Graudate_Course_in_String_Theory.pdf

(2726 KB) Pobierz
134090006 UNPDF
Introduction to String Theory
Angel M. Uranga
Contents
I Introductory Overview
1
1 Motivation 3
1.1 Standard Model and beyond . . . . . . . . . . . . . . . . . . . 3
1.1.1 Our Model of Elementary Particles and Interactions . . 3
1.1.2 Theoretical questions raised by this description . . . . 5
1.1.3 Some proposals for physics beyond the Standard Model 7
1.1.4 String theory as a theory beyond the Standard Model . 13
2 Overview of string theory in perturbation theory 15
2.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 What are strings? . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 The worldsheet . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 String interactions . . . . . . . . . . . . . . . . . . . . 19
2.1.4 Critical dimension . . . . . . . . . . . . . . . . . . . . 23
2.1.5 Overview of closed bosonic string theory . . . . . . . . 24
2.1.6 String theory in curved spaces . . . . . . . . . . . . . . 26
2.1.7 Compactication . . . . . . . . . . . . . . . . . . . . . 31
2.2 Superstrings and Heterotic string phenomenology . . . . . . . 34
2.2.1 Superstrings . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.2 Heterotic string phenomenology . . . . . . . . . . . . . 39
2.2.3 The picture of our world as a heterotic string compact-
ication . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.4 Phenomenological features and comparison with other
proposals beyond the standard model . . . . . . . . . . 42
3 Overview of string theory beyond perturbation theory 45
3.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Non-perturbative states in string theory . . . . . . . . . . . . 47
i
ii
CONTENTS
3.2.1 Non-perturbative states in eld theory . . . . . . . . . 47
3.2.2 Non-perturbative p-brane states in string theory . . . . 52
3.2.3 Duality in string theory . . . . . . . . . . . . . . . . . 55
3.3 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 What are D-branes . . . . . . . . . . . . . . . . . . . . 62
3.3.2 Worldvolume theory . . . . . . . . . . . . . . . . . . . 64
3.3.3 D-branes in string theory . . . . . . . . . . . . . . . . . 66
3.3.4 D-branes as probes of spacetime . . . . . . . . . . . . . 69
3.3.5 D-branes and gauge eld theories . . . . . . . . . . . . 71
3.4 Our world as a brane-world model . . . . . . . . . . . . . . . . 72
4 Quantization of the closed bosonic string 77
4.1 Worldsheet action . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.1 The Nambu-Goto action . . . . . . . . . . . . . . . . . 78
4.1.2 The Polyakov action . . . . . . . . . . . . . . . . . . . 78
4.1.3 Symmetries of Polyakov action . . . . . . . . . . . . . . 79
4.2 Light-cone quantization . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Light-cone gauge xing . . . . . . . . . . . . . . . . . . 80
4.2.2 Gauge-xed Polyakov action, Hamiltonian . . . . . . . 83
4.2.3 Oscillator expansions . . . . . . . . . . . . . . . . . . . 85
4.2.4 Light spectrum . . . . . . . . . . . . . . . . . . . . . . 88
4.2.5 Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.6 Final comments . . . . . . . . . . . . . . . . . . . . . . 90
5 Modular invariance 91
5.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Worldsheet coordinatization in light-cone gauge . . . . . . . . 92
5.3 The computation . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3.1 Structure of the amplitude in operator formalism . . . 93
5.3.2 The momentum piece . . . . . . . . . . . . . . . . . . . 95
5.3.3 The oscillator piece . . . . . . . . . . . . . . . . . . . . 95
5.4 Modular invariance . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4.1 Modular group of T 2 . . . . . . . . . . . . . . . . . . . 96
5.4.2 Modular invariance of the partition function . . . . . . 99
5.4.3 UV behaviour of the string amplitude . . . . . . . . . . 100
CONTENTS
iii
6 Toroidal compactication of closed bosonic string theory 105
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Toroidal compactication in eld theory . . . . . . . . . . . . 106
6.3 Toroidal compactication in string theory . . . . . . . . . . . 110
6.3.1 Quantization and spectrum . . . . . . . . . . . . . . . 111
6.3.2
0
eects I: Enhanced gauge symmetries . . . . . . . . 117
eects II: T-duality . . . . . . . . . . . . . . . . . . 120
6.3.4 Additional comments . . . . . . . . . . . . . . . . . . . 123
0
7 Type II Superstrings 125
7.1 Superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.1.1 Fermions on the worldsheet . . . . . . . . . . . . . . . 125
7.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . 127
7.1.3 Spectrum of states for NS and R fermions . . . . . . . 128
7.1.4 Modular invariance . . . . . . . . . . . . . . . . . . . . 133
7.1.5 Type II superstring partition function . . . . . . . . . . 135
7.1.6 GSO projection . . . . . . . . . . . . . . . . . . . . . . 135
7.1.7 Light spectrum . . . . . . . . . . . . . . . . . . . . . . 136
7.2 Type 0 superstrings . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3 Bosonization
. . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 Heterotic superstrings 145
8.1 Heterotic superstrings in bosonic formulation . . . . . . . . . . 145
8.1.1 Heteroticity . . . . . . . . . . . . . . . . . . . . . . . . 145
8.1.2 Hamiltonian quantization . . . . . . . . . . . . . . . . 146
8.1.3 Modular invariance and lattices . . . . . . . . . . . . . 148
8.1.4 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2 Heterotic strings in the fermionic formulation . . . . . . . . . 154
8.3 Spacetime Non-susy heterotic string theories . . . . . . . . . . 158
8.4 A few words on anomalies . . . . . . . . . . . . . . . . . . . . 159
8.4.1 What is an anomaly? . . . . . . . . . . . . . . . . . . . 159
8.4.2 Anomalies in string theory and Green-Schwarz mech-
anism . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9 Open strings 165
9.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.2 Open bosonic string . . . . . . . . . . . . . . . . . . . . . . . . 167
9.2.1 Light-cone gauge . . . . . . . . . . . . . . . . . . . . . 167
6.3.3

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin