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

Ilyas, Muhammad

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

Summary

This review aims at showing the light cone gauge quantization of strings. It is divided into three parts: The first part 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 an electric as well as a 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 theories being the same underlying M-Theory.

Excerpt

5.5

String coupling and the dilaton . . . . . . . . . . . . . . . . . . . . . .

5.6

S-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A Appendix A

The Massless States Of Closed String

B Appendix B

The Spinors Algebra In 2D

INTRODUCTION

General relativity and quantum mechanics were the two major breakthroughs that

revolutionized theoretical physics in the twentieth century. General relativity gives

the idea to understand of the large-scale expansion of the Universe and gives a small

correction to the predictions of Newtonian gravity for the motion of planets and the

deflection of light rays, and it predicts the existence of gravitational radiation and black

holes. It describes the gravitational force in terms of the curvature of spacetime which

has fundamentally changed our view of space and time i.e. they are now viewed as

dynamical [1].

Quantum mechanics, on the other hand, is the essential tool for understanding the sub

atomic particles and microscopic physics. The evidence continues to build that it is

an exact property of Nature [2]. The fundamental law of Nature is surely incomplete

until general relativity and quantum mechanics are successfully reconciled and unified.

String theory is a candidate which resolves this problem and a straightforward attempt

to combine the General relativity and Quantum mechanics. String theory is based on

the idea that particles are not point-like, but rather tiny loops (i.e. closed strings)

or (open) pieces of string i.e. the 0-dimensional point particle is replaced by a 1-

dimensional string [3]. This assumption leads to some features which are

General features

Even though string theory is not yet fully formulated, and we cannot yet understand the

detailed description of how the standard model of elementary particles should emerge

at low energies, or how the Universe originated, there are some general features of the

theory that have been well understood.

Vibrating string

String theory predicts that all objects in our universe are composed of Vibrating strings

and different vibrational modes of the strings represent different kinds of particles. Since

there is just one type of string, and all particles arise from string vibrations, all particles

are naturally incorporated into a single theory [3].

Gravity

String theory attempts to reconcile the General relativity and Quantum mechanics.

One of the vibrational modes of strings is the graviton particle, the quantum version of

the gravity, so string theory has the remarkable property of predicting gravity [3].

Unification of forces

There are four fundamental forces that had been recognized to exist in nature.

1. Electromagnetic force

2. Weak force

3. Strong force

4. Gravitational force

As the quantum version of electromagnetism describes the photon (a massless particle)

and its interactions with charged particles, while the Yang-Mills theory describes W

and Z bosons and gluons (the mediators of the weak and the strong nuclear forces)

and their interactions. All of these theories make a single theory named the Standard

Model of particle interactions, which is a gauge theory. The Standard Model of particle

physics does not include the graviton particle and its interactions. The graviton which

has spin 2 can not be described by the gauge theory. Since the standard model is

believed to incomplete due to it does not incorporate the gravity forces. String theory

is currently the most promising candidate to unify all the fundamental forces. This is

a general feature of the string theory [3].

Yang-Mills gauge theory

Standard model of particle physics describe the elementary particles in nature. It

reconciles the special relativity and Quantum mechanics [4]. And is based on Yang-

Mills theory having the gauge group

SU (3)

× SU(2) × U(1)

However it has some shortcomings. It does not include Gravity and it has about 20

parameters that cannot be calculated and we use them as an input. While string

theory predicts the gravity as well as describe all the elementary particles and has one

parameter, the string length. Its value is roughly equal to the typical size of strings.

Yang-Mills gauge theories arise very naturally in string theory. However, it is not yet

fully understood why the gauge group

SU (3)

× SU(2) × U(1)

Of the standard model with three generations of quarks and leptons should be singled

out in nature [5, 6].

Supersymmetry

A supersymmetry is a symmetry which relates bosons and fermions. There exists a non

supersymmetric bosonic string theory which is an unrealistic theory due to the lack

of fermions. In the order to get a realistic string theory which explains the beauty of

nature, we need a supersymmetry. Hence supersymmetry is a general feature of string

theory [3].

Extra dimensions of space

In quantum field theory, for point particle, we let the dimensions of the spacetime to

be four while superstring theories predict some additional dimensions of the spacetime.

The superstring theories are only able to work in a ten dimensions or eleventh (in some

cases) dimensions of spacetime. To make an ordinary four dimensional space time,

there is a straight forward possibility that is, the additional six or seven dimensions

can be curled up and compactified on an internal manifold having the sufficient small

size, which can not be detectable at the low energies. The idea of an extra dimension

was first introduce by Kaluza and Klein in 1920s. Their aim was to unify the electro-

magnetic force and the gravitational force. The compactification of an extra dimension

can be imagined as, let us consider we have a cylinder having the radius R. when the

cylinder is viewed from a very large distance or equivalently, when the radius of the

cylinder R becomes too small then the two dimensional cylinder will look like a one

dimensional line.

Generalizing this idea by letting the cylinder as a four dimensional spacetime and replac-

ing the short circle of radius R (compact space) by a six or seven dimensional manifold,

hence at large distance or at the low energies, the additional dimensions (compact man-

ifold) can not be visible. These additional dimensions or compact manifold are called

Calabi-Yau manifolds [3].

The size of the strings

Quantum field theory deals the particles as a mathematical zero dimensional point while

in string theory the ordinary point particles are replaced by a one dimensional string.

These one dimensional strings will have a characteristic length scale which is denoted

be and can estimated by the dimensionality analysis. As string theory is a relativistic

quantum theory which also includes the force of gravity, since it must involves the fun-

damental constants speed of light c Planck's constant ¯

h, and the gravitational constant

G From these, we can form a length, called the Planck length

1/2

= 1.6

× 10

-33

cm ,

The Planck mass becomes

1/2

= 2.176

× 10

-5

g ,

And similarly the Planck time

1/2

= 5.391

× 10

-44

s .

The Planck length scale is a natural guess for a fundamental string length scale as well

as the characteristic size of compact extra spatial dimensions. At low energies string

can be approximated by the point particle that explains why the quantum field theory

has been successfully in the describing our world. The relativistic quantum gravity

effects can be important on the above three scales of planks length, planks mass and

planks time [3].

BOSONIC STRING THEORY

The bosonic string theory is the simplest string theory that predicts and describes

only a certain set of boson. As the theory does not describe any fermions, so it is an

unrealistic theory but this theory is a natural place to start, because the same techniques

and structures, together with some additional terms are required for analysis of more

realistic theory (super string theories).

String can be regarded as 1-brane moving through a space-time which is a special case

of a p-brane, p-dimensional extended object. Point particle corresponds to 0-brane.

Similarly the two dimensional extended object or 2-brane are called membranes.

2.1

The relativistic string action

In quantum field theory, the action for point particle (0-brane) is proportional to the

invariant length of the word-line or particle trajectory. Similarly when we replace the

point particle by 1 dimensional string (1-brane) then the action is proportional to the

proper area of the word-sheet or the area swept out by the string in D dimensional

space-time.

N G

- T

(2.1)

The word-sheet is parameterized by the two coordinates

= which is time-like and

= which is space-like. For close string the will be periodic while for open string

will be have some finite value [3].

2.1.1

The Nambu-Goto string action

As one-dimensional string having

(, ) space-time coordinates tracing out a two

dimensional word-sheet. In the order to calculate the area of word-sheet, we will need

a metric.

Let

(i = 0, 1) denote the word-sheet coordinates and take the metric

having the signature (

- , +, +,..., +) with , = 0, 1,...., D - 1 , describe the

background geometry in which the string propagates then

- ds

= g

(2.2)

Here

= ,

= and the minus sign is used with ds

is for having real time-like

trajectory. In Minkowski flat space-time g

. The Nambu-Goto action then takes

the form

N G

-T

- det g

-T

(

X.X )

- (

)(X

)d d

(2.3)

Here

, X

and by the dimensional analysis T =

, T

is the string

tension and we will use natural units in which speed of light is 1.

2.1.2

Equation of motions, boundary conditions and D-branes

Let us start from the Nambu-Goto string action which is in the form of lagrangian

density

S =

d L=

dL(

, X

)

(2.4)

Where L is given as

, X

) =

- T

(

X.X )

- (

(X )

(2.5)

The equations for the motion of string can be obtained by the action principle.

S =

- X

(2.6)

As we will use the X

for string coordinate while for general space-time we will called x

Where

-T

(

X.X )X

- (X )

(

X.X )

- (

(X )

(2.7)

And

-T

(

X.X )

- (X )

(

X.X )

- (

(X )

(2.8)

For S = 0 we get some conditions (boundary conditions) and equations of motion,

which are, the equations of motion for relativistic string

. (closed or open) are.

= 0

(2.9)

And the boundary conditions are

A) The boundary condition for closed string is, as for the closed string the word-sheet

is like a tube (cylinder type), so their boundary conditions are

(, + 2) = X

(, )

(2.10)

B) While the word-sheet of open string is like a sheet, there are two types of boundary

conditions for open string

i) Neumann conditions

) = 0

(2.11)

ii) Dirichlet conditions

) = 0, = 0

(2.12)

Here

are the end points of the open string. No momentum flows off the ends of the

String by implying the Neumann conditions while the endpoints are fixed in spacetime

by implying the Dirichlet conditions. When the end points are fixed in space-time this

means that the string is attached with some physical object, which is called D-brane.

While when the Dirichlet conditions are applied to a subset of the d indices (spatial

For closed string

[0, 2]. While for open string [0, ]. The length of string is

These are the momentum densities, P

, P

. While P

is called momentum conjugate.

dimensions) of X

, i.e. to the indices p+1, . . . ,d, this mean that the endpoints of the

string are restricted to move on a p-dimensional hyper-plane. This higher dimensional

object is called a Dp-brane, where p indicates its dimension and D stands for Dirichlet.

It turns out that these objects should be considered to be dynamical as well [3, 7, 8, 9,

10].

2.2

Constraints and wave equations

As the word-sheet have the two parameters, (, ) by reparameterization (gauge) of

and of the word-sheet, simply using a class of gauges

which fix the parameterization

( and ) of the word-sheet [11] and give the some constraints equations which are

X.X = 0,

+ X

= 0

(2.13)

Both these constraints equations combine to give

(

± X )

= 0

(2.14)

Using these two constraints equation to simplify the momentum densities P

and P

which give

, P

(2.15)

Putting these momentum densities in the equation of motion which is

0, we get

- X

= 0

(2.16)

So by reparameterization we got the equation of motion just as a wave equation.

Reparameterization of word-sheet In static gauge (t = ), our solution is not fully explicit so we

use the more general gauges. A class of choices for is n.X(, ) = (n.p) , Where is equal to

two for open string and one for closed string and "

n" is chosen in such a way that (n.p) is conserved.

And the conservation of (

n.p) means n.P

= 0 for both closed as well as open strings And similarly

the associated parameterization gives n.p =

n.P

The string tension T

is equal to T

, and is the Regge-slope parameter.

2.3

Open string Mode expansions

Let we have a space-filling D-brane for this all the string coordinates X

satisfy the

boundary condition (Neumann) at the end points. And as the solution of a wave

equation can be written in the form of superposition of waves moving to the left and

right on the string.

(, ) =

( + ) + g

(

- ))

(2.17)

Where f

and g

are the arbitrary functions. As the Neumann boundary conditions

= 0 at the endpoints

= 0 , at = 0,

The Neumann boundary conditions at = 0 give

(, 0) =

( )

- g

( )) = 0

(2.18)

This equation means that the derivative of f

and g

are same then these two functions

are differ by some constant i.e. g

= f

, so replacing this and absorbing the constant

into the definition of f

. This give

(, ) =

( + ) + f

(

- ))

(2.19)

Now consider boundary conditions at =

(, ) =

( + )

- f

(

- )) = 0

(2.20)

Since from this equation we see that f

is a periodic with period of 2. so we can write

it in term of Fourier series

(u) = f

n=1

cos nu + b

sin nu)

(2.21)

Now integrating this equation and then putting the result f

(u) back into equation

(2.17) and simplifying, we get

(, ) = f

+ f

n=1

cos n + B

sin n ) cos n

(2.22)

Now replace the coefficients in above equation (2.22) by new coefficients which have

some simple physical interpretation.

cos n + B

sin n =

-i

- a

-in

(2.23)

Here * denote the complex conjugate and

2 factor is to make the a

dimensionless

and the physical interpretation of these new constant and their conjugate is, they

become annihilation and creation operators when considering Quantum field theory

Similarly f

in equation (2.22) has also a simple physical interpretation which is as

from the momentum conjugate equation

d =

(2.24)

This equation tells us that the quantity f

is proportional to the space-time momentum

which is carried by the string. From above equation (2.24), putting the value of f

with

the new coefficients equation (2.23) into equation (2.22), then we gets

(, ) = x

+ 2 p

- i

n=1

- a

-in

cos n

(2.25)

In the order to make the simple expression of above equation (2.25), we replace the

constant f

= x

and defining

2 p

= a

n and

-n

= a

n , n

And also

-n

= (

)

, using these new coefficients (modes) and simplifying by which

we get

(, ) = x

+ i

n=0

-in

cos n

(2.26)

If all the coefficients (oscillators) a

vanishes, then the equation represents the point particle.

a is oscillators while is mode.

