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Introduction to String Theory, Lecture Notes Introduction to String Theory Lecture Notes ETH Z¨urich, HS13 Prof. N. Beisert, Dr. J. Brodel¨ c 2014 Niklas Beisert, Johannes Br¨odel,ETH Zurich This document as well as its parts is protected by copyright. Reproduction of any part in any form without prior written consent of the author is permissible only for private, scientific and non-commercial use. Contents 0 Overview 6 0.1 Contents . .6 0.2 References . .6 1 Introduction 1.1 1.1 Definition . 1.1 1.2 Motivation . 1.1 1.3 Some Conventions . 1.5 2 Relativistic Point Particle 2.1 2.1 Non-Relativistic Actions . 2.1 2.2 Worldline Action . 2.2 2.3 Polynomial Action . 2.3 2.4 Various Gauges . 2.4 2.5 Quantisation . 2.4 2.6 Interactions . 2.5 2.7 Conclusions . 2.6 3 Classical Bosonic String 3.1 3.1 Nambu{Goto Action . 3.1 3.2 Polyakov Action . 3.2 3.3 Conformal Gauge . 3.3 3.4 Solution on the Light Cone . 3.4 3.5 Closed String Modes . 3.4 3.6 Hamiltonian Formalism . 3.7 4 String Quantisation 4.1 4.1 Canonical Quantisation . 4.1 4.2 Light Cone Quantisation . 4.2 4.3 String Spectrum . 4.3 4.4 Anomalies . 4.6 4.5 Covariant Quantisation . 4.7 5 Compactification and T-Duality 5.1 5.1 Kaluza{Klein Modes . 5.1 5.2 Winding Modes . 5.2 5.3 T-Duality . 5.3 5.4 General Compactifications . 5.5 6 Open Strings and D-Branes 6.1 6.1 Neumann Boundary Conditions . 6.1 6.2 Solutions and Spectrum . 6.1 6.3 Dirichlet Boundary Conditions . 6.3 6.4 Multiple Branes . 6.5 7 Conformal Field Theory 7.1 3 7.1 Conformal Transformations . 7.1 7.2 Conformal Correlators . 7.3 7.3 Local Operators . 7.5 7.4 Operator Product Expansion . 7.7 7.5 Stress-Energy Tensor . 7.9 8 String Scattering 8.1 8.1 Vertex Operators . 8.1 8.2 Veneziano Amplitude . 8.3 8.3 String Loops . 8.4 9 General Relativity Basics 9.1 9.1 Differential Geometry . 9.1 9.2 Riemannian Geometry . 9.5 9.3 General Relativity . 9.7 10 String Backgrounds 10.1 10.1 Graviton Vertex Operator . 10.1 10.2 Curved Backgrounds . 10.2 10.3 Form Field and Dilaton . 10.6 10.4 Open Strings . 10.8 10.5 Two-Form Field of a String . 10.10 11 Superstrings 11.1 11.1 Supersymmetry . 11.1 11.2 Green{Schwarz Superstring . 11.5 11.3 Ramond{Neveu{Schwarz Superstring . 11.7 11.4 Branes . 11.10 11.5 Other Superstrings . 11.11 12 Effective Field Theory 12.1 12.1 Effective Action and Compactifications . 12.1 12.2 Open Strings . 12.3 12.3 Closed Strings . 12.4 12.4 Relations Between String Amplitudes . 12.5 13 String Dualities 13.1 13.1 T-Duality . 13.1 13.2 Strong/Weak Coupling Duality: S-Duality . 13.4 13.3 Strong Coupling Limits . 13.5 13.4 M-Theory . 13.6 14 String Theory and the Standard Model 14.1 14.1 The Real World . 14.1 14.2 Geometry of Toroidal Manifolds and Orbifolds . 14.2 14.3 Calabi{Yau Compactification of D = 10 Supergravity . 14.4 14.4 String Theory as a Phenomenological Model . 14.5 4 15 AdS/CFT Correspondence 15.1 15.1 Stack of D3-Branes . 15.1 15.2 Anti-de Sitter Geometry . 15.2 15.3 N = 4 Super Yang{Mills . 15.3 15.4 Tests . 15.3 5 Introduction to String Theory Chapter 0 ETH Z¨urich,HS13 Prof. N. Beisert, Dr. J. Br¨odel 22. 12. 2013 0 Overview String theory is an attempt to quantise gravity and unite it with the other fundamental forces of nature. It combines many interesting topics of (quantum) field theory in two and higher dimensions. This course gives an introduction to the basics of string theory. 0.1 Contents 1. Introduction (1 lecture) 2. Relativistic Point Particle (2 lectures) 3. Classical Bosonic String (3 lectures) 4. String Quantisation (4 lectures) 5. Compactification and T-Duality (2 lectures) 6. Open Strings and D-Branes (2 lectures) 7. Conformal Field Theory (4 lectures) 8. String Scattering (2 lectures) 9. General Relativity Basics (2 lectures) 10. String Backgrounds (3 lectures) 11. Superstrings and Supersymmetry (4 lectures) 12. Effective Field Theory (3 lectures) 13. String Dualities (3 lectures) 14. String Theory and the Standard Model (2 lectures) 15. AdS/CFT Correspondence (2 lectures) Indicated are the approximate number of 45-minute lectures. Altogether, the course consists of 39 lectures. 0.2 References There are many text books and lecture notes on string theory. Here is a selection of well-known ones: • classic: M. Green, J.H. Schwarz and E. Witten, \Superstring Theory" (2 volumes), Cambridge University Press (1988) • alternative: D. L¨ust, S. Theisen, \Lectures on String Theory", Springer (1989). • new edition: R. Blumenhagen, D. L¨ust,S. Theisen, \Basic Concepts of String Theory", Springer (2012). • standard: J. Polchinski, \String Theory" (2 volumes), Cambridge University Press (1998) • basic: B. Zwiebach, \A First Course in String Theory", Cambridge University Press (2004/2009) 6 • recent: K. Becker, M. Becker, J.H. Schwarz, \String Theory and M-Theory: A Modern Introduction", Cambridge University Press (2007) • online: D. Tong, \String Theory", lecture notes, http://arxiv.org/abs/0908.0333 • ... 