McMahon D. String Theory Demystified

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186

String Theory Demystiﬁ ed

components of space-time supersymmetry, ﬁ rst in the context of the point particle

which gives the D0-brane action and then in the case of string theory. The discussion

presented in this chapter is far from complete. The reader who is interested in

studying superstring theory on a serious level is urged to consult the references.

Quiz

1. Verify that

θθε

→+

by calculating

δθ ε θ

Q= [,]

2. Calculate

{,}DQ

3. Find

ΓΓΓ

†

4. Compute

{,}.ΓΓ

CHAPTER 10

A Summary of

Superstring Theory

Our ﬁ nal foray into the details of superstring theory will be a cursory look at

heterotic superstrings in the next chapter. The heterotic string theories are a kind of

hybrid between the 26-dimensional bosonic theory and the 10-dimensional

superstring theory with fermions. This probably sounds intractable, but we will see

in a minute how this can possibly work out into a consistent theoretical framework.

Before we dive into heterotic strings, let’s take a step back and qualitatively

summarize the overall picture of string theory.

A Summary of Superstring Theory

Before jumping into some mathematical details about heterotic string theory, let’s

review the basic structure of string theory. This is a good idea because there are

several string theories with different states of the string. Five of the theories are

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String Theory Demystiﬁ ed

superstring theories, and we also saw that it is possible to construct a theory

consisting only of bosons. Actually you can list four different bosonic string

theories, which we will do here.

BOSONIC STRING THEORY

We began our look at strings by considering bosonic string theory. This is an

unrealistic theory because we know that the real world contains particles that are

fermions. Nonetheless bosonic string theory provides an easier framework that can

be used to illustrate the key ideas and techniques of string theory.

Some key aspects of bosonic string theory you should remember are

• It introduces the concept of extra spatial dimensions. In order to avoid

ghosts (states with negative norm) we were forced to accept that there are

26 space-time dimensions.

• The ground state (the lowest energy or lowest excitation mode of the string)

has a negative mass-squared

(/).m

1=−

′

This state is called a tachyon.

The presence of a tachyon in the theory indicates that the ground state

or vacuum is unstable. Note that in relativity, tachyons are particles that

travel faster than the speed of light. Therefore the tachyon is a physically

unrealistic particle. There is no known way to remove tachyon states from

bosonic string theory.

• Bosonic string theory always includes gravity. This is indicated by the

presence of a spin-2 state called the graviton. This is a hint that string theories

provide a framework for the uniﬁ cation of all known physical interactions.

• Bosonic string theories also include a state called the dilaton. This is a

scalar ﬁ eld which is denoted by

. It is related to the coupling constant g

via

g = exp

, where

is the vacuum expectation value of the dilaton

ﬁ eld. If you need to brush up on your quantum ﬁ eld theory, note that the

coupling constant determines the strength of an interaction. The dilaton

ﬁ eld is dynamical (it is space-time dependent), so in string theory we obtain

a dramatic result that the string coupling constant can be dynamical. The

dilaton is also known as the gravitational scalar ﬁ eld and may play a role

in the recently discovered nonzero cosmological constant.

Strings can be either open or closed and can be oriented or unoriented. If a string

is oriented, this means that directions along the string are unequivalent. So you can

tell which way you’re going along the string. By choosing open or closed strings

and oriented or unoriented strings, we can actually construct four different bosonic

string theories.

If a bosonic string theory has open strings, it automatically includes closed

strings as well. This is due to the dynamical behavior of strings. If a string is open,

it is possible for the endpoints to join together, forming a closed string state. Let’s

summarize the four possibilities for bosonic string theory.

If a bosonic string theory only includes closed strings that are oriented, then the

spectrum of the theory includes the following states:

• Tachyon

• Massless antisymmetric tensor

• Dilaton

• Graviton

Now, suppose that we only have closed strings, but the theory describes unoriented

strings instead. That is, we can’t tell which direction we are moving along the string.

In this case, the theory no longer includes a massless vector boson. We can summarize

the key aspects of the spectrum as

• Tachyon

• Dilaton

• Massless state which is the graviton

Now let’s turn to bosonic string theories that include open as well as closed

strings. Again, we can choose strings that are oriented and strings that are unoriented.

The oriented theory is characterized by

• Tachyon

• Dilaton

• Graviton

• A massless antisymmetric tensor

The closed string and open string tachyons are distinct. Choosing oriented open

+ closed bosonic string theory gives us

• Tachyon

• Dilaton

• Massless graviton

There is also a massless vector state for open strings, which can be oriented or

unoriented. So we see that all bosonic string theories are plagued by the presence of

a tachyon state. They have an unstable vacuum and do not include fermions. As a

result, we are forced to consider superstring theories.