This is the solution of wave equation and a simple expression for the open string.

Similarly we can get

± X

-in(±)

(2.27)

2.4

Closed string Mode expansions

Let us consider the general solution for the wave equation, which is

(, ) = X

( + ) + X

(

- )

(2.28)

Where X

is for left-moving wave while X

is for right-moving wave of the string.

The word-sheet of the closed string is a cylinder, so to describe the closed string, we

need some conditions, as the closed string have no endpoints but having the periodicity

condition.

(, )

(, + 2)

(2.29)

So by this periodicity, we can write

(, ) = X

(, + 2) for all and

(2.30)

Now let introduce two variables

u = + and v =

By putting these new variable in equation (2.28), which becomes

(, ) = X

(u) + X

(v)

(2.31)

Now by periodicity condition (, )

(, + 2), the above equation becomes

(u + 2)

- X

(u) = X

(v)

- X

- 2)

(2.32)

From this equation we can say that both X

(u) and X

(v) are periodic with the period

of 2. Therefore we can write the mode expansions

(u) =

-inu

(2.33)

(v) =

-inv

(2.34)

By integrating these equations, we get

(u) =

u + i

n=0

-inu

(2.35)

(v) =

v + i

n=0

-inv

(2.36)

Now putting these two equations (2.35) and (2.36) into equation (2.32) and solving by

which we get

(2.37)

Now by using this equality and equations (2.35) and (2.36), we get the equation (2.31)

(, ) =

+ x

) +

+ i

n=0

-in

(

+ ¯

-in

)

(2.38)

As from the canonical momentum conjugate equation, we can write

d =

(2.39)

(2.40)

This equation (2.40) differs by a factor of two from the corresponding open string

result and also tells the same idea i.e. the quantity

is proportional to the space-time

momentum which is carried by the closed string.

And let x

and x

be equal i.e. having one conjugate coordinate zero mode, without

any loss of generality,

= x

Finally using the above equation then equation (2.38) becomes

(, ) = x

+ i

n=0

-in

(

+ ¯

-in

)

(2.41)

Here we required these two relations for the reality of the above equation

-n

= (

)

, ¯

-n

= ( ¯

)

As when

= ¯

then the above equation (for closed string) is equal to equation (2.26)

which is for open string. Since the closed string can be viewed as a two copies of the

open strings. This is a complete mode expansion for a closed string [3, 7].

2.5

Light cone solution and Transverse Virasoro Modes

The light cone solution is, to represents the motion of string by using the light cone

coordinates and this impose a set of conditions which is called light cone gauge. The

light cone gauge is one of the choices among the general gauges. As we know that the

more general gauges

n.X(, ) = (n.p) , n.p =

n.P

(2.42)

Where is equal to two for open string and one for closed string. In the order to select

the light cone gauge, we need to impose the above conditions with a vector n

such

that n.X = X

, 0..., 0

(2.43)

n.X =

+ X

= X

, n.P =

+ P

= P

(2.44)

Now using this pair of equations (2.44) into equation (2.42), we get

(, ) = P

, P

(2.45)

This choice of gauge is called the light cone gauge. Let the transverse coordinates be

denoted as

= X

, X

, ...X

Here (l = 2, 3, ..., d). Now using the constraints equation (2.14) and expanding it into

light cone coordinates.

(

± X )

= 0

= (

± X ).(

± X ) = 0

-2(

± X

). (

±X

) + (

±X

)

= 0

(2.46)

Here we use the dot products of light cone coordinates.

Now using the equation (2.45) and calculating the equation (2.46), we get

± X

(

±X

)

(2.47)

So we have developed the relation between the light cone coordinates and the transverse

coordinates. And the mode expansions for open string ( = 2) in the term of light cone

coordinates are

(, ) = x

+ i

n=0

-in

cos n

(2.48)

And as our gauge conditions give

(, ) = 2 P

(2.49)

With position zero mode and oscillators for the X

are

= 0,

-n

= 0, n = 1, 2, ...,

Similarly

We assume that P

= 0, the vanishing of P

means that a massless particle moving in the

negative x

direction. While P

certainly satisfied P

(, ) = x

+ i

n=0

-in

cos n

(2.50)

These are the complete set of mode expansions in light cone coordinate [12]. Now using

equation (2.27) with =

- and = l, then we get

± X

-in(±)

(2.51)

± X

-in(±)

(2.52)

Now putting these two equations (2.51) and (2.52) into equation (2.47), by simplifying

we get

n-p

(2.53)

From above equation (2.53)

, we have the explicit expression for the minus oscillators

in the term of transverse oscillators

. This represents the full solutions [3, 11].

From the above equation (2.53), the right side has given a special type name, which is

the Transverse Virasoro mode L

, L

n-p

(2.54)

Now by using the equation (2.53) and the equation which we defined earlier that is

2 p

, then we get

= 2P

(2.55)

Now using the value of

from equation (2.54) then equation (2.51) and (2.47) are

written

± X

-in(±)

4 P

(

± X

)

(2.56)

A critical string only have transverse excitations, just like for a massless particle only has transverse

polarization states.

Similarly for closed string ( = 1) since the equation (2.47) becomes

± X

(

± X

)

(2.57)

And as like equation (2.56) we can also write for closed strings

(

+ X

)

= 4

n-p

-in(+)

(2.58)

(

- X

)

= 4

n-p

-in(-)

(2.59)

So we define the Verasoro mode for closed string

n-p

, L

n-p

(2.60)

Now using equations (2.58), (2.59) and equations (2.57), (2.41) we can easily finds

2 ¯

(2.61)

From this equation (2.61) we cal also write

n-p

And for n = 0 then ¯

= ¯

for equation (2.61), then, we have

= L

(2.62)

This equality is called level-matching and it means that one in the terms of right-moving

oscillators and one in terms of left-moving oscillators are equal as level matching.

In the quantum theory level matching implies that the states of right-moving excitations

of a string are equal to the states of its left-moving excitations [12]. We will explain

this later.

2.6

Quantization and Commutations relations

In the order to quantize our classical results we need to replace the Poisson brackets

by commutation relations and the fields by operators and then will solve the Virasoro

constraint equations, and will describe our theory in a Fock space that describes the

physical degrees of freedom [13]. Before we have written the classical equations of

motion in the light cone gauge. Now we will use (to quantization) the results of our

light cone analysis of the classical relativistic strings. Our Hamiltonian for open strings,

as we have X

= 2 p

so from this we can write

= 2 p

and in addition, as

generates X

translations so from this we can write Hamiltonian as

H = 2 p

For quantization and commutation relations, we will start from a set of operators.

(, ) , x

, P

(, ) , P

( ))

(2.63)

From this, we have the full set of basic operators of string theory because the above list

has the collection of all zero modes plus the infinite set of the annihilation and creation

operators [3]. First we will deal the open string and assume the space-filling D-brane.

Let postulate some commutation relations from the above set of operators

(, ), P

(, ) = i

(

- )

(2.64)

And

, P

-i

(2.65)

And all the other commutation relations are equal to zero i.e.

(, ), X

(, ) = 0, P

(, ), P

(, ) = 0

, X

(, ) = 0, x

, P

(, ) = 0

, X

(, ) = 0 ,

, P

(, ) = 0

(2.66)

From the commutation relation in equation (2.64) we can also write as

(, ),

(, ) = i2

(

- )

(2.67)

Similarly the other commutation relations from the above relations are

(, ),

(, ) = i2

(

- )

(2.68)

(, ),

(, ) = 0 ,

(, ), X

(, ) = 0

(2.69)

Now by solving these equations we get

(

± X

)(, ), (

± X

)(, ) =

±i4

(

- )

(2.70)

This commutation relation in equation (2.70) is useful to write down the commutation

relations for the oscillators. The classical modes

will become the quantum operators

with a nontrivial commutation relation. For commutation relation of oscillators, let

recall the equation (2.27), we can write

(

+ X

)(, ) =

-in(+)

[0, ]

(2.71)

(

- X

)(,

-) =

-in(+)

[-, 0]

(2.72)

Now let define an operator A

(, )

-in(+)

, A

(, + 2) = A

(, )

(2.73)

The periodicity is due to the definition of A

(, ) and from this definition we can write

(, ) =

(

+ X

)(, ),

[0, ]

(

- X

)(,

-), [-, 0].

(2.74)

So from this set of equation (2.74), there are four possibilities of commutation relations,

we can summarized as

(, ), A

(, ) = 4

(

- ) , , [-, ]

(2.75)

Now solving this equation (2.75) by using equation (2.73), we find

= m

m+n,0

(2.76)

This shows the commutation relation between the modes and also

commutes with

others oscillators and these are equivalent to an infinite set of annihilation and creation

operators. To see this let start from our defining oscillators i.e.

= a

n ,

-n

= a

n , n

(2.77)

As both (a, ) are classical variables and now they become operators. As those classical

variables which are complex conjugate of each other will now become the Hermitian

operators in quantum theory i.e. the operators x

and p

are Hermitian as

)

= x

, (p

)

= p

(2.78)

Similarly the above oscillators and modes are now operators and we will take these like

= a

n and

-n

= a

n, n

From this we have

(

)

-n

, n

(2.79)

From these conditions we can replace equation (2.76) as

-n

= m

m,n

, a

m,n

(2.80)

And similarly

= 0,

= 0

These commutation relations shows (a

, a

) satisfy the commutation relations of canon-

ical creation and annihilation operators of the quantum simple harmonic oscillator. So

from this, we have

are annihilation operators while

-n

are creation operators for

1. Using these Hermiticity conditions (2.78) and (2.79) we can write

(, ))

= X

(, )

Now to find the commutation relation between

and x

, for this as

2 P

and

using equation (2.64) and by simplification we get

, P

= i

(2.81)

By quantization, replacing the classical variables to operators, our mode expansion then

become

(, ) = x

+ 2 p

+ i

n=1

-in

- a

cos n

(2.82)

Here replace the modes by corresponding oscillators. And this is in the term of

annihilation and creation operators, expansion of the coordinate operator. Similarly

now quantize the closed string, the Poisson brackets will be replaced by commutation

relations and the classical variables by quantum operators. For commutation relations,

as the operators content of closed strings can be treated as the two commuting copies

the open string operators i.e. left moving and right moving. Similarly we can use the

almost same techniques to derive the commutation relations for closed strings. The

commutations relations for closed strings are

, ¯

= m

m+n,0

, ¯

= m

m+n,0

= 0, ¯

, ¯

m,n

, p

= i

, x

, ¯

= i

(2.83)

Our Hamiltonian for closed strings in light cone coordinates is, as we know from our

light cone gauge, as p

generates X

translation and also we have X

= p

so from

this we can write

= p

, so our Hamiltonian will be

H = p

There is only the factor of two in the open strings and closed strings Hamiltonian due

to the value of , which is two for open string while one for closed string [3, 14].

