INTRODUCTION TO SUPERSTRING THEORY
INTRODUCTION 1 2 HISTORICAL PERSPECTIVE 2 3 CLASSICAL STRING THEORY 5 4 QUANTIZATION OF THE BOSONIC STRING 19 5 INTERACTIONS AND LOOP AMPLITUDES 32 6 CONFORMAL FIELD THEORY 34 7 CFT ON THE TORUS 71 8 SCATTERING AMPLITUDES AND VERTEX OPERATORS OF BOSONIC STRINGS 94 9 STRINGS IN BACKGROUND FIELDS AND LOW ENERGY EFFECTIVE ACTIONS 98 10 SUPERSTRINGS AND SUPERSYMMETRY 100 11 ANOMALIES 118 12 COMPACTIFICATION AND SUPERSYMMETRY BREAKING 126 13 LOOP CORRECTIONS TO EFFECTIVE COUPLINGS IN STRING THEORY 155 14 NON PERTURBATIVE STRING DUALITIES A FOREWORD 175 15 OUTLOOK 207 REFERENCES 209 CONTENTS 1 INTRODUCTION STRING THEORY HAS BEEN THE LEADING CANDIDATE OVER THE PAST YEARS FOR A THEORY THAT CONSIS TENTLY UNI ES ALL FUNDAMENTAL FORCES OF NATURE INCLUDING GRAVITY IN A SENSE THE THEORY PREDICTS GRAVITY AND GAUGE SYMMETRY AROUND AT SPACE MOREOVER THE THEORY IS UV NITE THE ELEMENTARY OBJECTS ARE ONE DIMENSIONAL STRINGS WHOSE VIBRATION MODES SHOULD CORRESPOND TO THE USUAL ELEMENTARY PARTICLES AT DISTANCES LARGE WITH RESPECT TO THE SIZE OF THE STRINGS THE LOW ENERGY EXCITATIONS CAN BE DESCRIBED BY AN E ECTIVE ELD THEORY THUS CONTACT CAN BE ESTABLISHED WITH QUANTUM ELD THEORY WHICH TURNED OUT TO BE SUCCESSFUL IN DESCRIBING THE DYNAMICS OF THE REAL WORLD AT LOW ENERGY I WILL TRY TO EXPLAIN HERE THE BASIC STRUCTURE OF STRING THEORY ITS PREDICTIONS AND PROB LEMS IN CHAPTER 2 THE EVOLUTION OF STRING THEORY IS TRACED FROM A THEORY INITIALLY BUILT TO DESCRIBE HADRONS TO A THEORY OF EVERYTHING IN CHAPTER 3 A DESCRIPTION OF CLASSICAL BOSONIC STRING THEORY IS GIVEN THE OSCILLATION MODES OF THE STRING ARE DESCRIBED PREPARING THE SCENE FOR QUANTIZATION IN CHAPTER 4 THE QUANTIZATION OF THE BOSONIC STRING IS DESCRIBED ALL THREE DI ERENT QUANTIZATION PROCEDURES ARE PRESENTED TO VARYING DEPTH SINCE IN EACH ONE SOME SPECI C PROPERTIES ARE MORE TRANSPARENT THAN IN OTHERS I THUS DESCRIBE THE OLD COVARIANT QUANTIZATION THE LIGHT CONE QUANTIZATION AND THE MODERN PATH INTEGRAL QUANTIZATION IN CHAPTER 6 A CONCISE INTRODUCTION IS GIVEN TO THE CENTRAL CONCEPTS OF CONFORMAL ELD THEORY SINCE IT IS THE BASIC TOOL IN DISCUSSING RST QUANTIZED STRING THEORY IN CHAPTER 8 THE CALCULATION OF SCATTERING AMPLITUDES IS DESCRIBED IN CHAPTER 9 THE LOW ENERGY E ECTIVE ACTION FOR THE MASSLESS MODES IS DESCRIBED IN CHAPTER 10 SUPERSTRINGS ARE INTRODUCED THEY PROVIDE SPACETIME FERMIONS AND REAL IZE SUPERSYMMETRY IN SPACETIME AND ON THE WORLD SHEET I GO THROUGH QUANTIZATION AGAIN AND DESCRIBE THE DI ERENT SUPERSYMMETRIC STRING THEORIES IN TEN DIMENSIONS IN CHAPTER 