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String Theory Preprint typeset in JHEP style - PAPER VERSION Revised May 6, 2011 An Introduction to String Theory Kevin Wray Abstract: This set of notes is based on the course “Introduction to String Theory” which was taught by Prof. Kostas Skenderis in the spring of 2009 at the University of Amsterdam. We have also drawn on some ideas from the books String Theory and M-Theory (Becker, Becker and Schwarz), Introduction to String Theory (Polchinski), String Theory in a Nutshell (McMahon) and Superstring Theory (Green, Schwarz and Witten), along with the lecture notes of David Tong, sometimes word-for-word. Contents 1. Introduction/Overview 5 1.1 Motivation for String Theory 5 1.2 What is String Theory 8 1.2.1 Types of String Theories 8 1.3 Outline of the Manuscript 9 2. The Bosonic String Action 11 2.1 Classical Action for Point Particles 11 2.2 Classical Action for Relativistic Point Particles 12 2.2.1 Reparametrization Invariance of S˜0 16 2.2.2 Canonical Momenta 18 2.2.3 Varying S˜0 in an Arbitrary Background (Geodesic Equation) 19 2.3 Generalization to p-Branes 19 2.3.1 The String Action 20 2.4 Exercises 24 3. Symmetries and Field Equations of the Bosonic String 26 3.1 Global Symmetries of the Bosonic String Theory Worldsheet 26 3.2 Local Symmetries of the Bosonic String Theory Worldsheet 30 3.3 Field Equations for the Polyakov Action 33 3.4 Solving the Field Equations 36 3.5 Exercises 42 4. Symmetries (Revisited) and Canonical Quantization 45 4.1 Noether’s Method for Generating Conserved Quantities 45 4.2 The Hamiltonian and Energy-Momentum Tensor 48 4.3 Classical Mass Formula for a Bosonic String 51 4.4 Witt Algebra (Classical Virasoro Algebra) 52 4.5 Canonical Quantization of the Bosonic String 54 4.6 Virasoro Algebra 56 4.7 Physical States 58 4.8 Exercises 62 –1– 5. Removing Ghost States and Light-Cone Quantization 64 5.1 Spurious States 65 5.2 Removing the Negative Norm Physical States 68 5.3 Light-Cone Gauge Quantization of the Bosonic String 71 5.3.1 Mass-Shell Condition (Open Bosonic String) 74 5.3.2 Mass Spectrum (Open Bosonic String) 75 5.3.3 Analysis of the Mass Spectrum 77 5.4 Exercises 79 6. Introduction to Conformal Field Theory 80 6.1 Conformal Group in d Dimensions 80 6.2 Conformal Algebra in 2 Dimensions 83 6.3 (Global) Conformal Group in 2 Dimensions 85 6.4 Conformal Field Theories in d Dimensions 86 6.4.1 Constraints of Conformal Invariance in d Dimensions 87 6.5 Conformal Field Theories in 2 Dimensions 89 6.5.1 Constraints of Conformal Invariance in 2 Dimensions 90 6.6 Role of Conformal Field Theories in String Theory 92 6.7 Exercises 94 7. Radial Quantization and Operator Product Expansions 95 7.1 Radial Quantization 95 7.2 Conserved Currents and Symmetry Generators 96 7.3 Operator Product Expansion (OPE) 102 7.4 Exercises 105 8. OPE Redux, the Virasoro Algebra and Physical States 107 8.1 The Free Massless Bosonic Field 107 8.2 Charges of the Conformal Symmetry Current 115 8.3 Representation Theory of the Virasoro Algebra 119 8.4 Conformal Ward Identities 124 8.5 Exercises 127 9. BRST Quantization of the Bosonic String 129 9.1 BRST Quantization in General 129 9.1.1 BRST Quantization: A Primer 130 9.1.2 BRST Ward Identities 135 9.1.3 BRST Cohomology and Physical States 136 9.2 BRST Quantization of the Bosonic String 140 –2– 9.2.1 The Ghost CFT 146 9.2.2 BRST Current and Charge 151 9.2.3 Vacuum of the BRST Quantized String Theory 152 9.2.4 Ghost Current and Charge 154 9.3 Exercises 156 10. Scattering in String Theory 159 10.1 Vertex Operators 159 10.2 Exercises 160 11. Supersymmetric String Theories (Superstrings) 162 11.1 Ramond-Neveu-Schwarz Strings 163 11.2 Global Worldsheet Supersymmetry 166 11.3 Supercurrent and the Super-Virasoro Constraints 168 11.4 Boundary Conditions and Mode Expansions 172 11.4.1 Open RNS Strings 173 11.4.2 Closed RNS Strings 174 11.5 Canonical Quantization of the RNS Superstring Theory 175 11.5.1 R-Sector Ground State VS. NS-Sector Ground State 176 11.5.2 Super-Virasoro Generators (Open Strings) and Physical States 177 11.5.3 Physical State Conditions 180 11.5.4 Removing the Ghost States 181 11.6 Light-Cone Quantization 183 11.6.1 Open RNS String Mass Spectrum 184 11.6.2 GSO Projection 187 11.6.3 Closed RNS String Spectrum 190 11.7 Exercises 193 12. T-Dualities and Dp-Branes 197 12.1 T-Duality and Closed Bosonic Strings 197 12.1.1 Mode Expansion for the Compactified Dimension 199 12.1.2 Mass Formula 200 12.1.3 T-Duality of the Bosonic String 201 12.2 T-Duality and Open Strings 203 12.2.1 Mass Spectrum of Open Strings on Dp-Branes 206 12.3 Branes in Type II Superstring Theory 209 12.4 Dirac-Born-Infeld (DBI) Action 212 12.5 Exercises 214 –3– 13. Effective Actions, Dualities, and M-Theory 216 13.1 Low Energy Effective Actions 216 13.1.1 Conformal Invariance of Sσ and the Einstein Equations 217 13.1.2 Other Couplings of the String 220 13.1.3 Low Energy Effective Action for the Bosonic String Theory 223 13.1.4 Low Energy Effective Action for the Superstring Theories 227 13.2 T-Duality on a Curved Background 229 13.3 S-Duality on the Type IIB Superstring Theories 233 13.3.1 Brane Solutions of Type IIB SUGRA 234 13.3.2 Action of the Brane Solutions Under the Duality Maps 236 13.4 M-Theory 239 13.5 Exercises 243 14. Black Holes in String Theory and the AdS/CFT Correspondence 245 14.1 Black Holes 245 14.1.1 Classical Theory of Black Holes 246 14.1.2 Quantum Theory of Black Holes 248 14.2 Black Holes in String Theory 251 14.2.1 Five-Dimensional Extremal Black Holes 252 14.3 Holographic Principle 255 14.3.1 The AdS/CFT Correspondence 255 A. Residue Theorem 260 B. Wick’s Theorem 261 C. Solutions to Exercises 262 –4– 1. Introduction/Overview 1.1 Motivation for String Theory Presently we understand that physics can be described by four forces: gravity, elec- tromagnetism, the weak force, responsible for beta decays and the strong force which binds quarks into protons and neutrons. We, that is most physicists, believe that we understand all of these forces except for gravity. Here we use the word “understand” loosely, in the sense that we know what the Lagrangian is which describes how these forces induce dynamics on matter, and at least in principle we know how to calculate using these Lagrangians to make well defined predictions. But gravity we only under- stand partially. Clearly we understand gravity classically (meaning in the ~ = 0 limit). As long as we dont ask questions about how gravity behaves at very short distances (we will call the relevant breakdown distance the Planck scale) we have no problems calculating and making predictions for gravitational interactions. Sometimes it is said that we don’t understand how to fuse quantum mechanics and GR. This statement is really incorrect, though for “NY times purposes”, it’s fine. In fact we understand perfectly well how to include quantum mechanical effects into gravity, as long we we dont ask questions about whats going on at distances, less than the Planck length. This is not true for the other forces. That is, for the other forces we know how to include quantum effects, at all distance scales. So, while we have a quantum mechanical understanding of gravity, we don’t have a complete theory of quantum gravity. The sad part about this is that all the really interesting questions we want to ask about gravity, e.g. what’s the “big bang”, what happens at the singularity of black hole, are left unanswered. What is it, exactly, that goes wrong with gravity at scales shorter than the Planck length? The answer is, it is not “renormalizable”. What does “renormalizable” mean? This is really a technical question which needs to be discussed within the context of quantum field theory, but we can gain a very simple intuitive understanding from classical electromagnetism. So, to begin, consider an electron in isolation. The total energy of the electron is given by e2 E mˆ + d3x E~ 2 mˆ +4π r2dr . (1.1) T ∼ | | ∼ r4 Z Z Now, this integral diverges at the lower endpoint of r = 0. We can reconcile this divergence by cutting it off at some scale Λ and when were done well see if can take the limit where the cut-off goes to zero. So our results for the total energy of an electron is now given by e2 E mˆ + C . (1.2) T ∼ Λ –5– Clearly the second term dominates in the limit we are interested in. So apparently even classical electrodynamics is sick. Well not really, the point is that weve been rather sloppy. When we writem ˆ what do we mean? Naively we mean what we would call the mass of the electron which we could measure say, by looking at the deflection of a moving electron in a magnetic field. But we dont measurem ˆ , we measure ET , that is the inertial mass should include the electromagnetic self-energy.
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