McMahon D. String Theory Demystified

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216

String Theory Demystiﬁ ed

superpartners, the status of string theory will be put in doubt. On the other hand, the

discovery of superpartners does not prove string theory, but would be a good

indication that the theory is on the right track.

Compactiﬁ cation and Quantized Momentum

In this section, for readers who are curious, we brieﬂ y describe a different approach

developed by Gross, Harvey, Matinec, and Rohm where compactiﬁ cation is used to

construct the heterotic string. We follow the description laid out in Kaku (please see

References). In the light-cone gauge, the action for the heterotic string can be

written as

SdXXXXiS S

=−

′

∂∂ +∂∂ + ∂+∂

∫

−

πα

τσ

Γ ()

∑

⎛

⎝

⎜

⎞

⎠

⎟

The approach used here is to compactify the extra bosonic dimensions to generate

the group

⊗

. The extra 16 dimensions of the bosonic sector are compactiﬁ ed

on a lattice. As described in the previous section, the right-moving sector is

supersymmetric. The spinors

()

τσ

−

have 8 components (so

a =18,...,

Remember we are in the light-cone gauge, so only consider transverse components.

The index i is used for the space-time components, in the light-cone gauge

i = 18,...,

as well. The remaining index I is used to run over the lattice used to compactify

the extra 16 dimensions. So it runs over 1 to 16.

The physics is much the same as the previous analysis. Bosonic states

()

τσ

and

()

τσ

−

are included in the left- and right-moving sectors, respectively. The

right-moving sector also includes the fermionic component

()

τσ

−

, while the

states

()

τσ

are in the left-moving sector.

The action is invariant under the supersymmetry transformation:

δε

µτ σ

Xp S

Sip X

iia

=∂−∂

+−

−

()

() ( )

ΓΓ

εε

The following constraints are used to keep each component properly locked

away as a left mover or a right mover:

()∂−∂ =

τσ

0Γ

CHAPTER 12 Heterotic String Theory

217

The usual formulas for the modal expansions apply

xp i

τσ τσ

τσ

−

(

)

=+ −

(

)

∞

−−

(

)

∑

22 2

xp i

τσ τσ

τσ

(

)

=+ +

(

)

∞

−+

(

)

∑

22 2



xp i

τσ τσ

τσ

(

)

=+ +

(

)

∞

−+

(

∑

22 2



))

−−

(

)

=−∞

∞

−

(

)

∑

SSe

ain

τσ

Using this approach, we write down two number operators. The number operator

for the right-moving sector is

NnSS

⎛

⎝

⎜

⎞

⎠

⎟

−−

−

∞

∑

αα

For the left-moving sector we have



(

)

−−

∞

∑

αα αα

The mass can be written in terms of the canonical momentum

mNN p

=+−+

(

)

∑



Now, let’s see how compactiﬁ cation affects these results. It is easiest to

understand by stepping back to compactiﬁ cation of one dimension as we did in

Chap. 8. Kaku gives a nice example which we restate here. Take a single-dimensional

theory and let

xx R=+2

Consider a ﬁ eld deﬁ ned on this space

()x

. Since the coordinate is periodic, the

ﬁ eld must be also

φφπ

() ( )xxR=+2

218

String Theory Demystiﬁ ed

Now, we know that we can expand

()x in a Fourier series. That is, we can

expand it in p where p is the conjugate momentum for the coordinate x. The

expansion looks like

φφ

()xe

ipx

∑

Now, of course, the exponential function is 2

periodic. Calculating

φπ

xR+

(

)

φπ φ φ

ππ

xR e ee

ip x R

ipx ip R

(

)

∑∑

(

)

(

)

The presence of the extra term

ip R2

(

)

means that we must take

So, we relearn an important rule about compactiﬁ cation:

• Compactiﬁ cation quantizes momentum.

