or overlap conditions satisfied by the :',: function can be written after a conformal transformation as ' ,, condition for the light cone vertex function. Since the cOIltinlui~Y:QJ~::~~ conditions define the vertex, it must follow that the two vertices a conformal transformation.

7.6 Proof of Equivalence

3.15

write the CSV vertex function as

!Yew}

= cxp

(t f.

, .. I ,. .. 0... _ 1

a~~)a~:,~c,"~) 10) ,

(7.6.11)

(7.6.18) wiLh the assumption that we can impose the following continuity '00.00 this vertex function

I - z)-J J+ B(z)Pifl - t-II + C(z)Pjtz)) !Yesv} = 0 (7.6.19) unknown values of A, B, and C and where P is given by

L a~ l - ~ . 00

PI'{F )

=

..

(7.6.20)

~

",nm,;;og the various harmonic osciHators through I Vcsv ), we find the constraint s on A, B, and C:

A

C

--+ 8--=0.

z

I -

(7.6.21)

z

gives us wide latitude in choosing the values of A, B, and C. By looking at where these functions converge, we find that we can set Ill> I,B =Owhen lz- II < Izl,andA "'" Owbenll-zl > 1. we will select B = 0, wbich slillicave..<; a certain degree of Specifically, we choose

,

A=",~-:-: I - ~l.

8 =0,

(7.6.22)

1-,

C = _ '" -;-,__'c '.

Yl.

""

: goal is to show the relationship bern'ceo this vertex and the usual formalism, which maps the upper '''' ;00 through

half~p lane

into the light cone (7.6.23)

in the p-planc we parametrize the three-ming vertex function in reel;oo by the parameter 1/" where

o ~ '13 :::; fJ2rr, Ihrr

~

T/}

~

j[.

(7.6.24)

316

7. BRST Field Theory

and al

f3l

= la31'

R_ _ Vl -

a2

a3

,

ar+l

Yr = - - - ,

aT

• -or

in

e

,.

Given these definitions, we can rewrite the A and C coefficients

zr

01 d(l A = (1 _ Z)-1 dOl

l

(1 - yd,

C = 03 dz . Z d0 3 _ Now comes the key step. We know that P transforms under the ¢Pifff~ group as follows (see 4.1.7), where we have taken the exponential OJ'n ''' ~~jii variables: .

z

dP)h. P/L(p).

Pp.(z) = ( P dz

Thus, we can reexpress the continuity equation as

( (1 -

01 Z)-1

d(1 - Z)-I pp. do, 1

+ 03 Z

dz Pp.) d0 3 3

IV)

= O.

Using the transformation properties of P under the conformal have

(Pi(ol)

+

Pi!(lh») IV} =

gr(]IUP)~;~~

o.

But Or = ei IJr • This is the expression we desire. Bya conformal ·t [;:~~II we have created a new vertex that satisfies a different continuity TIn which is the a coordinate of three strings. Thus, this new velte:x~:tl.t3~ satisfies precisely the original continuity conditions of the light COlae;,: function between strings I and 3. Similarly, by taking different coti).'Q~ we can also show that the continuity conditions between su:ings 2> ~~l1::~ also satisfied. Thus, we have shown that, by a conformal can reexpress the continuity equation for the covariant Vellezianl(n~~"tll terms of the continuity equation in the covariantized light cone c~:tJ~e~ But since confoimal transformations are generated by the L" 0 means that, on-shell, the covariantized light cone vertex furlction:: t.:':" '''~" the same as the covariant Veneziano vertex (7.6.17). At this point, we must explain why there should be two versions of1tl1;~mJ. formalism., one based on the fully symmetric vertex and the other b~;~~:j(~ light cone vertex, such that they both have the same on-shell prc)pelrtieS:~

7.7 Closed Strings and SUperntrillgS

317

ii,.,,,, to ,,,,'mlbl, each other in crucial details. The CQvariamized tight cone for example, bas an additional four-siring interaction tenn. This is because the gauge for the covariantized light cone theory does "~~:~;~'i~' Jfwe for the light cone configuration, we " withOllltlIC addilion of the four-string and even then it onJy closes on-6hell. ,~mn"'t'ic theory, however, can be shown to be consistem without a interaction. Thus, these two theories differ in fundamenta.1 ways. nol all the details are clear, it seems that the resolution of the the second fidd theory, based on the covariantized light cone, an inrompiele thoory. This is because we must integrate over all err. In the old light cone theory, this integration was really equal to an integration over the momentum However, in the covariant theory, a, is a redundant parameter, meaning. Thus, the integration over Ot, produces an infinite oftbc !>ame thing. 11IU5, for example. in the zero slope limit produces an infinite number ofYanbo-Milis actions:

__11~ daF(a)l+ .. . 40" ~

,

(7.6.30)

j:::~::\;,;:nonsense. TIlls infi.n.i!e number of copies of the same thi.ng, due

it

over the fictitious parameter Ct , yiclds an infinite redundancy theory. Thus. there is a fundamental problem with the second eRST

resolution of this problem is that a must be expressed as a genuine and hence can be gHuge-fixed. However, within the present it is impossible to make a into a genuine gauge parameter. vastly increase the numher of fields in the theory in order to gauge moth" of the string, and this is possible only in the gcomcl::ric fomlalism

::',. Closed Strings and Superstrings u",... ly, closed sl::rings and superstrings are less developed than the opcn string. In the BRST formalism. for example, there exists con· confusion ovt:r how to write the correct theory. For clo.sed strings

..~g~h~OS~'c~o~.:;n~I~;n~g" comes /Jutwrong all wrong. [~:~:~i:;:'~:h~.~s "consistent truncation" of Q it has the

In fact, the naive ghost counting number, either or the Hilbert space

gn"<mooes." closed 51ring [ II , 39, 401. we have two duplicate copies of the generators Ln and i~. In the BRST fonuolism, the vacuum therefore

318

7. BRSTFieldTheory

possesses double the previous ghost number because 10} = I-} 1-),

-t - ! =

-'..: ·1,:,·:·:':'

(01 =(-1 (-I.

while Q remains with ghost number 1. This means that the naive ·a.£fl[~i actually vanish

because the ghost counting does not come out correctly, i.e., -1 This is clearly undesirable. Various solutions to this ghost cotmtirig:~i have been proposed, none of them totally satisfactory. For :'Z~·:,.hX artificially truncate the zero modes within Q, keeping its intact. This means that the zero modes

co; bo : co; ho will be arbitrarily removed and replaced by the smaller set A

A

co;bo

Qnow decomposes as coCK + iq + d + <5 + d + J -

such that the new operator Q=

2co(R +

R).

sm;~nf~i~~i!~11

Notice that the old zero modes have been dropped and a resembles the set used for the open string) has been inserted.. In order to have the theory independent of the origin of the u we must impose the constraint on the closed string:

K) 1<1» = O.

(K -

The fact that we have a smaller set of zero modes means that vacua found earlier for the open string case: I-} and 1+},

I}

= ¢

1-) + 1{r I+}.

Notice that this truncation was chosen so that we retain the identity:: ::::::::::Q.;S A

2

Q =0.

The action is (41]

Q]<1»

•

Now let us tum to the supersymmetric case, where the siITlati'Qp;=:!;:l1 worse. For the Neveu-Schwarz bosons, we actually have no nr('hfj~tm BRST zero mode ghost counting because the commuting ghC)Sls half-integral moded, and hence do not change the nature of '~"",... :."" The BRST vacuum still has ghost number - ~. Thus, the still (AI Q IA).

7.7 Closed Stri ngs nnd Superslringl;

3 19

""'''. for the Ramond fermions [ II , 41-53], thcre are severe problems intcgrally moded commuti ng Faddeev-Popov ghOSIS. Let Yo be tbe of the ghost fcrmion oscillato r. Then the re is an infinite number of vacua because this ghost is bosonic in nature, nOI fermionie. EDch o f sel of vacua i ~ labeled by Yo~

10) .

(7.7.1 1)

thllt the field functional must be power expanded into a series of w

"'{X. Yo) = 2:>.{X)y:.

(7.7.12)

•

'''::~';;:~ do we use?This is extrcmcly inconvenient, and itshows the lim-

;~ .

BRST approach. (A similar situation occurs in conformal field "picrure" changing opeT1itors. However, conformal fie ld theory is fonnalism , so all these pictutCS collapse. Here, the string fi eld by definilion, is off-shell, and all these vacua must enter into the

are several proposals for truncating the theory in the DRST approach. either trunclltc Q. keeping it nilpotent, or truncate the field 14» so have \0 sum over an infinite number of ghost vacua. Let us begin ,",,;j' , hy introducing a unified notation by which we can de.~c ri bc the for the Neveu-Schwarz and Ramand models simultaneously. Let the following [1 1]:

,n

A~ = (O!~.

w:).

B". = (b~, Ih).

e". =

(c~ .

(7.7. 13)

y;,),

1ft: '

labels both the usual bosonic string aod the antieommuti ng field reminds WI that the fields can be integrally or half-integrally mooed, and CN represent the ghost fie lds for theory. In particular, b~ and c" corresponding to the bosonic string, and {3,; and y;, label the fermionic field. where the dot again can be either integrally or ".~lIy moded. The commutation relations are

gh'"''

A ~ A N - (_ l)"m ANA:' = 1]j.I~o( M

OM eN

+ (_I)MNc",B,w =

oe M

+ N).

+ N),

(7.7.14)

is - I is M and N are fennionic and equal to

+I

,: ,L J ] = F1- / Lt:, then the BRST operator, in this notation, is

Q = C_NL ,'I + F/~: C_JC_JBK:' is equal 10 Iho usual Virnsoro operncors.

(7.7.15)

320

7. BRST Field Theory

Specifically, for the Ramond string, the operator is Q = coK

+ yoF + d + 8 -

2

2Rbo - 2Rflo + Yo bo, A

where

and F = Fo

+ f%[C:::jJ BM -

B_M(-I)MCM],

Z

where f equals 2 if M is bosonic and M /2 if M is interchanges fermionic and bosonic modes, and d, K. 8. R are counterparts for the Vuasoro case. .••. Fo is the usual Ramond operator, which satisfies the following ... relations that define the algebra ofSuperdiff(S): . {G m, G n} = 2Lm+n

+ i D (m2 -

!)8m.-n,

[L m • G n ] = (!m - n)G m+n ,

[L m • Fn] = (!m - n)Fm+nt {Fm• Fn} = 2Lm+n

+ !Dm 28m._n,

[L m• Lnl = (m - n)Lm+n

D 3 + 8m 8m.-n.

Notice that the algebra of Superdiff(S), which is usually of physical fields, can be realized on the ghost fields BN well The operators F. K, etc., satisfy, in tum, the following relations: d 2 = 82 = O. F2 = K,

2R =

[F,

Rl.

{d. 8} = F{F, R}.

These commutation relations, in turn, are sufficient to show operator Q is nilpotent. As we mentioned before, the problem with the Ramond se¢~~t:~ml the ghost mode Yo is integrally moded and satisfies c~rr;l~J~i:~~~mll ticommutation, relations, so there is an infinite number of . described by 10) for each n. One way out of this prob trarily truncate the zero modes of the operator Q while nilpotency. Now we reach the crucial step: as in the case of the bosonic cIq$~r.:I we will arbitrarily eliminate the bosonic ghost modes Yo. flo

Yo

7.7 Closed Strings and Superstrings

32 1

b'16)~o.

(01 "16)~ 1. , do:fine a modified

(7.7.21)

Q that simply banishes the zero modes

yo and f30

theory. We define

Q ~ i,F + (- I)" '(d + I) -

b,(FR

+ RFl.

(7.7.22)

counts the total number of fermion creation operators in a state. Q2= 0, which was the guiding principle in defining this modified the Ramood stales, we defi ne the GSO operator, which in tbis '"ti.on " (7.7.23)

refers only to nonzero models and

r l1 refers to the lO-diJl1ensiollaJ

matrix_ [0 this representation, the

usa

projection simply

states with the wrong :.pin statistics.

bi"a", Ram,ond superfield can be decomposed in this truncated Fock (7 .7.24) is a zero-form of chirality r

ll

= I and

I~) is a G =

+ I (-

I)-form

b'16) ~ o. (61 i, 16)~ I. .f'u"","daction nnw reads

(7 .7.25)

(7.7.26)

I = (WI r O. The action is invariant under (7.7.27) our previous deco mposition of 11.IJ ) in (7.7.24) yields

FI,,) +111 FR +RF I
(7.7.28)

ie.'''i'," has a[sa been found in other ways, such as making a cle\'er truncation of 1"It ) and preserving the original Q. This method the same result.

7. BRST Field Theory

322

The situation becomes much worse, however, when we UI:>\;W,iS-Jjef closed strings. In this case, the constraint [K - K] I\}I) = 0 runlS :l)~~lfffil [F - F] 1\11) = 0

which is a dynamical constraint. Hence, we are not able to apply U1l!U:I'lit.tl to the states. Let us, however, present one more truncatioil operation promise of being local in a and hence generalizes to II'ltel~ac1tioils recall the original ghost counting that was performed when the and f operation. The ghost numbers were carefully ch()se~~;:$'!'i gauge symmetry closed and the axioms of cohomology (7.5 .8) to ;. satisfied:

*

Object

Ghost Number

\11

Q

*

/I.

J It's easy to show that the above choice of ghost nwmbers: cohomological axioms that we listed before. Now let us generalize these rules for the open superstring to maintain the cohomological axioms and in addition wish conditions that two gauge transformations yield a third gauge with the same ghost number. Given these conditions, we quickly find that the * and the f tl"·;;.,t;:.~i;;;.;:it have the wrong ghost numbers. In fact, as we found before, th;;';:;;if6~ exactly zero. Instead of "truncating" the zero modes, as we us use conformal field theory to supply us with the missing These insertions from BRST ghost fields will be only at the do not lose locality in a. Thus, this truncation, which numbers of fields, is superior to the previous truncation, which ®i~~ modes. Several new features must now be added to make this sc::~~efil~llll (I) The superfield \II is now the sum of an NS field a and an R .. . :;:;;;;~:;m

\II = (a, 'ljI).

(2) Similarly, the gauge parameter A is also the sum of two piei~¢.~~~~m A = (e, X).

*

is now equal to ~ (because as before in (7.4.16), but now the sllIj~~~ ghosts contribute ghosts contribute Thus, we must invent a new mUlltltj~~~~m1

(3) The ghost number of

t,

t).

7.7 Closed String.<; and Superstrings

323

superstring, which will have ghost number! (which we will 0).

-!

because of the contribution from now has ghost number ghosts . We need to postulate the existence of a new which has ghost number - ~. b~l~:~~'~~~':;;;;: can construct the ghost numbers of all ofour fields ~, such that we satisfy closure afthe gauge transformation and axioms [17J: Object

Ghost Number

a

- j

',"

_1

,

0

,

- I +I

X

,,

Q

•

(7.7.32)

,1

0 X

I -I

-,,

Y

1 J

-,,

Y nre still unknown but they have ghost numbers to fill in the

'umb,,,,.

we have made an educated guess that successfully satisfies our "',urr'ptioru; •• " ",.foe,d w;,h "" task ofactuaUy calculating explicit these fields and operators @Md . . is not a problem \0 construct BRST fields with these ghoSLnumbers.

1-

Y~~';":~~~:~:~~~~:~:::~~~~~:~:~::;~~~:;;:~~OfthCm) it hns the correct . Then we can always apply the ghost operator on each state and obtain a generalized BRST field number. of course, is defining the operations f and 0. has only ghost number!, we need a new confonnal operator with I and eonfonnal weight O. Fortunately, conformal field theory opemtor:

,ue',,"

x = IQ.n

(7.7.33)

;""";"h,,,.on rule for gauge parameters must obey At ®Al -( I

#

2)= Al.

X operator, we ean satisfy this multiplication rule:

(7.7.34)

324

7. BRST Field Theory

By looking up the ghost number of each field and each op(~raj;Ot;·=~ all ghost numbers are correct. To see this, let us symbolically plication rule in terms of ghost numbers. We want to mtllti]Jly with ghost numbers ( - ~, - 1), such that we get back the (If we carefully write all the various ghost numbers in this mu1nP,Ui we find symbolically

+ 1_ 12-- -I ( _12' -1) €I (_12' -I) -- (I - 1 2 2

+ 12 - I' 1

=(-~,-I). Thus, we have closure. Next, we wish to show that A €I A, which occurs in the also has the correct ghost number of A. We wish to calculate the g1;i~)$t~~ of the following operation:

(a, 1jr) ® (s, X) = (X(a

* s) + 1jr, X(1jr * s + a * X».

By carefully counting the ghost number of each piece, we find'

(-!, 0) €I (- ~, -1) =

(1 -

4+ ! -

~ = 0+

4-

I, 1 + 0

= (-4,0), so we have closure again. Finally, we need to construct the new § operator that has 5'''Ji,,:~:t;'f:!1I' The naive integral operator has only ghost number - 4' so we nelllltlitl.'ti that has ghost number -1. Once again, conformal field theory such an operator:

This operator is called the "inverse picture changing operator" sense, the inverse of the operator X. We can show lim X(CT1)Y(CT2) = I, (JI--)o0'2

so that X and Yare inverses of each other. With this new integration operator is defined so that

f

(a, 1jr) =

f

on/~raj[m;:

Y (41i) a1jr,

i.e., the integration rule is the same as before except that we . Y(4n) at the midpoint. Notice that this operator is defined only of the string, so that we still maintain a theory local in CT. With these new ghost counting operators, we can now following action:

7.7 Closed Strings and Superstrings

5A::::QA+A@A-II.®A.

I","<e'g' of'thii~ fonna ,m

325

(7.7.43)

"th"

it generali7.es to the interacting case. arc local in u, so there is no problem in applying the continuity 00 interacting strings. say a few words about a promising formulation of suing field makes no mention afthe background metric and may give clues to

"~:-':~;:7; This new formulation is called the "pregcometrical" (.1 yields some surprising conclusions. the action

2

L ~ - 2 ¢l '" 38

¢l

* Cfl.

(7.7.44)

[...re, this action makes no sense. The action makes no mention of metric of ~'Pac(}--time. which is desirable, but two things seem

wrong with the action: it has no kinetic term and ils equations of triviaL To study this peculiar action, let us write the equation of

¢l*¢l=O.

~

(7.7.45)

, the previous isthe actually fora an infinite set ~~i;l~;~~fi§~~~~~O'~SO theoryshonhand is empty in point pal1icle to an infinite-component field theory, so it for a string theory. Assume, for the mom(:nt, that a classical exists for the equations of motion which is nonzero. It must satisfY $0>1< 410 =

o.

(7.7-'6)

expand the field around this classical solution ¢l

--+ ¢>o + g¢

(7.7.47)

this back into the Lagrangian. Because ¢o satisfies the classical of motion, we have the new Lagrangian based on a new classical

(7.7.48)

~th""!:

""IT" new. Wc still have not made any contact with a physical

us make the crucial assumption behind this approach. We an operator D exists tha t satisfies

D¢=¢lo*¢>- (- l)¢'¢>*¢>o.

,

(7 .7.49)

~l:;(~~D exists, then we can show that it is nilpotent. Now insert

~

back inlo the Lagrangian, which now becomes

L = <J:I

* D¢+ ~g(J)3.

(7.7.50)

326

7. BRST Field Theory

If we can make the identification that this D is equal to the "~,,..a·; then we have shown that the action is precisely the BRST action W,itb)if, term! Thus, the usual BRST action might emerge when we eXIj~i~:j1 new classical solution of the theory. The novel feature of this approach is that nowhere have we maIM:~Ei tion ofthe background space-time metric. The background OC(::tlT!::C:i>li;jliP. kinetic term, not the interacting term. In fact, the choice of the M~ metric emerges when we expand about a classical solution to · ~t~1il of motion. This is why this approach is called the "pl·egeOIne1tr.l~3.f:~: In principle, the geometric of space-time should emerge as one aml~ji: possible vacua. In practice, however, we must be careful to check for the CO'~$~Wi1 our approach. The key equation was (7.7.49). Under certain asS~~ appears possible to find solutions for this equation where D satiisfile~~!iii for the BRST string field theory. In this case, it appears that the usU:aU BRST string field theory is nothing but one solution of the cI>3 a~~11 solutions of this action presumably will yield BRST theories d different classical metrics. For example, let us choose

~

cI>o = Q~l,

len;-mrnQ:'~'~~~i

where Q is the BRST operator defined only on the midpoint (for open strings) and I is the identity operator (which e~l~ left half and right half of the string coincide and zero .) BRST Q operator is equal to QL + QR. Ifwe make the Ibs1titi.iI~~~ original action cI> = QLl +gcI>,

we find that we recover the original BRST action (7.5.7). (The of Q emerges in the manipulations because QRI = -QLI .) Surprisingly, these definitions can be shown to be consis u,;i·h: Hbli.::h~ five axioms in Section 7.5, so that the usual BRST string actioDi :S~~~ one among many possible solutions to (7.7.45). So far, it can be shown that flat space is a consistent solution . remains to be seen, however, what other kinds of classical bac;kgro,",ij;1 found as solutions to the equations of motion. Lastly, let us now discuss some of the new directions taken U'::'l;!:!:.l:l theory. The first approach is to correct a hidden defect in the theory describ~d earlier. The superstring field theory is _~ ... l". [57]. For example, when calculating the four-point scattering aiD:p~~jt1;(~ three-string vertices meet on the world sheet, such that two pic:ttu;·~~~ operators X collide, yielding a divergence. We can show, U"'ll!$;::J;!~ffll! field theory, that two picture changing operators, defined at the.~tij~1! yields an infinity. Not only are the amplitudes divergent when j'tlte~~~ij the moduli space of the disk, but the gauge symmetry itself

7.7 Closed Strings and SuptfStrings

327

""uco changing operators collide at the midpoint when multiplying two ,. lIm, . the problem is hidden within the algebra itself. have been some anempts to solve this problem by playing with the changing operators, by altering tho ghost number of the ~trings. un~v,~:~~"~~ disappeari. Since all pictures must necessarily yield the q superstring amplitudes, one solution is to change the picwhich the theory is formulated. In other words, the action, which is . may be defined in various pictures, but the on-shell S-matrix is

,

development concerns the closed string, perhaps the most urgent facing string field theory. Naively, we might start with a closed string such thlit all strings have the same lengths and then construct a '~";~~ Howevcr, Kaku and Lykken [58] have shown that the resulting ;d not fully reproduce the Shapiro-Virasoro amplitude, i.e.• there region of the integration region. They posrulated the existence "~:~~,~:~~iJ\teraction necessary to restore modular invariance and the il region. By unitarity, there arc therefore an infmite number diagrams necessary to restore modular invariance, i.e., the action must Kaku and gmups at MIT and Kyoto l59, 60, 61J established the 'p',Il",o,,'al closed string field theory. Not only is Ihe gauge invari~:~:;,;,:~o;:,::p.~olynomlal interactions of closed ~trings, the action is also 'fi that the complcte modular region is recovered, piece by integrating over the moduli space of N -point functions. A generalof ' hi·,' bosonic string field theory to the superstring case, however. has successful. Because tlle closed string field thcory ill nonpolynomial, of colliding pictun;: changing operators. Now, there which can collide sheet. development is the construction of a nonpolynomial closed string for the D = 2 clnsed string ease [62]. In fWO dimensions, string be ,cJ~nmul..ed in terms of a Liouville lhcory. In fWO dimensions, retains its hasic nonpolynomial structure, except that the states are i'"!,ler and the ghost insertions at the midpoints arc different. in order counting eoru;tmints. development concerns a geometric string field theory, one based posrulates which may unify all these various string field theories · geometric theory gauges the string length. so we can "interpolate" · Witten's formulation (based on fixed string lengths) and the light· formulation (based on variable string lengths). This unifies the two nU>aI'i,nu,. ,which are now seen to be specific gauges of the same theory. first BRST theory is defined in the " midpoint" gauge, interact at their midpoints, while the second BRST theory is "endpoint" gauge, wbere strings interact at their endpoints, as

328

7. ERST Field Theory

Lastly, we should conclude by saying that string field tneqry:::£@. evolving subject. As a result, it has not yet lived up to its proliDjl~¢::~am nonperturbative information about string theory. Although is defined independent of perturbation theory, it is still too form nonperturbative calculations. By contrast, M-theory has alt(~~ a wealth of information about the nonperturbative nature although in a nonrigorous fashion. It remains to be seen if M-theory can be formulated in a sec,()l:i!~ijji fashion, as in string theory.

7.8

Summary

The origin of the covariantized gauge approach was to cor[s~~im invariant under 00

lJ 1<1» =

L L_

n

IAn} .

n=1

We postulated this invariance based on an analogy with the which is also invariant under a field that rotates into ghost U"J"'''.' decompose the previous equation, we will derive precisely the ..:"",,'~.'!ol.'H the variation of the Yang-Mills theory . (<1>1 P[Lo - I]P I) ,

where P is a projection operator. By a power expansion, we can W~MII solution to this gauge problem (<1>1 (Lo - !LILI

+ ...) 1<1».

At the free level, this reproduces the usual Maxwell action. also be written to higher orders by brute force

+ L:'1(4Lo +!D+ L:' 1(6L o + 6)I1L2 + h.c. - L_2(4Lo + 2)(2Lo + 2)I1L2 + ... "

P(L o - l)P = Lo - 1 - 4L_ILI

where

11 = 2(l6L~

+ (2D -

lO)Lo

+ Dr l

and P is a p,ojection operator that eliminates ghost fields. N01~if:~@!' theory is nonlocal, meaning that we must incorporate awdliary·.~~:t9!1~ up the nonlocal terms. The necessity for auxiliary fields already shows up in the approach. Notice that the Paddeev-Popovgh,osts propagate in ....":.~~

pJ(8"

+ Br)el + P~(BT -

BaW 2 •

7.8 Summary

329

that our path integral must be altered to take into account these of freedom. The Hilben space of our field functional must now

1 = 10) IX)

J

DXDO (01 (XI,

(01 (XI " ) = " (X, 9).

(U6)

ri",""th,,, 0"' 'G ""n', functions must now be expanded to read /1 , 01)= =

f::"

J

J [Xi.', lx [ ..

DXDPDeDP,ex p [,

dad".(a,T)]

Z

D ¢I ¢l(X/,8;)4>°(X,.8j)exp i

9.

1

DXDBL(¢).

(7.8.7) quantization introduces a new nilpotent operator Q such lbat 00

..,

Q = co( Lo - I) + l)C_nL~

+ L_wc~J (7.8.8) (7.8.9)

"',,,eaolion can simply be expressed as L =

(I Q I")'

£1") = QIA).

(7.8.10)

of this to the interacting theory can also be performed function techniques. We need a conformru mapping that will half-plane into a fully symmetric configuration. One choice is the mapping of six charges [0 produce the three-string vertex

o

p

= "L.., 0i1ln(Z

1

- zJ) =

.

In z3 -,.

-

irr/2,

(7.8. 11)

Z +1

i""-l Ct") =
= I.

=a_2=a3- I ,

(7.8. 12) Z-t = -ZI. Z-l = -ZJ.

330

7. BRST Field Theory

This allows us to write an explicit formula for the vertex hlIlICh~Jnr

1V123) = exp

[!ct~nN~~ct~m + PoN;~ct~m + !Noo t P~<+.: tI'M·;::rn~ r=l

One major advance in the BRST theory is a fonnalism where "",,>ri;l be reduced down to the basic assumptions of the theory. It turns five axioms, which are simply postulated with no derivation, we c·;~:~i entire BRST theory: ( 1) Existence of a nilpotent derivation Q2 = O. (2) Associativity of the

* product [A * B] * C = A * [B * C].

(3) Leibnitz rule Q[A

* B] = QA * B +(_l)AA * QB.

(4) The product rule

(5) The integration rule

f

QA =

o.

With these rules, we can now show that the invariant action L

=

f

¢

* Q¢ + ~¢ * ¢ * ¢

which is a Chem-Simons term. ~}.~ill Lastly, we can show that the three-string vertex function is e< shell to the usual Veneziano vertex:

IiT 123) =

D [~e~L~n] V exp

I

123 } •

Thus {phy 1V123} = (phYI it 123) . The embarra~sing thing, however, is that we can also write the light cone vertex and show that it is also equivalent on-shel Veneziano vertex. Thus, we are faced with two equivalent functions, both of which produce the same on-shell vertices.. <. Although the open bosonic string was relatively easy to v language, the superstring and closed string have been much les

References

JJ I

51' app".clh . The problem is thatnarve ghost COWlting oflhe superstring string actions yields the wrong results:

<"'I Q 1<1» = O. to change the ghost count ing, there have been several proposnls to either the Hilbert space or the· Q operator so that the COrTect ghost is achieved. The problem with this Hpprooch is that the zero modes differently from the other modes, so that locality in u is probably and hence these approaches probably cannot be generalized to the promising approach is to introduce bosonizcd ghost operators that ee.orree! missing ghost numbers and insert them at the midpoints where lose u 10C3lity. .....e mention some new directions:

~

theory

'~~~i~~,:::~~~~string fieldterm of the apparently the kinetic actiOIl.

may be con-

superstring is actually anomalous because of colliding changing operators. This problem may be solved by changing the in which the action is defined. closed SIring fie ld theory (with fixed string lengths) is neces· The gauge-i nvariant string field theory action fo r strings has been explicitly constructed. However, it is nol generalize tbis to superstrings. for D = 2 closed string field theory bas been constructed which

~;~~~:;~LiOuvi [[e theory.

string field theory has been developed in which in the string ,n~: been gauged. so that Witten's string field 1heory emerges in a ~~ gauge," while the light-cone-like theory emerges in an ··endpoint

e(

is that it is formulated non-pcrturbatively action. However, the theory has not yet lived up to its promise. is well defined but is still too difficult to extract nonperturbative By contrast, M-theory, although it is ill defined mathematically, a wealth of nonpenurbalive infonnation. It remains to be seen if be formulated in a second quantized languagc.

OW" .0 fslr;". field theory

,Nuc/. Phys. 8267, 125 (1985). " 'dJ; Lykken, in Symposium on Anomalies. Topology, and Geometry, Scientific, Singapore, 1985.
332

7. BRST Field Theory

[4] R. C. Brower and C. B. Thorn, Nucl. Phys. B31, 163 (1971). [5] V. Kac, Lecture Notes in Phys. 94,441 (1979); see also B. L. t'ej,gjl'i:iit Fuchs. Soviet Math. Dokl. 27,465 (1983). [6] M. Kaku, Phys. Lett. 162B, 97 (1985). [7] M. Kato and K. Ogawa, Nucl. Phys. B212, 443 (1983). [8] E. S. Fradkin and G. A. Vilkoviskii, Phys. Lett. 55B, 224 ( [9] W. Siegel, Phys. Lett. 142B, 276 (1984); 15lB, 391,396 (I [10] T. Banks and M. Peskin, Nucl. Phys. B264,' 513 (1986). [11] T. Banks, D. Friedan, E. Martinec, M. E. Peskin, and C. R. Prsil'(';;" Phys. B274, 71 (1896). [12] W. Siegel and B. Zwiebach, Nucl. Phys. B263, 105 (1985). [13] B. Zwiebach, CTP 1308, October 1985. [14] K. Itoh, T. Kugo, H. Kunitomo, and H. Ooguri. Progr. Theoret; , (1986). [15] D. Pfeffer, P. Ramond, and V. Rogers, Nucl. Phys. B274, 131 [16] E. Witten, Nuc!. Phys. B268, 253 (1986). [17] E. Witten, Nucl. Phys. B276, 291 (1986). [18] A. Neveu and P. C. West, Plrys. Lett. 165B, 63 (1985). [19] A. Neveu, J. H. Scbwarz, and P. C. West, Phys. Lett. 164B, 51 ,.' , [20] A. Neveu and P. C. West, Nucl. Phys. B268, 125 (1986). [21] A. Neveu, H. Nicolai, and P. C. West, Nucl. Phys. B264, 173,(1,'986:m [22] H. Hata, K. Itoh, T. Kugo, H. Kunitomo, and K. Ogawa, Phys. 4"P',I,,;.:;!,: (1986); Phys. Lett. 172B, 195 (1986); see also I. Y. Aref'eva ' Theoret. Mat. Fiz. 67,484 (1986). [23] H. Hata, K. Itoh, T. Kugo, H. Kunitomo, and K. Ogawa, rnl~~)n 2360 (1986); Phys. Rev. D35, 3082 (1987); RIFP-674 (l ~~g:~i1ml (1986). : [24] H. Rata, K. Itoh, T. Kugo, H. Kunitomo, and K. Ogawa, Nucl. p~}i.fj::IJ~ (1987); see also N. P. Chang, H. Y. Guo, Z. Qui, and K. Wu, .........,' ,!-:J<'; 1986. (25] M. E. Peskin and C. B. Thorn, Nucl. Phys. B269, 509 (1 [26] S. Giddings, Nucl. Phys. B278, 242 (1986). :'I~ ' [27] S. Giddings and E. Martinec, Nucl. Phys. B278,91 (1986). [28] S. Giddings, E. Martinec, and E. Witten, Phys. Lett. 176B, 3(:.')':lHl ii [29] D. 1. Gross and A. Jevicki, Nucl. Phys. B282, 1 (1 QR7Y;::::j~ (1987). [30) S. Samuel, Phys. Lett. 181B, 249,255 (l986); N. Ohta, rnvs; :'J{.e.v.I (1986). [31] E. Cremmer, C. B. Thorn, and A. Schwimmer, Phys. t4~@~*1

v4Milii1

(1986). [32] E. Cremmer, Quantum Gravity-Integrable and Conformal In Conference, LPTENS 86/32, September 1986. [33] L. Caneschi, A. Schwimmer, and G. Veneziano, Phys. (1969). [34] S. Sciuto, Nuovo Cimento Lett. 2,411 (1969). [35] A. R. Bogojevic and A. Jevicki, Nucl. Phys. B287, 381 (1 [36] A. Neveu and P. C. West, Phys. Lett. 179B, 235 (1986). [37) For an alternative approach to string lengths, see A. Neveu aJi,n~;:~ Phys. B293, 266 (87).

References

333

1and B. Zwiebach, Nucl. Phys. 8282, 125 (87). and S. Raby, Nue!. Phys. 8278, 256 (1986). and R. Holman, Phys. Rev. Len. 58, J304 (87). Phys. Rev. l.eu. 56, 440 (1986); 56, 1316 (E)( 19lHi); S. P. de tdwis Phys. Lell. 8174,383 (1986); B188, 425 (1987); 8200, 466

';i:Ph'"J;;d s. Uehara, Pllys. Lell. 1688,70 (1986); Phys. LeI!. B173, 134

'ti'

. l.ell. 1738,409 (1986); Phys. Lell. 1798, 342 (1986). Date, M. Ounaydin, M. Pemici, K. Pilch, and P. van Nicuwenhuize'n, . Lell. 171B, 182(1986). :j[,z'''"~ A. Ne~u, H. Nicolai, and P. C. West, Nue!. Phys. 8276, 366 ~~''f!:,and

A. H. Zimennan, Nud. PIIys . 8269. 349 (1 986); Phys. Leu. (1986); Phys. Lell. 1668, 130 (1986); Phys. Lell. ]688, 75

" Phys. Leu. I 72B, 204 (1986). A';"d,Phys. /..elf. 1728,32 (1986); 1808, 45 ( \ 986); Nue!. Phy.!. 821$1 ,

,,",,"= and E. Maina, Phys. l.elf. 1808,53 (1986). and C.J. Zbu, Phy.f. l.ell. 180B, 50 (1986). "hi"O, "ulll, Phys. 8296, 333 (88). K. Ogawa, and K. Suchiro, KUNS-846 HE(TH) 86J06. OIl"". Pllys. Lett., 1689, 53 ( 1986). i and J. Distler, Nucl. I'hys. 8273, 552 (1986). 'ri~bn. Enrico Femli Institute Report No. 85·27, 1985; T. Yonca, Pro~et:d lhe Sevemh Workshop on Grand UnijicationJlCOBAN '86, 1byama, E. Winen. unpublished. ""• • ,. Itoh, T. Kugo, H. KUnitomo, and K. Ogllwa, Phys. Lett. 1758, 1]8 Horowitz. 1. Lykkcn, R. Rohm and A. Strominger, Phys. Rev. Lett. 57, See also K. Kikkawa, M. Macno, and S. Sawada, Ph},s. wt. 19711 ,

(~~:B6),

Phys. B3]4, 209 ( 1989). l ykken, Phys. Rev. 038, 3067 (1988). Phys. Rev. 041 , 3734 (1990); M. Kaku, Phys. Left. 2508, 64 ~di,"dB.

Zwiebach, Ann. Phys. 192, 213 (1989).

, H. Kunilomo, an d K. Suchiro, Phys. Leu. 2268, 48 (1989). Third Quantized Topological Gravity. in 20th Con! on Differential . Me/hods in Theo,.eli~al Physics (S. Catlo and A. Rocha-Caridi, - World Scientific, Singapore, 1991; Reeent Developments in String Field Theory, in ProcewlIgs Triesle Summer School in Physic.f, World Scientific, Singapore,

, Symmetries and 20 String Field Theory, in Proceedings of inlemalional 111gJ'ler Symposium, World Scientific, Singapore, oku, """in"". Confo rmal Fields. alld Topology, Springer-Vcrlag, New York,

,,"

Part III

Phenomenology and Model Building

,~.

8

ies and the n-::::ilnger Theorem

eycmd GUT Phenomenology ::~~~~;.n'" 'roily Imifi~lt'''"I'l'of,a1lkn'own interactions

to

satisfy

be based on simple phy!licaJ assunlptions, expressed in terms of a emllCl'l', which will allow no more than one coupling constant. n finite theory of graviry coupled to the minim.'l.i SU(3) ® model of particle interactions.

this book., we have amy begun to explore the first possibility "'~:~~~, that a second quantized field theory exists that is based ~J

principles. However, the developments we have prescOlcd in have been purely formal. Unless we can make contact with the : dala, then the theory, no matter how elegant, must be : . ' The true lest of a unified fie ld theory is that it can rep roduce the .', data at low energies. ,is that dimensional breaking from a iO-dimensional 10 four dimensions can occur only nonperturbativcly. To any in the perturbation theory, the dimension of spaC&-l.ime seems In general, fie ld theory is the only reliable formalism in which

because the first quantized fomtalisOl we do not yet understand how to in string theory, mainly because

I ·

still in its infancy. Thus, physicists have not been of any classical vacuum solution. Hence we will of quantum stability in Part m of this book and will ()(1 classical solutions of the equations of motion.

338

8. Anomalies and the Atiyah--Singer Theorem

Given this important restriction, it is surprising that initial a~!~~f.!~Si ploring the experimental consequences of classical string theory a wealth of new phenomenology that takes us beyond GUTs. will first ask whether the string theory is compatible with the r~sJ~lm: standard GUT theory. Specifically, we will ask whether it can r:b~~1 GUT theories ofSU(5), 0(10), or E6.In this respect, the string tl some measure of success. We will show in Chapter 10, for ex!lri):J~l~~:m E8 @ E8 heterotic string can easily be broken down classically , has crural fermion solutions and an acceptable GUT phfm.ome:no:l~~iilii However, we must also demand that the string theory go beyondtf:.·li;!;$t~ phenomenology of the GUT theory. Specifically, we must ask ~""'.:"'<'4N questions of the string model:

( 1) (2) (3) (4)

Can it explain three generations of chiral fermions? Can it explain the experimental results on proton decay? Can it explain the smallness ofthe electron mass? Can it explain the vanishing of the cosmological constant an~~Sli(q metry breaking?