2.7

Transverse Verasoro operators

As before we have write down the classical solution of motion of both open and closed

strings in light cone coordinates and then we solved for X

in the term of transverse

coordinates by using the constraints equation (2.47) i.e.

the in the term of

modes

as shown (for open string) in equation (2.53) so

, L

n-p

(2.84)

Where l (repeated index) is summed over transverse light cone direction. As before L

were transverse Virasoro modes but now it is called transverse Virasoro operators due

to the modes become operators. As the two operators (inL

) fail to commute for

n = 0, so L

is the only operator which needs a Normal Ordering. From the definition

of L

-p

(2.85)

p=1

-p

p=1

-p

Since the last term is not Normal Ordered, taking the last term and making it as a

normal order by using the commutation relation, such that

p = 1

-p

p = 1

-p

- 2)

p = 1

So L

then become

p=1

-p

- 2)

p=1

(2.86)

And also from the definition of L

(with normal ordering constant a ) in the term of

is as

2 P

+ a

(2.87)

From equation (2.86) and (2.87), we can say that

a =

- 2)

p = 1

(2.88)

By Zeta function we find the value of

p = 1

p and so

a =

- 2)

(2.89)

The Normal order L

is then become

= p

p = 1

-p

- 2)

(2.90)

This is all about the open string transverse Virasoro operators. Similarly for closed

strings, the transverse Virasoro operators are

, L

n-p

2 ¯

, ¯

n-p

(2.91)

With (L

)

= L

, (L

)

= L

-n

and ( ¯

)

= ¯

-n

due to (

)

-n

and ( ¯

)

= ¯

-n

Similarly we have (

)

-n

. As we shown that

and

commutes only when

(m + n) equal to zero but the case is totally change for

i.e. two Virasoro operators

and L

never commute for (m = n) . Now to find the commutation relations for

Virasoro operators lets begin from the commutation relation of Virasoro operator and

oscillator, which is

n-p

(

-n

m+n

- n

m+n

)

So we find

-n

m+n

(2.92)

This commutation relation is also valid for m = 0 . Similarly

, x

-i

(2.93)

Now let us consider the two Virasoro operators L

and L

, and the commutation

relation between them is not quit easy. So let the sum split as

m-k

k<0

m-k

We made the above L

as normal ordered for any value of m. So we can now evaluate

the commutation relation like

, L

m-k

, L

k<0

m-k

, L

Now evaluating then we get

, L

- k)

m+n-k

k<0

- k)

m+n-k

m-k

k+n

k<0

k+n

m-k

(2.94)

Here we have now deal with two different cases m + n = 0, m + n = 0 and . If m + n = 0

, the two in each term commute then their order is irrelevant so in that case, we will

switch the order in the last two terms of equation (2.94) and then replace the variable

k by k

- n. So we get

, L

= (m

- n)L

m+n

, m + n = 0

(2.95)

A mathematical set of operators L

with n

Z, satisfying the above equation (2.95)

defines the Lie algebra. This algebra is called the Witt algebra or the Virasoro algebra

without central extension. Now the second case m + n = 0, let n =

-m in equation

(2.94) then an extra contribution arises due to insisting the normal order. Such that

, L

k=0

- k)

-k

k<0

- k)

-k

k=0

m-k

k-m

k<0

k-m

m-k

(2.96)

For normal ordering, let replace k

-k in the second terms, k m + k in the 3rd

terms, and k

m - k in the last terms

, L

k=0

- k)

-k

k=1

(m + k)

-k

k=-m

(m + k)

-k

k=m+1

- k)

-k

(2.97)

Let assume that m > 0, (this argument goes the same way as in the other case) by

this all the terms are now normal ordered except the 3rd term for which

-m k 0.

Splitting the summation of 3rd term in above equation (2.97) and then simplifying, we

get

, L

k=0

- k)

-k

k=1

(m + k)

-k

+ (D

- 2)A(m)

(2.98)

Where A(m) is

A(m) =

k=0

k(m

- k) =

k=1

(2.99)

By using the Mathematical induction and Zeta functions. We get

A(m) =

- m)

(2.100)

Now putting the value of A(m) from equation (2.100) into equation (2.98) and solving,

we find

, L

= 2m L

- 2)(m

- m)

This completes the commutation relation of Virasoro operators for m + n = 0. Now

generalizing these two cases, we get

, L

= 2m L

m+n

- 2)(m

- m)

m+n,0

(2.101)

The second term of right-hand side of the above equation (2.101) is called the central

extension and a mathematical set of operators L

with n

Z, which satisfying the

above equation (2.101) defines the centrally extended Virasoro algebra. As the term is

said to be central because it is a constant and commute with all other operators in the

algebra. There is no central term for m = 0 and m =

±1.

These are the complete commutation relations, here i, j are the transverse indices.

Closed strings Virasoro operators and their commutation relations have the same tech-

niques as we have done for open strings [12, 15].

2.8

Lorentz generators and critical dimensions

Lorentz invariance of a string action allows us to find a set of conserve word sheet

currents, M

and the resulting word-sheet charges M

for open strings with

[0, ],

are

(, )d =

- X

(2.102)

Now putting the value of P

from equation (2.15) so we get

- X

(2.103)

Now using the explicit mode expansion for and from equation (2.26), by simplifying we

get

= x

- x

- i

n=1

-n

)

(2.104)

where the 1st two terms are due to orbital angular momentum while the summation

terms are due to the angular momentum due to excited oscillator modes [12]. This

equation (2.104) is the classical Lorentz generators of open strings in the terms of

oscillators. Similarly for closed strings, the Lorentz generator becomes

= x

- x

- i

n=1

(

-n

)

- i

n=1

-n

- ¯

-n

)

(2.105)

-l

is the most delicate quantum Lorentz generators in light cone gauge because, X

coordinate is a non-trivial function of the transverse coordinates X

and a consistent

-l

should be generates the Lorentz transformations on the strings coordinates. As

Lorentz algebra is not generally reproduces by generators , M

which implies that the

theory is not Lorentz invariant, so to make it invariant, we should

-l

, M

-j

= 0

(2.106)

As for point particle [J

, J

] = 0, as J is the Lorentz generator for point particle.

Since from above equation (2.104), we have

-l

- x

- i

n=1

(

-n

)

(2.107)

As we know that the Lorentz generators should be Hermitian and normal ordered but

as from equation (2.107) we write

-l

)

= M

-l

So making it Hermitian by writing

+ p

) instead of x

, we find

-l

+ p

)

- i

n=1

(

-n

)

(2.108)

Now it's fully Hermitian and it's also normal ordered because of

are normal ordered.

To make it complete Lorentz generator, we should put the definition of P

and

from

equation (2.87) and (2.84), so our Lorentz charge M

-l

becomes

-l

= x

4 p

+ a) + (L

+ a)x

2 p

n=1

-n

)

(2.109)

The above equation (2.109) is our Lorentz charges in light cone gauge, in the order

to make it Lorentz invariant, we should calculate the above commutation relation in

equation (2.106) and then simplifying, we got

-l

, M

-j

m=1

(

-m

)

(2.110)

With

which is

m 1

- 2) +

- 2) + a

(2.111)

As for Lorentz invariance, the above equation (2.110) should be equal to zero, so this

means

m 1

- 2) +

- 2) + a = 0, m Z

By solving this we get

D = 26

And a =

-1. This shows that the relativistic strings can be properly quantized in the

flat Minkowski space if the number of space-time dimensions is 26 [3, 13, 16, 17, 18].

A similar calculation can fix the dimensionality of space time to the value D = 10 in

super strings.

2.9

State space and mass spectrum

The mass of relativistic string (that perform an arbitrary motion) can be calculated

from the mass operators, which is the relativistic equation in light cone coordinates, as

-p

= 2p

- p

Now writing the mass operator in the term of Virasoro operators (open strings), as from

equation (2.55) putting the value of 2p

, with normal ordering constant (a =

-1)

we find

-1 +

n=1

(2.112)

As the sum inside the bracket in the Number operator, like

n=1

(2.113)

So the mass operator then becomes

(

-1 + N

)

(2.114)

And we can easily calculate the commutation relations of number operator with oscil-

lators, which are

, a

= na

, a

-na

(2.115)

Now let us define our ground state of quantum strings. We have started the quantization

from our basic operators as in equation (2.63) and from this we have canonical pairs

, p

) and (x

, p

) so we can define the ground state from this pairs, as it is usually

convenience to work in momentum space so the ground states is

, p

(2.116)

These are the ground states of string for all values of momenta indicated by the labels

and also called the vacuum states for oscillators in string theory. We can create states

from the ground states by simply acting the creation operator on the ground states. As

we have an infinite numbers of creations operator, for which we can write the general

basis state

| of the state space, so

| =

n=1

l=2

n,l

, p

(2.117)

Here

n,l

is the positive integer and is define as the number of times that a

(creation

operators) appears. And as the number operator acts on the basis state, so their Eigen

value will be

| = N

| , with N

n=1

l=2

n,l

Similarly for closed strings, the states becomes

, ¯

n=1

l=2

n,l

, p

m=1

j=2

m,j

, p

(2.118)

And the number operators are

n=1

And ¯

m=1

(2.119)

From the zero modes of transverse Virasoro operators, the level matching condition is

= ¯

. Some of states and the mass spectrum of both open and closed strings are

given in the below tables.

List of some open string states

List of some closed string states

Since from the closed string spectrum, we get a particle named graviton which shows

that the gravity emerge into string theory [3].

SUPERSTRINGS

The bosonic string theory has some unsatisfactory features, like the spectrum of the

closed-string contains a tachyon particle. Open strings spectrum also contains tachyons

which are unphysical because of implying instability of the vacuum. As the eliminations

of the open string tachyons has been well understood in the term of D-branes's decay,

while, the closed-strings tachyon has not been understood yet [7].

And also the spectrum of the open as well as closed strings does not contain fermions.

With out fermions, the theory is unrealistic. To make a realistic theory which describes

the nature and all particles (Bosons and Fermions), we required a supersymmetry, which

relates the bosons and fermions, and the resultant theory is called superstring theory.

There are two basic approaches to develop the super string theories [3].

1. The Ramond-Neveu-Schwarz (RNS) formalism, which is the supersymmetric on

the string world sheet.

2. The Green-Schwarz (GS) formalism, which is supersymmetric in ten-dimensional

Minkowski space-time.

These approaches are equivalent at least for ten-dimensional Minkowski spacetime. This

chapter will describe the RNS formalism.

This version of string theory will be constructed along the same line as for the bosonic

theory i.e. we will consider the classical action, the solutions to the equation of motion

and their mode expansions and then we will apply the canonical and light cone gauge

quantization procedures [19].

3.1

The super string Action

To get fermions in our theory we introduce a new dynamical world-sheet variables

(, ) , and

(, ) as like for bosons the dynamical word-sheet variable is X

(, ).

These classical variables are

(, ) with ( = 1, 2) are anti-commuting variables,

rather than commuting variables.

In light cone gauge we set X

proportional to and X

was solved for, in terms of

other quantities. While in the case of superstrings, this remains the same but in addi-

tionally the light cone gauge condition also set

= 0 and then allow us to solve for

. As X

and

, both gets contributions from transverse X

and

, So this means

that both contributes into the light cone Lorentz Generator M

-l

, And from this we

can fixed the space time dimensions, which is D = 10 for super string.

Now let us start from the classical action that describes the full set of degree of freedom.

As we will use the light cone gauge which concern with transverse field, so we can write

the action in the term of transverse field

S =

- X

) + S

(3.1)

With

(

)

(

)

(3.2)

Here S

action is the Dirac action for fermions, which live in the two dimension world-

sheet [7, 20].

3.1.1

Equations of motion and Boundary conditions

In the order to find the equations of motion and the boundary conditions, we will vary

the fields

in action S

. So we have

(

)

(

)

(

)

(

)

(3.3)

By simplifying we can get the equations of motion and the boundary conditions, as

= 0 which gives the equations of motion

(

)

= 0, (

)

= 0

(3.4)

This implies that

is the right moving while

is left moving such as

(, ) =

(

- )

(, ) =

( + )

(3.5)

Similarly from S

= 0, the boundary conditions becomes

)

) = 0

(3.6)

This should hold for both the end points,

= 0 and

= for all . As from above

equation (3.6) we can also write

) =

) for each end point and by this,

with out loss of generality, we take

(, 0) =

(, 0)

(3.7)

And the relative sign between

and

become important for other end of string. So

we have

(, ) =

(, )

(3.8)

Let we have a fermion field,

which is define over an interval

[-, ] then the

boundary conditions can be written as

(, )

[0, ]

-) [-, 0].

(3.9)

And finally, from the relative sign, we have the two conditions i.e. periodic and anti

periodic fermions

. which are

(, ) = +

(3.10)

(, ) =

(3.11)

These are the two sectors or boundary conditions for fermion field and we will call

them, Ramond (R) sector for periodic as in equation (3.10) and Neveu Schwarz (NS)

sector for anti periodic fermion

as in equation (3.11).

3.2

Neveu Schwarz sector

As Neveu Schwarz fermion is the function of

- and has changed the sign when

+ 2, so the mode expansion can be written as

(, )

rZ+1/2

-ir(-)

(3.12)

Here

is an anti periodic, for any r = n +

with n is an integer.

is an anti commuting `field which means that the expansion coefficients b

will be

then anti commuting operators, and for the negatively r like b

-1/2

, b

-3/2

, b

-5/2

,..., are

the creation operators, while for positively like b

1/2

, b

3/2

, b

5/2

,..., are the annihilation

operators. And these operators will also satisfy the anti commuting relation, like

, b

r+s,0

(3.13)

Similarly these operators will act on the ground or vacuum which we called Neveu

Schwarz vacuum or simply

|NS . And the states in the Neveu Schwarz sectors are

| =

l=2

n=1

(

-n

)

n,l

j=2

r=1/2,3/2..

-r

)

r,j

|NS p

, p

(3.14)

This is the full ground state which is in the product,

of the ground state |p

, p

for

-n

and

|NS for b

-r

operators.

For the NS sector the mass squared operator with out normal ordering, is given by

p=0

-p

rZ+1/2

-r

(3.15)

Now to find the normal ordering constant, we use the same procedure as we have done

for Bosonic string theory. As for bosonic oscillators,

each coordinate contribute

to the normal ordering constant, " a ". So we take this as

Similarly for NS sector fermions the contribution term in mass squared operator M

will be

r=-1/2,-3/2,..