For the heterotic string, we compactify each of the extra bosonic coordinates:

XX L

II I

=+2

Here,

represents the lattice spacing. If we span the lattice with basis vectors

then

LneR

∑

Here, the

are the radii of the compactiﬁ ed dimensions. Now, we use the conjugate

momenta

from the bosonic states, which are compactiﬁ ed as a generator of

translations. We take

to be the generator of translations along the lattice in the

Ith direction. The periodicity condition

XX L

II I

=+2

means that

ipL

⋅

CHAPTER 12 Heterotic String Theory

219

This can only be true if the canonical momentum has an expansion of the form

∑

where the

are integer coefﬁ cients of the expansion. This has the same form

obtained in the one-dimensional case, where we found p = n/R. So, to compactify

over a lattice you deﬁ ne basis vectors, then sum up over all directions dividing by

the radius in each direction.

Summary

In this chapter, we gave an overview of the heterotic string. Using the usual

approach to supersymmetry, we wrote down the action, applied anticommutation

relations, defined number operators, and applied the mass-shell condition. As

a result, we found a theory which contains the dilaton and graviton and their

superpartners among the massless states. Further analysis would show that

the theory is large enough to include the standard model. We can summarize

this as follows:

• The heterotic theory has left-moving and right-moving currents on the

string.

• The right-moving currents carry supersymmetry charges and give fermionic

states.

• The left-moving currents carry the conserved charges of Yang-Mills

theories.

After describing the basic machinery of the heterotic string, we illustrated how

compactiﬁ cation can be used to get rid of the unwanted 16 extra dimensions that are

added to the theory by including the left-moving bosonic theory.

Quiz

1. How many states are there for the left-moving sector of the form



−1

2. How many states in the left-moving states have the form



−1

220

String Theory Demystiﬁ ed

3. Consider the constraint equations in the compactiﬁ cation section. Why is

()∂−∂ =

τσ

the only constraint on the bosonic modes for the extra 16

dimensions?

4. Given the action

SdXXXXiS S

=−

′

∂∂ +∂∂ + ∂+∂

∫

−

πα

τσ

Γ ()

∑

⎛

⎝

⎜

⎞

⎠

⎟

, what are the

commutation relations satisﬁ ed by the bosonic modes?

5. Consider the description of the heterotic string in the compactiﬁ cation

section and calculate

(

)

=Γ S

CHAPTER 13

D-Branes

One of the most interesting developments in string theory over the last decade or so

was the realization that the theory could incorporate higher-dimensional extended

objects—that is, objects beyond one-dimensional strings. When these objects are

associated with Dirichlet boundary conditions we call these extended objects

Dp-branes, where p indicates the number of spatial dimensions it has. The word

“brane” comes about by analogy. In our everyday world of three spatial dimensions,

we are familiar with the notion of a membrane, which is a two-dimensional surface

that can separate two regions. The idea of a Dp-brane is to generalize this concept

to consider an extended object of p dimensions.

If the number of spatial dimensions of the D-brane is equal to the total number

of spatial dimensions in the entire space-time, we say that we have a space-ﬁ lling

brane. There are three pertinent examples we can think of immediately to illustrate

space-ﬁ lling branes:

• If space-time is just the three spatial dimensions and one time dimension

we are used to from everyday life and special/general relativity, then a

D3-brane would be a space-ﬁ lling brane.

222

String Theory Demystiﬁ ed

• In bosonic string theory, there are 26 space-time dimensions. So, a

D25-brane would be a space-filling brane.

• In superstring theory, there are 10 space-time dimensions. So, a space-

ﬁ lling brane has 9 spatial dimensions and is called a D9-brane.

Chances are if you’re reading this book you’ve completed calculus so you’re

familiar with the notion of a hyperplane. When ﬁ rst getting started, the best way to

think about a D-brane is

• It is a hyperplane-like object.

• The endpoints of open strings are attached to it.

This is illustrated in Fig. 13.1. Note, however, that not all D-branes are hyperplanes,

but this is a good way to visualize them.

The spatial dimensions not associated with the brane are called the bulk. The

volume of the brane is called the world-volume. Note that time ﬂ ows everywhere,

in the bulk and on the D-brane as well.

A model of our universe has been proposed where we live in a D3-brane and the

bulk consists of the remaining extra spatial dimensions. Perhaps the most fundamental

physical insight that has resulted from the study of D-branes is that

• The interactions of the standard model (electromagnetism, strong, and

weak forces) are constrained to the brane.