Although it is still too early to say, there are indications that is rich enough in content that it contains mathematical mech~~i~ may answer the above questions. Specifically, new to~,ol()gil:al<m~~!}i!lii enter into the picture, such that basic phenomenological cor~cejRt~rij generation number, are now recast in a new topological langmlg¢.;: is the crucial new mathematical jearure that allows us to go be),'Ott.(f:i GUT phenomenology. We will begin this chapter with a discussion of the isopin g.lr\J.:l:!lmt~m allowed in string theory. We will find that the properties of the S· ~.. ~·n~~~ as cyclic symmetry and factorization, provide only the we:akles~:i.411 on the gauge group of the theory. Then we will show that the c@i~~~. anomalies in string theory places stringent conditions on which may be allowed by the theory. Basically, the gauge group of a sup:¢X~OO theory must contain exactly 496 generators, which restricts us tOEiltlil~1

~~~;:c;~~~111

how the cancellation process takes place, we or To E8 understand 0 E8. venient to review some of the elementary properties of In particular, new developments in supersymmetry have now lU,"'f\I¥:'l'l' to prove the Atiyah-Singer index theorem from a simple ...,al"'~f"e tionally, the proof of the Atiyah-Singer theorem has been inacce:SS)J~

physicists because of the intricacies of the mathem~Sl~ti~c;al~~llill We begin oUr discussion of phenomenology by . model via Chan-Paton factors [1]. Ever since the early days it was known that isospin factors can be introduced trivially by a simple multiplicative factor. (In the next chapter, we will d~~:AA~ more sophisticated way to introduce gauge groups via cOlnpactlm Kac-Moody algebras.)

8.1 Beyo nd GUT Phenomenology

339

: h,u>-Paloo method produces the scattering amplitude T by si mply VeneLiano-Bom lenn A by lrdce of isospin malrices. both are manifestly cyclically symmetric, and the n summing over various

the

the

lati,"' ,flh,' exi,';"lllc&" 2,3, ... , N) = LTr(A,1.21.)"· AN)A(I, 2. 3, .. " N).

(8. 1.1 )

""'" the trace is cyclically symmetric regardless of the isospin group, wc ,~;IriCI'" 'os on the of group itself Thus. we wish to impose coaslrainlll will allow us to eliminate some of the nonpbysical

mat

choice me

'!!~::~::~!l~~~i~dwe firs t insert acomplete set ofinrenncdiate stotes

:,e

amplirude, We then demand;

"the amplirude T fuetorize explil:itiy; twisting of extcmal legs yield on eigenvalue of ± I; and the massless Yang-M ills particle on the external legs and in tile infactorized channel belong to the adjoint representa tio n of the gauge

begin by imposi ng the first condition, that the amplirudc factorize In Fig. 8.1, we divide up the N external particles into two clusters, and let the inlermediate line represcnt the particle X. The left-hand ."'.c"," the scattering of particles a into X, while the right-hand piece the particle X breaking up into panicles b. Following (5.1.7), we

=

p-1 3

x •

•

p

. Constraints imposed by unitarity. Isos pin can be introduced into the . multiplying the amplitudes by Chait-Paton isol:pin factoB, which must cOndiliolls arising from unitarity. The fuclonzed scattering arnplinadc '~:i~:,'~;:::: afGhan-Paton factof!li where the cxtemal 1ines are arranged in :Ie and counterclockwise order. (X represents a colla::!ion offactorileU

340

8. Anomalies and tbe Atiyah-Singer Theorem

find T(1, 2 ..... N) =

I

2

S -In x

:LT(a:-+ X)T(b x

:-+

X)

+ ...

':'

Now let us reexpress this factorization fonnula in terms amplitudes. Each of the amplitudes obeys A(l, 2 ..... N) ::;::

1 2

:L A(l, 2, ... , X)A(X, p. P +

s-m x x

Notice that the Veneziano amplitudes are automatically tactoriizaljI¢:i~it cally symmetric, so the factor T(a :-+ X) must contain two teI1~;i\~l~ cyclic and anticyclic ordering of the external legs. Thus, the nt'i\iiriii; T's must contain 2 x 2 terms altogether. Let us now write ...."".• "'.::~~~~ explicitly: T(I. 2 .... , N) =

1

2 {Tr(AI A2'" Ap_IApAp+1 ... AN_I

s -mx x A(l, 2, ... , p - 1, P. X)(X, P + I, '" .•

+ Tr(ApAp_I ... A2AIAp+IAp+2 ... AN) xA(p.p-I .... ,2.l.X)

x A(X. P + I, p

+ 2, ... , N -1. N)

+ Tr(AIA2 ... ApANAN-I ... Ap+l) X

A(l, 2..... p, X)A(X, N .... , p

.• <

+ 1)

+ Tr(ApAp_I ... AIANAN-l ... Ap+l) x A(p, ... , 1, X)A(X, N, ... , p + I)} .

. .

Next, we wish to simplifY this expression. We will now)D'[lPOS,tl~m' assumption, that the twist operator corresponds to

Q=(_I)N+I. Notice that if we twist all external legs, this changes the cyc~lic9t~!~ an anti cyclic ordering. By assumption, we are considering of massless vector particles, which have N = O. Thus, the tWI.sqm~ up a factor of (-1) for each external leg. Thus, the total con.ttil:im twist operator is

A(l, 2, ... , p, X) : : :; (_l)P+1 A(X, p, P - I, .... 2, Now let us 60llect all terms and arrive at some conclusions. ~~~!tl tenns, we find

Tr{[AIA2" 'A p - (-l)PApAp-l .. 'Ad X [Ap+lAp+2 ... AN - (-I)N-PANAN+I .. , "'p:l,'I 'U::@ X

A(I. 2 .... , X)A(X, P + 1, ... , N).

8,2 Anomalies and feynman Di
341

us insert a C()mplete set of isospin matrices A.. in the llbove equation. always do this, so we take (8.1.8)

Tr{lp'l ... AI' - (- 1)"A p

...

Ad

X (Ap+1 . .. AN - (_1 )"' -1')..,... ... AjH- l lJ

=

LT'UA, ... A, - (- I)'A,· .. A,P•• I

• (8.1.9) is ~~~~;;let us impose our third and fina l condition. The third cri terio n

:s

that the combination oflamlxla matrices in mebrackets must part of the algebra of the group: ).,=).,1"').,, - (-1)".1..1>"').1.

(8. 1.10)

is that the Chan--PntOD faclor, in order to preserve tbe adjoint for the massless vector particle, must have a gauge group for

combination of generators is also a generator of the algebra. . . U(N). (Acrually, also ruled out, but for other reasons. We cannot consistently couple closed superstrings with U(N) because N = 2 supcrgmvity cannot N = 1 mailer mulliplcts.) ~~~l~;;,thiS analysis does not yield any moreconstraints,!;O the model

y

. . We will now look al the question of anomalies, fix the gauge group 10 be either S0(32) or E~ ® E8. As in the ease '. theories, where the cancellation of anomalies between the quarks and a central role in model building, the cancellation of anomalies impOrllIl1t rule in fixing the gauge group of the string model.

""(lmath" and Feynman Diagrams have a very profound origin [2-4]. They shed light on the deepest of quantum field theories. anomal ies arise whenever the symmetries of the classical action do to the quanrum level. The classical symmetries do not survive the ~I,m,ing the quantum theory. There arc two types of anomaUes, in gnuge theories are actually welcomed. For example, a in scale invariance in QeD with massless quarks might be fe· masses. Thus, a breakdown in global scale through anomalies might be the origin of quark masses. Another

342

8. Anomalies and the Atiyah--Singer Theorem

global anomaly might be responsible for the breakdown ofU(N) variance that occurs in quark models ofQCD down to SUe N) ® would be phenomenologically desirable. For the superstring lll\"'l!~'k global anomalies would be undesirable. For example, if global olated the modular invariance of the multiloop amplitudes, disastrous consequences for the internal consistency andd l!:!I~rlll theory. Fortunately, it can shown that this global anomaly in II is absent in the string theory [5]. By contrast, local anomalies in gauge theory and in sUIlen;tringU1~;~ be canceled at all cost, or else these theories make no sense qu~mtilm:~iW;Ii ically. For example, the elimination of chiral anomalies is one ot1tl:i~j!Sli ways in which to build new models ofquarks and leptons. In the the quarks and leptons have chiralities that just cancel the superstring theory, local anomalies in conformal invariance and . . try must also be eliminated. The elimination of the conformal the dimension of space-time and also the fermion content of the elimination of chiral anomalies will fix the gauge group We begin our discussion by examining the simplest local . :;1.OCQ~ij chiral anomaly, which occurs because the process of . Pauli-Villars or dimensional regularization) does not respect Specifically, if we have a theory with the invariance

!fr

~

eiYse!fr.

then we would expect classically to have a conserved chiral .¢Wlftiji (1.9.8): BI1 J!1·s = BI1 (lirysy l1 !fr) = O. However. there are complications due to quantization. method, for example, inserts an imaginary massive particle ·1·t·Cl::tJ'!W to make all Feynman diagrams converge m'

1 ---,------,,p2 + m2

~

1 1 M 2 - m2 = ---,,-----::------::-----:,,.:,. p2 + m2 p2 + M2 (p2 + m2)(p2 +

Propagators. which usually converge as p -2 ,now converge as --,': ......., almost all divergent Feynman diagrams convergent. Once we ha're.:~~~ finite S-matrix, we set the mass M of the imaginary particle to.g",o::t~mt However, mass terms explicitly break chiral invariance:

B( lir!fr)

=1=

O.

Thus, the PauU-Villars regularization program does not respect and we expect that the divergence of the axial current is not cannot preserve both conservation of the current and regult;rr~'q.l Feynman amplitudes at the same time. But since the ref~ll:JI"i,~atiionpt! is more important (or else we have no theory), this means that we m1~ml current conservation.

8.2 Anomalies and Feynman Diagrams

343

'">~~I~. dimensional regularization will also fai l to preserve the chiDimensional regularization assumes that we can analytically the Feynman diagrams into complex-dimensional space by the

io

f

d·p

~

J

(8.2.5)

dClp.

to generalize the traces over Dirac matrices for vector particle couthis breaks down for axial vector couplings because there is no ",cioo of Ys in complex spac.c: (8.2.6)

regularization program breaks down for chiTa! fenn ions rcgulari7.ation cannOt generalize the Y$ matrix to a complex , therefore, that thedivcrgcncc (lftheaxial current is not conserved. of the axial current. The triangle diagram poses regularizabecause cach internal femion line, which circulates within the di,>J!l'un, converges only as p-l, which is not sufficient to give us graph and hence has ambiguities. By carefully regularizing the ,"'.,;.,,: we wind up with

aI' )'<.5 =

-\ _

16tr 2 ti

l'W1p

Tr(F P ) ,, ~

(8.2.7)

~P'

total derivative or Q topological term. givcn by -

\ --l£I'''~r Tr (A"a~Ay

4"

+ i A~A ~ Ay).

(8.2.8)

for the triangle graph was derived in four dimensions. However, can also be generalized to D-dimensional chiral theories. where groph is now replaced by a polygon graph. Let us begin hy writing

";~::" ~~, ~~~~ vector and axial veclorparticles interacting with Let us begin with /If - 1 external vector particles and one particle of external momentll ki, polarization (k). and isospin

fe

{r

.b,,md c arc interacting with a ferlll.lon of internal momenta pf. The propagator vertex (vector) _ vertex (axial) _

r ·p

--r-I)~b '

p

r· p.", r n+!r. p.".

(8.2.9)

344

8. Anomalies and the Atiyah-Singer Theorem

We assume that the external particles are on-shell, that the pollari~~~~ vanish when contracted on external momenta, and that the sum '0" f::thjf.~ momenta is zero, so that

kL = 0, k/ . {i(k i ) = 0, N

l:)i '

O.

'1=1

Then Feynman's rules for the polygon graph give us

G~

!

V

d pTr(AuL"'AaN)Tr

r · {I \ (nN r· Pip~2[l+rv+11', :"z-:.z,00< 1=1

Let us extract the most divergent part of this diagram. We nOI[lcf:mp,t;~ over the tenn containing the r V +1 can be explicitly carried out.

Notice that the anti symmetric matrix arises only when we take U'le:lltl1ai matrices multiplied by r D+I' Let us recombine the various tactc',rS:jjfiJiifj which must now be contracted onto the antisymmetrilccl~m~ia~tr~!ix.~~~~1~11 conservation of momentum reduces the number of ir factors by one. In general, the only terms that survive are ofthef", [email protected] G~

!

n N

dDp Tr('/l.aL .. . /l.aN ' )

_12 Co... Ill"'/lD rilL rill ~ ~

1=1

• .• kllD

•

Pi

Notice that there is now a direct correlation between the lines and the number of dimensions of space-time because the tensor is of rank D. Thus, the leading divergence must satisfy

D = 2N - 2.

d!i~ooli

In four dimensions, this means the triangle graph is divergent. In 1 this means the hexagon graph is divergent. Notice also that ~~f~flm explicit expression for the anomaly term. The external mOlm~:J:l~~:~f~ replaced by derivatives aIL' and the polarization tensor {/Li by the field ,AIL, so that we can replace the contractions ~~:~~i~~~ contraction bver the Yang-Mills tensor FlLv ' Thus, after I.
--+

{/L --+ k[/L{vl --+

a/l' A/l, F/l v ,

8.2 Anomalies and Feynman Diagrams

345

);,,;ngall faclors, we find [6-S] j(lJ2)D

. . a,pd =

: .

,T"" .....• FI"I'I ... F"O-'P.".

2D- 1Jl'{Ifl)D(i D )1

~

(8.2 .16)

I there is an anomaly in the theory of gravity coupled to chira! , which again is proportional to a total derivative or a topological couple external graviton legs to cbint! fCl1nions circuhlting ill of n single-loop graph. we can n.-peal our previous analysis with cW' d;i'gnnns. The major difference comes from three factors:

now must include the coupling of fermions 10 the energy-momenrum rather than the chiral CUlTcnt The gravilon coupling 10 fermions place through vierbeins (because GL(N) has no finite-dimens ional

represcuullions; sec Appendix). eXfernal polarization vector {,. now becomes an external polarization

ve,lex funct ions now contain higher tensor components. , the leading divergent pan ofilie diagram still contains the e"""I ~ ./I" co,,\rnotcd onto external momenta k/ and external polarization tensors again, it is trivial to recombine all factors lind show that the leading can be reassembled in x-space in terms of curvature tensors. For the coupling of an axial vector and two energy-momentum tensors

,d;n,m;oos y;elds (8.2.17)

;,d ;:~'?~:n~:~:":':; 'm". are two

total

derivative terms, in

X(M)=_ I j d 1x h R=2-2 g ,

(8.2. 18)

there

well-known

Ih". is the celebrated Gauss-Bonnet identity:

4K

w ill prove later, where x(M) is the Euler number of a twOmanifold M, R is the contracted curvature tensor, and 8 is the elosed manifold (i.e .• the number of holes or handles). This identity ib~,~~;~o;~;;'~~th::c scalar curvature in two dimensions is a Iota I derivative. IU we havc the identity t:'''og~ Tr(R,,~ R..,) =

a../i R!lIO"p + b.J8 R!~ + c.JR Rl

= total derivative

(8.2. 19)

numbers a, b, and c. the integral of 3 lotal derivative is zero. However, this is not true if has a boundary or nonrrivia! topology. In this case, the intcgral derivative picks up a term from the boundaries and the topologies. o,v" thc contracted curvature tensor for a closed manifold is e linear function of the genus, or the number of holes in the surface,

"'1e"",1

8. Anomalies and the Atiyah-Singer Theorem

346

which is a topological number. This is the origin of the anomalies, total derivatives, and topological numbers. Thus, both the gauge and gravitational anomalies can be delwe(1'ijt:i;im!1 man diagrams. However, for higher dimensions this becomes pr:i1ifim difficult. Indices proliferate very rapidly for higher d,·ummsllon:s. :f.~~~~W generalize these results for higher dimensions, we will derive from the functional formalism without appealing to Feynman diaj{!-III

8.3

Anomalies in the Functional Formalism

So far, we have been approaching anomalies in bits and ple:ceik would like to have is a systematic method with which to c~c~;:l~~~111 derivative terms for an arbitrary gauge group and arbitrary Let us begin with the functional expression for the fermion mte:tiitli the external vector or gravitational particles:

I

I

D1jJ Dlfre-s =

D1jJ D\lr exp [ -

{I

dDxlfr!O

+ rD+t<}£hlt.:.os:::I:i

where the covariant derivative explicitly represents the coupling f',i to the external Yang-Mills and gravitational field: ' 0 ::

P=

ri'(di'

',e::liii

ffi:

+ i)..a A~ + W~b M ab ),

where ri' are the gamma matrices in D dimensions multiplied by."Ujlial field, and M ab is a generator of the Lorentz group OeD - 1, 1). Notice that the functional integral is quadratic in the fer:!lliI?p:)ij¢P. the integral is a Gaussian. Using our results on integration uV'O',>,~;J:..\t!!ltl variables in the Appendix, we can now perform the integral andl i'J~~*~~II~ minant. Then we can reexponentiate the determinant back into tl to obtain a term g) that is explicitly dependent on the gatig¢:~ explicitly performing the Gaussian integration in (8.3.1), we fimi~~~1t determinant:

rcA.

eir(A,g)

= det{!(l + rD+I)J:'}'

where A and g are the Yang-Mills and graviton fields. We now u, sl'n:f!¢!~ detM = eTrinM •

(This is most easily proved by making a similarity transt'011lnatlQn:J~ nalizes the matr~ M. Then M reduces to a diagonal matrix with ,,', ' along the diagonal. In this form, the id entity is trivial to prove. the reverse similarity transfornnation to restore the M -matrix.) Th~~ take the log of (8.3.3):

rcA, g) = In{det[!(l + rD+1)p]} = Trln{~(l

+ rD+I)p}.

8.3 Anomalies in the Functional Formalism

347

us calculate the gallge variation of this field functiona l. Under the (8.3.6) ,d tb"" alO« a Taylor expausion in s, II field functional trallsfonns as qA~) == [(A" - D" e)

= l'(A,,) = r{A,,)

-! + ... +! dOxTreD,,~ +... d D x Tr D"e Sf

8A,

= rcA,,) +

f

SA,

dDx TreDIlJ 1d + ....

(8.3.7)

last expression for r(A~), we llsed the fact that the current generated transfonnation is given by J,

~

8rcA)

(8.3.S)

lA,

te'Nritethe G appearing in (8.2.13) in functional language:

~

8 G = D,,-r(A,,). 8A,

(8.3.9)

D"J" = G.

(R,3.1O)

::~:~:~~;~ ;~;:i:~'~i, meum a n.onvan.i~hing must also salisfy currl!n.t a self-consistency condilion, G. the Wcss[9l. Notice that the generator of a gauge transformation DA = -

,

~r

,

= D,,- . ,lAIl

(S.3.!!)

have a new way of writing G: G(A ) = ,l A rrA!').

(8.3 .!2)

know that the local gauge generators fonn a closed algebra.

rOAI' 01.,1 = 8"".

Thu~,

(8.3.13)

means that (he G's must slltisfy the constraint (8.3.!4) G as a variation otT(A,,), then it is obvious that the Wcss-Zumino condition is satisfied. However, if we mile the anomaly term in

348

8. Anomalies and the Atiyah-Singer Theorem

terms of curvature tensors, then it is highly nontrivial that this selt::¢4~~jii condition is satisfied. This then provides a powerful check on our co~iillJ~; Now that we have a functional formalism in which to discuss an(>tm us begin a discussion of the mathematics behind anomalies. Spl~cijti.qjlii will show that the anomaly can be written in terms of generalized chliFn~ classes that have been studied by mathematicians. Then we will sh()W)!fj elegant part of the theory, namely that the integrals over these cl;l~@.!!j classes yield various index theorems. Thus, our strategy is to "he".;;' Anomalies --+ Characteristics classes --+ Index th"n"'~"""ii:

8.4

Anomalies and Characteristic Classes

To study characteristic classes [10, 11], which will give us a S~E~t~=m ysis of all possible topological terms, we will use the language Appendix.) The theory of forms, in some sense, only produces re~i~l1~ili be derived using the usual analytical methods of calculus. dimensions, the number of indices rapidly proliferates beyond the powerful shorthand notation of the theory of forms allows us nipulate tensors of arbitrary rank in any dimension, which is standard tensor calculus. In Chapter lOwe will see that the theoryi'i;if.1mi the most convenient language for the theory of cohomology and .. Armed with the theory of forms, we now construct a series of classes that will allow us to write by inspection the set of aIllOp1olO;gJ,:qj for practically any dimension and any group. Let us now define an invariant polynomial that has the nTe>np1rru:~ilili~~e~ P(a)

=

Peg-lag),

where D! and g are group matrices. Examples of invariant po.lynortll~] Det(l

+ a).

Tr ea.

Now let n be a curvature two-form:

n =dw+w!\w that satisfies the Bianchi identities

dn +w!\ n - (-ly"n!\ w

= O.

In the Appendix we prove two important statements for .' mials:

[1] [2]

dP(n) = 0, p(n)=dQ.

for some Q. That is, an invariant polynomial based on cUJ'Valtun~f~ltJ closed and exact. The theory of forms allows an explicit COllS1J1X~I~~!II

8.4 Anomalies and Characteristic Classes exampla. in

349

me Appendix we show

Trn" = nd =

11 dlt~-'

Tr IA(dA +IAl)"-I } (R.4.6)

d{j)n _l.

,~:~~~:,~~~I'i~iP';:~~: of curvature two-fonns is exact and closed. This ~~ in the language of forms o f previous identities that were hand. f orcxampJc. in follI' dimensions we havc (&.2.7) and (8.2.17). dimension we have (8.2.16). This new form tU~_1 is called the mU,M Jun,,,, which we encountered in (82.R). is important, bccause earlier we showed that in two and fourdimensions polynomials in the curvature tensor that are rotal derivatives. But unable 10 construct these higher topological tcnn.<;. Equation (8.4.6) the solution to Ihc problem. the Janguageofforms, wccsn also find a convenient expression for the consistency condition. In D-dimensional space, let us formally 10+2. (Of course, this is actually D + 2 dimensions, but we will put aside this point.) This form, by is exact,

(hi,,,,,,,

"'ion,,

(8.4.7) is a Chem-Simons form . JO~2 is ruso gauge invariant because it forms. 1bc facllhat it is both gauge invariant and us to writa

en,,,,iyof,,,,,,""'x

S...lO+l = = = =

SlldwD+l dSIIWD~ l

2 d wD+l

O.

sec that the gauge variation of the Chern-Simons form is closed. We

(8.4,8) ~~~,, ~:~;:~Construclion is to find a fom) for lhe anomaly G s uch thai Itl

a gauge variation of some other form . We now write

G=

1

(8,4.9)

wv,

we can use Stokes' theorem to show

G=l

AI

lJ)o= (dWD =SII r,WD+l'

1E

" ~m.cc M and E arc related by M =

Jr.

i.I E.

(8.4,10)

350

8. Anomalies and the Atiyah-Singer Theorem

This is our desired result. We have now shown that the anc)1llilI:y:ten a gauge variation of another form. Thus, G automatically satisfjleslli(~~ Zumino consistency condition, because

DA,G(A2) = DA, DA 2

f

WD+I·

We will use this construction when we actually cancel the superstring. Armed with the theory of forms, we can now construct the . . polynomials at will. We will find that there are four Jlliljor characteristic classes:

(1) (2) (3) (4)

Chern classes; Pontrjagin classes; Euler classes; and Stiefel-Whitney classes.

Let us now restrict the curvature form 0 to be a member of "-''''l''':i::~ an arbitrary k x k matrix with complex elements. Let us define <.u""Ul;""·~ fonn over this group as

c(O) = det

(I + 0) _i 2rr

= 1 + CI(O) + C2(O) +"', .

where we have power expanded in powers of O. The terms in which eventually terminate, are = I, i Cl = -TIn, 2;r 1 C2 = 8rr 2 (Tr(O

C()

C3 =

1\

n) - Tr(Q)

i

- - 2 (-2Tr(Q 1\

48;r

- Trn

1\

TIn

1\

Tr(nn,

: :. Q 1\ Q) + 3 Tr(Q 1\ n) 1\ TrQi.

.: ....

1\ Trn),

where dCj(Q) = O.

Notice that the series eventually ends, because Q raised to a his.lt::l;!m power is equal to zero because the dx il are Grassmann-valued. , the Chern classes C;, we can also introduce the Chern character: ...

ChaJ"ll~.!cristic

8.4 Anomalies and

always diagonalize the matrix

Classes

3S J

n so thai it has only eigenvalues Xi

diagonal. Thus, we can a lways find a matrix S such that

x,

0

0 0 0

x,

SQS-l = (-21ri)

0 0

0 0

Xl

0

0 0

(8.4. 16)

0

'.1I;on, the traces and detenninants become trivial : c(Q)

=

• TI(I +Xj).

(8.4.17)

j-I

thai the topological invariant for SU(2) in fO llr dimensions is simply

clMI=det( 1 + 4~A"~) = 1+

8~2 Tr(F 1\ F).

(8.4. 18)

i to the Chern classes, there are the closely related Ponrrjagin . leI n be a member afO(k), and then define

pen) = del (I

- 2~ n) = I + PI + P2 + . ".

(8.4.19)

.

class, however, is defined slightly differently. It is not based on t::~ 'likC th e Chern and Puntrjagin classes. but on the Pfaffian. Let

a = 4a;1 dx l

1\

dx i .

•is<, it 10 the rth power: ct.' :::

r! (2Jr)' e(a)dx 1 " dx 2 1\

.. . 1\

dx b

•

(8.4 .20)

thai this is an antisyrrunctric matrix. We cannot diagonali.zc an an-

matrix such that its diagonal elements arc its eigenvalues

Xi,

'. '.c"n write it in the form

0

-x,

x,

0

0

0

0 0

0

X,

-x,

(8.4.21 )

0

';rian, e(a). which is built out of the antisymmelric matrix Ct . is called In terms of these x 's, the Pfaffian is equal to (8.4.22)

352

8. Anomalies and the Atiyah-Singer Theorem

We can also conclude the Pfaf'fian e(M) is the square root Pontrjagin class: p(M ) =

2 1 2 XjX2 .• , x2r

2() = eM.

Some examples ofthe Euler class are

I

D =2:

e(M)= 2Jr R 12 ,

D =4:

e(M)=

~2eabcdRab /\ Rcd , 32n

where

£

X(M) =

e(M),

which are precisely the identities that we mentioned earlier in ~~,"'~x (8.2.19). Finally, we want to list some of the other characteristic pollynoni~~li~til of interest. If Xi labels the eigenvalues of the curvature form n, Todd class = td(M) = Hirzebruch class = L(M) =

a roof polynomial

,.

= A(M) =

n n n.

Xi

1 - e-X; Xi

/ tanh x;

•

I -Xi 2 I

i

•

•

sinh "iX/

We will find. for example, that the A invariant polynomial plays in calculating the anomaly for spin-! particles coupled to Lastly, we have the Stiefel-Whitney classes, which, related to any curvature form. Instead, the Stiefel-Whitney class ~~::.f:I. in deciding which manifolds are spin manifolds, i.e., which OllC·:;,i::Y.< spinors. Many manifolds that at first glance seem to admit spinQ:t$:~W have a problem. We merely quote important features of the Stil~J~I~~m class index. The vanishing ofthe first Stiefel-Whitney class ind.ex,· ~~@ the manifold is orientable: WI

= 0

++ M

is orientable.

Most important, a manifold M can admit spinors if its second StH~t?'t~ class index is equal to zero: WI

=

W2

= 0 -> M is a spin manifold.

8.5 Dirac lndcll

353

we have 1il>1ed some of the imponant properties of chatacterislic

will show that the integrals over certain characlI:ristic classes yield indc.x theorems. One of the most important will be the Dirac index, . on a spin manifold. In facl, it is possible to reexpress the other invariant , of the Dirac index. Ifwc have a spin manifold in N-SVdce, then it is to define the Dirac cquillion in this space:

r A D",1/!=/>1/r=O,

(8.S. I)

arc the Dirac matrices (multiplied by the vierbein). Let us examine f'J given in (8.3.2) and its eigenvalues: i P 1/F~

t

= A,.1/I".

(8.5.2)

~~;f::'~::~~~~:~ fJ comprises those solutions which are Alln ihilated i.e., have zero eigenvalues. spaces, the spinor space actually divides in half,

on the eigenvalue of the operator: (8.5.3)

thcscsolutions positive and negative chirality solutions. In general, '~)~~:':~::',lh C chiralities always occur in pairs. lhis is because the

lu

the mass term of lhe Dirac equation, and massive fermion be s.plit into two Weyl fennions. which are always massless. for the zero eigenvalue case, the ehira litics do not have to occur in enn have unequal numbers of positive and negative dural states for ,m"h,c rcnnions. define II ~ to be the number of independent solutions with zero cigen(:::~,~~:~(negative) chirality. Let us also define the indl'x of the Dime r. d ifference between the numbers of independent zero modus and negative chirality: Index(fJ)

=

n+ - n~.

(8.5.4)

to note thar this number is a topological number; i.e., it remains continuous change in the manifold that does not change the . The Dirac index is a topological inv-drillnl because, aftcr a transitions wiU occur bernlceo zero e igenvalue cigrnvalue fermion states, but onfy pair.~ of nonzero eigenvalue call enler the zero eigenvalue state or leave il. lhis is because, in .bew"" ma.<;sive, chiral states must find a partner. Thus, the changes in the index occur only in pairs, and each pair contributes zero to Thus, the index must be a topological invariant (sec Fig. 8.2). For , if we have N pairs of remtions entering the zero eigenvalue state,

olndex(p) = on.;.. - on~ =

N - N = O.

(8.5.5)

354

8. Anomalies and the Atiyah-Singer Theorem

~tll

FIGURE 8.2. Pairwise emission or absorption into the ground enter or leave the ground state only in pairs with opposite chirality. ~ difference between positive and negative chiral states populating the V.l~..~.U:~~ remains the same. This constant is called the Dirac index.

There are several ways in which we can express this index. ~m~t¢:1ili:i of an operator is defined to be the set of states that it aru1ihilates:; write Index(p) = dimker(p+) - dimker(p_), where we have split up the Dirac operator into positive and nel~atl:ye:¢l Another way to write the index formula is lndex(p) =

L

(nl r D +1 In)

n

= Tr(rD+l) =

Ll- Ll

+ = n+ - n_.

Notice that each positive chiral state contributes + 1 to the "ULJ"'Ii:~~.JJ negative chiral state contributes -1, so the sum yields the ..... u, ...... Although the above formula is quite elegant, it must be car'efuln~i}ij Noticed that a careless definition of the index leads to me:aning~eS!$~

2:n+ - 2:n_ =

00 -

00.

Because the cancellations occur level by level, it is incorrect to::!SOO~m positive levels first and then subtract the sum against all nelgativ~ sum over all positive or negative levels is infinite by itself. is finite and well defined. The proper way to treat the index is to insert a factor that alI()MI::tfs;~ the sum over all Rositive and negative levels separately (which is vergent) as a function of a parameter. One simp Ie expression t"rt'~IA-.n Dirac index is Index(p) = Tr(fD+Je- PD \ where f3 is an arbitrary positive number and D2 is the operator. Because we are taking the trace, we are free to

,,,,,,,,r,,, .......

8.5 Dirac Index

355

"",u",,«I, so we can rewrite the trace in tenus of the eigenvalues il.i of

Index(p) = L Sign(i)e-P"",

signU):::::

±.

(8.5.10)

the massive eigenvalues come with equal numbers of positive and . ,hi,,1 solutions, they cancel because they contribute with different hence do not occur in the sum. TIle zero eigenvalues ni:/:, however, pee""" come with equal numbers of positive and negative chiral hence we have once again Index(p) = LSign(i)e-P)'" =

2),1;+ + (-n;_)] + L[e-,II~·

- e- PA ;]

'-..#1

).,-(1

(8.5.1 1)

:::::f1+-n_.

"::'i~,fermions can enter and leave the zero eigenvalue state only in

Ii , the difference of the positive and negative chiralities are not byth'presence of nonzero eigenvalue femlions (which always occur hence their contribution to the index cancels exactly). Thus, the " ..,,,with nonzero eigenvalues do not affect the index at all. not an academic question because it bears directly on the anomaly ",m"'pl", we know that the chiral currcnt can bc v,Titten as (8.5. 12) '. Ihis is conserved if we use the equations of molion. The covariant . isospin indices) is equal to

Dp.JP.·5 = Qp.Jp. .3 + [A p.. jp.s ]

= *y,p~

- (I'*)y,,,.

(8.5.13)

, this is exactly equal to zero because yp. Dp. W _ O. We must be to includc thc contribution ofthc vac uwncxpcctation value, which may not vanish. Let us define

SAx , y) = 101*(x),,(y) 10), yp. DIlSF(X, y) = 8(A. y),

(8.5.14)

vacuum expectation value of the divergence of the current . Then

Dp.J IlS = 2 Tr(ysPS,.(x. y»).;,=y = 2 T r(Yst(.l. x».

(8.5. 15)

356

8. Anomalies and the Atiyah-Singer Theorem

The last expression, which contains o(x, x); makes no sense . ize the integraL This is to be expected because we saw earlier that· . graph was divergent and required careful regularization. One standard method of regularization is called the heat kernel Dl¢~ra; it inserts a converging factor into the expression:

D/lJ J1 5 = lim 2 Tr[yse-rD\Ii(x, Y)}x=y, r--+O

where the trace is taken over spinors, and we have regulated the ':·k"j~f~$SI that it is finite (for positive T). But notice that this expression for the anomalous term in the ";":,~' H;;;, servation expression is precisely the value of the Dirac index. u:,,:,,·:,.,:,», established a direct link between the Dirac index and the nOl'lCClnSl~'ffi~ the axial current. When we evaluate this trace for an arbitrary spin manifold, (8.4.19) and (8.4.26), ·e

we:$]m

Index(p) =

L

A(M}

112 .. + 5760 (7 Pl - 4p2) + .. ',<

= I - 24 PI

where the dimension of space-time is a multiple of four. In we have D =4:

1 PI = 24. 18rr Index(p) = - 24 2

f

Tr{R 1\

For the case of the spinor interacting with aYang- Mills field in metric, the index becomes an integral over the product of two m¢!!m~ roof genus coming from the gravitational part and a term cOlniiii~:l~1 gauge part: IndexCP) =

L

A(M) ch(V}.

Notice that this formula can exist in even dimensions, not . four. For D = 2 we have Index(p) =

ForD =4wehave Index(p) = .. dim V 2 24·grr

f

1Mr

Cl (V)

= _i 2rr

Tr(R 1\ R) - - 1 2

8rr

f f

Tr F.

Tr(F 1\

For the case of interest, six dimensions, we have Index(p)

=

48 .lgrr2

f

TrCF 1\ F 1\ F) -

~

J

Tr{F 1\ Tr R ...:.

8.6 Gravitational and Gauge Anoma.lies

357

the Atiyah-Siogcr theorem can be written a.s the integral over forms, one for the gravitational part and one for . For completeness. let us list the four classical complexes and index theo rems that rcsuil from them: Index '[beorem

Index

Index(d + 8) = X(M) reM) = f L(M ) " Index(8) = f td(M) 1\ ch(V) Index(p) = f A(M)" cb(V)

Gauss-Bonnet Hirzebruch Riemann-Roch A roof gcnu.s

0",'"

(8.5.22)

is the Euler characteristic, tll is the Todd characteristic. ch is the Chern .

~::~(::~:::i:we double the value ofthc curvature form. ch is the usual en, and is the complex conjugate of at when we use coordinates l. Z to describe the two-dimensioJUli surface. (We .,.,;11 complex manifolds a11ength in Chapter 10.)

1"

a

. ' Gravitational and Gauge Anomalies wc have erected a powerful apparatus by which we can cOQSl.nlct

mo.ru,ai using the method of characteristic classes, let us invcstigatc in theories where spinors are coupled to gauge and gravitational

;:~::~ :~~',"::::::::~Y~~~(:::'~t:;::::~ and Witten [12}. Let us begin ~I some elementary features ofspinors. even dimension D = 2k., the Dinlc matrices are complex 2k_ matrices. In general, these Dirac matrices split into positive and ehiral representations. In an odd number of dimensions D = 2k + I, matrices are also 2i -dimensiona!, and there is only on~ rep re!)endo not have negative and positive chiralities in an odd number of (and hence no anomaly). Therefore, we will restrict our attention number of dimensions. (8.6.1)

r1+1=

+1

r;J+' =-1

+ 2.

if D =4k

ifD = 4k.

(8.6.2)

I has eigenvalue ± I if D = 4~ + 2. Thus, under a combined CPT chirality ofa particular state does not change. Positive chiea! states into positive ones, and the same for negative states. We find CPT(D = 4k

+ 2};

,,+ ). [ ,,+ v--"'#

(8.6.3)

358

8. Anomalies and the Atiyah--Singer Theorem

Thus, it is possible to have unequal numbers of positive and ne!~ati~:jtf:l states, and hence an anomaly is possible. However, r D+ 1 has eigenvalues ±i if D = 4k. Thus, under a C states with r D+l = i are transformed into states with r D+ I clural states are converted into negative ones, and vice versa:

P{:Wfi

'if!+ -H- 'if! - , { 'if!- -H- 'if!+ .

CPT(D = 4k):

If D = 4k, then positive and negative chiralities occur in gravitational interaction will not lead to anomalies. In summary, gravitational anomalies are possible only if D = . Now let us analyze Dirac, Weyl, or Majorana spinors. Weyl .. be defined in any even dimension. This is because the operator can be defined in any even dimension. However, it turns out spinors can exist only in 2, 3, and 4 (mod 8) dimensions. ~~~~f~~l2t~r~OOIi that are simultaneously Majorana and Weyl can be defined only '''':'''::~.I'!~~ dimensions. We can summarize this: Spinor

Dimension

Dirac Weyl Majorana Majorana-Weyl

AnyD D = even

D

= 2, 3, 4 (mod 8) D = 2 (mod 8)

Our particular case of interest is 10-dimensional In 10 dimensions, three types of internal particle may gravitational anomaly: (1) spin-4 fermions interacting with N external gravitons; (2) spin-~ fermions interacting with N external gravitons; and (3) anti symmetric fourth-rank tensor fields interacting with N .....",,4+;"i;

It is obvious why the spinors must contribute to the anomaly. agators converge only as I! p, and the usual Pauli~Villars and di#~~ regularization methods fail because we choose chiral fields with The self-dual fourth-rank antisymmetric tensor is also added markably, there seems to exist no naive covariant action aSS:OCllatf)d tensor. It seems to be well defined in the light cone gauge but not C()~t~~g so we suspect that it may also have an anomaly. We begin with ~ spin-4 particle coupled to gravitation: D

pa

S = / d x e e 1{ii ya Dil

[40 -

Expanding around fiat space: ella

= liil a + hpa + ....

y5)]

'if!.

8.6 Gravitational and Gauge Anomalies

359

interested in the lowest order coupling of the vierbein field with the

LI = -~ilIIlU¢-Yllau(l - y S)1/r, £.1 = -i6(h~,A," .... )¢-I-J'~~t( 1 - Y')1If.

(8.6.8)

".Iy,w,,,,,. [,ole
[te~m, .ho,,,v,,,,, is a four-point Pcynman graph, me sea· gull term. it seems hopelcss to calculate the single fennicn loop diagram with ",_._" number of graviton legs, but there are tricks we can use to reduce ",ol," simple problem. Specifically, using simple symmetry arguments,

that we can reduce the problem to the sCQrrering ofa scalar charged intersecting with a constant electromagnetic field. Fortunately, the

QED a

aknown

o~,~:::::;;~,,:;oa~~IW'S in is well--studicd problem with 1 the problem is to reduce the complex problem to one ': already known. . also generalizes for the other two cases. We arc also interested in of internal spill- ~ and antisymmctric tensor fields with eA1emai Symmetl)' arguments again a llow us to reduce the problem to a much The c irculating spin-~ particle can be reduced to an internal vector .0. ,mo the aflth:;ymmetric tensor field can be reduced to a sp in- ~ particle:

"0" .

spin

t and spin 2 ....:,. spin 0 and spin I,

spin ~ and spin 2 -+ spin I and spin I.

(8.6.9)

antisymmetric tcnsor and spin 2 ....:,. spin! and spin I.

)","anm, "cu\''-', it is easy to con~tTUct the polygon graph, We know that for the scattering of spin-! particles can be represented as

h/2

..

==:

i21J:+ J Ml R(f:(il, pUl)Z(t1i ). pU»,

(8.6.10)

. : M = regulator mass and R(t(i )

{j))

,P

=

_F;J1.' .... . /UJ.lp(l 1£(I) III III

.•. p(2.1:+1)£(21+I). ..... _1

Il. l , ]

(8.6.11)

the ith external gravi[on's polarization can be given by FY )

II"

=

E(i)t;(f) Il ~.

(8.6. 12)

equation. notice that we have the freedom to choose the polarizajib external graviton to be the product of two spin- I polarizaJion I. Th[" [,th" k"y the original scattering amplitude particles in tenns of a reduced one with spin-I particles. O~~~;~;I;; has now been reduced to the scattering of a propagating d particle (oj charge ~) iflleracting with a constant e/ectromag-

,.",,1\0'" '" ,,,,wn"

360

8. Anomalies and the Atiyah--Singer Theorem

netic field, which was computed by Schwinger decades ago field be represented by

[13].l:;.:~~U~(i

Earlier, in (8.3.5), we used functional techniques to show that t~J~illl term was related to the logarithm of a determinant of a propagator. that

Then Z is Z

1

= -lndet(-D Dil + M2) vol Il

1

= --1 vol 0

00

-ds TreseD~ D"}. e-sMl .

s

'

Schwinger proved that _1_ Tr esD~D~ = _1_ eB vol 4nsinheBs' where B is the effective magnetic field ofthe EM field and e is the cl:l~it:g~~ will present a more general proof of this at the end of this chapter... :Alfliffti we cannot diagonalize the antisymmetric Maxwell tensor, we can alV;1a&'~~ it as

o o o

o o

-X2

0

Then the anomaly contribution equals ..

As expected, this is just the index of the Dirac operator, or the iI'IteJgt;a.;J!J1~ Next, we have calculate the anomaly for a spin-~ fermion intc@jg coupled to an arbitrary number of external gravitons. Once again, tne:,1l'l'li to use symmetry arguments to reduce the problem to a much sirrlph:r:Q~ action for a spin-~ particle coupled to gravity is

to

I IlV).p,lr 5 D ,/. L -- -'is 'I' Il Y Yv ). 'I' p'

8.6 GravltBtional and Gauge Anomalies

361

U131lhe spinor Vr/l has a vector index /.L as well as a spinor index. which PP'""'.' Again, let us c"tr.lct only the lowe!>t coupling limns: LI

iltl'· 1 ="ih ¥r"YCI(JII"2(l

L2 =

S

- y )01,." ,

/6(h~a"hTa)~ .. rCl"·~( 1 i

L)

_ YS)l/r",

(8.6.20)

_ (J"h"")¥ray ~ ,,, ...

= 2"(Ja h"v -

.. 'g'''", we can also write the general one-loop amplitude as 1m =

221:+1 i

Ml R(lj). pU')Z(£li). p(J).

(R.6.2I)

Z represents the scattering of an internal charged vector meson intcra constant electric field. leI us write an effective theory of charged 'me• .,,, that reproduces Z to onc loop: ~d'

lo

Z => Trln H = _

_

,

Tr~ -·H,

+ ~iAa)2q,,, + ~jF"v¢".

H¢ .. = -(aa

(8.6.22)

(8.6.23)

'en",;,,,,,,,,,,,," be performed, and we arrive at

n.

24"+1

::: _i(21fr2k-1 R(.s{i1.

pi})

"'<1

!x-

2t+1

2

L(2coShXj - I).

~

smh"2 XI

(8.6.24)

) ..0

last anomaly leon we wish to calculate is the contribution from an "::~~~.;:;':~':~n:sor field. When we calculate lhe anomaly associated with n: however, we must be careful becnuse there is no covariant associated with this particle. Let us begin with an anrisymmetric tensor 4k + 2 dimensions and its curvature: FI" "'r'l>~' =

al' l

A../"_...... , + cyclic perm.

,

(8.6.25)

""""" the constraint that it is self-dual: F

-

I'I"'U,,,, - (2k

+ I)!

t;.Ul.O.!·- .......,

"' - ""~I

F

",u,···",,·,·

(8.6.26)

fe~~:;~ :~~:~'~:. By the Bianchi identities, we can show tbat this relation ;j,

to the equlItions of motion. In other words. tbe naive action L __ p1

","',,,''',

(8.6.27)

both self-dual and antidual states, not jllst the self-dual ones. Thus,

': ':~'~~,~~';ln fact, we can prove that a covariant action that propagates bl stlltes does not ex.ist (14]. V~;:~, ~:'o:~ use a trick to calculate the scattering amplitude. Although

d(

seem to be a covariant action. the Feynman rules for such a

362

8. Anomalies and the Atiyah-Singer Theorem

particle can be written covariantly. The trick is to use a spinor nel[9':tij: the antisymmetric tensor field. The energy-momentum tensor for this field is

1 "I "'1l2k 1 2 TILv = (2k)! FP.al·'·"lk Fv - 2(2k + I)! gp.u Fot \'''''2k+1' Fortunately, all the Feynman rules for this antisymmetric F tenso~:~~m. even if its action is not known. Now we will rewrite this antisYIIUlq¢tt}iii in terms offermion fields by a trick. Let us define N

N " ' Tr (r P.1/l.2"·P.. )otp FP.tlll"·P.. • = 2- /4 '~

A.. ~,,,{J

11=0

We can also invert this A.."P • FILIP.l " ' /L. -- 2- N / 4(r" . .. ·Ill ){Ja~

The advantage of using this embedding is that fermion fields are ~~~!~ to use than antisymmetric tensors. The two-point function is .. IJ:'

(¢"p(q )¢y~( -q)) = (2 q 2) - I«y5 y /LqlL)"y (ySyp.ql' )(J~

+ q2 S"Y

"",<.......",.~.

The final amplitude can be written as

h = -i M 222k+ 1 R(s(i),

p(j))Z, 2 where Z represents the coupling of a charged scalar with externlal:Ji!W;[~

Z = Trln(iyP. DIL

= --1 10 2

+ i M)

00 . dse- sM1 Trexp(-s(-Dp.DIl

+ irllv F

0

We know, however, that Tre-isr.,F., = 22k+~

2k+!

Il cosh~xi' i=~

The final result for the anomaly contribution from the an1tis)7IIllrq:~I~~1 field is 2k+1 !x, 2k 1 1 fA = !i2 + 2;r-2k- R(s(i),p(j)) 2 ~ cosh!xi. ;=1 sinh 2' X i

n.

Of course, we.also have to calculate the anomaly C~~~jl~:~;~'~:~~1 gauge fields. The calculation there is almost identical to the done, except now we must have n gauge particles coITesiponding;:~g generators. Let us now put (8.6.18), (8.6.24) and (8.6.35) together, . contribution from the mixed anomalies coming from the gauge seC;1~~~i:l venience, we will write the total anomaly contribution in terms

8.6 Gravitational and Gauge Anomalies

363

earlier in (8.4.7) was a convenient form for the anomaly because show that the anomaly term manifestly satisfied the Wess-Zumino condition. We have

"".cy

-I = 720

I

Trr-+ 24.4& TrF"'TrR2

__1_ Tr p2

256

+ (n

(_I Tr R4 + _I (Tr R2l) 45 36

-~96) [ 2. ;835 Tr R~ + 4. ,1080 TT R2 Tr R4

(8.6.36)

1

+

1 (TrR2i +_TrR2TrR4+ 1 1 _CTrR2)1 8·1296 384 1536 ' represents the curvarure tensor, and F !.he Yang-Mills is our final result. At first glance, ilsppears as if ooly a series of call make such a horrihle object vanish. However, a series of miracles I.CCU'ffi. SpecifieaUy. we need 10 show that Bstring modeJ IS consistent following:

represents

496, which kills half the lemu; in the anomaly; ,itlih,mnainin,g nonzero tenus can he regrouped into a factorized form ; ," this reduced, factorizoo form ean be canceled by other terms coming an effective point particle action t:.S. -:~~~li~(a:;lIlhrce conditions can be imposed simultaneously. ~I (0 is easy LO satisry iflVe set n = 496. Then a large number vanish in (8.6.36). condition (il) is considembly to satisry, but we can show ",naiini"g terms in (8.6.36) fac lorize into the product of two tenns if be reexprcsscd in terms ofTr p2 Tr F C and (Tr F2)3 , i.e.,

harder

, Tr F6 = _ Tr p2 48 I 112

....

Tr p4 _

1

crr

pl)l. (8.6.37) 14,400 is satisfied. then we have !.he further reduction:

1\

(Tr R"

+ kTr p 2}XS<

(8.6.38)

,naly (: .6.:'6) would vanish if k = -110 and if

F' - -'-(Tr p2)' __ I Tr F2 /\ Tr R2..1.! Tr R4 nOlI 2~O '8

+ ...!..(Tr R2)' • 32

(8.6.39) it seems remarkable that under any circurn:;tances can we satisfy by having these rigid relations hold. condition (II), we must check which Lie groups arc compatible

~

:~E~:~J~:::·~~;(;8.•~6~.3:~7'~).,:N~:O::t~i,:~,~:th~atlfwe the now traceconvert over thethis gauge mato traces we

will use the symbol "tr."

364

8. Anomalies and the Atiyah-Singer Theorem

The proof that groups exist which satisfies (8.6.37) is stnl1gI1tt()~iW we know that, in the fundamental representation ofSO(l), the matmi~F~ represented as an anti symmetric I x l matrix. In the adjoint replres@::~~ can be represented as

Fab.cd = !(FaCOl>d - Fbcl'J ad - Fadl'J hc + FbdO ac )' We can insert explicit expressions into the trace calculation of F6,,'li~mB Tr F6 = (I - 32)tr F6

+ 15 tr F2 tr F4.

In order to satisfy the factorization condition, we must be able tOI~Atl.Wi number of terms, including the sixth power of the curvature this is possible if I = 32, which has precisely 496 generators. back again to SO(32). The proof that E8 ® Es also satisfies (8.6.37) is a bit more lllV'Ul~.t:U" to know whether or not the Clebsch-Gordon coefficients of the us to write independent invariants that allow us to contract Mathematically, we need to know if there are independent CasinW!~~ of orders four and six. Fortunately, there is a theorem which homotopy group X 2n-1 (E8) contains the integers, then there is Casimir operator of order n. (Homotopy is a way of geller.atlllg~q~ classes whose members are continuous maps, rather than spaces It can be shown that the first homotopy groups satisfying this coli{ij~~i and XIS:

x/(Es) = 0

for 3 < i <

Thus, the only independent invariant of interest is of order two, • ~{@~I that Tr F4 and Tr F6 are not independent and can be rewritten in tet,~.~ The last step is to actually calculate the coefficient appearing in T;:R~:J~ Since we are only interested in the overall coefficient, we can alvv~y'~~j§ this by taking a specific representation of a subgroup within £8 and the adjoint representation 248 can be decomposed, under "n,·,'." "·: sum of120ffi128. Thus, (8.6.37) can be proved for £8 ® £8 as (The equation can also be shown trivially for U(1)496 and £8 ® We are not quite finished yet. We still have to satisfy "'UI.IUJUo/~I~:: show that there exists an effective low-energy counterterm :$lm kills the anomaly. If we take naively D = 10 chiral su~,erg7'a'tli~ 496 super-Yang-Mills fields, we find that the Xg term does not chiral D = 10 supergravity must be ruled out as an acceptable qwm~~1 However, the superstring theory has new coupling terms in the zerQ~J~~ that can cancel the remaining terms. As a low-energy approximation, superstrings can have m()I,'¢:~~~ action than appear in the D = 10 chiral supergravity action. At :t1.iM9l seem swprising, because super gravity is the low-energy limit . or. ~j. theory. However, even at the low-energy levels, there is a diffe~::fi.~~ over an infinite number of Reggeon states will, in general, S\ltfP~ fl .

8.6 GT
365

f~:~:~Il'::ap;~hs~,:th~~ao we would naively expect. (For example, the Yang-~ i with up 10 four fields in its Lagrangian. Yet the first siring theory predicts only thrcc-Reggeon couplings. Where is the four-particle vertex? h comes trom the swn over an infinite number stales, which will give us effective four-particle states in the low'il,",L In general, high<.... trees and loops give us all the terms necessary the Yang-Mills and gravitation theory.) we can show thai there exists an effective tree-level tcOD thai will anomaly term. To show this.. let us first wri te the 12-form as /'2 - (Tr R2 - tr p l)XS'

(8.6.43)

be rewri tten as

('.6.44) . of the identities:

Tr RI:;: dW1L. tr FI :;: dlJJ]v, UJ1Y = Ir(AF - t Al), (lJJL

(8.6.45)

= Tr(wR -lc.i).

OW)y

=

O(flll

= -d~l'

-dwJy,

(8.6.46)

terms are Chem-Simons thrcc-fonns corresponding to either

i (Y) or the Lorentz (L) cOfUlcctions. We would now like 10 12-form 1\2 that we have carefully cak:ulaled in tenns of 8 new CtJLO. so that the Wcss-Zumino consisteocy conditions are satisfied. our suategy is to find the G term by constructing Wl0 and

1\2 =

dWI\ .

G = 8"

lwu = l

dWl0

=

£

Wl0·

WII .

(8.6.47)

out that. working from 8 12-form down to a la-form, there is a of arbitra riness in the final result. It is not hard to show that, !rI, we ean work down to an J I-form and then to a IO-form. We ,"Ole II" final result: WII = ~(Wll-wly )X8+ J(TrR1 - trF2)X7

,""ou,"

+ o:d«(Wll. -

Wn )X7 ).

(8.6.48)

.i",naroitr,""co,ost'"' and also :=

(J + a)(Tr R! -

tr F 2}X6

+ (~ -

a)(wR - Wly)X 1.

(8.6.49)

366

8. Anomalies and the Atiyah-Singer Theorem

where we have the following expressions for X6 and X 7:

dX7 = Xg,

JX7 = dX 6 • Combining everything, we noW have an expression for G: G=

(~ + a)

f

(W31. - W3y) dX 6

+ U- a)

f

(WzL - Wzy

This is the factor G that we wanted to calculate. Now the qU!~stiQ~tjlf or not the D = 10 supergravity coupled to the super:¥y;,::;:w. contribute an anomaly tenn that cancels G. It is not hard, "nT'''''':'' identities, to show that the addition of the fonowing effective ac.t1lQW;W a new contribution to the anomaly term that will precisely cancel: G~~::] AS"'-'

f

BXs -

(~+ a) f (W3L -

W3Y)X 7 ,

where a is a constant, the W3 terms are Chem-Simons tenns;::"",:.,w;~ connection, and B is the two-form B M N that appears in the supersymmetric Yang-Mills theory with D = 10 supergravity Manton action). A careful comparison shows that they yield an:mi;!jiji theory if we choose the variation of the B field as

JB =

wiL - nl

y ·

It is important to notice that this is not the usual variation of the supergravity theory. Thus, the Chaplain-Manton supergravity a~~!9.~3.m with anomalies. This is discouraging, until we realize that in SUt!¢tsi~ we have effective terms arising from loops and the sUlnrrIati.on9Y:~1fjm number of resonance states. Thus, it is possible to have ~"'.~'~~~ B field and still have an acceptable supergravity theory. however, is to explicitly show that this cancellation takes is one thing to show that the B field might have the correct v;~a~:Iml anomalies canceL It is another thing to show that it actually Surprisingly, the cancellation of anomalies for the entire s~~~;1 out to be much simpler than the cancellation of anomalies for tl. supergravity system!

8.7

Anomaly Cancellation in Strings

We saw that a series of miracles happens to cancel all

an4:)Ilt~);i~&:

n = 496 and the gauge group is equal to 80(32) or Es ® E g•. ·1Jj~ro however, were formulated only in the low-energy apI)ro;rurta~~J redone for the general case. Surprisingly, because the m9~~ tower of resonances in a compact fashion, the calculation is much easier than the calculation for the supergravity the,oiiYl

8.7 Anomaly cancellation in Strings

367

us begin with the Pauli-ViUars regularization method, replacing the propagator with a massive one: I -....

Fo

J Fo+ im ,. Po - im Lo+m

(8.7.1)

~~[2;~1~~~~~;~~~:i~!~~~~~~

f:

regularization procedure

any other regulariLa-

theofanomaly violates chiral invariance. Here, the addition the massoccurs Icnn chiral invariam:e. regularization., the single-loop hexagon diagram that violat:es parity

" 1

T(O)=

d pT,

(Po-V(I)···-V(6)r" 1'0 -) , Lo

(8.7.2)

Lo

(8.7.3)

+ f' l l) is !he projection operator that selects out the "even G stat',soflhe NS-R model: [-II = rUrJ. I'd = (_I)L:,
(8.7.4)

~;I:~~~~:~::~O" , there is actually no anomaly. because the lheory

lim (1'(0) - T(m)].

(8.7.5)

.-~

of large m. the dependence on the Innss should drop

Qllt

and

result. Let us now perform all the traces over the Dirac matrices. the amplitude thai is left is k)

,,(1 1 d pTr

£(~. k)

=

1

1.0+m2Vo(J)··· Lo+m2VO(6)rd

EI"l'l"'jI!~I'.'! ... ",{i'··· (fsk~ ' .. . k,?

)

' (8.7.6)

(8.7 .7)

m.ust be exercised in taking the limil as the mass goes to infinicy. we must add togelher the contributions from the planar and the single-loop graphs. We find ot({. k) /.\

Ii

o /_\

d vj 8(Vi-t 1 - VI) (Ol Vo(k l , 41)'" Vo(k,. z,) 10) .

k)1 n

(8.7.8)

1

t({.

o

dVI(}(VI +1 -11,.)(01 Ve(k .. Zl)'" VO(k 6• z,)IO).

I_ I

368

8. Anomalies and the Atiyah--Singer Theorem

Putting everything together, including the isospin factors cOIniIlgj~t~ill Chan-Paton factors, we have G ,..., (n

x

+ 32l)Tr().. \A.2 ... A.6)S(S, k) e(Vi+\ -

Vj)

11 D

dv,

(01 Vo(k l , ZI)'" Vo(k 6, Z6) 10) ~<

To have a vanishing anomaly term, we need only show that n + 'l
1= {

~

-1

Usp(n), U(n), SO(n).

(3) Most important, the factor 32 comes from several sources.T , ,~~ factor of 32 in the Jacobian because the nonorientable ', tegration from 0 to while the planar graph had an int'egr'4ijj~naM

!,

A

to 1. Furthermore, the integration region was only as b:~~~illl there was another factor of 32 coming from the 32 ways in. ',' number of twists can be placed on siX internal propagators.T:Jj~~:J a cancellation, we must fix

11

1= -1 ~ gauge group is Soen), ' .. . + 32l = 0 ~ gauge group is SO(32).

In summary, we have an anomaly-free theory with Chan-·Pi:j.l(9.l\:;tJ we choose SO(32) as our gauge group.

8.8

Summary

We have used Chan-Paton factors and the anomaly cancellation t(:q'9f~!JJ group of string theory. First, the Chan-Paton factor is "111.'1-'1), ~¢:.::tm. various group gerierators that multiplies the Veneziano-Born tePtm~'~?:W

T(l, 2, 3, ... , N) = LTrO.1A.2A.3·· ·A.N)A(I, 2, 3, ... , penn

Unfortunately, the only restriction that unitarity places on the gm~, must be Usp(N), SO(N), or U(N), for arbitrary N.

8.8 Summary

369

more stringent restrictions on the group are found when we demand b~::,~~bc~c'~n,:o:~malY-fTCe. In general, an anomaly occurs whenever a clas:5 Lagrangian does nOI survive the quantization process.

anomaly arises, for example, because the method of regulariza· or dimensional regularization) must necessarily mvan ance. the divergence of the axial current, instead of being zero, is

- I j,"-.l 11/4 = - 62 £ "'."PFIW F"p. o

(U2)

I " a total derivative or a topological term, given by jJd

= _811"2..,ua,8y Tr(Aaff"Al' + ~AIlA,8Ar)'

(8.8.3)

"",,,,I, the anomaly in a higher dimension will be proportional to lerm. For example, using

:

th~

<J

theory of forms. we can show that

~~;:;~~!r~;~::~~: tensor can be written as the divergence of another we can show Lhal the trace over curvatures can be writ1en of a Chem--Siroons form: TT

n~ = nd 10

1

dU,,- l Tr tA(dA

+ rA 2t- J }.

(8.8.4)

of these invariant polynomials takes us to thc theory of charac. TIlece are four classicai characteristic classes. The Chern class

t( c(Q) = de! (I + 2~Q) = I +CI(Q) + Cl(Q} + ....

(8.8.5)

"",g;'n class is defined foc O(k) as p(Q)=det(/-

2~Q) = I + PI +P2+····

(8.8.6)

class is defined in lenns of the Pfaffian:

ex' = r! {2rrYe(ex)dxl I\dXl" ···l\d.x 1r .

(8.8.7)

is the Stiefel-Whitney class, which cannot be written in tenns forms. However, we will find the Stcifel--Whitney class to be when we are ana!y.zing spin manifolds. In particular, the vanishing two indices yields an orientable spin manifold on which we can

WI

=0

WI

=

-to>- M is oricnlable,

(LI;!

= 0

-7

M is a spin manifold.

(8.8.8)

370

8. Anomalies and the Atiyah-Singer Theorem

The index theorems, in turn, are usually written in terms of invariant polynomials: Todd class = td(M) =

x· . I _ ~-Xi '

TI

n [

Hirzebruch class

= L(M) =

~Xi '

I

I

a roof polynomial = A(M)

= IT .. zx; smh

i

.

i:Xi

theore~~1

The most useful of the index theorems is the Dirac index concerns the number ofpositive chirality zero eigenvalue solutu:ms:tt);tlJ; equation minus the negative chirality solutions. Using the sigma model, we can show the following: IndexCP) = TrC _l)F e-

rD2

=

~! d DxfJjJ.JjJ.5

=

(2n

. )(1/2)D

_1-

.

:. ··!i~i.

!TreFdeCI/2[(!R) - lSinh! ·.· •.

..

,2

2 :. __ '"

Armed with this theoretical apparatus, we can calculate the galJig~. ~ia itational anomalies found in supergravity and in string theory. In,¢'1*-.~B.g gravitational and gauge anomaly contributions arise because the • can be either: (1) a chiral spin-! fermion; (2) a chiral spin-~ fermion; or (3) an antisymmetric tensor that has no covariant action.

The calculation of the anomaly contribution is carried out Feynman diagrams. The calculation, however, simplifies we can make certain assumptions about the polarization tensor lines, thus reducing its spin. Thus, the difficult problem of COI1I@~~ the various indices is reduced to a much simpler problem of COill®:l~ a lower-spin particle. The final results are

8.8 Summary

371

everything together, we find the total anomaly contribution from and gravitationa l sectors: ~I 6 -TrF

720

I

,

+ 24·48 TrF'TrW

",la,u,ly,we can set this 10 zerO if we make a few assumptions. first. we 10 be 496. Then we assume th~t we can facto ri ze the anomaly into ,duo' of"vo tenns:

III - (Tr R2 XK =

+ k Tr p2)x 8,

2~ Tr p4 ~ 72~O(Tr F2)2 1

2:0 Tl' F2/\ Tr R2

(8.8.13)

1

+ 8 Tr R4 + 32(Tr R2)2. 10 supergravity is immediately ruled out because the id"nrity canna! be satisfied. Hoy.'ever, string theory has one big adover :mpergravity. The presence of higher spin fields in superstring thai the zero slope limit of the theory does not have to reto supergravity. In particular, the couplings o f the B field in arc such thatlhey can. in principle, cancel the tenns shown

,atu.1 proof, however, that the anomaly term disappears must be carried the full single- loop hexagon graph in the string formalism, We use a 'V~;';,:d;~: regularization on the intennediate lines and then sum over nonorienlable loops. The anomaly is proportional to

iii

.,

I1n€(C,k)

/ d ,"(1 pTr Lo+m

2Vo(l) ' "

16)) rJ , Lo+m 2VO(

('.'. 14)

(8.8.16)

372

8. Anomalies and the Atiyah-Singer Theorem

Finally, adding the planar and nonorientable diagrams together J 5

yi~r~M~® . . ~~jmm

'0 ~~(n-,+~:.a)lt(..tJA2· .. ::

11 uVi i= l

x 9(IJi+J -

Vi)

(01 Vo(k J, ZI)'"

For this term to be zero, we must have II = 32

l= {

~

-1

Usp(n), U(n), SO(n).

Thus, the gauge group must be 0(32).

References (1969).

1. E. Paton and H. M. Chan, Nucl. Phys. B10, 51 S. L. Adler, Phys. Rev. 177,2426 (1969). J. S. Bell and R. Jackiw, Nuovo Omento 60A, W. A. Bardeen, Phys. Rev. 184, 1848 (1969) •. [5] E. Witten, in Symposium on Anomalies, Ge(nnet~ A. Bardeen and A. R. White) World Scientific, [6] P. H. Frampton and T. W. Kephart, Phys. Rev. 028, 1010 (1983). [7] P. K. Townsend alld G. Sierra, Nucl. Phys. [8] B. Zumino, Y. S. Wu, and A. Zee, Nucl. Phys. [9) J. Wess and B. Zumino, Phys. Lett. 37B, 95 (I [10] T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. [11] C. Nash and S. Sen, Topology and Geometry New York, 1983. (12) L. Alvarez·Gaume and E. Witten, Nucl. Phys. [13] 1. S. Schwinger, Phys. Rev. 82,664 (1951). [14] N. Marcus and J. H. Schwarz, Phys. Lett. [15] M. B. Green andJ. H. Schwarz,Phys. Lett. [16] M. F. Atiyah and 1. M. Singer, Ann. Math. 87, (1971). [17] L. Alvarez-Gaume, Commun. Math . Pllys. 90, [18] D. Friedan and P. Windey, Nucl. Phys. B235 [1] [2] [3] [4]

9

Strings and "fication

Compactification problem facing string

is the question

reduction can be made, the theory lacks any real contact quantities. breaking canbe dODe within the framcworkoffie ld theory, the best we can do is look at dassical solutions where spontaneous moos"""h", alrcadytaken place. In this chapter thc heterotic :.tring, which incorporates the groups Eg ® ER that arise when we compactify a 26-dimcnsional space down ~w

in the previous chapter, the cancellation of anomalies allows for of either 0(32) or E8 ® £8 . We saw, however, thai thc Chan,,"md does not allow for the possibility of cx~ptional groups. Thus, use yel anotber method, arising from compactification on a self-dual achieve an £8 ® £8 model. Although the heterotic string is a closed , it contains within il the super-Yang---Mills field. whieh usually s :,,~;~:::t~~ string sector for Type I strings. ~E

string takes advantage of the deceptively simple identity

26 = 10 +16.

(9.1.1)

that a 26-dirnensional string, when compactified down to a JOstring, leaves 16 extra dimensions that may be placed on [be root E8 13 £8, which is known to yield an anomaly-free lheory. TIlis was made by Freund Pl.

374

9. Heterotic Strings and Compactification

The heterotic string takes advantage of the fact that the CIO:S~tJi::S1 two independently moving sectors, the right-moving and lett-Illlovifilf: depending on whether they propagate as functions a +r or a - r. 1·; f1.f.l::lrreti string makes fundamental use of this splitting. The word "~'~~~i!~lll "hybrid vigor." This means that the asymmetric treatment 0 movers creates a hybrid theory considerably more sophisticated.fuil··ifj.:i6.~~ Type I and IT superstrings. It has been shown that it is has .... . anomalies, and is finite to one loop. Before we begin a discussion of the heterotic string, hmNe"et., descnoe the process of compactification by describing the sin}pli~~1:1OH case, that of a scalar particle in a periodic one-dimensional that we make the identification

x=x+2nR, where R is the radius of this space, which is just the real axis one-dimensional lattice r oflength 2n R:

divli:l~j;1;:

Rl

Sl =

r'

A field defined in this periodic space must therefore satisfy

¢(x) =
+ 2n R)

which means that it can be power expanded in periodic

¢(x) =

L
eiPX

eiB~en~(~~111

,

"

where

n

p- - R' where n is an arbitrary integer. Thus, we see that the m01mf!ntiim:.;i:l to x becomes quantized in tenns of integers. This is a feature Cflfliif.l\f compactifications. . Now let us generalize this to a particle living in five .. dilneJns.i,o~g~~9Iru such that the fifth coordinates has curled up and become :~~~~'~T:~.m~m scalar field that satisfies the massless Klein-Gordon equation:

Os
where Ps = nj R. Notice that each eigenfunction can ch~lllg'~fU~~~ "mass" of the Klein-Gordon operator: Os = D4

+ a~ = 14 -

p;.

9.1 Compactification

175

I conclusions can be drawn from these simple examples: comp
~~~~~::,;:;~s~P~"~c.~e~.~~,~~;~~:~ti~~~~

(enns of periodic eigenfunctions coordinate. [n one dimension it is simply sines or In higher dimensions we may have spherical harmonics.

let us consider the ca<;e of compactifying theories of nigher spin, S""",'I relativity, which w ill generate a Maxwell field out of the fifth

.0,.II:y, the idea of compacrification first came from KaJuza (2--41. who :: ]e\ler to Einstein in 19 19 proposing the idea of combining Maxwell's i Einstein's general relativity by expanding spac()dimensions. Kaluza proposed 10 write the metric tensor as

-

-("" "") lis"

gAB -

_

....

•

(9. 1.10)

'I'

w" d"fu" the five-dimensional metric in terms of the Maxwell field A" fout-dimensional metric tensor g"u:

8s11 = g",

:=

8J1~ = gu~

+ 1(2 A"A~.

KA""

(9.1.11)

moment, let us assume that the fiOh dimension is not experimentally because it has curled up into a very small circle. Thus, the fifth is periodic: Xs =Xs +2iTR.

(9.1.12)

traveling a dis tance 'Dr R in the fifth dimension. we arrive al the , mng pnint. The original five-dimensional Riemannian manifold is to have ~'P l il up according to

Rs -

R~

x St.

(9.1.13)

us assume that the radius of the fifth dimension is so smaU that it

'n,,,,,,,,d. Thus, we can set

as

-4-

O.

(9 .1.14)

376

9. Heterotic Strings and Compactification

With this added assumption, the equations simplify enonnously.. of the metric, for example, is usually given by 8gjtv = ajtAv + avAil + .... When the circumference of the fifth dimension is small, we have •. 8g sll = al'As

+ ....

Written in terms of (9.1.11), this reduces to 8AI' = K- 1al'A s , which is precisely the U(l) gauge variation of the Maxwell ___._ .....~_''''''' is only one action that has this U(l) symmetry and has only two;:· we thus conclude that Einstein's theory, written in five :: to Maxwell's theory coupled to a four-dimensional example, we can explicitly calculate some of the Christoffel sYIn~i~l&: approximation:

r SIl •v =

!K(aI'Av - avAil) + ... = !KFl'v,

We can thus explicitly reduce out the five-dimensional Einstein approximation, and we find . -1 ~R -AB L = -v-g ABg 2k2 = 2k2'V -1 I-::gR gJl.V_! I-::gF Fl'v+ .... -15 I'V 4'V -15 I'v

Thus, a genuine Maxwell field emerges in four-space when we COlli:pi~ml five-space theory of gravity. Similarly, we can generalize this result to higher manifolds, N -dimensional manifold peels off a smaHer compactified dimension P: RN -+ R N -

P X

Kp •

re~iilif~~~I~~

When this is done, we can reanalyze the gauge group that this breakdown, and we can derive the orthogonal groups or groups. In this way, we can show that Yang-Mills theory emergl~S::~~ higher-dimensional gravitation theory. Although this Kaluza-Klein formalism elegantly unites both the Vl\iiiw. and gravitational theories in one framework, there is a severe drawbac1i~~ approach that dates'back to Kaluza's original proposal, namely, fifth dimension suddenly curl up into a tiny balI? Klein's su~~ge:stij~m:::$!: curled up quantum mechanically into a ball of radius equal to I.A,:·I:
9. 1 Compactlfication

~

377

::.!:::~!;r.~~!~~::~~ compactificalion in the fr-.amework [5], which mustquestion undergoof the reduction from 26 or 10 down to Let lL'; first study the compactification of the ith coordinate string:

X/Ca, r) =

Xl

+ 2et" pi r + L ~a~ COSflae-inT •

(9.1.21)

•., n

, the periodicity of the ith coordinate enforces Ibe condition that quantized according to integer multiples of the integer Mi

''''lun.;,

I

P =

M,

R;'

(9. 1.22)

Ri is the mdius of the ith compactified coordinate. This is exactly as In gencral, the radii of the compactified dimensions do not have to same. As before, we also find that the mass spectrum of the theory by the presence of the compactified dimensions. By analyzing the lo"i"., we scc that the masses are given by (9.1.23)

N is the mass operator: (9.1.24) the

llillSS

spectrum is shifted by the square of an integer, as in

the closed string, however, we have an additional contribution to the . the extra complication that the closed string can Loop N; times the ilb dimension, much as a rubber band is wound around tube. It is important to notice lhat this configuration, wbich cona new term to the Hamiltonian, has no COWltCrpart in point particle p",tifi"tion . We thus have, for the closed string, Xi =

Xi

+ 2tx' pi t' + 2NI Ro

+ (a ' /2)1/2 L ~(a~e-2In(T-~J + a~e-21n{T+"')).

(9.1.15)

.", n

have two integers M; and Nil which represent the effect of compact ifor the closed string. The integer N I, in fact, desc ribes the soliton state wound around the compactified dimension an integer number of that this soliton stale is stable because of topology.

378

9. Heterotic Strings and Compactification

The shifted masses are now given by

1 IO-D 2 !clm ;;;;: 2" L(a t2 M;;R2

+ R2N;;a t2 ) + N + N,

;=1

where N and N are energy operators for the two different sectors string. In this discussion, we have compactified the 10 a torus. However, this is phenomenologically undesirable, be(:am~e::j:t:l us with N ;;;;: 4 supersymmetry and without any chiral fermions .• desirable to compactify on more sophisticated surfaces, such as a Lie group (see Appendix). We now turn to the heterotic . compactify on the lattice of E8 0 E8 or Spin(32)/Z2.

9.2

The Heterotic String

Let us now begin a discussion ofthe heterotic string of Gross, H " ...."',";.~4iq and Rohm [6]. We saw in the last chapter thatthe anomaly either 0(32) or £80 E8. However, we also saw that Chan-Paton compatible with exceptional groups. Thus, we are forced to use the of compactification to generate the isospin group Es 0 E8. The heterotic string is a closed string with the unusual feature that f·M",..,., compactification of the left- and right-moving sectors separately. l#:JtM:M moving sector, which is purely bosonic, let us begin with a '1.6--di!~~~j1 string where 16 of the dimensions have been compactified on a ., . 16-dimensionallattice we will eventually choose will be the lattice • or Spin(32)/Z2.) The right movers, on the other hand, are supersymmetric; i.".•. ;",,~_ tain the GS Majorana-Weyl fermionic field sa(r - a), where tlW.:miJ~ runs from I to 8 (in the light cone gauge), and also the field Xi: Left movers

Right movers Xi(r - 0') S"(r -(I")

where I, which labels the directions of the 16-dimensionallattLce;is~~1111 1 to 16, and i, which is the space-time index, runs from 1 to 10light cone gauge. Notice that the space-time Xi appears in both in the left- and rign.~~ sectors. Putting everything together, we now have the light cone

9.2 The Heterotic String

379

t

2, Jd' 1"do (a.. Xia~Xi + a~xJaaXl + iSy-(o, + a",}s). 4Jfll'

1=;

0

(9.2.1 ) action, we have to enforce the constraints that place the various fields

proper sector.

(a, - ao)x' y+s=

~ 0,

t(l tYltlS=O,

(9.2.2)

I y+ = .Ji{yO + y'). action is explicitly supersymmetric under the variation

.sX1 =. (p+) - lf 2ey 'S,

I

llSIJ = i(p+)-J{2Y_YI
(9.2.3)

us now Vlrite an explicit breakdown of the theory in terms of normal (notice that we have slightly changed the nomlalization of the spinors

of(3.8.8)). Xi(r - 0') =

ix' + ~/(t" -

ro

0)

+ Ii L

~=1

,

a" e- linfr - al ,

n -i

¢O

X i (,

+ 0') =

Ixl

+ Ipl(r + 0') + ~l L n=1

an e- 2in(<"+ol, n

S"("r -a) =

X'(r + /1) ==

(9.2.4)

Xl

ro

-I

" .. I

n

+ plCr + a) + 4i L Q'''e- 2in(T+a ).

nil"ly, we can read otfthe canonical commutation relations orlh.e theory from the action, From these, we can, in turn, write the canonical nutati,,. relations between the various oscillators: [x l. pl]

== i}jiJ,

[a~. a~] = [&~ . a~] = no", _mol) , [a~, ii~] = 0, -I -Jj , ,,, [ an,a", =n"._m •

IS:, S~I =

(9.2.5)

(y+II}·b o", ..",.

::: h is the chirality projection operator h =

~ ( l + YII)'

. ODe commutator, however, that is rather subtle. Notice that we have compactified degrees offreedorn to he left-moving. Thus, the

380

9. Heterotic Strings and Compactification

quantization of these modes must take into account this will be an inconsistency within the theory. Fortunately, the onlY~Q:~iii made by this constraint is to multiply the canonical commutation half: ..

[Xl, pi] = !i8 lJ •

. . . . .,.'..' '.0 This can be explicitly checked by using the Dirac bracket formulat~rf':' constraints or by checking that the newly defined commutator is cciM~t:~. the constraints.) Let us define the number operator for each sector: 00

( i j . 1 S-Sn, ) N = " ~ ct_nctn T in - nY n= !

"-j 00

~

N

~i = ~(ct_nctn

-I-I + ct_nctn)'

n=!

;:

This means that we can categorize the oscillators in both sectors · . Left movers

Right movers

a' a N

ai

sa

l

N

With these definitions, we find, as before, that the mass can be wq~~ '0 ' . ' • • _._~",.

I

16

2

1=)

±m2 = N + (N - 1) + - I)pJi. In addition, there is one extra constraint coming from the factth~· a closed string: the theory should be independent of the origin .. in (2.8.7). Thus, if Vee) is the operator that rotates a closed a parameter, then we must satisfy the following for physical stat~$;~ V(8) Iphy) = ]phy} .

For the heterotic string, the rotation operator is

Vee) =

exp

[2ie ( N -

N+ J -

~ ~(pli)

j.

This operator, acting on an arbitrary function of a , produces the V(8)F(a)V- 1(8)

To ensure that constraint:

Vee)

= F(a + 8).

1 on physical states, we must satisfy . 16

1~ I 2 N = N - 1 + - ~(p ) . 2

1=1

9.2 The Heterotic String

381

to now, we have not endowed lhc 16.dimcnsional lattice with any . To fix the theory, we take the space to be

= _R"

T 16

(9.2.14)

A

A represents an as yet unspecified latLice in 16 dimensions . ::' us span this 16-dimensionallattice by basis vectors: •

>

,

..

(9.2. 15)

e...

that we wilt compactifY the l6-dimensional space with a lattice simply thai if we walk in the direction l,J that is specified by onc oflhe lattice we cvcntually wind up at the same spot. With this basis, we want the coordinate to be periodic along any ofthese basis vectors:

"of""""',ri"g

+ 2nLI,

Xl = Xl I} =

1

"

,,2

I_ I

~Lni l!:RI'

(9.2. 16)

. the n l arc integers and RI are the radii of the various cOlDpactified Thus, if we wander off in any of the directions labeled by thc wind up back at the original position. This places a nontrivial the momenta. We know from ordinary quantum mechanics that . in the Ith direction. Thus, periodicity means operator, acting on states, must have an eigenvalue (9.2. 17)

~~:::~::;;~:~~:g~e:~nerates the displacement in (9.2.16) and makes a trans(I ' I the torus in tbe lth direction, until we wind upat the . J spot.

the canonical momenta must be defined as

" ,-,

pi = ..n'Lmlej' / R1,

(9.2. 18)

is an integer and the e' define the dual lattice A * such that

, PiJ

",., L.,e;e D

=

,-,

j

•

(9.2.19)

we should mention that the lauicc over which we compactify not totally arbitrary. First of all, it must be an even lattice because

'~~~:~::/(9~.;;2.13) due to demanding 0' independence of tho Hilbert U(B) operator in (9.2.11) can be set equal to ODC on the only if Hpj)' = integer, i.e.,

N

(pl )2 = even integer.

(9.2.20)

382

9. Heterotic Strings and Compactification

Therefore the winding numbers L 1 must lie on an integer, even The metric tensor for the lattice corresponding to a Lie group 16

gij

" / f = '~eiej' 1=;

we find therefore that the metric must be integer-valued and gji is ev~:n~:lSil the lattice must be self-dual (i.e., the lattice vectors are equal to . on the dual lattice) because of modular invariance, which will be(:pnijl apparent as we discuss the first loop diagram in Section 9.6. Thereru:·'e:i;i~ even, self-dual lattices. They exist only in 8n dimensions. In there is only one, r 8, the root lattice of Eg.-In 16 dimensions, meif~::~M two of them: ....

which fixes the lattice to be either £8 ® £8 Dr Spin(32) / Z2. Thus, : : . .. have very little choice in the selection of th~ gauge group. The metric for Es is as follows: 2

-1

-1

2 -1

-1 2 -1

-1 2

-1

-1

2

-1

-1

2

-1

-1

2

-1

-1

2

There are several ways in which to describe the root lattice for £Jl!.';'W~'W simplest is as follows. We have 84 root vectors given by

1:Si#j:s7, and 128 root vectors given by !(±e[

± e2 ± e3 ± ... ±

e8)

and 28 vectors given by

.-

± ei ± eg .

Including the eight members of the Cartan subalgebra, that giv'e~;:~I~:1t:~ 248 generators, which form the adjoint representation of the gr()"iip~:;:~ Later in this chapter we will also use the concept of simply . i.e., groups that have roots of the same length. The simply laced g.r-i;tii~ the type A. D. and E.

9.3 Spectrum

383

us brieRy summarize the kinds of Lie groups we wi lt encounter in our ,ssion of the heterotic string: Lattice

Roots

Even Self·dual Simply laced

{p''f = even; (g,j = even) A=A·;(tletg,j=l) Same length; (A, D, £)

lattices are important because the U(9} = 1 condition forces us to lattices, and modular iovariance (as we shall see in Section 9.6) to have self-dual lattices:

~' ~,en

UfO) = I _ Even lattice, Modular invariance .-,. Self-duallatticc. dimensions, the only even, self-dual lattices arc the root lattices for and Spin(32) / Z2.

now investigate the spectrum of the theory. First, we note that the no",,,,itl,i'n the theory are explicitly space-time supersymmetric. The of spacfHime supersymmetry can casily be constructed from the j:::,'~~~:fi:~e~l:d~'~. This. as we saw in carlicrchaplers, allows us 10 construct !r i w ith the sYlD.O'Ictry. We found earlier in (3.8.22) that the supcrsymmetric operators can be written as

(9,3, L)

• acts on the right movers. Wc can easily show, given the canonical Ubtion relations, that {QG, QIII = -2(hy" P,,)"".

(9.3.2)

,&"Il= the left-moving sector, we nOlicc that isospin is introduced in an different fashio n than the Chan-Paton factors. Contrary to the usual ' Pa'Lon factors, which simply multiply the amplitudes by the trace of a generators, the isospin factON emerging from the compactific8tion ,,,oiannorr.ic,,LI,, "p,mdthe number of isotopically allowed slates. Norbe index I is placed on the hannonic oscillator itself, which greatl y tbe number of allowed Slates of the Fock space. ";:~~~;:<; the twO sectors don', really communicate with each other, )I:: Fock space of the heterotic string will be the product of the righi-moving sectors:

(9,3.3)

384

9. Heterotic Strings and Compactificatioll

At the lowest level, let us first investigate the right-moving sec:tor; 4@Ji* in (3.8.13) and (3.8.14), the eight states in the fermionic la} or, ll:~~@iW ground state of the superstring can be written,in terms of each vw,a;t' la}R =

~(Yi sot li)R'

li R ) =

i T6[SO(Yf, Y+W la}R'

Thus, we have a total of8 + 8 = 16 states in the right-moving Now let us investigate the left-moving sector. We have -+ 8 states, -+ 16 states, -+ 480 states.

.

'

(It is very important to realize that this particular representatioIl9::fj'fijiijiJ trom breaks manifest E8 ® E8 symmetry. To seethls, notice that the'':te.l~H. operators, which correspond to the dimension of the root lattice n+', tl'lob: but these 16 operators do not form a representation of the group. Si~~It} vectors Ipl ; (pli = 2) correspond to the 480 roots of length this also does not form a representation of the group. Only and 480 and form 496 states do we finally construct the adjoint repr#~i'fi of the group, At the massless level, it is still possible to recon:lbiIu~:tli~:&i states into E8 ® E8 multiplets, but it becomes increasingly pr()hIl)itf~~~tJ!dR mass levels. Later, we will give an argument, based on demonstrating how we can show that the entire spectrum is, in !Clef.,:! symmetric.) At the massless level, we obtain the spectrum by multiplying left movers. The total number of states in the lowest level of the h~!~'li'~~~II is thus the product of the left and right movers: 16 x 504 = 8064 ' breakdown of these states is now given as follows. Of the 8064 st~i¥111 that 128 belong to the N = 1, D = 10 supergravity (in the light,

supergravity -+ &~l 10}L x

Ii or a}R .

If we break this down into distinct states, we find graviton -+ &~ IIO)L x Ijh antisym. tensor -+ &~l 10) L x Ij) R

.

+ (i -

~ j),

(i ~ j),

gravitino -+ &~l IOh x la) R •

Other states belong to the super-Yang-Mills theory defined on:l!.R:JIi!9:~~ example, the super-Yang-Mills multiplet (A~, 1fra) is in the adjJoinli::~~ tation of the group, which has 496 states. These 496 isotopic staties by the 480 + 16 states contained in the left-moving sector: ::::::::7&

&~110)L

+ Ipl;(pl? = 2}L'

9.3 Spectrum

385

"',,em,dtilplot can be represented as 496 x 8 states: super-Yang-Mills -) [a ~ 1 )Oh

+ Ipl)l) x

Ii ora)~.

(9.3.9)

we have exactly JO-dimensional supergravity aud Vang-MiUs ~

o.