7 Introduction to String Theory Chapter 1 ETH Z¨urich,HS13 Prof. N. Beisert, Dr. J. Br¨odel 22. 12. 2013 1 Introduction 1.1 Definition String theory describes the mechanics of one-dimensional extended objects in an ambient space. (1.1) Some features: • Strings have tension: (1.2) • Strings have no inner structure: but not (1.3) • Several pieces of string can interact: ! (1.4) • Strings can be classical or quantum: vs. (1.5) • Strings can be open or closed: vs. (1.6) 1.2 Motivation Why study strings? Extended Objects. We know a lot about the mechanics of point particles. It is natural to study strings next. Or even higher-dimensional extended objects like membranes. (1.7) particle string membrane 1.1 These objects are snapshots at fixed time t. Introduce the worldvolume as the volume of spacetime occupied by the object: (1.8) worldline worldsheet worldvolume The worldsheet of a string is two-dimensional. In fact, there is a great similarity between strings and static soap films. Quantum Gravity. String theory offers a solution to the problem of quantum gravity (QG). Let us try to sketch the problem of quantum gravity with as little reference to quantum field theory (QFT) as possible. There are two established classical1 gravity theories: • Newtonian Gravity (non-relativistic) • General Relativity (GR, relativistic, geometry of spacetime) We know that nature is quantum mechanical, therefore gravity must also be quantum for consistency with the other fundamental forces. In practice, the effects of QG hardly play a role except for considerations of the early universe and for black hole radiation. Field quantisation introduces quanta (particles): • electromagnetism: photon • strong nuclear forces: gluons • gravity: graviton • matter fields: electrons, quarks, neutrinos, . These particles interact through vertices (Feynman rules) which can be composed to more complex interaction processes (Feynman graphs). The Standard Model (SM) of particle physics has relatively simple set of rules (qualitatively) S = + g + g2 : (1.9) Here represents a coupling constant. 1Here and in the following the term \classical" will refer to the absence of quantum effects. Classical theories can be either non-relativistic or relativistic. 1.2 Conversely, Einstein gravity has infinitely many vertices which are governed by Newton's constant G p S = + G + G + G3=2 + G2 + :::: (1.10) In fact, we can introduce additional couplings ck: p 3=2 G + (G + c4) + (G + c5) + :::: (1.11) This is perfectly consistent with the assumptions of GR, except that the additional terms introduce higher-derivative corrections to the Einstein equations. Classically 2 we do not need the ck, but in QFT we do. The point is that loops in Feynman graphs generate divergences, e.g.: = 1: (1.12) In QFT we have to sum up all competing processes, e.g.:3 2 3 (G + c4) + G + G + :::: (1.13) In this sum, we can absorb the divergence into a redefinition of the (new) coupling 3 constant c4 = −G 1 + c4;ren. This process is called renormalisation. All is well now, the divergences are gone, but there is no good way to set the renormalised c4;ren to zero (or any other distinguished value). Unfortunately, 2A general principle of QFT is that we need to include all permissible interaction terms which are not excluded by some principle, typically symmetries or a power counting scheme. 3A curious fact is that quantum gravity does not produce a divergence in the one-loop graph (G2 term). 1.3 4 cancellation of all divergences requires infinitely many ck's. The quantisation of Einstein gravity introduces infinitely many adjustable parameters. All parameters have to be known (measured) in order to have a predictive description of nature. This renders the theory non-predictive! The only good prediction is at sufficiently low energies much below the Planck scale: There the theory is approximated well by GR with only G as the coupling constant. What does string theory have to do with it? Quantum string theory turns out to contain particles which gravitons in many ways. Moreover, string theory does not generate divergences; it is a finite theory! Finally, string theory has just two fundamental coupling constants. Is all well now!? Almost, many more couplings may be hiding in the description of the vacuum state which is relevant when actual physics is to be addressed.
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