• Gravity can escape from the brane. Gravitational forces are distributed in

the brane and also throughout the higher dimensions. Hence, the strength of

gravity is diluted by the higher dimensions. This explains why its strength

is so different from that of the other known forces.

For simplicity, we will discuss branes within the context of bosonic string theory.

D-brane

Open string with ends attached to

D-brane

Space-time outside D-brane is the “bulk’’

Figure 13.1 A D-brane is a hyperplane-like object to which open strings attach.

CHAPTER 13 D-Branes

223

The Space-Time Arena

The easiest way to describe a Dp-brane mathematically is to use the light-cone

gauge. To specify the D-brane, we need to choose which coordinates will

satisfy Neumann boundary conditions and which coordinates satisfy Dirichlet

boundary conditions. To use the light-cone gauge, we also need to define light-

cone coordinates that will satisfy Neumann boundary conditions, these will

include:

• Time

• One spatial coordinate, which we choose to be

(, )

στ

For a Dp-brane, we let

ip= 2, ,…

in the light-cone gauge. Then as usual we

deﬁ ne:

(,)

στ

(13.1)

Neumann boundary conditions can be written as

∂=

σπ

(13.2)

So, the coordinates chosen to satisfy Neumann boundary conditions are

XXXip

i+−

=(, ) (, ) (, ) , ,

στ στ στ

2 …

(13.3)

Let us suppose that the D-brane is located at

. That is, letting

ap d=+1, ,…

(13.4)

The remaining spatial coordinates will satisfy Dirichlet boundary conditions. We

use

ap d=+1, ,…

to denote these coordinates. In bosonic string theory we take

d = 25

while in superstring theory

d = 9

. So the Dirichlet boundary conditions will

be applied to

Xapd

(, ) , ,

στ

=+1 …

(13.5)

224

String Theory Demystiﬁ ed

Given xx

= , the Dirichlet boundary condition can be written as

XXxapd

aaa

(, ) (, ) , ,01

τπτ

===+…

(13.6)

Notice that we can also specify the Dirichlet boundary conditions by deﬁ ning:

δπτ τ

XX X ap d

aa a

=− =+(, ) (, ) , ,01…

(13.7)

Then we could write the Dirichlet boundary condition as

= 0

(13.8)

The coordinates are divided into two groups and given labels depending on boundary

conditions that are applied:

• The coordinates with indices

=± =, , ...,ip2

are called NN coordinates

since they satisfy Neumann boundary conditions at both ends.

• The coordinates with indices

ap d=+1, ...,

are called DD coordinates

since they satisfy Dirichlet boundary conditions at both ends.

A simpliﬁ ed illustration of the boundary conditions is shown in Fig. 13.2. To

summarize, a Dp-brane is located at

and has extension along the

directions.

D-brane

Directions normal to D-brane-Dirichlet

boundary conditions

Directions along D-brane-Neumann boundary conditions

Figure 13.2 A visualization of the boundary conditions and an open string.

CHAPTER 13 D-Branes

225

Once again we apply the quantization procedure, considering the bosonic string

theory case. The ﬁ rst step is to write down the modal expansions. These will be

different depending on which coordinates we look at because now we have NN

coordinates and DD coordinates. The ﬁ rst step is to write out the modal

expansions.

Now let’s recall the open string modal expansion, which can be written in the

following way:

Xxpi

µµµ

στ α τ α

( , ) cos(=+

′

≠

−

∑

22 ))

(13.9)

Taking the derivative of this expression with respect to

we ﬁ nd

∂=−

′

≠

−

∑

µµτ

στ α α σ

Xi en

(,) sin( )2

Clearly this expression satisﬁ es

∂=

σπ

which are the Neumann boundary conditions. So, we take the modal expansion for

the

to be

Xxpi

iii

( , ) cos(

στ α τ α

′

≠

−

∑

22 ))

(13.10)

For the DD coordinates, we really have two requirements that have to be satisﬁ ed.

We need the summation over the modes

∑

≠n 0

in the expansion to vanish at

σπ

= 0,

This indicates that we should use

sin( )n

instead of

cos( )n

which is in the

usual open string expansion. However, we also need

XXx

(,) (,)0

τπτ

Looking at the usual modal expansion, this tells us that we must take

→

Quantization