>t~~~I~~\~~';~~~~~:~;::~::: ::I~~ 16 to yield an irreducible representation ~I This means that, in general. the Fock space of the helerotic string manifestly recombine into E, ® E, multiplets. At the next level, for , the states form irreducible representations of the algebra only in the

of the calculation. . the next level, where m 1 = 8, for the right-moving sector we have 128 '. and 128 fcrmions : 128 bosons

I a~dj)/C· I

. I 28 'icnmons

~I Ib)II'

a~ l la)II' N .J_I I i)~ .

(9.3. 10)

that we have fonned bosons out of the tensor product of two fermions

above equation. ,lcft-.no·" " 8 "''',''~. considerably more involved, with a total of73.764 ' . We bave a tolal, therefore, of256 x 73,764 = 18,883,584 states! The

"" scalars -+

Ip' , (pl)2 = 4)L -+ 61,920 slales, al. 1 Ipi , (pi)2 = 2)L -+ 7680 states, a~la:1 10h -+ 136 states,

(9.3.1 1)

a~lIO)L -+ 16states, of 69,752 scalar states. (To count the states in Ipl. (pl)2 = 4), we "Iact thaI the number of points oflengtb squared equaI to 2m for integer lattice r l6 is 480 times the sum ofllie seventh powers of the divisors for this states, there are 480(1 + 21) = 61.920 states r6J.) it is not obvious ataIl tbatthese 69,752 scalarstate.s can be rearranged of E, ® £8 multiplels. However, a careful anal)'sis shows that the)' •. b,ok,," down into the following £8 ® EB multiplets:

1)+ (1. 3875) + (248. 248) + (248. I ) +(1.248)+ (I. 1) + ( I , I). (9.3. 12) vectors in tbc lcft-moving sector arc arranged as

I

ci'_1 p I , (p')' = vectors -+

[

2)/. -+

3/W0 statCS,

a~2 10) I. -+ 8 stales,

a~IQ~[ jO}L -+ 128 states.

(9.3.13)

386

9. Heterotic Strings and Compactification

and we have the tensor state

a~la~l 10) L ~ . 36states. The total number of states in the left-moving sector is thus 73, At higher levels, the degeneracy climbs exponentially as d(M) ~ e<2+...tilrrRM.

It was not obvious that these states can be reformulated E8 ® E8 multiplets. At higher levels, it seems like a hopeless that higher masses all can be rearranged into irreducible repiffit~ of the group. What we need, of course, is a higher "vrnrrlp.fr'{.i-:'itH this to all orders. We will do this when we apply Kac-l'tlO()dy Section 9.7.

9.4

Covariant and Fennionic formulations

Although we have analyzed the heterotic string's spectrum in me~·-:ItJ~.f: gauge, we can also write down an explicitly covariant quantized action. We begin by writing the right-moving SUI)efl;yn:$J~ tor in covariant form. Let Pi' and QU represent two of the geller.~~!~~ super-Poincare group. Let us generalize the operator that tions in the x direction. An element of the supertranslation I!fciui:i:::ia: by

where () is a lO-dimensional spinor. This operator will shift a . the coordinate by X and a fermionic coordinate by (). Now . 1, 2)

IIO' =

h-1ouh

= (oaXIL -

i8ylloaB)PIl

+ BaBQ .

•.••

Now the right-moving GS action can be written as SR =

f d2z~eTr(naIIp)e:eap + f + f d2zeA++(e~naf.

d 31;t:aPy Tr(nuIIp ': l:~:~i[;:::::@

The first term on the right is the usual quadratic term of the GS • ~.'f¢JJ![@ second term is the nonlinear term of the GS action written as a We':~~m term. It is a three-dimensional integral over a surface whose b011Mt~~ cides with the world sheet of the string. The sum of these two :~:~mH alternative formulation of the GS action. The third term of enforces the right-moving constraint.

9.4 Covariant and Fermionic Formulations

387

the left-moving action that contains the isotopic sector can also

a choice of using eitherfennions or boson!. from the of Lie groups that the generators of an algcbru can in terms of the product of either bosonic or fcrmion ic fields. Thus,

51. =!i

f dlle", /aoifi'e~,

(9.4.4)

Ihis time I = I to 32. which labels the fundamental representation It is important to note that these fcnnions transfonn as Lorentz

The I index is an inlernal index. We can, of course, choose either

:

;~~~~:~:~;O~'~N~,~.ve:i~u-1>:' chwart.: (antiperiodic) boundary conditions on oscillators. we can represent the same [cft.moving isotopic (9.4.5) = J. 16. This, of course, yields the light cone representation we used when we broke Lorena covariance. (It may seem strange at first to left and' sectors labeled by ±. which seems to single out a oon",.,tiO? io cwo dimensions. However, the above fonnulation reparametrization invariant because the ± directions are placed in thc space. Thus. two-dimensional repammetrizalion mwriunce remains let us write the entire space-{ime action (minus the isospin part) that both left- and right-moving sectors in a covariant fashion:

/ d 2 (:e

H(e; iJg x"i -1; ljrl-' p - e': ilg W" + ~ i(Xgp' pit w,ll)ajl X",}. (9.4.6)

<etion has ;'N = ~" supersymmetry: !e~

=

il:p_~o,

6X .. = -2Vg E. 8X" = ;E1/II', l!l/J/I. =

(9.4.1)

{a.. x" +; x.. \11'" p"]t.

us, instead of choosing the light cone gauge, choose the conformal the heterotic action reduces to

(9.4.8)

of all this was to show that a covariant version of the heterotic exists and thltt we can use either fcnnion or bosonic fields to rcpresen! port of the field". We are not bound to a light cone formulation in on"",," fields.

388

9. Heterotic Strings and Compactification

It should be noted, however, that even in covariant •...,LU"'U<~U"":. string appears a bit awkward and contrived. Perhaps a future theory will allow a more elegant formulation.

9.5

Trees

Trees are constructed in the heterotic string almost exactly the ",.·'a·:ii::tl~ constructed for the light cone superstring, except now the veltlces:~[6 and right components and we also must take into account the corrip~(ifUIl on a 16-dimensionallattice. t1:L~ji~iml To describe the supergravity multiplet, we will introduce UtL U to represent the spinor field and p/lU to represent the of the graviton. In the on-shell light cone gauge, we will particles to be massless and transven;e:

p+ll = pl1+ = k/lp/LV = 0, U+<1 = ktLUM = y+U/l<1 = ~(1 - YlL)U/lU = O. In analogy with the superstring light cone vertex developed in C~~Ii~m can construct the vertex functions for the heterotic string based on .ilj.MlJijj the vertices in (3.9.2), except we need an extra insertion of PI' ttHclijlltl:! Lorentz indices. In addition, the vertices are the direct product and right-moving vertices. We now choose for the boson and vertices for the supergravity sector:

VB

PtLv(k)

supergravity:

1"

da 8 11 pVik-X,

= 111: daFUpvUaV(k)eik.X,

VF

where we use the definitions in (3.9.6):

k+ = 0, B i = pi

P'. =

+ ~kj Rij, i

dX = -Ip'' + dCr: - a) 2

L' eOl'

n#O

2'In (t-C7 ) ,

n

Rij = kSyij- S,

" Fa

= ~i(p+)-1/2[Sy . P _ !:RijkiSyj:]a,

pi = ~ pi

+ L ti;,e-2in (T+al. 11#0

(Notice that we have explicitly chosen the frame where all k+ where we have a large simplification. The calculation for nonzero

9.5 Trees

389

""ml,I;,,,,,,'" because it requires making a Lorentz transformation with Lorentz generators M- 1 .) vertices for the gauge mesons can al!':o be written down. Oue compli· however, is that the 496 vector mesons in the adjoint representation

gauge group can be broken down into 16 " ncuemr gauge bosons us members or the Cartan subalgebra, and 480 "charged'" corresponding to K' with (Kif = 2. The neutral ones are defined (9.5.4) cbarged ones by (9.5.5)

must have two types of vertices for the gauge bosons. The vertices mesons are essentially the same as beforc. except ilia! we index for an internal one:

v~ =p~(k)

vJ = pI =

f' d(1F"PIU(JI(k)ejJ;~X",

dX' = pI -Id(r + cr}

,,::~";~:~:r.;;::::v;:,:rtices 1<

1"daBjiplt!k~X~,

(9.5 .6)

2::ale··li~(r.:.."). nf.{l

"

for the emission of the charged gauge particles,

internal momentum K f, can be: found

v:

=

p~(k)

lif

daBIJ. eit_X-: e2iK'X':C(Kj,

Vf = Io n daFtJUU(k)it. X~:e2iK'X':C(K), (9.5.7)

ith.none,11 ordered part of the venex comes from me left-moving sector define C later. way of checking to see thai we have the correct form for the vertex is to explicitly operate on them with the supersynunetry operator to verify that boson vertices tum into fermion vertices and vice versa. the proof, which follows the proof given in Section 3.9, is rather and will not be presented.

of the system can also be found easily by generalizing the I :;;~~J.:~a:~,:;,'; We will. however, find it convenient to absorb the a ':c of the vertex function and put it into the propagator. This be done because the a displacement matrix. as we saw earlier, is

390

9. Heterotic Strings and Compactification

given by U(a) and the r displacement operator is generated by Hamiltonian: 16

H = ~p2 +2N +2(N -1)+

L(pii. [=1

The vertex function, before this extraction, is V(k, r) = eiHr

If(

daU(a)VoUt(a)e- iHr .

The vertex function that we want is

V0 = V(r =

0" :-

0).

The propagator, which now absorbs the a and r integrations, oeC;IQJ:m

6. =

00

1

dr

o

=

r 1

If( 0

da

_e- Hr U(a) IT

d2Zlzl(l/4)p2 _2ZNZN-I+(lJ2)LI(pl)2

1zl<1

=

~8 (N - N + I - ~2 LCPl)2) . H I

The above propagator is just what we might have expected. tioo simply enforces the constraint (9.2.13) that makes the of a rotation in 0". and the poles are given by the inverse of ll;l.1'<',:.111 Hamiltonian, ' The N -point function can therefore be written as (0, kll V(k2)6. ... 6. VCkN-d 10, kN) .

Let us now construct the amplitude for the scattering of four bosons with momenta k;, POlariza:ion Pi, and charge K j , wherel:
Lki = 0, ;=1

K; = o. The calculation of the four-point function is long but

9.6 Sing\e.Loop Amplitude

K

= -~ [stP, · PlP'l' P"

391

+ SIIP2' P1Pr . p" + IIIP, . P2P3' p.l

- ~r{PI · k 4 Pl . 11. 21>2 · P. +P1·kJP, · k,P , 'Pl

+ Pr - k.}P4 · k 1Pl ' Pl + Pl' ~Pl' k,p, . p.l - ~1(p, ·k,~ · k 2 PJ · p, +PJ ·x..PI ·K.2P2 ·P.

+ Pl·I4PI • k,P) ' P4 + P3' k , P4' k 2P2' p.] + PJ' k.P2· kiP, . P. + PI ·k.Pl ·k)p)· P. + P3 ·k2P. ·k,P, -Pl] .

- !U[PI . klP,, ' k,P) ' P2

(9.5. 14)

is a phase factor and

s = - (k , +kd. r = - (11.2 + k))2, II

= -(k\ + *))1, s+t+u=O,

S = (KI

+ Kd ,

(9.5 . 15)

+ K)2, U = (K, + K )}1, T = (Kl

S+T+U=8.

Single-Loop Ampli tude ill ',",of "heth"o,),. however. is the calculation ofthe single-loop diagr.un : where we demand Ihat the theory be fini le. It is now a straightfor'"'''''' to wrile the single-loop diagram in terms of these vertices and

A""

= T,(6 V(N)6 .. · 6V( I».

(9.6. 1)

consider scattering by charged gauge mcsons. • the trace calculation is loog but slnlightforward. After the trnce is we have

A.ioop "" i K

f Ii d2zilwl-l[_4n]S w-I

f(W)-24

1.1

)( n

1 9<j~'

In Iwl [x(eJI. wW'f2Jt··tj (';(Cj/, w}]"'r K, I. .

(9.6.2)

392

9. Heterotic Strings and Compactification

where

[ln2IZI]

w =ex

p 2lnlwl

X(z,) .1,(- -)

=

'I' Z, W

L(w, Zi. Kd

"" ..;z n

l- z

(l-wmz)(l-w", , ' : (1-

m=1

. ::.', '

[ln2z] l-z n"" (0- wmz)(l-w ,,;z w

x e p 2 In w

=L PeA

m=I

(1 _

·'

m

m

.

exp [41n W (p - t lln~ Qi)2] , ;= 1

nw

and where i- I

Qj ==

LK

j ,

j=1

loz)

= L - 2., j=1 rr I j

Vi

lnw

i=--

2rri'

ejl

= ZIZ;+I ..• Zj'

where the sum over PEA represents the sum over all points in'::'tl}im K is a kinematic factor identical to the one found for the cause the final result bears so much resemblance to the amplitude, it is not hard to show that the amplitude is mvari,ant.:W'i:4] ::::::~:m

J Vi ~ + I, l Vi ~ Vj + i. Vj

Slightly more difficult is the proof that the integrand is j'IMl.ljWi;@~ + 1 and i ~ - i -\ • which is necessary to prove modular ttwwa, Fortunately, most of the terms in the integrand are j'(lerrtic~IH~ in a single-loop amplitude in (5.5.1). However, we must vm"'I1'.: ~ of the terms that are different, i.e., the factors dependent on As before, the theta function, if we make a modular trrurlstiq:mt~ arguments, transforms as follows:

i

8

1

.(

V ...

ci+d

ai+b)= (ccr

+d

S

i

+ d)l/2 exp (-irrcv2): - d a + . ... .';~:.:~...,u.

where SB = 1. Since the X function can be written in terms .. : we have

ar+b) = cr+d

1

la+dl

:'.. ' x(vli). :

. ::

9.6 Singl e-Loop AmplilUde

J9J

"rt;,;o,n nlDction few) also transforms as f(w ):::: (i / T)'fl w -

' [2 W, I/ 24 f(w'),

lnw T :::: -

- 1

- 2lTi

r

In w'

-:::: -

-f-

2lri

(9.6.9)

important transformAtion is the one for L;

(-2>1' ) ' ~ exp [2>1"w( t;:"Q.lntn '-nri )'

'"

;: L(w. ZI . K,·)::::

W

+

tn

P-

l.i

2 ~W (t,Q'ln"Y]·

(9.6 .10)

identity depended crucially on the lattice being self-dual, which is of the strongest arguments for this restriction on the Janice. Let these factor.; togeth.er:

n

1Jr(i'i j ;l r l K,.K' L(t ,vj)

:~. (w'r' f(ii/)- 24 e2[ f1 ,,( -~Ij ":9~!>4

::

::·

.::

·· · ··

- I

L (T

'

T) (iT s exp -VI

T

( (. )') ilT

T {;; Qli'ii

.

(9.6. 11 )

factors in (9.6. 111 iliat apparently violate modular invariance, metors of T and the e~poneutia l s involving r. Fortunately, both sets ",n;sh. The factors of r cancel because £\\'0

us with r '2-4-1

:::: I.

The exponential faclOrs also vanish because

eliminate the factors of f and thc exponentials of t in (9.6.111, we this combination is modulnr invariant. "''",,. of the alDplirudc under v, -4 VI + 1 and Vi -4 V; + T means truncate its region of inlcgration: 0< 1m VI < 1m" -! ...-:: Re v,....-::~.

(9.6.12)

9. Heterotic Strings and Compactification

394

Also, as in the case of the closed single-loop diagram under r ~ r + I and r -+ -1/ r, we are free to choose a hlI:ldalll'¢:iiUti {

-!2 -< Rer < ! - 2'

Irll.

,

Once again, the choice of this fundamental region allows potential singularity at r :::: 0, and hence we have a finite On(Hol()lj:.~ In this discussion, modular invariance was crucial in showing u.~~=I tude, like the usual superstring, is finite. We can simply choose ~ region where no singularities are present. However, the calcula~i~~;ijfi precisely why the theory is modular invariant. Let us amIlYi~e,j~.~ case and isolate the point at which modular invariance comes matters, let us consider the vacuum one-loop amplitude with Let us first define a function F (which appears in the trace q~~i,ijjij

(9.6.1)):

L

F(r, X) =

e-irrr(L-XY,

LEA

where we sum over lattice sites, X is an arbitrary 16-(lirrlensi(JtUalf:v(i~i root lattice space, and each site is labeled by integers n,:

L=

Lllief.

Notice that F is periodic: Fer, X)

=

F(r, X

+ ed

because this shift can be simply absorbed into the integers n;. this function in terms of its Fourier transform F:

L

Fer, X) =

e- 2iM .X FCr, M).

MeA"

Notice that, because F is periodic, the M's must necessarily

lattice: 16

MI =

L m e7 i

1

•

;=1

Now take the reverse Fourier transform to find F in terms of dl6X FCr M) _e2ircM·X F(r X) .>.

M

f

'-.JT8T

"

where is the volume of the torus. Now insert the eXT)re~)si(jnJ-ot:f.~ previous equation. Perform the integration, and we arrive at for F:

- M) = -1r - 8 em · M'/ r. F(r,

M

9.7 £1 and Kac-Moody Algebras

395

would likc 10 insert this identity into the expression thai actually the one-loop vacuum c.1leulation. We must calculate the traee over l.:.~!:,~'~~;; which contains the piece (pI? Thus, in the lrnt:e calculation M f appears: f(r) = ,"

L

e - lttrL.L.

(9.6.21 )

'" 14). we have F(r, 0) = 1(,), ......

;. ..... substitutc the expression for

fer)

= -

=

F. and we have an eJtPression for f:

I ( I)' L

./fiT

- -

'

(9.6.22)

e'tf.,.M/ r

MEr. '

~r(-D·

(9.6.23)

is ooLhing but the usual f defined in (9.6.21) ovcr the dUll/lauiee, key result. that the modular transformation, _ - l iT has il t/redua/lanice. Thus. for modular invariam:ewedemand Janice be sclf-dual. In fact.., this is the origin of the condition that the self-dual. Here, we see the tight link between self-duality (which us cithcr E, ® £8 Spin(32)/Z2) and modular inNlh.ough the original choice oflhese groups came from the anomaly "~;:~;we scc that we need precisely these groups for modular invariance lit' . of the nmplirudc.

";:;';;:;;;;,;we::,:s;ee

restricts to

re-

or

; £8 aDd Kac-Moody Algebras i; wo ""V Ih" thespcctrum ofstates was nol manifestly £s®E~ invariant.

that

the lowesT-lying stales could difficulty were we able to show in irreducible representations of the group. Because the number rapidly proljfcrates into tens of millions, it becomes prohibitive (""'Hnl!' the SIllies into E. ® £8 multiplets. section we will use the techniques of Kao-Moody algebras (8--10] in Cllapter 4 to show, to all orders, thai the heterotic string docs :.have a spectrum that is E. ® E, symmetric. We will use the vertex .. defined in that chapter to geoerate the representation of a .K.1C'all!eb" (which wo rks only if the lattice is even. self-dual, and the level I).

to

;:~~~:~~:;~ that we desire is the Chcvalley basis. where the 496 up into 16 mUlually commuting generators (which form

i)J

396

9. Heterotic Strings and Compactification

the Cartan subgroup) and the 480 root vectors. Notice that obey the condition

fnf1:,:-I:/;'i·'....t-::,

[pI, pI] = O.

Trivially, we can always represent the Cartan subalgebra as sinlply~f~j~11 commuting vectors pl. More difficult, however, 1s the COllsttllC1ti(;jifl!ij~ root vectors. The simplest operator with 480 states is the vertex 01p()rator,J.~l~111 write in terms of a line integral surrounding ilie origin: E(K) =

J 2d~

V(K, z)C(K).

r

nlZ where the vertex function V is defined over the lattice (rather time):

tlL~~%iI~~

and where

(Kif = 2, Z=

e2i (r+
where the cocycle is as yet unspecified. Notice that the definition, point in 480 directions in the 16-dimensional root Iattt~:l We now demand that the 16 elements of the Cartan subal:gel:'~'i~g. 480 elements of root lattice space obey the commutation rel:ati()n$o.~::" This will, in turn, fix the C operator. We demand 2::m~ s(K, L)E(K [E(K), E(L)]

=

{

:1 pI

+ L)

if(K

+ L)2 =

if K +L = 0, otherwise,

and [pi, E(K)] = K' E(K),

where B( K , L) are the structure constants of the algebra with In some sense, we have done nothing yet. We have simply commutations of the algebra of E8 ® Eg in the Chevalley V""",,,, known, and then demanded that our ansatz satisfy them. The that a solution of these equations actually exists which fixes, For the moment, let us drop the factor C and see if we:'!;:;@;:.~ commutation relations. It is easy to show ' V(K, z)V(L, w) =

which is valid for

Iwl

<

IIII

(wZ)-(l/2)K.L(Z _ w)K.L:e2jK'X(z)+2i~.A'.W'(

Izl·

9.7

E~

and Kac-lv1oody A!gebras

397

the w integration is performed such that Izi > Iw l in the first tcon and in the second. By carefully exam.ining the 1af.1 expression, we find satisfied all the identities of the ChevaUey basis eKcept that the are all wrong. Anticommutators arise instead of commutator:;, That we had to introduce the factor C(K). The condition we must impose is C(K)C(L) = ,(K , L )C(K

+ L).

(9.7.9)

operator is called a "cocycle" or ''twist'' and was added into the of the generator E(K) in orde r to get the right statistics. Acting on ;ofmc>m'''"' P • we can show that C( K) has tbe property C(K)lp) = ,(K, p) Ip).

(9.7.10)

. demand that this law be associative, then we have a restriction on . By acting OD states successively by the C's and demanding arrive at ,(K , L),( K

+ L, M ) = , (L , M),(K , L + M ).

(9.7.111

these the two cocycle conditions. Several explicit representations of the phase exist. We CaD always choose s( K , L),(L, K ) = (-I)" ', .(K , O)=-,(K , -K ) = I.

(9.7.12)

"I :~~::~~:i we show that the generators of the Lie algebras can be ::cj of the Cartan generators HI and the eigenvectors Eu. We now exact correspondence between the operator!> appearing in the heterotic the generators of a Lie group:

Heterotic

Canan-Weyl

p'

H,

E(K) K'

E.

(9.7. 13 )

a

:: last step in thc p roof is to notice that thc 16 pI and the 480 E( K) ", thc vertices of the thea!)' (e.g., see (9.5.6), (9.5.7)), so therefore the ', ,' consists of these operators acting on the vacuum. But since these together make up the 496 generators of £8 ® E8, the entire spectrum be arranged according to irreducible representations of E~ €I Eg.

398

9. Heterotic Strings and Compactification

9.8

Lorentzian Lattices

So far, we have been discussing heterotic strings in which ,. ' ••. compactified from 26 dimensions down to 10. Then, , ..........''''''' ventiona1 wisdom, we must compactify ,the 10-dimensional SPil'¢':~ii D-dimensional space-time. Yet another intriguing DUl.5Ulllll;Y]lS compactify the 26- and lO-dimensional spaces down to D al1lrle~!$iOijji the very beginning, bypassing the intermediate stage. This is ....... 4I;1l Namin [11,12], which yields gauge groups of rank 26 - D, wJj,~~lI;:ti than the £8 ® £8 considered so far. Let us begin with the two sectors that are 26- and lU-dlImens:iOll:ani compactify them both down to D spacertimedimensions. sector has 26 - D = P dimensions, and the right-moving sec:tor'ha:U:tj%,~ dimensions. Now parametrize the compactified dimensions as ' •: : : •

XA =

qA

+ 2L ACt' + a) + L ~a:e-2ill(I+(1) ..LO . nr

xB

=

qB -

2[8(r - a)

2n

..

, .. . . , ....

+ L ~a:e2in(r-(1),

,

n=f.O

2n

"

'

where A ranges from 1 to p and B ranges from I to q. the relation p = q + 16, which guarantees that the number Oflun:(~jjg space-time dimensions for both sectors is D. Now impose the constraint that Lo - Lo annihilates states, '<;'i f~:H,;~~ preferred a origin for the closed string. This yields the COlldl1tJ.or;\~:

t(e - p) = N -

N

+ 1=

integer.

The mass relationship is now given by

!m

2

= ~k2 +

!P + N + N -

1.

Notice that these equations all reduce to the usual heterotic p = 16, q = O. Now let us check that the final result is modular invariant. new restriction on the lattice, which at this time is still loop calculation (9.6.2) done for the general case is almost identii~,~; for the usual heterotic string. A careful analysis of the terms , ., generalized case contains the new factor '

L e-l1tf(L-r:!~1

Ki V;/ff+i1tr i}.

Now calculate the change in this factor under the It follows that we pick up a new phase factor

tra~Sj'onna1~011i'~~~;;

9.8 lorenttian Lattices

399

to cancel this term, we are forced to have (L~} -

(llfi =

even integer.

(9,8.6)

extra minus sign, the mctric on the lattice is "Lorenttian" rather (For the usual heterotic string, L is zero, so we never sec thc behavior of the lattice.) when weappty the modular trnnsfonnation T -+ - r - I , we find thlll must be self-dual. In summary, modular invarianc.c at the one.loop is preserved if we have an even, self-dUll/. Lorentzian lattice. ,th;,""",,alily. it can be shown that such Lorentzian lattices do indeed when we satisfy the condition p - q = an, for integer n. For our case, caD also calculate the number of parameters in this lattice. It can be that the Lorentzian lattice is unique up to an SO{p, q) transfonnation. mass fonnulas we bavc derived are invariant only Wider SO{p) and the total number of parameters is dim SO{p, q) - dim SO{p) - dim SO(q) = pq.

(9.8.7)

the total number ofparamcters that chnracterize the lattice is p(p - 16). have considerab ly expanded tbe nwnber of groups that are available . :. by this method of compactification. The total rank of the group is now . : D. This means that we can have groups as large as SO(52 - 2D). In :. dimensions, this means S0(44), for D = 4, we can also have : . £ 6 ® £, ® SU(2), or Es €I E~ ® G, where G can be either SO(4) or . We can also have Et €I E. ® G10-J, where G must be simply laced. this we may conclude that Lorentzian lallices are an entirely new Ute heterotic string. Actually, this is 110t qui te right. It rums represent the Lorentzian lattice within the conventional .. ,pict,""oftb" hc"',oricslri"g. Consider, for example, the followingstri.ng '. in the presence of background fields:

(9.8.") goes from 1 to 16. Bl j is an antisymmetric tcnsor, and 6~ is an

[~~~.~

~

tensor in 2-space. Now assume the 81l> Bj j • and Af fields are by constant background fields. Notice thai the total number . in this approach is easily found by counting the number of modes within these fields:

;q(q+l) 16q

I

(9.8.9)

iq(q - I) number of pq parameters. Now quantize this system, assuming that approximated by

Xi = 2all '

+ q'(r).

(9.8.10)

9. Heterotic Strings and Compactification

400

Now substitute back this expression into the action, quantizing<, the presence ofthese constant background fields. The net effect '" ', ', ground fields is to add pq new parameters .into the ' ,, , which is precisely the number of new parameters intrr(O~d~u~ce~:d~':l':~!IE pactification on Lorentzian lattices. It can ' be shown, fi conventional compactification in the presence of these ba(:kgrOtmd:Th fact, equivalent to compactification on Lor~ntzian lattices. In summary, the Lorentzian lattice not only is equivalent LU'<1LUe;::e: tional compactification scheme when we include the presence ~ ... ,.."",~",.. fields but also provides a very convenient framework in which tO~i~~~ enormously large class of compactifications.

9.9

Summary

The anomaly cancellation requires that we have either -v--, M:::li{'&i However, the Chan-Paton factors are incompatible with ex(~p@l~:;ii Thus, we have no choice but to consider compactification as the::\Si:i which we can genemte exceptional groups for the superstring. '" The simplest possible compactification process in one diIl!lelllsJ~!:rejj that we make the identification

x = x +2]( R. A one-dimensional scalar function must then be expanded in teri:rilt1iffj eigenfunctions:

¢(x) = L¢neiP-", n

where n P = 'R'

We see that momentum is quantized by taking periodic b01~nclat'lr:C(Jj] The heterotic string, by contrast, compactifies 26 d~'~~~~:~i~~~.:jlll dimensions, leaving a 16-dimensionallattice. It is a closed arates the right- and left-moving sectors. The right-moving se"t¢t;:~ I O-dimensional and contains the fermionic superstring and the 1:)'~~~~tm The left-moving sector was originally 26-dimensional but wa,S:' ,~~i~~~ down to 10 dimensions, leaving a 16-dimensional theory define(l::~.~~1 and a lO-dimensional bosonic string (which combines with tbe¢',i~~ bosonic string''to complete a bosonic closed string). The finalil·.(~,r.~~ light cone gauge, is

S = --1 - , / dr 4]((¥

!orr du ((JaXi(JaXi + L16 (JaXlaaXl + i 0

1= 1

9.9 Summary

401

,,,.eenl'o,,,e the constraints

(a r y+ S =

-

a" )X l

= 0,

!(l + rll)S = O.

(9.9.5)

aclion is explicitly supersymmctric under the variatlon

I

;;X; = (p+)-L/2 sy i S,

oS" = i(p+)- L/l y_ y,,(8r

-

O,,)XI'C.

(9.9.6)

we analyze the spectrum of the heterotic string, we must take into the quantization of the momentum and also the winding of the : ""m,d the compactified dimensions. This yields the conditions on the

im2 = N

" l l. + (N - \) + -I 2:)p

(9.9.7)

2 1="

;'~~~:':."~~:,:~:'::~~s::p::e~'~ttum~,ariscs because we want to make the closed ro:ation. The opemtor ilia! generates this

:.i

(9.9.8)

N ~

N - I + -I L" (p')'. 2

(9.9.9)

1=1

ftmctions can also be conslrUCted for the heterotic string. They are. simply left- and right-mm-jng products of the usual slIpt:;r.;tring verwere constructed earlier .in Chapter 3. We choose the ansatz for the im"ily multi plet:

VB = Puv(k) VF =

1"

1"

duB"pve'H,

daFaPvUV"(k)eiA-X,

(9.9. 10)

(9.9.11)

402

9. Heterotic Strings and Compactification

Fa

= 4i(p+rl/2[Sy. P - t:RjjeSyj:t.

pi = !pj + L&~e-2jn(r+".). n#O

With this vertex function and the usual propagator (plus COrl$tn~~~J[Mi calculate the four-point function for the scattering of four ll1A~$~f~

~lIillrnli

bosons:

where K is a complicated function of the polarizations and defined as S = CKI

S; .. ".·Z;",.= ..

+ K 2)2,

+ K 3i, (Kl + K 3i,

T = (K2

U=

S+T+U=8. Similarly, we can define the one-loop amplitude, which caI(bi~tii shown to be modular invariant. Thus, we have the freedom singularity at i = 0, which means that the theory is one-loopI ~~}~~ii~ When analyzing the spectrum of the theory, we found til hibitively difficult to construct the spectrum in terms of E8 ® "-:'<1:.:_ The proof that the spectrum can be represented in terms·· representations of that group can most easily be constructed l1,,;,ii:O':;'&';' Moody algebras. The Kac-Moody generators have the relation: I f Til - I'flilTm+lI [Tm' 11 -

+ k rnadj am.-n· ~

Notice that this appears to be an ordinary Lie algebra that has over a circle. We can also rewrite this algebra in terms of representation, in which the generators look like [Rjce), Rj(e)] = o. [RI(8), Ea(8 ' )] = -2nS(e - 8')ll'.jEaCe), and [Ea(e), E_",(e')] = 2nS(8 - e')

L alRiCe) + 2rri8'ce -

and [Ea(e). E{3(e' )]

={

2nS(e - e')E

0

a+

(3(e)

if ll'. + ,t3 E r, otherwise.

Using vertex operators, we can explicitly construct a replresenti~ff~ Kac-Moody algebra. Thus, since the vertex operators generate .

References

403

theory, the spectrum itself mus t b~ invariant under Kao-Moody trans~::;;:~:. Therefore, even though the spcctnun is not manifestly Es ® E, we have establishcd thai it acruaJly is synunetric under tha t group. coropacti.fication schemes more general than the one presented are by using a different compactificatioD scheme down to D space- time

w

!~;~o;.~:B~Y~,direCtlY eompactifying the left -moving sector from 26 down ~.

di mensions and compactifying the right-woving sector from to D dimensions, we can bypass the inte rmediate step of the E. ® ,,,,,mt"i' string. The new feature of this comp:1etification scheme is that invariance restricts us to Lorenrzian lattices, Le .• lattices where the

~:,~::~::;~;:~:f signs. It can be &hown that such Lorcntzian lattices arc ~

defined. This thus gives us an entirely new class ofheterolic- rype

with gauge groups as high as SO(52 - 2D). Although these models diffe rent from the standard heterotic string. it can be shown that, background ficld s gji' AI. and Sf}, wc can gencrnte these new

O. Freund, ITP prcprint, \984. Kaluza, Silzllngs/er. PFE'USs. Akad. fJ1.1s. f
CHAPTER 10

Calabi-Yau SpaCeS and Orbifolds

10.1

Calabi-Yau Spaces

Although the heterotic string is a great advance over the usual forml;i}ijlfj string theory, the question still remains whether we can break ~~~r~111 to four dimensions and do rigorous phenomenology. The answer, is no. At present, our arsenal of techniques is still too primitive question of whether the theory undergoes spontaneous dl·lffie:nsl.0nl~ We will have to wait until further developments are made in or M -theory before any conclusion can be reached concerning tht( . of the theory. . Surprisingly, it turns out that rather mild restrictions on sufficient to obtain reasonable phenomenology. Although none so far agrees totally with the minimal model SU(3) ® SU(2) ® remarkably close with just a few assumptions on the classical vac;uit Unfortunately, however, we find an embarrassment of riches. to be thousands or even millions of possible classical solutions, an{Ht:i~ to be seen how to choose one vacuum over another. Thus, au.J.'~~J phenomenology can be performed on the string, we still have development of a nonperturbative formalism before any definiliv¢,: can be made cOllcerning the true vacuum of the theory. In this chapter, we will thus assume that the cOInpflctificfltioofi~~~!.!~ formed and will discuss two methods of writing the classical vai:fMM string theory:

(1) Calabi-Yau spaces. We require that N = 1 supersymmetry inf..otU@(ft! sions be unbroken. This simple assumption forces us to cOllsil:letm~g

10.1 c:alabi-Yau Spaces

i

405

. This. in tum, forces the six-dimensionaJ

to be a Calahi-You manifold (I. 2]. We compactify on tori divided by the ac tion of a discrete group.

us to break the gauge group and arrive at different low-energy (Orbifolds are probably spccial1imits of Calabi-Yau

spacc.~,

Ihis is not totally clear.) us begin, however, by taking the zero slope limit of the theory, which D = 10 supergravity coupled to Ea €I £ 8 sllper-Yang-MilIs theory and make some reasonable assumptions about the breaking

Horowitz, Strowger, and Wilten (I , 2] made (he following a.bout the zero slope limit:

the IO-dimensional universe has compaclified down to a product of

four- and a six-dimensional universe: ( 10. 1.1 ) 'h'" tl" M4 universe is a maximally symmetric space, i.e.,

R

Ru....' = 12(gll.. 8.., - 8u,801Q)

( 10.1.2)

K is compact. (The asswnption Ihat the four-dimensional manifold is '~~:::~:~ restricts the manifold to be either de Si tter, anti·de

0' II an N = 1 local supersynunctry remains unbroken and has slll'Vived compaclification. jru" .ome of tt,e t,os,onicfi"" " can be sct to zero: (10. 1.3) "~::';:,~assumption about N = I supersymmctry, in particular, is espebecau se it places nontrivial constraints on the structure of the If supersymmetry is unbroken, then the super~yt:nJ11etry generavanish on the vacuum ID) (see (8 ,8.4». The variation ofa ferm ion supersymmetry, is given by

t.

6"

~

(,Q, ,,].

( 10. 1.4)

''''"m expectation value of this equation vanishes if supcrsymmetry is (since Q armibilates the vacuum):

(018y, 10)

~

O.

( 10. 1.5)

the classical limit, me variation of a ferm ionic field and its vacuum value are the same:

'V' - (01S" 10).

(10. 1.6)

406

10. Calabi-Yau Spaces and Orbifolds

Thus. if N = 1 supersymmetry survives compactijication. then ojthefi,mionjields must be zero:

tM:~

n the supersymmetry p..Jar:atOi However, ill select aa~~~:~::~gt='11 subset unetry is preserved. Ins is equal to [3]

- 90/r

kJ

)sHjkl

+ '" ..

This, in tum, places nontrivial restflc:t\o:ns Originally s was an arbitrary spinor supersymmetry be exact, means that we many possible s such that N == 1 SUI,en,y Now, the variation of the D = 10 _I

8l{!i =

/C

DiS

K

-1

+ 32g 2
C,X = 4g,.j¢rr} F,~s +. Q

OA. =

re six-dimensional md't9~ In addition, we h.",,,,·:1'h-,;,:

-1 t;;

v2¢

(I' &¢)e

_.

+

where the Roman letters i, j, k, a, b, c drop higher four-fermion interaction identity, which is satisfied exactly:

dH == Tr R /\ RNow let us impose the second and restrictions on our parameter s. Our varia

the system, t::;p~\;.L~j~;:;j constant spinor, on the spin '-'VL,W~\~l;l;l,fj the spinor invariant. f:tt~f1j travel around inl· i~r~fwl we obtain the v:

These are highly nontrivial constraints statement that forces e to be a covariar In particular, it places enormous constn covariant derivative and hence the space The first equation in (10.1.10), for eXaJ of the spinor s through a distance leavt we can perform two such displacement: For example, if we differentiate (10.1.l0 spinor around a closed path. We find , ~.

when we make a~~AA~M that the manifold K

Die = 0 --"" [£

--"" Ri This implies that the spinor remains uncl around a closed path. This, in turn, impl

10.1 Calabi-Yau Spaces

407

:'. last condition, in particular, is important because it says that the metric describes a flat four-
10' -+ 10

+ tl,'"~[D,", D~]e,

(10.1.13)

the area of the closed path is proportional to tl,mn. Thus, we see that simply rotated from its original orientation, with Lhc rotaLion matrix the commut<:ltor of two displacements, which is the curvarure tensor. S -)

Ue,

(10.1.14)

(10.1.15)

us make severol consecutive closed paths, ca\;b lime lea.ving and reW the sllI1le point. In general, each time we make an arbitrary number paths around the sam\; point, we wind up with the original spinor small rotation. Thus, the set of all such rotations forms a group: ( 10.1.16) is called the ho!onCJlIIY group. let us apply this result to our special elise. For the manifold K 6 , an

",()UP

";;~:~n:(h~;'~"~':(1 f(

connection that is an 0(6) gauge field . A spinor = 8 elements. However, we know that 0(6)

~

SU(4).

(10.1. 17)

',ihcs,,,i,ghl elements in a spinor of 0(6) can also be rearranged according (10.1.18)

SUe4), these two objects transfonn as spillors of opposite chirality. We the spinor e have positive chirality. This eliminates halfthccomponcnts, it transforms as the 4 ofSU(4). lh" we are investigating, however. is not nn arbiufry spinor but rhe condition

""in,,,

(10. 1.1 9)

e = Ue,

( 10.1.20)

the spinor remains the same after being transported around a closed

..QieU4"]I?tion now is; What is the subgroup of 0(6) mat leaves invariant the II.. The answer is a familiar one, taken directly from the theory of

408

10. Calabi-Yau Spaces and Orbifolds

Higgs breaking. There, we also want to knpw the answer to the qU.¢~~!ti is the largest group that will leave a constant spinor or vector ~nv~$ii The answer this question, note that we can always, transformation, put S in the form

o o o 60

So far, we have done nothing. We have simply taken an arb,itnlW:s~j§ put it into this form by an SU (4) rotation. But now it is obvious .,. group U that will keep such a spinor invariant is the subgroup matrices within SU (4), that is, SU(3). Notice that we now reSltric:nUI~;rJl to be block diagonal:

0)

= (SU(3) VOl

.

We can now trivially satisfy 6 = V 6. it is important to note :tI'i'~'(:t.::f~ is quite generaL We chose a particular representation of 8 ' ,,: general result, which works for any represen41tion of s. In conclusion, we say that the existence of a covariantly c" ims,~~ forces the holonomy group to reduce from 0(6) to SU(3). We SU(3) holonomy. The logic that we followed can be summarized as

N = I SUSY -+

= 0 -+ SU(3) Holonomy.

DiS

This is an important result, but it is rather useless. Very little manifolds with SU(3) holonomy. In fact, none are known _:~T,"·:"'~ this important piece of information cannot be used for phlenc.m~:li,Q~OO: ever, there is still hope, because we have not yet exhausted ~¢;:jm~ that can be derived from our principles. We have not by any means exhausted all the implications variantly constant spinor in the theory. We can, for example, the following object out of the spinor field, which transforms •••, on the space:

J}

= -igikefkjs.

Using a few identities on spinors, we can show that

rJP - -8 m n......

P m-

Whenever we can construct such a tensor J that maps the spa,~::~~ and satisfies J2 = -1, we say that the manifold is almost COI'niJJ~ dimensions this tensor statement simply says that there exists a ri''ffl~ml

10.2 Review of de Rahm Cohomology

409

-1. which is trivial.) This tensor is a very interesting object iflhe is covariantly constant For example, by differentiation we find

=::

D""r• =0.

( 10.1.26)

m.~~·

lffiIt the metric. in addition to being Ricci flat. is Kiihler. (These . wi ll be defined shortly.) Lastly. we can also defUlc a one-form:

r =r'!,J:dx"',

(1O.1.27)

r!p is the Christoffel symbol. We find 1.hat this form satisfies dr~o

(10. 1.28)

stales that thc fil1;t Chern class vanishes: CI = O. .",,,,",,ry,, retracing thc logic of our assumptions, we have

1 SUSY

-+

Dit = 0 -+ K is Ricci fiat, Kiihler, with vanishing first Chern cln.ss. (10.1.29)

" !~"" advantage oftms new rcsull is that many Ricci-flat, Kahler manvanishing first Chem class arc known. Thus, instead of relying on holonomy. which is a dead end, it is much preferable to rely on these Kahler manifolds.

",ii::;~~,:th~"at remains unresolved is the relationship between these two Fortunately. Ca labi colljccluccd (and Yau later proved) the

(c

that [4-6]:

A Kahler manifold of vanishing finit Chern class Kahle.r metric ofSU(3) hu/cmomy.

(Calabi-Yuu).

"a,dm"'" Q

manifolds for phenomenological purposes. ::;%I:I~d!~:::~S~~~~~:~~~::~:~~:;~~~ (at least in principle) to : actually construct such Calabi- Ya u manifolds. it is important

",;ew, few elementary facts about algebraic geometry and cohomology.

make a digression and discuss some simple properties of K1i.h1cr Metrics in the language of cohomology. We will find that IIUUly results from cohomology theory can be incorpor.l.ted directly into phenomenology.

Review of de Rahm Cohomology in the Appendix, the theory of diffL'fCnlial forms begins with an that is nilpotent:

d = dx"a". d~ =0 .

(10.2. 1)

410

10. Calabi-Yau Spaces and Orbifolds

We say that an N -form w is closed if

dw = O.

closed:

Similarly, we say it is exact if there exists an N - 1 form a w = dCi.

exact:

Notice that the set offorms is a subset of the closed forms: exact forms C closed forms. In three dimensions, these statements can be summarized by thewelti statement that the gradient of a scalar always has zero curl: .

A = V .¢

~

Vx A = O.

In fact, the theory of forms allows us to simply reexpress many in ordinary tensor calculus in three dimensions. In Maxwell theory, we say that two field tensors Ap and same physics if they differ by a total derivative: AIL - BIL = alLA

~

AIL '" BtL"

This, of course, is the essence of gauge theory. In this malthf:IrultiCan~ we will say that two forms wand Wi belong to the same eqtliV;ll¢jl"¢it~ they differ by a closed form. w-

Wi

= da

~

w ...., w'.

In Maxwell theory, we want to construct the set of all •..e_t.~l.,l1.Vlll.tili~:tlmfi the theory of forms, we do this by defining the set HP(M) to be: 1~¢::1~~ closed p forms modulo exact forms, i.e., H

P(

M)=

closed p forms . exact p forms

This defines the pth de Rham cohomology group HP(M) 4.t!Ji~~~~ manifold M. Notice that the cohomology group counts how many i'llldel~n.9~j~t::t.~ forms can be defined on a particular manifold. Cohomology is on the local structure of the manifold. However, as the name is a duality between cohomology and homology, which is based 9i;i;:~ properties of a manifold: co'homology -+ local properties of a manifold, homology -+ global properties of a manifold. • To make the relationship between homology and cohomology • let us define how to establish this duality. We begin by writing +h ... ·i'ritOi a region C defined on a manifold M, where C can be a line, swf~~~~~

10.2 Review of de Rahm Cohomology

4 11

think of this as a map that takes 0 p form w and manies it with a s urface D;~:d:;~ a real nwnber. Thus, we can view thls operation as the "scalar

,;r

fonn with a swface:

l

(Clw) =.:

(10.2.&)

w.

w~~' ~':;':~eth~ e familiar Stokes' theorem., generalized to N-dimensional

in the language of forms;

·8

( dw

Jc

~

( '".

(10.2.9)

lK

C is an N·dimensiona l manifold and ac is defined as the "boundary" . :: Now let us rewrite Stokes' theorem so thai it is reexpressed as a scalar Stoke.'1'theorem:

(Cldw) = (aClw).

(10.2.10)

we nave established thai the dual of the cohomology operator d is equal boundary operator:

d

+t

u.

( 10.2.11)

a

of this duality, we suspect that the boundary operator also defines to the cohomology group, which we call homology. '" P"'c;"., let us clarify what we mean by the boundary operator a and by C in tenns ofsfmplcxes. Let us define me line segment thai extends PI to P7. as a I-simplex. The I-simplex has a definite orientation

[ree,io", line segment = [PI , P1 J = - ( P2' III J.

( 10.2.12)

define a triangJe as a 2-simplex. which is labeled by lhn:e points or

( 10.2.13) . tbat the 2-simplex is cyclic in the thn..-c points, bllt flip s sign if thc '. are arranged anticyclically. Obviously. we can generalize this concept . an N -simplex:

N -simplex = (PI, ... , PH+I ].

(10.2.14)

te Inat the vectors fonned by taking differences of the points PI must be independent, or "lse the

sim pl~

collapses to a lower dimension.)

412

10. Calabi-Yau Spaces and Orbifolds ,

Now let us define the boundary operator that maps m-,slIIlpU~J(¢'~:ftr~ I)-simplexes. For example,

al[PI. P2] = [pd - [P2]. where [p] is a point. Thus, we define the "boundary" of a un(~ :~~:~@{(i the two endpoints. Acting on a triangle, the boundary opleralot:~tAi segments: a2 [pl. 0, P3] = [PI, P2]

+ [P2. P3] + [p3, pd·

Thus, the boundary operator acting on a 2-simplex (a triangle) it up into its three edges, which are composed of lines or I-sirritll~j It is easy to check that the bOlmdary operator is nilpotent. Fn~','~'!i.;.~ the simple case of triangles we have P3] + [P3, pll} = [pd - [P2] + [P2] - [P3] + [P3] ,

a.a2[pl. P2, P3] = =

al {[PI. P2] + [P2.

o.

Thus, we have the important result that the boundary Op(lratoIiW~(iJji

a2 =

O. '

This can be generalized to the arbitrary case. Let us now Ue1ill1I;E:m boundary operator acts on an N -simplex: N+I

aN[PI •... , PN+I] = L(-IY[PI, ...•

Pi, ... , PN+d,

;=1

where we omit the points that are hatted. Thus, the boundary 0'. 'p:~j~~ N -simplexes into (N - I )-simplexes. It is not difficult to aN-IaN =0

for the general case. In analogy with the theory of homology, let us define the set (jn~~ be the set of simplexes that obey

az =

O.

For example, take the two-dimensional doughnut or torus. Th1ete:}itj types of cycles that we can draw on this surface. For ex~unI)le;)J~:~ we saw that we can draw two types of cycles on the dOllgtLDutWl;'!:r:~ continuously shpmk to a point. However, there is also the cyc:let.l:l:itt: a closed line on the surface of the torus that can be COIltinuOllSl.Y. point. We want a method by which we can eliminate the secon(ft~~i Let us define the set of boundan'es B as the simplexes that c:j~:~~U as the boundary of some higher simplex C: '

B = BC.

41 J

10.3 Cohomology and HomolollY

that the set of boundaries is R sunset of the set of cycles: boundaries C cycles.

(10.2.22)

say lhat two simplexes are in the same equivalence class if they differ only bo'n,'",y. Thus, we want to extract tile sci of a1l inequivalent simp lexes, by defi ning the pth homology group:

Hp(M ) =

p-cycles . .

p -boundancli

( 10.2.23)

',U(,W5 us to eliminate the redunda.1t cycles on tile torus that can be conshrunk down to a point, keeping only the two independent cycles ~n<'i,,'e the torus. Thus, the concept of homology is a narural one.

Cohomology and Homology the relationship between the homology and cohom ology groups for a ,act maw' . Because of the duality that exists between them, we can they have the same dimension. Let us defi ne the Berti numbers to be im<,n,iion of the H 's: (10.3.1) be a set of cycles defined on a compact manifold M. Let Wq be a set forms defined on M. Then construct the matrix Opt : O(c p > wq ) = [ <,

w,.

(10.3.2)

is called the period matrix (see (5.6.3J) and (5 . ! I.S}), and it can under quite general conditions that it is an invertible matrix. Thus, x N matrix with nonvanishing detenninant. "Rut if the period matrix matrix in N dimensions, then the dimension of the closed fonns w'l ,I to (hed"ne,,,;;on of the cycles C p, which proves that the Betti numbers same for the cohomology and homology groups. This is an extremely result, because it means that we can usc either local (cohomology) (homology) properties ofa particular manifold to calculate its Betti Q

. also important to realize that the Betti numbers arc topological num· . on the overall topology of a manifold. Thus, any linear of the Betti numbers is also a topological number. In particular, is the Euler number: x(M) ~

" L (- I )'b,. ,="

( 10.3.3)

414

10, Calabi-Yau Spaces and Orbifolds

To understand the properties of these Betti nUI11bers, we will haiie;:~;i duce a few more operators, including the Laplacian, Let us also initooo Hodge star operator, which converts a p-form into an (N -

*(dX i1

1\

dX il

1\ '"

1\

dx ip )

..;g

--'---13·iJi2 ....i.

(n _ p)!

.

'.+1'1'+2''''.

d X·i +1

d X I p +2

1\

1\ • •• A ··'u...

where eijkl .. , is the totally antisymmetric tensor in N dimensions.

* *wp =

j;l\.i:~ff.l

~.·.·r. ~;timm

(_l)PIN':'PI Wp ,

In N -space, the product of a p-form and an (N - p)- form Yt~jl~i form, which is proportional to the volume d N x, Thus, we can 'It¢gr4ii the product of a p and an (N - p) form to obtain an ordinary nw:n!1.!~ti;:t: now define yet another inner product: i' n

(Ci p I

d/J p) =

£

Ci p 1\

*/J".

Now that we have introduced the definition of the scalar pr<X1UlCt~.ttU~~ us to define the dual 15 of the derivative d:

(ct p I d/Jp-I) = (8a p I /Jp-l). Explicitly, the adjoint of d is given by

0= (_l)Np+N+I *d

*.

We also have

88 =

o.

Notice that the adjoint reduces by 1 the degree of a differential totmi:] raises it. Let us now define the Laplacian as ~ = (d

+ 8)2 =

8d

+ d8.

Let us define a p-form w to be harmonic if harmonic:

f1w = O.

Let us define a p-form to be coclosed if coclosed:

8w = O.

Let us define a p-form to be coe:xact if it can be written as coexact:

w

for some (p + I)-form a. We now arrive at an important theorem:

= Oct

! 0.3 Cohomology and Htlmology

WI' = da,. _1 + S/Jp+1

+ Y,..

41 S

(10.3.14)

Yp i$ a hannonicform. . a powcrful result, because we can show that each cohomology class only one hannoDic form. To sec this, first construct the inner product a fom} and its Laplacian: (w Illw) = 1&0(2

+ (dW(2.

( 10.3.15 )

the st..,telllent that a fonn is hannonic (which sets (w I llw) = 0) ~ui""Icnt to stating that it is exact and coexact (because (8w1 1 = 0 and = 0). In fact, it can be shown that /).w

ut iI·dw

= 0 iff

&tJ = dw = O.

( 10.3.16)

= 0, this means that dS/J = 0, and so 1l{3 = O. Therefore ( 10.3.14) w=da:+y.

(10.3.17)

, every cohomology clas.( cQntain$ one harmonic repN!$(!ntative. fact thai there is one harmonic representative within each equivalence of ex.act fonns gives us an aJtemative way to define the Betti numbers. ""also say thatlhe Bctti numbers count how many independent harmonic there are on the manifold. We have the following equivalent description Belli numbers: number of independent closed foons. number of independent cycles,

( 10.3.18)

number of harmonic forms. we are frce to use either of three equivalent formalisms to calculate thc . numbers. last formulation of Betti numbers, in terms of independent harmonic gives us an altemative definition. The set of harmonic forms can be preSs,. as the. kernel ofthc Laplacian (i.e., the fo rms that are sent to zero). we can define the Beni numbers as

dimke,,',.

bp =

I

dim Hp. dim H P.

(10.3.19)

e :;~:'~I:::';~:1~ to study the properties of some oflbese Betti numbers. ~1 always define tbe sca lar product for an N ·dimcnsional manifold:

(10.3.20)

416

10. Calabi-Yau Spaces and Orbifolds

Consequently, these two spaces contain the same number : elements. Thus,

b p = dim HP = b N -

p

= dim HN -

p•

This is called Poincare duality. We will usually take bo b N = 1. Another way to prove Poincare duality is to notice that if a is harmonic, then it bas a dual that is also harmonic: Llw = 0

Ll

-+

*w = o.

But because the number of independent harmonic forms is "''iLl<1~;:j~:J;j number, and because the Hodge star operation converts P~78~~~I~1 p )-forms, we must have Poincare duality. _ Lastly, if we have a product manifold, then the Euler number manifold is the product of the Euler numbers of each 1ll""-'UV1"L• .

X(M x N) = X(M) x X(N). Ifwe write this out in terms of Betti numbers given in (10.3.3), Mj~j~ii

bk(M x N} =

L

~p(M)bq(N).

p+q=k

These are called the Kunneth formulas for product manifolds. · .. Let us now take a few simple surfaces and calculate their Bei~~~~ (1) Two-Torus

A two-torus can be cut up into cycles in two independent ways. T~i~: j'Ji:tlm of the independent one-cycles that we can draw on a torus is LU'-\''';:+l'r-y.-0 b l = 2. Also, by Poincare duality, we have bo = b 2 = 1, { b l = 2.

T2:

Thus, the Euler number (10.3.3) for a torus is

X (T2 )

= 1-

2 + 1 = O.

(2) Riemann Surface As we saw in Chapter 5, there are 2g ways in which we can divide U·Iv..~;~~ surface of genus g into independent cycles. Each hole or handle. cycles. Thus, b l ;.= 2g. We then have . bo=b2=1, { b = 2g. l

Thus, the Euler number for a closed Riemann surface is

X(M) = 1 - 2g + 1 = 2 - 2g.

10.3 Cohomology and Homology

on the sphere

41 7

S,.., of coW'Se., can always be collapsed down to a point.

there are no independeot cycles that we can write. Thus,

bo = bH = I. [ b, = 0 otherwise. number is therefore =

X(5 H )

I~

if N .is odd, if N is even.

( 10.3.29)

(10.3.30)

,,,,joel,,, given tha t (10.3.3 1) use the product rule 00 Euler numbers to show that the Guler number

X{T2) = X(SI)l == 0,

(l0.l.l2)

agrees with our earlier calcuJalion. 'rod.uct of Spheres 0.3.24), we can show thai the product S3 x S3 has

bo=b.= I, bJ =2. [ b, = 0 oLhcrwisc.

S) x S3 :

( 10.3.33)

that the Euler number is zero, as expected from the fact challbe Euler

three-sphere is also zero: (10.3.34)

mil",I",. we can take the product $2 x S1

X $2.

which has the Betti numbers

bO=b.=1. S2XS]XS]:

b2 =b.=3.

[ b, = 0 otherwise.

( 10.3.35)

its Euler number to be X(S2

x 52 x S2) = X(S2)3 = 8.

( 10.3.36)

( 10.3.37)

418

10. Calabi-Yau Spaces and Orbifolds

It is a simple exercise to show that the Kunneth relations for prcxl1i¢.dgal in (10.3.24) yield

Putting these Betti numbers into the Euler number, .we find

.. .

as expected. (6) Six-Torus For the six-torus T6, we have

T6

= Tz

X

T2

X

Tz.

Therefore we can use the Kunneth relations to solve for the the product manifold. It is easy to show that

JJ"'L";'lWUl

bo = b6 = 1, b l = h5 = 6, b2 = b4 = 15, h3 = 20. We see that the Euler number is equal to 0;

This is also true because each circle within T6 has Euler number v.:... .."'~ simple rules, we can obviously generate the Betti numbers of

(7) Real and Complex Projective Space The space C PN is formed by taking ordinary complex (N +. :: making the identification Zp

= AZp

for some nonvanishing complex number A. This last cotldi1:jorlfedm~~~:~1 (N + I )-space to complex projective N -space. C PN is a g~lleria~fj~'fm! projective space PN , which is created by identifying points in real by Xp

= kxp

for some nonzero real k. We can also construct it by taking the s·1m~r:~~ identifYing antipodal points. Some examples of real projective .i!i.' ~4:;j~

10.4 Kahler Manifolds

419

space are

CP, = S2. P, ~ SO(3) ~ SU(2)/Z,. CP _SU(N+I) U{i\')

N -

( 10.3.45)

.

. numbers arc given by

b, ~ I. bN ~ 0 (N even). hi = 0 oilierv.'ise,

I

x(PNl

= 1 (N even),

~

I (N odd). ~

( I OJ.%)

0 (N odd),

(10.3 .47)

Kahler Manifolds our discussion has revolved mainly around real manifolds. However, generali:.:c quite easily to complex manifolds. d~~:::~N:,-d~~imetlSiOnal complex manifold, we want a 2N real manifold :g of the usual definition ofa complex number: l = x +iy. = -I. If we have a 2N~dimensionaJ real space, what we want is a ,": 2N tensor fie ld J/ thai can replace i , such that J2 =- 1

(lOA. I)

question. so that we can deflne a complex nu mber as

z=.l+Jy.

(l0A.2)

ini'bldl, """' have such a tensor fie ld J are called almost complex. ~"'cc" we want more tha n J USt the existen ce of the tensor field J. We to diagona!i7..e it. If we usc complex coordinates l
=

XQ

+ iy",

Za =x
( 10A.3)

like to make a change of coordinates such thai .~" Jnb = 1o".

( IOAA)

10. Calabi-Yau Spaces and Orbifolds

420

, The question is: Can we always diagonalize J in a nellgh.bolrhoiijd:~f.~ using a coordinate transformation: z~ = Z~(Zb)

(which is not a function of z)? Such a coordinate a~~~~I:ci~I~1 holomorphic, which is the generalization of the concept of ferent points p, we of course will have to patch them tog;ether:l~~~~i:~ holomorphic transformations until we can cover the entire surta¢!~::~ Ifwe can always find such holomorphic transformations that ciW!Jfn J around a neighborhood of any point p. of the manifold, '",'re,,!~,:i:?1: manifold has complex structure and is a complex manifold. Thus, we have the analogy One dimension

N dimensions

i

Jba = iSi> " . .c:J holomorphic z = X + Jy

analytic z = x + iy

(Intuitively, the process of patching together neighborhoods ~~!:1! agonal resembles the "elevator problem" in general rel:ativritv. :.... ~";"'" coordinate transformation, we can always locally transform 1O,e::'Q symbols to zero at any specific point, which corresponds to ....!'~!~~ an elevator without gravity. In general, we cannot globally tra.i~$ts!OOi Christoffel symbols away entirely, or else the space is fiat, but",,'.'~r¢l:iiJ point on the manifold enter the elevator frame.) . At first, this definition might seem verbose fora rather However, certain 2N -dimensional real manifolds can be criterion. For example, the spheres S2N are not complex IIl3lni1uI~~:'~ S2, because S2 = C PI). Thus, it is not at all obvious that a real manifold can be rewritten as an N-dimensional comple~XX!;[;fjiJ Now let us discuss differential forms on these complex n complexify our basis differentials as

= dx j + idyj.

dzi

Now the concept of a p-form can be generalized to a (p, W

=

W,r]'k ···U.l "j-k- ···u-

di

j

u

1\ dz ..• dz 1\

dz i

1\

dz1 ... dz u ,

with p unbaqed and q barred indices. We can now <1etme derivatives:

-

a

I '

-W"k "Ok- i rJ ...,rJ "·u

a-W

d i d- u aad-I zl z W'jk ... ;ijk ... Z 1\ ..• 1\ Z.

=

az

dz

dz' ... 1\ dz u '

aw =

!i

1\

~~~

10.4 Kabler Manifo lds 'rop,,'i~;

42!

011 th", two derivatives are

a' ~ a' ~ 0,

aa + aa = 0,

{ d= Hi:l+a).

(10.4.10)

, form now takes two indices (p, q) to label it. Thus, we can now define a

of

on we now

:;~~~,~l~~~:~::~I"~,,~ead basedand d, , based de on Ralun a. We cohomology can define closed exact for a

f.

as before, except we now use

ainstead of d:

Hp.q{M) = &_ closed (p , q)-form. exact (p, q)-form

a

(lOA. II)

fo", ,,,"'" also construct the adjoint operator and the Laplacian. In the case, we actually have two Laplacians: 6~

oot + &ta. --t -aa + i}ta.

_

6~ =

(10.4.12)

have the complex version ofthc Hodge theorem:

Every complex (p, q )-form has a unique decomposition (J)

=

+ 8{3 + atr.

Cl

(10.4.13)

is a hannonic form

= 0

t:.s
are (p, q - 1)- and (P, q

+ I}/orms, respectively.

"SO have the generalization afthe Beni numbers, which are now given

dimHr(M)=hp.q.

( 10.4,14)

,I"ion to the Hetti numbers is bIt =

-

L

.....

o h,we, by complex conjugation,

h P·q =

h P·

q

.

(10.4.15)

h~'P

(10,4, 16) define a Kahler manifold. If we have a complex manifold with

metric gij. then we can ah..'Uys construct the form (10.4.17)

422

10. Calabi-Yau Spaces and Orbifolds

This is called the Kahler fonn. A complex manifold is called .L~~"~J.::

dO. =0. A Kahler manifold has a Kahler fonn that is closed. A Kahler manifold, because of the above definition, has a la:r:ge::,!:lnt1 of beautiful properties which make it among the most elegant .,~'' ,~''::~:;:~i,Q manifolds. We simply list some of its properties: . (1) We can show that the Hermitian metric of a Kahler manifold can::tii!:: in tenns of a derivative of a single function. the Kahler po:teritiat?~m

a2 ¢ gij = ozlai j

'

(2) It is easy to show that a Kahler manifold satisfies, by exr>licitcaJCl ~a

20d =

= ~il'

This is a powerful identity because it states that the VlIl:10l1S:::]:;i1 that we can fOlm on a Kahler manifold are all the same. ~~II! confusion when using different Laplacians for the Kahler ' (3) If a manifold is to admit a Kahler metric, then the even co~~!ijj the Betti numbers must be greater than or equal to one, an(Vl:tsk~J:ti numbers must be even: @~rs

V"1.1 ~er

b2 p > 1, { b2n + 1 = even,

--+

for integers p and n. (This simple criterion rules out a large "l' ''1':~l;i:~~J folds. For example, this rules out S3 x S3 as a manifold that metric, because b2 = O. It also rules out P N but allows C PN '.:: (4) If the J::' tensor is covariantly constant, then the metric . ... converse is also true for complex manifolds.

va J% =

0

B-

Hermitian metric is Kahler.

(5) If the torsion fonn that we construct on a complex manifold v~l~~t~BfI the metric is Kahler. (6) The only nonvanishing Christoffel symbol of a Kahler roanm;)m:~

r abc =

cj;

g gat.h·

The only nonvanishing component of the curvature tensor

R:

bc

= -

r:b,c

and other curvature components related by symmetry conjugation. The contracted Ricci tensor becomes

R -- -

ab -

a

2(lndet g) ---=,'-

ozaaib

10.4 Kahler Manifolds

423

(10.4.26)

a direct consequence, we can show that Kahler manifolds have U(N)

In facl. this caD be used as an alternative definition of Kiihler C;~~:~~.. Notice that lhe holonamy group is the rotation group generated closed paUlS around the manifold. which is a function of ( 10.4.27) M ;j

is Lbe rotation matrix for some 2N-dimensiona! tangent space

The coefficients of this rotation, we sec, are

funCl io n~

ofwc 3nti-

Riemann tensor R"bed. which CW1 1l0W be viewed as an element a 2N x 2N antisymmcnic rotation matrix, Th\ls, in general, we have holOGamy for a general manifold. However, if the manifold is

we can show that che restriction on the curvature tensor we found reduces the SO(2N) rotation matrix down to U(N), which is a ofS0(2N). The restriction that (10.4.24) is the only oon7.ero aC the curvature tensor breaks SO(2N) symmetry.

manifolds. which have U(N) holonomy, can be further restricted if demand that they have vanishing first Chern c1a~s. In this casc, U(N) '~;:~:~:~r:educes 10 SU(N) hnlonomy. In fact, we have the theorem of D which works for all N, thaI a Kahler manifold oj vanishing Chern class always admits a Kahler metric ofSU(N) holonomy. In , it can be shown tbat the vanh;hing of the first Chern class is equiv-

!:;~C:~':'~~:~\I:/a

Ricci-flat mC!:tric. Thus, wc will use these two eoncepts

.1.;" now be instructive to consider explicit examples ofKiihler manifolds. ,em""" Sumce . oriented Riemann surface admits a Kahler metric. Because the line of any Riemann surface can be put in the form (10.4.28)

is Kabler because we ean always find a Klihler pOtential ¢ such thaI

a' azaz¢ =

e".

(10.4.29)

surfaces are trivially Kahler because any two-form on a Riemann including the Kahler (onn, is clO.'led.

424

10. Calabi-Yau Spaces and Orbifolds

(2) Complex N Space Complex N space eN is trivially Kahler because its standard

Imlfe1i~

ds 2 = Lldlf. i

can always be put into Kabler form. (3) Sphere We note that SN, in general, does not admit a Kahler metric. T'~f$i:~~~i the even Betti numbers h2p for p equal to a nonzero integer .~'t~~~!m.1 equal to zero. Hence, they cannot be greater than or equal to vu,,,,,.·.,,. one of the conditions on a Kahler manifold. The exception i$ d'iit:1~ S2, which has no even (nonzero) Betti numbers. To show n.···.',K;';':"'; admits a Kahler metric, notice that a two-sphere has the line e'llettJ&1J dx 2 + dy2

2

ds =

(1

+x2 + y2)2

.

From this, we can write the Kabler form Q

= !i dz 1\ dz 2

(1

+ zz)2

if we put it into complex form. This Kahler form is exact, and li¢j~~ admits a Kahler metric. Although S2 is Kahler, we can sh()V;('itij~ therefore it is not Ricci flat. We notice that Sp x Sq is a complex manifold if p and q However, this does not mean that they are Kabler. In fact, S3 are both complex manifolds, but they are not Kahler. is Kiihler (but is not Ricci flat). (4) Complex Projective N -Space To prove that C PN is Kabler, we note that it is possible to a Fubini-Study metric on C PN : Q =

!iaaln (1 +

W"~f'ilill

tziZi). r=]

(5) Complex Sqbmanifolds of C P N It is trivial to see that complex submanifolds of C PN are also K.aJ~~t~ use precisely the metric defined for C PN for the sul)m;am·told,)~x! be careful to take only the components of the metric tensor to the submanifold. Since the original metric was Kabler, th ..;·itM'i> submanifold (which is the same metric) must also be L,."'...." •.~;

lOA Klihler Manifolds

415

,<>-ton" T2 acrualJy has vanishing first Chern class c) == O. We can also the four-dimensional Lorus T4 is Kabler. Finally, we can show that ,·t'oru" T6 is both Kahler and Ricci flat. Thus, compactification on the . appears to have the desirable fcature that N = 1 supersymmclIy is . However, che drnwback oflhe six-torus is that too much symmetry ~"'ed . Tn fact, N = 4 supcrsymmctry is preserved on T,» making it

,ptable from the point of view of phenomenology. now summarize some of these results in a table:

R-Kiililer mean~ Ricci flat aod Kahler, Y, (N) means yes (no), X equals andn == 1,2, J ... the condition of being Ricci flat places even more restrictions for example, it can be shown that a Kahler manifold complex dimensions has c] = 0 if and only ifthere exists a co\'nriantly nOIlV"dnishing hoiomorphic three-form w. This shows, for example, S2 x S2 does oot admit a Ricci-flat Kahler metric. (We know thai this hilS b3 =: O. Thus, by definition,mere are no harmonic three-fonns on mil'o"li.. But this also means that there are no holomorphic three-forms, , it cannot admit a Ricci-flat Kiihler metric.) ,i,"p'" cOI1$equenlZ of This is Ttl::l.l ::I.oy IUlnnonir. (1'.0) in thr~r. dimensions can be multiplied by w, thus creating a hannonie (10.4.34)

'ntity is true because w is covariantly constant and hence acts like under the laplacian. Thus, a harmonic form remains a hannonic multiplication by w. Also, a (P . O)-form becomes a (0, 3 ~ p)-form . we are contracting with the Hermitian metric tensor gat; and not gu/). in tun" allow~' us to eliminateafmost all the various Hodge coefficients. various reflection symmetries and the previous symmetry, we can It ooly·" •.. •. /12.1 • II 1.11 survive as independent components of a manifold holonomy. Of these, we c"n also eliminatc h 1•Ofor the foll(lwing

426

10. Calabi-Yau Spaces and Orbifolds

reason. We know that the Laplacian can always be expanded 04i@~:ffill £!"d = _'12

+ curvature terms.

Acting on a one-form, the various curvature tl;:nsors reduce to tl,lMjij~i But the Ricci tensor is zero on a Ricci-flat manifold. Thus, form must also be covariantly constant. This means that the ~"'¢t!it:1J:\J a harmonic one-form must be zero: b l = O. But it also mea~Oij~~j by (10.4.15). Putting everything together, we can now sholvi : fil¥:~: manifold,

x=

L (-l)P+qhp,q = 2(hl,l - h

2 , 1).

p.q=n

This last identity for a Ricci-flat metric will become extremely ~~~II we discuss the generation problem. It turns out that h 1.1 and the number of positive- and negative-chirality fermions we "ciitt:dil~i manifold, so that the above equation states that the gellerlltiOlillj[ijffi the Euler number generation number =

! IX I.

Thus, we have a topological derivation of the generation

10.5

Embedding the Spin Connection

Armed with some of these elementary results from algebraic 19tJIQ~~~ now return to string phenomenology and apply these results. >,. , Earlier, we saw that the condition of N = I ' ', existence of a covariantly constant spinor, which in tum ' dimensional manifold K was Kahler, Ricci flat, and had N = 1 supersymmetry -+ covariantly constant -+ Kahler, Ricci flat, SU(3) holonomy.

Now we wish to exploit the remaining condition, which is the 13i~~Mi (10.1.9): ' '

Tr R A R =

1 30

Tr F A F.

This is actually a strange identity, because we have the R.H!m~uw;,t~ left and the Yang-Mills tensor on the right. The equation u::~:ill identity, dev~~d of any content, but in the presence of our a! d¢ = H =0,

we find that the equations are actually quite difficult to s~~;=r.il preservation of this identity is nontrivial because the system is 4 One attractive way to satisfy this odd identity is to set pi:tJr:t:1~~~ E8 gauge fields equal to the Riemann spin connections, whjlch::

10.5 Embedding the Spin Connection

427

. This produces a nontrivial link between the spin connection and ",~-M' gauge field. We can perform this embedding as follows on the

(10.5.3) is the spin connection, which occ upies pari afme gauge field matrix. embed the spin connection fie ld into the Yang- Mills gauge field, we .find a .subgroup of the gauge group that contains SU(3). This means, of , that we are breaking the original gauge symmetry of the Yang-Mills simplest decomposition is

(10.5.4) check, however, that the C lebs.:h·.(Jordon coefficient:; afC such that . idenlilY is satisfied exactly. In particular, we must show that we the factor of in the Bianchi identity (10. 1.9). kn"wthaJt' ,e Yang-Mills gauge fie lds arc in the adjoin t representation has 248 elemenlS. We must now find a breakdown of these 248 representations of SU(3) ® E 6 • which is always possible to do.

io

w ,,,' """,'0

248 = (3. 27) Ell (3. 27) Ell (8, 1) Ell (1, 78).

(10.5.5)

(iO.5. 1) is satisfied, let us convert from the adjoint rcpresenlation 10 the 3 CilJ,1f A is a generator of SUP), then we wish to fin d the between the trace of this matri x squared in the 8 representation '3 represe ntation. The answer is ( 10.5.6) focus on the SU(3) content oC(10.S.5). Notice that we have 27 sets transforming in the 3 e'3 representation of SU(3), and also one sel . But the sum of the rrace of the squares of the OCtcts, as we saw in be multiplied by 3 when we convert to the 3 @'3 representation matrices. Thus, the total redundancy in the 3 63 '3 representation

27+3xl=30 fac tor of 30. Because of spin embedding, where thc curvature tensor in the same spaee as the Yang-Mills tensor, we enn now satisfy missing factor aOO emerges when we count the specific in (10.5.5). E, reasons,

,

group, by contrast, docs not have complex representations, to describe crural fermions. but £6 does. The 27 multiplet, in multip let for fermions for model building with £,.

428

10. Calabi-Yau Spaces and Orbifolds

The group E6 is also good from the point ofview onow-energy because the 27 of the fermions can form a supersymmetric mtlltij~J~~~ 27 of the Higgs. :::m::1m In summary, we have the phenomenologically acceptable cOlil¢l~Mii Tr R

A

R=

1 30

Tr F

A

F ~ spin embedding -+ 27 tenmo'ti$~:

Because the fennions are now in the 27 representation, question: How many generations are there? The GUT, as we with the problem of generations. There was no reason for as!mntiP:~:ij~ one generation. In the superstring picture, the situation is precisely the now show that we get too many generations!

10.6

Fermion Generations

One of the most powerful applications of algebraic nomenology of string theory is the calculation of the from topological considerations. To calculate the number of generations predicted by the the:o~M¢o."¢::u.1iij calculate the number of massless particles. The 1O-dimensional ~.',l,~~~~m operator for the particles in question becomes, after c~;~~~~.~lific~~l~~1

6t tw(n>Jem~crolrllpmmUl.bJ..Lhuur'ad.s;.y~\mm~j.$l\!'2:~. 010lfr = (04 + 06)lfr =

o.

In general, 0 6 will have eigenvalues denoted by m 2 , that is, so that our wave equation becomes

(04

+ m 2)lfrm =

O.

We are interested in the massless sector in four dimensions, so only the zero eigenvalues of the 0 6 operator. Thus,

04¥r

=

06¥r =

O.

This is an important equation because it has two mt1erpretati()ns. of course, that the four-dimensional fermions are massless. means that ¥r is a harmonic form in six dimensions. massless modes in four dimensions will be related to the forms that we can , write for the six dimensions manifold. Wei$a~Je (10.3.18) that the number of harmonic forms of degree p is ecm~n~ Betti number. Thus, topological arguments alone should give.~..H~m ofgeneratio1lS! In summary:

m 2 =0-+

{

¥r is harmonic in six dimensions,.....•• 1jr is massless in four

' .:

10.6 Fennion Generations

429

expect that the number of generatioos is a topological number because Dirac index theorem.. We know that the solutions of the D irac equation in ,i,,,,,,,,I, have zero modes:

i(P}'"

~

O.

(10.6.4)

index of this operator is equal 10 the difference between the positive ""tiyo chiralilies of the zero modes: ( 10.6.5)

index is a topological quantity defined on II spin manifold, so we that it can be related to the chara~teristic classes we found earlier. In discussing SU(3) holonomy, we will find thai

,in"

i"d,,

Ind..(P} ~ Ilx(M)I.

(10.6.6)

is also equal to the generation number, because we will be

only fcnnions of one specific chirality. Thus, the precise relation generation number and the Dirac index, or the Euler number, generation nwnber = 4Ix(M)I.

(10.6.7)

this, C()nsidcr a manifold of SU(3) holonomy. with Betti (Hodge) that have two indices p , q. The Euler number can be written as

,

X(M) ~ L(-I)''''h".

(10.6.8)

bc

i down, which i:ifi;~~~~O:;fh:Ih;:'~:~~~~:~~""~d~~1~;~~~~:, : ~~can r related Betti numbers. We find. if compare the hclicityi1ctcr-of WE

~nme"i'

pairs with their multiplicity, (2.~) (~. I) (l,~) (!,O)

ho.c, hO.h

(2hl.c+hc.I),

(ho.o

(10.6.9)

+ h l . l + h2.l).

oaI:y" the spin-~ fennion sector, given by C!. 0), wc find that their number equals ferotion multiplicity = ho.c + h l . l

+ h2.1.

(10.6.10)

il:j:i~;~i~; x:'~~v:~. is too large. We want only the subset of this figure ()f

the 27 and 27 of the spin- ! fcnnions. in (10.5.5). we found that the Z48 of EB ® Es can be decomposed 3) and (27.3). It can be shown that the multiplicity associated with

430

10. Calabi-Yau Spaces and Orbifolds

each of these two representations is equal to

(27,3)

h 2•1 ,

(27,3)

hl.l.

Thus, the generation number is equal to

#(27) - #(27) = h 2 •1 - h l •l , where # represents the multiplicity of the 27 representation. But in turn, is precisely half the absolute value of the Euler number, (10.4.36). The relation between the generation number and the Euler mun~~tiii surprising because there is no reason to believe that any """"'l\Jlli~I).~'~ between the two. The generation number is a function of the Yarlg-MIU group E8 ® E8, while the Euler number is a function OOf~!th~:e~n~~~111 Naively, we do not expect the two to be correlated. But the n the two is established because we performed spin embedding, gauge group and produces an intimate relationship between the grQuV:·@jj K6 and the gauge group. Thus, the essence of this important result of compactification and spin embedding. Next, we would like to calculate the Euler number, and hence tt·1~:~~ of generations, for a series of Calabi--Vau metrics. We showed earlier that a submanifold ofthe Kahler metric on it has the same Hennitian metric gil, is also Kahler. Thus, we wanflhi.~l6'ii! the set of submanifolds of C PN that have vanishing first Chern This can be accomplished by placing constraints on the z's by polynomials to zero. Consider the submanifold of C P4 obtained by the constraint 5

Lzi = o. 11=1

It can be shown that this manifold has vanishing first Chern

clal~fC8ro:it:

:tr~~~111

X(M) is equal to -200. Let y(N;dl.dl •...• dtl

represent the submanifold obtained by setting k homogeneous P~l~I;1 degrees d; to zero within C PN • Fortunately, the fonnula for tl: class, not just the first Chern class, of these sub manifolds is knj~w)Q:;::~!~ holonomy, the total Chern class equals (1 C

=

k

+

J)k+4

fT=1 (1 + d;J)

'

where J is a certain two-fonn obtained by nonnalizing the ~Q[i~iill C Pk+3 • By expanding this fonnula and then setting the first,

10.6 Fennian Generations

431

, 2: d; =k+ 4.

(10.6. 16)

h., 1

we find only five possibilities with vanishing first Chern class: X(Y(4:.5) = -200, X (Y(3;l.4)) = - 176, X(Y(5;3.])

X(Y(6:l .2.2) X(Y(l;2.l.2 .2)

= - 144, = -144,

( 10.6. 17)

= - 121t

the generation numbers are unacceptably large! This is phenomenologundesirable because we know ftom arguments from llucleosynthcsis, and asymptotic freedom in QeD that we want very few such as three or four.

, we can still reduce the generation number and make it much . Let us divide the origmanifold Mo by a discrete symmetry group G that acts freely au the '. . (i.e., no fixed points), yielding a manifold M . If the number of disgenerators of G is n( G), then the Euler number of the original manifold by the discrete group G is o

X (M)

= X(M,)

,,(G) ,

M,

M =G'

(

10618) ..

( 10.6.19)

consider the previous example of the submnnifold of C P4 with

,

2: zi=o

(10.6.20)

thai this polynomial i:;; invariant under the following synunetry mhO." --+ (ZS. Zlo Z2 •.•• , Z4). 4 (Z I, Z2, . .. ,Z;) --+ (ZI, aZ2 . 0:2 z), ... ,0 '::5), (ZI, <:2 ••.• , Zs)

(10.6.21)

a is a fifth root of unity. The discrete symmetry group is 25 X Zs

( 10.6.22)

has 25 generators. Thus, the Euler number of this new manifold is -200 X(M)= =-8 25

(l0.6.2J)

432

10. Calabi-Yau Spaces and Orbifolds

which predicts four generations. Many other types of models are possible with reasonably nwnber:

x(

)

= -16,

Y(7;2.2.2.2) ) X Z2

= -16,

Y(5;3.3)

23 x 23 X(

22 x Z2

Y(7;2.2.2.2})

X(

Z8 X

= -16' '

(Y~~5}) =

-40.

The point here is not that we have obtained the correct phlenc)lll(~~~iij~ point is that with a clever choice of a discrete group, it is reals0I1$.I¢:itl might be able to write a model with an acceptably low number qn~~~(ffi We must stress that the very existence of chiral fermions in sup~~@j is something of a miracle. In standard Kaluza-Klein thl"nn there are serious obstructions to constructing a theory with four dimensions. In fact, an acceptable supersymmetric Kahu~a-Kl.j~~ with chiral fermions does not exist. Thus, it is quite rernarkal)le ,~,!,..~.;~,.,.; obtain any chiral fermions at all with superstrings. Lastly, the choice of a nonsimply connected manifold surprising, but it turns out to have other ph4~nc)m(~no'10~~lcallly features.

10.7

Wilson Lines

Our previous discussion of embedding the spin connection into rne':g;m in (l0.5.3) broke the group E8 down to E6 ® SU(3), which has ex~i~~ good phenomenology because the fermions are in the 27 re~lre~;erilrati Our next step is to break E6 further to obtain the minimal orrl11t1 SU(2) ® U(l). The problem here is that most naive methods of breaking E6 ~Q'Wi will necessarily break N = 1 supersymmetry as well. Howe-ver .:~~iOO trick that we can still use on manifolds that are not simply cOImeQti!j~~ In general, we know that the path-ordered product of el:~,:NifJI group around a loop is gauge-invariant. We can express this as

:::::i:lm

, l'

U

~ P exp (1 All dX

Il

)

,

where P represents the ordering of each term with respect to C. For small paths. we find that U is proportional to the eXI)oti.lei!,~W curvature two-form.

10.7 Wilson Lines

433

F;"

when the curvature vanishes. we expect that the Wilson loop unity. This is because we can always shrink lhe closed path C to a If the area tensor of the small closed path is given by 6."u. then the loop becomes

(10.7.2) ",o'er, if the path is not simply connected. this argument fails. Thu..., we vanishing curvature tensors, yet the Wilson loop does nOI have to be

[ 1,7J. i is ideal for our case, because we are now considering nons imply conmanifolds.IJU does not equal l , then the group £6 hreaksdown to Ine that commutes wilh U. tbat

£6

contains 8 maximal subgroup:

(10.7.3)

C is the strong color group and Land R represent left and right weak breaking can be accomplished by choosing one element, Uo of £(,. that

Va = 1.

(10.7.4)

element generales the permutatioDgroup

Z~.

We now want the subgroup

that commutes with Uo. I.II general. this will be a group of rank six. We ,:foce,e,"'pl" , choose a specific element of £ 6 Co be

u, = (.) x

( ~o :0 ~

~ -2

) x

( ~0 0~ ~s ) .

( 10.7.5)

the matrices represent the clements of the three SU(3)'s within £6. and . r, 0 are all nth roots ofumlY. We then have the following breakings for choices of these elements:

=!

->

y!e= 1-> SU(3)c®SU(2),.®U( I)0U( I)0U(I), SU(3)c ® SU(2), ®SU(2),,, U(I)" U(I). (10.7.6)

selecting out different elements of £6. we can get different groups. For

,

E

" Z ". =SU(3)c"SU(2),®SU(2).®U(I)"U(I). ; , j x Zs

( 10.7.7)

"pl;c'. form of Uo is

( ~:, o 0

~ . )( ~';.

a - 2J

0

0

o o p-v

)

(10.7.8)

434

10. Calabi-Yau Spaces and Orbifolds

Yet another choice is to fix the discrete group to be Z3, in which C·ili0~IMI

E

.

~ = SU(3)c ® SU(3) ® SU(3). Z3

cJl~~I~~m~

In general, these solutions that contain the minimal model also gauge interactions. In fact, there are 27 subgroups of E6 that can YJ~#l:i~~ the minimal model, and aU of these have unwanted U(1) gal:lgegr~~ni~ survive the breaking. These, of course, must be further ,",,,'JUWlJ.a,'".L,l~ through another mechanism such as exploiting any ~'flat" Cllf,ectf9:i3~~:ii1 superpotential.

10.8

Orbifolds

Although there are still problems with the Calabi-Yau cornpatetu method, it is rich enough, in principle, to provide qualitatively tI1e:~j~~ for breaking the gauge group down to the minimal theory. In practice, however, Calabi-Yau manifolds are quite runlCUu. and only a few of them are actually known. We would like to h~'iI:~:~ fiat-space solutions to investigate. Unfortunately, the simplest .tot;i~4! pactification is unacceptable phenomenologically for, among N = 4 supersymmetry survives after breaking. If we start persymmetry in 10 dimensions and compactify down to four wind up with N = 4 supersymmetry. (If we start with N = 1 10 dimensions, the 16 supersymmetry generators QU after . i .,. ,.'~~.!-:.~ become Q f3, where fJ is a spinor in four space-time dimensions ~ from I to 4. Thus, compactification of N = 1 supersymmetry nn'wn' number of dimensions always leads to an extra O(n) symmetry.) .'.~ij~j~g However, by constraining the toroidal compactification in a .'q

rigid fashion, we should be able to reduce N = 4 ~~~:ri;~·i·J. ~ . ~i~l~a 1 supersymmetry. The proposal of Dixon, Harvey, Vafa, and ." compactify on an orbifold [8, 9] obtained by taking a manifold ' •• out by a discrete group, such that there arefixed points (Le., . M~~. change under the transformation). An orbifold, because of the sin~!~tjj these fixed points, is not a manifold, but apparently strings can p · t;o.P:~(gl these orbifolds without difficulty. The advantage of orbifolds is
= e21Ti / n z

for some integer n. This divides the complex plane into 11 eqlJiv~ue~lt#~ sectors. Notice that this identification divides up the plane witttt:l~~~~ symmetry group Zn.

10.8 Orbifolds

435

'0';« that the origin is a fixed point under this tnulsformation; Le., the maps into tbe origin under this rotation. Now if we were to slice up thc plane, cxtract just one of these triangular sectors, and then ",-rap up according to the above identification, we arrive at a cone. Thus, the or orbifold is nothing bUI a two-dimensional space divided by the action discrete group

"'"«"

cone

R' = orbifoJd = -Z.

(10.8 .2)

origin, which is a fixed point, now becomes a potential singulari ty. Thus, is nol a manifold. If we follow a path around the origin, the total angle ""''''''' is nol 360 degrees, but 360 degrees divided by fl. us take another simple example. LeI us start with the two-torus defined

"'king the identification .t=z+ l , Z= Z

+ i.

(10.8.3 )

,dii,,;,lcs the complex plane into an infw..ite number of squares of width I edges are identified with each other. Now let us create an orbifold out tWO-IOrus by dividing by 2 1• which is generated hy reflections

P(,) = -,.

( 10.8.4)

construct the surface

T,

(10.8.5) P an orbtfold. Notice that the reflections leave four point" invariant, are the fi;o.:ed points:

,1.

o.

~( I

+0.

while

( 10.8.6)

of these fixed points lie on the edge of the unit square, one Lies the square. does this orbifold look like'! Under the action of the reflections, the within the square are funber identified with each other, making the Now imagine two smaUer divide up into smaller squares of width each of width Place them directly on top of each other. Now four edges of these two squares together, forming a closed surface. og;oaJlly. it is the same as the surface of a square beaubag. that this surface has four singulari ties, corresponding to the four

t.

t.

~:~:!;f~::~.ilf,:~:we followed B path around each of these fixed points, would be I SO degrees.

~

, to convert this orbifold back into a nonnal manifold calculate its Euler number. Let us cut off each of the four fixed Tl , divide by Z,. and then resew back a small patch at each of these "",
436

10. Calabi-Yau Spaces and Orbifolds

out these holes, sewing the squares together to form the back four patches, we obtain a manifold topologically The Euler number of a disk d is equal 1 and the Euler torus is O. Therefore, the Euler number of the torus minus lUUI£:![l(eQ equal to

to

X(T2

-

4d) = X(T2 )

-

4X(d)

= -4.

Ifwe divide by the action of 2 2 , then the Euler number of the res~~j~i IS

X (T2

;24d ) ~ -2.

Finally, by gluing four disks back onto the surface, we must "'U~knjl~J the Euler number: X(

T2-4d Z2

+ 4d)

0-4 = -2

+4 =

2.

This checks with our intuition, because this manifold is sphere S2, which has Euler number 2. Now, we would like to generalize the previous example by 9' ~I§I~ complicated compactified surface in six dimensions. Let us the complex plane by making the following identification: Z= Z { Z = Z

+ 1, + er i / 3 •

The first transformation divides the complex plane into an . narrow vertical strips. The action of both transformations plane into an infinite number of equilateral triangles. The gion" T of this space consists of two of these equilateral back. This space, because it consists of an infinite number of is invariant under rotations by 120 degrees:

Thus, this space has Z3 symmetry. Under this rotation by 12()4~~~1l are three fixed points, or points that are left invariant: . ;-.

z = 0,

2e21Ti /6

.j3'

Notice that the fundamental region T, which consists of two eqi»t~~in gles, contains three fixed points, one at the origin and the otherp':~~:5M two equilateral triangles. '

10.8 Orbifolds

437

let us perfonn the folding operation on this space T and obtain an T z- -. Z,

(10.8.13)

, Z is not a manifold because the three fixed points (two of which occur the region T itself) arc potentially singular. generalize to the six-dimensional case, we might consider simply taking . product of three of these complex spaces:

(10.8.1 4) has 3 x 3 x 3 = 27 fixed points.

space has the great advantage that the N :;::: I supersymmetry in 10 "',iioru".n be broken down to N = 1 supcrsymmetty in four dimensions. ," compactify on the space:

M4

X

Z,

(10.8.15)

thaI the O( I0) group is broken down as follows: SO(IO)::J SO(4)® SO(6).

(10.8.16)

50(6) = 5U(4), we can show that the original QO of N = I, D = 10 ~;:;~. is now broken down into four spinars transforming as the ~ under 5U(4). Ifwe compactify naively on TI>. then SU(4) is and each o rlhe four supersymmetry generators in four dimensions r%:::,~:;'~:""":·;ves. However, if we start with an orbifold by divjding by ~c ZJ as beiongingEo Ihc SU(3) s ubgroup ofSU(4). Thus, by Zl necessarily breaks SU(4) symmetry, since there are no ilirce~ion.' representatio ns of SU(4). But only one of the four components 4 rurvives the division by ZJ, RIld hence only N = 1 supcrsymmetry , whic h is fortunate . •• umn",cy, orbifolds can be used to break the overa ll symmctry of the because only symmetries that commute with the discrete group survive. be used to break the ga uge group as weU . :: g be a spt."eific element oftbc JO·dimensional gauge group such that

'!",,""'oa.

(l0.8. 17) integer m. Then we demand tbat the s tates of the theory Bre those that with the combined action of

g x Z ...

(10.8. 18)

we have a mechanism for breaking both gauge and supersymmeuy

= 1, which is an element of SU(3). the s tates of the theory he invariant under the combined

we might take g3

438

10. Calabi-Yau Spaces and Orbifolds

."

operation yields the breaking of E8: ~

E8 ® E8

Es ® SU(3) ® E 6 •

Now let us calculate the Euler number on this surface and of generations. Once again, we must cut off the 27 fixed pOilllts divide by Z3, and then sew back on 27 patches. The Euler number of the surface T is zero, so the manifold T . disks d located at the fixed points has Euler number

X(T - d) = -27. Ifwe divide by E 3 , the Euler number becomes -9. Now we mUlst.~~~. disks d. This time, however, each disk has SU(3) holonomy and'~:'~~j~:wi 3, so the total Euler number is X

(T-d) + - = Z3

27XCd)

72.

Thus, we have 36 generations offermions. The previous example was only a toy model in six dirneIlSi()ns,.:lt;~lii however, to construct orbifold models that have much fewer ge~i~~iHii low as two or four.

10.9 Four-Dimensional Superstrings In model building using orbifold compactification to produce fOl~·~!im~ stringent constraints must be met, such as modular invariance an(Ufii~1fJii:i1 conditions. These conditions are nontrivial, because modular up the boundary conditions. For example, when studying modular invariance, we can six-torus T6 divided by a discrete point group P = Zn such boundary conditions for the orbifold T6/ P are as follows: X (0'\

+ 2](, 0'2) =

hX(O'l, (2).

+ 2Jl') =

gX(O'l, (2),

X(ul.O'2

where h and g are elements of P of order n, that is, gn the usual bosonic or fermionic boundary conditioIlS, we have ." to ±. However, when compactifying on orbifolds, this generalized. The presence of g and h, of course, can break the overall theory, both for space-time and for the internal group. The ~c;~~~al the compactification is the subgroup that is unaffected by this subgroup that commutes with g and h. To study how the one-loop trace is affected by the co~n~~ct!~~~~1 diagonalize g and h in terms of their eigenValues, g. h ~

10.9 Four-Dimensional Supel"l'itrings

439

so that they are of order fl. For simplicity, we take the following odid,.y condition: X(a

+ 2:rr) = e2>riv X(a}.

(10.9.2)

that the Fourier decomposition of the string modes is now shifted. we must now expand the string in terms of a new set of X(o-) = p"'''n~' of the

Li(m+.. p X m. '"

(10.9.3)

t.' w ithin the Fourier modes changes the trace ca1culation one-loop amplitude in a subtle but important fashion, which wiJl put

rl~:'~:~~:;::~I~' on the Vi · calculation., the presence ofll enters in the zero point energy. For

;I

bosonic string, for example, the unregularized Hamiltonian contains ~ Ln a_nan + anl"L n . This, of course, has infinite matrix elements be nonnul ordered. T~ehn i cally speaking, normal orderi.ng the ere' ..d ."ni,h· operaEOrs creates an infinite zero-point energy given by This infinite zero-point energy can be handled in several wa)<-s (e.g., . theory to be Lorentz invariant), but it can be shown that the same resul ts as zeta function regularization. Toe zeta function

(10.9.4) ""','".. is analytic in $, and we can analytically continue the zeta fuoction and show that {(-I) = - I~' This is the desired answer, because is the analytic continuation of I::'l n. usc zeta function regulariutiuD to compute the contribution coming shined modes. We use the fact that I

"2

L M,ZM

II = L(II

+ ar' t~-l =

-

1 24

+ ~a( I -

a).

(10.9.5)

,,~z

- a) factor is the new contribution to the zero-point energy in the trace arising from the shifted modes of the orbifold. '" "onoiil",h., h'>I"cot,i·,"tcin.g, 'Nh,,, "" trace over the single loop yields containing the left- and (ighHnoving energies:

(10.9.6) r

+ /I, the expression remains invariant if we (10.9.7)

'~~;l~~~,for the right-moving sector as e2"hl and the two sets ;e as e 2Jt iv " and eZ>riVl. for the O( 16) 00(16) subgroup

440

10. Calabi-Yau Spaces and Orbifolds

of E8 ® E8, then from (10.9.5) we have the following zer'O-t:IOlllt;:e,fili to the ground state energy:

" Vj(Vi ER = 2:1~

-

1) +

1

12'

I

EL

" = 2:1~Vli(Vli -1)+ ,

1

12

+(1

#

2).

Since v = r/n, we can put everything together until we have> constraint:

L::rl = Lrfj + Lrii · j

j

which is true mod n for odd n and mod 2n for even n. Now consider the effect of these constraints on the Cotnp;il¢~tM~ cussed earlier with the Z-orbifold, Le., T6/ Z3,SO that the order 3. This means that 3Vli and 3V2i can be set to be lattice v¢~:tQi'$:) The constraints from modular invariance tell us that v~,2i sh(lIW~!:1.1¢ some integer. But since 3V1,2i is a lattice vector of E 8 , we can l\Wl~ be ~ times some integer. Finally, we know that any point in space containing the E8 lattice is within a distance of I from This means that we can always choose vr 2i < 1. Modular invariance is so rigid that there are only five tent with the constraint (10.9.9) on the various v's. Each of v's breaks the symmetry group Es ® E8 down to the sullgr10titn mutes with the g and h twist factors. It is not hard, given the: for the v's, to calculate what this subgroup is, Because 3 w~~m lattice vector, the subgroup that survives after symmetry brciatiift group that commutes with these lattice vectors. We simply q)lj~tt~ solutions [8, 9]: al'

(1)

The solution has no chiral fermions and hence is unIlhy'sicaI (2) 8

L V~i =~, i=1

S

Lvii =0, 1=1

Vii V2i

1

I

2

)

= ( 3' 3' 3' . .. , = (0, ' .. , 0).

10.9 Four-Dimensional Superstrings

leaves us with the group

E~

441

® SU(3) ® E8 •

(10.9. 12) VII::::

(i.~.··

'-'1; = (~.

.).

o... ,).

leaves us with the group E7 ® U(l) ® 80( 14) €I U(I}.

V J/

yields

E~

=

l'J.,'

=

(t. i, ¥.... ),

(10.9.13)

® SU(3) ® £6 ® SU(3),

,

" , L

.

(10.9.14)

V 21=9' I

I

I

I

2

)

= ( 3' )' 3' l' :3"" , Vu = 0, ... ).

Vii

a.

leaves the symmetry group SU(9)® SO(14)® U(l), (The symbol ... series of zeros.)

;'P"""'"

lot;" groups we have constructed so far with orbifolds do not resemble Model al all. The gauge groups are still too large, and there arc . Models of this type often have 27 generations because conSTruct string fields that are twisted around lhc 27 fixed of the orbifold, yielding a redundancy of 27. However, we can further the gauge group and control the number of generations by postulating ';'~~~.~~ Io~f background gauge fields (WilsOIl lines) that correspond to ).r loops on the torus. This can, in principle, yield models v.1rh generations and the gauge group SU(3) ® SU(2) ® U(I)n . As in .. i"us discussion, the gauge group that survives wiU be tbe subgroup Es that commutes with the Wilson line. In the case of orbifolds with lines. the gauge group that survives is the subgroup of Es ® Es thai with P = Zl as well as the Wilson line.

442

10. Calabi-Yau Spaces and Orbifolds

For example, define the Wilson integral parametrized by

f i

at .:

A!/1 dxll- = 2JrA!/ ell- 2Jra!I ' 1'-

where i = 1- 6 and I = 1 - 16 and ef define the lattice of the in our constraints \UOL ,"'!:n:t;1l we have now introduced a new vector to construct new solutions. We can show that 3a/ and 3v I are vector. Under a modular transformation 'f ~ -1/ 'f, the E8 ® E8 l@~¢~:~~ transform as

a/

pi ~ pi

+ Vi + 11 ,a! '

where 11, = 0, ± 1. Different values of n, will yield different twl,.~~~~~ Repeating all the arguments we used earlier, we can also show thaL~f:Ij(~;~~ transformation 'f ~ 'f + 3 results in the constraint [10, 11] 3(v l + n,a!)2 = 2m for some integer m. This is the desired constraint emerging *. p.ili;~il@' invariance, which is much less restrictive that (10.9.9). An extraordinarily large number of solutions are now !-'V"'''IVI'',,: cause of the presence ofthe a{ term. However, let us select one that yields several desirable features, such as three generations. the following set of values for the and

vi

at:

Vi = ( and

t,

a: = (

(~, ~, ~. ~. 0, 0, 0), (0,0,0,0, 0, ~, 0,0),

1_(

(0,0, O. O. 0, O. 0, ~),

a3

-

(t, t, ~,~, 0,0,0, t)·

We can check that in the untwisted sector, the subgroup of E8 with the action of the Wilson line is given by

that:~j~ml

[SU(3) ® SU(2) ® U(1)5],

while the other E8 is broken down to SU(2) ® SU(2) ® U{I)6. Furthermore, we should note that the number of generations duced to three. (This is because the 27 fixed points can be furtherdi three sectors of nine fixed points if there is a Wilson line in one complex dimensions. Adding two Wilson lines creates nine sec;t(')~::ij~ fixed points. A careful choice of the Vi and will kill all but one ing us with three generations.) As a further bonus, this model has thei4el

at

10.9 Four-Dimensional Superslrings

443

'p,.rty thai extra colored triplets, which might mediate fast proton decay, are limitation oftllls model, however, is that there are too many U( I) factors, 1 is bad for phenomenology. In facl, the Wilson liDe technique does not the rank of the group (because we are dividing by a discrete group), still have gauge groups of rank 8, which is too large. These eXlra U(l) , moreover, must be checked to see if they are anomalous.

far, none of the solutions obtained with orbifolds [12-25] (or Calabi-manifolds) has exactly the desired low-energy s{tuClurc. There are s till bloms getting the right generation number, the right gauge group at low , acceptable values for proton decay, etc. The point, however, is that ~,::;~:::: of COOlp3ctification b.-ives us the ability 10 construct potentially iii of solutions that mny be compatible with modular invariance. is required, of course, is a systematic way to compute all four,,",,;0,,1 physically relevant solutions to the string equations. So far, the of the work has been to postulate a particular compactificatinn scheme :. then check for consistency with modular invariance. Recently, there has considerable work in precisely the other direction, i.e., starting with invariance from the beginning and then searching for all possible schcmes compatible wirh it. at the outset demands the fo llowing set of critcria for relevant model; namely, the model must be: (a) tachyon-free , . anomaly-free, (c) modular-invariant, (d) supersymmClric, and (e) fourSo far. there are only preliminary results toward this ambitious but the early results havc been encouraging. we want is a way to calculate all coefficients C within the one-loop (5.9.6) for each spin structure. For example, in Sections 5.9 and 5. 11 the one-loop and multi loop spin structures and how they changed . Nowwewant a systematic studyofrhe effect of invariance on the multiloop !':pin structures {26-31]. Let us begin with "I'.os;oo of the string amplirude in tcrms of all possible spin structures:

''''';<'"',!.

(10.9.22)

a and b label various spin strucrures over thea andb cycles, each element a is equal to either 0 or 1, C is a coefficient tha t must be determined, A's represent the amplirude for cach spin structure. For each fcmion. are 22: different spin s tructures on a Riemann !':urface of geuus g. us rcwrite the equations of Sections 5.9 and 5.11 in :l more systematic . For example, the theta functions must satisfy (5 . 11.5) when we make , Z) transformation on them. Ol.'mandiog modular invanancc at the level means that C must satisfy the change in boundary conditiorts by (5.9.7) and (5.1 1.l 7). In this new notation, we can rewrite these

444

10. Calabi-Yau Spaces and Orbifolds

constraints

C[ a ] = b

_e(i1l/8lEa,c [

a ,. ] a+b+l'

and

c[: ]=

e(i1r/4)L
] ,

where the sum L is taken over f fermions (taking left movers), each af takes on value 0 or I. and vector sums are When we demand modular invariance at the multiloop level, further restrictions on the C's. For example, unitarity re~~~i;~~~~l~~~BI factorize the amplitude describing a genus g surface into the 1 tori; i.e., by (5.1.7) we should be able to slice up a genus different genus I surfaces by inserting a complete set of In our new notation, this means

C [:: ... :: ] =

c [ :: ] ... C [

:: ].

As we saw in Chapter 5, Dehn twists can mix up the (a, b) g surface, so we must also demand (at least at the two-loop M'e(ill'/4lI:a,aic [

a ] C [ a' ] =

b

b'

c[

b

a

+ a'

] C[

b'

where 8 equals + 1 (-1) if the state is a space-time fermion (bo.son};: What is remarkable is that a rather simple solution of these ·Ci~t~f1:~1fj possible. First, let us make a few definitions: (1) Let us replace the coefficient C [ : ] with the eqtlivalent ex:prc:ssil\)jj:l[~* where a (13) is the set offermions that are periodic around the4(t~r.jM (2) Let us introduce a simple "multiplication" rule: af3 = a U 13

-

a

n 13.

This means that a 2 = a - a = 0, the empty set. Let F equaltl ·· .i.·e:~1~ fermions. Then the product Fa equals the complement of is, Fa = F - a. (3) Let us introduce Sa, which equals + 1 for fermions and - I (4) Also n(a) i~. the number ofleft-moving fermions nL(a) LLJ.l'''U~~::n!~ of right-moving fermions nR(a). (5) Let us define Sa = e j ll'n(al/8 and the parity operator ( -If fora sp.tti.~ll~ a that obeys (-1)u1/l = -1/I(-1)U

if 1/1 belongs toa.

( -1 1/1

if 1/1 does not belong to a ·

r = t/I( -1)'"

10.9 Four-Dimensional Supemrings

445

, A spin structure (a I f3) is compatible with supcrsymmetry if (-I)"TF(-l)"

= B"TF •

( 10.9.29)

where TF is the genemtor of supersymmctrlc transformations in thc 5upcrconforTIlai group. : Define Y as a collection of sets with the above supersyuunetric prop::: erty (which forms a group closed under our definition of multiplicatioo :: «(0.9.27). Define::: to be a subgroup ofY that satisfies (10.9.30) ': : where C(0 1m"" 1.

that we have our definitions, let us Statc the constraints arising from MU' E""">use n(a» I'"~ m>ultipl>, !ht, wehavc lheeonstraints basis elements bi of 8 :

air."1

'oy,man"" .

II(b l )

= omod 8.

n(b/n bj )=O mod4. n(b l n h} nb t il b,) = Omod2,

( 10.9.31)

..,llas the cODStraints emerging from modular invariance: ifa. {3 E 2 . th .

I P) ~ [

C(a

I fJ) = e!I1~C({J I a). I cr) = GFG"C(cr I F),

C(a C(a

±I

C(a

I p)C(a I y)

~

0

5.C(a

o

eI'WlSC,

(10.9.32)

I py) .

• {to y wi thin the collection 8. we have the constraints coming from the confonnal anomaly cancelwhich must always be strictly observed. Let us usc the fenn ionic (not

o~,:; ;:~~>:;';~ti~·~~)~Of the lattice compnctification. This was used for the

:t1

. We musl carefu lly include the fcnnionic cOlltncution

conformal anomaly in the supcr-VirdSOro algebra. It is rather remarkthat a representation of the fermionic partner T,,· to the energy-momcntum 1>, g;'ven strictly in teons of the adjoint representation of fermions:

",T.cru,

(10.9.33) we calculate the anomaly arising from this term, we find thai the ru~~~:contribution to the anomaly is ~ N, w here N is the number of !,Il in the group. left-moving seclor, the number of compaetificd dimensions is 26 - D , D is the number of spaec-iime dimensions. We represent the eompactby fermions in the adjoint representation. In this seclor, the three

"',to,

446

10. Calabi-Yau Spaces and Orbifolds

contributions to the anomaly are XI-'

left movers:

{

b, c ghosts 1{ru

Since the sum of the three contributions to the anomaly must.•• ~·\t~~=WI obviously have N = D) fennions in the set 1/fu. Now let us analyze the right movers, where the anomalies mllst::M!:l:l:zero:

2(26 -

right movers:

XI',1{r1'

{

b'Ct~'Y

I {-2:;11 I. 3D/2

--+

The condition for zero anomaly is therefore N = 3(10 - D). tn:i;iJ@jiil (D - 2)NS-R fennions (in the light cone gauge) contained in D - 2 + 3(10 - D) fermions for the right movers (in the light COl1ll;1:g~ summary, we must have the following number of fennions in O[{tet::lQ:~ml the anomalous term in the Vrrasoro algebra: left movers --+ 2(26 - D), { right movers --+ D - 2 + 3(10 - D). The anomaly cancellation is automatic if we have this many left- and right-moving sectors. Now that we have explicitly expressed all our constraints, our follows: (a) We first calculate the total number of space-time fermions 1/f.~.'~II fermions 1{rG that satisfy (10.9.36). which removes COlltOrm:~~,: from the theory. Then we calculate the total number of spin . arise from this set, which we call F. (b) We next randomly choose a collection of spin structures as within B that includes F, the complete set. (c) We then check for closure under multiplication (10.9 .27), whlCti;~ the full group B compatible with this initial choice. (By chclosj;IjgJi,p initial sets for 8, we can reproduce different cOlnpactifi(:ation::~M Cd) We then calculate the square matrix C(a I /3) defined ov¢nll~ that satisfies our constraints. Some arbitrary phases are it·ltrOlq\tl~~:4i~ fashion, whichallow us to generate more than one solution fqr:M(#.f. We will now discuss some of the properties of the soluti~.tiltI!~ equations. Remarkably, we find that all consistent solutions nelc~~@tnl gravitons, dilatons, and antisymmetric tensors. We also find thaLttl~¢:l. ofthe massless spin-~ field is sufficient to prove the absence ofta¢ft}1~ the vanishing of the cosmological constant at the single-loop l<:iV"h".'"

J 0.9

Four·Dimensional Superstrings

447

'~:~:~f;I:e;sult", and they show there is tremendous self-eonsistcncy within ~ i that yield phenomenologically desirable results. let us discuss 11 few specific solutions to these equlltions. We will discuss the lO--dimensional Type II theory (without compactification) in cone gauge, after we have imposed all the light cone constraints on _ :: fennions. Then the fennions consist oflbe eight-component right-moving -:. fermions!XR = lfr l - B and the left-moving fermions (Xt. = lit l - ', internal fermions are set to zero, t/td = O. simplest choice is

(10.9.37)

8 = /0, Fl .

,""mma' ely. a carefu l analysis oflbe sp ectnun of lhis theory s hows that hID; tachyon:> and hence is unacceptable. choice

(10.9.38) more than one solution. depending on the choice of certain pbases {~~~ the C(a J /3) matrix. Tnc two cnoiccs reproduce the Type llA and ::: ITS theories. the four-dimensional cOOlpactifi ed theory, we must choose a different 2. In four dimensions, the spaco-time fermions are represented by bHmovin.g transverse l{r1-2 and left-moving According to the internal degrees offreedomcan also be represented by 18 = - D ) fcrm ion.<;, in wh ich case we have )..1-18. for the right movers ~1_18 for the left movers. We will regroup thesc 18 internal fennionl'i ,~;:~~::v;~~as (x', yl, l'), where I :::: 1. ... ,6 and represen ts six SU(2) U1 Wc then ChOOlOie the subset !X = (1/11 - 2, ,1 - <» . Then the choice

1/11-2.

S = {0, 01 , Fa, FJ

(10.9.39)

p'",du",~ several N

= 4 s upcrsymmctrie four-dimcnsional lheuries, with 'ymmetry SU(2)'. SU(4) ® Sum,.1 SU(3) ® SO(5).

, let us analyze the IO-dimensional heterotic string in the light cone where we have the transverse right-moving spaeo-time spinors O1R = .,,,' II,,, 32 :::: 2(26 - D) intemalleft-moving spinors Ci!L :::: Fa R = )..1 _ 32.

3 = {0, F l

( 10.9.40)

an SO(32) theory with t3cnyons and no massless fermions, which .

I

8 = {0, !XR , at., F } lhc usual Spin(32)j 22 heterotic string.

(10.9.4 1)

448

10. Calabi-Yau Spaces and Orbifolds

(3) Ifwe define t¥1 = p"I-16} and a2 = {J...17-32}, then the set

has two possibilities, depending on the choice of phases.

:]]1111

leads to the standard supersymmetric E8 ® Es theolY, and leads to the (nonsupersymmetric) SO(16) ® SO(16). (4) After a four-dimensional compactification, we can again : ' : ' , right-moving internal fermions as (Xl, yl, Zl) for I == I ' ,,' a = (1/11-2, ZI-6). Then the simplest choice S = {0, a, Fa, F}

yields an N = 4 supersymmetric theolY with SO(44) gaulg¢:,SW This is one ofthe solutions found by Narain using even selr-o'Q'lt~i lattices. Clearly, there are probably thousands of different sets of" under the multiplication rule (10.9.27) and satisfy the modul This formalism gives us a handle on the problem of cOIlstructingi:~~~ classification of all possible compactified solutions compatible wij~n:n invariance. Before concluding this chapter, it is worth pointing out other
Wrm

Tr gqHLqHR. The calculation ofthe trace can be done exactly as before, except noj;;4~~~~~ are different. As before, we find that the condition for modular '.~,i.)" r~~m level matching;' n(E L - E R ) = Omod 1.

If the eigenvalues of g are represented by e2rr1r;lll, this condition i!' jl\fe!~ strictions on the sum r}. The resulting constraints on the identical to the ones found for the symmetric orbifold (10.9 .9), exc:¢p~~~

Li

10. 10 Summary

449

sectors are now treated differently, depending on the strucasymmetric orbifolds are more difficult to work with i a hu-gc class of asymmeoic orbifolds can actually that are compatible with modular invariance. (tn practice. the orbifold is sometimes embedded in a larger symmetric orbifold, the calculations an: simpler, and then truncated at the end to retricve the ,",,<:
:

:~\::"': cnonnous class of modular invariant compactifications in four This will be of great help towurd understanding tbe complete set I relevant superstring compactifications.

Summary seen thai Ii few physically reasonablc assumptions about the com. process have led to a wealth of phenomenological predictions. no model has yet been proposed that can successfully predict all properties of the low-energy particle spectrum, we are encourby Ihl:: qualitative results we bave obtained. Let us now specifically tbe logical sequence by wbich given assumptions led to ec.rtain choice is 10 compactify the lO-dimcnsional space on thc However, compaclifying on this surface transfonns N = ",''"'"",''1' in 10 dimensions into N = 4 supcrsymmetry in four diThus, wc must search for new assum ptions that will be morc ;o""e<>olog·j "Uy acceptable. began with the following assumptions: lO-dimensional universe has comp.1clified to M to

--10

M~ X

K6 ,

(10.10.1)

M4 is a maximally symmetric space,

R

RJl.WJfJ = i2(8J1.agl'6 - gJl.fJ8"",),

(10.10.2)

K is compacL N = t local supcrsymmetry remains unbroken afier the compacrifica-

of the bosonic fields can be SCI to zero: (10.103)

e ;~;~:.~~~~, that N :: 1 supersymmctry survives the compactification, tr assumption . It implies that the variation of the spinors must be

10. Calabi-Yau Spaces and Orbifolds

450

zero I

81{1A = -DAe = 0, K

8X a

= fij

F;je = O.

The existence ofa covariantly constant spinor e places an en()rol(l,~gj of constraints on the manifold. In particular, if we once agam Die = 0 -?- [Dj, Dds = 0, -?- Rijkl rkl c = O.

Thus, we can show that the space is Ricci flat. Second, we cal'LSl the space has SU(3) holonomy. We deflne the h010nomy OTr.,,.... generated when we take a spinor around successive "'v.,,,upal:hs.al'~i@;i Normally, in six dimensions the holonomy group is SO(6), ·or.·,.r>i:f;.'i SU(4). But if e is covariantly constant, it means that c-?- Ue. By an SU(4) transformation, any spinor can be brought into

o o o eo This, in turn, means that the U matrix rotates only the three the column matrix. Thus, U must be a member of SU(3), and SU(3) holonomy. In general, spaces with SU(3) holonomy are notoriously difft~~ij~ with. However, we can show that the space is Kahler and so us(Hl::pj theorem by Calabi-Yau. We can always define the tensor

J ji =

. ik-r

-lg

e

kje,

which satisfies J2 = -1, just like i 2 = -1 for the usual dejnnitid~:~i~i111 numbers, and so we can use this tensor to define a complex ~~~:~1;1~)~~~1 we know that this tensor must be covariantly constant (because e . constant), and thus the space is Kahler. Fortunately, Kahler manifolds that are Ricci flat (and hence ha;1f.1M!9! first Chern class),can be shown to be equiva1ent to spaces of and are easy to construct. Thus, we have the logical sequence Die

= 0 -?-

K is Ricci flat, Kiihler, with vanishing first Cb.ert'iii}X

The problem, however, is that now we are flooded with polten'oat1)rl of manifolds preserving N = I supersyrnmetry! :;:;::~::--M

10.10 Summary

fur. we have not broken the £ 8 ®

451

E~

gu uge synunetry. Our next step is thai the Bianchi ide ntities, which use
Tr R

A

R = l~ TrF 1\ F.

( 10.10.10)

''2:~:~:St:~;:~I~;:d~iffiCUlt to satisfy. On tbe left, we have the C\JIVaspace, and on the right we have the Yang-Mills

:Fe

forms. In order to solve this !1I.ther strange-Iook.ing equation, wc will the spin connection into the gauge fie ld, thereby mixing the two manand breaking the original gmlge symmelI}'. Because we have SU(3} in the Riemann sector, we will have the following breaking in the when we mix the two sectors:

I;""hi' identity _ $pin embedding -+ SU(3) ® £6@E•.

(10. 10.11)

gives rise 10 desirable resu lts. We know tha t E8 by itselfdoes representations and hence is not an acceptable candidate fo r building, but E. does. In fact, the decom position under this breaking of elements of the adjoint of l!:1 becomes

248 = (3. 27) Ell (3, 27) Ell (8, 1) Ell ( I, 78).

( 10.10. 12)

: i$ gratifying, because the 27 is the most suitahle rep resentation fo r tbe '. leptons in GUTs with 1::4 , Thus, we It.ave A R

=

~ Tr F A F -+ spin embedding -+ 27 fermions.

(10. 10.13)

next question is: How ma ny generations of the 27 do we have? No r,,",e li""ce."" n number has nothing to do with the gauge group. In GUTs, entirely di5tincL We can have an arbitrary number of exact c()pies of i in GUTs. However, in string theory the process of s pin produces a constraint on the generation number. Because of spin the gauge fennions are now linked to the K6 manifold. The gennumber can be viewed as a topological number, bccause the di fferen ce positive and negative Chilli! zero e igenvalue solutions of the Dirac is a topological numbe r. Specifi cally, generation number = ~1i«M)I.

(1 0 .10. 14)

ulee oo""lb" ofRicci-ftat Kllhler manifolds is usually quite we can always reduce thil! numbe r drascically by taking a subman iexample, we can divide by 8 discrete group thaI preserves a certain 'O"ual in the coordinates. A good examp le is C p~ divided by Zs x 2 5• not connected. This mllOifold has four generations. the model even further to the Standard Model SU(3}® N = I supersymmctry. Normally, Ihis is quite schemes break the group down to the standard model supcn;ymmetry. One solution to this problem is to use the o f Wilson lines. We saw earlier that the manifolds K6 that we are

452

10. Calabi-Yau Spaces and Orbifolds

considering are not simply connected in order to produce a number. For nonsimply connected manifolds, the Wilson loop

U = P exp

(l

A/LdX/L)

does not necessarily equal 1 even if the curvature tensor because we cannot always shrink closed looped into points. break E6 symmetry down to a subgroup G by choosing an elelll~.D.:~:j~ that G commutes with all elements of this element. For eX~IIDJ'le;;it:.wt:ui element of U that is Z5 x Z5, we find

E6 = SU(3)c ® SU(2)L ® SU(2)R ® U(l) ® U(l). Z5 x Z5 Unfortunately, we see that, in addition to arriving at the stand:arili)~~ also have unwanted V(l) groups. Thus, Wilson lines -+ SU(3) ® SU(2) ® U(lt . Orbifolds are another way of compactifying the ably the limiting case ofa Calabi-Yau space. Orbifolds are a torus T6 and dividing by a discrete group Zn, which allows

T6 Zn (These fixed points apparently do not spoil the properties of the Nontrivial constraints are placed on orbifolds by modular j"lllV~j®m-{ mixes the boundary condition nontrivially. The boundary orbifold:

+ 2IT, (12) = hX«(1J, (12), X«(1], 0"2 + 2IT) = g X«(11, (12), X(O"J

'~~::~e~i~;llllli

(where hand g are elements of Zn) can break the model. The subgroup of the space-time group and th~ internal vives the process is the sub group that commutes with g and h. Ih\'~(I~}lO)1j g and h to equal the elements e 2Jriv, , where VI = ri / n for some 41tl~~flj the zero-point energy of the Hamiltonian is shifted by an am.OUl[H to Vi(Vj - 1). Because the trace over the single loop is serls"l~~l~:~~~fi~~1 point energy, this means that modular invariance places a n eigenvalues:

where ri are the eigenvalues from the right-moving sector, ai1<MI1~~M of eigenvalues '1.2;; are from the left-moving E8 ® E8 sectot.:J~':~~ Wilson lines, then the group that survives the compactification that commutes with g. h, and the Wilson lines.

References

453

p~:;:;:::,;:~~~:.\::,~ there are many solutions to this compactification, some

and the group SU(3)®SU(2)® U(l)n. which yields too I) factors Unfortunately. the method of Wilson lines docs not change rank of the we will have too many U(l) factors. over Calabi- Yau spaces is that they arc sim~ of them can be cx-plicilly constructed. Unfortunately, I there is no systematic way to catalogue these """,II, of solutions. step in this direction is to usc modular invanance and the absence of '~Y''"s and anomalies to systematically derive all possible solutions. We with the amplitude wrilten as a sum over spin strucrures:

til

(10.10.20)

we dem&nd that the C's faCtori7.e properly, have modular invariance. and models that have no tachyons or anomalies. Remarkably, it is possible the constraints on C and obtain solutions. The simple ones reproduce of the known compaclifications, but thousands of other compactifications . . This yields the hope iliat we may be able to exhaust . a realistic model.

'Y ;~~~~~:~;~';::7;~~;:~large classes of fo ur-dimensional soluh . The heterotic string is an example of an compl3.ctification, i.e., treating and right-moving sectors dif. Asynunelric orbifolds are the largest class of orbifold s found SO far

left-

using

. ,:::;;;.~~::;:::~S~ii~gn~ificanf[Y with the work of others different types of ~f addition to asymmetric orbifo[ds, we should also menpossibility of non-abelian orbifolds. i.e .• orbifolds based on dividing finite groups such as the crystal groups. The advantage of constructions is that they help us eliminate some ofthc unwanted U(I) and also yield three and four generations. See [40].)

G. Horowitz, A. Strominger, and E. Witten, Nl./c/. Phys. 8258,46 in Unified String Theories (edited by M. Green and D. Gross), , Singapore, 1986. S. Manton, Phys. Lett. 120B, IDS (1983). Geometry und Topology: A Symposium in Honor Qf S. University Press, Princeton, NJ, 1957 . . Yau, Proc . Natl. Acad. Sci. U.S.A. 74, 1798 ( 1977). . YlIU, in Symposi!lfn orr Anomalies, Geometry, and Topology (edited by W.

Bardeen and A.lt While), World Scientific, Sinilapore, 1985. Hosomni, Pl!y~·. Lett. 126B, 303 (! 983).

454

J O. Calabi-Yau Spaces and Orbifolds

[8] L. Dixon, J. Harvey, C. Vafa, and E. Witten, Nuc!. Phys. B261, B274, 286 (1986). [9] C. Vafa, Nucl. Phys. B273, 592 (1986). [10] L. E. Ibanez, H. P. Nilles, and F. Quevedo, Phys. Lett. 187B, 25 [I I] L. E. Ibanez, J. E. Kim, H. P. Nilles, and F. Quevedo, Phys. Lett. ... (1987). . [12] D. X. Li, Phys. Rev. D34 (1986). ... [13] D. X. Li, in Super Field Theories (edited by H. C. Lee), Plenum, .. 1987. •••. [14] V. P. Nair, A. Sharpere, A. Strominger, and F. Wilczek, Nue!. Phys.> . (1986). .. :~:~mim [15] B. R. Greene, K. H. Kirklin, P. J. Miron, and G. G. Ross, Phys. (1986); Nucl. Phys. B278, 667 (1986). [16] A. Strominger, Phys. Rev. Lett. 55,2547 (1985). [17] P. Ginsparg, Harvard preprint HUTP·86/A053, 1986. [18] S. Karlara and R. N. Mohapatra, LA·UR·86·3954, 1986. [19] M. Dine, V. Kaplunovsky, M. Mangano, C. Nappi, and N. Sieberg, B259, 46 (1985). .. [20] S. Cecotti, J. P. Deredinger, S. Ferrara, L. Girardello, and M. KOlIlCft!qeIl~ Lett. 156B, 318 (1985). ·::::7":;::;:;:; [21] J. P. Deredinger, L. Ibanez, and H. P. Nilles, Nue!. Phys. B267, [22] 1. Breit, B. Ovmt, and G. Segre, Phys. Lett. 158B, 33 (1985). [23] A. Strominger and E. Witten, Comm. Math. Phys. 101,341 (l [24] G. Segre, Schladming Lecture Notes (1986). [25] H. Kawai, D. C. Lewellen, and S. H. H. Tye, Phys. Rev. Lett. 57, 12·•2.(.<)::",j(4l~ Phys. Rev. 034,3794 (1986); Nucl. Phys. B288, I (1987); Phys. L.'Ct!~g~~ (1987). [26] W. Lerche, D. Lust, and A. N. Schellekens, Nue!. Phys. B287, [27] 1. Antoniadis, C. Bachas, and C. Kounnas, Nuc!. Phys. B289, [28] 1. Antoniadis and C. Bachas, CERN-THA767/87, 1987. [29] C. Kounnas, Berkeley preprint UCB-PTh 87/21, 1987. [30] 1. Antoniadis, C. Bachas, C. Kounnas, and P. Windey, Phys. (1986). [31] K. S. Narain, M. H. Sarmadi, and C. Vafa, Nue!. Phys. B288, 55 [32] R. Bluhm, L. Dolan, and P. Goddard, Nue!. Phys. B289, 364 (J [33] L. Castellani, R. D' Auria, F. GHozzi, and S. Sciuto, Phys. Lett. 1 [34] L. Dixon, V. Kaplunovsky, and C. Vafa, SLAC-PUB-4282. [35] V. G. Kac and 1. T. Todorov, Comm. Math. Phys. 102,337 [36] P. G. O. Freund, Phys. Lett. 151B, 387 (1985). [37] A. Casher, F. Englert, H. Nicolai, and A. Taormini,Phys. Lett. 1621[s·,. [38] F. Englert, H. Nicolai, and A. Schellekens, Nue!. Phys. B274, 31. [39] D. Lust, Nue!. Phys. B292, 381 (1987). . [40] N. P. Chang a,nd D. X. Li, "Models of Non-Abelian Orbifoldst . •. 87-15. ..

Part

IV

M-Theory

1AP"rER

11

Theory and Duality

Introduction now, none of the methods discussed in this book have provided us with insights inlo nonperturbativc string !heery and its true vacuum. There in our understanding aftbe string perturbation !hooey because of P'~~:::~, associated with supermoduti. Funhennore. string pcrturnation I~ yield the true VACUum of ::;tring theory. Conformal field theories,

they give us

Ii

powerful tool to classify mi ll ions of possible

vaCUIl

:t~Ih::e1:o:i"'1.::ore~:Sll~·1I defined perturbativcly and are unable to probe the . String field theory, although it is defined independently

f

. currently too difficult to solve for the nonpcrturbative of the most exciting new approaches to nonperturbativc string theory M-theory and d uality {J -<», which in fact force us to reconsider the I role played by strings in supcrsymmetry. Not only have they given us results concerning the nunperturbative behavior of string theory, I forced us 10 reconsider previously discarded theories, including membranes of various dimensions (called " p -branes") which I duali ty allows us loshow thai self-eonsistcnt superstring theories arc nothing but different of a single theory, called "M-thcory." In this revised picture, the string theories are nothing but differeut vacua of 8 single theory. While

~

~~"~~Ih~:e,O~'l';~O~O~'~Y:P;'~:O::b<~~S~thecorrelouons vicinity ofacross each VftCtla, duality different vaCtla.allows us

and duality have ye t to give us definitive information four-dimens ional vacua with N ::: I supersymmetry, they have already "'mIlch of the nonperturbative uaturc of string theory in 10, 8, and even

458

II. M-Theory and Duality

6 dimensions, giving us a complex web of dualities betweeDJl~1111 compactifications. Furthermore, M-theOlY indicates that the "true home" of actually be the eleventh dimension, where we find new, exotic .O.~j~m super membranes and 5-branes. ~~llt~'111 Just a few of the surprises and insights of M -theory are as " • All five string theories, which on first glance have ent®J¥~~m properties and spectra, are now seen as different vacua ot1:Ii,¢:;$iUiiE • The strong coupling limit of Type IIA and E8 ® E8 netero,tlcRfrlfli is revealed to be an II-dimensional theory, whose lOV\f-e:n~tg given by II-dimensional supergravity. Compactification circle (line segment) yields Type llA (heterotic) string the()tv,: • The complete action of M-theory is unknown, but is belieYI~¢:l~ membranes (2-branes) and 5-branes. Closed strings in low'er::tlu can be viewed as compactifications of these membranes. • A vast network ofperturbative and nonperturbative aUi:UHlesl~l.t) dualities) has been established between different string~~;~11 dimensions, for the first time giving us insights into the behavior of string theory. • Exact statements about the spectra and properties of pendent of perturbation theory, can be made by analyzing 1:4~;:p.r~~~ "BPS saturated states," which obey nonrenormalization '~;~IIII BPS states can be enumerated from group theory alone b: central charges in the supertranslation algebra. • The nonperturbative regime of string theory yields membranes, called "D-branes," which play an essential Open strings with Dirichlet boundary conditions can eri~~,:::~~:~Mii branes. Surprisingly, these D-branes allow us to use field theory techniques to calculate highly nonlinear intl::ral~ij~~l • By counting D-brane states, we can derive the BekerlS~~~l@. entropy formula for black holes using statistical mech~lDi(:!i:l:1~ • II-dimensional M-theory, in the infinite momentum to a IO-dimensional theory of point-particle Dirichlet 00 limit. which is called M(atrix) theory. Remarkably, this··'· based on a simple U(N) quantum mechanical Yang-Mills .... order, with a finite number of degrees of freedom.

11.2

Duaijty in Physics

Duality is actually an old phenomenon. Maxwell's equations, fot'~mI invariant under the following duality transformation: E -+ B,

11 .2 Duality in Physics

B

~

-E ,

459

(11.2.1)

e _ g.

(J 1.2.2)

g represents the charge of a monopole. Dirac's theory of monopoles that

eg =

(11.2.3)

2Jrfl .

represents an integer. This leads us to a curious conclusion, that theory of clectric charges defined in the strong coupling region • ,b",o,"" lilfge} is equivalent to Ii weakly coupled theory of monopoles becomes weak) end vice versa. The strong coupling region of electro· which is normally prohibitively difficult to analyze, is now revealed :':::~:,%,theory monopoles coupling. Although this result n: to be a curiosity, it represents one of the few instances in where the nonperrurbative region can, in principle. be analyzed to be deceptively simple. to gcn.eralize this (0 Yang-Mitis theory and to supersymmctry, have been mixed. There exists a classical formula from the theory (classical solutions of gauge thcory with both electric and magnetic os) ""to; fl

n

of

al weak

'h"

(1I .2.4) is a constant, and

k

e and g represent the electric and

"}"''ll,,,.N,,ri,. that Ihis gauge theory solution obcys the relationship

it was conjectured that th.is may, in fact, be an exact synunerry of [7-8]. However, attempts to prove this kind of relationship at the have floundered over the years. tantalizing clues were noticed over the years which pointed to the of duality. It has long been suspected that the so-called BPS relageneralize tnc above constmint, are unrcnonnalized by quantum to supcrsymmetry and hence may hold exactly in the completc led some to suspect that perhaps duality may be preserved non· in a cenain class of supersymmetric Yang-Mills theories, such theory. i years ago that the D = II supergravity equations of toroidaUy compactified, obeyed peculiar "'hidden" symmetries relations. But the origin of these duality relations were totally so they were thought to be a curiosity, rather than B fundamenta l

'",oen,tly have all these speculations concerning duality and nonperturtheory been pulled together to give a comprehensive picture, giving ·""dented insight into the nonpcrturbative structure of string theory.

460

11.3

11. M-Theory and Duality

Why Five String Theories?

Duality is emerging as the key to understanding one of the deEi'p string theory, such as why there are five of them. It's puzzling that be five perfectly finite, seif-consistent field theories which comb with other quantum forces. For example, in the Green-Schwarz formalism, notice that ways in which we can construct a supersymmetric Nambu-Goto action. For a supersymmetric theory, we are alHJW<;U duce a 32 component Majorana spinor, which can be de(;OIl[lp()sed 16-component Majorana-Weyl spinors BU. (Introducing more ponents will generate massless spin 3 fields, which are probably mC:Qn:M:t1l quantum mechanically, and also isomultiplets of gravity, which •• v .... ~.:~,w. the equivalence principle.) IfB± is a Grassmann field, ofeitherpm;ithri:l':ijffi tive chirality with 16 components each, then we can construct a sUJ:lers::Yiljiij generalization of 8j X 1': IT) = 8 XIL - ie, r lL 8 B, j

j

which is invariant under the global supersymmetric transformation · • 80

S,

isrlLB.

8XI'

If we have only one chiral fermion B+ on the world sheet, N = 1 supersymmetry (which describes the heterotic string). If and (}: of opposite chiralities, we can combine them into single 32 cQi~ftij Majorana spinor B with N = 2 supersymmetry. This will give string theory. If we have Bland of the same chirality, then this gives us "~U~"'. string theory with N = 2 supersymmetry. We then find three to construct a global invariant n;:

0;

8i Xfl -

BiX

It

-

aixlL -

i8+rlLiW+ i8r lL ai (}

lIA,

i8jk8~rltBjB!

lIB .

heterotic,

Then the first part of the superstring action S 1 can be generalization of the Nambu--Goto action: ,,-

81j =

.

.

niITj/L.

Sl = -T

f

2

d uJ-det gu'

We obtain five superstring theories in all, since there are two tYP·e~:Jm erotic strings, and the Type I string is derived from the Type rill rn".nM

I 1.3 Wby Five String Theories?

461

wodd sheet parity even sector (and adding open strings and Chan-Paton to cancel anomalies). A~~~~~ this Nambu--Goto term is explicitly invariant under a global supcr-

in

transfonnation, it is nOI locally invariant, and hence has the wrong of degrees of freedom. To remedy this, we must add a Wess-Zumino

. We define an invariant form hwl.: W' dO+r IJ. d8+

hwz =

I

nf' dor~ .r!l dO

heterotic.

.

ilA,

Slj nl ' dO~ r l • d8~

lID.

(11.3.5)

is the product of all r matrices and SI; is a two-by-two matrix which .1, th, Pauii matrix cr,. hWl can be expressed as hwz = db. Ifwc carefully b from hwz. then the Wess--Zumino-like pan of the action is given by r

ll

(11.3.6)

