Matrix Models of String Theory

422 Pages·2018·21.808 MB·English

by  Prof Badis Ydri

Matrix Models of String Theory

Matrix Models of String Theory
Badis Ydri
Annaba University, Annaba, Algeria
IOP Publishing, Bristol, UK
ªIOPPublishingLtd2018
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ISBN 978-0-7503-1726-9(ebook)
ISBN 978-0-7503-1724-5(print)
ISBN 978-0-7503-1725-2(mobi)
DOI 10.1088/978-0-7503-1726-9
Version:20181001
IOPExpandingPhysics
ISSN2053-2563(online)
ISSN2054-7315(print)
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PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London
IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK
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To my father for his continuous support throughout his life …
Saad Ydri
1943–2015
Also to my …
Nour

Contents
Author biography xiii
1 Introduction 1-1
References 1-4
Part I String theory
2 String theory 2-1
2.1 Actions, symmetries and solutions 2-1
2.1.1 Polyakov action 2-1
2.1.2 Boundary conditions and symmetries 2-3
2.1.3 Closed string solutions 2-5
2.1.4 Open string solutions 2-9
2.2 Canonical quantization and Virasoro algebra 2-10
2.2.1 Canonical quantization 2-10
2.2.2 Virasoro algebra 2-13
2.3 Spurious states and critical strings 2-17
2.3.1 Spurious states 2-17
2.3.2 Critical strings 2-19
2.4 Lightcone gauge quantization 2-20
2.4.1 Critical strings 2-20
2.4.2 Rigorous proof 2-23
2.4.3 Spectrum 2-23
2.5 Exercises 2-24
References 2-42
3 Polyakov path integral 3-1
3.1 Gauge fixing and Fadeev–Popov ghosts 3-1
3.2 The energy–momentum tensor 3-4
3.3 Quantization of the ghosts 3-7
3.3.1 The open string 3-7
3.3.2 The closed string 3-8
3.3.3 Virasoro generators 3-9
3.4 BRST symmetry 3-12
3.4.1 Gauge theory 3-12
3.4.2 General case 3-13
vii
MatrixModelsofStringTheory
3.4.3 String theory 3-15
3.5 Exercises 3-22
References 3-22
4 Introduction to conformal field theory 4-1
4.1 The conformal groupsSO(p + 1, q + 1) 4-1
4.2 The conformal group in two dimensions 4-3
4.3 The energy–momentum tensor 4-6
4.4 The operator product expansion 4-8
4.5 Conformal field theory and BRST quantization 4-18
4.6 Representation theory of the Virasoro algebra 4-27
4.6.1 Virasoro algebra revisited 4-27
4.6.2 Overview of representation theory 4-28
4.7 Theorem 4-31
4.8 Exercises 4-32
References 4-35
5 Superstring theory essentials 5-1
5.1 The superparticle 5-1
5.1.1 Bosonic particle 5-1
5.1.2 The superparticle 5-2
5.1.3 Action 5-3
5.1.4 The κ-symmetry 5-5
5.2 The Green–Schwarz superstring 5-6
5.3 The Ramond–Neveu–Schwarz superstring 5-9
5.3.1 Supersymmetric action on the worldsheet 5-9
5.3.2 Energy–momentum tensor and supercurrent 5-10
5.3.3 Super-Virasoro constraints 5-14
5.3.4 Boundary conditions (Ramond and Neveu–Schwarz) 5-16
5.4 Canonical quantization 5-18
5.4.1 Commutation relations 5-18
5.4.2 Ramond (fermionic) and Neveu–Schwarz (bosonic) open string 5-20
sectors
5.4.3 Super-Virasoro algebra 5-23
5.5 The light cone/path integral quantization 5-25
5.5.1 The light cone quantization and critical dimension 5-25
5.5.2 The GSO conditions and superstring spectrum 5-26
5.6 Other very important topics 5-31
viii
MatrixModelsofStringTheory
5.7 Exercises 5-31
References 5-34
Part II Matrix string theory
6 A lightning introduction to superstring theory and some 6-1
related topics
6.1 Quantum black holes 6-1
6.1.1 Schwarzschild black hole 6-2
6.1.2 Hawking temperature 6-2
6.1.3 Page curve and unitarity 6-5
6.1.4 Information loss problem 6-5
6.1.5 Thermodynamics 6-7
6.2 Some string theory and conformal field theory 6-7
6.2.1 The conformal anomaly 6-7
6.2.2 The operator product expansion 6-9
6.2.3 The bc CFT 6-12
6.2.4 The superconformal field theory 6-14
6.2.5 Vertex operators 6-14
6.2.6 Background fields 6-15
6.2.7 Beta function: finiteness and Weyl invariance 6-17
6.2.8 String perturbation expansions 6-19
6.2.9 Spectrum of type II string theory 6-20
6.3 On Dp-branes and T-duality 6-23
6.3.1 Introductory remarks 6-23
6.3.2 Coupling to abelian gauge fields 6-24
6.3.3 Symmetry under the exchange of momentum and winding 6-25
6.3.4 Symmetry under the exchange of Neumann and Dirichlet 6-27
6.3.5 Chan–Paton factors 6-28
6.3.6 Electromagnetism on a circle and Wislon lines 6-29
6.3.7 The D-branes on the dual circle 6-31
6.4 Quantum gravity in two dimensions 6-33
6.4.1 Dynamical triangulation 6-34
6.4.2 Matrix models of D = 0 string theory 6-35
6.4.3 Matrix models of D = 1 string theory 6-37
6.4.4 Preliminary synthesis 6-38
6.5 Exercise 6-39
References 6-39
ix