CHAPTER 10 A Summary of Superstring Theory

189

190

String Theory Demystiﬁ ed

Superstring theory is a generalization of bosonic string theory which extends the theory

to include fermions. There are ﬁ ve different superstring theories. We use the word super

in our description of them because all ﬁ ve theories are based on a theory of physics

known as supersymmetry. This theory is characterized by the idea that each fermion has

a bosonic partner and vice versa. Some examples are given in Table 10.1.

The existence of supersymmetry is a good indirect test of string theory. For string

theory to be true, supersymmetry must exist in nature. At the time of writing no

super-partner has ever been discovered, so supersymmetry either doesn’t exist in

nature or it has been broken. One way it could be broken is that the superpartners

are extremely massive. This means it would take high energies to see them. The

Large Hadron Collider (LHC) set to begin operation in 2008 may be powerful

enough to detect supersymmetry.

So, superstring theory includes supersymmetry, which allows us to introduce

fermions into the theory. It also includes ghost states, which are removed in an

analogous, manner to what we saw in bosonic string theory. When the ghost states

are removed we arrive at the second general characteristic of superstring theory:

• There are 10 space-time dimensions.

There are two ways to introduce supersymmetry into string theory, reviewed in

Chaps. 7 and 9, respectively:

• The RNS formalism adds supersymmetry to the worldsheet.

• The GS formalism adds supersymmetry to space-time.

We can still characterize superstring theories by noting whether or not they

include open and/or closed strings, and whether those strings are oriented or

unoriented. In addition, a superstring theory can be characterized by the number of

supercharges used in the theory. This is done by saying that a theory with N = m

Superstring Theory

Table 10.1 A listing of some particles and their postulated super-partners.

Partner Superpartner

Photon (spin 1) Photino (spin 1/2)

Graviton (spin 2) Gravitino (spin 3/2)

Quark (spin 1/2) Squark (spin 0)

Electron (spin 1/2) Selectron (spin 0)

Gluon (spin 0) Gluino (spin 1/2)

CHAPTER 10 A Summary of Superstring Theory

191

supercharges has N = m supersymmetry. Finally, we can characterize each

superstring theory by the gauge symmetry that it admits. All superstring theories

eliminate the tachyon from the spectrum and include a graviton, so superstring

theory naturally describes gravity.

TYPE I SUPERSTRING THEORY

Type I superstring theory can be characterized as follows:

• It includes both open and closed strings.

• It describes unoriented strings.

• It has N = 1 supersymmetry.

• It has SO(32) gauge symmetry.

In addition, type I superstrings can have charges attached to their ends called

Chan-Paton factors, a topic we will explore in a later chapter.

TYPE II A

Type II A theory describes closed, oriented superstrings. We can summarize the

theory as follows:

• It only includes closed strings.

• It has N = 2 supersymmetry.

• It has a U(1) gauge symmetry.

Since this theory only has a U(1) gauge symmetry, it is not large enough to

describe all the particle states seen in nature. It can describe gravity and

electromagnetism, but cannot describe the weak or strong forces. The theory has

two supercharges, and

ΘΘ

and

have opposite chirality. Practically speaking, this

means that each fermion has a partner state with opposite chirality.

TYPE II B

Type II B theory also describes closed strings, also oriented. Although it includes

fermionic states because it is a superstring theory, it has no gauge symmetry and so

can only describe gravity. Like type II A theory, it has N = 2 supersymmetry, but

ΘΘ

and

have the same chirality. This remedies the difﬁ culty in type II A theory

in that the fermions described in type II B theory do not have partners of opposite

chirality. But the lack of a gauge group indicates the theory cannot be the whole

story as far as a uniﬁ ed theory of physics.

192

String Theory Demystiﬁ ed

HETEROTIC SO(32)

There are two heterotic theories that both describe closed, oriented strings. A heterotic

theory is a kind of fusion between bosonic and superstring theory. The left movers

and right movers are treated using different theories. We describe modes moving in

one direction using bosonic string theory, and describe the modes moving in the

opposite direction using N = 1 supersymmetry. The extra 16 dimensions of the bosonic

theory are regarded as abstract, mathematical entities rather than actual space-time

coordinates (like superspace). There are two heterotic theories, both with large gauge

groups that can describe all particles in nature. The ﬁ rst has SO(32).