the string action equals SI + !h. Vbe.n .,' include both types of heterotic s1rings (with Ea ® £8 and SO(32) melry) and also the open and closed string sector of Type I strings, then ",un aU five superstring actions. (Usually, when we refer to superstring wc !irc actually referring to thc Es €I £3 heterotic string.) and similarities of these theories can be seen by analyzing sector of these theories, which reduces to supergravity in lO The five superstring theories, and their supcrgmvJty reductions, bY'

'T)'1" ItA string theory reduces to N = 2A (nonehiral) supergravity. UB string theory reduces 10 N = 28 (chiral) supergravity. ® Eg heterotic SLring theory reduces to N = I supergravity coupled to €I £8 Yang-Milis multiplet. ,,,,,,.,, heterotic string theory reduces to N = I supergravi[)' coupled to Yang-Mills multiplet. theory, whieh contains both open and closed strings, reduces = 1 supcrgravity coupled 10 an SO(32) Yang-Mills multiplet.

that each of the five superstring theories are significantly different othcrs. For example, Type I theory is based on both closed and open while the others arc based on only closed strings. Similarly, the gauge of the five string theories, except for the last (WO, arc markedly dif. their low-energy structure. The mystery of why there should be five g ",ch of which appal"entlyunifies gravity with the other quantum ,is solved when we begin to analyze various dualities (see Fig. 1 Ll).

""0"''''.

462

11. M-Theory and Duality

FIGURE 11.1. The web of dualities in M-theory. By compactifying WtItel~ii to 10 dimensions, we can derive Type llA strings and the het,ernti. compactifying down to nine dimensions, all five string theories can beltmi~~m!;§

11.4

T -Duality

To understand how unusual T -duality is, recall the fact that, in or~MiW particle fie1d theory, compactification of one dimension leads momentum

n R for a circle of radius R and an integer n. If we take the limit p=-

find that the momentum becomes continuous, and hence we retij~;\i:it:j uncompactmed theory. But now take the limit R ~ O. Here we . momentum becomes either 0 or 00, and hence the col:np:actifie(l; effectively decouples from the theory. We see, therefore, that are entirely different in point particle field theory. Now consider compactifying, say, the ninth dimension, of the theory. Now an additional complication appears which did not point particle theory: the string can wind around the sion. We know that the momentum operators take the following . compactified theory:

(Pl.. PR ) =

(2~ + mR,

n 2 R - mR) •

where the II, as usual, arises from the Kaluza-Klein excitations df'~~ but m labels the number of times the string winds around the c:·lrc.J~~;::::~I Notice that the mass spectrum for M2 is invariant under [9]:

1

R +'>-2R when we interchange n

+'>-

m.

11.4 T-Duality

46J

'unusual symmetry, one which links the large-scale behavior structure. Unlike point particle field theory, the differentiate bern:een these two regions. Notice that this dual ity which intCIchanges winding modcs with K.a1uza- Klein modes, is a result of the geometry of string theory and does not appear in point thcories. rewrite tills duality transformation in the familior language o f ~~'::;;:~nfield theory, this transformation is equivalent to making the

ax ~ ax, &x --+ -ax.

( 11.4.4)

sign change for onc set of movers, When we apply thi::; same du. theory in theNeveu-Scbwat"'.l-Ramond !,N8--R) fonnalism, fiod

( 11.4.5)

-, this transfonnatioll of the ninth lcft-moving oscillator also reverses the ,olfth, IO-dimcnsional lcfi-moving chirality operator which is constructed the fermionic zero modes: (11.4.6)

the chirality of the left-movers. The Type IIA spinors, defined in tcmlS ,,;"tive and negative chiralities, are transformed by T -duality into a theory the same type of chirality. i.c., the Type IlB theory. In nine dimensions, the followi ng duniity emerging [10): T:

IlA~rm.

(11.4.7)

use the symbol ~ to represent symbolically the duality relationship.) therefore view Typc IlA and Type UB as merely two extreme points tbe same continuum of vacua labeled by R. As R -+ 00 or R _ 0, we the usual Type II string theories in 10 dimensions. . the five supe rstring theories have now been reduced down to fOUf. I ,oolh" T -duaJity can be found by analyzing the 1::8 ® £8 and the SO(32) string. COllsider first the Narain lattice. Let us compactifY the bctdown to d dimensions. This rne3ns we must compactify 26 - d dimensious and 10 - d right-moving modes. Lei us eornpaclify a lattice r I. and r /I. We recall that by constructing the heterotic string amplitude and demanding that il satisfY modular we have the additional constraint:

ft.·rt.-rll· r R=Z, Z refers lo lhe integers.

( 11.4.8)

464

11. M-Theory and Duality

The fact that there is a minus sign in the metric means that this . lattice. Modular invariance also demands that these lattices be vV<,U;-:'SI! and Lorentzian, but that still leaves considerable freedom in Ch()QsItlM;i}i tices. In fact, we can always preserve the previous constraint bY>f9.1inm lattice by SO(26 - d, 10 - d) and still satisfy modular mv(rriaJlc¢:~jii configuration representing a different vacuum [11]. The group SO(26 - d, 10 - d), however, is still too large to tvn,ifu'~k;';'~ space of possible string vacua. There are still redundancies. The n:llI'~~~m is still invariant under separate SO(26 - d) ® S0(10 - d) rotllttiffiiNi separately on the lattice of the left- and right-moving modes. Finally, we must divide out by the action of T -duality to space of inequivalent vacua. The discrete group T = SO(26 - d, 10 - d, Z)

merely reshuffles the lattice points md hence does not ChEt1i~,~jtiMl physics. This is the generalization of R ~ R md simply mElp,$:~9 into other equivalent vacua. :: We find, in summary, that the moduli space which describest')~:~':j~Pi1 distinct, inequivalent vacua of the heterotic string is therefore

1

. modulI space

= M26-d.lO-d =

SO(26 - d, 10 80(26 _ d) ® S0(10 - d) ®

where the number of independent parameters in Mm,n is mn. (In the litemture, M26-d.lO-d is often written as SO(26 - d, 10 - d, Z)\SO(26 - d, 10 - d)/SO(26 - d) ®

~,~)n~~Njm

but for clarity, we shall use the notation which simply divides space by all the redundmt factors.) This symmetry must also be reflected in the space of ,,,. a,lq.l, effective supergravity theory, since the vacuum expectation scalar fields labels the possible vacua. For example, in the compactify c dimensions, then the scalar particles have the follov@lg~!itl components given by the reduced internal metric gil' intemal!i.,.j~.~,~~ two-form Bi}, and internal Cartm gauge fields Ai, where i '~e:~p , ' OC.sJ1 compactified coordinates ', I

gij:

2"C(C + 1),

Bij:

2"c(c - 1),

Q

cn •

Ai

.•

1

for a total of c2 + en scalar particles, where n = 16 for the het:ercttic.J;lWi~ is precisely the dimension of the coset moduli space SO(c + n, c)j~S:~OC~

11.5 S-Duality

465

. So the scalar fields parametrize the moduli space of possible vacua, as

importaOl lesson is thai the vacua transformed by the T -duality group - d, 10 - d, Z) contains both the E8 ® £8 heterotic string as well RS S()(3:2) heterotic sIring as extremal points in the same space. Symbolically,

T:

£s 0 Es

+l'

50(32).

(11.4. 13)

we now have gone from four superstrings down to three. can pCrfOlUl a similar analysis of the lattice compaclification of the I

string. where we compactify simultaneously in both the left- and righlsectors. NOT surprisingly. W~ find tlHlt the T -dllidity erOllp is llgaio T = 50(10 - d, 10 - d, Z). However, me full moduli space of

:~::~:~ vacua for Type IlA and Type liB theories is considerably roore

~

than simply a combio!:ltion of the orthogonal groups because of of extra scalars, as we will see in the next chapter.

S-Duality we have only analyzed T -duality, which is perturbative in the string constant Wld hence sbeds no light on thl: nonpenurblilive nature: of them;;",. More interesting is S-duality. which links the weak coupling of one theory to the strong coupling region of another. Since S-duality ;""entiy nonperrurbarive. we expect completely new surprises to emergc

will find that the dilaton field is central to this discussion of nonpereffects because its expectation value is related to the string coupling The first quantized string Lagrangian contains not only the usual ld,,,,,,;b;,,g the area of the world sheet, but also a coupling to the Riemann of the world f;heet

(11.5.1) '''the fact tial the Eulcr number for a two-dimensional Riemann surfaee is given by

x ~ -' j 'd2aFsRm, 4n ,re"oce if we make the substitution ¢ -+ ¢ e- s gains II new term

(11.5.2)

+ (¢}, the Euclidean path(11.5.3)

_ _g;, -2+n, where

/I

is multiplied by the coupling constant is the number ofbowl(laries or external strings. lfwe

466

11. M-Theory and Duality

place e{.p) at each vertex function of the Riemann surface, we can factor into the string coupling constant g. by redefining it as '.

g. =

e{.p)

Thus, the coupling constant is related to the vacuum eXJ)ectatlon t~~~Jj dilaton field. The key point is that S-duality, which changes (¢) into connects the strong coupling with the weak coupling region. LeI::~:~ this works in several examples.

11.5.1

1Ype 1IA Theory

S-duality reveals a profound, counterintuitive result when app 'Prl :tl:i·:1l strings, where we find a mysterious II-dimensional structure eml~t~~Wi We begin our discussion by considering the original 11:~lli pergravity theory [17], which was historically dismissed nonrenormalizable and could not accommodate chirality. We alyze this theory from the perspective of duality, adding in will cure both ofthe original problems. The bosonic version of this theory contains the II-dime:nsjionaJ)p,YJ~ as well as the third-rank, antisymmetric tensor A¥ N p: .;!i]~a~

s = /(11 ~

f

dllx

{R [R + 112iF12]

+ (72)2 _2_SM1Ml ... M!1F FA} M, ...M, M 5... M8 M9 MloMu ..

: ::.

', :.

;:::m:::m

where FM, ... M4 is the field tensor constructed out ofanljS)i'IllInetriz,e~!f¢ffii of AM1M1MJ' Now compare this with the effective Type IIA action in fields, once we integrate out over higher fields. We couple the the massless fields ofthe theory and treat them as background ne[<1i!i:;:'m:~ to the graviton gil v and dilaton ¢, we have an antisymmetric, secon:lt~r~Wl B /LV coming from the product of two Neveu-Schwarz fields. (We bosonic sector of the closed string in the comes from the product of two NS operators or two R Op(~ra1ton;,¢j~tijj the left and right movers, i.e., the bosonic spectrum is sp1mnled •.Q)l~;w.~ NS L ® NS R or RL ® RR. R-Rstates exist for Types IIA, IlB, there are no R-R states for the heterotic string.) In fact, both Type IIA and Type I1B have the same ma:;sle:>snl;!~~ NS-NS sector Ns-NS:

{¢, gllv, Bllv } •

But the R-R sector for the Type IIA theory contains the ad(li~'g~~11

R-R:

{GjL, AjLvp} .

11.5 S-Dua1iry

467

let US write the bosonic part of the effective action of Type ITA theory lowest order} in tcnns of massless fields. which is just IO·dimensional nchirnlsupergravicy with N = 2 supersymmetry. If we let K = dC, H = and G = d A, then the action for the massless fields (10 lowest order) is

s=

f

d'"x{J

g'-"[R+4Id~I'-1IHI']

- J dlKI' + ,iIGI']l + I~

f

GAG

A

8.

( 11.5.8)

ago, it was noticed that if we compactify the: I I-dimensionsl su· theory on
O~,~~:~~'. ~m::etrie tensor 8MN and the three-fonn lil fields : gMN -

A MNP

-

AMNI'

into the following

(8,.~, CIt' q,).

(A IIUP ' 8.
(11.5.9)

the r.ldius of the eleventh dimension is given by ~J1: precisely, the II-dimensional metric tensor and
be ,Iocomp""d " ds 2 = 8MN dxMdx N = e-24>!lgll~ dx ll dy~

+ e~{3 (dy -

dXIlC~)2.

A = ~ dXIl A dx~ Adx""'lIvP+ !dx" /l.dx" AdyBII ... (11.5.10) first, we may eriticize this resuit because II-dimensional supergravo:r;;:~~~!:o~n a circle has Kaluza-Klein states which are not secn in lO Type lIA string theory. Howe'o'er, as we will see later, Kaluza-Klein states acruaJly do exisl in lO·dimensions, disguised as

"'~':~:i~~~:::~::,When we compare lhe Kaluza-Klein slates coming from

it

i the solitonlike states arising in lO·dimensions, we will soe are ide.ntical. In other words, the lO-dimcnsional Type nA (with its litllles) is actually an I l-dimensional theory in disguise! IlA theory expressed in terms of massless fields is highly the CUIVature tensors, while II-dimensional supcrgravTherefore, we expect that Type IIA string theory, rewritten as ~:;;;;:,:;;::~I theory, includes I I-dimensional supcrgrnvity only as its i::! limit. then led to postulate the existence of an entirely new II-dimensional called M-theory. Our conclusion is then as follows: there exists a new

468

11. M-Theory and Duality

II-dimensional theory, called M-theory, containing 1 gravity as its low-energy limit, which reduces to Type llA Kaluza-Klein modes) when compactified on a circle. :spleCIJtiC3lH5l/ililt:i coupling limit of Type IIA is M-theory. This is truly a remarkable result [18-19]. It indicates that the stl;t~H~~all limit of IO-dimensional Type IIA superstring theory is coupling limit of a new II-dimensional theory, whose 10\1f-elller.gy:liWWi by II-dimensional supergravity. More precisely, we have

Rll = (gsi /3 • From this, we see why previous efforts failed to notice this ~e;~VJ~~m.1 dence. Using perturbation theory around weak coupling in ..q.l~"'Hj Type IIA superstring theory, we would never see II-mme~Jp.p;it which belongs to the strong coupling region of the theory. large, we enter the strong coupling region of Type IIA strilljt:=' Rll --* 00, so the II-dimensional nature ofM-theory We also see indications that M-theory is much richer in string theory. In M-theory, there is a three-form field A MNP , to an extended object. We recall that in electrodynamics, a the source of a vector field All. In string theory, the string acts a tensor field 8 1lo • Likewise, in M-theory, a membrane is the so~tf¢.~~:~ij Although strings cannot exist in II-dimensions, we see that om!n7:~~ can, such as supergravity, supermembranes, and its dual, (We will explicitly construct the actions for these super chapter.) All these astonishing results, however, raise more For example, although M-theory explains the strong coupling b¢~~W.~ IIA superstrings, it does not tell us what the full nOlllp()1)'l10IJniaka~f~ii II-dimensional theory is. We only know its low-energy dimensional supergravity. At present, there is no known way derive the entire nonpolynomial theory. Also, we know quantization of these membranes and 5-branes in 11 cannot make definitive statements about their interactions. . .. ::,:::::::::Y"::: Ironically, 1 I-dimensional supergravity was previously ::t.~·:.·j~~ physical theory because: (a) it was probably nomenormalizable (Le., there exists a

COlmJj~

seventh loop level); n~~~~iii (b) it does not possess chiral fields when compactified on (c) it could not;reproduce the Standard Model, because it·C'
11.5 S.Ouality

469

tensor which render the theory fm ite. The question of chirality is solved although I I-dimensional supergravity cannot give us chirality when '~~~:c: on a manifold, M-Ihcory gives us chirality when we compactify I : which is not a manifold (such as line segments). And the problem is too small [0 accommodate the SlHndard Mode! is solved when ..';~~~ the theory nonperrurbatively, where we find £8 ® £8 symmctry Ct when we w mpactify on line segments. can also sec why physicists missed this rather simple correspondence years. Because the coupling conSlant gTOWS with the radius of compe,nub",j·,m thcory will ncver reveal lhis eleventh dimension to in the expansion. This correspondence between 11- and 10"""k,",1 physics is inherently n nOllpcrturbntive one. We represent this relationship symbolically as S:

M on SI ++ IlA.

(1 1.5.12)

Type lIB Theory ,S· ,j",,,·,y between Type HA and M-meory is so remarkable that it must be in as many ways as possible. Bccauseofthewebof S- and T -dualities, a number of self-consistency chccks we can make. Type lIB strings are linked to Type rIA strings by T -duality in dimensions, there should be a uontriviaJ link between I I-dimensional

'';;~~:i~~~od Type fIB

:i i

strings

(to lowest order). thc fields coming from the NS-NS seclor, thcre are also Type coming from the R-R sector, which consist of anothe r antisymmet-

ie~:::;;:~,:"::n:~sor 8~u' in addition to a scalar field l , and a fourth rank, ;y

CI'Utrp:

(ll.5.13) cffective ac tion for Type ITR theory, whcn all higher modes have been ,~ •..•• out, is (to lowest order) S=

f

dJO.t .j g

-

~IH' -

{e-U IR + 41d¢12 - ~IHI2] 1

IH ll - 6101M+12J - 4 8

f

C+

21dll 2

1\ H 1\

H '.(l1.5 .1 4)

have dropped all fermio n terms and all higber terms in the curvature, H = dB and H' = dB', and where M = dC and M is self-dual. (In 10" 00, we have deliberately neglected the rather subtle point that there simple covariant action of a self-dual antisymmetric field.) "0 show S-dunliry, we will make the substitution gl'u '""7 e-'H2 gl'"'

oro"

, us

470

11. M-Theoty and Duality

-

f c+ /\

~e~IH' -IHI2} - 4~

H /\ H'.

Let us compactify the Type lIB theory on a circle of radius RIo. If ' about M -theory is correct, then this should be equivalent to the COlllpiwti ofM-theory first on a circle of radius R II , and then on a circle ofIltdilij we compare II-dimensional supergravity compactified on S1 ®, IIB theory compactified on S" we find that they are identical, 'w¢!:m~ following identification: Rll

gIIB

= -. RIO

But notice that the SI ® SI forms a torus, and the modular arrmh: Cif:jOO is SL(2, Z). In particular, a torus described by Rill RIO is '''''''~'''J'::~~~:m described by RIO I R II. But this means that the Type lIB theory d'~S~i~ coupling constant g is equivalent to the Type IIB theory del;crilbeldbov.::ljl~ the theory is self-dual. This is a nontrivial prediction ofM-thp'''M)f.' To check this prediction, we first notice that the Type IIB actilQiiis::i; invariant under an SL(2, R) symmetry given by, r-+

aT: +b

cr+d

,

where r

= l + ie-~,

where a, b, e, d are real numbers obeying ad - be = 1, where field, and where we simultaneously make the transformation on ~~::~r:~~

More specifically, we can show this invariance by

M=

e~ (1~12 ~),

A=

(:

!),

which transforms under SL(2, R) as

M = AMAT. We can also put H and H' into the colunm matrix iJ, such

/H=(H) H'

,

Then the Type lIB action (after a simple rescaling) which invariant under SL(2, R) is given by

S=

f

MP dlOx .yI- g (R - l..iJT 12 (J.VP MiJ

+ !Tr(W'Ma M- 1) 4 (J.

,

"

11.5 S-I)uality

471

the fennionic terms, the self-dual tensor, and the higher

this classiclIl symmetry of the lowest order action is quantized. R) reduces to the subgroup SL(2, Z) , so S-duality is given by

s ~ SL(2. Z).

(11.5.24)

is a~tonishing is tha t this S-duality relationship is defined as an abstract '~;:!~~between fields in the Type lIB theory, but is defined geometrically M over physical space, which s hows the power of this formalism. set I = 0, this SL(2, Z) synunetry, as a sub.~et . contains the impClI1anl ",Iano.

¢....,.. -¢.

(11.5.25)

words, the strong coupling of Type lID theory, whic h is normaUy the realm of pClturbalive methods, is now reveille{! to be another Type string, as predicted by M-theory. Thus, Type ITB string theory is self-dual. is indeed II surprising result, demonstrating that S-duality can establish .ti,msh'lp between the strong coupling of one string theory w ith the weak o f anottler. We find

S:

liB

.f+

lIB.

(11.5.26)

M-Theolyand Type lIB Theory also establish a direct relationship between M-theory and Type rm although it is an awkward ooe. Defore, we cumpactified M-theory on is equivalent to Type IIA theory compactified on SI (which in tum to Typt ITB theory). The 12 of M-thcory compactification, we saw cao be parametrizcd by an SL(2. Z) modular symmetry, which in tum to givc a geometric explanatioo oftbc SL(2, Z) symmetry of the Type 'lh,,,y. Unfortunately, this relationship between M-theory compactilied on Type lIB oompnctified on S1 only holds in nine dimensions. In order to the Type liB theory in 10 dimensions. we necd to let RIO go to zero is equivalent by T -duality to letting thc radius of compactification of . UB theory go to infinity). This means that M-thcory compactified on a . :: of zero area and fixed shape is equal to Type nB theory in 10 dimensions, . is a rather fl:wkward relationship. symbolically represcnt this limit, then we have :wc 1"

"'>1:r

Ti

M-theory on

1

M-theory on

s,

Ti

+f"

llA .

B

TlB.

(\ 1.5.27)

we may puzzle over several strange mcts. First, it appears odd

1~;.~~~:~':nb~' supergravity cannol reproduce a chiral theory after but M-theory can. In fact, the lack of a chiral interprel8-

la

472

II. M-Theory and Duality

tion for ll-dimensional supergravity was an important reason Ior:OM:! eliminating it on physical grounds. For example, M-theory compactified on T2 reduces to D = .-,:..:-~~,;.= supergravity coupled to a tower of KK states plus higher terms. ~1't~il that we let the area of the torus go to zero for fixed shape, we n.,,,nv.e'.j,, supergravity theory. So starting from a nonchiral theory, we chiral one. Where do the chiral states come from? The answer is that M -theory has higher corrections to II-dimeW~~)iI pergravity. In particular, M-theory also contains massive "LaLt;:;. wrapping up a membrane on T2 • In the limit that the area of thetQi~ii to zero for fixed shape, we find that these massive modes ~~~:l~IJI such that the effective theory reproduces D = 10 chiral Type lIB •... theory. The lesson here is that M -theory contains, in additioo:n~i~t~0~aI~~le1~i:111 supergravity, higher p-brane states which cure the many "c mer theory. These higher p-brane states, in particular, are im])ortanUtl:W1 all five superstring theories into one. A second mystery is related to the fact that the Type UB two background fields given by the tensor fields Band B'. It that the Type lIB string acts as a source for the B field, via .... sij 8; Xli 8j XV Bliv , and hence has a charge under this field. Type.: . cannot act as sources for the B' fields as well. But SL(2, Z) . and B', so there must be another object, besides the Type lIB g~ir~~I~ a charge under Bf. But what is it? Like the Kaluza-Klein modes ~ compactifying M-theory down to Type IIA theory, we will ~;tjrq;11 are new BPS states which carry charge under Bf and yield the symmetry. Before ending our discussion of Type lIB theory, let us factors of cp which appear in the various low-energy effective theory. We recall that the Euler number for a Riemann X = 2 - 2g. For a sphere with no handles, g=O and the Euler path integral over e- s picks up a factor of g;X, which gives us This means that the effective string action defined on the . accompanied by a factor of g;2 = e-zq,. Indeed, we find this . string actions for fields coming from the NS-NS sector. For the ..... lIB actions, these tenns include the R, cp, and H fields. However, the massless background fields coming from the not have any cp dependence at all, since these fields do not the string world sheet. They cannot couple to the string world usual way, yia products of ai X Ii. Instead, they couple via spinor as ST1t11t2 .4 n S. In general, this means that there will be world sheet. This also means that the tenns in the Type II actions R-R fields do not contain any cp dependence. Thoughout this discussion, we will find that the R-R sec:to~:)~~:~ theory contains highly nontrivial information concerning the nOll~¢iW.Wj!lm structure of string theory.

11.5 S-DuaJity

473

E8 ® E8 fIeterotic String .. far, by considering N = 2 superstring theories, we have reduced five ~r.;tri,"gs down to three. But what about dural superstring theories with = J? In particular, what about the £8 ® E, heterotic string and the '"ypc I Ooce again, wc find many surprises [20). begin by oompac(irying the sri II-mysterious I I-dimensional theory, low-energy sector is II-dimensional supergravity, On a finite line scgi.e., we first compactify aD SI, giving us Type IIA theory, and Lhen thc action of 'L}.. This discrete symmet ry acts on SI by the following:

·__ X".

dividing by ~, we also reduce the N = 2 symmetry of the T)pc HA by half: down to N :.= I. We also wish 10 keep the bosonic Siaies which under 'Lt, which include: the 10-dimcnsionai metric gil"' the scalar and the antisymmetric tensor A,...II. Notice that these bosonic fields = I supersymmetry make up precisely the spectrum of the EI ® EB string (minus the gauge multiplet). by tbe line segment introduces anomalies into the theory, which be eaneeled because M-theory is anomaly-free. To solve this problem, naturally introduce two additional pieces to the action, each one associated the endpoints of the line segment, which form (WO hYJ>Crplanes. We recall must have 496 vectors in an N = 1 theory to kilt the anomalies. This be solved by having 248 vectors defined on each hyperplane. We know E. can give us precisely 248 slales, and we also know that ~ symmetry thaI we have equal gauge groups on the ends of the line segment, so tbeory has gauge symmelry E, ® £3 or SO(32). In other words, we anomaly by placing a super-E. Yang-Mills theory at each of the located at the endpoints of the line segment. so that the resuiling E B® E. symmetry. (At present, p~isely how the super~ Yang-Mills emerges from the compactification of M-theory 00 a line segment is understood.) we find that the larger the string couplillg constant, the l:Jrger the between these two hyperplanes, i.e., R = gl/J. Thus, to any finite perturbation theory, we will never see tbe direct cffr..'Cts ofthese This is the reason why siring theory previously missed these and the nonpcrturbalivc J>tructure of the heterotic string. ! this symbolicnl ly by slating;

S: M-theory on SI/'L}.

++

EI ® £8.

(11.5.28)

Type 1 Slrings ••'. .ave reduced Ihe five superstrings down to lWO. We are still left with I strings, which, unlike the others, have both open and elosed strings. I . obtained by comparing Type I and S0(32) heterotic both have SO(32} symmetry. Althougb they realize the 80(32)

474

11. M-Theory and Duality

symmetry in entirely different ways (Type I strings via ChaJl-':Pl,\jt~i~ii and SO(32) strings via the Frenkel-Kac construction}, they ~~~~:::!~g low-energy sectors. We suspect that they may be S-dual [2 As before, let us now make a prediction about this theory. Let us first compactify M-theory on Sl/Zrz with a We arrive at the Es0Es heterotic string. And now let us with radius given by R, which breaks the symmetry down to We have then compactified M-theory on a cylinder, with raO'lUS: L. Now let us compare this resulting theory with Type 1 and the compactified on a circle of radius R. The effective low-energy action is given by

Notice that the cfJ terms appearing in the Type I effective lOVlr~tiififfi~ have the correct character. The R and dcfJ terms appear mJ;lltil[1H!~~:a~jj~ as expected, since they are defined on the sphere. (However, field is missing.) The F term is accompanied by a factor of e~jfW~[. Yang-Mills terms are associated with the open string sector, w.l)ltcJi:ls::M on the disk (with Euler number I) rather than the sphere. comes from the R-R sector, and hence has no rp dependence. the Type I action have the correct world sheet structure and rp del?¢~{(~@jl The SO(32) heterotic action is given by

S=

f

d 10x,J=ge-2
[R + 41drpf -

~IHI2 -

clTr IFI2].{ ::::::::::W

Notice that this action has the correct ¢ dependence. Since the he(¢~~ has no R-R sector, we find that the entire SO(32} action is mt!mi?i.~gl factor of e-2
SO(32)

+*

L

1 US-Duality

475

,m."rrin2 these two theories (which were derived by simply reversing the compactificn.tion ofM-theory) we find

8,

R = i / KSO(ll) = L'

( 11.5.32)

the previolls case of M-tbeory compactified on a torus, we found that

:. Oludular symmetry of tbe torus could be used to show the equivalence of

TIB theory with coupling constant g with Type l1B theory with coupling i/g. This gave us the self.<Juality of the Type lIB theory. However, case of M-thl!:ory compactified on the cylinde r, Ihere is no counterpart mndular symmeLry of the torus, so we do not expect Type I theory 10 I However. what we find is that Type I theory is dual 10 S0(32) strings. precisely, we have the following transformation which cOIwerts the Ttheory into the SO(32) theory: g,,~ -+ t-fl g..... ,

¢ -+ - l/!, H' -+ B,

A

-+

a'A.

(1 1.5.33)

can also fmd a relationship between M-theory and S0(32) theory in di~;~S~:~! although it is rather artificial Like the relationship bctwcl.'Tl til nod Type HB theory. we have to take II rather awkward limit. We with the fact thaI M-thcory compacrified on a cylinder. i.e., [S, 17-2 ] €I S1, 1050(32) string theory on S,. To retrieve the SO(32) tht:ory in 10 .em;ior• • we have to take the 7,ero area limit for fixcd shape. So we have M.theory on S'/"[,2 ++ E8 ® E8• M·lhcory on Tl 17~ ++ SO(32).

(11.5.34)

addition, M-theory sheds some light on the reason why Type I theory both open IUld closed strings. The Type J dosed string theory. we can be viewed as II truncatioD of the l)rpe rIB closed Siring. We note 11",T"De HS closed suing tbeory, because it contail1S left and right movers same chirality, is invariant under the world sbcct parity operation fl, exchanges Icft and right movers, and sends (I into -q. We can therefore out ofthc Type lID theory the subseclor which is parity even under n. means that the Type r closed string is an "orientifold" of the Type liB This lrWlcation gives us the N = 1 unoriented Type 1 closed SIring. the two-fonn Bllu found in the Type II theory couples 10 the string . ' '" X " ' .' x ' , which has odd parity under the world sheet parity operalion is hence projected out Thus, Spo cannot couple 10 the Type I closed Because II strings contain this term c'ja; XpaJ X ~ BII.' it means fPC II SIring" h.,ve a conserved charge under this two--fonn, i.e., the Type act as sources for the two-fonn field, as in ordinary gauge theory.

476

It. M-Theory and Duality

However, the absence of Bit" for the Type r theory means that .' strings do not have a conserved charge, i.e., they can break . TIlls is the reason, in fact, why the Type I theory closed strings because they do not have a conserved charge under Bltv ' (In the same way that the electric charge of a point particle surlace integral over a sphere of the dual of the Maxwell ~:~~?ff~W:JiI associated with the string is equal to the surface integral of the uU'H,:<;I!~, tensor over a seven-sphere. If a closed string were to break, H.,,:...:·~"c,: sphere could slide off the string, and the surface integral For Type II strings, where the charge is conserved, the charge carl1l~~;~~ equal zero, and hence the Type II closed string cannot break. HIiIW~~~~:i not true for the Type I closed string, which has no such cOflsel"Vej:i¢llm it can break.) Lastly, there is one more symmetry, called U -duality, wn.ere:1VItn,~~ large radii and strong coupling is dual to a theory with small .~~,":~.~!'!I coupling. Summarizing all three, we have S -duality:

¢ -+ -¢.

T -duality:

1fr -+ -1fr. 1fr -+ ±¢.

U -duality:

where 1fr is a scalar field whose vacuum expectation value COlltr()ls:J!'i~f~1I of compactification. In the literature, U -duality [23] also the minimal group which contains both S- and T -dualities, I.e., .. :.: U

~

S® T.

We will discuss U -duality more when we analyze BPS

1l.6

Summary

Let us summarize our results so far. We have found that S, T. provide a powerful way in which to analyze the perturbative and tive structure of string theory. In particular, they show that all theories are really the same theory, and that they may ultl'[nately,:Jl!l~:;Jl into a theory called M-theory. Although the action for M-theory to the lowest level (where it reduces down to I1-dime:nsjollillsijM~ the theory has already given us unparalleled insight into the nODlp~j!~ behavior of string theory in various dimensions. Our discussion began with T -duality, which is based on the. momenta of the compactified closed string is given by .. n (P L , PR ) = (2~ + mR. 2 R - mR) , where the n, as usual, arises from the Kaluza-Klein excitations & ..r:Uij¢l1 but m labels the number of times the string winds around the c':if'"~l~i;~~1

11.6 Summary

477

mass spectrum for M2 is invariant under 1 2R'

R_-

( 11.6.2)

we interchange n +7 m. :R,wnillo' in the language of super conformal fie ld theory, this Ir.m sfonna~ . equivalent to

ax -+ ax.

ax -+ -ax. WL - -'IlL.

WR - '/IR.

(\1.6.J )

j" i.n hun, reverses the sign of the chirality operator on onc set of states, and

changes the chir.tlity o ra Type fiA theory inio a Type liB theory. Thus, the result that T:

fiA_TTB .

(1 1.6.4)

five superstring theories have been reduced down 10 four. Similnrly, we ""'OW that the rn1l heterotic strings are '{-dual . Naively, the moduli space for the heterotic string compactified down to d dimensions is given - d, 10 - d). We arrive at this resuh by analping the onc-loop demanding that Ihey be modular invariant. is 100 large. We can stiUdivideoutby SO(26-d)and SO( 10the mass operator is separately invariant under these groups. Lastly, divide out by the 7 -duality grou p SO(26-d, 10 -d, Z)which merely the lattice points into each other. This 7 -duality group shuffl es the ,lb"O,rt into the 80(32) theory. Then the full moduli space is given by: .

SO(26 - d , lO-d)

moduli space = SO(26 _ d) ® SO( 10 _ d) ® T '

T =SO(26-d, IO - d , Z).

(11.6.5)

( II .6.6)

five superstring theories bave been reduced down to three. Although T.dualities were interesting when they were first discovered, the real "''as made when the S-dualities were first discovered. Although the Shave not been proven rigorously, they are believed to be true because independcm checks. stan by reanalyzing the II -dimensional supergravity acrion

s= ~fdllxIJ-g [R+ l~lFf] <"

(11.6.7)

478

II. M.Theory and Duality

where FMI ___ M4 is the field tensor constructed out of of AMI.Ml.MJ' It was noticed years ago that this action, when of vanishing radius, reduced down the Type IIA supergravity compactification of the 11 ·dimensional theory, we find the cort~~.p.:~~iiIi1 with the lO-dimensional Type IIA supergravity metric ds 2 = gMN dx Mdx N = e-14>/3 gJJ.v dxJJ. dyV

+ e41>t3 (dy -

dxJJ.CJJ.)2 ,

A = ~dx!L A dx v A dx PA!LVP + 4dx" A dx v dy B!LV •• Let K = dC, H = dB, and G = dA. Then we have the Type .

s=

f

dlOX{J

- J

ge-2
g[IKI2+ 1121G12])+

1~4f GAGA

cOlnsidel-ed~l~~1

Although this result was known for many years, it was for many reasons. First, II-dimensional supergravity is a sick thei[)r.v;::~ this result was valid only for Type IIA supergravity, not the theory. TlUrd, there were Kaluza-Klein states arising from the that had no counterpart in either Type IIA supergravity or All these problems can be solved if we introduce M-theory. 'Ui~,",.1 nesses of II-dimensional supergravity can be solved if we low-energy sector ofa higher theory. Second, the complete Type ity action should arise if we compactify the higher terms of the 1 theory. (We can, in fact, work backward and define the higher dimensional supergravity as emerging from Type IIA superstring lastly, the Kaluza-Klein states of the compactified 11 be viewed as BPS saturated states of the superstring. Similarly, we can derive the E8 ® Es heterotic string by theory on a line segment. Then an E8 symmetry emerges at line segment, giving rise to E8 ® E8 symmetry. (Ironically, 1 supergravity was, in part, abandoned because it could be shown UJaL'. I.lL could not produce a chiral theory after compactification on a rrui.ti.itJ~~ we see that II-dimensional supergravity can, in fact, yield chi.ni:l : m~~ compactifying it on spaces which are not manifolds.) When we compare the two actions, we find the cOlTe~;pondlmc:e,: ;.

RII = (g)2/3.

So we see that the strong coupling limit of 1O·dimensional Type theory yields the 1 i-dimensional M-theory. Other dualities can be found as well. We first notice that the is self-dual under the discrete group SL(2, R). In addition to thf'. -:f. p.ir~!(.J1 B!LV arising from the Neveu-Schwarz sector, we also have the :: : .

11.6 Summary

479

coming from the Ramond-Ramond sectoT. These two tensor fields can combined into Ii single column vector. !ice this, we introduce:

M =

e~ (1

~),

1 /

( 11.6.11)

transforms under SL(2, R) as

M=AMAT.

(11.6. 12)

also put H = dB and H' = dB' into the column matrix

H. such th at (11.6.13)

the Type ITB action (after Ii simple rescaling) which is manifestly ,,,;an. under SL(2, R} is given by :;: =

,

J

dlOx';=g (R -

l~iI:"f>MiI/J"P + ~Tr(aItMa.. M-I»),

( 11.6.14)

,::;.~~;:;~ve dropped the fcnnionic tenus, the self~ual tensor, and the higher also notice that SO(32) heterotic strings and Type I stri ngs bave the same ;rtw,ba,:;ve gauge group. The effective low-encrgy bosonic Type J action is

=

Jdxl0xJ_g l e-:!.i> [ R+4Id.p12]_e~ TrIF12_~I H '12 1 ·

( 11.6.15)

SO(32) heterotic action is given by

s=

f

dIOx..;'-ge-2·[R+4Idtf;lz-iIHI2-a'Tr lFll ].

,"",,,;,," !h,,~ two theories compaetified on Sl

(11.6. 16)

with M-theory compaetificd

Sl/'L) and then by SI, we find R 81 = 1/8so(31) = L·

( 11.6. 17)

summary, when compactified on a circle :>'1, we found the following \lu" !;.;,, in nine dimensions: T:

!

Type IlA Es® Es

+->-

Type nB,

+->-

S0(32).

(11.6. 18)

; '>S. ,m turn, reduces the various supcrstring theories dov.n 10 two cmegories,

with N = 2 supe rsymm~try (the 1'ype II tl1eories) and those with N = 1 heterotic theories).

480

II. M-Theory and Duality

We also found the following S-duality relations in 10 diIllen.sions:!: S:

{TYPe lIB Type I

<:+ <:+

Type lIB, SO(32).

jllll:ill n

The last relation is particularly important because it links superstring theories with the N = 2 theories. When going from 11 to 10 dimensions, we found the S-...."'"'u"".,

S:

M-theory on S.

<:+

Type IIA,

M-theory on To2

<:+

Type lIB,

M-theory on S./~

++

E8 ® E8,

<:+

SO(32).

M-theory on

Ti /~

Lastly, although these duality relations are quite impressive, we m~i&f:j challenge many ofthese results because of their lack of rigor. We tablished the S-duality transfonnation to lowest order in the Although this is still an unsolved problem, we have confidence in because of a number of self-consistency checks we can make, the pelling being the analysis of BPS saturated states and p-branes. to these topics.

References . For reviews ofM-theory, duality, and membranes, see [1-6]. [I] J. H. Schwarz, Lectures on Superstring and M- Theory Dualities, rA:~I~ijm1\ijJij School, World Scientific, Singapore, 1996. [2] J. Po1chinski, TAS] Lectures on D-branes, TAS] Summer Sclt(jQFi Scientific, Singapore, 1996. [3] P. K. Townsend, to appear in the Proceedings o/the 1996 ICTP SU1ntniilt~~ in High Energy Physics and Cosmology, June 10--26, 1996. [4] MJ. Duff, Supermembranes, TAS] Summer School, World Singapore, 1996. [5] C. Vafa, Lectures on Strings and Dualities, Feb. 1997, [6] P. S. Aspinwall, hep-th/96 I 1 137 . [7] c. Montonen and D. Olive, Phys. Lett. B72, 117 (1977). [8] E. Witten and D. Olive, Phys. Lett. B78B, 97 (1978). [9] K. Kikkawa and M. Yamasaki, Phys. Lett. B149, 357 (1984). [10] M. Dine, P. Huet, and N. Seiberg, Nucl. Phys. B322, 301 (1989) ..•••. [II] K. Narain, Phys. Lett. B169,41 (1986); K. Narain, H. Sarmadi, Nue!. Phys. B27.9, 369 (1987). [12] For references, see A. Giveon, M. Porrati, and E.Rabinovici, 77 (1994). [13] A. Font, L.lbanez, D. Lust, and F. Quevedo, Phys. Lett. B249, 35 [14J S.1. Rey, Phys. Rev. D43, 526 (1991). [15] J. H. Schwarz and A. Sen, NucZ. Phys. B411, 35 (1994); Phys. (1993).

References

48 1

utt.

8329,217 ( 1994) A. Sen, I"'~"'al. J. Mod. Phys. A9, 3707 (1994); Phys. E. CremmeTlind 8 . Ju lia,l'h~ !..en. 80B, 48 (1978); Nucl. PhY$. 8J59, 141 ( 1979). E. Willen. Nucl. PIry$. 8-433, 85 (1995).

P. K. TOWMCOO. PIryJ. Ult. 8350, 184 (1995). P. Horava and E. Witten, Huel. Phys. 8460, S06 ( 1996); Nucl. Phys. 8475, 94 (1996). E. Witten, M.Jcl. Phys. 8433. 85 ( 1995). J. Polchinski and E. Witten, Nucl. PhjJS. 8460, SZS (1996). C. M. Hull lind P. K. Townsend, Nuel. Phyl. 8438, 109 (1995).

CHAPTER 12

Compactifications and BPS States

12.1

BPS States

As remarkable as these dualities may appear, we can still they are accidents of our low-energy expansions or represent sorne.tbttjrg:aii about the theory. We must establish other consistency checks on ~t;:~g::1l[gru turbative results. Perhaps the most persuasive evidence comes called "BPS saturated states," which allow us to probe nOltlpf~rtu~~~I~W~ nomena because of the nonrenormalization theorems. In this crulllt~tt~:~~ see show these BPS saturated states allow us to establish a web o~:~~~~~ dimensions down to D = 4. To begin. let us analyze one of the simplest examples ofa field theory in 1+ 1 dimensions. We start with the Lagrangian L = -!(8,,¢i

+ 4tr"a"l/! -

!V2(¢) - !V'(¢)tl/!,

where V(¢) = }..(¢2 - a 2 ) and 1/1 is a Majorana fermion. supersymmetric, with two chiral supercharges given by Q± =

f

dx(¢

± ¢')1/1± T

V(¢)l/!~,

where l/!± are the left- and right-handed components ofl/!. write the supersymmetry algebra Q~ = p+, Q~ = p- ,

{Q+, Q _ } = T,

12.1 BPS States P± = Po

± Ph and

483

T is the central ch8Igc. From the algebra, we find

(12.1.4) comes the key point. Since the square of the supercharges is always

, this means thai we can sandwich this relationship between eigenstates the particle is at rest.. and we find ( 12.1.5) is the Bogomol'nyi bound, which establishes a relationship between the and charges of these states. In this chapter, we will be interested in slates which saturate this bound, Le., fo r which the im,"quality becomes """I;ty. We find the equality is satisfied when we sandwich this equation two states Is) which obey (Q + ± Q_) I s} = 0, i.e., for states which .·,nnih under precisely half of the supercharges. These are the BPS staleS •

. simple example in 1 + 1 dimensions illustrates thc main features wbleh ! also find in String theory. We will find that BPS saturated states for preserve half of the Sllpersymmetry generators, and most important, .b'h'~th," the BPS relationship is preserved to aJl orders in perturbation due to the supersymmctry nonreoormaHzation theorems. let us ger.eralize these results hy considering the super translation in four dimensions

(12.1.6) = 1. 2 .... N , which contains N(N - 1) central charges contained within t. matriees U and V. In particular, for N = 4 and N "" 8 we have J2 and 56 central charges, respectively. Historically, central charges were largely ignored, since they describe massive states N the<:ries. But for our diseu:)sion, they will be crucial in finding saturated states. before, we can sandwich this algebra between eigenstatcs of Po, where ."',un" that the state is at rest Then the expectation value of Po yields the term M. This algebra now reduces \0 M ~ klZI

(12.1.7)

constant k, where 121 symbolically represents the charges that we from U and V . In other words, the masses alld charges are now related by gives us a powerful relationship between the masses :~;.~:;;~:;';T:'h~;:':;result i theory and the charges ofthosc states. For the states which this relation, it is believed that supersymmetry protects them from ~n'nu.nnalliz<~ by quantum effects, so the relation holds nonpcrturbatively. n:'ason why they play such an importanl role in establishing .h,d;~/ o]f d,mlil] The jXlint here is tbat the BPS relationship is probably

484

12. Compactifications and BPS States

satisfied nonperturbatively because of nonrenormalization a powerful constraint on the properties of dual theories.

12.2

Supersymmetry and P-Branes

Now generalize this relation for the arbitrary case. As we will of p-branes alters the relation to

{Qu, Qti} = (r/LC)()(p P/1

+L

(r/LI ... /LPC)()(ti Z/1, ... /Lp.

p

In the original work of Haag, Lopuszanski, and Sohnius, wnlq.l1(Q~te the representations of supersymmetry algebras, these higher cerltr;:(n~.il dropped because they implicitly assumed only the existence of ' But if we allow higher p-branes to exist, then we necessarily na,re;·tl['j ;~f, higher terms to the super translation algebra. Previously, there were serious objections to higher p-t.ralles.m~t~i the ll-dimensional supermembrane (2-brane) theory was reJ,e~fl~~;~: reasons: (a) it was unstable; (b) its three-dimensional action had ultraviolet divergences; (c) even its free action was unsolvable; it was quartic and .~·.o'-J.': But M-theory forces us to reexrunine this from an entirely new . before that M-theory can be represented, to lowest order, as 1 supergravity. To higher orders, however, the action is um'l\.. ~U!'y 11 dimensions, the only known generalization of II-dl:·mensJOmUi is the supermembrane, which might provide a clue as to the theory. In particular, under a double-compactification (reducing of space-time and one dimension of the world volume), the 1 supermembrane reduces down to the usual Green-Schwarz string:: To begin our discussion of membranes, consider an coupled to a point particle source (i.e., a zero-brane). The coupling is given by

f

D

d xA/1PJ.,

where the source j It is given by: rex)

=

f

dT 8

D

(XJL -

X/1(T») a,X/1(T),

where X /1 ( T) labels the location of a point particle. This relllro(iucesJ~: formalism of Maxwell's theory of point particles.

12.2 Sup~rsymmetry and P-Branes

485

L~:~:':; for a string (a I-branc), the suing variable X" couplcs to a

is

background second-rank tensor

/

j""(x) =

f

B pv

as follows:

dDXB,,~j"v.

dt da OD (XI! - X,,(t,

(12.2.4)

a») fYo;X1'OjX

U •

( 12.2.5)

~rting the currcaL into the coupling, we find the usual lerm coupling Ihe

to the massless background field

f

drdarPo.. X"o j X " B,,~.

( 12.2.6)

, this can be generalized 10 describe a gauge theory with a massless, p + I rank antisynunetric gauge potcntial coupled 10 a p -brane. the p-branc variable X/.(a;), where a/ labellhe coordina tes of volume swept out by the moving p -brane. coupling is given by

(12.2.7)

)"." .... ,',," =

f

dl' t.l u/i D (xr• - X!.(Ui») Ei<---i,. ,

aj, X'"

... OJ,...., XI'~' l . (12.2.8)

COIlStruct the field lensor associated with the p-brane potential: ( 12.2.9) construct the dual theor)', we introduce another field lensor F' such that the dual to F:

·F

l,tt-hood side is a D -

(12.2.\0) ",···",".1 -- F' ""'." ••1" (p + 2) rank tensor. The right-hand side is a q + 2

-. tensor. Since these two numbers must be equal, wc therefore have the

p+q=D-4,

(12.2. 11)

us the dimension q ofLhe dual ofa p-brane. For example, in four the dual oran eiectroo (O·brane) is another O-brane, the monopole. i the dual of a string is a 5-brane. Likewise, in Il -dimensions, of a membrane is a 5-brane.

486

12. Compactifications and BPS States

Lastly, we can construct the electric and magnetic charges these p-branes. We find Q£

~

1

*F.

SD-p-2

QM

~

rJ

F.

Si>+2

where we have encircled the p-brane by a hypersphere. (The . . electric versus magnetic charge, as we see here, is a bit arbitrary, reverse them by taking the dual.) One simple guiding principle which determines which is to analyze the central charges appearing in the Surprisingly, the supersymmetric algebra allows us predict VVlll",U appear in which dimensions, even before performing any For example, in Green-Schwarz superstring theory the u~ •• v ... only up to a total derivative. Normally, this total derivative is if we calculate the currents corresponding to supersymmetry, algebra of the currents do not close in the usual fashion. In PaJ~~~ total derivative term contributes precisely the central charges ·fQ4fP;ij:=;W super translation algebra. By an explicit calculation, we find :~~=IIII charge contribution of a p -brane is given precise Iy by a pth rank field ZI1I ... I1" appearing in the algebra. This is the key point: by central charges ZJLI ___ l1p in the algebra, we can read off the sDl~ctJra in the theory. This is quite remarkable, because we now know nonperturabative p-branes in the theory without ever having tO~~11111 Now consider the fact that the largest number of components generators that can appear in the superalgebra is 32. In II can construct Majorana spinors with 32 components. There sh()ull~;.: (33 x 32/2) possible terms on the left- and right-hand side of the algebra, including the central terms, is given by {Q"" Qp} =

(rM C)"'fJ PM + (rMN C)",p ZMN

+ (rMNPQRC)u,8 ZMNPQR. Each central charge term on the right-hand side corresponds to •.•. Counting states, we find that they contribute 11

+ 55 + 462 =

528 states

as expected. Thus, we expect to find a membrane and its dual, a .. dimensions. ;. As a check on this formalism, we note that this membrane must ...•. third-rank tensor. Ifwe examine the currents of ll-dimensional we have a third-rank tensor field AM N p, which couples to a 2-brane Now reduce the algebra down to 10 dimensions, in which component spinor will become two Majorana-Weyl spinors,

<.

12.2 Supersymmetry and P-Branes

487

',::~:~~~::;,~w::',~now have two spinaes Q~. one for each chirality. If they have

PI

in 10 dimensions. then they can be recombined into 3 single givi ng us the algebra for the Type 1lA superstring:

+ (r"C).,z + (rMfllct,ll' 2M + (r"' . . C)ajJ 2 M.... + (rMNf'a rII C) .. " ZUNI'Q + (rM"'PQR"C)"" ZMNl'aR'

IQ • . Q,) ~ (rMe)., PM

:ocmtings~,,,s.

( 12.2.15)

this reduces to

10+ I + 10+45+210+252 = 52R states

(12.2. 16)

expected. These, in tum, correspond 10 p = 0, I, 2, 4. 5 branes for the ""UA theory. Notice that the S-brane of the I I-dimensional theory reduces 4-branc and 5-brane of the IO-dimensional theory, and that the 2·brane I I.dj~nsional

theory reduces to the I-heane and 2-brane of the 10",e,,,;(m.1 theory. (NoLice thl:lt PM of the 1 I-dimensional theory splits off a scalar P l1 and a vector in the IO-dimellsional theory. Viewed from . of the eleventh dimension, we see tha t they correspond to . modl.."s. However, viewed from the tenth dimension, they are O· . soliton-like states coming from the R- R sector. This i puzzle mentioned earlier, that KalUZB-Klein modes Il-dimt:nsional supergravity compactified on a circle correspond excitalions of the Type rIA lO-dimensional string.) summary. if we let (PI. Pl.) represe nt dual p-branes carrying (eJec;. O,",PC'";';;) charges, rc~pectively. then the Type lIA electric and magnelic can be arranged according to

R-R , (0. 6). R- R , (2.4). [ NS-NS, (1.5).

(12.2.17)

;m; I,,'ly, for the Type JI B string. we have two chira l spinors Q~. where the is given by

(12.2.")

P is a chiral projection operator, and the tilde refers to traceless SO(2) teosors. Counting slates, this gives us

w.,.;,

10+(2 x 10)+ 120 + 126+{2 x 126)= 528 states

(12.2. 19)

488

12. Compactifications and BPS States

as expected. This, in tum, corresponds to p = 1,3,5 (and also p corresponds to an instanton). Comparing this with the anti symmetric fields of the Type lIB we find two second-rank tensors, one from the NS-NS sector, tne:-nNtA the R-R sector. In summary, we find that the Type IIA theory must have p-branes, while the Type lIB theory must have odd Dirichlet dimensionally reducing the various charges ZJ1!' .. J1p' we can various p-branes appearing in lower dimensions as welL It is reIlDar this classification for various dimensions can all be done at the grc,tiri~tl level without constructing the p-branes.

12.3

Compactification

Now that we have assembled the machinery of BPS saturated .... these results to discuss compactifications for D < 8. As we and lO-dimensions, the nonperturbative dual of a superstring superstring. We find that a similar situation happens in lower< ... dimensions. . In order to systematically compare the various strings various dimensions, it is necessary to calculate the number tries which survive the compactification process. To do this, concept ofholonomies. After compactification, we wish to find now many spinors constraint

c~::*jll

Die = O.

For each covariantly constant spinor which satisfies this have one generator of supersymmetry which survives the process. We recall that the group generated by successive cOlnrrlUt:ln~~i;i~~ called the holonomy group. Once we know the holonomy tn'r'l'...... pactification, we also know how many supersymmetries ha\'¢:::~wm~ compactification process. This counting is especially easy pactification, where the holonomy is equal to one, and all ~~~~reml generators survive the compactification. To see how this toroidal compactification takes place, we";~1 range the 32 fermions contained within Q", in IO-dimensions .' dimensions, where i counts the number of supersymmetries... In eight dimensions, a Majorana spinor is not possible, tUt~jt~6 is possible with eight complex components, or 16 real c~nr(d~:lljl components can be rearranged into Q~ so we have N = string and N = 2 for the Type II strings.

12.3 Compactificmion

489

:: In six dimensions, a Majorana spinor is again not possible, but II Weyl spinor po,,;·',"e with four complex components, or eight real components, so we 8 x 2 = 16, i.e., N = 2 is the maximum number ofsupersymmeoies for . and also 8 x 4 = 32, i.e .• N = 4 is the maximum number for the Type II strings. : In dimensions, a Majorana or Weyl fipinor has four components, so 4 for the heterotic string and N = 8 for the Type II strings. ·. W'e '''" summarize some of this in the following table, where D represents heterotic

number of uncompactificd dimensions of spaco--time, S i~ the minimal ro:<~~:,:o'~f;a spinar in D dimensions, and N represents the number of .p generators after compactification on torii. D

S

10 16

Heterolic Type II N= 1 I

N = 2

9

16

N

8

16

N= I

N = 2

7

16

N = 1

N=2

~

N

~ 2

(12 .3.2) 6

8

N = 2

N=4

5

8

N=2

N=4

4

4

N =4

N

3

2

N=8

N

2

I

N = 16

N = 32

~

8

= 16

of compactifying on toni, consider the complex manifold K,. It is a fiat, Kiih l~r manifold with SU(2) holonomy, and hence it is a Calabi-Yau 1 but in only two complex dimensions. It tbe reforc breuks j ust half the ",nun,",,' genemtors. (For example, while toroidal compactification of IIA on T4 gives us N = 4 in six dimeruions, K J compactitieation only us N = 2 in six dimensions.) the Ileterotic string after compactificatioo on K], this compactification only half the remaining supersymmetries, so we have N = 1 in six be summarized in this table. Let D be the number ofuncompactiand let N be the number of supersymmctry generators after

490

12. Compactifications and BPS States

successive compactification on Calabi-Yau manifolds with with two complex dimensions (as in K3) and three complex GlDl(,lp.$I(ii K3 ® T2)' D

Heterotic Type II

10

N

=

I

N=2

6

N

=

1

N=2

4

N=2

N=4

Compactification on a Calabi-Yau manifold with SU(3) N = I in four dimensions for the heterotic string, as de~;rre:d reasons. It preserves only one-fourth of the supersymmetries. Furthermore, the holonomy group ofa seven-dimensional G 2 • This can take us from II-dimensional M-theory down to which is physically relevant. As a first step toward constructing new dualities in we must make sure that the number of supersymmetry seemingly different theories matches. For example, we see candidate is Type II string theory compactified on K3 being string theory compactified on T4 • We will see that this conjectlllr~~!f$

12.4

Example: D = 6

There are several ways in which to construct these dimensions, such as:

12.4 Example: D = 6

49\

us now analyze various dualities that exist in D = 6. For example, if we apthe mcm,bn"" around a one-dimensional space we call M\ . which can be a circle Sl or a line segment Sd~. we arrive at a string theory (either theory or the f8 ® £8 heterotic Siring). Then we can compactify this four-dimcnsionaI manifold, M 4 • and arrive at a six-dimensional theory.

,,"on,orn,

~

~;'jfi;::w~,~~ca~n~:"~Vr'~'~":~'hfi;~SJo~ro~'j:':'~w~e~c'~n

4

p-branc around a q dimensional wrap the 5-brane around M , then we can compactify this string in six-dimensions. By duality, olher, one being fundamental

two string theories should be the other being solitonic. . us tlIke Ml =: .'i\ or SJ~, and M4 = T4 or Kj . Then we arrive at

possible siring theories by compnctifying the membrane on M[ «> M 4 • . should be dual to the four string tlleorics we obtain by compactifying 6-b,canc on M4 ® M, . we wrap the membrane on SI and St/Z, we obtain Ty;>e llA and the ~E. h,,,,,·,,;, strmgs., respectivcly. Similarly, if we wrap the 5-brane around 14 and K J, we obtain Type IIA and the heterotic !ltring. respectively. can then place the variou!I D = 6 dualities on a chart L1- 2]:

,. (I. 1) ( ! , 1) (2. 2)

, SI

K,

5 1 /Z:

1~ T~

51

Type IIA heterotic Type IlA

heterotic Type lIA Type IIA

supcrsymmetries thllt survive in four

D ~ 6. N

~

(2. 2) Theory

-:- lei us examine the dualities among cornpactified string thcories in more -: there arc a bewildering number of duality groups that enter into thoory, it will be useful to use !he old supergravity fotmalism gtt,ac.lfwe compactify the II-dimensional supergroviry action, we find effective, low-energy action is genericaUy given by (12.4.1)

q,1 arc various scalar fields which take values in a target space

i

}vf

~:~·~gij(rP}. and Flp." represent the abelian field strengths. We will be

the equations of motion for the scalar fields 4/, which arc invariant symmetry group G, such thal}vf is the b.omogeneous space G / H, H is the maximal compact subgroup of G_ the years. the physical significance of the group G appearing in sutheory was something of a mystery. Nonetheless, G was carefully

492

12. Compactifications and BPS States

cataloged for various dimensions and supersymmetries. For the following table for supergravity in various dimensions Yang-Mills fields, which forms the low-energy sector of the theory [3]

D

Supergravity Group G

T-Duality

10

0(16) x SO(1, 1)

0(16, Z)

9

O( 1, 17) x SO(l, 1)

0(1,17, Z)

0(1, 17,

8

0(2,18) x SO(l, 1)

0(2,18,Z)

0(2, 18,

7

0(3, 19) x SO(I, 1)

0(3,19, Z)

0(3, 19,

6

0(4,20) x SO(1, 1)

0(4,20, Z)

0(4,20,

5

0(5,21) x SO(l, 1)

0(5,21, Z)

0(5,21,

4

0(6,22) x SL(2, R)

0(6,22, Z) 0(6,22, Z)

3

0(8,24)

0(7,23, Z)

2

0(8,24i l )

0(8,24, Z)

0(8, 0(8,

where D represents the number of uncompactified symmetry group of the equations of motion for the scalar script (1) appearing for the heterotic string compactified down . •.. to the affine group.) Notice that once we make the transition from compactified the quantum heterotic string theory, we find that the group . a rather straightforward way to become the T -duality and of string theory. In this way, we find that the duality groUp!n1:jr::~. theory provide a convenient explanation for the group G, whii(fh:I~~~1 mysteriously in supergravity theory. This process is also there are no R-R fields in the heterotic theory, which when we compactify the theory. This means that the various boson fields yields the massless scalar fields qi The situation with the Type II theories, however, is this table, where the group G for the supergravity theory is

12.4 Example: D = 6

493

D [3l D

Supergravity Group G

T -Duality

V-Duality

lOA

50(1. I)/Z,

1

1

lOB

SL(2. R)

1

SL(2. Z)

9

SL(2. R) x 0 (1. 1)

Z,

SL(2, Z) x Z2

&

SL(3. R) x SL(2, R)

7

SL(5, R )

0(3 ,3, Z)

SL(5, Z)

6

0(5,5)

0(4.4. Z)

0(5.5, Z)

5

£6(6)

0(5, 5, Z)

E6I,6)(Z)

4

E 7(7)

0(6,6,Z)

E 7(1)(Z)

3

Es(s)

0(7.7. Z)

E • .,(Z)

2

1::9(9)

0(8.8 , Z)

E 9(9)(Z)

£111(10)

0(9,9. Z)

E 10(Io)(Z)

0(2,2, Z) SL(3. Z) x SL(2. Z)

we find many R-R fie lds contributing to the massless spectrum, which supergravity theory, and hence the theory is more , \ve' can usc the Type II supcrgravity theories investigate the T - and U -duality groups. consider the D = 6. N = (1, 2) theory, which we obtain by the membrane on Ts (or by compactifying the Type IIA theory at the in different ways yields powerful checks on the theory. compaclitying the Type IIA theory on T4 , we

'1'~:;£~:~O;;~I:~.!4, Z). But the compactification ofM-theory - yields the rc i modular group SL(5, Z). To reconcile these twO faCTS, ,ho,, ,, the smallest group whicb contains both as subgroups, which is 5, Z). This, in tum. is consistent with the S0(5, 5, Z) V-dllality symwe obtain by analyzing I I-dimensional supergravity compactified to six dimensions. also use this discussion to generalize this result for other dimensions. ". compactify II-dimensional supergrovity on a (c + I )-dirnensionai torus, :: 5, then the moduli space of the torus is given by SL{c + I , Z). But if

494

12. Compactifications and BPS States

we compactify the 10-dimensional Type lIA string on a c amle~i~m~ then the T -duality group is SO( c. c. Z). Again, to reconcile theise; ~W~:til~ observe that they are subgroups of the noncompact form of £c-~!\,j!;>:}';:W.bl the U -duality group, which is a symmetry which combines bOl:ttPI'¢.i1tiit)i and nonperturbative states [3]. Thus, we find that

c > 5. In this way, we can use the low-energy sector ofM-theory, 1 supergravity, as a guide in which to construct the string UUIU' u:y,:J! various dimensions.

12.4.2

D

= 6, N = (1, 1) Theories

Now let us analyze the dualities of the D = 6, N = (I, 1) theorii\l$/If.6J,~ pIe, membranel5-brane duality yields the fact that the IIA on K J is dual to the heterotic string compactified on T4 • If we compare the two theories, we find that they have the limit: D = 6, N = 2 supergravity coupled to 20 abelian multiplets. Since the six-dimensional super-Yang-Mills scalar fields, we have 4 x 20 = 80 scalar fields altogether, moduli of the possible vacua. In addition, there are four tained within the supergravity multiplet, so there are 20 + 4 vec:tot: altogether. Now let us analyze the spectra of the compactified heterotic .~,·.~~~1Wti theories. We recall that the heterotic string compactified dovvn to stx:.9il1tool via T4 has a moduli space of vacua given by SO(4.20) SO(4) ® SO(20) ® T which is 4 x 20 = 80 dimensional, as expected. The also has 80 scalar fields. To see this, we note that the metric 4 x 4 = 16 scalar fields after compactifyi,ng four a·mumsILon:s.: E8 ® E8 Yang-Mills field A~ gives us 4 x 16 = 64 scalar yields 80 scalar fields, as expected. We have therefore the following scalar states; gil v: :

10 scalars, 6 scalars,

A ~:

64 scalars.

B llv

Also, A~ yields }t) vector fields. When we combine this fields contained within gIL" and B llv , we wind up with 24 UO) expected. We also note that the N = I heterotic string has Qa with 16 ' six dimensions, the Weyl spinor has eight components, so it can : • . •. into Q~ with N = 2, or (1, I) in this case.

12.4 EKample: D = 6

495

In analyzing the D = 6 theories, it is useful to wri te the spcetra as massless '':::~'~~;:~ of the little group in six dimensions, which is 5U(2) ® SU(2). w mul ti plicities of the various states undc r the little group , we find fo llowing group representations for the bosonic stales:

(3, 3), tensor : (3 , I)oc(l , 3), vector : (2,2), scalar: (I, I).

gtm'itoo :

(12.4.7)

we decompose the variollS 10-dimensional fields according 10 Ihis six"""ie,",ll classification by the linle group, we find

g,.' B,,'

A;, .'

(3,3) + 4(2 , 2) + 10(1, I), (3. 1)+(1,3)+4(2,2)+6(1, I). 16(2, 2) + 64(1. 1),

( 12.4.8)

(1, 1),

.ie. yi" lds 80 scalar fields described by ( I. 1).24 vector fields descri bed by

: 2), and one dilaton. as expected. when we comp::uc this with the Type TIA theory eompactified on K 3. find many similarities. For example, we have the same nwnber of vector All from the IO-dimensional theory contrib utes one veclor fie ld. The

~:~~:;,~eo~:n;:cn~;!butes 22 vectors. There is also one CKtra vector because Al' up si . Splice is dual to a vector. There are thus I + I + 22 = 24 vector fields, as eXfX.'Cted. also no te that K) prese-rves half the supersymmClries after compactifiwe have N = (I, I) instead of (2, 2), which agrees with tbe previous supersymmetry we found fOT the heterotic string compnclified on T,.

"'~,:~:;fe~'rmiOniC fields of the two thcorics match.

H·

we encounter some p roblems when analyzing the moduli space the fields of lhc two theories. The hetcrotic and Type !LA theories have :ce,,,fields,with the heterotic theory possessing £.® E, Yang-Mills but the Type theory has notbing like this in comparison. We seem far too lew fields in the Type ITA theory. However, this problem is once we realize that the moduli space of the Type 11 theory receives ';bu,i·,ns from several different sources, including the XJ manifold and licalar fie lds. so that the IWO lheories in six dimensions actually have 'isel'y ",. 80 scalar fields. this. Ict us first anaJyzc the moduli space of X J. which is

SOp. 19) R "SO(3)"SO(19) ' t

(12.4.9)

57 di mensional p lus one additional dimension for the ofthc space. Thus, the IO-dimensional mctric gp.u can be decomposed

496

12. Compactifications and BPS States

into a six-dimensional metric plus an additional 58 scalar fields to the moduli of K3. To find the moduli space of Type IIA theory compactified on have to factor in the contnbution from other fields. The reason is also have the additional contribution of the bosonic fields ¢, B/J-v in addition to the graviton, each of which may potentially .. moduli space. . .. To calculate the number offields that can live on K 3, we note that tl;i~[Uft of massless bosons on K3 is equal to the number of harmonic p-foj·~~~~11 This, in turn, is given by the Betti numbers for the manifold, whiCCh~~i~~~~1 hI = b3 = 0, hi = 3, Hi = 19, and b4 = 1, i.e., there are tl two-forms and 19 anti-self-dual two-forms that can be defined OIj(wi~:~ There are thus 3 + 19 = 22 generators. These 22 two-forms l1V:):J.tg::~ transform as scalars under the six-dimensional Lorentz group. In Qm;~l!:~~ the B IlV field reduces to the six-dimensional tensor plus an fields, while the other fields contribute nothing. Thus, aI together, Type IIA string theory compactified on K3 has )~4~2~ scalar fields, the same as in the previous case. We find therefore the following number of states:

gil": B Il ":

C/J-vp: All:

¢:

(3, 3) + 58{l, 1), (3, 1) + (1, 3) + 22(1, 1), 23{2, 2), (2,2), (I, I}.

In fact, the moduli space of Type IIA compactified on K3 is same as the one for the heterotic string compactified on T4 • two effective low-energy supergravity theories, we find that the same. We find therefore that

12.4.3 M- Theory in D = 7 Similarly, we can perform the same type of analysis in seven can "delete" or "litt" the action of Sl on both sides of the in six dimensions to obtain yet another dual relationship in This means analyzing the duality between compactifying thlee):M~~1 and the heterotic string on T3 by removing SI. Notice that bl T -duality group SO(3, 19, Z) and the same moduli space SO(19) ® T, as expected.

12.5 Example: D =4, N = 2and D = 6, N = I

497

This can be summarized as:

I

liAon KJ

£8 ®

M-theory on K J

Es®Eg.onTJ

Es

00

T4 ,

(12.4.12)

015"'1,,;'in11Iy. a very large web of compactificalions down to various dimenhas now bccn found using membranc/5-branc duality, where the dual of string nuns oul to be the compactification of another type of a powerful check on the seif-consistcncy of this approach, insight into the previously mysterious strong coupling theory. : We caution that we arc still far away from analyzing reductions down to D = . N = 1, whicb is the physically relevant compaetifieatioll. However, new , results have already provided insight into D = 6, N = I and D = 4, . := 2 compactifications, which should provide a guide for the much more . caseofD =4,N = I. smnmary. a va~t number of discrete dualities can be constructed in various in,:~~:~~;; One starting point is to analY7.e the continuous dualities that were ~, calculated years ago for ordinary supergravity theories, Bnd then discre te subsets and combinations of them wbich may pcrsi!'it in a theory.

",";tri,,,

Example: D

= 4, N = 2 and D = 6, N = 1

Mnl)1 plly"ic
CVVe

Ie;::t::~~:;~e,~~.~~~,;'.~%,!~~;. ~d;l~o D = 4, N = 2. will concentrate :~ latter.) We can i the D = 4 theory by compactifying the D = 6 for most cases. We will find already at this level that highly nontrivial ~It" m,,,g., . which we expect to carry over into the physically relevant case =4, = I. begin by compactifying the 1::8 ® £8 heterotic string dowo to D = 4, 2, where we need a six-dimensional, Ricci-flnt complex manifold wilh holonomy. We can show that there is only one choice: K 1 ® T1 . We will that this theory is dual to the IIA tbeory compactified on a Calabi-Yau ~ .•:.I_ with SU(3) bolonomy. The analysis is much richer than the previolls 6 case, and provides clues as to what we will find in the more complicated ofD=4.N=1. first notice Lhat the heterotic string compactified on Kl €I T2 is complibecause of the nontrivial CUIVature of the manifold and the presence of YWlg-·M,iU, fields. For example, we know from the Bianchi identity that

dH =TrF

1\

F-TrR

1\

R.

(12.5.1)

498

12. Compactifications and BPS States

If the left-hand side integrates to zero, then we can compute the the right-hand side, because they are equal to the second Chern cla:s'$' i ~:f"dt: respective spaces. After integrating the previous equation, we find sYi:~J'1~~~1 that

We know that the second Chern class for K3 ® T2 equals 24, so by :th~i:miii identity, this means that Cl for the gauge space E8 ® E8 is also , , ,, : ~~ the integral over F A F is equal to the instanton number. Thus, 'Vi'":¢" ':;Q 24 instantons. Because the gauge group is E8 ® £8, we have a '::,: to distribute these 24 instantons. In general. we can have 12 - n one E8 and 12 + fl instantons in the other. Thus, the string , interested in are indexed by the integer n . This theory, in tum, is dual to the IIA theory compactified on :, Yau manifold. But since there are possibly an infinite number manifolds, we must restrict our choice. In particular, we wish Calabi-Yau manifold with an integer index n, the counterpart n:t':ij;:~:::i which describes the way the instantons are distributed for the he~t#gi{~~ We will first make two simplifying observations. We first n,lij"t"e;:;a; particular class of Calabi-Yau manifolds can be written as KJ ":':.......... i.e., K J fibers defined locally over complex projective space • •~:~~:~~; space. This means that there is a distinct K3 defined at each " We represent this symbolically as K3 over PI. Similarly, we also note that One particular class of K3 tum, be written as elliptic fibrations. The elliptic fibration can by placing T2 fibers at each point on the complex projective base space [7). For this particular class of K 3, we write it K3 = T2 over Pl. This means that this class of Calabi-Yau manifolds can be . defined locally over p], and K3 in tum is defined locally over an(J.m~it::l~.@ itively, this means that the Calabi-Yau manifold can be written Sylp.J~i~Jji~ T2 defined locally over a new manifold F, which is equal to P, dejm¢:i;l:~ij! over another PI' But we also know that the set of PI fi brations over another PI as the Hirzebruch space Fl!' where n is related to the the manifold. For example, for n = 0, we have Fo = P, i8l Pl . This process can be expressed symbolically as i

K3 over P, = [T2 over PI] over PI = T2 over [Plover Pdn

= T2 over Fn . Puning everything together, we now suspect that the £8 ® £8 het4t~~~ compactified on K3 ® Tl is dual to the IIA string compactified on

12.6 Synunclry Enhancelnenl and Tensionless Strings

499

nanifoldgivcn by the elliptic nhentioD ofFft (8]. So we have S : Es 0 £. on Kl ® 1i

~

tIA on [Tz over

F~],

( 12.5.4)

the integer n describes how thc instantons are disrributcd (for the ~"roli" "tring), or the index of the Hir.£cbruch space (for lhe Type I1 A theory). more careful analysis of the moduli space, thc BPS spectra. and the

~

