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TitleLight Cone Gauge Quantization of Strings, Dynamics of D-brane and String dualities.
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Table of Contents
                            Abstract
INTRODUCTION
BOSONIC STRING THEORY
	The relativistic string action
		The Nambu-Goto string action
		Equation of motions, boundary conditions and D-branes
	Constraints and wave equations
	Open string Mode expansions
	Closed string Mode expansions
	Light cone solution and Transverse Virasoro Modes
	Quantization and Commutations relations
	Transverse Verasoro operators
	Lorentz generators and critical dimensions
	State space and mass spectrum
SUPERSTRINGS
	The super string Action
		Equations of motion and Boundary conditions
	Neveu Schwarz sector
	Ramond sector
	Super transverse Virasoro operators
	Counting states
	Open superstrings and the GSO projection
	Closed string theories
		Type IIA Superstring Theory
		Type IIB Superstring Theory
		Heterotic superstring theories
	Type I
	Critical dimensions
D-BRANES
	Tachyons and D-brane decay
	Quantization of open strings in the presence of various kinds of D-branes
		Dp-branes and boundary conditions
		Quantizing open strings on Dp-branes
		Open string between parallel Dp-branes
		Strings between parallel Dp and Dq-branes
	String charge and electric charge
		Fundamental string charge
		Visualizing string charge
		Strings ending on D-branes
	D-brane charges and stable D-branes in Type II
STRING DUALITIES
	T-duality and closed strings
		5.1.1 Mode expansions for compact dimension
		Quantization and commutation relations
		Constraint and mass spectrum
		State Space of compactified closed strings
	T-Duality for Closed Strings
		Type II superstrings and T-duality
	T-duality of open strings
		T-duality and open strings
		Open strings and Wilson lines
	Electromagnetic fields on D-branes and T-duality
		Maxwell fields coupling to open strings
		D-branes with Electric fields and T-dualities
		D-branes with Magnetic fields and T-dualities
	String coupling and the dilaton
	S-duality
Appendix A The Massless States Of Closed String
Appendix B  The Spinors Algebra In 2D
                        
Document Text Contents
Page 1

Light Cone Gauge Quantization of
Strings, Dynamics of D-brane and

String dualities.

Muhammad Ilyas1

Department of Physics
Government College University

Lahore, Pakistan

Abstract

This review aims to show the Light cone gauge quantization of strings. It is divided

up into three parts. The first consists of an introduction to bosonic and superstring

theories and a brief discussion of Type II superstring theories. The second part deals

with different configurations of D-branes, their charges and tachyon condensation. The

third part contains the compactification of an extra dimension, the dual picture of

D-branes having electric as well as magnetic field and the different dualities in string

theories. In ten dimensions, there exist five consistent string theories and in eleven

dimensions there is a unique M-Theory under these dualities, the different superstring

theoies are the same underlying M-Theory.

1ilyas [email protected]
M.Phill thesis in Mathematics and Physics Dec 27, 2013
Author: Muhammad Ilyas
Supervisor: Dr. Asim Ali Malik

i

Page 2

Contents

Abstract i

INTRODUCTION 1

BOSONIC STRING THEORY 6

2.1 The relativistic string action . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 The Nambu-Goto string action . . . . . . . . . . . . . . . . . . 7

2.1.2 Equation of motions, boundary conditions and D-branes . . . . 7

2.2 Constraints and wave equations . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Open string Mode expansions . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Closed string Mode expansions . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Light cone solution and Transverse Virasoro Modes . . . . . . . . . . . 14

2.6 Quantization and Commutations relations . . . . . . . . . . . . . . . . 18

2.7 Transverse Verasoro operators . . . . . . . . . . . . . . . . . . . . . . . 22

2.8 Lorentz generators and critical dimensions . . . . . . . . . . . . . . . . 27

2.9 State space and mass spectrum . . . . . . . . . . . . . . . . . . . . . . 29

SUPERSTRINGS 32

3.1 The super string Action . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Equations of motion and Boundary conditions . . . . . . . . . . 33