-r

r=1/2,3/2,..

-r

- (D - 2)

+1/2

(3.16)

And as

r=1

r =

odd

r +

Even

r =

odd

r + 2

r=1

From this we write

odd

+1/2

r =

r=1

r =

(3.17)

So the above equation (3.16) then becomes

r=-1/2,-3/2,..

-r

r=1/2,3/2,..

-r

- 2)

(3.18)

above equation (3.18) gives the idea that the contribution term for NS fermions is

N S

And the full normal ordering constant for M

is written as

a = (D

- 2)(a

+ a

N S

) =

-(D - 2)

(3.19)

For D = 10 then from above equation (3.19), a =

and hence the mass squared

operator becomes

+ N

, with N

p=1

-p

r=1/2,3/2,..

-r

(3.20)

As the fermionic oscillators are contributing half integers N

to so by this we get some

of the first few states in this NS sector are as

, N

= 0 :

|NS p

, p

= 0 , N

: b

-1/2

|NS p

, p

, N

= 1 :

-1

, b

-1/2

|NS p

, p

= 1 , N

-1

-1/2

, b

-3/2

, b

-1/2

|NS p

, p

(3.21)

The ground state (N

= 0) of the NS sector is a tachyon. The first excited states

) are massless and are eight states labeled by transverse index l.

As from this list we have both boson as well as fermions. As from above list (3.21)

States which have an even fermion number are bosonic while odd fermion number are

fermionic. So let us define an operator which distinguishes bosons and fermions from

one another. This operator will be called as (

-1)

, where F stand for fermion number.

The operator has plus one value on the bosonic state while minus one value for fermionic

state. The fermions ground state is then written as

(

-1)

|NS p

, p

- |NS p

, p

(3.22)

Similarly the operator (

-1)

acts on the state, as in equation (3.14)

(

-1)

| = -(-)

r,j

(3.23)

Hence this result allow us to take (

-1)

operator as an anti commuting with all other

fermionic operators, like

(

-1)

, b

= 0

(3.24)

The fermionic or bosonic character of the states is so far restricted to the (, ) world-

sheet [7, 21]. We will discuss later that these states are fermions or bosons in spacetime.

3.3

Ramond sector

Now the Ramond boundary conditions (3.10) in which field

is a periodic and can be

expanded in the terms of oscillators

(, )

-ir(-)

(3.25)

is an anti commuting, so the Ramond oscillators d

will be also anti commuting

operators. And the negatively like d

-1

, d

-2

, d

-3

,..., are the creation operators, while

for positively like d

, d

,..., are the annihilation operators. Similarly the Ramond

oscillators d

will satisfy the anti commutation relations

, d

m+n

l j

(3.26)

As from this (3.26) the zero modes, d

should be treated with care. It turns out the

idea that these eight operators will have to organize by the simple linear combination

of the four creation and four annihilation operators. Let us we have the four creation

operators out of these eight operators

(3.27)

As being the zero modes, these creation operators will act with out changing the mass

squared state. Let we have a vacuum

|0 , since the creation operators in (3.27) can

give the 2

degenerate Ramond ground states. Since we have total 16 ground states in

which eight of them will have an even number of s acting on

|0 , while the other eight

will have an odd number of s acting on

|0 . Let the eight states which have an even

number of creation operators is

with a = 1, 2, ..., 8 then

|0 ,

|0 .

(3.28)

Similarly the other eight states which have an odd number of creation operators is

with ¯

a = ¯

1, ¯

2, ..., ¯

8 then

|0 ,

|0 .

(3.29)

Hence

and

states makes the complete set of the degenerate Ramond ground

states and can be denoted as

with A = 1, 2, ..., 16 . The state space of Ramond

sector can be written as

| =

l=2

n=1

(

-n

)

n,l

j=2

r=1/2,3/2..

-m

)

m,j

, p

(3.30)

Just like as in NS sector, the Ramond sector has also an operator,(

-1)

which is anti

commuting with all the other fermionic oscillators, including the zero modes

(

-1)

, d

= 0

(3.31)

Conventionally, we declare

|0 to be fermionic, like

(

-1)

|0 = - |0

(3.32)

The operator gives the idea that (

-1)

states are fermionic while

states are bosonic.

In the R sector

, the mass squared operator without normal ordering can be written

p=0

-p

-n

(3.33)

Similarly for R sector, the contribution term in M

n=-1,-2,..

-n

n=1,2,..

-n

- 2)

(3.34)

Since the above equation (3.34) gives the idea that the contribution term for R fermions

Hence the contribution term for R sector a

is equal (with opposite sign) to the contri-

bution term for bosonic a

. So the total normal ordering constant becomes zero, thus

we can write the mass squared operators as

-p

+ nd

-n

(3.35)

This implies that the Ramond ground states are massless and some of few excited states

in this R sector are as

= 0

= 1

-1

, d

-1

= 2

-2

-1

, d

-1

{

-2

-1

, d

-1

} |R

{

-1

-2

} |R

{

-1

-2

} |R

(3.36)

As we have separated the states into the two groups which have an identical number of

states, the left of the bars states gives (

-1)

-1 (fermionic states). While the right of

the bars states gives the bosonic states i.e. (

-1)

= +1 . And hence for each fermionic

state, there is a corresponding bosonic state which is the supersymmetry [7, 21]. But

however, this supersymmetry is on the world sheet and the space time supersymmetry

will be arise by combining the states from both, Ramond and Neveu Schwarz sectors.

3.4

Super transverse Virasoro operators

For quantization of superstring theory, we needed the super Transverse Virasoro op-

erators which are the generalization of the Transverse Virasoro operators. As before

we find the transverse Virasoro operators for bosonic string theory, the similar calcula-

tion will be required for the superstring [13, 22]. By including the supersymmetry, the

Virasoro operators will become the super Virasoro operators and can be written as

= L

(a)

+ L

(b)

NS sector

= L

(a)

+ L

(d)

R sector

Here L

(a)

is the transverse Virasoro operator which can be written as

(a)

n-p

And the L

(b)

is terms is due to NS sector while L

(d)

term is for R sector, which can

be written as

(b)

rZ+1/2

(r +

-r

n+r

and L

(d)

(m +

-n

n+m

These are helpful in quantizing the theory.

3.5

Counting states

In the order to count the number of states at any given mass squared operator. For

this we need a generating function which contains the information about the num-

ber of states.

As for bosonic string theory (open string), we have the oscillators

, a

, ..., a

), then the generating function can be written as

f (x) =

n=1

- x

(3.37)

As there are 24 transverse light cone directions for each oscillator, since each will have

a generating function. So the complete generating function for bosonic string theory

can be written as

f (x) =

n=1

- x

)

(3.38)

This is the generating function for the bosonic open string or simply for N

. Since the

mass squared

, is more physical than the number operator N

. So we need to

define the generating function by using

instead of N

. As for bosonic open string

= N

- 1 then for this the generating function will be obtain by dividing the

above generating function (3.38) by the one power of x. Hence the generating function

for the bosonic open string theory, f

(x) can be written as

(x) =

n=1

- x

)

(3.39)

We can also expand it as

(x) =

+ 24 + 324x + 3200x

+ 25650x

+ ...

(3.40)

These are the states of the mass squared operators i.e. 24 massless states of photon

and so on.

Similarly the generating function for Neveu Schwarz is written as

(x) = 1 + x

As there are 8 transverse light cone directions for each oscillators b

-1/2

, b

-3/2

, ... , since

each will have a generating function, and also the M

= N

for NS sector, so the

complete generating function for NS sector can be written as

N S

(x) =

n=1

(1 + x

)

- x

)

(3.41)

And by expanding we find

N S

(x) =

+ 8 + 36

x + 128x + 402x

x + ...

(3.42)

These are the states of the mass squared operators i.e. 8 massless states.

Now for Ramond sector, as M

= N

, since the generating function becomes

(x) = 16

n=1

(1 + x

)

- x

)

(3.43)

The overall 16 factor is due to each combination of the Ramond oscillator which gives

rise to the 16 states as by acting on each ground state. By expanding, we find

(x) = 16 + 256x + 2304x

+ ...

(3.44)

We notice that the Ramond coefficients (3.44) are actually double to the corresponding

Neveu Schwarz coefficients (3.42). We will discuss this later.

3.6

Open superstrings and the GSO projection

As from the Ramond sector, we see that the word sheet has a supersymmetry i.e. the

ground state which is divided into two groups of

and

, which built from the

zero modes d

, and has an equal number of fermionic and bosonic states having the

opposite value of (

-1)

As the zero modes, d

carry the Lorentz index vector and can transform under Lorentz

transformation but the Ramond ground states (

) do not transform like a

vectors but transform as a spinors. Which shows that the index a, ¯

a are the spinor

indices but are different spinors (and different fermions).

This means that there are two types of fermions in R sector, but both of them gives the

opposite values of (

-1)

and with two types of fermions we cannot get any space time

supersymmetry because we identified

as a space time bosons, but bosons cannot

carry any kind of spinor index.

Now to solve this issue as, by projecting the spectrum of R sector in a specific way in

which we choose only those fermions which have (

-1)

-1, by this we can get space

time fermions, and this projection is called Gliozzi, Scherk, and Olive (GSO) projection

[3, 7, 21]. Hence the resultant projected R sector is then called the R

- sector (R minus)

which have (

-1)

-1. Similarly, the , is the set of states for which (-1)

= +1. So

the generating function for R

- sector then becomes as

R-

(x) = 8

n=1

(1 + x

)

- x

)

(3.45)

By expanding, the power series of (3.45) is then

R-

(x) = 8 + 128x + 1152x

+ ...

(3.46)

So we see that there are eight fermionic massless states in R

- sector. Similarly doing

the same thing for NS sector, hence, we find sector states for which (

-1)

-1 and

N S+ sector which have (

-1)

= +1 . Since the N S

- sector contain a tachyon, hence

it is useful to take the full open superstring by combining the set of states from the R

sector and N S+ sector.

Now to find the generating function for N S+ sector f

N S+

(x) by taking account of GSO

projection which eliminate the contribution of even number of fermions, so we get the

generating function for N S+ sector as

N S+

(x) =

n=1

(1 + x

)

- x

)

n=1

- x

)

- x

)

(3.47)

For supersymmetry, we need to have f

N S+

(x) = f

R-

(x) so this means that

n=1

(1 + x

)

- x

)

n=1

- x

)

- x

)

= 8

n=1

(1 + x

)

- x

)

(3.48)

This identity is being proved by Carl Gustav Jacob Jacobi in his work on elliptic

functions, which is published in 1829. While in our case, for the basis of supersymmetric,

it is a key equation.

3.7

Closed string theories

As we know that the closed (bosonic) strings can be obtained by the combining the

left moving, and the right moving copies of the open strings. Similarly the closed

superstrings can be obtained by combining the open superstrings. Since there are two

sectors (NS and R) for open superstring, so there are four possibilities for the closed

string sectors, like

(NS, NS), (NS, R), (R, NS), (R, R)

(3.49)

As in the case of open superstrings, the space time bosons arises from NS sector while

the space time fermions arises from R sector. Similarly, in the case of closed superstring,

we can get the space time bosons from (NS, NS) as well as from the (R, R)

sectors

while the space time fermions from (R, NS) and (NS, R) sectors.

The (R,R) sectors are "doubly" fermionic and thus the spacetime bosons

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 +

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

= M

+ M

(3.52)

Here M

and M

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 + ) : ¯b

-1/2

|NS

-1/2

|NS

, p

(3.53)

(NS + , R + ) : ¯b

-1/2

|NS

b R

, p

(3.54)

- , NS + ) : |R

a L

-1/2

|NS

, p

(3.55)

- , R + ) : |R

a L

b R

, p

(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

(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

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 +

Right sector :

NS +

(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 + ) : ¯b

-1/2

|NS

-1/2

|NS

, p

(3.60)

(NS + , R

- ) : ¯b

-1/2

|NS

b R

, p

(3.61)

- , NS + ) : |R

a L

-1/2

|NS

, p

(3.62)

- , R - ) : |R

a L

b R

, p

(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.

(R - R ) sector: This sector gives the space time bosons and in the case of type IIA

superstring theory, the massless bosons contain the Maxwell field (eight states) and the

anti-symmetric gauge field with three index (56 states), while the type IIB superstring

theory, the massless bosons contain the scalar field (one state), the Kalb Ramond field

(28 states), and the totally anti-symmetric gauge field with four index (35 states) [3, 7].

(R - R) the massless fields of type IIA : A

, A

(3.64)

(R - R) the massless fields of type IIB : A, A

, A

(3.65)

These are the some massless field of R sectors.