11 GAUGE AND GRAVITATIONAL ANOMALIES ARE DISCUSSED IN PARTICULAR IT IS SHOWN THAT THE SUPER STRING THEORIES ARE ANOMALY FREE IN CHAPTER 12 COMPACTI CATIONS OF THE TEN DIMENSIONAL SUPERSTRING THEORIES ARE DESCRIBED SUPERSYMMETRY BREAKING IS ALSO DISCUSSED IN THIS CON TEXT IN CHAPTER 13 I DESCRIBE HOW TO CALCULATE LOOP CORRECTIONS TO E ECTIVE COUPLING CONSTANTS THIS IS VERY IMPORTANT FOR COMPARING STRING THEORY PREDICTIONS AT LOW ENERGY WITH THE REAL WORLD IN CHAPTER 14 A BRIEF INTRODUCTION TO NON PERTURBATIVE STRING CON NECTIONS AND NON PERTURBATIVE E ECTS IS GIVEN THIS IS A FAST CHANGING SUBJECT AND I HAVE JUST INCLUDED SOME BASICS AS WELL AS TOOLS SO THAT THE READER ORIENTS HIM HER SELF IN THE WEB OF DUALITY CONNECTIONS FINALLY IN CHAPTER 15 A BRIEF OUTLOOK AND FUTURE PROBLEMS ARE PRESENTED I HAVE ADDED A NUMBER OF APPENDICES TO MAKE SEVERAL TECHNICAL DISCUSSIONS SELF CONTAINED 1 IN APPENDIX A USEFUL INFORMATION ON THE ELLIPTIC FUNCTIONS IS INCLUDED IN APPENDIX B I REDERIVE THE VARIOUS LATTICE SUMS THAT APPEAR IN TOROIDAL COMPACTI CATIONS IN APPENDIX C THE KALUZA KLEIN ANSATZ IS DESCRIBED USED TO OBTAIN ACTIONS IN LOWER DIMENSIONS AFTER TOROIDAL COMPACTI CATION IN APPENDIX D SOME FACTS ARE PRESENTED ABOUT FOUR DIMENSIONAL LOCALLY SUPERSYMMETRIC THEORIES WITH N 1 2 4 SUPERSYMMETRY IN APPENDIX E BPS STATES ARE DESCRIBED ALONG WITH THEIR REPRESENTATION THEORY AND HELICITY SUPERTRACE FORMULAE THAT CAN BE USED TO TRACE THEIR APPEARANCE IN A SUPERSYMMETRIC THEORY IN APPENDIX F FACTS ABOUT ELLIPTIC MODULAR FORMS ARE PRESENTED WHICH ARE USEFUL IN MANY CONTEXTS NOTABLY IN THE ONE LOOP COMPUTATION OF THRESHOLDS AND COUNTING OF BPS MULTIPLICITIES IN AP PENDIX G I PRESENT THE COMPUTATION OF HELICITY GENERATING STRING PARTITION FUNCTIONS AND THE ASSOCIATED CALCULATION OF BPS MULTIPLICITIES FINALLY IN APPENDIX H I BRIE Y REVIEW ELECTRIC MAGNETIC DUALITY IN FOUR DIMENSIONS I HAVE NOT TRIED TO BE COMPLETE IN MY REFERENCING THE FOCUS WAS TO PROVIDE IN MOST CASES APPROPRIATE REVIEWS FOR FURTHER READING ONLY IN THE LAST CHAPTER WHICH COVERS VERY RECENT TOPICS I DO MOSTLY REFER TO ORIGINAL PAPERS BECAUSE OF THE SCARCITY OF RELEVANT REVIEWS 2 HISTORICAL PERSPECTIVE IN THE SIXTIES PHYSICISTS TRIED TO MAKE SENSE OF A BIG BULK OF EXPERIMENTAL DATA RELEVANT TO THE STRONG INTERACTION THERE WERE LOTS OF PARTICLES OR RESONANCES AND THE SITUATION COULD BEST BE DESCRIBED AS CHAOTIC THERE WERE SOME REGULARITIES OBSERVED THOUGH ALMOST LINEAR REGGE BEHAVIOR IT WAS NOTICED THAT THE LARGE NUMBER OF RESONANCES COULD BE NICELY PUT ON ALMOST STRAIGHT LINES BY PLOTTING THEIR MASS VERSUS THEIR SPIN M2 J 2 1 WITH 1 GEV 2 AND THIS RELATION WAS CHECKED UP TO J 11 2 S T DUALITY IF WE CONSIDER A SCATTERING AMPLITUDE OF TWO TWO HADRONS 1 2 3 4 THEN IT CAN BE DESCRIBED BY THE MANDELSTAM INVARIANTS S P1 P2 2 T P2 P3 2 U P1 P3 2 2 2 WITH S T U P I M2 I WE ARE USING A METRIC WITH SIGNATURE SUCH AN AMPLI TUDE DEPENDS ON THE AVOR QUANTUM NUMBERS OF HADRONS FOR EXAMPLE SU 3 CONSIDER THE AVOR PART WHICH IS CYCLICALLY SYMMETRIC IN AVOR SPACE FOR THE FULL AMPLITUDE TO BE SYMMETRIC IT MUST ALSO BE CYCLICALLY SYMMETRIC IN THE MOMENTA PI THIS SYMMETRY AMOUNTS TO THE INTERCHANGE T S THUS THE AMPLITUDE SHOULD SATISFY A S T A T S CONSIDER A T CHANNEL CONTRIBUTION DUE TO THE EXCHANGE OF A SPIN J PARTICLE OF MASS M 2 THEN AT HIGH ENERGY AJ S T S J T M2 2 3 THUS THIS PARTIAL AMPLITUDE INCREASES WITH S AND ITS BEHAVIOR BECOMES WORSE FOR LARGE VALUES OF J IF ONE SEWS AMPLITUDES OF THIS FORM TOGETHER TO MAKE A LOOP AMPLITUDE THEN THERE ARE UNCONTROLLABLE UV DIVERGENCES FOR J 1 ANY NITE SUM OF AMPLITUDES OF THE FORM 2 3 HAS THIS BAD UV BEHAVIOR HOWEVER IF ONE ALLOWS AN IN NITE NUMBER OF TERMS THEN IT IS CONCEIVABLE THAT THE UV BEHAVIOR MIGHT BE DI ERENT MOREOVER SUCH A NITE SUM HAS NO S CHANNEL POLES A PROPOSAL FOR SUCH A DUAL AMPLITUDE WAS MADE BY VENEZIANO 1 A S T S T S T 2 4 WHERE IS THE STANDARD FUNCTION AND S 0 S 2 5 BY USING THE STANDARD PROPERTIES OF THE FUNCTION IT CAN BE CHECKED THAT THE AMPLITUDE 2 4 HAS AN IN NITE NUMBER OF S T CHANNEL POLES A S T X N 0 S 1 S N N 1 T N 2 6 IN THIS EXPANSION THE S T INTERCHANGE SYMMETRY OF 2 4 IS NOT MANIFEST THE POLES IN 2 6 CORRESPOND TO THE EXCHANGE OF AN IN NITE NUMBER OF PARTICLES OF MASS M2 N 0 AND HIGH SPINS IT CAN ALSO BE CHECKED THAT THE HIGH ENERGY BEHAVIOR OF THE VENEZIANO AMPLITUDE IS SOFTER THAN ANY LOCAL QUANTUM ELD THEORY AMPLITUDE AND THE IN NITE NUMBER OF POLES IS CRUCIAL FOR THIS IT WAS SUBSEQUENTLY REALIZED BY NAMBU AND GOTO THAT SUCH AMPLITUDES CAME OUT OF THE ORIES OF RELATIVISTIC STRINGS HOWEVER SUCH THEORIES HAD SEVERAL SHORTCOMINGS IN EXPLAINING THE DYNAMICS OF STRONG INTERACTIONS ALL OF THEM SEEMED TO PREDICT A TACHYON SEVERAL OF THEM SEEMED TO CONTAIN A MASSLESS SPIN 2 PARTICLE THAT WAS IMPOSSIBLE TO GET RID OF ALL OF THEM SEEMED TO REQUIRE A SPACETIME DIMENSION OF 26 IN ORDER NOT TO BREAK LORENTZ INVARIANCE AT THE QUANTUM LEVEL THEY CONTAINED ONLY BOSONS AT THE SAME TIME EXPERIMENTAL DATA FROM SLAC SHOWED THAT AT EVEN HIGHER ENERGIES HADRONS HAVE A POINT LIKE STRUCTURE THIS OPENED THE WAY FOR QUANTUM CHROMODYNAMICS AS THE CORRECT THEORY THAT DESCRIBES STRONG INTERACTIONS 3