Strings Branes and Superstring Theory - S. Forste

Strings Branes and Superstring Theory - S. Forste -...

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arXiv:hep-th/0110055 v3 3 Jan 2002 Strings, Branes and Extra Dimensions Stefan F¨ orste Physikalisches Institut, Universit¨ at Bonn Nussallee 12, D-53115 Bonn, Germany Abstract This review is devoted to strings and branes. Firstly, perturbative string theory is introduced. The appearance of various types of branes is discussed. These include orbifold fixed planes, D-branes and orientifold planes. The connection to BPS vacua of supergravity is presented afterwards. As applications, we outline the role of branes in string dualities, field theory dualities, the AdS/CFT correspondence and scenarios where the string scale is at a TeV. Some issues of warped compactifications are also addressed. These comprise corrections to gravitational interactions as well as the cosmological constant problem.
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Contents 1 Introduction 1 2 Perturbative description of branes 4 2.1 The Fundamental String . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Worldsheet Actions . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1.1 The closed bosonic string . . . . . . . . . . . . . . . . 4 2.1.1.2 Worldsheet supersymmetry . . . . . . . . . . . . . . . 7 2.1.1.3 Space-time supersymmetric string . . . . . . . . . . . 10 2.1.2 Quantization of the fundamental string . . . . . . . . . . . . . 13 2.1.2.1 The closed bosonic string . . . . . . . . . . . . . . . . 14 2.1.2.2 Type II strings . . . . . . . . . . . . . . . . . . . . . . 19 2.1.2.3 The heterotic string . . . . . . . . . . . . . . . . . . . 25 2.1.3 Strings in non-trivial backgrounds . . . . . . . . . . . . . . . . 28 2.1.4 Perturbative expansion and effective actions . . . . . . . . . . . 36 2.1.5 Toroidal Compactification and T-duality . . . . . . . . . . . . . 42 2.1.5.1 Kaluza-Klein compactification of a scalar field . . . . 42 2.1.5.2 The bosonic string on a circle . . . . . . . . . . . . . . 43 2.1.5.3 T-duality in non trivial backgrounds . . . . . . . . . . 46 2.1.5.4 T-duality for superstrings . . . . . . . . . . . . . . . . 48 2.2 Orbifold fixed planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.1 The bosonic string on an orbicircle . . . . . . . . . . . . . . . . 51 2.2.2 Type IIB on T 4 / Z 2 . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.3 Comparison with type IIB on K 3 . . . . . . . . . . . . . . . . . 57 2.3 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.1 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . 61 2.3.1.2 Quantization of the open string ending on a single D- brane . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3.1.3 Number of ND directions and GSO projection . . . . 67 2.3.1.4 Multiple parallel D-branes – Chan Paton factors . . . 68 2.3.2 D-brane interactions . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3.3 D-brane actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.3.3.1 Open strings in non-trivial backgrounds . . . . . . . . 79 2.3.3.2 Toroidal compactification and T-duality for open strings 85 2.3.3.3 RR fields . . . . . . . . . . . . . . . . . . . . . . . . . 91 i
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CONTENTS ii 2.3.3.4 Noncommutative geometry . . . . . . . . . . . . . . . 93 2.4 Orientifold fixed planes . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4.1 Unoriented closed strings . . . . . . . . . . . . . . . . . . . . . 98 2.4.2 O-plane interactions . . . . . . . . . . . . . . . . . . . . . . . . 101 2.4.2.1 O-plane/O-plane interaction, or the Klein bottle . . . 101 2.4.2.2 D-brane/O-plane interaction, or the M¨ obius strip . . 107 2.4.3 Compactifying the transverse dimensions . . . . . . . . . . . . 112 2.4.3.1 Type I/type I 0 strings . . . . . . . . . . . . . . . . . . 113 2.4.3.2 Orbifold compactification . . . . . . . . . . . . . . . . 116 3 Non-Perturbative description of branes 124 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.2 Universal Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.2.1 The fundamental string . . . . . . . . . . . . . . . . . . . . . . 127 3.2.2 The NS five brane . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.3 Type II branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4 Applications 138 4.1 String dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.2 Dualities in Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.3 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.3.1 The conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.3.2 Wilson loop computation . . . . . . . . . . . . . . . . . . . . . 149 4.3.2.1 Classical approximation . . . . . . . . . . . . . . . . . 149 4.3.2.2 Stringy corrections . . . . . . . . . . . . . . . . . . . . 154 4.4 Strings at a TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.4.1 Corrections to Newton’s law . . . . . . . . . . . . . . . . . . . . 165 5 Brane world setups 167 5.1 The Randall Sundrum models . . . . . . . . . . . . . . . . . . . . . . . 167 5.1.1 The RS1 model with two branes . . . . . . . . . . . . . . . . . 167 5.1.1.1 A proposal for radion stabilization . . . . . . . . . . . 171 5.1.2 The RS2 model with one brane . . . . . . . . . . . . . . . . . . 174 5.1.2.1 Corrections to Newton’s law . . . . . . . . . . . . . . 175 5.1.2.2 ... and the holographic principle . . . . . . . . . . . . 179 5.1.2.3 The RS2 model with two branes . . . . . . . . . . . . 181 5.2
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