3.7.3

Heterotic superstring theories

There are two types of Heterotic superstring theories. These are the closed superstring

theories. in the Heterotic string, we will combine the left moving bosonic open string

with the right moving open superstring as like the type II closed superstring theories

which arises by the combination of both left and right moving copies of the open super-

strings. As there are 26 space time dimension of open bosonic string theory, ten of them

are matched by the right moving open bosonic coordinates with the open superstring

[3]. The extra 16 left moving coordinates can be described a torus with a very special

properties to gives a consistent superstring theory. There are precisely two distinct tori

which have the require properties and they will corresponds to a Lie algebras SO(32)

and E

× E

. By this we can get a consistent theory which lives in a 10 dimensional

space time. Heterotic superstrings theory comes into two versions, E

× E

type and

SO(32) type. These groups characterize the symmetries which exist in the theories.

The group SO(32) is the group generated by 32-by-32 matrices which are the orthogonal and

having a unit determinant.

is the largest exceptional group, here E is for the exceptional.

3.8

Type I

We have discussed the oriented string theory

in which the operator X

(, ) involves

a parameter

[0, ] i.e. both the type II superstring theories and the Heterotic

superstring theories are the theories of oriented closed strings. We can also construct a

theory that will be an unoriented string theory. For this, we define an operator which

can reverse the orientation of the strings [3, 7]. The unoriented strings can be obtained

by restricting the oriented strings spectrum to the set of the states which are invariant

under the action of . Unoriented strings are not the strings with out the orientation

i.e. they viewed like the quantum superposition of states which as a whole are invariant

under the action of . Hence, we can imagine the unoriented state as a superposition of

a string states and the same states with opposite orientation. A supersymmetric theory

of both open and closed unoriented strings is called the Type I superstring theory.

These are the five different superstrings theories which can be relate by the dualities.

3.9

Critical dimensions

For bosonic string theory, we have fixed the space time dimension by using the com-

mutation relation of the Lorentz generators, M

-l

, M

-j

= 0

The same idea will be used here as we used before in bosonic theory [7]. Now will take

the super Generators, by the same phenomenology we can construct the super Lorentz

generators. For NS sector the super Lorentz generator becomes

-l

= x

4 p

+ a) + ((L

+ a)x

2 p

n=1

-n

)

2 p

r=1/2

r(L

-r

- b

-r

)

The orientation is defined as the direction of increasing of .

And similarly, the super Lorentz generator for R sector can be written as

-l

= x

4 p

+ a) + ((L

+ a)x

2 p

n=1

-n

)

2 p

m=1

m(L

-m

- d

-m

)

By calculating the above commutation relation for N S sector the super Lorentz gener-

ator, we get the non vanishing result which is

-l

, M

-j

m=1

(

-m

)

with

m 1

- 2) +

- 2) - 2a

As for Lorentz invariance, the above equation should be equal to zero and for this

the dimension of space time, D = 10 and the normal ordering constant, NS sector

. and by the similarly method, for the super Lorentz generator R sector

= 0.

D-BRANES

A Dp-brane (with p spatial dimensions) is an extended object. The letter D stands

for Dirichlet. When p is equal to the total number of spatial dimensions, then this

type of Dp-brane is called space filling brane. In bosonic string theory, D25-brane is a

space filling brane because p is equal to the total number of spatial dimensions. The

endpoints of open strings must lie on the D-brane.

Similarly, when the Dirichlet conditions applied to the subset of the d indices of X

i.e. to the indices p + 1, ..., d, this means that the endpoints of the string are restricted

to move on a p-dimensional hyper plane. This higher dimensional object is then calling

a Dp-brane, where p indicates its (spatial) dimensions and D stands for Dirichlet.

It turns out that these objects should be considered to be dynamical as well.

4.1

Tachyons and D-brane decay

In bosonic string theory there is a particle having imaginary mass, called tachyon. The

open string is attached to a D-brane. The tachyons make the D-brane unstable [7, 23].

In the order to study the tachyons and D-brane decay, let start from the scalar field

because the field associated with the tachyons are the scalar field and the Lagrangian

for a scalar field can be written as

L = -

- V ()

Here V () is the potential of scalar field. From scalar field Lagrangian, we can find the

equation of motion

- V () = 0

As the scalar field potential can be written in the form of mass, as

V () =

As when M

> 0 then the potential V () will have the stable minimum at = 0 while

when M

< 0 then the potential V () will have the unstable maximum at = 0. For

simplicity, let us consider the field only depends on time, then the equation of motion

becomes as

(t)

+ M

(t) = 0

Now when the mass squared is greater than zero then the solution of the above equation

become

= A sin(M t + a

)

As due to the stable point, the scalar field could be sitting at = 0 forever and will be

simply oscillate whenever it is displaced [24]. Now consider the imaginary mass squared,

like M

, where is positive, in this case the equation of motion becomes as

(t)

(t) = 0

The solution of this equation become as

(t) = A cosh(t) + B sinh(t)

Let the solution (t) = sinh(t). As when time is zero, (0) = 0 but when time goes

to infinite then (

) = . This can be imagining as the field is rolled up. By

using the trivial solution, we can say that the tachyon will stay at = 0 but any small

perturbation could make it to roll-off. The point = 0 is an unstable point for tachyon

and tachyon cannot stay here for some definite time, which shows the instability. As

the mass squared of tachyon in an open string theory is

Since the potential of the free tachyon becomes

f ree

tach

() =

As the presence of the tachyon is the signal of instability of the open string theory. As

the open string is attached to the D-brane, since we can say that there is instability in

the background of space filling D-branes. In bosonic open string theory, the space filling

D-brane is D25-brane. As it is a physical object and its have some energy density T

[7].

Tachyons are the states of an open string which is attached to D-brane this means that

the tachyons are the excited states of D-brane and by this, we can say the tachyons

states will lower the energy of D-brane. Explicitly, the existence of tachyons means

that the space filling brane, D25-brane is unstable.

4.2

Quantization of open strings in the presence of

various kinds of D-branes

4.2.1

Dp-branes and boundary conditions

Considering a Dp-brane and let introduces the spacetime coordinates x

, with =

0, 1, 2, .., 25 for bosonic string theory, splitting these coordinates into two groups i.e.

tangent to the brane and normal to the brane. As the tangential coordinates includes

the time coordinate as well as let some spatial coordinates p , then the normal to the

brane includes (D

- p) coordinates. We can write it as

, x

, .., x

tangential coordinates

, x

p+1

, x

p+2

, x

p+3

, .., x

normal coordinates

(4.1)

For simplicity we let x

= ¯

, with (a = p + 1, ..., d). We can also write the above

equation (4.1) for string coordinates X

(, ) in similar fashion, like

, X

, .., X

tangential coordinates

, X

p+1

, X

p+2

, X

p+3

, .., X

normal coordinates

(4.2)

Since the open string attached with the Dp-brane and the end points must lies on this

Dp-brane then the normal coordinates of string to the brane will satisfy the Dirichlet

boundary conditions such that

(, )

= X

(, )

= ¯

, a = p + 1, ..., d

(4.3)

The coordinates X

will call DD coordinates due to these satisfying the Dirichlet bound-

ary condition on both the end points. The end points of the open string can move ev-

erywhere along the tangent direction to the brane. By this the open string coordinates

which is tangent to the D-brane will satisfy the Neumann boundary conditions, as

(, )

= X

(, )

= 0 , m = 0, 1, ..., p

(4.4)

These call NN coordinates due to both of the open string endpoints satisfying the

Neumann boundary condition. Hence we can summarize this as

, X

, .., X

NN coordinates

, X

p+1

, X

p+2

, X

p+3

, .., X

DD coordinates

(4.5)

Now in the order to use the light cone coordinates we need to have at least one of the

spatial NN coordinate from which we define X

coordinates. So this means that we

will assume p

1 and our analysis will not apply to the strings which will attach to a

D0-brane. Hence in light cone coordinates, we may write

, X

}

, i = 2, ..., p and a = p + 1, ..., d

(4.6)

4.2.2

Quantizing open strings on Dp-branes

As before we have quantized the open string in the presence of the space filling D-

brane and now we will quantize the open string in the presence of the Dp-brane and

to study the effects on string spectrum states due to the presence of the Dp-brane. As

the NN coordinates X

(, ) will also satisfy the same conditions that were satisfied

by the light-cone coordinates X

(, ) of the open strings which were attached to the

D25-brane. Since our previous results are useful for the coordinates X

. Let start from

equation (2.47) and let l

(i, a) then we finds

±X

(

± X

)

+ (

± X

)

(4.7)

As X

(, ) are satisfying exactly the same conditions which satisfied by X

(, ) so by

this we can write

± X

-in(±)

(4.8)

Now to investigate the normal coordinates X

(, ), as these coordinates are normal to

the brane will satisfy the wave equation. Now solving the wave equation for X

, we

get

(, ) = ¯

n=0

-in

sin n

(4.9)

And by this we can easily finds

n=0

-in(±)

(4.10)

This is same as equation (4.8) but differ by a minus sign and a zero mode which is

absent here in equation (4.10).

We can easily quantize this string which is attached with a Dp-Brane. As P

then the non vanishing commutation relations are

(, ),

(, ) = i2

(

- )

(4.11)

Similarly we can easily find that

= m

m+n,0

(4.12)

The mass squared operators becomes as

-1 +

n=1

m=1

(4.13)

Here ¯

is not an operator but a number.

Here the repeated index will be summed, the normal ordering constant is (a =

-1),

and the critical dimensions are same as for D25-brane. As l

(i, a) since our ground

state will not remain the same as D25-brane. Since for Dp-brane, the ground state will

be labeled by p

, p

and then becomes

, p as with p = (p

, ..p

). Now the state

space are written as

| =

n=1

i=2

n,i

m=1

a=p+1

m,a

, p

(4.14)

The associated wave functions take the form

1...

(, p

, p)

Since some of the fields that satisfy M

0 associated with Dp-branes are, the scalar

field (tachyons), the massless Maxwell field (photons) a

, p and similarly the os-

cillator that acts on the ground states from those coordinates normal to the Dp-branes

are a

, p , these are the massless the Lorentz scalar (due to the index ) which are

normal to the branes.

4.2.3

Open string between parallel Dp-branes

Now considering the open strings which is extends between the two parallel Dp-branes.

Let the first Dp-brane is located at x

= ¯

'while the second one is at x

= ¯

. The

classes of the open string which are supported on the particular configurations of the

D-branes are called sectors. In the presence of a two parallel Dp-branes, an open string

can begin from one brane and end on the other brane. There are four sectors in our

case of two parallel Dp-branes.

Consider the sector, in which open strings begins on the brane one and end on the

brane two. For this, the boundary conditions for the DD strings coordinates are

(, )

= ¯

, X

(, )

= ¯

, a = p + 1, ..., d

(4.15)

Similarly for X

which are normal to the branes and will satisfy the wave equation. By

solving the wave equation for X

, we find

(, ) = ¯

+ (¯

- ¯x

)

n=0

-in

sin n

(4.16)

From this we get

-in(±)

and

(¯

- ¯x

)

(4.17)

Similarly the mass squared will becomes

- ¯x

-1 + N

(4.18)

Where

n=1

i=2

m=1

a=p+1

(4.19)

To develop the ground state for our parallel Dp-branes, we need additional labels which

distinguish the various sectors [7]. These additional labels will be two numbers. The

first number will denote the brane on which the open string end point = 0 lies while

the second number will denote the brane on which the other end point = lies. Since

the ground states can be written as

, p; [ij] and the four sectors are

, p; [11] , p

, p; [22] , p

, p; [12] , p

, p; [21] .

(4.20)

Here the first two sectors are those which we discuss earlier i.e. ¯

= ¯

. Let us

first discuss the [12] sector, the mass squared operators for the lowest value of number

operators becomes

, p; [12]

, M

- ¯x

When the separations between the two branes vanished then the field associated is a

scalar tachyonic field. From this we can find the critical separation which is

|¯x

- ¯x

| = 2

Hence the ground states represents the massless scalar field for the zero separation while

for large separation, the ground states represents the massive scalar fields. The next

states (due to the number operators) are

, p; [12] , M

- ¯x

These are the (d

- p) Lorentz scalar which is normal to the branes.

, p; [12] , M

- ¯x

These are the (p + 1)

- 2 = p + 1 massive states. As we know that the massive gauge

fields has more degree of freedom than the massless gauge field. Hence from this we

have one massive vector as well as (d

- p - 1) massive scalars.

We have obtained a very interesting situation, as the separation between the Dp-branes

goes to zero and coincide but still they are distinguishable and we have four open string

sectors. The massless string states that represents a strings extending from Dp-brane

one to Dp-brane two, includes the massless gauge field and (d

- p) states of a massless

scalars. This is same content as that of a sector in which strings begin and end on

the same Dp-brane. Whenever the two D-branes coincides, we get four massless gauge

fields. From the world-volume of two coincide D-branes we indeed to get a U(2) Yang-

Mills theory.

