##### 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

BIBLIOGRAPHY 93

[12] G. ’tHooft, Introduction to string theory, 2004.

http://www.phys.uu.nl/ thooft/lectures/stringnotes.pdf

[13] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vol. 1 and 2,

Cambridge University Press, 1987.

[14] D. Lust and S. Theisen, Lectures on String Theory, Springer Verlag. Berlin,

1989.

[15] E. Kiritsis, Introduction to Superstring Theory, Leuven: Leuven University

Press, 1998. hep-th/9709062.

[16] T. Mohaupt, Introduction to string theory, 2003. hep-th/0207249 v1.

[17] A. M. Uranga, Introduction to String Theory, Xavi, 2010.

[18] J. Scherk, An introduction to the theory of dual models and strings, Rev. Mod.

Phys. 47, 123 (1975).

[19] A. Restuccia, J. G. Taylor, Light-cone gauge analysis of superstrings, Phys. Rep.

174, 283-407 (1989).

[20] D. Bailin and A. Love, Supersymmetric gauge field theory and string theory, IoP,

2004.

[21] S. Forste, Strings Branes and Superstring Theory, 2002. hep-th/0110055 v3.

[22] M. Kaku, Quantum field theory.. A modern introduction, Oxford University

press, 1993.

[23] A. Sen, and B. Zwiebach, Tachyon condensation in string field theory, J. High

Energy Phys. 0003, 002 (2000). hep-th/9912249.

[24] G. B. Arkfen. Mathematical methods for physicists, 5th Ed. Academic Press,

2001.

Page 98

BIBLIOGRAPHY 94

[25] C. Rebbi, Dual models and relativistic quantum strings, Phys. Rep. 12, 1-73

(1974).

[26] C. P. Bachas, Lectures on D-branes, 1998. hep-th/9806199.

[27] Bruzzo, Gorini, Moschella (eds.), Geometry and physics of branes, IOP, 2003.

[28] J. Ambjorn, Y. M. Makeenko, G. W. Semenoff, and R. J. Szabo, String theory in

electromagnetic fields, J. High Energy Phys. 0302, 026 (2003). hep-th/0012092.

[29] A. Giveon, M. Porrati, , and E. Rabinovici, Target space duality in string theory,

Phys. Rep. 244, 77 (1994). hep-th/9401139.

[30] S. Alexandrov, Matrix Quantum Mechanics and 2-D String Theory. hep-

th/0311273 v2.

[31] D. Lust, String Landscape and the Standard Model of Particle Physics, 2007.

hep-th/0707.2305.

[32] V. d. Schaar, String theory limits and dualities, (Ph.d Thesis, 2000).

[33] J. A. Nieto and C. Pereyra, Dirac equation in (1+3) and (2+2) dimensions,

2013. arXiv:1305.5787v2

[34] P. B. Pal, Dirac, Majorana and Weyl fermions, Am. J. Phys. 79, 485-498 (2011).

arXiv:1006.1718v2

[35] A. Iqbal, lecture notes on Topological strings, LUMS, 2010.

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

BIBLIOGRAPHY 93

[12] G. ’tHooft, Introduction to string theory, 2004.

http://www.phys.uu.nl/ thooft/lectures/stringnotes.pdf

[13] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vol. 1 and 2,

Cambridge University Press, 1987.

[14] D. Lust and S. Theisen, Lectures on String Theory, Springer Verlag. Berlin,

1989.

[15] E. Kiritsis, Introduction to Superstring Theory, Leuven: Leuven University

Press, 1998. hep-th/9709062.

[16] T. Mohaupt, Introduction to string theory, 2003. hep-th/0207249 v1.

[17] A. M. Uranga, Introduction to String Theory, Xavi, 2010.

[18] J. Scherk, An introduction to the theory of dual models and strings, Rev. Mod.

Phys. 47, 123 (1975).

[19] A. Restuccia, J. G. Taylor, Light-cone gauge analysis of superstrings, Phys. Rep.

174, 283-407 (1989).

[20] D. Bailin and A. Love, Supersymmetric gauge field theory and string theory, IoP,

2004.

[21] S. Forste, Strings Branes and Superstring Theory, 2002. hep-th/0110055 v3.

[22] M. Kaku, Quantum field theory.. A modern introduction, Oxford University

press, 1993.

[23] A. Sen, and B. Zwiebach, Tachyon condensation in string field theory, J. High

Energy Phys. 0003, 002 (2000). hep-th/9912249.

[24] G. B. Arkfen. Mathematical methods for physicists, 5th Ed. Academic Press,

2001.

Page 98

BIBLIOGRAPHY 94

[25] C. Rebbi, Dual models and relativistic quantum strings, Phys. Rep. 12, 1-73

(1974).

[26] C. P. Bachas, Lectures on D-branes, 1998. hep-th/9806199.

[27] Bruzzo, Gorini, Moschella (eds.), Geometry and physics of branes, IOP, 2003.

[28] J. Ambjorn, Y. M. Makeenko, G. W. Semenoff, and R. J. Szabo, String theory in

electromagnetic fields, J. High Energy Phys. 0302, 026 (2003). hep-th/0012092.

[29] A. Giveon, M. Porrati, , and E. Rabinovici, Target space duality in string theory,

Phys. Rep. 244, 77 (1994). hep-th/9401139.

[30] S. Alexandrov, Matrix Quantum Mechanics and 2-D String Theory. hep-

th/0311273 v2.

[31] D. Lust, String Landscape and the Standard Model of Particle Physics, 2007.

hep-th/0707.2305.

[32] V. d. Schaar, String theory limits and dualities, (Ph.d Thesis, 2000).

[33] J. A. Nieto and C. Pereyra, Dirac equation in (1+3) and (2+2) dimensions,

2013. arXiv:1305.5787v2

[34] P. B. Pal, Dirac, Majorana and Weyl fermions, Am. J. Phys. 79, 485-498 (2011).

arXiv:1006.1718v2

[35] A. Iqbal, lecture notes on Topological strings, LUMS, 2010.