~~~i:~ac~~'i:O:ns bear oul this conjecture. Other interesting features of lhis which we only briefly mention, arc symmetry cnhancemenl suings.

Symmetry Enhancement and Tensionless Strings

Type l1A

h:,;:~:;:;r.!:'~;";;: strings l:ompactified on K 1 and heterotic strings on T4 has many other interesting properties. For example, the group of the compaclined nelerolic Siring in six dimensions is U( I )2•. o~ev<'r,we CRn easily modify the theory so the gauge group becomes larger.

)ll

a cer1ain choice of the root vectors, we can cnchanee the Lie group so

~

:~~::~,:~:~symmcuy oflhe type A-D--E. For example, for the lefi·moving

hclerotic stri ng, tbe root vectors belong to the lattice A 20, 4. whose group is the familiar T -duality group 0(20 , 4, Z). If a root Q' obeys a 2 = -2, then we can calculate its corresponding mass, and that it is massless IlDd corresponds to a charged vcctor particle. Thus, the roots of the lattice are chosen such lbal lhey obey this relationship, more massless vector gauge particles corresponding to an enhanced given by the classification A-D--E. enhancement in the heterotic theory compactified all T4 is not SQ obvious is how the dual oompactificJ Type IIA string this gauge enhanccmcnt when compactified on K}. At first, it seems to obtain an enchanced symmetry of the type A-D-E from comllA string theory. Remarkably, a careful analysis shows that the manifold on which we compactify the Type 11 string can have orbwhich cmibil cer1ain types of singularities. By classifying the possible gul.ru-lli" of these Calabi- Yau manifolds, we find that they can be calCaccording to precisely the same A-O--E classification. So symmetry takes place for both the compllctified heterotic string anJ the Type reasons. This is yet another confinnation of J uality. A:~~.~stran gc property of six-dimensional heterotic strings is that they tl phase transitions. Let us analY.le the D = 6, N == (1 , 0) theory, is chirnl. Because of this, we must carefully analyze the chiral anoma· ~,~ may enter into the calculation after compactification. These dural must be canceled, which places more restrictions on the model. calculation shows that the anomaly cancellation is possible if the

~

500

12. Compactifications and BPS States

anomalous eight-form Is factorizes into the product of two totJr~~9~~U 18 -+ X4 1\ X4 , where

CJt

where R is the Riemann tensor, Ct labels the gauge group, Mills two-form, and Va and Va are constants given by the , be shown that the kinetic energy terms for these gauge fields " following combination: ', :~~~~~~~:":J:::;:;

})vae-

va1~i~1111

BU;

where Va is positive. notice that the kinetic energy term negative. This term vanishes if we have the following Cotldl1tlOlit

e2¢>O = _ V" Va for some value of r/J = r/Jo. The vanishing ofthe kinetic energy term means that the th~:or:yis!ffiil~~ a phase transition. Since the Yang-Mills kinetic energy T"nm " ,,,, by 1/ g2, it means that the coupling constant diverges. This a1S{'r••~~::®i interpretation. Before, we saw that there can be dyonic strings P'.,:t·t: ~"#~ theory with electric/magnetic charges given by (p , q). The ', dyonic strings is given by Tp.q

= pe- rP

+ qerP .

It can be shown that (p, q) is proportiona1 to (Va, Va). This tension of these dyonic strings vanishes at the singularity. These ' called "tensionless strings." (For example, for the group SO(Il), we have V = 1 and therefore that the tension vanishes for the (1, - 2) dyonic string.Fd!r:lli~ 1 E 8 , we have IJ 30 and ~, so we have a tensionless string trir.:f.Fle:

v

=

v= -

dyonic string.) It ttJrns out that for a wide variety of compactifications, we strings. However, there is one compactification in which this appears. For the case of the E8 ® Es string compactified on

X 4 = Tt R2 - ~ Tr

X4 =

Tr R2

fu~:lt~~jwl~m

Fr - i Tr Fi,

+ (1 - ~~)

Tr FI2

+ (1 - ~;)

where 1, 2 label the two Es groups and the instanton 1l] = 12-nand1l 2 = 12+n.Inotherwords, VI = lJ2 = kandvi This means thatfor III = n2 = 12, we can never have tensionless

12.7 F-Thcory

501

embed instantons symmetrically into the two E~ 's, then we have a smooth ,",;,'ioo between weak and strong coupling, without any phase lransition. lasl feature of six-dimensional string theories is the existence of bctduality. If we let the area of the torus 72 in KJ ® T2 go to we recover the D = 6, N = i theory compaetified on just K J. This that the heterotic string compacrified on K 1 is duaJ to the limit of IIA compactified on T2 defined locaUy over F" when the area of the tOnlS ,",,,I,,,. infin ity. This later theory can be defined ill several ways, which in gives us IlA theory being dual to yet another heterotic string. Combining "" ""0 heterotic string theories, we lind H new heterotic/heterotic duality in we find that several unexpected phenomena arise when prohing dualities of the D = 6, N = I theory and D = 4, N = 2 theory. This, indicates that the physically relevant D == 4, N = I theory is sure to Sllmn1l1ry.

F-Theory the unification of the five superstring theories is quite remarkable, but are many loose ends. In particular. it secms odd that M-theory, existing : II-dimensions, cannot be directly reduced down to Type IlB superstring :. ill iO-dimensions. Instead, we must first compactify M-theory via T20 . then compare tliiS to the Type JIB theory compactificd on SI. Since our goal : fmd a single thCOI)' whose vacua yield the various strings and p -branes, : would prefer to have a theol)' which can be compactified do\VIl directly to IO-dimensional Type IIB theory without taking B strange limit. intriguing fac t is that the Type liB theory has S-duality given by . Z), which is also the moduli group of the torus. So far, this SL(2. Z ) "::~:has been as pwely mathematical symmetry, without any ac significance. However, we can also posrulate that this SL{2, Z) from the compactification of a genuine space-time symmetry on 72 • case, we postulate the existence ofa new 12-dimcnsional theory, with (10. 2). This is caJled F-theory [9].

treated a

;

~~~~:~~ 12-dimcnsionaJ symmetry have long been studied in terms of

Earlier, we saw that the maximum nwnber of components that 32.lfwc use the usual LorcntzilLll metric, then I I-dimensions maximum Dumbcr of dimcnsions with 32 Majorana spinors. However use the non-Lorentzian space-time group O( I 0, 2), then we find that ",jonma-\\lcyJ spinor has 32 components, not 64 [IO,II}. Thus, F-theory be such an outlandish idea. essentilll idea is that F-theory compactificd on T2 is identical to Type as lheorigin of the SL(2, Z) modular symmetry. Ifwe n,'~:~,~~~~':~serves Type IIB theory on some manifold B. then we can reexpress this

502

12. Compactifications and BPS States

in terms ofF-theory compactified on a larger manifold A, properties. Recall that the SL(2, Z) transfonnation operates on th¢ I;matQWiW~ axion defined at a local point in space4ime. Thus, there is a Seplarai«nru: transfonnation for each point on B, which means that there must:ii~;~~ defined at each point on B. Specifically, if z is a point on the ~;~~!~III the complex structure of the torus is given by r(z) for each point that a fiber bWldle has a topological structure quite different manifold T2 ® B.) This means that A must have the structure of . ., not a product manifold. The fiber is T2 and the base manifold is~ ' t~~!~~m every point on B we have a separate T2 • In summary, we say that F-theory compactified on A is jdenti\~m::~ IIB theory compactified on B, if A has the structure of a fiber 'buildl:j;f:afjj~ elliptic fibration of the manifold B. We write this symbolically A = T2 over B.

We can also use F -theory to explain the S-duaH ty between the and the Type I theory. Let us compactify F-theory on T2IZz, SO(32) theory. As before, this corresponds to compactifying on '" we have lost the SL(2. Z) symmetry of the torus. This means :b :j:;i:;:t;J two inequivalent theories if we take R ~ 0 or R ~ 00. So SQ,(~~~j~ of being self-dual as Type lIB theory, is dual to another theory, T' ,' . "",;·'r:"\iI(1iI\ We can also exploit the fact that K3 = T2 ® T2IZ2 for a K 3. If we again "lift" T2 from our previous duality theory compactified on a certain type of K3 that gives us the compactified on T2 • We summarize this symbolically with: F-theory on T2 F~theory on A F-theory on T2I'Lz F-theory on K3

++

lIB, lIB on B, SO(32),

+?

E8 ® E8 on T2 ,

+? +?

for A being an elliptic fibration of B. There is no doubt that F-theory has proven to be a t():p.H~ the web of dualities. However, there is some doubt as to how ... really is. It may indicate that the origin of M-theory really lies: . dimension, or it may be just a convenient mathematical trick. . U"",LUL

12.8

Example: D = 4

Although D = 4 theories are difficult to construct, we can e~uW*~:~1 simpler case of D = 4 and N = 8, which is found by ord~~~!~: compactification. Although this theory is not realistic, we will

12.8 Example: D "" 4

503

,count BPS slates. We know from our previous discussion of superalgebras there should be 56 central charges for an N = 8 theory in fou r dimensions.

start by compaclifying l l-dimensional supergravity down to four meas;a", ·a . T, . us analyze the number of gauge fields in four dimensions, and hence the of electric charges appearing in the BPS relation. The metric tensor gives us sevefl vector fields, while the A1,uh gives uS 7!/5! 21 = 21 vector There arc thus 28 electric charges in the tour-dimensional theory, as ,,~.~d from our earlier discussion of the 0 = 4 super translation algebra. remaining 28 magnetic charges comes from the dual theory.) us compare this with the cha rges coming from just the supcrsymmetty ,b,•. ~; ; D'" we compactifY on T7. the supersymmclry algebra gives us 7 + 7 ,,"a magnetic charges. (This comes from the seven charges contained PM as wcll as tbescvcn charges eontained within the five-form 2 .. /1 ...../.) n;~:~;;: :c~here are 21 electric charges obUl ined by wrapping the membrane, j 21 magnetic charges obtained by wrapping the 5-brane. (This is thc number of space-time scalars we can construct out of two-form and five -form ZMNOI'Q in seven dimensions is 7! / 2 1 5! = 21). There lhus 56 electric and m.1gnetic charges nltogether, as expected from the

"',..0

fonnulation of II-dimensional supcrgravity compactificd down 10 D = iup"grnvity, in tum , should be S -dual to Type llA string theory compaetified us check this. Naively, we might expect that the moduli space of for this string theory is described by 0 (6, 6}/ 0(6)@0(6), and indced :' T-duality group is 0 (6, 6, Z). But this group is 100 small to describe the .' space uflhe theory. Thus. we must also analyze the scalar contribution the R~R seClor.

In~:~'~:;D~:~,~w;e: :,re~;;n~terested in the U -duality grou p, which wc can find by ~ contain both the T -duality group and the S-duality We find that E7(?\(Z) , which i!': a di:;crete !':ubgrnup of Ihe the moduli spacc of s upcrgravity vacua, contains both Z) and the T -duality group 0(6. 6. Z} as subgroups

E",,(Z) OJ SL(2, Z) ® O(~, ~, Z). ~,o.d ud '''h'' the U-duality group

(12.&.1)

is E7(7)(Z ), A careful analysis of the

salurdted states will confinn this conjecture. us analyze the vector fields left over after compactification and benee they couple to. If we compactify from 10 to 4 dimc!lSions. then g,.. and 3nlisymmClric tensor B ila in the Neveu-Schwarz sector 6 + 6 = 12 gauge vectors. These couple to 12 elecrric charges J Z under Ihe T -duality group (If 0(6, 6, Z). But at this point, pcrtum.llion theory fails to acco unt for all the cbarges ofthc theory. theory gives us only 12 electric charges, bUI we know that they have 28 electric and 28 magnetic charges.

504

12. Compactifications and BPS States

The addition ofR-R fields only makes the situation worse. The ~~i~~i fields C,.,. and A,.,.ab, in principle, contribute 1 + 15 = 16 gauge H"j~,l,I~;;::au~ problem is that they couple to the string world sheet via the field tenl$Ql~~1 than the potential. Thus, they do not couple to any electric or E;u"u ....:\:J] This means that we are still missing 16 electric and 28 magnetic c~11 The final resolution of this puzzle comes in when we analyze tlJ perturbative theory, which will indeed show that there are new p-l,~~~jHW?a called D-branes, which couple to the R-R fields in precisely the Now we see the resolution via nonperturbative effects. is necessarily a nonperturbative symmetry, puts the missing 16 in the same representation 56 as the 12 charges found earlier in theory. We ,find thatthe 56 can be decomposed withrespectto SL(2, as U-< . .

56 -+ (2, 12) EB (1, 32).

J~;II

The essential important point is that the T -duality separately NS-NS 12 and the R-R 32. There is no mixing between these tWij::Sjftj But U -duality places all of these states into the same multiplet "'''',;'.lll nonperturbatively mixes all these states together, :J:J:::=:@ With the addition of these R-R charges, we find the BPS IIA string are satisfied. The 12 + 12 charges originating in sector can be constructed in terms ofK.aluza-Klein states and of the compactified string. The 16 + 16 charges, which mysterious, come from the R-R sector from a new type of D-branes. This gives us 28 + 28 electric and magnetic charges, For the heterotic string, the analysis is a bit different. The S is SL(2, Z), the T -duality group is 0(6,22, Z), as expected, buf:t1ift duality group is only SL(2, Z) ® 0(6, 22, Z). The dimension ot.:lJ~~ space of vacua is thus 22 x 6 = 132. This can also be seen by d~~~~ the fields of the heterotic string to yield the scalar sector. If a, b dimensional torus, then the metric gab gives us 21 scalars, the tW(~fQJ gives us 15 scalars, and the 16 gauge fields A~ for the Cartan £8 ® £8 or SO(32) gives us 6 x 16 = 96 scalar fields. Adding,tf'~~~~~I find 132 scalar fields. as expected from duality. '{ Now let us count the number of U(l) gauge fields and heII9~:~n~~~ij of electric/magnetic charges. The metric gila gives us six the two-form B,.,.a. The E8 (8) E8 gauge field A~ gives us 16 total of28 gauge fields, or 56 electric charges altogether, as

12.9

Summary

So far, many ofthese relationships relied heavily on the of string theory. To obtain more confidence in the approach,

12.9 Summary

S05

n~:::.::::,:~:: analysis of the theory. This is given by the BPS saturated are believed 10 be nonrenormalizcd because of the nonrcnormal-

at

theorems of s upcrsymmctty. These BPS states, in turn, are related to

,,";",ence ,ofp-brane Sl
,

",o,d",lIy, the second term was dropped, since it violated the well-known theorems of 5upersymmetry, which made the tacit assumption thal only .;;;~,~~~c~~~ existed. NO\\I tlult we arc: dropping this tolcit asswnption, we ,w presence of p-branes wb..ich saturate !.he BPS relationship. Wldcrstand the dynamics of p-branes, consider 3 p + I rank·lensor ;;""",n"";o gauge potential coupled to a p-bmnc. The coupling is given

(12.9.2)

( 12.9.3)

xI'(crj) is a generaliz."llion oflhe string variable usuaUy found in siring lo,,,,,n,{ru
aiL, A l'l ...II-", + permutations.

(12.9.4)

" ,,,,lin,trod",,, another field tensor F' such that the dual of F is idcnUfic..-d (12.9.S)

F' is the field tensor corresponding

(0

yet another tensor potcntial

",,,<,,,d;,," to a q-brane, we then have thc condition

p+q=D - 4

(12.9.6)

gives us the dimension of lhe dual of a p-branc. consider the fuH I I-dimensional super algebra, including the central

IQo. Q6J = (rAt c).." PM + (J"'MNC)CI} L.MN

+ (rMNI' QR c).,6 ZUNI'r:lR'

(12.9.7)

central cbarge term on the right-hand side corresponds to a p-branc. mting '''''"'' we find that cach term connibutes

II

+ 55 + 462 =

528 stales,

(12.9.S)

506

12. Compactifications and BPS States

where the 55 states correspond to a membrane, and the 462 to a 5-brane. What is rather remarkable about this simple analysis is that we c~H~ljfm the existence of the complete set of BPS p-branes for the th"",,TV ttiii having to construct them! . For example, consider the 1O-dimensional Type IlA algebra:

{Q". Qp} =

(rMctll PM + Z + (rMC),,; ZM + (rMN + (r MNPQ C)"1I ZMNPQ

+ (rMNPQRctp ZMNPQR. Counting states, this reduces to 10 + 1 + 10 + 45

+ 210 + 252 = 528 states.

By analyzing these equations, we see that there should be BPS sta.l(e:~~ p-branes. If we analyze the lO-dimensional Type 1m algebra, we find

{Q~. Q~} = 8ii (prMC)"/J PM

+ (prMC)ap ·Z~ -+ fiU~r":'NPC) J""t MN.e.+ Sij (~rMNPQR + (prMNPQRc) "Il Z+jj MNPQR'

Counting states, this gives us 10 + (2 x 10) + 120 + 126 + (2 x 126)

= 528 states,

as expected. This, in tum, tells us that there should be odd p-brane IIB string. Of what use are these p-branes? We will find that they in defining the nonperturbative structure of the theory. For ex~IIliJ"J~j(~~ the long -standing puzzle of what were the sources for the R-R 1teJ!as:;:~ that the Type IIA theory had massless R-R fields given by { the Type lIB theory had R-R fields given by {i, B'jJ.v. CjJ.vap string variable XJ.I' it was impossible to construct a source Thus, the string had a net charge under N3-NS fields, butna(1u~~ under the R-R fields. Now we see that the Type IIA(B) thp,iftt-::·trl nonperturbative states given by even (odd) p-branes which c~i:~jg~l~[ll for the R-R fields. ,.

This is important for the Type lIB theory, for example, [email protected]~iw]11 symmetry rotates the tensor field B into B', so there must be carry the charge associated with B'. In this chapter, we have seen how the dualities found in hig:ll¢ji)>>j~ naturally lead us to dualities in lower dimensions. In PaIl1C·U:l~f,,: the nonperturbative region of one compactified string rnp,nrv

12.9 Summary

string theory.

507

string theory

,o~:::~~:~:~'lk~'::~~:,~ For example, Type IIA rn on K J is dual to the heterotic smng compactified on T... .: There are man}' ways to see how this duality works. The simplest way is : compare two seemingly unrelated string theories which have the same lowSpecifically. we shall analyze the supersymmetry generators survive the compactificalion process. lfwc begin with a supcrsymmctry Q" in 10 or 11 dimensiODS and then begin 10 compactify it down . , we find !hat it decomposes into Q .. .I. where a tabels the 'in:.~:;:~e~ in a lower dimension and i labels the number of supersynunetly m In this way, once we know the holonomy group of the manifold o

•

which we arc compactifying a theory, we can calculate the number of ,.",,.,="t'Yg~,,,",,,!

N tlla! survives the comp:lctific:ltiol1 process.

particular, case D = 6 has been analyzcd cxtcnsively. We begin with fact that M-theory in II-dimensions contains both a membrane and its dual, '·b'~ne (wh,;ch is given to us by analyzing the BPS supersymmetry algebra). can compaclify both the membrane and 5-branc on a five-dimensional 'c.~;,f.:~:~ by product of a one-dimensional space M, and a four~. space M 4 • The resulting theories in six dimensions should be to each other, sinee they were dual to eaeh other in !I-dimensions. can let Mf == S, or Ml = S';~. Also, we can let M4 = Xl or M~ = 74. way, we now have four ways in which La compactify the membrane and In this way, we now obtain four possible dualities. interesting case is D = 6and N = (I, I), which establishes II duality. Evidence for this duality is given by analyzing the stnlcrure of both theories, which are not obviously the same. . space of the heterotic string compactified on T4 is given by the

the

SO(4.20) 80(4) 0 SO(20) 0 T'

(12.9.13)

is 4 x 20 = RO dimensional. This, in tum, must match the numher of

",::c~:.~r:h::'~'P~1:;:~I!'~:~TO see Ihis, we note that the metric gil-v

= 16 afit;rCOUlpactificatioll, Similarly, the Yang--Mills field A~ gives us 4 x 16 = 64 scalar field.,. The sum 80 scalar fields, as expected:

B"v yidd 4;<. E~

g"U:

10,

BII-":

6,

(12.9.14)

A:: 64. \I,e can calculaLe the gauge group which survives tbe compactification yields 16 vector fields. When we comhinc this with eight vector j, ,oor"a;.nod within gil-I' and BII-v, we wind up with 24 U{ I) vector fields, as can put this altogether by analyzjng the Iinle brrouP of the six-dimensional which is given by SU{2) ® SU(2). We fmd the following on-shell

508

J2. Compactifications and BPS States

decomposition of the various fields: gllv:

E llo : A~:

!/J:

(3,3) + 4(2, 2) + 10(1, 1), (3, I) + (1,3) + 4(2,2) + 6(1, 1), 16(2,2)+64(1,1), (1,1),

which yields 80 scalar fields described by (1, I), 24 vector fields de$¢dlj!Ji; (2, 2), and one dilaton, as expected. Now compare this with the Type lIA theory cQmpactified on K3. TtiiM·~ ing of scalar states is much trickier, because we must carefully an!llYli moduli space of K3, given by R+

SO(3, 19) ® SO(3) ® S0(19) ,

which is 3 x 19 = 57 dimensional plus one additional di!jnerlSiC)jf~tQf:: volume of the space. Thus, the lO-dimensional metric gjJ.v can be dec.QmiIi:( into a six-dimensional metric plus an additional 58 scalar fields. Similarly, All contributes one vector field. TheCllup tensor conltl11i~t~ vectors. There is also one extra vector because AjJ.uP in six-dilmelnsi4:>ti.~~:$t is dual to a vector. There are thus 1 + I + 22 = 24 U(l) ve(~tm' U~i'lm expected. We find therefore the following number of states:

gjJ.v : ¢ : All: BjJ.v: CIlVP :

(3,3)+58(1,1), (1 , 1),

(2,2), (3,1)+(1,3)+22(1.1), 23(2, 2),

so there are 80 scalar states described by (1, 1) and 24 vector states ·.. by (2, 2). ... . Lastly, we observe that K3 preserves half the supersymmetries pactification, so we have (1, 1) instead of(2, 2), which agrees with (1, I) supersymmetry we found for the heterotic string Thus, the fermionic fields of the two theories match. In summary, we find that the low-energy actions of the two the same number of fields, the same moduli space, and the relationship when we let ¢ -+ -cp. This result can be genemlized by analyzing the supergravity for various dimensions spanned by the scalar fields. We expect thh:a~~t!iij~;~~ uli spaces for supergravity theories can be generalized to the n for superstrings by making the duality groups discrete. Fortunately, spaces for II -dimensional supergravity compactified down to VaJ:IOllS::{Um sions have been cataloged long ago, and represent the first step in estiil?11J

12.9 Summary

509

relationships between tv.·o dissimilar string theories. By comparing the space, the supersymmetry group, and BPS slates, we can therefore "".bli,h a number of dual relationships. Much more difficu il (and more interesting) are !he compactifications to =' 6, N I or D = 4, N 2, which represent the cutting edge of research. ]."ull> for these two eascs will shed much light 011 the physically relevant case D = 4, N = I. Not surprisingly, we find tbat these oornpactifica tions are

=

=

:j,il~~~::~!~. bccausc oftbe in[rOduction ofCalabi-Yau manifolds. the case of D = 4, N = 2. By checki ng the supersymmetry chan,

::;

can see that the heterntic ~tring can yield this symmetry if we rompnclify a six-dimcnsional manifold with SU(2) holonomy, of which there is only choice: K 10 T2. Using the Bianchi identity, we can show thai

dH=TrPJ\F+TrR J\ R.

(12.9. 18)

:WI,,, the left-hand side is zero, and we integrate over the right-hand side, we that the second·Chern class ~ defined over the gauge space £ . 0 E, is ~qua l

to the second-Chem class defined over the manifold K J ® T2• which is 24. satisfy this relationship, we must have 24 inslantons for the· gauge group. must howe 12 - It instantons for one £8. and another 12 + /I insumtoos for

other Ea. We suspect that this theory is dUlil to a Type n theory compactified on n c:,~~~~:~manifold with SU(3) bolonomy. labeled by some inde:o: n. Fortu~: is n simple way in which to construct such Calabi-Yau manifolds. make the oh.~ervation that a certain class of K3 manifolds can be writas elliptic fibrations, or, crudely speaking, manifolds with lorii T2 deftned ."""h point of a base manifold. In thi:; case, the base manifold is given by P I. Then wc can write tbe following sequence o f relationships between fibe r

K} over P I = [T2 over P.] over P I

= T2 over [P lover PI In = T2 over F ft •

(12.9.l9)

last step involves the fiber bundle defined by P I fibers defined over a base . given by PI , which is given by the Hirzcbruch space Fn. Thus, the defined by compactifying the heterotic string on KJ ® Tl the index labeling Hirzebrueh spaces Fn when we compaclify the If string on n particular Calabt.-Yau IlUInifold. we have £8 ® £8 on K) 0 T2 we have 12 -

-H-

IlA on [T2 over Fnl.

n instantons in one £8 sector and

(12.9.20)

12 + 11 instanlons in the

510

12. Compactifications and BPS States

Many of these results can, in turn, be derived by postulating the . ' a 12-dimensional theory which, when compactified, becomes Type $~$I··· theory. We recall that we can define an SL(2. Z) symmetry at eacliiip.~r· the manifold, which d~fines a fiber ~un.dle. We. there~ore define F-th~9~r.~~. theory when compactdied on an elhptlc fibratlOn, YIelds the Type II:$.:tfr We can summarize many of these relationships via .... .. . F-theory on T2

*+

lIB,

F-theory on A

*+

F-theory on TdZ2

*+

lIB on B, SO(32),

F -theory on K 3

+*

E8 ® E8 on T2 •

for A being an elliptic fibration of B. Whether F -theory is fundamental theory remains to be seen.

References [1] M.1. Duff, M-theory (the Theory Formerly Known as Strings), hep,-JIV9:1;i 1996 '-:-:-:':':-:.:1 [2] 1. H. Schwarz, Lectures on Superstring and M- Theory Dualities. Tl~i:/ School, June 1996. [3] C. M. Hull and P. K. Townsend, Nucl. Phys. 8438, 109 (1995). [4] P. S. Aspinwall, K3 Suifaces and String Duality, hep-th/9611] [5] M. Duff, R. Minasian, and E. Witten, Nucl. Phys. 8465,413 (I [6] E. G. Gimon and 1. Polchinski, hep-th/9601038. [7] G. Aldazabal, A. Font, L. E. Ibanez, and F. Quevedo, hep-thl9602o.St [8] D. R. Morrison and C. Vafa, Nuc!. Phys. 8473,74 (1996). [9] C. Vafa, Nucl. Phys. 8469,403 (1996). [iO] M. P. Blencowe and M. J. Duff, Nue!. Phys. 8310,387 (1988). [11] T. Kugo and P. K. Townsend, Nue!. Phys. 8266,440 (1983).

13

olitons, D-Branes, d Black Holes

Solitons seen thai these BPS saturated states are essential to prove the duality . we have conjecrured. The important point is that it is possible aU of these p-brnne stales as solitons, solitonlike solutions (called

or as supcrsymmetric actions. Solitons are II powerful way in which the noopcrturbattve ~tructurc of II theory, so it is not surprising that and soliton like objccrs will play an im portant part in filling oul the 'S ,,,.,osof lh, theory. Even if we start with a (hoory which consists purely of we are forced to admit the presence of these membrane sLlItcs because are solutions 10 the classical equations ormation. , start with D = 11 ~upergre.v ity, we can construct classical solutions . i correspond to membranes as follows. We break-up the vector X'" = , y"'). with JL = O. 1, 2 representing the membrane coordinates, and m =

... 10.

rne D =

I I supergravity equations of motion are satisfied by the

'owing mctric corresponding to a membrane 11 J:

=

I + y~ )'" (dy'l + idOD , (I + ky~ )-'" dx" dXIl- + (k

(13.1.1)

is the volume form for the S, sphere, and the four·formfield strength to me dual of me volume form on S7. (This solution. it can be I diverge nt at the origin, meaning that it probably corresponds fundllmental solution.)

mil"l,y. the 5-brane soliton of the D = I I supcrgravity theory is given the vector X M = (x", y''') where 11. = 0, 1,2,3,4,5 and m =

512

13. Solitons, D-Branes, and Black Holes

6, ... , 10. Then the metric tensor is given by [2]: ds 2 =

(1 + k~)-1/3 Y

dxi-Ldx ll

+

(1 + k~)2/3

(d/

+ y2dQU

Y

••

.....

and the four-form field strength is proportional to the volume 'V,~:'~1:I;:~~ (Unlike the membrane solution, the 5-brane solution is finite at thfi~J!tgjM~~ it is probably not fundamental.) Thus, the p-brane states form an integral part of the web of BPS st~jt¢'§~11 ing with a D = 11 supergravity theory, we necessarily introduce the ... and 5-brane solutions (as demanded by the central charges of the alg¢.lit solitons or solitonlike solutions. This is an essential point. The ne,;.>::jjJjifj phy of M -theory is that the various strings are just different vacua()t~~:~OO theory. Perturbation theory just probes the vicinity of each vacua, t:i.!W~~ take us from one vacua to another. However, when nonperturbative introduced, then we necessarily find p-branes emerging. Thus, sense have lost their central role in this web of dualities (other mBIq
s = ~! dDx"; 2K

g

(R _!(a¢)2 _ 2(d +1 1)!

e-a(d)q, F2

2

) p+2 '

where Fp+2 is the usual antisymmetric field strength corresponding .. which couples to the p-brane. We wish to couple this to the p-brane action. We introduce X Il (;), which is now a function of the variables l; which p-brane world volume

S" = T / dP+1l; -

1

(p

(~..;ggjjajXMajXN gMNe04>/(p+l) +

p;

.., 2 •••••.p+1 (l X' M ) (J. X Mp+1 A '" . "p+1 M, .... ,Mp+1

+ I)! 6· •

where gU is the metric on the world volume of the p-brane. represents the generalization of the Nambu-Goto action for a last term represents the coupling of the massless p + I rank field to the p-brane variables. To solve these coupled equations, we start with the ansatz ¢ = A i-L1.JL2 .. ·JLp+1 -- -deCI(g11" )6JL,JLl .. ·i-LpH'

ds 2 = e2A d x JL dX Il

+ t?B dym dym,

1

13.2 SupermembrancActions

513

11 = 0, I, 2, ... , p represent the p world volume parameters, and = p + 1, P + 2 .... , D - I. Similarly, we split the p-branc coordinates as (XM = XI', Y"'),and XI' =~I' . and ym = constant. TIlen a solution can be given as D-p-3 A = 2(0 2) (C - a¢o/2).

p+l B = -2(0

,¢ a

3)(C .-a¢ol2).

p

a'

= .(C - a",f2)

+ a¢ol2.

a' = 4 - 2{p + I)(D - p - 3)/(D - 2),

,-c =

I

e-(J4Jr,/2

+ klyp+l,

e-a~!2 _ (K1T Irr)

In y ,

D - P - 3 > 0, D - P - 3 = 0,

(13.1.6)

(13.1.7)

k = belT I(D - p - 3)QO-p-2. Q is the volume ofa hyperspbere. and is the vtlcuum expectation value of ¢. :: From this, for example, we can calculate the electric charge of the p-brane QE

=

;'1

">12K

e-~~~ F

= .../2/{T(_llO- I'-I )(!'+21.

( 13.1.8)

SD_.,-,

The point is ilial even if we originally started with a theory of strings. we ,:::~a~~~,;are forced to introduce solitonlike p-brnnes intu the theory. For :ir below 10, we find a large number of p-branes, so it is important we systematice.lly calculate their action and theirpropcrtics. ln thc process, will find many surprises.

Supermernbrane Actions to constructing the solitons as classical solutions oflhe equations of we can construct the locally supersymmetric actions for these various '. lct us first count the physical degrees offreedom for the p-brane. Becausc p-brane action is defined in a p + I world volume with reparametrization . the coordinate X I' has JJ - P - I physical degrees offi"eedom. To a supersymmetric theory, this in tum must equal the number of dcgrees ,,,do,n within the spinor

D - P - I = ~MN,

(13.2 .1)

the dimension ofthe spinor and N is the number of supcrsymtheory. We have to divide by 4, since, by local kappa invariancc.

514

13. Solitons, D-Branes, and Black Holes

we halve the number of fermion fields, which is halved again wnen::l~ffi1 shell. This relation is easily satisfied for the string (in 10 membrane (in II dimensions) where the right-hand side is equal tice that it fails for the D-branes and the 5-brane in 11 au'nellSlIJnSi'j\It).W such as vectors and tensors on the world volume, will have to be correct this defect.) In particular, we find the following possible solutions [4]:

p = 0: p = 1: p=2: p = 3:

D = 2,3,5,9,

p = 4:

D = 9, D = 10.

p = 5:

D = 3,4,6, 10, D=4,5,7,1l, D = 6,8,

To write the supermembrane action [5], we start with a ization of the original Green.-Schwarz action. We begin with .. XIl(aj, a2, ... , ap+d defined on the (p + I)-dimensional VYv..... a p-dimensiona1 membrane moving in space. As before, we generalized derivative as . IT; = OiXIl - iOrlLOie,

e

illl

where is a spinor defined in D-dimensional space. combination is invariant under the global supersymmetry transtiotjija~~ oX Il = ierlle,

oe =

8.

Then the first part of the action is given by a simple Nambu-Goto action S[ = -T

!

dP+1aJ-det ITi . IT j

.

Although S[ is trivially invariant under a global supersymmetry traJli.sf!Q~~1l it fails to transform correctly under a local supersymmetry ~:~~I~I hence the number of fermionic and bosonic degrees of freedom .... :. To correct this problem, we introduce a second contribution ~:(t;I~~:~gg which is a Wess-Zumino tenn. We begin by introducing an h defined by c.

h=

2'ip.

-

ITllp ... ITIl! der Il! ...Ilp de.

Notice that this p-fonn h is invariant under the previous glolb~l(~~til metry transformation. The point of introducing h is that we can no,&::1l'!p' a (p - I)-fonn b, where h = db,

13.2 Supemlcmbr.me Actions ,.". ." demand that dh = O. Because dn" = i dBf"

(dBr'"

d(J)(dOr "'_.", dO) =

515

d(J, we have (13.2.8)

0.

: This, in ttlrn, forces us (0 have

{r'" P)I"" (r" '''·'''P)y~) =

(13.2.9)

0,

,h"co 1D is the chirality projection operator, if 0 is a chual spinor. For the case = I, this yields the well-known constraint that D = 3.4, 6, 10, which gives the Green-Schwarz string. This new identity, however, forces us to obey 8 constraint., which is given by contracting the identity with ( r ~)"fI. After 3 • J we lind once again thaI D - P - 1 = M N /4, as before. : Thcn the action is given by (13.2.10)

!t we write this out in detail for the supennembmne, we find for {he WcssS2 = -

i; /

dlu

{(s/}tBrl'ui1;O) [n j n;

+ in j'O r a1o U

- (lxe r'a)o)(iir"a,Olll.

(13.2. 11)

. let us cbeck that this action is locally supersymmemc. Let J(J be· at thill point Then we find I _ ll 6b = - nil, ... n \ i do r 1'1 ..1,~ 8(J. ( 13.2. 12) pi the case p = 2. we find: oS = 2iT / d l uo8

"" iT

!

(.J

ggil r i - Is;jkriJ) [hO

d l u 60(l - cr)J l:gll r ,ajlJ.

,~

= 6 I1-g eIJtn''n"n' i I r ,,~p' v

w,,,,,, therefore that lJS = 0 jfwc choose c = ±l,

! ~ (I

( 13.2. 13)

± r)'(u).

( 13.2. 14) ( 13.2.15)

± r) is a projection operator.

~

~~~~t~~'~~i~,~th~,~t,~thi'~inijtro~!d::"~,~ti~on~O~h~WCSS-ZwninO

locaUy supersymmetric, termsuch intothat the of the 32~componen t spinor. (We can by going on-shell. leaving

516

13. Solitons, D-Branes, and Black Holes

us with the desired eight components to match the eight cOlnpI:ment$, bosonic theory.)

13.3

Five-Brane Action

By counting states, we find that the 5-brane does not satisfy the usual '. of p-branes. To construct the 5-brane action, we must introduce a "'4'''',,,,",= rank tensor which is described by a self-dual field tensor Fmnl . Let us count the number of physical modes

-2)

(D-p-l)+ ( P 2

=~MN,

where the first two terms represent the string and two-form degrees respectively. The solution for this is p = 5, giving us a 5-brane. This poses new problems, however, since in general a covariant " a propagating self-dual field is extremely difficult to construct. ' theorems exist, in fact, which actually show that it is impossible covariant action for a propagating self-dual field under certain ' Solutions to this long-standing problem have been proposed, but an infinite number of auxiliary fields. Recently, however, a COllVllncj~$.:~$!j1 for the 5-brane has been written involving just a single auxiliary U<;JI1i,q;1:I7J1o We start with a tensor field on the world volume Amn. Its field tenlsOld~iil by Fmnl = 2 (a[Amn + amAnl + anAlm) . Then its dual F*[mn is defined as

1

F*lmn _

- 6../

g

s[mnpqr F pqr·

We will also find it crucial to introduce a scalar field a eliminated via gauge fixing) and the tensor

Fmn =

I

J(a ma)2

* I Fmn[a a.

The bosonic part of the 5-brane action can be written as L

=

LJ =J-det (gmn+iFmn),

".

L

2

= ~ ..;=g F*mnl F. 4 (a,a)2

aa P

nip·

This action is invariant under the usual variation of an (for small parameter ¢,,):

.5 AmI! =

! a[ma ¢nj,

oa = O.

antisynrnti¢~ij,9jl!

13.4 D·8ranes

is

517

surprising. But what is unusual is thaI it is a[so invariant under a which allows us to gauge the a field entirely away

DOt

,,;''';0''

I1A"'/I Sa

=0

2(ta)1 (F",npaPa - V",n) ,

=,p,

(13.3.6)

(13.3.7) last variation is thc

to establishing a covariant action. Although the

tl~1~,~',~~';"~::~:, 1 ;~':::~~:~CI~:~:;"IC., .",,,",,lIly we wish to eliminate the physical states of the theory.

,:3

chcck that this theory docs in fac t yield a .st::lf~clual te nsor field , we can the Dction in fiat space, in which we obtain

~W"

,,,,,,,,,d

(13.3.8) we derive the constraint F"'~I - F:'nl

=0

(13.3.9)

d~::;;~i'~~we see that the tensor F"ml is self~dua1.

:S

we fmd that the above action can be made supersymmetric with . addit ion of spin or fie lds 8. :: have now found 5~brane solutions in two dilTcrent ways: via solilon '" w·e,,,,I!y"';OO and also via a supersyrrunetricaetion. There confusion. however, over which extended objects are"fundamental" I 0'''' ,n,"so i ." whieh diffcr by whether they arcdivergcnt at the Under a duality transformation, in fClct, we can turn one into the other. example, we found e'drlier that the counting of BPS Slates requires both states and winding modes, the latter whieh are considered to be solitonic. T -duality converts winding modes into K.K modes, so what is fundamental what is solitonic is a matter o f tIlstc. However, one classification that is used is 10 consider the tensions for the various p-branl.!il. The tension ' Ih, fim ,j""",,, 1 suing is of order I, the tension for the D -branes is of order and the tension for the D = 11 . solitonie 5~branc is of order 1/ g: .)

,I;,,",, oj'Ibe

wl'klh

D-Branes we will explore in more dctail the D·brane [8. 9}. Thcre arc important for introducing this new kind of object, which shed much light on the t theory.

518

13. Solitons, D-Branes, and Black Holes

First, we notice that we need a new type of object to ,",VUIl./iQ; background fields. So far, the strings that we have analyzed only to the NS- NS background fields, not the R-R fields. The R-R · . couple to the string world sheet. In fact, to any finite order . theory, the string has no charge under the R-R background H"'l'-\:>~,: need a new nonperturbative object, called D-branes, which can for the R-R background fields. We recall that the tensor field B /LV background field arising .... sector couples directly to Type II strings, so that Type II strings .• ::. under this tensor field. However, the R-R fields couple to the ... their field tensors F/-tl .... /-t ' . So the NS-NS and R-R bac~kgJroundfi#lt~:~i to the string via

where Sa are the standard spin fields defined on the string . the string has no charge under the R- R fields. This poses duality will in general mix the two sets of charges. For example, ••... that the Type IIB string has two massless tensor fields B and B'.::.·~ '(!'tf: duality group SL(2, Z) turns one into another. But since the TYiPi.tUB carries no charge under B', it means that we must introduce : : ~·H·..ij.j~~:~ the D-brane, to carry this charge. For example, if (I, 0) reJ)re~;tm't,$JW Type lIB string which has charge one under the NS-NS field but: Z¢j(~:~ under the R-R field, then an SL(2, Z) rotation will in general .tr:@~!f{iJOO (1, 0) string into a (m. n) string, which can never be seen to anyfj·:j~1~jw~m~ perturbation theory. Second, there is a surprisingly large family of such U-I)ralles, ~ij.J~j~ Ill .

>

mensions. If we analyze the super translation algebra in I 1 ;~IIIII recall that we only have p = 2 and p = 5 M-branes. But if we •. : •• super translation algebra, then the p = 2 and p = 5 sources in .. :. decompose into a very large family of p-branes in lowe~~~·:. :·~. '~ . ~illl of which correspond to D-branes. In general, we will have D-branes for the Type IIA string, and odd dimensional IIB string. We noticed earlier that there was a simpJe relationship the dimension of a super p-brane. However, the D-branes pactifying M-branes do not, in general, obey this relationship . . to introduce a new vector field in addition to the p-brane generalize the brane-scan. A vector field on the (p + 1)-dimensional world volume p - I degrees of freedom to the physical states. So we must formula D - P - I = M N to the following

!

D-brane:

D - 2 = !MN.

Notice that the p has dropped out of the calculation. Compaffil~H~ the R-R states generated by Type IIA, lIB, and I strings, we hn,1··rnllN'

13.4 [)..Hranes

5 19

match them perfectly. This allows us to complete the coullting of BPS

which is a powerful check 0 0 the self-consistency ofour nonpemtrbative of duality. D-branes were discovered by analyzing T -duality in Type I when we made the standard T -duality transfonnation. closed

were mapped into closed strings. We found that (13.4.3) aT-dual ity transformation. So T -duality simply linked d ifferent closed , However, jfwc try to perform a T -duality lransfomlation 00 Type we find that the boundary conditions interfere with the analysis. the Neumann boundary condition at Ihe end of an open string, duality transfonnation, turns into a Dirichlet bowtdary cond ition. us compactify lhe JLth dire<:tion. The standard open string expansion of transforms into X/J as follows: (13.4.4) (13.4.5) X• I'

• :: xI'

+i

. , PI' let

1n ( Z ')

' fl L,,0 ~

2 ..

a m.1' m

(13.4.6)

«(:-'" _z-m) ,

(13.4.7)

olher words. the Neumann boundary condition for X JI becomes a Dirichlet , ndary condition for

XI'; (13.4.8)

= 0 at the endpoints, where

XI' is thc dual string.