Let we have N Dp-branes, for this the sectors will be labeled by the pairs [i, j]

, here

i = 1, 2, ..N . The [i,j] sector will consist of the open strings which will start on ith and

end on jth brane and we have N

sectors. The interaction of strings can be visualized

as The first open string will join with a second open string and will form another open

string. For this, the end point of first string ( = ) will join the beginning point of the

second string ( = 0). And the new string (became from these two) will begin at the

beginning point of the first string and will end on the end point of the second string

[7, 25]. If the strings stretched between the D-branes, then the first string will from

The discrete labels

i, j are used to label the branes, and the various open string sectors are

sometimes called Chan-Paton indices.

the sector will have to join by the second string from sector to gives the product open

string in the sector. The interaction can be written as

[i, j]

[j, k] = [i, k], here j is not summed

If the N Dp-branes are coincides, then the N

sectors results in the N

interacting

massless gauge fields. Since from this N coinciding D-branes will carry U(N) massless

gauge fields.

4.2.4

Strings between parallel Dp and Dq-branes

Now considering the two parallel D-branes, but having different dimension. Let we have

two D-branes which are Dp-brane and Dq-brane and assume that p > q. From this,

there will be a p-dimensional hyperplane parallel to the Dp-brane which will contain

the Dq-brane. This means that we will have some common tangent as well as normal

directions for both branes. For strings coordinates X

, we can write for parallel Dp-

brane and Dq-brane with p > q

, X

, .., X

common tangential coordinates

, X

q+1

, X

q+2

, .., X

mixed coordinates

p+1

, X

p+2

, .., X

common normal coordinates

(4.21)

Here the mixed coordinates will satisfy the Neumann boundary conditions on the start-

ing Dp-brane while the Dirichlet boundary conditions on the ending of the Dq-brane

and these coordinates will call ND coordinates. Similarly for the strings coordinates,

we can also write

, X

, .., X

NN coordinates

, X

q+1

, X

q+2

, .., X

ND coordinates

, X

p+1

, X

p+2

, .., X

DD coordinates

(4.22)

In light cone coordinates, we can write these as

, X

}

(4.23)

Where i = 2, ..., q , r = q + 1, ..., p and a = p + 1, ..., d . Now let for the position of the

Dp-brane is while for position of Dq-brane is ¯

and ¯

. The boundary conditions for

the ND coordinates X

are

(, )

= 0 , X

(, )

= ¯

(4.24)

The expansion of X

(, ), by using the same procedure, can be written as

(, ) = ¯

+ i

odd

-i

cos

(4.25)

The Hermiticity of the above expansion, X

gives (

)

. And by the above

expansion (4.16) we can find

± X

odd

-i

( ±)

(4.26)

The non vanishing commutation relations are

(, ),

(, ) = i(2 )(

- )

(4.27)

And by the similar method as we have done for space filling D-Brane, we can also find

m+n,0

(4.28)

Now as we have three types of coordinates like NN, DD, and ND or DN. The normal

ordering contribution term for both NN and DD coordinates is same because of all

oscillators are integrally moded

= a

(4.29)

However, for the ND or DN coordinates, the contribution term can be calculated as

odd

- q)

(4.30)

Since this means that the contribution term for ND or DN is

= a

(4.31)

Similarly we can easily find the total normal order constant, which is

a =

[q + 25

- (p + 1)] +

- q)

-1 +

- q)

(4.32)

Now the mass squared operator then becomes

- ¯x

- 1 +

- q)

(4.33)

With

n=1

i=2

odd

r=q+1

m=1

a=p+1

(4.34)

And the ground state are now labeled as

, p; [12]

, p = (p

, ..., p

)

(4.35)

The labels of the ground state indicates that the corresponding fields will be living in

a (q+1 ) dimensions of the space time i.e. the fields will live on the Dq-brane world-

volume, which has the lower dimensions. The ground states having N

= 0 corresponds

to a scalar field on Dq-brane. This scalar depends upon the separation of the branes.

4.3

String charge and electric charge

4.3.1

Fundamental string charge

As we know that, for the point particle, the word line is one dimensional and Maxwell

gauge field A

carrying one index and the point particle carries electric charges whenever

it interacts with Maxwell field, in which the particle couples to the Maxwell field.

Similar idea used for string charge, the string should couples to a new kind of gauge

field. This gauge field is the Kalb Ramond anti symmetric tensor (of rank two)B

which is the massless field and arisen in closed string. The complete dynamics of the

string coupling to Kalb Ramond field can be written in the term of action is

S = S

str

d dB

(X) (

)

x H

(4.36)

Here S

str

is the string action the second term is the action term due to string charge

while the last term is the action for the Kalb Ramond. And k is the dimensionful

constant which makes the action dimensionless i.e. ([k

] = M

-D

). The H

is the

field strength of B

which can be written as

(4.37)

In the second term of the above action (4.36), B

(X(, )) can be written for the

general space time B

(X) as

(X(, )) =

- X(, )) B

(x)

(4.38)

Now the variation of above action (4.36) gives us

S =

x B

(x)

- j

(4.39)

Here j

is the (anti symmetric) current

d d

- X(, )) (

)

(4.40)

By the variation principle, we find

= j

(4.41)

The tensor j

is conserved quantity like

= 0

(4.42)

As the above equation give the idea of conservation of j

but there is a free index,

which means that j

is the set of conserved current labeled by () a free index. And

the charge density, as the zeroth component of j

, since the charge density (¯

) of the

Kalb Ramond field is then j

, here k will be running over the spatial value. Similarly

from the above equation (4.42), we find

= 0 and from the charge density we can

easily find the charge of string, ¯

Q as

Q =

x j

(4.43)

To understand it more, using the static gauge X

= = t and then simplifying the

equation (4.40), we get

(x, t) =

d x

- X(t, ) X (t, )

(4.44)

From this, we get the idea that the charge density is tangent to the every point on

string and lies along orientation of the string.

In the order to write this in more explicit form that the string density depends upon

the orientation, for this let a long static string stretched along the x

, then the string

can be describe as

(, ) = f () , X

= X

= ... = X

= 0

(4.45)

Here f () is the function of with the range of

- to , this function will be either

increasing or decreasing. Now using the above equation (4.45) and solving the equation

(4.44), then we get

, ..., x

; t) =

sgn(f )(x

)(x

)...(x

) =

sgn(f )(x

)

(4.46)

Here sgn(f ) is for the sign of f . This result show explicitly that the charge density j

depends upon the orientation or the sign of f ().

4.3.2

Visualizing string charge

In the order to visualize the charge of string, let us consider we have a static string

for which j

= 0 ( i,k are spatial components), since for a static string, the equation

(4.41) becomes as

= 0

(4.47)

By this H are to be time independent so H

ijk

= 0 and the other equation is

okl

= k

(4.48)

Now let us introduce a new vector B

. This is called field strength dual to H and

which can be define as

0kl

klm

(4.49)

Here

ijk

is the anti symmetric Levi-civita symbol. Since the above equation (4.48)

then becomes

(

× B

)

= k

(4.50)

Now from this equation we can write

× B

= k

(4.51)

This equation is the Ampere's equation, where B

is the magnetic field. Integrating

the above equation (4.51) over a curve of a two dimensional surface S. so we get

.dl =

.da

(4.52)

By this equation, we get the idea that the curve is link to a string and the strings

number

N which is associated with is define as

N =

.dl =

.da

(4.53)

Here

N is the number of strings which is linked by the curve .

4.3.3

Strings ending on D-branes

As by the quantization of closed strings, there arises a massless field called Kalb Ramond

field which lives over the all spacetime and the string can couple electrically with it to

get charged. Now considering the charged string which attached to D-branes, for this,

the action for the string couple to Kalb Ramond field, is written as

d d

(X(, ))

(4.54)

Here

is the two dimensional indices , = 0, 1, anti symmetric with

= 1. Now

using the above action and calculating the variation in S

, which we find that

d d (

(

)

(

))

(4.55)

Here we use

(X) =

(4.56)

The arguments of are the string coordinates X(, )due to the arguments of B

As from equation (4.55), the two total derivative in which the time derivative give no

contribution, since this means that are vanishes at the end of time. For a closed string

both the derivative vanishes while for the open string, there introduces a boundary

contributions terms, which in not vanishes. In the order to find B

for an open

string, let the string coordinates along the brane are X

and the string coordinates

normal to the brane are X

. Since we can write this as

= (X

, X

) , = (m, a)

(4.57)

If this D-brane is Dp-brane, then (m = 0, 1, ...p) and solving the equation (4.55) for

= (X

, X

) coordinates, so the coordinates X

are DD for which

= 0 and

the limits we take the

[0, ] then we get

(4.58)

Here we get the two boundary terms and hence, the gauge invariance has failed. From

this we got the idea that the string charge conservation failed at the end points of the

string. In the order to restore the gauge invariance, we should add some terms which

give the electric charges to the end points of the string. then the Action can be written

S = S

d A

(X)

d A

(X)

(4.59)

As whenever we vary

(i.e. B

), then we have to also vary Maxwell

field A

on D-brane, like wise A

, and the two terms have opposite sign which

show that the end points of string, which lies on Dp-brane, are oppositely charged. This

action (4.59) restored the string charge conservation and we got the idea that the string

end points have the opposite charge due to the Maxwell gauge field A

and by this we

restore the conservation of charge invariance.

The Maxwell action is also not invariant, as it is proportional to F

, like

-B

(4.60)

To make it invariant, we should add B

into Maxwell field F

such that

+ B

= 0

(4.61)

This

is called the field strength on D-brane and the Lagrangian density on D-brane

is proportional to

, so expanding it

(4.62)

As the last term has some interesting physical significance, expanding it as

-F

+ ...

(4.63)

In the order to understand the physical significance of this term, let as the action for

string charge, second term in (4.36), can be written as

x B

(x) j

(x)

(4.64)

This equation tells us that string charge density j

couples to B

. Since anything

couple to B

will carry string charge so it means that F

will represent the string

charge on D-brane but as F

is equal to the electric field (F

= E

), which gives the

idea that the electric field lines will carry the string charge on D-branes.

4.4

D-brane charges and stable D-branes in Type

There are also other extended objects rather than strings that carry charge i.e. Dp-

branes. To study the charge of a Dp-branes, let start from our previous concept of

charge that the string carry charge when they couple to Kalb Ramond gauge field, so

this coupling is written as

d d

(X( ,))

(4.65)

Now generalizing this idea like, a Dp-brane will be electrically charge if it coupled to a

massless anti symmetric tensor field of indices (p + 1 ). As Dp-brane has (p+ 1 ) di-

mensions and is parameterize by and the set of its coordinates (

, ...,

). Let the

space time coordinates which describe the position of this brane is X

, ...,

)

with = 0, 1, ...d, and the anti symmetric tensor field is describe by A

...

(x) then

the generalize coupling from (4.65) is written as

d d

..d

...

, ...,

))

(4.66)

Kalb Ramond gauge field is the only massless anti symmetric tensor in bosonic closed

string theory and there is no other massless anti symmetric tensor which means that the

Dp-branes cannot be charged in bosonic string theory. While the type II superstring

theories have some additional anti symmetric massless tensor in (R - R ) sectors, as

listed in (3.64) and (3.65)

(R - R ) the massless fields of type IIA: A

, A

(4.67)

(R - R ) the massless fields of type IIA: A, A

, A

(4.68)

This means that A

coupled to the D0-branes and A

coupled to the D2-branes while,

coupled to the D1-branes and A

coupled to the D3-branes. And the field A does

not couple to any D-brane (it coupled electrically to the object called, D-instanton).

Summarizing these as

T ypeIIA : D0, D2,

(4.69)

T ypeIIB : D1, D3.

(4.70)

These branes are the stable charged and cannot decay into closed or open string states,

while the bosonic D-branes are unstable due to the existence of tachyons, and carry no

charge. Similarly in type IIA superstring theory, the Dp-branes with even are stable

but are unstable in type IIB superstring theory while, in type IIB superstring theory,

the Dp-branes with odd p are stable but are unstable in type IIA superstring theory.

All the stable D-branes of type II superstring theories are charged and the Dp-branes

which is not appeared in the above lists (4.69) and (4.70), like D4, D6, and D8 of type

IIA superstring theory while the D5, D7, and D9 of type IIB superstring theory, means

that these are carrying the magnetic charges for either (R - R ) gauge fields in listed

above in (4.67) and (4.68) or for the other (R - R ) states that we not includes in the

discussion [3, 7, 27, 28].

The (electric) charge of the Dp-brane has the simple physical significance, when the p

(spatial dimensions) are curled up to a circles and Dp-brane is wrapping around, the

resulting compact space. Since here, p compact space directions will lies on D-brane

and others space time directions, which defined here, the lower dimensional spacetime,

will be normal to the brane. Hence, the lower-dimensional observer which has only

access to the non compact directions and will see the brane as a point particle.