HETEROTIC E

× E

Similar to Heterotic SO(32) theory but has the gauge group

Dualities

The state of string theory at this point appears to be a random mess, but the discovery

of a set of dualities which relate the ﬁ ve theories amongst themselves saved the day.

The fact is that the ﬁ ve theories are all related to one another, and we can transform

between them. This has led physicists to believe that there exists an underlying

theory. The ﬁ ve superstring theories arise as different aspects or solutions of the

underlying theory. While some aspects of the potentially underlying theory have

been characterized, the actual underlying theory remains unknown. It goes by the

name of M-theory.

T-DUALITY

We have already studied one duality in detail in Chap. 8, T-duality. To review T-

duality relates a theory with a small compact dimension to a theory where that same

dimension is large. T-duality relates string theories as follows:

• It relates type II A and type II B theory.

• It relates the two heterotic theories.

T-duality can be summarized by saying that if we transform from a small to a

large distance scale we exchange momentum and winding modes (and vice versa).

T-duality relates type II A and type II B theory in that if we move from small to

large distance in type II A theory, the theory is transformed into type II B theory and

CHAPTER 10 A Summary of Superstring Theory

193

vice versa (or switch momentum and winding modes). The same holds for the two

heterotic theories. This means that type II A and type II B are really the same theory,

and the two heterotic theories are really the same theory.

S-DUALITY

The second big duality that has been discovered is S-duality. Remember that a coupling

constant determines the strength of an interaction, and in string theory the dilaton

ﬁ eld determines the value of the coupling constant. String theories have different

coupling constants that are weak or strong. By letting

ϕϕ

→−

where

is the dilaton

ﬁ eld, since the coupling constant is deﬁ ned from

g = exp

, we see that we can

transform a large coupling constant into a small one and vice versa, changing a strong

interaction into a weak one and vice versa. This is what S-duality is about. S-duality

brings type I superstring theory into the fold. That is, under S-duality

• Type I superstring theory is related to heterotic SO(32) superstring theory.

• Type II B is S-dual to itself.

So, a strong interaction in Type I superstring theory is the same as a weak

interaction in heterotic SO(32) theory, and vice versa. In other words, the two

theories are really the same theory at different coupling strengths.

Quiz

1. T-duality relates string theories in which way?

(a) It relates strong interactions in type II A theory to weak interactions in

type II B theory.

(b) It relates strong interactions in heterotic theory to weak interactions in

type I theory.

modes in two different theories.

(d) It only relates large and small distance scales.

2. The most signiﬁ cant difference between superstring theory and bosonic

theory is

(a) Bosonic theory has 16 extra space-time dimensions.

(b) Superstring theory eliminates tachyon states and incorporates fermions

into the theory.

194

String Theory Demystiﬁ ed

(d) Superstring theory has an unstable vacuum.

3. The difference between types II A and type II B theory is

(a) Type II A theory is nonchiral and type II B theory is chiral.

(b) Type II A theory only describes open strings.

(d) Type II B theory is nonchiral and type II A theory is chiral.

4. The dilaton ﬁ eld

(a) Is only found in bosonic string theory.

(b) Is related to the coupling constant, but only in heterotic theory.

(d) Is known to be a mathematical trick not found in nature.

5. Physicists were excited by dualities because

(a) They add fermions to the theory.

(b) They show that the ﬁ ve superstring theories are related, so are different

aspects of an underlying, unknown theory.

different aspects of an underlying, unknown theory.

6. The number of space-time dimensions in string theory

(a) Is ﬁ xed at 26 by an ad hoc assumption.

(b) Is ﬁ xed at 26 for superstring theories and 10 for bosonic string theory,

because this eliminates ghost states from the theory.

because this eliminates tachyon states from the theory.

(d) Is ﬁ xed at 10 for superstring theories and 26 for bosonic string theory,

because this eliminates ghost states from the theory.

CHAPTER 11

Type II String

Theories

In this chapter, we will review the states of type II A and type II B superstrings. We

will do so using the worldsheet supersymmetry plus Gliozzi-Scherk-Olive (GSO)

projection approach because it’s a bit simpler, so we will review some of the

discussion of Chap. 7. In the next chapter, we will brieﬂ y discuss heterotic

superstrings.

The R and NS Sectors

To introduce worldsheet supersymmetry we began with the action [Eq. (7.2)]:

dXXi=− ∂ ∂ − ∂

∫

σψρψ

µα

αµ

()