Matrix Models of String Theory

by Badis Ydri

Overview

Written for postgraduate students as a pedagogical introduction to string theory.  Extending beyond an introductory review of the subject, it encompasses key analytical and numerical tools, as well as useful physical models in applications. The book is augmented with numerous codes in addition to problems and exercises.

Product Details

ISBN-13:	9780750317245
Publisher:	Iop Publishing Ltd
Publication date:	12/31/2018
Series:	IPH001
Pages:	500
Product dimensions:	7.22(w) x 10.42(h) x 1.13(d)

About the Author

Badis Ydri is currently a professor of theoretical particle physics at the Institute of Physics, Annaba University, Algeria. He is also a research associate at the Dublin Institute for Advanced Studies, Ireland, and a regular ICTP associate at the Abdus Salam Center for Theoretical Physics, Trieste, Italy.

Table of 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

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 groups SO(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 k-symmetry 5-5

5.2 The Green-Schwarz superstring 5-6

5.3 The Ramond-Neveu-Schwarz super string 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 sectors 5-20

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

5.7 Exercises 5-31

References 5-34

Part II Matrix string theory

6 A lightning introduction to superstring theory and some related topics 6-1

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 be 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

7 M-(atrix) theory and matrix string theory 7-1

7.1 The quantized membrane 7-1

7.2 The IKKT model or type IIB matrix model 7-6

7.3 The BFSS model from dimensional reduction 7-9

7.4 Introducing gauge/gravity duality 7-12

7.4.1 Dimensional reduction to p + 1 dimensions 7-12

7.4.2 Dp-branes revisited 7-13

7.4.3 The corresponding gravity dual 7-16

7.5 Black hole unitarity from M-theory 7-19

7.5.1 The black 0-brane 7-19

7.5.2 Supergravity in 11 dimensions and M-wave solution 7-20

7.5.3 Type IIA string theory at strong couplings is M-Theory 7-26

7.6 M-theory prediction for quantum black holes 7-28

7.6.1 Quantum gravity corrections 7-28

7.6.2 Non-perturbative tests of the gauge/gravity duality 7-31

7.7 Matrix string theory 7-34

7.8 The black hole and confinement phase transitions 7-37

7.8.1 The black-hole/black-string phase transition 7-37

7.8.2 The confinement/deconfinement phase transition 7-39

7.8.3 The mass gap and the Gaussian structure 7-44

7.8.4 The large d approximation 7-46

7.8.5 High temperature limit 7-48

7.9 The discrete light-cone quantization (DLCQ) and infinite momentum frame (IMF) 7-49

7.9.1 Light-cone quantization and discrete light-cone quantization 7-49

7.9.2 Infinite momentum frame and BFSS conjecture 7-52

7.9.3 More on light-like versus spacelike compactifications 7-53

7.10 M-(atrix) theory in pp-wave spacetimes 7-56

7.10.1 The pp-wave spacetimes and Penrose limit 7-56

7.10.2 The BMN matrix model 7-61

7.10.3 Construction of the BMN matrix model 7-61

7.10.4 Compactification on R × S₃ 7-68

7.10.5 Dimensional reduction 7-75

7.11 Other matrix models 7-83

7.12 Exercises 7-84

References 7-84

8 Type IIB matrix model 8-1

8.1 The IKKT model in the Gaussian expansion method 8-1

8.2 Yang-Mills matrix cosmology 8-4

8.2.1 Lorentzian type IIB matrix model 8-4

8.2.2 Spontaneous symmetry breaking and continuum and infinite volume limits 8-8

8.2.3 Expansion 8-10

8.2.4 Role of noncommutativity 8-12

8.2.5 Other related work 8-17

8.3 Emergent gravity: introductory remarks 8-17

8.3.1 Noncommutative electromagnetism is a gravity theory 8-17

8.3.2 Seiberg-Witten map 8-21

8.4 Fuzzy spheres and fuzzy CPⁿ 8-24

8.4.1 Co-adjoint orbits 8-25

8.4.2 Fuzzy projective space CP₂ 8-27

8.4.3 Tangent projective module on fuzzy CP₂ 8-28

8.4.4 Yang-Mills matrix models for fuzzy CP_k 8-30

8.4.5 Coherent states 8-31

8.4.6 Star product 8-34

8.4.7 Fuzzy derivatives 8-37

8.5 Fuzzy S⁴_N: symplectic and Poisson structures 8-38

8.5.1 The spectral triple and fuzzy CP^k_N: another look 8-38

8.5.2 Fuzzy S⁴_N 8-42

8.5.3 Hopf map 8-48

8.5.4 Poisson structure 8-49

8.5.5 Coherent state 8-50

8.5.6 Focal flatness 8-51

8.5.7 Noncommutativity scale 8-52

8.5.8 Matrix model 8-53

8.6 Emergent matrix gravity 8-57

8.6.1 Fluctuations on fuzzy S⁴_N 8-57

8.6.2 Gauge transformations 8-59

8.6.3 Emergent geometry 8-62

8.6.4 Emergent gauge theory 8-65

8.6.5 Emergent gravity: Einstein equations 8-68

8.7 Emergent quantum gravity from multitrace matrix models 8-73

8.8 Exercise 8-73

References 8-74

Appendix A Algorithms and Monte Carlo codes for the matrix models of string theory A-1