Let (x

, ..., x

) be the compact directions and (X

, ..., X

) be their corresponding coor-

dinates of brane. If compact directions are curled up to the circles of radii (R

, ..., R

)

and assuming the parameters

[0, 2] then

, ...,

) = R

, k = 1, ..., p

(4.71)

Here the repeated index k is not summed. This represent the wrapped Dp-branes and

the coordinates X

are running from zero to 2R

. Let the non compact dimensions

be the X

with m index is for the non compact directions, then

, ...,

) = x

( )

(4.72)

This equation shows that, Dp-brane appeared as the point particle to the observer

which has the lower-dimensions. These assumptions made the equation (4.66) as

d d

..d

...R

12...p

(X(,

, ...,

))

(4.73)

Since the will take the values over the non compact directions, = m, then the above

equation (4.73) becomes

d d

..d

...R

m 12..p

( ), X

(

)

(4.74)

Finally, let the part of A...field which is independent of compact coordinates i.e. A

m 12..p

( )),

then the above equation (4.74) becomes

-R

...R

d d

..d

m 12..p

( ))

(4.75)

By solving this integral, we find

( )

p/2

( )) =

)

(4.76)

Here V

is the volume of compact space, V

= (2R

)...(2R

) , And l

is the string

length l

= ( )

1/2

. And the gauge field ¯

( )

p/2

( )) = A

m 12...p

( )).

The equation (4.76) recognized that the coupling of the point particle to the Maxwell

field ¯

which means that the Dp-brane appeared as a charged point particle. From

this the charge Q of the brane is

Q =

)

(4.77)

The charge Q depends upon the volume of the branes [7].

STRING DUALITIES

Duality is generally used for the relationship between the two systems which have very

different descriptions, but identical physics. There are two types of dualities in string

theory, T-duality and S-duality.

In many cases, the T-duality implies that the two

different geometries of the extra dimensions are physically equivalent i.e. the circle of

radius (R) is equivalent to the circle of ( R

-1

) radius. As there are five superstring

theories which looks like the different theories from one another, but the T-duality

relates the two type II superstring theories and the two Heterotic superstring theories.

Similarly, S-duality relates, string's coupling constant g

to (g

-1

), in the same ways as

like the T-duality. S-duality relates the type I superstring theory to Heterotic SO (32)

string theory, and type IIB superstring theory relates to itself.

To deal with the T-duality, we should first discuss the effects on the string when one of

the spatial dimensions has curled up to compact space.

5.1

T-duality and closed strings

5.1.1

5.1.1 Mode expansions for compact dimension

Let us consider one of the spatial dimension be curled up,

in free bosonic string theory

i.e. the X

dimension is curled up into a circle of radius R. Now we are going to check

the effects of this on closed string [7, 29], as before we have taken the closed string

Where some author writes T for target `while some use it for toroidal and S is for Strong coupling.

By identification we can compact a dimension i.e. x

x + 2R

periodicity condition as

(, ) = X

(, + 2)

(5.1)

This periodicity condition is used whenever the closed string moving in a non compact

dimensions. As usual we will use the light cone coordinates. When there is a compact

dimension, let say X

X and the light cone coordinate are

, X

, ..., X

, X

(5.2)

Then the periodicity condition becomes as

X(, ) = X(, + 2) + m(2R)

(5.3)

Here m is the number, called the winding number, and defined as, the number of times

that the string winds around the circle of compact dimension and its sign depend upon

the direction of winding. For other non compact dimensions, = 25 the equation (5.1)

holds. Let defines the winding w in the terms of the winding number m and the radius

of the compact space as

(5.4)

By this the above periodicity condition in equation (5.3) becomes as

X(, ) = X(, + 2) + 2 w

(5.5)

The mode expansions for the this equation (5.5) can be similarly derived as we have

done before, the expansion become as

(u) =

u + i

n=0

-inu

(5.6)

(v) =

v + i

n=0

-inv

(5.7)

And from these two equations we get

(5.8)

As we see that ¯

is equal to

for zero winding. By calculating the momentum along

the compact direction like

P =

(

)d =

( ¯

)

(5.9)

Form solving these two equations (5.8) and (5.9), we get

- w)

(p + w)

(5.10)

Using these new definition by which we get

X(, ) = x

+ p + w + i

n=0

-in

(

+ ¯

-in

)

(5.11)

This is the mode expansion of compact dimension and by this we can get easily

X + X =

-in(+)

(5.12)

- X =

-in(-)

(5.13)

5.1.2

Quantization and commutation relations

For quantization we adopted the similar method like the modes becomes operators and

starting from the commutation relation between the momentum P

(, ) and the string

compact dimension, X which is

[X(, ), P

(, )] = i(

- )

(5.14)

Similarly the other commutation relations, will become by same method as we done

before for non compact dimensions, as

[ ¯

, ¯

] = [

] = m

m+n,0

, [

, ¯

] = 0

(5.15)

Now by the explicit form of ¯

and

, we get

[p, w] = 0

(5.16)

Since ¯

and

commuting with ¯

which means that

[p, ¯

] = [p,

] = [w,

] = [w, ¯

] = 0

(5.17)

And from equation (5.14) we easily find that

] = [x

, ¯

] = i

(5.18)

And using the explicit form of ¯

and

we find

, p] = i , [x

, w] = 0

(5.19)

As we see that the winding w commutes with all the operators which appear in X.

This gives the idea that the winding w is a constant or just a constant number. More

physically the winding w is an operator which gives the Eigen values of w corresponding

to various possible windings of strings. The quantization is only possible for those closed

string sectors which has some fixed winding w. As along the compact dimension, the

string would act as like a point particle that is moving on a circle

since the momentum

will be then quantize and can be written as

p =

, n

(5.20)

Here n is named as Kaluza Klein excitation number. Since both the operators p and w

have discrete values.

5.1.3

Constraint and mass spectrum

For the mass spectrum, we begin from the Virasoro operators in which the sum over

transverse l is splits into a sum over i and a term due to compact dimension, as

+ ¯

By winding, we loss and gain some states for string, while for particle, we just lost states when

we compact a dimension by identification because the particle cannot wrap around circle to gives new

states.

+ N

(5.21)

All the oscillators of X

and X will contribute to the number operators ¯

and N

[2]. Thus we find that

- ¯L

(

- ¯

) + N

- ¯

- pw + N

- ¯

(5.22)

Since L

- ¯L

does not vanish here which imposed a constraint that

- ¯

= pw

(5.23)

As both the number operators having the numbers Eigen values, so by the quantization

of both p and w makes the above equation (5.23) more physical as

- ¯

= n m

(5.24)

This gives the level matching condition and now the mass squared operators, M

- p

becomes as

= p

+ w

+ ¯

- 2)

(5.25)

This means that both the momentum p and the winding w give the contribution to the

mass squared operators [3, 7].

5.1.4

State Space of compactified closed strings

As there are additional terms in mass squared operators so for the ground states we

will add the additional labels such that

, p

; n, m

, n, m

(5.26)

And the state space can be constructed by applying the creation operators on the

ground states like

r=1

i=2

i,r

s=1

j=2

j,s

k=1

l=1

, p

; n, m

(5.27)

And the number operators which will act on above state space are

r=1

i=2

i,r

k=1

, ¯

s=1

j=2

s ¯

j,s

l=1

l ¯

(5.28)

The mass squared operators can also be written as

+ ¯

- 2)

(5.29)

This is the mass squared operators for both none zero momentum and winding for the

modified level matching condition N

- ¯

= n m

5.2

T-Duality for Closed Strings

As the mass squared operators depend upon the compactified radius R and there is a

remarkable property that if we replace the radius R by a radius ~

R = R

-1

and n by

m then the mass squared operators remain the same, this symmetry is called T-duality

for the closed string and the radii R and ~

R are dual radii.

-1

(5.30)

Hence the mass squared operators for both the dual radii are same as

(R; n, m) = M

R; m, n

(5.31)

This gives the idea that if we start a theory having a small compactified radius R, we

can transform to a dual theory which have the large radius ¯

R [7].now for a complete

dual theory, let us define a dual coordinate (compact coordinate) operator

( + )

- X

(

- )

(5.32)

The mode expansions for dual coordinate ~

X are

X(, ) = q

+ w + p + i

n=0

-in

( ¯

-in

)

(5.33)

By comparing this with equation (5.11), we find that

(5.34)

The dual momentum ~

becomes as

X =

(

)

(5.35)

The commutation relations for ( ~

X , ~

) can be calculated by the same manner as we

have done for (X ,

). The Hamiltonian for both (X ,

) and ( ~

X , ~

) is same and

written as

H =

+ p

+ w

) + N

+ ¯

- 2

(5.36)

In the order to make a map between these two (X,

) and ( ~

X, ~

), the above (5.34)

transform as

~¯

(5.37)

This map makes the physical equivalence of the two different theories and hence, T-

duality is the symmetry, which exists between the different string theories. A complete

summery is given in the following table (5.1)

Table: (5.1) complete dual theory

5.2.1

Type II superstrings and T-duality

Let us consider X

coordinate is curled up into a circle of radius R in type II superstring

theories and the T-duality transformation carried out on this coordinate as

and X

-X

This interchanged the momentum and the winding numbers similarly, the word sheet

fermions will also transform under T-duality as

and

This means that the T-duality reversed the chirality of the right moving ground states of

Ramond sector. As the relative chirality of both left and right moving ground states is

the thing which distinguished the type IIA and type IIB theories. If one is compactified

on a circle of R, let say Type IIA, then the T-duality gives the type IIB which will be

compactified on ~

R [3].

5.3

T-duality of open strings

5.3.1

T-duality and open strings

In the case of open strings, the situation is a little bit different when we compact a

dimension because the open string cannot winds around the compact dimension. Let

we have a D25-brane and a compactified X

circle having the radius R, the open string

has quantize momentum p

but no winding but after the T-duality transformation,

we will have a D24-brane and a compactified X

circle of radius ~

R. The Dirichlet

boundary conditions will impose a zero momentum constraint but the open string will

have winding now. The open string mass squared of the two theories will coincide when

R = R

-1

, because of the momentum states contribute to M

in the first theory in

the same way as the open string winding states contribute to M

in the second theory.

By permitting the duality conversion to change the D-brane, we can write it as

(D25; R)

(D24; ~

Now to find how the T-duality in the case of open strings, let us start from the assump-

tion that we have space filling D25-brane i.e. the open strings have NN-type coordinates

and let the compactified dimension X

(, )

X(, ), for this the mode expansion

can be written as

X(, ) = x

+ i

n=0

cose

-in

(5.38)

With

2 p =

(5.39)

Now separating the above string coordinate (5.38) into left moving and right moving,

X(, ) = X

( + ) + X

(

- )

(5.40)

Where

+ q

) +

( + ) +

n=0

-in

- q

) +

(

- ) +

n=0

-in

+in

(5.41)

Here q

is an arbitrary constant. For the T-duality transformation, let the dual coordi-

nate is

X(, )

( + )

- X

(

- )

(5.42)

For this ~

X the mode expansions are

X(, ) = q

+ i

n=0

sin e

-in

(5.43)

This equation is some thing like equation (4.16) in which a string is stretched from one

D-brane to another D-brane, and hence, there is a correspondences between the above

equation (5.43) and (4.16), and gives the idea that the coordinate ~

X is of DD type i.e.

The end points are fixed as

X = 0 for the end points, = 0 and = . The open

string stretches when goes from 0 to such as

X(, )

- ~

X(, 0) =

(

- 0) = 2 ~

(5.44)

Since n will have any possible integer values, the physics behind this are that, infi-

nite collections of D24-branes having uniform spacing 2 ~

R along the compactified X

direction. These configurations will be physically equivalent to the single D24-brane

which is at some fixed positions on the circle of radius ~

R. We notice that the T-duality

maps the Neumann boundary conditions into Dirichlet boundary conditions as

X = X

( + )

- X

(

- ) =

X = X

( + ) + X

(

- ) =

(5.45)

T-duality has transformed the open string with Neumann boundary conditions, on a

circle of radius R to an open string with Dirichlet boundary conditions, on a circle

of radius ~

R. Now let we have D25-brane in a word in which k spatial dimensions are

compactified into circles, Then the T-duality will transformation on each circle will give

a physically equivalent world in which we will have a D(25 - k )-brane and each circle

will be replaced by a circle with dual radius. And generally, if we have a Dp-brane

which is stretched around a compact dimension, T-duality along this direction will give

a D(p - 1 )-brane at some fixed position on a circle of dual radius [7].