";:':~~:~;'~::~:~;:i';;.~~.s~th~c~,~crf;o::':",;is~a:;n open string such that the endpoint is

v'

means the

string, th is that endpoints dual open string lie on a fixed (24 I)-dimensional hyperplane. But .' is a theory of quantum gravity, the dynamics of the theory " e\'enrually this nyperplanc move with time, we therefore are forced io~~:'~ a new objcct: (! new type of p-brane on which Strings can end , the ~( or D-brane. I

+

with Dirich let boundary conditions on (,~,,:~_~~~~::~~tlh~;C~,:T~:-dual If westring integrate

:" !

cia 8".X2' =

X2~(JT) - X2S (O) = 2;ra'p1.S = 2Tuin / R = 2TrflR. (13.4.9)

we have used

k=

,,,tion for the

R. But since we have compactified the twentydual theory, it means both of the must ct.' /

that

ends

string

520

13, Solitons, D-Branes, and Black Holes

lie on the same D-brane. This can be generalized for the out of D total dimensions are compactified, then T -duality leads ~ ..U'.""J;I p-brane with p = D - 1 - k. Notice that T -duality converts a Neumann boundary condition I'U· I~· ~~~r.; let condition. This, in tum, can explain how T -duality changes the. D-branes in Type IIA and lIB theories. We saw earlier that Type has even (odd) Dirichlet p-branes, Ifwe make a T -duality even and odd p-branes must tum into each other. The way this ~~~fj follows. If we start with the Type IIA theory and compactify and make a T -duality transformation, and if a D-brane is wri:lPf,ed;:~ circle, then one of the Neumann boundary conditions of an opl~n;::sttiti~fi into a Dirichlet boundary condition, so the world volume drrnet$lQ D-brane decreases by one. In this way, we can go back and forth. and·odd D-branes via T -duality, ~~~~€~t;~il~~ Now consider the case where we have open strings with '.:' For example, we recall that the Type I string incorporates because the open strings have isospin matrices attached at functions appear with an explicit Lie algebra matrix )",fj in '''''''--'';: rise to a trace over these matrices Tr)., a ).,bA. cAd for scattering am:pUt~ will find that the usual Kaluza-Klein formula for the momenta, significantly altered by the presence of gauge fields. Consider an open string moving in the presence of a massless> . case the U( N) Yang-Mills field. Choose a classical 1 Mills field equal to a pure gauge field All = _iU- all U, such 15 1S (iX 0 I {21fR ... eiX ON/21rR) • U -- di age

where U is a diagonal N x N matrix but takes different Yi11IUl;lll:::~::: along the diagonal. The presence of this classical corlfig~iti(J,i't U(N) symmetry do'wn to U{l)N. Notice that U is not periodic under the transformation X In fact, we pick up an additional phase factor for U U -+ diag (iO,

'"

2S

l~lfllll :

e iON ) U.

Now consider the effect of this phase change on a vector ve~~g~tl:~ which is labeled by a Chan-Paton factor lij}, where these fundamental representation ofU(N), so i = 1,2, ... , N. The pick up a phase e iOl , the jth element picks up the complex the vertex function picks up a phase,

"

lij} -+ ei (6;-6 j )lij}.

The vector state Ii}} also bas momentum p2S associated picks up a phase factor under X 25 -+ X 2s + 2n R. Normau~(.)~~:~~i momentum p picks up a phase factor ei pa under the shift x -+ . therefore, that the additional phase picked up by the gauge neJc:1)1~~~ 8; j(2n R) to the momentum.

1].5 D-Br!lllcActions

52J

possible momentum for the vertex funct ion is now equal to

order to determine where tbe open strings end, take the integral 1"".JI". We now find X25(1I') - X 2S (O) = (2rrn

+ 8j

-

O;)R

( 13.4.14)

the open string no longer bas to begin and end on the same O-brane. We find open strings coding and beginning on N distinct parallel D-bram."S, atcd a t p,,,ilio,,, (J, R, with a network of open strings connecting them. The group associated with this is not U(N), but U(I) ..... it is possible II) restore U(N) invnriance. Now let Ihe D-branes into one O-brone. Then we have 8; -+ ()j. and the full U(N) is restored. The vector state Ii j) 00 longer picks up 8 phase facto r the . transformation . . r(:storation ofU(N) cno also be seen by analyzing the spectrum oftbe Nonnally, the sp~trum of the open string is given by

M2 = (p2j)2 + ~(N - I)

(13.4.15)

(2>,"

(13.4.16)

a'

+

Hj 2:Jra'

e,)k )' + !c(N _ I), a'

N

is the nwnber oJ>l."r'ator o f the harmonic oscillators. If we analyze 'he lying states for N = 1 and n = 0, we find that new massless vector emerge as 01 ....,. 9;, which are required to complete the SpeCtrum of tbe U(N) gauge field . when the hyperplanes are separated, we have N massh:ss vector fields, f~;~~';; of the hyperplanes. But as the hyperplanes converge, the number JlJ vcctors rises to N2, giving liS H(N) symmetry. The lesson here the gauge group associated with N parallel and distinct D-br.rnes is • which becomes U(N} as the D·brancs become coincident. Th is will crucial when we discuss bound states of N Dirichlet O-branes.

D-Brane Actions 'a,,;oo for O-branes can also be constructed in several ways. we might repeat the same steps used in ordinary closed string theory effective low-energy actio n (9]. Since we W'e working with an open massless sector is given by a vector and a spinor. The coupling of to the background vector field is given by the following boundary

522

13. Solitons, D-Branes, and Black Holes

term:

+

!

25

ds

L

o

P

Ai

A;{x , ... , x )8n X ,

;=p+1

where the massless vector field only depends on xO • ... , x P, the cQc'm describing the p-brane. We compactifY the coordinates labeled by portant point is that A; describes the fluctuations transverse to~.·~.~~ijll Because of this, the p-brane is dynamical, rather than being a it plane. As before, we integrate out the higher string modes, beta function relation

fJ

= O.

Since the resulting theory must be conforroally invariant, we "al.i·.UU! these beta function equations as the equations of motion of the The action for the D-brane is then chosen so that the fJ = 0 equations of motion. By explicitly performing all these steps, we find the D-brane by the Dirac-Bol'11-Infeld action T

f

dP+laJdet(Gij

+ Fij -

Bij),

where P' represents the value of XI-' at the boundary of the Dirichlet conditions, and where G and B are the usual terms forUl¢; gl-'u' There is yet another way to construct the action which is m()~4n~~~ the p = 2 M-brane action of M-theory and dimensionally ;~IIIII one dimension [10, 11]. The resulting lO-dimensional theory both the usual Type IIA string as well as the p = 2 D-brane. Thus, a Kaluza-Klein mode arising from the cOlnpacti~~;~~~91 dimensional super membranes becomes a Dirichlet 2-brane in 1 •• We recall that the supermembrane action in 11 dimensions. contained the term gijnpnj.M, where M is an 11-di.menslon~11 separate out the eleventh dimension, and define I{J = XII, then UL".eu.UL

nJ . M.--* gijn~n· g ij nM 1 . I J.I-'

+ igij (a·A. - ier ll a·e) (a·A. -: I'V J'V. I

Now we wish to replace the I{J field everywhere with a vector a;1{J = L;. This is accomplished by adding to the action a L.a~rJ:Jjji~111 A. J

13.5 D-Brane Actions

523

,. we eliminate the Ai field from the action., then it enforces the constrain! . - aJ Lj = 0, wllleh is solved by setting L, = Ji ¢, so we can replace OJ¢

l..; everywhere. :: So far. we have done nothing, since by eliminating Ai we reuieve the original . Now, leI us take a different patb and eliminate L, instead. Then the ,levan, tenns in the action are (13.5.6)

(13.5.7)

(13.5.8) ( 13.5.9)

(13.5.10)

aji! and bl ) are complicated funclions of nand e, arising from tbe original found in the membrane action, in which the l X rI components have been removed.

a

I

tballhc D·brane action is the usual Nambu-Goto action U{ I) Maxwell field temior. Tt has the srructurc of a Bom-

The pre,;",,, caicaiation was done in flat space with all the fermionic fields Morc interesting is the case when the D-brane moves in the presence "ekgco<mdl fi.,lds . Tne same calculation can be done with background fields fcrmionic fields to zero). Thcn we obtain, for the bosonic action - T

f

d p+l(Jj det(Gij

+ Fij

B I) .

(13.5. 11)

G and B are the usual terms formed from the pull-back to the membrane i.e., GIj = OIX" OJ x ~'1,,"' (Before, we found that this tenn in the action function of the Dirichlet boundary Icnn in the effective ficld theory i"',I;,m. We see that this has been transformed into the p-brane coordinate this fonnalism.) fact that FJ) - Bij occurs in this particular combination can be seen fact that tlle gauge tnmsfonnation of the B field., whieh usually canto zero, now picks up a boundary term, which must be canceled by a re'l)on.dio,g change in the vector field

r

BI)

--7

BI)

+ a/X) -

AI -+ Ai - Xi·

Bj'xf'

(13.5.12)

524

13, Solitons, D-Branes, and Black Holes

The most immediate application ofD-branes is to analyze theory compactified on a circle is equivalent to Type IIA string to the proof was the statement that the Kaluza-Klein modes compactified M-theory correspond to BPS saturated modes in 1 . We saw earlier that the momentum of the eleventh dimension .\'<:1:.,.,\• ..,,, according to N

PH =

R'

The N = 0 term corresponds to the ordinary string, the N =< sponds to a single Dirichlet O-brane, and the higher N modes '~il'~wjfij ¢'

bound states of Dirichlet O-branes, "'~,:§i;~ Now consider the case of N parallel and flat D-branes, with~ attached between each pair. Each end of the open string has a VP'l ..g!~l;I associated with it. Now let the N parallel D-branes gra.dwilly :tI:1~iWei single hyperplane, Then we find that additional massless vec:tor: ~4j{~j emerge, which yield a gauge theory based on D(N) [12]. Now re-do the calculation of the D-brane action using the /3 ~=n{~~tlmm If the branes are widely separated, then we arrive at the usual D( •.. ,.......~.. Infeld action. But now perform the same f:3 = 0 calculation, let1tltti~W!@: slowly coincide. We then find that coincident D-branes are def;~tj~9j~ dimensional D(N) super-Yang-Mills theory reduced down to of the D-brane

S=T

f dP+laTr(-~FJ.luFJ.lv+iWrJ.lDJ.lIJ1)+.".

..

(In general, there can be other terms involving higher powers F and its derivatives.) Notice thatthe fields are only functions not the entire lO-dimensional space. This means that many out

Fmn = amAn - anAm + i[Am, An], Fmj = &mXj + i[Am. Xj], Fij = i[X j • X j ], where i. j represent the modes i = p+ I. p+2, .... 9, andtheUt:,tit:;m,ll are represented by m = 0, 1,2, .' .• p. Consider the case, p = 0, representing a point particle. H''';~··i:'·.HiieF. O-brane, this means that all the membrane coordinates have .. for one: ..

Fij

i[Xi,Xj],

FOj

DoXi = aox j i[xj,e],

Dje Doe

+ i[Ao. xj],

-

aoa

+ i[Ao, a].

13.6 M(atrix) Models and Membranes

525

the action reduces to

~:

S= T

f

dtTr O
i()T DoO

+ ~ ([Xi, XiJ}2 + eT y / [X j. 8]) . ( 13.5. 19)

remarkable feature of this U(N) action is that we can now begin to multi-Dirit::hlet p-branc scattering amplirudes. Notice that if the rnafor the theory are block-
M(atrix) Models and Membranes of the most important applications of D-brane physics is to elucidate the of M -thcory. We say earlier how complicated M-theory was, with 2and 5-branes. But one of the most remarkable conjecrurcs about the

O~~~'~~:~~~;S::'I~h,~31~:il reduces 10 a simple man;lt modd expressed in )f in a certain limit. begin with the observation made earlier that M-theory compaetifi(:d on is S-duai to Type IlA string theory. The supersymmetric translation in 11 dimensions contains the term PM , which der.:omposes into PI' and compaclification on a circle. Notice that Z corresponds to the charge of IJlC .l l" Dirichlet O-brane, given by the Kaluz.a....Klein quantization condition N PI' = -

(13.6. 1)

R

~~~::;:N~.~Ai;S~W~~'.:S3I:;w~ =~:~~~;:lb~C;,KaI~I~uza-KJejO stales of

the

eompactified the nonperturbativc 10-dimensional

~i

O-brane statcs. take the infinite momentwn limit, i.e., let Z = Pll -+ co. In this , the Dirichlet O-brane tenn dominates over nil other tenns in the super ' I i algebra, so that the theory red uces 10 that ofa Dirichlet O-brane. This m omentum limit, in mm, can be realized by taking the limit N --+ co I

-!Io

00.

OI;,:cth31 a single Dirichlet O-branc has'momentum given by 1/ R, and that Slate of N DiricWet O-branes (which coincide) has momentum N / R. we can show that this bound Slate of N Dirichlet O-brones is mechanics (not quantum fie ld theory). corresponds to N = 0: This is because s trings cannot be the R-R fields, which are presented by the central charge z. lus, w.:~, the following conjecture [13,14]:

I"d to

526

13. Solitons, D-Branes, and Black Holes

Conjecture. The infinite momentum limit of M-theory is eqlifY<~@i~ N -+ 00 limitofN coincident Dirichlet O-branes, given by U( Mills theory. Since the full action ofM-theory is totally unknown, it seems ulous that such a complex theory can be represented (in this by such a simple theory: large--N point particle superthe fact that physics simplifies in the infinite momentum tTl'I,mp: li;(c~: reasons why it was introduced years ago. Large classes vanish as 1/ Pll as Pll -+ 00. For example, Feynman diagrams corresponding to particles," ated out of the vacuum are suppressed in the infinite means that the vacuum is trivial in this limit. diagrams in the infinite momentum limit resemble the "old-jta~Ilj:Q~iifi turbation diagrams found in nonrelativistic field theory. For we exploit the fact that Feynman diagrams involving eve:rytJtlijj!it: Dirichlet O-branes are suppressed as I! P11 in the infinite mClmEmn Pll -+ 00. In particular, the string, because it is not charged lffi~[~t~ field, has PI] 0 N and is suppressed in the infinite m()m(~rt~~ To see this, let Po label the momenta ofa collection of these vectors equal P. Then each momenta Pa can be

= =

Po = YJa P + P1.a such that

L1Ja = 1,

LP1.a = 0, o

Now let us boost the system of particles such that P -+ 00. H"~",~n large P, we see that al11)o are positive. The point of taking this large P limit is that we can write

Eo =

!p~+m2

- 2 +m2 =1JoI PI+ P1.0 _ u+O(p-2). 21Jal P I Let us now analyze the Feynman diagrams which appear in the~:tfiltilll

~:~::;~:~IIIII

infinite momentum limit. We see that the covariant ble nonrelativistic energy denominators found in nl,c{\nr'phriui,,,,ri', theory. Let us focus on the differences in energy. If two particles;'have positive and equal 1Ja, we see that ference goes to zero as 1/ P. But now analyze what happens difference between a particle with positive 1Ja and another vv ..~'.':~~ :~~~~ (which occurs because we must integrate over all momenta The energy difference grows as P. Thus, this energy 1/ P -+ 0, so that particles with 1)0 < 0 decouple from the

IHi M(atrix) Models and Membranes

527

,o;:~:::'::~ is thai Stales with neglltive or vanishing 'I~ have energy dcnomwhich make them decouple from the theory, leaving only the states positive 'Ia. But in matrix models, the on ly states with positive "fIQ 8re states with II!omentum PI I = N / R, where N > O. TIlliS, M-thoory in infinite momenmm limit reduces to a theory of particles with positive 11,,, . states with p<)sitive PI I, wbich are the Dirichlet o..brancs. These states, in , are described by super- Ynng-Mills theory defined on the world volume D-bmne. The states with vanishing or negative I)Q are the string states anti-Dirichlet brancs. which decouple from the theory because of infinitc

:p

Onc nontrivial aspect of this limit is that although we can simply drop the with strings and anti-D-branes because they don't have positive 'Id, the ",ory",",m.,·· that these particles must exist in M-theory. This identificatiun is made all the mOTe remarkable since super-Yang-Mills is defined only in 10 dimensions, yet the N -+ 00 limit of this theory II-dimensional theory (M-theory). Somehow the N -+ 00 limit ,,",'''p,,,,iI,Jo the emergence of the eleventh dimension. see evidence of Ihis simple yet nOll trivial conjecture beeau!)e the Yangfield has 16 fermionic fields. When we take compositcs of this, we find = 256 states, wbich is precisely the nwnber of states found in supergra\;ty also find coDvincing proof of this conjecture when we investigate the ofDiricWet O-brancs, and find they exactly reproduce the scattering l-dimcnsional supergravitons 115-18}, which is a highly nontrivial result_ example, let the scattering take place in the eighth and ninth spatial reo~oo,. where eight represents the horizontal axis and nine the vertical Before impact., Jet's say the two particles arc coming at each other aloog parallel. horizontal paths along the eighth axis. They are moving toward other with relative velocity v. But their trajectories are off-<:enter by "";0,[ distance, the impact parameter b, in the ninth direction. Then the ~~~b~o~tween these two Dirichlet O-branes in the 8-9 plane is given by

',n,,,;;,,,

l~.~:::~.~ space,

X 22

is a x matrix. such that the diagonal elements represent of the two Dirichlet O-brones. Lei Xi = at + ~y l , where B' the location of the two classical Dirichlet O-branes, and y ; represents '~~~:l~:, fluctuations around the two Dirichlet O-branes. LeI's say thnt the ~! element B t I represents one Oiriehlet O-brane, and 8 n represents the Then we can say that

if

o ) = "2, vtcr),

-vrj 2

o -b/2

) == tbcrl.

(13.6.5) represents the Pauli matrix, since the coordinates are 2 x 2 matrices for '."'"pU(l) x U(I). The other components of B/ are set equal to zero.

528

13. Solitons, D-Branes, and Black Holes

Now power expand the super-Yang-Mills action around configuration B. We find the following action written in terms

f

Sy = i

dT Uyt(a; - r2)y{

+ t Y4(a; -

r2)Y4

·l i·i·~ ..·~jll~

Of,.p .~~;i~::III~I~

+ !Y;i:li1i:i:}:::%.

Since Bi represents the classical position of the string, it only c·!I)nt~lj~ the mass term of the action expressed in the quantum variable the first line of this equation give us the kinetic and mass terms If A is the zeroth component of the gauge field, then we can the action for this field SA

=i

f dT(~AI(a;

- r2)Al

+ tA2(a; -

r2)A2

+ tA <::.

+ 2gab3afB~AaYh + heabcarY~AbY: _

Ingf: U3x sbex

'\/5

BI3 A aAbyi _ ~2 gabx .,",cdx A ayibAe Yi) ·· c d •

By explicitly inserting the classical value of Bi into these we can now read-off the free-field part of the action in terms of with mass given by yi and A. There are three isospin of the gauge fields yi and A, and there are 10 such gauge fields, 30 bosonic fields altogether. By diagonalizing the mass matrix, following arrdDgement of the boson and fermion fields: • • • • • • • • •

16 bosons with masses r2; 2 complex ghost bosons with masses r2; 2 bosons with masses r2 + 2v; 2 bosons with masses r2 - 2v; 10 massless bosons; 8 fennions with masses r2 + v; 8 fennions with masses r2 - v; 2 complex ghost hosons with masses m 2 ; and 1 massless complex ghost boson.

Ifwe functionally integrate out over a boson field with mass the following detenninant emerges in the path integral:

Da = det( -ar

+ r2 + a).

By including all these massive fields, we find that the functional modified by the determinant D4D4 D tot = D 0-6D-1D-l 2v -2v v -v'

13.6 M(atrix) ModeJs and Membranes

529

We can lift each D into the aelion of the path iniegral by D- 1 = e-k'II D • the effective action gets modified by

(13.6.10)

S_S-logD~.

We can define the potential associated with the scattering process by

log Dtol = / dTVc1f(b

2

+ Ih· l ).

(13.6.11)

Putting everything logether, after a bit of algebra we find 4

Verr = _ISv 16r'

+0

(C:). rll

(13.6.12)

This in tum, is precisely the number obtained from ordinary supergravity I which gives us confidence that this formalism is correct In fact, this has accualiy been done to two-loop levels, with prefect agreement i the supergravity result [ I *1. For gencral (p - p}-brane scattering, we have

avO

V~ff=--,-+O r -p

(V' ) -,-,- . r

-P

(13.6.13)

This formula has several interesting consequences. Notice that the potential function of velocity, and vanishes for zero velocity. This reproduces the between two static D-brancs, which is often found in the of solilons. For example, if we take two parallel and stalic hyperplanes Op,,,,,,nling two D-branes and calculale the interaction betweell them, we can

..

~~~,~;~~,'lh~'~fi~u~c;.n;~~~u~·:O~O:;S;O~f,;~:;o~p~.~n~:superstring which connects the two Ity~

the bosonic and fennionie string modes eaclt other out. and we find that there is no forcc between the two static '!"''I'''''ne D-br.mes. important, we find that the effective distance which a D-brane can ,ob" is proportional 10 thc velocity, which means tllst D-brancs may be able 'pn,be much smaller distances than the tmditional string. It is usually thought ,"me conventional string cannot probe any smaller than the Planck length.

~

:~~;;:~~~;~:

we can argue that a ncw''uncertainty principle" is emergusing strings to investigate distances smaller than the i has led some to state Ihatlhere is an ultimate distance, the "an,,, length, below which physics cannot probe. .. The velocity dt:pendence of the D-brane scattering potential, however. seems .. indicate that at low velocities, we may be able to probe distances much " 11,,,h,m the Planck length. For cx.ample, rrom the scattering potential given we can extract out the impact parameter b of the scattering process, turns out to be on the order of .,fU. So the effective distance p robed by D-brane goes as the square root of the velocity. Whether this suggests that a deeper physical significance remains to be seen. '· b,,",,s

h"v.

530

13. Solitons, D-Branes, and Black Holes

Lastly, we also see evidence ofthe M(atrix) model conjecture in . II-dimensional membranes which appear in M-theory can also beY~~G~ collections of Direchlet O-branes. This is a rather remarkable result;~$~~W lack of a suitable quantization scheme for the Il-dimensiona1 one of the many reasons why it was abandoned in the 1980s. membrane action (even at the free level) is highly nonlinear and quantize. Worse, the Hamiltonian does not admit stable states [19], tion is also ultraviolet nomenormalizable. For all these reasons, 11-di)~iJ$U membrane theory was considered an oddity, rather than a funld8Jn¢!i~l:~ The new interpretation of the II-dimensional membrane the:or)n~i~Jijij arises in the large N limit of Direchlet O-branes, and hence all the membrane theory can, in principle, be removed. Let us begin by quantizing the p-brane action, which can be with the string, either in the first or the second-order formalism. order formalism, we begin with the Nambu-Goto action, ;~U;i$} terms of derivatives of X1'-' We construct the canonical m01me:ntlliQl:;;

8£

PI'-=-8XI'- .

By an explicit calculation, we find that plL exactly obeys constraints: pP.OaXI'- = 0, P; = T2 det OaXI'-ObXIL'

where a = 1,2, ... , p. This means that the Hamiltonian of the i.e., it consists entirely of the sum of these constraints:

H

= Ao (p; + T 1dettla XlL o"X IL ) + AjPJ.l..O/Xp..

In the Gupta-Bleuler formalism, we can take the gauge

W11I;;'C.

Ao = 1. The problem with the resulting Hamiltonian is that .. Ht
the string variables. This is the principle reason why the qUlmt~ijQltt.: membrane remains one of the outstanding problems ofM-me:on The problem with quartic interactions persists in all cat'e"">. light cone gauge. For example, in the second-order formalism, contribution from the world volume metric gab. We will volume metric by separting out the zeroth time index. Let a, b represent the spacelike indices on the world volume, and let: 2, ... , p represent the usual world volume indices. Then ...

.__ (>Wgil -

2

[ -h

-w

+ habuau"] -lh

ab U

b

_w- 1habUb h ab

!·'1illllill

)

'

where hab represents the metric over the 2 x2 spatial co()rdinaltes:;N9~~~lJ'@] have exchanged the (p + 1)(p+ 2) /2 components of gij for the w, uo, and the p(p + 1)/2 coordinates hab' (This particular pat~~E

13 .6 M(atrix) Models and Membmncs

531

taken from the ADM variables used to quantize ordi nary gravity in the

~

~,::::~;~f01;""~;'~I~i,~m~o:h~:'~~"'~d;;W~~.:";::th~:cnofthetheusual shift and we lapse funclions.) metric tensor, find, as usual, . In this way, we recover the first-order formalism. ifwc let p =- 2. If we introduce the notation

If. g)

~

,·'aJa,.

( 13.6.18)

the gauge-fixed membrane action for the bosonic coordinates is

J

s~ H d, d'O' [(a"X' - {w.X'))'-j{x' . x l llx'.X' I] . (13.6.19) If we add in the fcrmionic variables, then the fu ll Hamiltonian is H =

f dZ/YHpl+~ hdl/ttYoyabl/t+-k {X f .X'l lx '.xJ I1 ,

( 13.6.20)

Yo = a.x'y' and y = U/2)e ob 8"X I 8b X'ylJ. : At mis point, we again see all the pathologies of the membrane action. we notice thai the interaction lenn is quartic in the striog variable X ' , that it is extremely difficuil io quantize the theory. The free theory of is thus highly nonlinear and basically intractable. This alo ne has research on membranes. more important, we see thaI there are directions in which this term actually vanishes. Since this term corresponds to the surface area a mem brane, we sec that it vanishes jf the membrane degenerates into II. infinitely thin line wilh zero area. we imagine thaI the membrane looks like II. porcupine with millions of "qui Us" emanating from it, we see that the wave function of a membrane " leak" out from these quiUs, and hence thc wave function is not stable. :, " 'e also have the problem that the theory is ultraviolet divergent on the world "~;~ which means that quantization of the lhcory is problematic. iru.ighl into this problem a m be seen if we compare mis theory with :5 action of ordinary super- Yang- Mills theory defined with only one time LeI us introduce AI = f GA~I , where -c" is a generator of U( N) . ..nme action becomes

.,dl;n,"e. ~

Jd, I(D,A' )' -1 [A', A/l[ A'. A'l + i~ Do'" + h'[ A'. '" J). T,

( 13.6.21 )

:

;~~I;~L' we see a strong resemblance betwccn lhe two theories.

we wish to make the following correspondence between U( N) theory and membranes: AI

J

--jo

Xl

•

[' J~{.}.

dT Tr -+

f

d

2 /y.

532

13. Solitons, D-Bmnes, and Black Holes U(N) -+ w(oo).

The key is that we have taken the limit as N goes to infinity. U(N) becomes U(oo), which (if we carefully take the limit in a becomes w(oo), which corresponds to area-preserving spherical membranes and torii. The two-dimensional co()rdj.na1:e the membrane thus emerges out of the index a of the U(N) grOUll~ Now we have a regularization limit in which we can analyze of the action. The Yang-Mills theory we have written, for finite in that the quartic term vanishes if A [ takes values in the ofU(N), since the members of the Cartan sub algebra commute selves. Thus, the "quills" have now been replaced by members subalgebra. Since the theory is unstable for finite N, we can now' full membrane theory is also unstable if we let N -+ 00. Although this result is unfortunate, we now have an entrrely terpretation from before. In the matrix model interpretation, the~··.t:~·:~'~jEI seen as merely a certain collection of Dirichlet O-branes. The'.' the membrane is now seen as a simple consequence of the . dition usually found for solitons. Thus, the matrix model has interpretation of the membrane as a peculiar bound state of T'\: -':-'-' (The 5-brane, which also exists in M-theory. is more subtle investigated. )

13.7

Black Holes

Because string theory is a quantum theory of gravitational . should be able to use it as a guide to probe delicate qu~:Stl4:m:s: hole physics. In particular, there is the long-standing problem Bekenstein-Hawking area-entropy relationship for black holes l":'\J!V4:·~X statistical methods. Although entropy is rigorously defined using:::SUl] methods via S = k log W, this definition requires a detailed the counting of the quantum states of the black hole, which been beyond the reach of physicists. In the past, thermodynamic, rather than statistical, arg;un:Lents:fi;~Y4i~ used to heuristically define an entropy for black holes. First, we
where K is the surface gravity of the black hole. In particular, is always positive, at least classically. For example, if a black nQl¢::3~!W a nearby star or two black holes collide, then the mass of the expands and the area always increases. Thus, from purely formata.;'~~

13.7 Blsck Holes

533

suspect that there might be a relationship between thc area ofa black hole its entropy, whieh should also increase. In particular, we wish to exploit the relationship between this equation and the I"" of thermodynamics

dE = TdS,

(13.7.2)

wtte"' dS is [he change in the entropy and dE is the change in energy. Notice we have to define what we mean by temperature for a black hole. At . temperature fora black hole may seem odd, since nothing, can escape a bl ack hole, since its escape velocity equals the spcod . Although black hole~ have classical gravitational fields strong enough any !':trny piece..<; of cosmic matter, fhey mu!':1 also ohcy the laws of mechanics, which allows for quantum tunneling through the potential . fields ncar a black hole are !':trong enough convert virtual pairs, which are created in the vacuum, into real ones. In the that clcctron-positron pairs can be created from tbe vacuum via strong electric fields, the black hole's gravitational field is strong to create panicle production from the vacuum. this means that quantum black holes ffillsl slowly rad iste or , with time, leaking their cncrgy into spacc. It can be shov.'T1 that . radiation emitted from such a black hole is that of a black radiator with a definite temperarure. Black holes can therefore be treated thermodynamic objects with a well-defined temperature. We find T =

Kj~Tt .

(13.7.3)

This yields the Bekenstein-Hawking relationship, which states that tbe 'I",py S of a black hole is equal to one-fourth tbe area of its event horizon

S = ~A.

(13.7.4)

The analogy with thermodynamic.~ beromes even stronge r for spinning black If a black hole has angular momentum. J and the angular velocity ofthe .on,,." is f2, then we have dM

= (KjSTt)dA

+ f2dJ ,

(13.7.5)

has a striking resemblance to lhe usual Jaw of thennodynamics

dE = TdS- PdV.

(13.7.6)

The defect in this derivation is that the ilTguments are only semiclassical thermodynamic, while a complete derivation would necessarily count the wrn...,n states of the black hole. Since the entropy is usually defined statistithere have been anempts over the years, none of them rigorous, which tried to derive lhis relationship from statistical counting arguments. ; Recently, M-theory methods have been used \0 shed light on Ihis problem Since BPS states are found frequently when analyzing black holes, it is that we can use them to calculate their entropy. We know that D·branes

534

13. Solitons, D-Branes, and Black Holes

in particular can act as sources for R-Rfields, and that the cOlmtm:grQf~:mii for certain cases can be established. For many configurations of black holes, we find that the horizon is zero, depending on the charges of the black hOle.lI!n~· ~~i~;~M=I~ nonvanishing area, we start with the Type IIB action in five d

S=

f

~eD d5xJ 16rr _

{R -

g

('VifJ)2 -

~('V Di 3 -

2D

!e-2D+2¢> iJ2 _ !e-2D-2¢> H2 _ e

+

4

4

-

4

G 2 } --.

Notice that there are three U(l) gauge fields: G = dA, which CQ~t,i#.~J~~ij the usual Kaluza-Klein field, (H+)ll v = Hllvs, and H_ = .,I¢+D;:":U:::'~;~ corresponds to the five-dimensional duaL We can then define three charges, two R-R (electric and Irullgij,.t}m~}.i~ and one K-K charge

Iml!

QI =

~

{ e- D+2¢> *H+,

is]

8

-1-1

Q2 -- 4

rr

2

S3

p = 2rrn L

e-D-2 *H-,

=~ l6rr

1

e3D *G.

S3

In five dimensions, we can find a simple solution to these sembling a Reissner-Nordstrom black hole (a black hole with Unfortunately, this black hole solution has zero area for the event'the momentum P is zero. (In order to find a black hole solution area, all three charges, including the momentum P, must be noiiieJXti); means that we must perform a boost of the Reissner-Nordstrom: momentum P, parametrized by a variable a . Then the boosted Reissner-Nordstrom solution is given by

+ sinh '~

2

r2 _ r2

a +

r

2 -

dt dxs

[1 _ a- l a)] + (1 - ~;) (I _:.~) -[ + +

2

2

(r:coSh

r

Sinh

dx 2

_ r2

-I

d?

5

r2

dQ~

where r _ and r + represent the inner and outer event horizons of thp:.. ",,,,-,, When a = 0, we obtain the usual generalized Reissner-Nordlstrloiii:'tUet$!

13.7 Black Holes

535

I be inlerested in the case where the black hole is"extremal." (For II black with charge, we have the relationship between the mass M and the charge M > Q. in suitable units. For an extrema! black hole. At = Q. For our wc have an extremal black bole when r + = r _ = ro.) For this metric, we find the following Charge relationship QIQ5 = r:r:V/gl.

(13 .7.1 2)

The momentum is given hy

NV

P = 2g2 Sinh{2a)(r~ -r:).

( 13.7.13)

will shortly take the limit a -+ 00 as r _ -+ r +, such that P remains Then the energy E and area A become

RV

[2(,! + ,i) + cosh2a(,i .- ,:,)] ,

E = 2g2

,!Rcosh

A = 4rr 1

(1

Jr~ r:. -

(13.7.14) (13.7.15)

'0<;" that the area of the e~'Cnt horizon A

vanishes if we take r _ -+ , + for a, which fo rced lL<; to make tbe boost [0 obtain a nontrivial resul t.) :: The area of the event horizon can be simplified as (13.7.16) : The Hawking radiat ion temperature for this wetric can be shown to be

T =

1,2

v +

,2

21l"r: cosh

-

(13 .7.17) (1

Putting everyth ing together, we flDd

s= Since P =

A =2rrJ 4

r V

6g ;R =2rr.jQ1Q5 PR.

(13 .7.18)

n/ R on the compaetified circle, we fmd our fi nal result for the (13.7. 19)

So far, all of our arguments have been semiclassical, totally ignoring the me,111 problem of acrually computing the statistical mechanical origin of entropy. All we have done is recast known results in tcrms of the string >ckgrrmnd equations_ task is now to find 11 statistical derivation of this re:ationship using wm.nun counting arguments. The key observation is that we know how to the BPS D·brane Sl
536

13. Solitons, D-Branes, and Black Holes

that there are D-branes of odd dimensionality. We focus on brane (a string) and the Dirichlet 5-brane compactified on a of volume (2rr)4 V and a large circle given by circumference If we wmp the Dirichlet I-brane around a circle of radius R, a charge Ql = 1. Likewise, wrapping a Dirichlet 5-brane arclunlttafOtif~E gives us a charge of Qs = 1. By wrapping multiple D-branes atQ~j~(;~ the circle or the four torus, we can generate field COIlIi~~lticIllS ~IIII charges Ql and Q2· ~. As we mentioned earlier, the interactions of D-branes is gi\ren:ltir. strings which stretch between them. The counting of states is Ult:ll;!:!l2~ by the open strings which terminate on Q 1 Dirichlet I-branes anl()d\·~·'[~mll 5-branes. For the moment, let Qs = 1. Then the states of ft··· dominated by open strings which are free to move in the other of the four-torus. This is the key to the entire problem: the C01111ting,:9f:~i states reduces to counting Q 1 ordinary strings which are free dimensions, i.e., the states include 4QJ bosonic strings and thi>,;;,.:,,:,n;;; partners. Fortunately, the counting of string states is wellll"nr,UTn' :we;:;tm for ordinary string theory, that the degeneracy ofhannonic os(;illa~(jt:~~i a conformal field theory at level Lo n, for large n, and for ceIllttf.ilr¢.Ij~ is given by

=

den, c) --+ exp(2rr.jnc/6)

for large n. Thus, we need to calculate c for these Dirichlet I-branes. In a . formal field theory with N B species of bosons and N F llOlllll''-'ll't'::!>;::~~ by

C=NB+!N F , where NB = 4Ql. We thus find that den, c) ~ exp(2n'JQln)

for Q5 = 1. But we know that the system is dual between the I-brane and the final result should be written in terms of the product Ql Qs. can relax the condition Q 5 = 1, and write the answer in terms Ql Q5· Thus, the final answer is given by S = logd, or S = 2rrvQl Qsn, which agrees precisely with the previous result, thereby C01:rli.rmit~:@OO~~

A/4.

We see that the key to this calculation was the fact that th":N1Ul statistical states was dominated by counting the states of a DiJricililim~RM@ (a string), which is well known. Although this result was rI"'T~,ti"rI.,·t'i11 dimensional black hole using Type liB strings, the result is pr(~~r]mm~

13.8 Summary

537

and has been shown to bold down In four dimensions for a variety . Rnd backgrounds. (In fom dimensions, tbis calculation the introduction of new objects, sucb as KaJuza--Klein monopoles.) the calculation can be carried out for rotaling black holes.

Summary we have seen the pervasive influence (If p-brones throughout M-lheory, • ."h have allowed us to determine !.he nonpcrturbativc behavior of many theories. In this chapter. we construct the explicit representations of p-branes. :; For example, we emphasize that we arc forced to include the presence of p-I",.o> in string theory. Even if we originaUy start with a theory purely ,eventually we must solve for the equations of motion!';, which allow and solitonlikc objects. Specifically, we can write tbe membrane )Iut;o" of Il-dimensional supergravity by introducing the following ansatz:

:! ds

2

=

(I

;~

+

r

21J

dx" dx" +

(I

+

;~

yo

(dj

+ jdS1;) ,

( 13.8.1)

I.L = O. 1. 2. m = 3..... 10, dQ 7 is the volume form for thc 57 sphere,

the four-form field strength is proportional to the dual of the volume form

5,. Similarly, the 5-'oranc solution of II-dimensional supergravitycan be written

2 (I + ~,~rlJ3 dx"dx,,+ (I + ~~yjl (di +jdQn.

:: ds =

( 13.K2)

the four-form field strength is proporuonaJ to the volwne form on S4. ~~ Not surprisingly, we can now present the general case of the p-branesolution '. D dimensions. Wc start with the Ilction '.

5= _1 !d{;X,J R(R~t{a¢l2K

1

2(d+l)!

e- a4> F'f.....2 )

,

(13.8.3)

r '

Fp+z is the usual amisymmetric field strength. We couple thi!'; to the p-brane action

5p = T _

f

d P+ 1/;

(p

(_~JYyiJ&jXMaiXN8MNea(d)4IJ(p+ 11 +

fl- XM,.. I A ) +I I)! Si;il.--.i.....;a.,XMI , _ .. ',-1 MI •.... M,..."

p;

1.JY (13.8.4)

yli is the metric on the world volume of the p-branc.

solve these coupled equations, we start with Inc ansatz

ds 2 ::::'eV·tLxJ1.dx,. +e2R dy m dym,

(13.8.5)

538

13. Solitons, D-Branes, and Black Holes

where /.l. = 0, 1,2, ... , p and m = p + 1, p + 2, ... , D - 1. ~m111~tl~ split the p-brane coordinates as follows: (XM = XI', YTJI ), and XI' ym = constant Then a solution can be given as D- p-3 A = 2(D _ 2) (C - ar/Jo/2),

p+l B = - 2(D _ P _ 3) (C - a¢o/2),

a

'2¢ =

a2

'4(C - a¢o/2) + a¢o/2,

a 2 = 4 - 2(p + I)(D - P - 3)/{D - 2), and k yP+I' K2T e- ao/2 - In y,

e-ao/2

e-c

=

+ --

D - P - 3 > 0, D - p - 3 = 0,

J!

where k = 2K2T I(D - p - 3)QD- p-2 and Q is the volume ofa So far, we have only constructed classical solutions :~~~:~~~l~d!~N;M~:~ITi membrane and 5-brane. To construct the explicit supermembrane (1\i~,I;!;I1! introduce the generalized Nambu-Goto term SI

= -T

f

dp+1aJ-det

TIl' TI j ,

where nf' = &/XIL-iePai 8. Thistermbyitselfisnot so we must introduce a Wess-Zumino term. We first introduce the i h = -2,TII'P ... ni
S = -2T 2

f

*b = -

(p

2T

+ I)!

f

dP+lasil .. .iJ>+ lb·

.

" ·.. ' .... 1

.

Written out explicitly, the Wess-Zumino term for the membrane " ,. S2 = -

i; f

d 3a

leePer I'vo/8) [TIiTIk + iTIier ok8 V

- (i)(erl'oj8)(er ak 8)]) , V

where

13.8 Summary

539

We see therefore that 8S = 0 if we choose

,=

c= ± l , where

(I

±

(13.8. 14)

r),(a),

!(l ± r) is a projection operator.

T he I I-dimensional mtmlbrane and the 5-brane arc sometimes called thc M-branes ofM-theory. However, we are also interested in the lower p-branes f<,uod in the BPS states. These include the D-bmnes. A Dirichlet p-brane (i.e., Dirichlet p -brane) can be defined as a p-brane which open strings can end. The action for these D-brnnes differ from thc us ual onc, in that wc can also add a vector fi eld A which propagates on the " " dd volume. Tbe addition of this vector field changes the usual counting of bome-scan, which now becomes D-brane: D - 2 = ~MN.

(13 .8.15)

is easy to see that the various BPS p -brancs found by decomposing the '"l"'rtom,;I,,'·,m algebra can be satisfied by tills condition. Historically, D-branes were first discovered by making aT-duality tmnsfor~::~o, on open strings. We that a T -duality transformation on closed results in

recall

:

a

( 13.8. 16) However, for an open string, this same T -duality changes the Neumann >,u"d"", condition for X I' into a Dirichlet bOlWdary condition for XIJ.:

a"x" =

0

....:,.

.

a,x .. =

(13.8.17)

0,

0 at the endpoints, where X~ is the dual string. TIle action for the D-brane can be found in sevenll ways. As with the ordinary

XPo

:::::

:: !

!

we Cffi first

'~:~,~'~~'~~,~th~':,~:~,funct.ions, which are then set to zero. Then it is an easy ~: : action whose equations of motion arc precisely f3 = O. The resulting action lor the massless fields is 3

/ d aJ-det(Gij

+ FiJ

HI})'

(13.8.18)

W~::l

fo

= XI< at lheboundaryofthe p -brane, and G and B are the usual terms from the pull-back to the membrane surrace, i.e., Gil = adlJ.alf~glJ.~' We can also derive the D-brane action in another Wlly. by taking the 11membrane action and then compaetifying it on a circle. We obtain

One oflhe most surprising results ofD-brane physics is the M(atrix) model

~

~'1~!:~~~t~h;,:t::~M-theOry

in the infinite momentum limit is equivalent to in the large N limil. Since M-thcory is known to contain and 5-bra.ccs, this is a truly remarkable simplifi cation, since the

540

13. Solitons, D-Branes, and Black Holes

action for Dirichlet O-branes is given by a sUI)er.-qulantuIll-ine<;halnitial;;$ij which is not even a quantum field theory. The motivation for this result can be seen by analyzing the sUJ)eI';:traffi algebra for M-theory. We recall that after compactification, the· operator PM becomes PIL and PI" where

Pll = Z = NjR. In 11 dimensions, this last term is the usual Kaluza-Klein state. B,tij~JJt:ti mens ions, this corresponds to the Dirichlet O-brane state, which the algebra in the infinite momentum frame. There are several tests of this conjecture. The simplest is to spectrum of the theory_ The super-Yang~Mills theory has a 1 spinor, which can be used to derive 2 8 = 256 states, which massless states of supergravity. Second, we can explicitly calculate Dirichlet O-brane and show that they match those of supergravity scattering. The ·aqllQiji;1Qe?J coincident D-branes is given by the super- Yang~Mills theory redti¢e~: world volume ofthe p-brane

S=T

f

dP+1a Tr

(-~ FpuFIJU + j$r lL DIL '11) + ....

What is highly unusual is that this action is also applicable to fields which are not coincident, so this action can be used to scattering. We use this fact to calculate Dirichlet O-brane scattering. with the action for N Dirichlet O-branes .

This action not only describes N overlapping Dirichlet U-blraIll¢S'i: describes the scattering process of several Dirichlet O-branes. For the gauge field is block diagonal, obeying the the gauge group

then we can use this to represent multi-D-brane scattering. This can be used to explicitly solve for two Dirichlet O-brane scaLtt~,@] first power expand the X field as follows: Xi = Bi + y'gyi w the classical confi~ation and yi represents the quantum The power expansion yields

Sy =

if

dT

2 (12yi(a 1r

-

r2)yi

+ 1 yi (a 2 -

122r

_Jggah gcbx Bi yiyi yi 3abc

_

+

1 yl a 2 yi . 223r?:.

r2)yi

!rgab'" gcdx yi yi yi yi) 4 abed'

..

I J.8 Summary

541

A is the zeroth component of the gauge field, then we can similarly wri te

n.",jo" for this field SA. = j

J

+ ~A2(a~ - r l)A } + t A3a : Al + 26ahl U, S;A .. Y: + -IRra, Y~Ab Y: dr (jA ,{a: - r2)A,

(L3.8.24) It is now a simple matterto functionally inlegl'3teoul the y' quantum fields 10

: Lo'west order. Since the 8 ' are classical fields, the above action can be wriltcD as of Ilua,""ti·,o. cubic, and higher tenns. The quadntlic terms can also be so they represent the propagation of free particles with a certain For a single field, Ute integration over the quadratic part yields the usual

D" = det(-o. +rl +a) .

(13.8.25)

including all these massive fi elds, we find that the functional inlcgral is no
-' 0 -211 - ' 0'0' O,at = 0 0-' 0 2~ V - V'

( l3 .8.26)

explicitly decomposing the above detenninants, we can then extract the

"mlril,utioo of these determinants 10 the action. We find YeW

== -

~~~: + o(~~),

( 13.8.27)

matches the result found in ordinary supergravity scanering. Lastly, we fi nd that D-brancs allow us to solve a long-standing problcm black hole physics, which is to find 3 statistical mechanical derivation of Bekeoste~Hawking radialion fonnulA , which stales that the entropy of a hole is equal to the one-fomth the area. The fact tha t the entropy of a black hole is proportional to the area i~ related fact tha t the black hole gobbles up new interstellar matter, which makes ".re. increase, rather tha.n decrease, in size. Like entropy, the area of a black always increases. Although thennodynnmic arguments have been made plausibly give rise to this relation. a statisticaJ derivation necessarily nv(,l ~,sa quantwn theory of gravity, which was al\\'~dys beyond reach. The introduction of D-branes changes aU of this, allowing us to count the :'aI'~i'n a black hole. Lei us start with a Type lIB theory compactified to five-dimcosions. Tbe effective action for the fields are

"""u,,,,

(J 3 .8.28)

542

13. Solitons, D-Branes, and Black Holes

We can integrate over the fields to obtain three charges which black hole

1 -1-1

Q1 = ~8 Q2 = 4

p

rc

e- D+2"'*H+,

SJ

2

SJ

e -D-

2

9 *H-,

= 2rcn = ~1 L

16rc Sl

e3D *G.

In general, we need all three charges to be nonzero in order to . nonzero area for the black hole. For the Type lIB string, there are. in odd dimensions, specificaUy Dirichlet I-branes and Dirichlet we compactify on a four-torus and a circle, we can wrap the JJ1JL1"1.uQ~ around the circle (giving us Q, = I) or the Dirichlet 5-brane (giving us Q5 = I). Let Q5 = 1. Let us wrap Q1 Dirichlet I the circle. Then we effectively have Q, Dirichlet l-branes (strings) free to oscillate in four dimentions. Thus, we have a sigma model 4Q, dimensions. Fortunately, we know how to count the number of states in a For large n, the degeneracy of states of the string is given by d(ll, c) --* exp(2rcJllcj6)

forlargen. For our case, c is equal to No + !NF , where N8 = 4Q\. ~.. @l..~W the system is dual, the charges must come in the combination QI Qs.,{~@f::~~ can relax the condition Q 5 = 1. Since S = k log W, we can now put everything together and get th~:.$~~i \T,i .

which is precisely one-fourth of the area. Although we chose Type lIB theory compactified down to 5 this result has been genemlized to a wide variety of other kinds such as four-dimensional black holes and spinning black holes.

13.9

Conclusion

At present, string theory and its recent reformulation, M-theory, most promising hope for a unified theory of all physical forces. string theory combine gravity with the other quantum forces, it enough, in principle, to explain all the symmetries found in respect, the theory has no rivals. There are, however, a number tests that the theory must pass:

13.9

CanchL~jDn

543

• The outstanding problem of string theory is to find the correct vacuum which describes our physical universe. Among the mill ions of vacua discovered in four dimensions, one must be found w hich can rcproduce the Standard Model with symmetry SU(3) ® SU(2) €I U( I) coupled to gravity. • Many of the marvelous propcrti(..'S of superString cheory come from local supersymmetry, but eventually supersymmetry must be broken irthe theory is to describe the physical universe. So far, no convincing method has been proposed to yield supersymmetry breaking. • The theory must explain the value of the cosmological eooslanl, wbich is known to be vanishingly small. This problem is closely lied to super· symmetry breaking, since the cosmological constant is zero in a locally supersymmctric theory. The key to all of these problems may lie in a nonpcrturbalive formulation ofth. th,,'''Y. At prescnt, each of rhe various proposals has problems. Conforfield theory, for example, is bnsieal1y a ~r1uroat ive fonnu lntion of String n~::~ and eallnot give us nonperturbative infonnation. String fi eld theory, alhi it is defined independent of perturbation theory, has proven too difficu lt fonnulale and solve. So far, our best hope bas been with M-theory. From an ll · dimensional pcr'p,";,c,all five superstring theorie~ can now be viewed as different vacua of same theory. A vast web of duality relations has established a nonperrurbalink between different string theories. The strong coupling limit of many ;trio. th"o
pto/"",hj,,g its complete sfructlire in 11 dimensions. So fiu, all we know is it reduces to ll·dimcnsional supergmvity in its low-energy limit. "a,>liz,atit'" aiM-theory. Unless M-theory is quantized. we will never underits spectra and rigorously understand its properties. Unfortunately, even membrane actions cannot be qllantized with kilO"",," techniques. if3 interactions. At present, v(..T)' little is known about memExci tation~ of O-branes may be given in tenns of strings, there are large gaps in our understanding of how membranes interact with other. ~~;:::;,~,~~g.~ th;~e of dualitie!J' down to = 4. Although a considerable In ha... b(..'Cn done on dualities in D = 8 and D = 8, very little still known uoout duali ties in D = 4 with N = I supersymmetry, which is

web

D

~:~;;:;;r!\~:~~~::m:iw~re~~st~.~!~ give us

M· theory still cannot explain why superinto the cosmological constant.

544

13. Solitons, D-Branes, and Black Holes

However, given the astonishingly rapid rate at which the theory ' . oping, there is considerable optimism that the mystery behind . soon be revealed. Perhaps some of the readers of this textbook will...,,,,-:~,,,,,,w those who accomplish this feat.

III

References [I] M. J. Duffand K. Stelle, Phys. Leu. 8253, 113 (1991). [2] R. Gueven, Phys. Lett. B276, 49 (1992). [3] For a review, see: M. J. Duff, R. R. Khuri, and 1. X. Lu, Phys. (1995). [4] A. Achucarro, J. Evans, P. Townsend, and D. Wiltshir, Phys. Leu. ( 1987). [5] B. E. Bergshoeff, E. Sezgin, and P. K. Townsend, Phys. Lett. U~t~7,. · ri);·I,::t~ Ann. Phys. 185,300 (1988). [6] M. Aganaic, 1. Park, C. Popescu, andlH.Schwarz, hep-th/97011 [7] P. Pasti, D. Sorokin, and M. Tonin, Phys. Rev. D52, 4277 (I: th/970 1149. . [8] 1. Dai, R. G. Leigh, and 1. Polchinski, Mod. Phys. A4, 2073 (I [9] R. G. Leigh, Mod. Phys. Lett. A4, 2767 (1989). [10] P. K. Townsend, Phys. Lett. B373, 68 (1996). [II] C. Schmidhuber, Nucl. Phys. B467, 146 (1996). [12] E. Witten, Nuc!. Phys. B460, 335 (1995). [13] T. Banks, W. Fischler, S. H. Senker and L. Susskind, Phys. Rev..: (1997). •• [14] For a review of matrix models, see: A. Bilal, hep-thl97 101 36. [15] U. H. Danielsson, G. Ferretti, and B. Sundborg, hep-thl9603081 [16] D. Kabat and P. Pouliot, hep-th/9603127 (B.9) [17] M. R.Douglas, D. Kabat, P.Pouliot, and S. Shenker, hepl-thI96ID80IA'K [18] K. Becker and M. Becker, hep-thl9705091. [19] B. de Wit, M. Luscher, and H. Nicolai, Nud. Phys. B30S[FS23], [20] I. 1. Bekenstein, Nuov. Cimento Lett. 4,737 (1972); Phys. Rev. D7, Phys. Rev. D9, 3292 (1974); Phys. Rev. D12 3077 (1975). [21] S. Hawking, Nature, 248, 30 (1974); Comm. Math. Phys. 43, 199 . Rev. D13, 191 (1976). [22] C. Vafa and A. Strominger, Phys. Leu. B383, 44 (1996). [23] G. Horowitz, gr-qc/960405I. [24] J. M. Maldacena and A. Strominger, hep-thl9603060.

Appendix

the mathematics of superstring theory has soared to such dizzying we bave included this snort appendix to provide the reader with 8 understanding of some of the concepts introduced in this

thai we must necessarily sacrifice a certain degree of i rigor if we are to cover a wide range of topics in this appendix. the interested reader is advised to consult some of the references later for more mathematical details.

A Brief Introduction to Group Theory . group G is a collection of eiemenJS gi such tbat: There is an identity ciement 1. There is closure under multiplication:

Every clement has an inverse: gl X g,-, = I .

Multiplication is associative: (8, x gJ) x 8t = Ri x (gi x 8t). Groups come in a variety offarms. SpecmcaUy, we have the discrete groups, a finite number ofelemcnts, and the continuous groups, such as the which have an infinite number of clements. Examples of discrete

Appendix

546

(1) The Alternqting groups, Zn, based on the set ofpennutations (2) The 26 sporadic groups, which have no regularity. The interesting of the sporadic groups is the group F], cornmont "Monster group," which has 2 46 .3 20 . 59 .7 6 • 112 . 13 3 • 17· 19 . 23 . 29 . 31 ·41 .

elements. In this book, however, we mainly encounter the continuous have an infinite number of elements. The most important of groups are the Lie groups, which come in the following four "la";:il,\~\W:ijJ series A, B, e, D when we specialize to the case of compact, An = SU(n + 1), Bn = SO(2n + 1), en = Sp(2n), J)~1r...sQ(2{'J.. J,

as well as the exceptional groups: G2; F4; £6; £7; £8;

of which E6 and E8 are the most interesting from the sta:nClI'olllti phenomenology. Let us give concrete examples of some of these groups b~~11111 set of all real or complex n x n matrices. Clearly, the set of invertible matrices satisfies the definitions of a group and hence group GL(n, R) or GL(n, e). The notation stands for a generalline:l¢:~ n x n matrices with real or complex elements. If we take the subset· or GL(n, C) with unit determinant, we arrive at SL(n, R) and group of special linear n x n matrices with real or complex eleme:p:~k

O(n) Now let us take a subgroup of GL(n, R), the orthogonal group consists of all possible n x n real invertible matrices that are Ox OT

= 1.

This obviously satisfies all four of the conditions for a group. matrix can be written as the exponential of an anti symmetric m"tri,i';' 0= eA.

It is easy to see that OT

=e AT =e -A = 0-1 .

A. I A Brief Introduction to Group Theory

547

general, an onhogonaJ matrix has ~n(1/ - I)

!

~~::~~:l:;~~::.;:~~:. Thus, we can always choose a set of ~fI(n - 1) linearly matrices, called the generators Ai, such that we can write any

"

nf2". r~ - 1)

O=exp

[ L

]

/J..;

.

(A. 1.6)

1'-1

real numbers pi are called Ihcparametcrs oflhc group, and there are (hus I) parameters in O(n). The number of parameters of a !..ie group is !:~::!,~:::dimension. The coounutator of two ofthe..~c gcncldtors yields anolher (A. 1.7)

the

f's

arc calle
I constants detennine the algebra completely. Notice that if we take cyclic combinations o f three commutators, we get an identity

(A.1.8)

.~~::f:~:~I~~~ these commutators, we find that the combination identically ~:~

to zero. This is called the Jacobi identity and must be satisfied for the to close properly. By expanding oul thc Jacobi identity, we now !:Lave ; . constraint among the commutators that must be satisfied, or else the group " ~oes

ncot close (A. 1.9)

~

A

~o:ri:~~;:~'b~'~ :se~~,~O~f~O::"!~h~Og~O~n~':1 ~11U~,:trices closes under multiplicalion. I this particular parametrization of the I

group, with generators and parameters, closes under muhiplicalion.

(A.I.IO) the Baker- Hausdortl theorem shows that C equals A plus B p lus multiple commutators of the A And 8. But since the A and 8 the Iacobi identities, the set of all possible multiple commutators of 8 only creates linear combinations oflbe generators. Thus. the group ,I"'esum!" multiplication. N olice that the structure constants of the algebra form a representation, called rep1"f!.~enlatjon. Ifwe write the strucrure constants as a matrix:

IS = U.*)t;. the s tructurc constants themselves form a represen tation.

(A.I.!!)

548

Appendix

We can always choose the commutation relations to be [M a". M Cd ] = 8ac M"d _ 8ad M"c + oM Mac _ 8"c Mad for the anti symmetric matrix M ab . One convenient representation ofthe algebra is now given by . (Mab)u '"'"' o~

oJ - or oj

which, we can show, satisfies the commutation relations of the ~~gt~~1~rn[]1 Let us define a set of n elements Xj that transforms as a VP(,~Ih;" group

x; =

OijXj.

In general, we can also define a tensor T/l I .i-'2.i-'h .•.• Jl-P

of rank P that transforms in the same way as the product of P x/l i

:

T "1."2 . ···.VN =0 "I .PI 0 V:!.#2 ···0VP.iJ.P T./l!.i-'2 ••..• i-'P"

In addition to the vector and tensor representations of O(N),.·· spinor representation of the group. Let us define the Clifford {fa.

r"} = 28ab .

Now define a representation of the generators in terms of numbers.

..!.. [r a

M ab =

rh]. 4i ' The Clifford numbers themselves transform as vectors [M ah , r C ]

= i(8 r b _ obT UC

U

).

In general, these Clifford numbers can be represented by 2n x 2n

.. ·

(ra)a,IJ

for the group O(2n). Therefore, a spinor T/la that transforms 2n elements and transforms as ,I,' _ ( M,b p.b) "f'a -

e

,f,

a,IJ "f',IJ,

where the M's are written in terms of the Clifford algebra and are parameters. , For the group O(2n' + 1), we need one more element. This IS

r2n+1 =

rl r2 ... r2n.

We can easily check that this new element allows us to COJlStruct: matrices for O(2n + 1).

Al A Brief Introduction to Group Theory

549

now tly to construct invariants under the group. Orthogonal >mlations preserve the scalar product XiX/:

= invarianl

(A.L23)

= x 0'0 x = x,x/_

(A.!.24)

XiX;

XjX; "

invariant can be wrineo (A. US)

the metric is Oij' In principle, we could also have a metric with alter"tingpositi've and negative signs along the diagonal, T!ij. which would create parameter space that is noncompnci. If we have N positive Rod M negaelements in "fIij. then the set of matrices that preserve this foim is called VV·,M), (OT)ijJ')jkO~1 = 1711. 7/1/

=

e(j)~I)'

(A.1.26)

~:(i)=±1.

the elements of E arc positive, this gives the group O(n). If the signs arc It"n"u"',thcn the group is noncompact. Special cases include projective group:

0(2, I),

Lorentz group:

0(3, I),

de Sitter group:

0(4, I),

an ti-dc Sitter group:

0(3,2),

confonnal group:

0(4,2).

(A 1.27)

For example, the de Sitter group can be eonstnletoo by taking the generators 'U'14. I) and then wriring the fifth romponenl as

(A.1.28) the algebra becomes [P",P' J ~M·',

0(4, I ):

I

[pO. , Mb~] = pbf/cr _ per/'b, [M o.b , Mcd] = f/~( MtlJ - ....

(A,1.29)

that this is almost the algcbra ofthc Poincare group. In facI, ifwc make substirurioo

po.

---.)0

±r po. ,

(A 1.30)

the omy commutator that changes is

[pO. pb ] = --'-- MOb ,

r2

'

(AI.3I)

550

Appendix

r is called the de Sitter radius. This means that if we go around a cm.~~::ffl de Sitter space and return to the same spot, we will be rotated by a f;"~f~~ transformation from our original orientation. Notice that if r goes to mtl4H¥f;~i!~ have the Poincare group. Thus, r corresponds to the radius universe such that, ifr goes to infinity, it becomes indistinguishable frori1~~l~~ four-dimensional space of Poincare. Letting the radius go to infinity IS:j:ia1jfio! the Wigner-Inonu contraction and will be used extensively in theories. After the contraction, the de Sitter group becomes the

SU(n) The group SU (n) consists of all possible n x II complex matrices determinant and are unitary:

vv t = 1. The notation stands for special unitary n x II matrices with complex Any unitary matrix can be written as the exponential of a Ht = H:

We can show that

V t =e _iHt =e -iH = V-I . Let n elements in a complex vector Uj transform linearly under sl)r@~~~~iliW

The n complex vectors Ui generate thefundamental representation nH'h"·",;;.liill~~ Then an invariant can be constructed: U7U; = invariant.

If u; = Viju j, it is easy to check that

'*i

I

i

u u = u

t) V *(U i

*i

re:r~:t~~il~i

= u Ui· The metric tensor for the scalar product is again 8ij. If we \\ some of the signs in this diagonal matrix, the groups that would pre:s,~t.i~ metric are called SU(N, M). An example of this would be the SU(2, 2). .. Any n x n complex traceless Hermitian matrix has n 2 - 1.: elements and hence can be written in terms of n2 - 1 linearly' matrices Ai. Thus, any element ofSU(n) can be written as ij

jkUk

A.I A Brieflntrodllclion to Group Theory

55l

that the group closes underthis ;~r:~!:~~::·~~~:S:;~;~~:l'::~~~::::i~~;:l~;~ and thllt we can the algebra of the group as (A.1.39) knowledge ofthc s tructure constants determines the algebra completely. can also construct representations of SU(n) out of SpinOIS. If we have group O{2n). then SU(n) is a subgroup. ff we construct the clements

(A. 1.40)

: wh". rlJ 3re Grassmann variables. then the generators ofSU(IJ) can be written

}... =

LATJ (AQ)JiA~.

(A.1.41 )

J"

" "0'. we have an explicit representation fo r the inclusion SU(n) C 0(2n).

(A. 1.42)

symplectic groups are define<:! as the set of 2n x 2n rea l matrices S thaI

'""""'" ,m antisymmctric metric 11: (AI.43)

(A. 1.44)

'1;}

=

I

0 - I

0

0 0

0 0

0 0

0 - I

0 0 I 0

(A lAS)

.:~~:~i;~llhcre is a series of "accidl!tllS" that allow us to make local ~I between groups. For example, 0(2) is locally isomorphic to

0(2)

~

U(I).

(A 1.46)

552

Appendix

To see this, we simply note the correspondence between a 0(2) and U(l):

c~sB

( -smB

SinB)

*+

ei9 •

cosB

Thus, they have the multiplication law 81 + 82 = 83 • Another accident is 0(3) = SU(2).

The easiest way to prove this is to note that the Pauli spin matrices . . complex matrices with the same commutation relations as the Thus,

where the matrix on the left is a 3 x 3 orthogonal matrix and right is a unitary matrix. Another useful accident is

0(4) = SO(2) ® SU(2). To prove this, we note that the generators Mii of 0(4) can be dJ.vide,~,:1lj~ sets:

and B = {M12 - M34, M31 - M 24 , M Z3 - MI4}.

Notice that the A and B matrices separately generate the algebra

that [A,B]=O.

~~IIII

Thus, we can parametrize any element of 0(4) such that it product of 0(3) and another commuting 0(3). Thus, we have-j element of 0(4) can be split up into the product oftwo elements of a set ofSU(2) groups. Unfortunately, these accidents are the exception, rather than the groups. We list some of the accidents: dim = 3

SU(2, c);."-' SO(3, r) "" Usp(2) "-' U(l, q) "" SL(l, q), . SU(l, l;c) '" SO(2, l;r) '" Sp(2, r) "" SL(2, r). • dim = 6 80(4, r) '" SU(2, c) ® SU(2, c),

SO"(4) '" SU(2, c) ® SL(2, r),

A.! A Brief Introduction \0 Group Theory SO(J. I ; , ) - SL(2. e) . SO(2. 2;d - SL(2. ,) ® SL(2. ,).

553

(A.U5)

SO(5. ,) - USP(4). SO(4. I ; ,) - U'p(2. 2), SO(3 . 2; ,) - Sp(4. ,).

(A. 1.56)

SO(6. ,) - SU(4. e) . SO(5, 1;1')'- SU-(4) --- SL(2. q) , SO'(6) - SU(3 . I;e). SO(4. 2; ,) - SU(2. 2; c).

(A. 1.57)

SO(3. 3; ,) - SL(4. r).

SL(N. q)

~

SU'(2N).

U(N . q)

~

U'P(2N) ,

Sp(N, q )

~

U'p(2N),

(A. I.'")

O(N. q) ~ SO'(2N).

, for AI

.:s

6. we have Spin(3) = SU(2), Spin(4) ~ SU(2) ® SU(2). Spin(5) = Usp(4),

Spin(6)

~

(A.U9)

SU(4).

is the SCI of ali" x n matrices wil.h unit dctcnninant that can have or quncemionic clements q. Quaternions are ~nemliz3tioDs of numbers such that any element can be written as (A.I. 60)

.h",e 0" c's arc real numbers and 90 = I ,

q~.2.1 = -f, qlq2 = -'I2Ql = Q3.

q2q1 = - q]q2 = q" q]ql = - qlq) = th ·

(A.1.61)

554

Appendix

Cartan-Weyl Representation In general, the exceptional groups do not have such a nice tenns of subgroups of GL(n). We will instead use the methods Caftan. Among the generators A; of a group, let us select out those er~m!r6:~~ which are all mutually commuting: [Hj , H j ] = O.

This is called the Cartan subalgebra. The number of elements in .. subalgebra is called the rank r of the group. Let us call aU the of the algebra E. What are the commutation relations between . other elements E? In general, we cannot have the commutator and E yielding another H, because this would not satisfy the ties. Therefore, the commutator of H and E must yield another always rearrange the rest of the elements E of the algebra so that eigenvectors of the H's. We will denote them by

E"" where 0: is called a root vector in r -dimensional space. In vectors live in r-dimensional space. Notice that if the group has N then there are N - r elements in E. Thus, there are N - r root r-dimensional space. We can always take linear combinations of .. E's until we have the eigenvalue equation .. OPT1L ........

[HI, E",] = aj E",.

By the Jacobi identities, the other commutation relations become

[Ea, E- ",l = ai H;, [Ea, E.a] = N a ..a E a+f3' The N's are the structure constants of the group. They are given N;f3

= !n(m + l)O!;Cf;.

The symmetries of the N's are given by N a .f3 = -N(3.a = -N_a.-.a.

N"'.(3 = N.a.-a -{3 = N_ a -{3.Ct.

Dynkin Diagrams

IIIII

Let us make a few d~finitions concerning the roots. Each root vec:tof an r-dimensional space. Thus, we can choose a subset of them r such that any other root can be written as a linear combination: r

P = LC;Ci;. i=l

A. I A Brief Introduction to Group Theory

555

root is callcd a positive root if the first non7.ero c in the above equation is . . A simple root is a positive roOf that eannot be writlen as the sum of positi ve roO IS. lust AS the structure eonstant specifies the a lgebra exactly, " nbc said of the Conan matrix, which is an r x r malrix defined by

th,,,,m,

A . - 2 (ai ,aj ) IJ , (lXJ>aJ)

(A. 1.69)

wh'" thc Cli arc the r simple roots. The diagonal clements oflhe Cartan matrix:. by construction, are all equal to But the matrix is not necessarily symmetric. In fact. it can be shown that the possible values oftbe off-diagonal matrix elements arc 0, - I, -2. -3. ";'~::~S; thc Cartan matrix defines the group enti rely. some simpl e properties ~i Cartan matrix can be used 10 define the group grnph.ically. The most ,m.","" n' oflhese is the Dynkin diagram. For a gro up of rank r, let us draw r dots. Each simple root is thus represe nted dol. Now connect the ith and jth dot with n lines, where n is equal to the of the off-diagonal elements

';001""

(A. 1.70)

ag'''" is a serie!' of dots connected by 8 series of mul ti ple lines, diagram. The powerofDynkin diagrams is that they uniquely the structure of any Lie group, and thus we can visually distinguish the

,

Since the elements of the Cartan 1Th1trix are related to the scalar products on root lanice space, we can also represent n in terms of the anglc () between rool vectors: n

B

o

90° 120'

I 2

1)5~

3

150"

If we specialize to compact, real fonns. we have the following Oynkin (see Fig. A.I):

""',n,,,

An = SU(n

c,-cJ . B.

+ I).

I,Sijj,Sn+l.

= SO(2n + I), ±e;

±Cj,

± Cj,

= Sp(2,,}. ± ej ± eJ'

C~

±2c;,

556

Appendix

•

•

•

•

•

an-I

an

•

•

• an_I

an

•

•

An • a,

a2

Sn • a,

012

en •Ci,

a2

On •

Ci,

a2

•

an

~~. a n_ 2 Ol"

G2

--===--...

F4 ...

_~.~--I~--~.--~.

E6 ...

-_.~- _~.~~I~-. ---e.

E8 ••

••

FIGURE A. I. Dynkin diagrams for various Lie algebras. The number resents the rank of the group. The number oflines connecting the the angle between two root vectors. Dynkin diagrams give us a cOTlvel~i4'I~t;i~~ of visualizing the entire structure of a Lie algebra. :::::;:;:w.l~

Dn = SO(2n), ±el ± ej' G2,

e; - ej' ±(e/ + ej - 2ek), 1 < i # j # k ::: 3, F4, I
± e4,

E6,

±e/±Cj,

1
A,2 A Briefl ntroduetion to Genera! Relativity

557

: where we have an even number of + signs in the las\ expression . .

,

E"

l :::i#j~6,

wh"e we have an even number of + signs in the last expression.

E ••

±e;± ej ' !:::ii=j:::7, ~(±CI ± e, ± ... ± el) ± ~el.

wio«e we have all even number of + signs in Ihe Inst cllprcssion.

A Brief Introduction to General Relativity ~~Sim~:o~s~,J~~,e:~ rcparametrization of space-time, a general coordinate l! is given by

xl' = il'(x}. ond:

(A.2.1)

this reparnmelrizalion, we use the chain rule 10 fmd thllt the differentials derivatives transform as

p'''i',"

dxll

ax' dj U = __ ai~ .

a

ax"

jj

(A.2.2)

say thaI the differen tial dx ll transforms contravar;ollt!y and [he derivative transforms covarianrly. In direct analogy, we now define vectors that a lso
AI'

ai" = axil- A",

8 ',," =

ax" B" ax"

.

(A.2.3)

lellsor T:'~.:: simply transfo mts like tbe produci of a series of vectors. The , ,,,,b,,, of lIldiccs on a tensor is called the rank of Ihe ten~or. II is now easy 10 show that the contraction of a covarianl and a contravariant '",«yields an invanalll: (A.2.4)

558

Appendix

We can show that the partial derivative of a scalar is a genuine ve~~tGif.t

ax" all ¢ = -a v ¢' axil The fundamental problem of general covariance, however, "Ui'~i~~1=~ru the partial derivative of a tensor is not itself a tensor. In order torj situation, we are forced to add in another object, called the C:hristoffe!f:11111 which converts the derivative into a genuine tensor: .... VIlAv = allAv

+ r~vA)..

We demand that

ax).. ai"

(VilAS = --V).A". .•••• ..... axil ax v This, in tum, uniquely fixes the transfonnation properties of the, •.

symbol, which is not a genuine tensor. c·,J~I#~~~1111 We can, of course, now define the covariant derivative of a tensor: V

VIlA = allA

v

-

r~).A\

as well as the covariant derivative of an arbitrary tensor of rank r. So far, we have placed no restrictions on the Christoffel space-time. Now let us define a metric on this space by defining dl¢i~¥~[t'~~ distance to be

ds 2 = dxllg llv dx v ,

where the g is the metric, which transforms as a genuine se(:ond-I·atlJk!~ii~W[ Now let us restrict the class of metric we are discussing by de:Jliqj.i:tJ covariant derivative of the metric to be zero:

VJLgv). = O. Notice that there are D x kD(D

+ 1)

equations to be satisfied, which is precisely the number of ele:m(lrl,~~jl~:: Christoffel symbol if we take it to be symmetric in its lower II'l(J1c:es.' can completely solve for the Christoffel symbol in terms of the

r IlV - g,,!3rJL".!3' et

-

f'/LV.!3 = k(a ll g v!3

+ av 81l!3 -

afj8Ilv)'

Notice that we assumed the Christoffel symbol to be symmetric' .. indices. In general, this is not true, and the antisymmetric Christoffel symbol are called the torsion tensor: r;v = r~v

- r~Il'

A.2 A Brief Introduction to General Relativity

559

In flat space, wc have the equation

[a""

Jvl =

o.

(A.2.14)

Since the derivative of a field generates parallel displacements, intuitively this equation means that if we parallel transport a vector around a closed curve in flat space, we wind up with the some vector. In curved space, however, this is not obviously true. The parallel displacement of a veClor around a closed path 00 a sphere, for example, leads to a net rolalion of the vector when we havc completed the circuil. The analog of the previous equation can also be found for curved manifolds. . We can interpret the covariant derivative as the parallel displacement ofa vector : and the Christoffel symbol as the amount of derivation from fiat space. If we : now paraHeJ displace n vector completely around a closed loop, we arrive at (A.2.15)

(A.2.16)

lei us now try to tonn an action with this formalism. We first note that the : volume of integration is not a true sl;alar: (A.2.17)

:: To create an invariant, we mUSl multiply by the square root of the determinant :.

the metric tensor: (A.2.1 R) product of the ffi'o is an invariant

FgdDx = invariant.

(A.2.19)

•. ~~t;~:::lhat the square root of the metric tensor does not transfoml as a scalar, ~· I of lbc Jacobian factor. We say that it transforms as a density. Notice that the curvature tensor has two derivatives. In fact., it can be shown the contrncted curvature tensor (A.2.20) islhe only genuine scalar we can write in terms of metric tensors and Christofsymbols with two derivatives. Thus, the only possible action with two

" (A.2.21)

560

Appendix

This formalism, however, cannot be generalized to include treat the transformation matrix axjJ.

oi v as an element of GL( D), we find that there are no finite-dimensional' resentations of this group. Thus, we cannot define spinors with . . alone. To remedy this situation, we construct a flat tangent space on the manifold that possesses O(D) symmetry. Let us define tangent space with Roman indices a, b, c, d, . ... Let us define the matrix that takes us from the x-space to the tangent space and

eal1 ea_ v e
gp,v,

= gap ep'

eUe ab = oab. a

....

We can now define a set of gamma matrices defined over either the ·••••.. the base space: . :::.

= yl',

yaeajJ.

{yl', yV} = _2gIlV.

Thus, the derivative operator on a spinor becomes yaeal'al' = yl'ojJ. =

p.

With this tangent space, we can now define the covariant spinor '!fr V Il '!fr = 011 '!fr

aer1Vat1:v(n;u~t:

+ w~b (J'ab '!fr,

an.~ :~~~~

where (J'ab is the antisymmetric product of two gamma matrices .i.: called the spin connection. Notice that the spin connection is a true .t·.¢•.!n~iit:~w.i:=i the J.t index. Under a local Lorentz transformation, the field

1/1 ~

VJl.1/I ~ eMVp.'!fr.

eM 1/I,

We can also use the 0(3, 1) formulation of general relativity with Christoffel symbols. We can define 'i1/L = aJl

+ w~b M llb ,

where M's are the ge'nerators of the Lorentz group. Then we can [VJl'

where

~] = R~~Mab,

A.l A Brief Introduction to the Theory or Forms

56 1

R:,t

Notice that this tensor yields an alternative formulation of the curvature te nsor. We also demand that the covariant derivative of the vierbein be equal to

7.cro: (A.2.31) lfwe antisymmetrize this equation in iJ,V. the Christoffel symbol disappears. Notice that the spin oorwcction has

Ox iD(D-l)

(A.2.32)

componenl,,_ This is precisely the number of componcnts in the antisymmetrized version of thc above equation. Thus, we can solve exactly for the connection in terms of the vierbcin. The Christoffel symbol and the vierbein arc very complicated expressions of each other. Given these constrai ned expressions for the Christoffel symbol and the connection fields, we can now show the relationship between the curvature tensors in the two fonnalisms:

(A.2.33) .

Ifwe take an aroitnlry spinor and make a parallcl transport around a closed ~ circuit with area ll.b. we have

(A.2.34) Notice that the a"" matrices arc the generators of Euclidean Lorentz transformations O{D). Thus, after a parallcl displacement around a closcd path, the spinor is simply rotated from its original orientation by an allgle proportional to

"'R"' jW.

(A.2 .35)

Ll

Notice that we can make an arbitrary number of closed paths starting from a single point. Each time, [he spinor performs a rotation. Notice that this forms B group. In fact, the group is simply O(D), which is called the flO/onumy group.

A.3

A Brieflntroduction to the Theory of Forms

: Let us define a on~form A by

(A.3.1)

A = A,.dx".

: where A" is a vcctor field and the differenlials dx" are now anticommuling:

dxfJ

1\ dx"

= -dx~

dxl'

1\ dx"

= O.

1\

dxl'.

(A.3.2)

562

Appendix

Let us define the derivative operator as d = dx ll 8w

Notice that because the derivatives commute

so therefore

where d is nilpotent. Let us now define a two.jorm: F = Fllv dx ll 1\ dx".

Notice that the curvature associated with a vector field is a two-form: .... F = dA = dXll81lAv dx v

= t<8ll A v - 8vAIl)dxll I\dx". Because the d operator is nilpotent, we have dF=d 2 A=O.

Thus the Bianchi identity for the Maxwell theory. expressed in terms is nothing but the nilpo~ency of d . A form w is called closed if dw = O.

closed: A form w is called exact if

w = dQ,

exact:

for some form Q. Thus, the curvature form F is exact because it can as the divergence of the one-form A. It is also closed because of the ..... . identities. . We can also combine this with a local gauge group with generators,· A = A~Au dXIl.

Then the curvature form is F = dA

+ A 1\ A.

Furthermore, the gauge variation of the Yang-Mills field under

is

A = AUAa

.' 8A = dA

+ A 1\ A -

A 1\ A.

Inserting the variation of the field A into the curvature F, we find 8F = F 1\ A - A 1\ F. )

A.3 A Brief Introuuction to the Theory of Fonns

563

Thus, the variation of the actiuD is zero:

oTr( F2) "'" 2 Tr(F 1\ .1\) -

2 Tr(A

1\

F} "'" O.

(A.3. 16l

Let us now write the anomaly term F F in the language of forms. The divergence of the axial c urrent is al!>O the square of two curvutures, wh.ieh is also a tolal derivative. In the language of forms, we find that this is an exact form:

(A.3.17)

where (A.3 . IS) is a three-fonn which we calI a Chern-SimollS form. In tum, its gauge variation is equal 10 another form that is also exact;

1lI]

Olll) = T r(dA I\dA}

=

(A.3 . 1.)

dw!.

where

(A.3.20)

WJ.=Tr(A l\ dA} .

We also note that these identities apply equally well to Yang-Mills theories as to general relativity. For gravity, we have the gauge group 0(3. 1). In other words. gravity has two g!lUge invarisoces, the general covariance of the coordinates x and the local Lorentz transfonnations of the tangent space. We will explain this more later. Tn general, an N Joml is defined as PIA WN -_ wil'I" ...!'N d"'d .t " .t "

" d x "'··-.

... "

(A.3.2 1)

AJI of the above equations. of course, can be derived w ithout the use of the theory of forms. However, forms give us a powerful shorthand that allow us to manipu laLC complex mathematical quantities. Notice, forexampte. that Stokes' j. theorem. expressed in the language of forms, now becomes

,

1" 1,. dt,) =

(A.3.22)

cu.

where aM is the boundary oflbc manifold M. Some simple properties of these forms are WpAW" = (-I}P
+(-

I )" llIp

1\

dwq .

(A.3 .23)

Let us also introduce the Hodge star operator, which allows us [0 take the of a p-fonn and convert it to an n - p-fonn in 11 dimensions: *(dxl" 1\ • .. 1\

dx!' ~ )

=

Ig11! 2 (n _ p )!

E, "' ···j.L~, dXil~ - l 1\ . .• 1\ ~w·.

dxl'·.

(A3.24)

564

Appendix

Some properties of the star operation are

**w p == wp /\

(-l)p(n-p)w p ,

*w p == roq

/\

*w p '

tha~~ ~~:~t;:?;~:.:t~;b~~~~:~er~:~n!~:. ~~e:;~:ei~C¥'. polynorrual as one .........................::$j~

that satIsfies

P(Ci) ~ peg-lag). Let us start by defining a homogeneous invariant polynomial of d¢.gr.~~~f.i]~:X which is dependent on the forms (Xi : .:-:-:::.:-:.:.:.:.:••••~.....:-:-:

P ;:;;; P(al.

0:2, ..•

t

Ctr ).

Let us differentiate this polynomial, being careful to differentiate e~9&:rii.r=$l separately within P: dP

=

l L ~ ~ ( ~ 1)d1+ ..d - P( C¥l, ••• , d Cii,···.

Ci r ) .

] .:s:i:S:r

Each time the derivative d passes over a fonn ai, it picks up a factoc·:j}>:-:.;.:.:.:.z....:.z.:-:

dai == (dOli)

+ Cljd( ~ l)d•.

N ow assume that -+

0:

g -1 o:g .

For g close to unity, we can always write

== 1 + W, 80: == -(J) A a + ex /\ w. g

_..... __ .. .. .... .

::::::::::::::::::::&.~~::::::~

and let u,s cal~late the variation of a homogeneous polynomial P of9~~t~fn~]~~t~ under thiS shlit: .<>
8P(a1

1

a'2""

1

exr ) = 0

=

~

~ (-1)

dl+ .. +d· l::::::::::::::~::::::::::::::~ ,- [P(ar. 0:2.···, OJ 1\ ai, ··:~::
I ~i :;:r

>:::::: ;:::::::::::~::::::::~ ...... .....................................'" ~.-

- (~1)d. peal, a2, .. ,. (Yi

1\

w •... , aT)]'

The trick is now to add the contribution of both d P and 8 P:

dP

+ 8p =

L (_l)dl+'+d peal, Ct2, ... , Dcti, "', i

Ci r ).

l
Now set each of the ctj to be a curvature form that obeys the Bianchi ~4~~ijtr.~~::::::::~ Dai == O. Thus; we find >~::::::::::::»~::::::::: dP(Q)

as we claimed.

== 0

AJ A Brief mtroduction to the Theory ofFonns

565

The second part of the proof is a bit more involved. Let us define

n =dW+WAW,

+ W' 1\ w'.

(A.3.35)

pen') - pen) ~ dQ

(A.3.36)

0.' = dul Our plan is now to show that

for some form Q. First we wall! 10 wrile a curvature fonn that Allows us to interpolate continuously between n and n'. Let

w,=w+t1]. , 1] =W - ( 0 .

(A.3.37)

Notice iliat the variable t allows us 10 interpolate between the two fonns:

t=O-+w.=w. t = 1 -+ WI = w'.

(A.3.3S)

It is easy now to find the curvature form that interpolates between the Iwo, as 8

function of f:

=dw, +w, I\W"

(A.3.39)

r = 0 -+ Q, = Q.

(A.3.40)

Q,

where Notice that the fonn varies between n and Q ' as t varies from zero to one. Now let us write the form q such that (A.3.41)

q(f3.a) =rP(/3,a.a,,·((r - I)-times)···a).

(J is repeated (r I)-times in the invariant polynomiaL This fOnD q will be the key to showing that P is an eXile! form. By the reasoning given earlier, if we differentiate q, we find

where the fonn

d'l{f/, Q,) = r cJ p(rl, 0, ... «r - 1) -lil1le~) . .. n,)

= q(D1], Qr> - r(r - I)t P(1], 0, A '1 -

1]

A 0" Q" ...• f.l,).

(A.3.42) But we also have the identity, due to the fact that q is an invariant polynomial,

2q(TJ 1\ 'I,

n,) + r(r

- t )P('I,

a, 1\ 1) -

1) 1\

n" ... , a,) = O.

(A.3.43)

In the previous equation, we used

8= I

+ 'I.

(A.3.44)

Putting both equations together, we find

dq(l'J, f.!,) = q{D1), Q,) + 2,q(TJ

1\

11,

n,) =

d pen,) dt

.

(A.3.45)

566

Appendix

Thus, we arrive at

1 1

o

dP(nr) = P(Q') _ P(Q) dt

= d

11

q(w' - w, Qr)dt = dQ.

We thus obtain our final result:

Q=

11

P(Q') - pen) = dQ,

q(w' - w, Q,)dt + closed forms.

Thus, an invariant polynomial based on curvature two-forms is both <:.•J·:·Ql~~ and exact.

A.4

A Brief Introduction to Supersymmetry

In the late 1960s, physicists tried to construct the master group allow a synthesis of an internal symmetry group (like SU(3)) and the . ... or Poincare group. They sought a group M that was a nbntrivial . internal group U and the Poincare group: M-:JU®P.

Intense interest was sparked in groups like SU(6, 6). However, the ' np()ssibh~;. Coleman-Mandula theorem showed that this program was it are no unitary finite-dimensional representations of a noncompact either: (1) the group M has continuous masses; or (2) the group M has an infinite number of particles in each' •... representation. .. Either way, it is a disaster. However, it turns out that supergroups or Lie groups allow for an evasion of the no-go theorem. . The work of Lie and Cartan concerned only continuous simple the parameters Pi were real. However, if we allow these Grassmann-valued, we can extend the classical groups to the Two large infinite classes of groups we will be interested Osp(N / M) and SU(Nj M) groups. Let us begin with the group O(N), which preserves the invariant: . O(N):

XiXj

= invariant,

and the group Sp(M), which preserves the form: Sp(M):

(jmCmn8n

= invariant,

A.4 A Brief Introduction to Supcrs)1l1metry

567

where the C matrices are real antisymmetric matrices because the (J/ are Grassmann-valued. The or,hosymplectic group is aow dcfmcd as the group that preserves the sum: Osp(N / M):

XjX,

+ (J",Cm ~ (J~ =

(A.4.4)

invariant

Notice that the orthosymplcctic group obviously contains the product

O'p(N I M ) " O(N ) ® Sp(M).

(A.4.S)

The simplest way to exhibit the matrix representation of this group is to usc the block diagonnl form

Osp(N I M) ~

It O(N) 8

A

SP(M)

I

(A.4.6)

with simple restrictions on the A and B matrices. Similarly, the superunitary groups can be defined as the grOllps that preserve the complex fonn

(Xi r xioij

+ (8"'y e"g"m.

gmll = ±8m".

(A.4.7)

The bosonic decomposition of the group is given by SU(N 1M)" SU(N) 0 SU(M ) ® U(i).

(AA.8)

Let us write the generators of Osp{ 1/4) as

Mit = (PI'" Ml"v, Q",)

(A.4.9)

which have the commutation relations

(A.4.lD) Written out explicitly, the commutato rs involving the supersymmetry generator

(Q" Q,) [Q •• P,] [Q",

~

2(y'C).,P,.

~

o.

M"v l = ((J/J")~QIi"

(A.4.11 )

PI' = -;0"

(A.4. i2)

.' ~: What we want is an explicit representation of these generators, in the same

. way that

is the generator oftranslalions in x -space. Now we must enlarge the concept to include the supersymmetric partner of the x-cooroinate. Let superspace as the space created by the pair

(A.4. D )

568

Appendix

where Btt is a Grassmann number. Let us define the supersymmetry

a

.

Qa = --- -l(y I1 B)aa/l.

aBtt

where B is a Grassmann number. We choose this particular cause the anticommutator between two such generators yields a as it should, {Qa, Qp} = -2(yl1C)ttp iaw

Notice that eQ makes the following transfonnations on superspace x l1

--7

x l1 - ieY/lB,

Btt

--7

B",

+ Ba.

Notice also that we can construct the operator D tt

a . = --- + I (y11 a/lB)",.

aBa This anti commutes with the supersymmetry generator

(Qa, Dp}

= O.

This is very important because it allows us to place restrictions on sentations of supersymmetry without destroying the symmetry. This us to extract the irreducible representations from the reducible ones. Let us now try to construct invariant actions under supersymmetry., L,;:eHii~::::::: define the supeifield V as the most general power expansion in this sUl,eqsptlW-j:::::: V(x, 8).

Then a representation of supersymmetry is given by oV(x, B) = Vex

+ ox, B + oB) -

vex, B) = eaQa vex, B).

Notice that this definition proves that the product of two superfields superfield VI V2 = V3.

Thus, we can construct a large set of representations of supersymmetry' : ' simple product rule. Now let us calculate the explicit transformation. fields. We will sometimes find it useful to break up the into two two-component spinors, because of the identity; 0(4) = SU(2) 0 SU(2). Using indices A and .4., A = 1,2, we will write a Majorana terms of its SU(2) x SU(2) content: a

X =

(~:),

- = (XA, -X- A) . Xtt

AA A Brief Introduction to Supersymmcrry

569

Ifwe invert this, we find

xA = X,i

~{1

+ Y5)X ,

= ~(l- Ys )X,

(A.4.2J)

ond

£12

= I.

(A.4.24)

In this notation, the covariant derivatives can be written as

(A.4.25) where (A.4.26) The real vector superfield V can be decomposed as Vex, 8, 0) = i8X !ie2(M - iN) + ~i,§2(M

ix'e -

c-

px') - i0 20()..1 -

+ it'Fep. -

~i

- !0202(D

+ ~DC).

~i

+ iN) - Bu"eA".

px) (A.4.27)

We can now read olflhe supcrsymmetry transformation parametrized by { on these 16 fields:

8C == ~Y5X'

+ y~Ng ~ i(A - i h).

Sx = (M 8M

8N = (ys(l - i

- iy"(A".

#x).

+ Y5iJ"C)C (A.4.2S)

8AII = i(YII).. + {allx. 8).. == -ju"~{aIlAv - Yss D, 8 D = - j~ PYs)..,

We call it a vector s uperfield because it conrains a vector particle in its rep· resentation (not because the superfield itself is a vector field undel' the Lorentz group), In general, vector fields can bccomplcx and thcy are reducible. To form irreducible representations, we will find it convenient to place constraints on them that do not des troy their supersymmetric nature. The constraints must therefore commule with the s upcrsymmetry generator. Notice thai, because D.. an ticornmutes with the s upcrsymmctry generator, we can impose this derivative on a superfield and still get a representation

570

Appendix

of supersymmetry. Let us now try to construct various rep'res:enltaHQru~:::glt;:; supersymmetry based on this simple principle. We can impose DitrjJ=O.

A superfield that satisfies this constraint is called a chiral S1J11prhp'dN:i;iU~~: that it has half the number of fields of the original superfield but transforms under the group correctly. A chiral superfield has the rjJ(x, f)) = A

+ 2(hfr -

8 2 F.

The variation of this superfield can also be read off the variation

8¢ = -i[¢, {Q + Q{]. We easily obtain OA = 2{1{r.

01/1 = -{ F - i a/! Acr/!{ , 8F = -2i a/! 1{rcr/!{ . We could also try other combinations of constraints, such as DArjJ = 0

on a chiral superfield. We find, however, that the combination constraints imposed simultaneously implies that rjJ is a constant. Another constraint might be DADAri> =0.

This yields the linear multiplet. (Unfortunately, actions based on ally equivalent to actions based on chiral superfields, so we learn Another constraint might be [D A• DiJ]rjJ =

o.

Unfortunately, this yields a constant field. We might also impose DiJD ADArjJ = O.

This again gets us back to the chiral superfield. Finally, we might aJS'[HI:v<::::]~~~~m - 2

DAD ¢ = 0

~;;

wem~~l~~~~r]~l

for real ¢. This actually yields an entirely new superfield, which to build the Yang- Mills action. In summary, the only new fields that transform as irreducible replreserl~~ij~~.§~j~~~j~j~ of supersymmetry are the chiral superfield, the vector superfield, andt·l~~~ij~ili~~~~~~j Mills superfield. The other combinations that we might try are either I redundant to the original set. Now let us discuss the problem of fonning an invariant action by d·i.~.t.iffrr:f.I:~:~~:~:~~: Grassmann integration.

A.4 A Brief lnlroduction to Supersymmetry

571

Integration OYer theR Glllssmann variables must be carefully defined. Ordinary integration over real numbers, of course, is translation-invariant :

i:

dx¢(:c) = [ : dxl/J(:c

+ c),

(A.4.38)

where c is 0 real displacement. We ""ould like Grassmann-valued integmtion to have the same property

J

d8¢(8) =

J

d8¢(e

+ c).

(AA.39)

If we power c:<paod this function q1(9) in a Taylor series, we have the simple expression ¢(8) = a + bO.

(AA.4<J)

If we define

lO = / dB, II =

!

(A.4AI)

d(J(),

then translation invarianee forces us to have

/ dOq,(O)=alo+b / l =(a+bc) lo+b/ l . Thus, we must have 10 = 0 and

(AA.42)

'I

can he normalized to one:

10 = 0, (A.4.43)

f l = I.

0'

J

dB = 0,

Jd88 =

I.

(AA.44)

In other word1:i, we have the curious-looking identity

(A.4A5)

/ dO= a°09' With these identities, we can show that

/D

d9; de; cxp

[l",~ 81 Afj9J ]

= det(AI).

(AA.46)

Thus, in general, invariant actions can be formed (see (A.4.27), (A.4.30))

J

d-9d 4 xV(x,8)-+ D-term,

/ d 1 () d- x¢(x, 9) -+ F -term.

(A.4 .47)

572

Appendix

The first integral only selects out the D~term of the superfield. integral only selects out the F ~tenn of the chieal superfield. In geJler:al;:we:i5i these" F" and" D" terms. We can check that these are invariant aCtllOnJ~;: 8

8 / d xV = / d

8

xs a Qa V = O.

This is because the integral of a total derivative, in either x or e Now let us try to write simple invariant actions based on F and D terms. The simplest invariant action is called the Wess-Zumino mCloe.l·t

8

S = / d x¢¢

+ (/ d 6x [lLifJ + !mifJ2 +

:t ¢3] +

h.c') .

Written out in components after performing the () integration, it "VJllL
S= / d x

{-~(a"A)2 - ~(aIlBi - ~Xy/.t a/.tX + ~F2 + !G2 }.

Notice that we have now constructed an action with an irreducible tion of supersymmetry with a spin-O and spin-~ multiplet: 0). To the (1 , ~) multiplet, we need the following construction for the given by

C!,

LUCI"'- VY

where -2

WA=DDAV. W..i = D2DJiV, DAWB=O,

where V is a real vector supermultiplet, which transforms as

8V = A - A, V is real, but A is chiral: A + =

-A. Under this transformation, we 8WA = 0,

so the action is trivially invariant under both supersymmetry and U (1) . iJ.'.~i!tl::::::::::: g' .

invariance. Notice that the vector supermultiplet contains the ~~~:~'~"~~rm::~::::~::~ All' while the chiral supermultiplet A contains the gauge parameter A. out in components, this equals

S

=! ~x "

(_!F2 4 /.tV

l1 !iiry a" 1fr + !D2) 2 2

which is invariant under

8A"

= sYp. t/!.

8t/! = (_!a/LV F/.tv + Y5D)S. liD = sY5y"8"t/!.

A.S A Brieflntroduclion 10 Supergravity

573

The next multiplet we wish to investigate is the (2,~) multiplet. Historically, it was thought that the Rarita-Schwinger theory was fuodamenta Uy flawed because it permitted no consi~tent couplings tn other field.~. However. physicists neglected to couple the Rarita-.~chwingcr field with the graviton. The inconsistencies aU disappear for this multiplet.

A.5

A Brief Introduction to Supergravity

There are at least four ways to formu latc supcrgrnviLy: (I) Components---this method relies heavily on trial and error. However, it yields the most explicit fonn of the action. (2) CU£V'd.turcs--this method stresses the group theory and the analogy with Yang-Mills theory. (3) Tensor caJculus---this yields precise rules for the multiplication of re presentations of supcrsymmetry. (4) Superspaoo---this is the most elegant formulation of supcrgravity; however, it is also the most difficult. Thc higher N superspace formu lations of supcrgravity arc still not known because the torsion constraints are too dlfficult to solve.

We will concentrate on the method of curvatures because it resembles the Yang- Mills construction we have been using so far. Since Osp(1I4) has 14 generators, let us define the 14 connecfionfields of Osp(1/4) as

h~ =(e;.,lJ):~. ;P~).

(A.5 .1)

Then the global variation of the cotwection fields is

'h'1"= f'RC e ,/e 1,£.

(A.S.2)

e A = (e~, suI> , g~).

(A.S.J)

Q

where

The covariant derivative is now given by

VJ>=a,,+h~MA =

a,.. +

e: P", + w~1> Mub + 1it

1'- -"

Q"

(a> b).

(A.5.4)

The fields transform under a local gauge transfoftlliltioll as

6h~ = i1l'e

A

+ h!eC f£s'

(A.S.')

Now form the COmml!tator of rwo covariant derivatives [VI" Vv] =

R:.M. .

(A.S.o)

574

Appendix

where

R: LJ

= afi.h~

- avh~

+ h~ h~ f2B'

In component fonn, we have

R~LJ(P)

== alLe~ + w~bee ~ (11- B- v), ch - (II ~ V) Rab(M) = ap.. Wah + wacw J.L \J /.L l.! II

f"V

R:v(Q) = OM 1ir~

?

+ if vU)~)rrab ~ (J.l 8- V).

The variation of the curvature can easily be shovro. to be 8R:v == R!v sC f~B' The action for supergravity is now given by

S=

f

4

d xs

I.I.vprr {R

(ab

~v M) Rpa(M

.. ':::::: {~~~~~~~~~~~~~~~~~~~ ()fJ( ) .......;...."""'.,.........• + R~v Q RfJa Q YSC}f~~tJ~~~~~~~t;

()Q'

)cd Cabcd

(A)$~ ~ Q:j~~~~~~~~~~~~~;

If we make a variation of this action, we find that the action is no(:t~~li~~~~~~~~~~~~~~~ invariant unless we set ".<::::::}:::::::::::::::~::~ ::: :::~:~~~~~~:~~:~:~:~~~~

'.~. ",jI" ~ .. J ..,jI..J ......tl..........J.~

Rll-v(p}a

= O.

The action is invariant up to the term v

8w: R~LI(P).

..

However, since we imposed this constraint (A.5.11) from the beginnin·· .. find that the action, indeed, is fully invariant under the transfonnation./: This constraint appears to be highly unnatural until we reaJize that it { same as the vanishing of the covariant derivative of the vierbein in (A.~;~," Thus, we will choose the vierbein to have zero derivative in order to hay¢: fina1 invariant of the action. The final action is I ~""jf' 5Da'Ppe l/r Il vup L -- - 2KI 2e R - 2'f'j.tYvY •

Unfortunately, the situation for higher N actions is. much less c1eru.:-;:: superspace method still has not been solved for the higher N, but the if.\ supergravity has been constructed using a trick: by extending the dime~~iQ~~~~~~~~~:=::=~:: of space-time to 11, we can construct N = 1. D = 11 supergravity. compactify in order to reduce down to N = 8, D = 4 supergravity. The starting point fdr constructing the II-dimensional supergravity)~::~~~:::::::::::::::: realize that we need equal numbers of bosonic and fermionic fields. SY::'mAl:~:~:~:~:~:~:~~: and error, we see that the following choices yield equal numbers of :fie14'~:(~~~~~~~~~~~~~~~~~~~~~~: .

e~

= ~9 x

WM = !(9

0 - 1 = 44 components,

x 32 - 32) = 128 components,

::::~::~ ::~:~:~:~:~:~:~:::~~~::

....: .~ ..~..~..~~~~~~~~~..r~..r::;~~~,

.. ':.:

. .. .. ... .. ..

...

....

~: ~ ~ ~ ~ ~ ~ ~ ~.~ ~.~

. . . :~ ~::~:~:~:~:::::::~:~::~

A .S A Brief Introduction LO $upergravity

Am

(~)

=

575

(A.5.14)

= 84 components. where thc vierbein is transve~ and traceless and r . 1/t = O. Then. by brule force , Crcmmer, Julia, and Scheck proved that the following action is invanant in I I dimensioD.ll: I ! t! R 2/C

L= -

, - JI je1/!

rO N' D N['j(W + W-)1 1/1,.. - :rs" eFMNI'Q

~: e(;j, M rH"'I'QRS'/ts + 12~N r PQ I{tR)( F + F)N I'Q R (A.5.15)

where ,lj e~ =

"A

MNP

f

tKijr AI{tM.

= - ';;ijrIMNl/tI'J'

,,'/1M =K

-1'

DM(W)r)

(A.5.16)

..Ii (.I'QRS + 288 r Itt

-

srI'

Q.wr

QIIS ) - . ry f l'QRS.

and where (A.5.17)

is the curl of A M NP and FHNPQ is the supercovariantization of FM NI'Q. where we choose {r A, rB) = 2'r~8, we antisymmetrize according to r d = Hr Ar n - r Br A), and indices ABC are flat and M N P are curved. The N = 1, D = 10 supcrgraviry can be found by truncating the earlier aClion. The spinor decomposes into a pair of Majorana---Weyl gravitinos and a pair of Majorana--Weyl spin . ~ fermions. The vicrbein decomposes into a lO·dimensional vierbcin and a scalar fiel d ¢, while the antisymmelric tensor field decomposes into a. boson field BMN • Thus, the sel of rcduced fields is where

F.I.I NN,l

fe~ ;¢ ; B~lNf,

11/r.\l;.\J.

(A.S. 18)

Our fi nal action in 10 dimensions is

e- I L = -

2~1 R - j~MrMNI' DN1/I,.

_ !ir'" D M )., 2

-

-

!q,-l!1 H~NI'

~(¢-taM¢)2 16K2

_

l../2~M rN rM.\(¢-laN¢) 8

+ ~t\" q, " U4 H NPQ(!i'M r MHI'QR1/!R +6!ir N r l'l/tQ - Ji.j,Mr ,'II'QrMA)+".,

(A.5.1 9)

576

Appendix

where A

8e M =

~

uifJ

../2K_ = --3-ifJl}).. •

8BMN =

8)..

K - 4 2'7)r 1/IM.

../2 3/4 (_I}r M1/IN 4ifJ

../2_) ,

_

- I}rN1/IN - T7)r MN )..

3../2 I (r . oifJ)7) + gifJ1 3/4 r MNP I}HMNP + ... . = --g-ifJ-

81/1m = K- 1D M7)

+ -;; ifJ- 3/\rZ PQ -

98Zr PQ )7)HNP Q + ... ,

and H = dB, and where ... means four-fermion Fermi-type terms th~(~fjI~l~l~ml~ are omitting. This is the low-energy limit of a Type IIA string (becalls.¢:::tTie~:::::::::::i: fermions have opposite chiralities from the dimensional reduction). Thus ·.th"~:.: are no anomalies from this theory because there is no chiral . The Type lIB theory, however, cannot be derived from dimensional tion because it has fermions of the same handedness (and hence can .. anomalies). Type lIB has no covariant action at all (but has on-shell equlatJ:(?Ji;$: of motion and also a well-defined light cone action). Next, we would like to couple supergravity to Yang-Mills matter. The Stiti¢r.~::::: Yang-Mills multiplet by itself is given by {A~; Xa},

where a represents the elements of the isospin group. Notice that we have ecjila:I;: numbers of fennions and bosons on-shell. The final action is equal to e- I L = e- I Lso(HMNP) - !ifJ- 3/4F%rNFMNa - !XarM(DM(w)xt - kKifJ-3/8XarMrNP(F~p

+ F",vP)(1/IM + 1~.J2rM)..)

+ 1~.J2KifJ-3/4XarMNPXaHMNP - 15~6.J2K2xarMNPXa1jrQ(4rMNPrQ __ I K2X-ar

512

MNP

+ 3r Qr MNP ))..

xajrMNP).. __ I K2X-ar 384

MNP

XaX-brMNPX

.

(A.5}f~~@mm@~: where the surprising thing is that we must modifY the condition H = dS";t,::::::::: that~· ..

H = dB -

2-1/2KW3.

where al3 is the Chern-Simons term al3

= Tr(AF - tgA3).

A.6 Notation

577

The variation of B under a gauge tTaILsfonnation is now oB == T I/2K Tr(A dA), 8H =0.

(A.5.25)

The action is invariant under oA~ = ~¢'118i1fMx n, g

8X ==

_ ~q.-318rMNF:fN'1+ :K[3iXd'1 _ ~irMNXur ., _ 2 MN"

(A.5.26)

..!...)"rMNI'QX"r "i !4 ' MNPQ', •

T hc transformations of the supergravity fields are the same as before (with modifi ed N field) and the new pieces that must be added to the transformation law are {j' ).

=

./i KX."rM11"f'x"rUNp·"" 432

8' BM N = r t/2 K¢3 18ijr[M XdA~J'

(A.5.27)

(/"'M = - 2~~KiQ r NPQx Q (rM N I'Q - 58MNfI'Q)'1.

,

We may nsk the obvious question: Are there supcrgravily theories beyond N = 81 The answer is probably no, This is because the supers>TIUDC1ric generators of the Osp(N /4) group have spin-to r[we take the maximum helieity state of the graviton and then operate on it with all possible Q's, we find that the series eventually mus t terminate or else we create particles of spin-~ and 3: QuQ~ · ·· Q~

Igraviton).

(A.S.2S)

Theories ofspin-! and 3, although they can be constructed as free theories, are thought to be inconsistent when coupled to other panicles. Thus, since thcre are eight half-steps between 2 and - 2, we find that N can only equal eight in the above series. Thus, 0(8) is the largest supergravity theory thllt does not have spins greater thlJl 2.

A.6

Notation

For the purpose of clarity, we have deliberately dropped lhe normalization factors appea ring in thc functional path integrals and the N -point amplitude A 1/ . This should not be D problem because they can be easily restored by the reader. In our units, the Planck length and Planck mass are equal to

L = rfiGc-1J = 1.6 x 1033 em, M = lFicG - t ] = 2.2 x to-I gm = 1.2 We set c and h = I.

X

10 19 GeV/c.

I

· i,

I

·\. · '.

578

Appendix

We use the space---time Lorentz metric (-, +, +, ... +).OntheworldsnE:er~::::::::: we use the two-dimensional metric ( -, +), where the first (second) index retem:~:}~:~:~ to r (a). OUf two-dimensional matrices are I

pO

=(

-i)

0 i

0

'

where -

0

1/f = ljJp . In general, we will use kJL to represent the physical momentum and Pl1 " "~(;)::::::::::::~ represent the operator whose eigenvalue is the momentum. However, as in .."1:,,:.0:.:.-:.'.:-:.: literature, we will often drop this distinction and use both interchangeabl}io:::: We use the following choices for the Regge slope )~ ::::::::::::::::::

c/ == ~ for open strings, (X'

==

i for closed strings.

Whenever possible, we use Greek letters to denote curved space indi'c"~¢"!~~::~:::~:::~:: We use f.i.. and lJ to denote 26- and 10-dimensional Lorentz indices in space. We use Greek letters, a, fJ to denote two-dimensiona1 curved sheet indices. Flat space indices, either in Lorentz or in two-space, are in Roman letters, a, b, c. When the context is clear, we \vill sometimes use metric ( +, +, ... , +) for the fiat tangent space. We choose our gamma matrices such that

{r A, r B}

== -2g A B,

r D +1 = for] ... ID-l. In an even number of dimensions, they are nonnalized so that

(rD+1i

== +1 if D == 4k + 2,

/~I!~~.~.~l

(r D+] i == -1 if D = 4k. When the ganuna matrices appear with more than one index, we take the _~~ of all antisyrrunetric combinations of indices. For example, we nVlljWll'j~lf~~~:j~ the gamma matrices so that

these

rAB

= ~[rArB _

rErA].

We define the light cone gamma matrices as :.,-'

r+

==

['- =

_l_(rO

-J2

+ r D- 1),

JzC['O -

['D-I).

When specializing to the case of 10 or 4 dimensions, we will often just us~:" symbol y f-L • " "",", :}}~::~=~{~~:

References

579

When we make the reduction down to SO(8) for the light cone fonnalism. we will use the direct products of2 x 2 Pauli matriccs: y' = (

~y" y,:,o ) .

yJ = -iT] ® 1"2 0 1"2 . y2 = il@fl ®f2. yl=il® Tl®rZ' y4 = i r l0rz01. yS = iTl®TZ® I, y6= ir z®l®rl' y7 = irz® I ®T~.

YS=l®l® l , , Ij _!( I j _ l'wl' -

2

>'w,;>'''b

j i) >'d,lYilb'

where 1"1 BTe the Pauli matrices, and each gamma matrix is a direct product of

three 2 x 2 blocks.

References For an introduction to group !hoary, see: [I ) R. Gilmore. /J# Groups. Lie Algebra, and Some oj Their Represelllations, Wilcy-Inlerscience. New York, 1974. (2J N. Jacobson, Lie Algebras, Dover. New York, 1962. (3 ) H. Georgi. Lie Algebros in Particle Physics. BenjaminICummings. Reading, MA, 1982. (4) R. N. Calm, Semi·Simple Lie ,Algebras llIId Their Represen/atiotU. BI".'f1jamin/Cummings, Reading, MA, 1984.

For an introduction to gcncl1I.l relativity, see: [5] C. W. Misner, K. S. Thome, and 1. A. Wheeler. Gravitation, Freeman, San Froncisco, 1973. [6] S. Weinberg. Gravitation and Cosmology, Wiley, New York, 1972. [7] S. W. Hawking lind G. f . R. Ellis, The Large Scale Structure o/Space-Time,

Cambridge University Press, Cambridge, 1973. (8] R. Adler, M. Bazin. and M. Schiffer, IlIlrodllction l(J Ge'lf~ral Re/(l/ivity, McGr.tw· Hitl, New York. 1965. [9] M. Canneli. Group ThcOIY and General Relativity, McGraw-Hill, New York. 19 77 .

For an introduction to tile theory offorms. ~ee: {IO] T. Eguchi, P. B. Gilkey, nnd A. J. /lansen, Phys. Rep. 66, 213 (1980). 111 ] C. Nash and S. Sen, Tbpology and GeamerT}' Jor PhysicLw', Al:adelllil;

New York, 1983 .

», ~~.

580

Appendix

{l2] C. von WestenholtZ Difforential Forms in Mathematical Physics, t

Holland, Amsterdam~ 1978, [13] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Ac~ya/~·fnR:ef.::::::: Index Theorem, Publish or Perish~ Wilmington, DE. E14] S. J. Goldcerg, Curvature and Homology, Dover~ New York, 1962. For an introduction to supersymmetry and supergravity, see: [15J P. van Nieuwenhuizen, Phys. Rep. 68C, 189 (1981). [16] S. J. Gates, M. Gtisaru, M. Rocek, and W. Siegel, Superspace or One n:lf:~'~~:m and One Lessons in Supersymmetry, Benjamin/Cummings, Reading, MA, i: [17] M. Jacob, ed., Supersymmetry and Supergravity, North-Holland~ .

]986. [18] P. West, Introduction to Supersymmeny and Supergravity, World Scie~ti:~·]~:~:~::~~~: Singapore, 1986. :::::::: [19] 1. Wess and J. Bagger, Introduction to Supersymmetry, Princeton Unive#j Press, Princeton, NJ, 1983. ::::::::: [20] R. N. Mohapatra, Unification and Supersymmetry, Springer- Verlag, New rq~~~::::::: 1986.

,,'

.

. .. . .....

Index

(\clion, firt! qU(lntize(\ bo~onic

poinl particle, 25

bosonic string. 52 confonnal field theory. 152, 158, 175 first order fonn, 28, 58 ghosta:lion, 152, 158--1 59. 175 Hamiltonian, 28. 58 heterotic string, 378, 400

second order (onn, 28, 58 superpanicle, J 03 superstring (conformal gauge), 105 supcrsl:ing (OS), 126, 138 superst:ing(N~R), 11 9,1 36.152 Action. second quantized

BRST string field theory, 300, ]03 light cone fie ld theory, 259, 263 , 278 point panicle, 40, 252 pregeometry, 326 superstring, 28 1, 324 Almost complex structure, 419 A~. 546 Anomaly

cancellarion in strings. 366-368 characeristic classes, 352 chiral, 347- 348 confonnal anomaly, 211

graviutional,357_JS$:

hexagon diagrams, 344, 37 1 A-roof genus, 352, 369 Atiyah--Singer Theorem. 338, 357 Automorphic fux.ti on

multiloop amplitude, 204-207 single loop ampli tude, 184

Betti number, 413, 4 15 Bianchi ide nti[)" 36 1,406, 427,562 black holes" 458., 5 11, 532

Bw. 546 Bosouizatl0n, 143, 163, 169-1 70 BPS re larions., 459 BPS stales, 458, 472,482, 505, 5 11,

m

BRST, first quattized bosonic siring, 70, 97 conronnal field theory, 160 current, 160-161 fermionic string, 121- 122 point particle, 30-33 BRST, second quantized

ax iomatic formulation, 309, ]30 point particle, 32-33 prcgoometry, 326

superstrings, ) 18 Calabi-Yau manifold, 405 Cartan subalgcbra, 168, 389, 554

· 582

Index

Cartan-Weyl representation, 168-169, 554 Chan-Paton factor, 339--341, 368 Characteristic classes, 348 Chern class, 350, 369 Chem-Simons form, 349, 576 Chiral invariance, 342-345 Closed form, 410 Closed strings field theory of closed strings, 317-319 modular invariance, I 97 multi loop amplitude, 209--210 single loop amplitude, 195 tree amplitude, 87-90 Cn ,546 Cocycle, 168, 171 Coherent state method, 82, 195 Cohomology de Rahm, 409-412 Dolbeault, 421 Hodge theorem, 229, 421 Compactification,374 Conformal anomaly, 217 Conformal field theory fermion verte~ 165, 175-176 multi loop, 234-236 operator product expansion, 147 spin fields, 143, 155-158, 169--170 superconforrnal ghosts, 158 Conformal gauge mu Itiloops, 211- 214 single loop, 60, 98 Conformal ghost, 70, 121,158-165 Conformal group, 549 Conformal Killing vector, 214 Conformal weight, 144, 158-165, 175 C Pn , 418, 424 Currents, 41, 105, 119, 160, 342-344,347,355 Curvature tensor, 6, 13, 405~ 559 D-branes, 458,504,51 1,517, 518, 521 , 539 DDF state, 94 Dehn twist, 198, 213, 228--229 de Rahm cohomology, 409-410

de Sitter group, 549 Differential forms, 409--411, 414-416 Dilaton, 192 Dirac index, 353-356, 370 Divergences bosonic single loop, 190-191 .. • closed bosonic loop, 198-200·.· fermionic single loop, 200 .••• • multiloop, 204, 210; 226 Selberg zeta function, 226, 24 ••

<•

Dn , 546 Dolbeault operator, 42 I Duality dual amplitudes, 73, 81 incompatibility with string theory, 249 duality, 457 Dynkin diagram, 554-557

E6 ,557 E6 subgroup,427, 432 heterotic string, 382 Kac-Moody, 395-397 metric, 382 modular group, 394-395 E 7 ,557 Es anomaly cancellation, 364 Dynkin diagram, 557 Energy-momentum tensor conformal field theory, 145, I 152-153 ghost, 159 point particle field theory, 42 superstring, 106-107, 119 Virasoro algebra, 58 entropy, 533,541 Euler Beta function, 15,81 Euler number, 413, 429 Exact form, 410 F -theory, 501-502, 510 F L- 2 formalism, 113--114, 137-1 F4 , 557 Faddeev-Popov quantization conformal field theory, 148, 1 162 multiloop, 215, 240

lndcx point particle, 30-32 string fie ld tbeory, 297-299 supcrconfonnal ghoSl, 121 J:ennion vertex, fi~t quanlizcd confonnal field thoo!)" 165- 166. 175-176 GS, 134 NS-R. 126 Fermion vertex. secol\d quantized BRST.323 light cone. 282-284 Feynman diagrams point particle, 4, II, 50 string, SO- 51 First quamization, bosonic sIring BRST, 70-71. 97 Gupta- Bleuler, 96 light cone, 67- 70, 96 problem.~, 247 First quantization, point particle BRST, 30, 34 CoulC)mb, 28-29 GlIpta-Blculer, 29 First quantilation, superstring BRST, 121 Gupta-B!cu[er, 120 light cone, 120 Fock space, 61 Forms, Iht:ory of, 562 566 Four-dimensional supcrslrin~, 443 Fubini-Sludy metric, 424 G l ,557 Gaus.~-Ilonnetlheorcm, 162, 345 General relalivity, 557- 561 Genel'Htion number Calabt-Yltu manifolds, 432 Euler number. 429-431 orbirold phenomenology, 44 \ Ghost number current, 160 second quantized superstrings, 322-323 Ghosts t1ccoupling,9G-94 faddeev-Popov, 31-33, 122, 160, 322-323 G-parity. 124 -126

583

Grand Unification Theories GUT, 8. 12, 338 hierarchy problem, 9

0""""""

integration, 570 nwnber, 32-33, 571 Grassmannian, 227, 235-237, 242 as modeJ, first quantization, 192 GS model, second quantization, 280-286 GSO projection a ' pllrity, 123-126 modular invariancc, 222 Gupta-Bleulcr quantization bosonic strings, 96 point particles, 29, 43, 46 superstrings, 120 Hamillonian first quanti~cd, 28. 58 second quantized, 259 Harmonic foml,4 14 Hannonic oscillarors, 31, 61-64, 18-80, 185-186 I [eterQtic string covariant fonn, 386-3111 divergences, 391-395 E. ® Eh 382. 386 397 motIular invariance. 395 single-loop amplitude, 39 1-394 spectrum, 38}-386 supersymmctry. 383 trees,388-391 Hirota equatio[l.$, 234 Hirzebruch class, 353, 351-]70 Hodge thoorcm. 229, 42 1 Ho]omorphi<: factorizati on. 221, 232-234, 241 Hoionomy group SU(3),408 SU(N),423 Homology, 410 Homology cycle, 208, 228 Hyperbolic. 202

I ndex theorems Aliyah-Singer, ]38, 357 Dirac, 353-370 Ga~on~, 162,345

584

Index

Index (continued) Hirzebruch, 357 Riemann-Roeh, 162, 357 Infrared divergences, 190-191 , 19&-200 Internal syrnrneuies Chan-Paton factors, 339-341, 368 via compactification, 381-383 Invariant points, 201 Kac-Moodyalgebra conformal field theoty, 170 E8 x E8 , 395-397,403 SO(IO), l70-173 Kadomtsev-Petviasvili hierarchy, 234-236 Kahler manifold, 419, 422 Kahler potential, 422 Kaluza-Klein theoty, 375 Klein bottle, lSI Kunneth formula, 416 Laplacian, 414, 422 Lattice even, 381-383 Lorentzian, 39&-399 modular invariance, 395 self-dual, 382-383,393 Lie algebra accidents, 552-553 Dynkin diagrams, 554-557 Kac-Mood~ 170,395-397,402 types, 546 Light cone, first quantized bosonic suing, 67-70, 96 supersuing, 128 Light cone, second quantized action, 259, 263, 278 four suing interaction, 275-280 supersyrnrnetry, 280-286 vertex,262-264 Lorentz group in light cone gauge, 69-70 .~ Lorentzian lattice, 39&-400 M(atrix) model, 539 M(auix) models, 525 M(atrix) theoty, 458

M-theoty, 457, 458, 467, 539 Majorana spinor, 567, 568 ·· Majorana-Weyl spinor, I . 280-281 Mandelstam map, 267 Manifold almost complex, 419 complex, 419 Kahler, 419, 422 Riemann surface, 200, 21 Mapping class group, 212 Mass-shell conditions, 65, membrane, 507 membranes, 457, 490, 525, Modular group heterotic string, 393 single loop amplitude, I SL(l, Z), 197 Moduli space, 212, 227 Multiloop amplitudes closed suing, 20&-210 constant curvature Dehn twists, t98, 213, divergences, 238 Grassmannians, 227, £5:>--'£',>:t~ 242 modUlar group, 212, 226, open bosonic, 200 Schottky groups, 200, 205 Schottky problem, 228 Selberg zeta function, theta functions, 184-185, 195,201,222,L.;)u- L.;) 235-236,241-242 Multiplier, 202, 225 Narain lattice, 463, 507 Neumann coefficients, 270 No-ghost theorem, 90-94 Nonorientable, 181, 190, 367) NS-R model, 10&-111, 119·,< • 136-137, 152, 319 0(6),407 Off-shell,247-248 O(N), 546-550 One [oop amplitude bosonic, 189

Index closed, 195 hctcrotic, 394 slope renonnalization, 191 superstring, 192 On-shell,247-249 Orbifold asymmetric, 448 g(''llcration number three, 441 modular invariance, 438 Wilson lines, 442 Z-orbifold,437

0",(112),1l6 Osp(Sj4), 12 Ol>-P(N j M), I I, 12 Partition function, 187, 190,205 Path integral anomaly, 34 6 muitiloops, 181,210 point particle, 18-25 sIring field theory, 224 trees, 72-78 Period matrix

cohomology, 413 multiloop, 229 Schottky problcm, 229 single loop, 209 Pictures confonmd field theory, 162 F J an d Fl pictures, 1\3---11 4,

137-138

picture changing operators, 32J-324 Planck length, J 0 Poincare duality, 41 6 Polyakov action, 57, 2 10 Pontryllgin class, 35 1 Pregeometry, 32~ Prime fOITTI, 235, 242

5135

Ricci flat, 406, 409, 425 Ri emann-Roch theorem confonnal field theory, 162 index theorem, 357 moduli spacc, 212, 227 supcrmoduli space, 224 Riemann surface, 200, 210, 423 Riemann tensor, 558 S-dua1ity, 465, 4 76, 50 1, 503

S-T-U dual ities, 458 Schottky group, 200, 205, 240 Schottky problem, 227, 2H- 237, 242 Second quantization advantage over first qua ntization, 35,44 BRST field thcory, 291 light cone field theory, 254-290 point particle, 34--44 Selberg zcta fWiction, 225-226, 241 Shapiro-Virnsoro model amplitude, 87 modular invariance, 197 multiloop. 2 10 problems with BRST string field Iheory, 317 si ngle loop. 195 Simplex, 411--412 Singularities hexagon graph, 344, 371 multiioop, 205-207, 2 J 0, 226 single loop, 190-191, 198-200 SL(2, R), 85, ISO, 184 SL(2, Z), 197 SO(N),546 SO(8), 2RO, 579 SO(IO), 161l, 170,175, 176 50(32),364

SO(44),399 Q uadratic diffe ren tial, 215 Quantum gravity, II , 15 Quatemion, jj2-jj4 Ramond sector, 107,--111, 136-137, 31' Regge behav ior, g I Regge slope, 55 Regge trajectory, 55, 62

Sp(N),546,55 1,555 Sp(2g, Z), 229 Spectrum bosonic string, 63 closed string, 65 OS su perstring, 13 1 heterotic string, 384-386 NS-R superstring, 110, 136-137 Spi n(N ),553

586

Index

Spin(32), 373 Spin embedding, 426-427 Spin fielcL 143, 155-158, 169-170 Spin manifold, 352 Spin structure multi loop, 222 single-loop, 222 Spinor covariantly constant, 407, 450 Dirac, 358 Majorana, 127,358 Majorana-Weyl,358 Weyl,358 Spurious states, 66 Stiefel-Whitney class, 350, 352, 369 String field theory BRST,291-331 closed string, 317 duality, 249 equivalence of, 311 four-string interaction, 275-280 light cone, 254-290 objections to string field theory, 248,286 superstrings, 280-286, 324 Witten's, 308-309, 324 Strings, types four-dimensional, 443 heterotic, 383-388 Type r, HAB, 130-132

SU(N), 550-551 SU(N 1M), 566 SU(6, 6), 566 Sugawara fonn, 172 Superconfonnal ghosts, 121, 158, 164,322-323 Superconfonnal group, 109, 122, 137,320 Supercurrent, 106--107 Superfield chiral,570 vector, 569 Supergravity ":i N == 1,10,388,574 N =8,11,574-575,577 problems with finiteness, 12 problems with phenomenology, 12 Yang-MiIls, 576 Supennoduli space, 162, 223

Superspace, 567 Supersymmetry two-dimensional world sne~el.llJ.L.-Lrln Super Yang-Mills, 573, 576 symmetry enhancement. 499 Szego kernel, 235

T -duality, 462-464,476,492, • 519 Tachyon, 61-62, 114, 192 Tangent space, 560 Teichmuller parameter, 203, 212-215,240 Teichmillier space, 212-215 tensionless strings, 499 Theta function heterotic string, 392 multiloop, 201, 230-231, 235-236,241-242 single loop, 184-185, 188, 1 spin structure, 222, 231, 242 Todd class, 353, 357 Trees bosonic, 72-78 confonnal field theory, I 69 heterotic, 389 supersymmetric, 111-117, 1 Twist operator, 85 Type I, lIAB, 130-132, 181, 576 U -duality, 476, 492, 503 Unitarity multiloop, 178 second quantization, 35, single-loop, 178, 181 UsP(N), 341, 546

£""-.<0,""

Veneziano amplitude, 81 Verma module conformal field theory, 150

Index covariant smng field meory, 295 Vierbein, 560 Virdsoro a1gehra, 57, 147-148, 172 Wess-.Zumino action, 572 WCSlt-ZumillO condition, 349, 363 Weyl scaling, I I H, 21 1 Wi lson lines, 433, 451 Yang-MiI1~

theory, 8,3 J 0, 562-563

Zero mode problems, 31~322 Zero norm state, 66-67 Zeta function regularization, 439

587

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