3.2 Neveu Schwarz sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Ramond sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Super transverse Virasoro operators . . . . . . . . . . . . . . . . . . . . 40

3.5 Counting states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Open superstrings and the GSO projection . . . . . . . . . . . . . . . . 43

3.7 Closed string theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

ii

Page 49

45

3.7.1 Type IIA Superstring Theory

In the order to get the closed superstring theory with a supersymmetry, for this we

need to truncate the above four sectors in equation (3.49). Since we can truncate as

Left sector :


 NS +

R −


 Right sector :


 NS +

R +


 (3.50)

By this we find the four sectors which is called the type IIA superstring and these

sectors are

(NS + , NS + ), (NS + , R + ), (R − , NS + ), (R − , R + ) (3.51)

This is the type IIA superstring and for this, the mass squared is written as

1

2
α′M2 = α′M2L + α

′M2R (3.52)

Here M2L and M
2
R are denoting the mass squared operators of the left and right sectors

respectively. Now listing some of the massless states of the varies sectors, which are

(NS + , NS + ) : b̄l−1/2 |NS〉L ⊗ b
j
−1/2 |NS〉R ⊗

∣∣p+, ~pT 〉 (3.53)
(NS + , R + ) : b̄l−1/2 |NS〉L ⊗ |Rb̄〉R ⊗

∣∣p+, ~pT 〉 (3.54)
(R − , NS + ) : |Ra〉L ⊗ b

l
−1/2 |NS〉R ⊗

∣∣p+, ~pT 〉 (3.55)
(R − , R + ) : |Ra〉L ⊗ |Rb̄〉R ⊗

∣∣p+, ~pT 〉 (3.56)
Hence, there are 64 bosonic states in (3.53) due to the index and having eight values.

These are just like the bosonic closed string theory massless states and carry the two

indices. So we get the graviton (35 states), the Kalb-Ramond field (28 states), and the

dilaton (one state).

(NS + , NS + ) massless fields are : g�� , B�� , φ (3.57)

There are 64 fermionic states in each of the states (3.54) and (3.55) due to Both the

states given in (3.54) and (3.55) have included a Ramond ground state, so these states

Page 50

46

are space time fermions. And give the total of 2 x 8 x 8 = 128 fermionic states. Similarly,

the states given (3.56) having the two R ground states, which have doubly fermionic,

so these will be the space time bosons which have 8 x 8 = 64 massless bosonic states

and together with the bosonic states in (3.53) gives the total massless, bosonic states

as 64 + 64 = 128 of the type IIA superstring. The space time bosonic states match

with the space time fermionic states, which is the supersymmetry of the theory.

3.7.2 Type IIB Superstring Theory

A different superstring theory arises by truncating the four sectors which is given in

(3.49) such that

Left sector :


 NS +

R −


 Right sector :


NS +

R −


 (3.58)

By this we find the four sectors which is called the type IIB superstring and these

sectors are

(NS + , NS + ), (NS + , R − ), (R − , NS + ), (R − , R − ) (3.59)

Now listing the massless states of this type IIB superstring theory, which are

(NS + , NS + ) : b̄l−1/2 |NS〉L ⊗ b
j
−1/2 |NS〉R ⊗

∣∣p+ , ~pT〉 (3.60)
(NS + , R − ) : b̄l−1/2 |NS〉L ⊗ |Rb〉R ⊗

∣∣p+ , ~pT〉 (3.61)
(R − , NS + ) : |Ra〉L ⊗ b

l
−1/2 |NS〉R ⊗

∣∣p+ , ~pT〉 (3.62)
(R − , R − ) : |Ra〉L ⊗ |Rb〉R ⊗

∣∣p+ , ~pT〉 (3.63)
(NS - NS ) sector: This sector is same for both the superstrings, type IIA and type IIB.

(NS - R ) and (R - NS ) sectors: these sectors give the space time fermions and contain

a spin 3/2 gravitino (56 states) and a spin 1/2 fermion called the dilatino (eight states).

In the case of type IIB, the two gravitinos will have the same chirality while for the

type IIA superstring, they will have opposite chirality.

Page 97

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