5.3.2

Open strings and Wilson lines

To study the effects and its dual picture, of open strings on D-branes having the gauge

fields which are characterizing by holonomies. Let us start from a compact dimension

on circle and on this, a flat potential

will have non trivial physical effects as like

Aharanov-Bohm effect. As when the component of the gauge potential along the circle

of compactified dimension takes none zero constant values, it gives the holonomy W or

Wilson line, which can be written as

exp(iw) = exp iq

dx A

(5.46)

Here A

is the gauge potential along the compact direction x. As w = q

dx A

lives on

a unit circle due to compact dimension, so it means that w

[0, 2] and w = q dx A

The potential which gives the vanishing field strength

is some thing like the angle such that

w = q

dx A

(5.47)

Since the Wilson line W

exp(i) is a gauge invariant and the gauge equivalents

will gives same holonomy W . For a constant gauge potential A

the above equation

becomes

q A

(5.48)

Now let us consider the open strings which have the opposite charges on the end points,

so that the string will act as a neutral and the Wilson line will be no effects. If a D-

brane wraps around the compact dimension then the mass-squared operators can be

written as

= p

- 1) , p =

(5.49)

In case of open string which having opposite charges on their end points and which

lies on the same Dp-brane then shifts in momentum

is as p

- qA

+ qA

= p. Now

lets us consider the two Dp-branes and the string stretching between these two, as each

Dp-branes has own Maxwell field. Let the negatively charged end points lies on first

Dp-brane and the positively charged end points lies on the second Dp-branes then the

momentum will be shifting from p to p

- qA

+ qA

and can be written explicitly as

(5.50)

The mass squared operator then becomes as

- (

)

- 1) , l Z

(5.51)

then as a result the effects of holonomies will be cancel out and the equation

will be reduces to equation (5.49). The T-duality picture of the two Dp-branes having

the different parameters of will be consisting the two D(p - 1 )-branes having the

different (angular) positions due to the values of [3, 7].

For point particle the addition of Wilson line make a shift in

p as p-qA.

5.4

Electromagnetic fields on D-branes and T-duality

5.4.1

Maxwell fields coupling to open strings

In the presence of background Maxwell fields, the string end points will couple to

Maxwell potential A

. As we have written this coupling terms in equation (4.59),

adding this to the string action

S =

d dL(

X, X ) +

d A

(X)

d A

(X)

(5.52)

Now let us consider those background fields which has constant electromagnetic field

strength F

, for this let define the gauge potential

(x) =

(5.53)

Putting this gauge potential in the above action (5.52) and then by using the variation,

we find the boundary conditions as

+ F

= 0, = 0,

(5.54)

We get this by taking the usual Dirichlet boundary condition X

= 0 for the coor-

dinates normal to the brane. There are the background electromagnetic fields which

changed the boundary conditions. Now using the explicit form of the

and simplify-

ing, we find

- 2 F

= 0, = 0,

(5.55)

This is the boundary condition for open string in the case of background field.

5.4.2

D-branes with Electric fields and T-dualities

Let us consider a D-brane which wraps around x

compact dimension and carrying

constant electric fields along x

such that

25,0

= E

(5.56)

The background field will be purely electric if F

-F

) and purely magnetic if F

Since the boundary conditions (5.55) becomes as

- 2 F

25,0

= 0

- 2 F

25,0

= 0

(5.57)

As X

-X

and let X

X then the above equations becomes

- E

X = 0

- E

= 0

(5.58)

Here

E is the dimensionless electric field

E 2 E

(5.59)

Now writing the above equations (5.58) in a more beautiful way, for this let

(

) and by simplifying, we get

-E

(5.60)

This is the desired form of the boundary conditions (5.58) in the form of matrix. And

the duality relations becomes

X =

X ,

X =

(5.61)

In the order to find a dual description, let us consider a D(p - 1 )-brane which is

moving with a constant velocity along the compact dimension. Let we have two frame

of reference S and S . The S is the rest frame of D(p - 1 )-brane while the relative

parameter boost of these two are = v/c, where is the speed of moving brane. Let the

string coordinates in S are X

and ~

X then the Lorentz transformation can be written

= (X

- ~

X = (

-X

+ ~

(5.62)

Now using the duality relations and simplifying, we find

(5.63)

These are the boundary conditions for the dual D(p - 1 )-brane. Comparing these to

equation (5.60) then we get

E = 2 E =

(5.64)

This shows that a moving D(p - 1 )-brane with the boost parameter = v/c is actually

the T-dual to the Dp-brane which is wrapped around on the dual circle having the

electric field

E = along in the direction of circle. Since the T-duality relates the

Dp-brane with electric field is physically equal to the moving D(p - 1 )-brane with no

electric field [7, 28].

5.4.3

D-branes with Magnetic fields and T-dualities

Now we will consider the magnetic fields in background. For this, let a Dp-brane having

its two directions lies on (x

, ~

) plane, ~

is compactified to a circle of radius ~

such

that both x

and ~

give a cylinder of circumference of 2 ~

. Let the open string

coordinates are X

and ~

, which are Neumann. After T-duality on ~

, gives X

as a

Dirichlet, since the dual picture is now a D(p - 1 )-brane which is stretch along X

some fixed position X

= 0. Now the magnetic field on this Dp-brane is let F

= B.

For this the boundary conditions (5.55) becomes as

- B

= 0

(5.65)

Here

B is the dimensionless electric field

B 2 B

(5.66)

And similarly we can write these boundary conditions as

-B

(5.67)

These are the boundary condition in the presence of background magnetic fields. In

the case of dual picture, let a D(p-1)-brane which have tilted on the cylinder, then the

boundary conditions can be written in the form of X

and X

that is rotated by an

angle . The and are Neumann and Dirichlet and the proper rotation can be written

cos

sin

- sin cos

(5.68)

Using the duality relation and simplifying, we get

cos 2

- sin 2

sin 2

cos 2

(5.69)

Now comparing this result with the boundary conditions (5.67), first let the diagonals

elements which gives

- B

1 +

= cos 2

(5.70)

Simplifying this, we get

B = ± tan , we will take the negative sign such as

B = 2 B = - tan

(5.71)

This equation gives that the zero magnetic field cannot rotate the D-brane and it will

required an infinite magnetic field to rotate it. But putting the

B = - tan , we can

easily confirm the off diagonals. In equation (5.71) the negative sign is necessary for the

confirmation of the off diagonals elements. Since the titled D-brane is the dual picture

of the D-brane which have magnetic fields [7].

5.5

String coupling and the dilaton

The massless scalar field dilaton has an interesting property such as its expectation

value can controls the coupling of string and this coupling is a dimensionless number

which can set the strength of the interactions of the strings.

In string theory, the string coupling is not a constant and can be written in the form

of dilaton field. The closed string coupling g

can be evaluated from the dilaton field

(x), such as

The string coupling g

is the not an adjustable parameter but a dynamical parameter in

string theory. The dynamical nature of coupling is an ideal property for the unification

of all the other interactions. Whenever the string coupling is small then the amplitudes

for the string interactions can be easily calculable by using the Riemann surfaces. The

Riemann surfaces can also allow us to understand the infinites which come in the

amplitudes of general relativity [3, ?].

5.6

S-duality

Another kind of duality which is called S-duality which relates the string coupling g

to 1/g

as likes the T-duality which relates the radius R to R

-1

. The S-duality

relates the type I superstring theory to the SO(32) Heterotic superstring theory, and

the type IIB superstring theory to itself. As when the coupling g

is small then we

use the perturbation theory but whenever the coupling g

is large then we can use the

S-duality i.e. the large coupling g

of the type I superstring theory is equivalent to the

weak coupling g

of SO(32) Heterotic theory. S-duality tell us that how these three

superstring theories behave at strong coupling but when the coupling is large, in Type

IIA and E

× E

Heterotic, then both of them developed an eleventh dimension of size

the l

. The 11th dimension is a circle in the type IIA and a line interval in the

Heterotic

Under the T-duality, one can get easily ~

M-theory is a new type of quantum theory which lives in 11 dimensions of spacetime. At

the low energies, it is approximately equivalent to supergravity, a classical field theory

lives in 11-dimensions of space time, but the M-theory is much more than supergravity

[3, 7, 30, 31, 32].

Appendix A

The Massless States Of Closed

String

As the massless level of closed string, the general state of fixed momentum. We write

it as

l,j

, p

Here R

are the elements of an arbitrary square matrix of the size (D

- 2). Any square

matrix can be decomposed into its symmetric part and its antisymmetric part as

+ R

) +

- R

)

+ A

And the symmetric part S

can be written as

- 2

with S

The 1st term is traceless as

- 2

= S

- 2

S = 0

Hence R

is decomposed into a traceless matrix plus a multiple of the unit matrix. Let

say the traceless part of R

is ^

and S =

D-2

then we have

= ^

+ A

+ S

Hence we decomposed the Matrix R

into a symmetric-traceless part, an antisym-

metric and a traceless part. Since we split the total massless states into three linear

independent states as

l,j

, p

(A-1)

l,j

, p

(A-2)

S a

, p

(A-3)

The equation (A-1) is similar to the quantum theory of the free gravitational field states.

Hence equation (A-1) represents Graviton states. The equation (A-2) corresponds to

the one-particle states of the Kalb-Ramond field, an antisymmetric tensor field B

with two indices. The equation (A-3) corresponds to a one-particle state of a massless

scalar field. This field is called the dilaton [7].

Appendix B

The Spinors Algebra In 2D

As the Clifford algebra satisfies

{

} = 2

In 2D it gives

= 0.

Lorentizian:

In this case we have

= +1 ,

-1 ,

= 0

Now we can choose the representation of these operators as a matrices such that

0 1

1 0

-i

-1

And the generator of the spin(1, 1) is then given by

[

] =

with

= e

Given a vector v = v

+ v

the Lorentz transformation is then given by

By simplifying, we get

+(v

)

-(v

-v

)

And hence

cosh

sinh

cosh

Thus the Clifford algebra generated by

{1,

-i

} is the same as a 2 × 2 real

matrices and the even part of the Clifford algebra Cl(1, 1) is generated by 1 and

both of which are the diagonal matrices and therefore Cl(1, 1)

even

. Because

the

is real and diagonal therefore the irreducible representations of spin(1, 1) are

one dimensional (Weyl) and real (Majorana). They transform as

(Positive chirality)

(Negative chirality)

Euclidean:

In the case of Euclidean, we have

= +1 ,

= 0

Now by the similar way we can choose the representations of these operators as matrices

0 1

1 0

-1

The generator of spin(2) is then given by

[

] =

-i

With

= e

-i

cos(

)

- sin(

)

sin(

)

cos(

)

Given a vector v = v

+ v

then the spin(2)

By simplifying, we get

cos

sin

- sin cos

Thus the Clifford algebra generated by

{1,

-i

} is the same as a 2 × 2 real

matrices and the even part of the Clifford algebra Cl(2) is generated by 1 and

since (

)

-1 therefore Cl(1, 1)

even

= C and the spinors transforms as

cos(

)

- sin(

)

sin(

)

cos(

)

We can also form the irreducible representation by taking

+ i

- i

transform under Spin(2) as

(Positive chirality)

-i

(Negative chirality)

The chirality is the eigenvalue under

-1

and

-i

Thus in both Lorentzian and Euclidean signature, we have chiral (Weyl) spinors. The

supersymmetry generators are spinors and in the two dimensions we can choose p

Weyl spinors of positive chirality and q Weyl of negative chirality. This gives us (p, q)

supersymmetry in the two dimensions [33, 34, 35].

Bibliography

[1] B. Schutz, A First Course in General Relativity, 2nd Ed. Cambridge University

Press, 2009.

[2] F. Halzen and A. D. Martin, Quarks and Leptons, Wiley, New York, 1984.

[3] K. Becker, M. Becker, J. Schwarz, String Theory and M-Theory, Cambridge

University Press, 2007.

[4] F. Mandl and G. Shaw, Quantum Field Theory, Wiley, New York, 2010.

[5] D. Griffiths, Introduction to Elementary Particles, 2nd Ed. WILEY-VCH, 2008.

[6] M. Saleem and M. Rafique, Group Theory for High Energy Physics, CRC Press,

2013.

[7] B. Zwiebach, A First Course in String Theory, 2nd Ed., Cambridge University

Press, 2009.

[8] J. Polchinski, String Theory, Vol. I and II, Cambridge University Press, 1998.

[9] C. V. Johnson, D-branes, Cambridge University Press, 2003.

[10] K. Hashimoto, D-branes: superstrings and new perspective of our word, Springer,

2006.

[11] P. Goddard, Goldstone, J. Rebbi and C. B. Thorn, Quantum dynamics of a

massless relativistic string, Nucl. Phys. B56, 109 1973.

[12] G.

'tHooft,

Introduction

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.

[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.

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Keywords: STRING THEORY LIGHT CONE QUANTIZATION Superstring theory Type II superstring theories Quantization of strings M-theory String Duality T-duality Closed string theories Supersymmetric string

Author

Muhammad Ilyas (Author)

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

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