McMahon D. String Theory Demystified

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196

String Theory Demystiﬁ ed

In the light-cone gauge, the fermionic part of the action assumes the form

SiTd

=⋅∂+⋅∂

−+− +−+

∫

σψ ψ ψ ψ

()

In type II string theories, we only consider closed strings. Therefore, we apply

periodic boundary conditions. There are actually two possibilities. Periodic boundary

conditions are known as Ramond (R) boundary conditions:

ψσ ψσ π

µµ

() ( )=+2

(11.1)

Antiperiodic boundary conditions are called Neveu-Schwarz (NS):

ψσ ψσ π

µµ

() ( )=− +2

(11.2)

Remember that a closed string has independent left-moving and right-moving

modes. We can apply either periodic or antiperiodic boundary conditions to the left-

and right-moving modes independently, which will give us four possibilities, as

discussed in Chap. 7.

THE R SECTOR

Now we review the R sector in more detail. The left

and right

−

modal

expansions are

στ

µµτσ

µµ

∈

−+

(

)

−

∈

(

)

(

)

∑



in−−

(

)

τσ

(11.3)

Quantization proceeds using the anticommutation relations:

dd dd

mn mn mn

µν µν µν

ηδ

{}



(11.4)

These relations are augmented, of course, by the usual bosonic commutation

relations. We also have the number operators:

Nndd Nndd

() ()

=⋅ =⋅

−

∞

−

∞

∑∑

and





(11.5)

The total number operator for the left-moving and right-moving sectors is given by

adding the bosonic number operator

αα

=⋅

−

∞

∑

to Eq. (11.5). We obtain

NNN



()

(11.6)

Taking

n > 0

,we can deﬁ ne creation and annihilation operators as follows:

•

n−

acts as a creation operator by adding n to the eigenvalue of

d()

•

acts as an annihilation operator by subtracting n from the eigenvalue

d()

The states are constructed in the usual way by using a fock space (or number

states). Letting

n > 0

, the ground state is annihilated by the bosonic and fermionic

annihilation operators:

µµ

d000==

(11.7)

for the right-moving modes, and similarly for the left movers. We can construct an

arbitrary state by tracing on the ground state multiple times:

−−

∏∏

()()

(11.8)

For states in the right-moving sector, the expression for the left-moving sector is

similar.

Now consider the special case of

. The anticommutation relation is

µν µν

{}

(11.9)

This is almost the same as the commutation relation obeyed by the Gamma matrices

(i.e., the “Dirac algebra”):

{,}ΓΓ

µν µν

=−2

(11.10)

CHAPTER 11 Type II String Theories

197

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String Theory Demystiﬁ ed

This suggests that these operators are related to the Gamma matrices using

µµ

= id2

(11.11)

This tells us that the states in the R sector are space-time spinors. We can write the

ground state as

where a is a spinor index that ranges over a = 1, …, 32. As we have seen earlier, this

is because a general Dirac spinor has 2

D/2

components where D is the number of

space-time dimensions. Since D = 10 for superstring theories, there are 32

components. The state

is a 32-component Majorana spinor.

Now recall that the chirality operator

ΓΓΓΓ

11 0 1 9

= …

acts on states

deﬁ nite chirality according to

−−

=−

(11.12)

States with deﬁ nite chirality are Majorana-Weyl spinors, which have half the

number of components, (16 in this case). We can write the state

as a direct sum

of positive and negative chirality states:

000

=⊕

+−

(11.13)

This gives the state [Eq. (11.13)] 16 ⊕ 16 = 32 components. The states

are

space-time fermions. However, they have the bizarre property that

is bosonic

and

−

is fermionic on the worldsheet.

THE NS SECTOR

In the NS sector, we have the modal expansions of the left- and right-moving

fermionic states given by

στ

µµτσ

µµ

∈+

−+

(

)

−

(

)

(

)

∑



rrZ

∈+

−−

(

)

∑

τσ

(11.14)

CHAPTER 11 Type II String Theories

199

The expansion coefﬁ cients satisfy the anticommutation relations:

bb bb

rs rs rs

µν µν µν

ηδ

{}



(11.15)

The number operators are

NrbbNrbb

()

=⋅ =⋅

−

∞

−

∞

∑∑

12 12





(11.16)

This allows us to deﬁ ne number operators for right- and left-moving modes:

NNN

()



(11.17)

Letting n > 0 :

• b

-r

acts as a creation operator, increasing the eigenvalue of N

(b)

by r.

• b

acts as an annihilation operator, decreasing the eigenvalue of N

(b)

by r.

Again we construct the states using a fock space. The ground state is annihilated

in the following way:

µµ

b000==

(11.18)

An arbitrary right-moving state is given by acting multiple times on the ground

state as

−−

∏∏

()()

(11.19)

The NS sector results in states that are space-time bosons.

To get the most general space-time state, one forms tensor products of the left-

moving sector and the right-moving sector. Each sector can be an NS state or an R

state independently of the other, so we have four possible states overall. If both

sectors are

states, then the state of the string is

=⊗NS NS

left ri

200

String Theory Demystiﬁ ed

This state is a space-time boson. If both states are R states

=⊗RR

left right

This is also a space-time boson (this is a state called a bispinor, since it is constructed

out of two spinors).

As described qualitatively in Chap. 7, we can also have left-moving and right-

moving modes from different sectors. Of course, there are two possibilities:

ψψ

=⊗ = ⊗RNS NSR

left right right

left

These states are space-time fermions.

The Spin Field

In supersymmetry, one can move between fermionic and bosonic states using a

supercharge operator. We now construct such an operator that allows us to move

from an NS (bosonic) to an R (fermionic) state.

Let

where

k = 15,...,

be complex bosonic ﬁ elds. We can deﬁ ne the spinors

in terms of these bosonic ﬁ elds using a process called bosonization. Remember from

Chap.7 that the

are space-time vectors, so have 10 components (each of which is

a spinor). These are deﬁ ned in terms of the bosonic ﬁ elds as follows:

=±

ψψ

910

=±

ψψ

We can construct a spin ﬁ eld operator by taking the product of these quantities, that is

Se e

ai i

±±

φφ

12 52//

(11.20)

CHAPTER 11 Type II String Theories

201

This is a 32-component SO(10) spinor. It acts like a supercharge, taking the bosonic

state

to the fermionic state

(11.21)

Now we have everything in place to describe the differences between type II A and

type II B string theories.

Type II A String Theory

The key aspect of type II A string theory that should stick in your mind which

distinguishes it from type II B string theory is opposite chirality. That is, for a state

⊗

and

will have opposite chirality.

First, let’s note that the total fock space is constructed as follows. First, we form

direct sums:

NS R NS R⊕

()

⊕

()

left right

Then we form the tensor product:

NS R NS R⊕

()

⊗⊕

()

left right

The physical state space is constructed using GSO projection, which will cure

the odd problem of the states differing as being fermions or bosons in space-time

and on the worldsheet described earlier, as well as allowing us to remove tachyons

from the theory. First, we construct an operator which will count up the number of

n−

excitations in a state:

Fdd

=⋅

−

∞

∑

(11.22)

It is easy to see that

()−1

tells us if a state has an even or odd number of

n−

excitations. Now suppose that

is a state with an even number of

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String Theory Demystiﬁ ed

oscillators and

is a state with an odd number of d

oscillators. It so

happens that

ΓΓ

11 11

ψψ φ φ

==−

(11.23)

Now deﬁ ne



ΓΓ=−

1()

(11.24)

Then if



ψψ

the state

has an even number of

n−

excitations. On the other hand if



ψψ

=−

the state

has an odd number of

n−

excitations.

Type II A theory is characterized by states with opposite chiralities. So, the GSO

projection of the right movers has the opposite sign of the GSO projection of the

left movers. This means that the chirality of space-time fermions will turn out to be

opposite. Speciﬁ cally note that

ΓΓ

11 11

()

=−

()

left right

(11.25)

in type II A theory. Now consider the spin ﬁ eld S

. Since

=±

and

where

is bosonic, we can consider the action of

on the

spin ﬁ eld itself. In type II A theory

ΓΓ

11 11

SS S S

aa a a

==−and

(11.26)

The action of GSO projection is to take the 32-component spinor into a

16-component Majorana-Weyl spinor.

CHAPTER 11 Type II String Theories

203

Type II B Theory

We have seen that type II A theory is characterized by opposite chirality. Type II B

theory is characterized by the same chirality. That is, for a state

⊗

and

will have the same chirality. The left and right movers will have

the same GSO projection:

ΓΓ

11 11

()

left right

(11.27)

The spinor ﬁ eld satisﬁ es

ΓΓ

11 11

SS SS

aa a a

==and

(11.28)

The states

00 00

12 12

RNS NSR

left right r

⊗⊗

−−//

µµ

and

left iight

also have the same chirality.

The Massless Spectrum of Different Sectors

We conclude the chapter by noting the spectrum of states seen in the different

sectors, and describing how these are different in type II A and type II B string

theories. For those with a background in differential geometry or general relativity,

forms are odd in type II A theory and even in type II B theory.

THE

NS〉 ⊕

NS〉 SECTOR

Type II A and type II B string theories have the same NS-NS sector. The spectrum

of states in the NS-NS sector is

• A scalar ﬁ eld

which is the dilaton described in the last chapter (1 state)

• An antisymmetric gauge ﬁ eld with 28 states

• A symmetric traceless rank two tensor

µν

(this is the graviton) with 35 states

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String Theory Demystiﬁ ed

THE

NS〉 ⊕

R〉,

〉 ⊕

〉 SECTOR

The states in these sectors are superpartners of the dilaton and the graviton. Type II

A and type II B theories have the same states, but the states

NS R R NS⊗⊗,

each have the same chirality in type II B theory and have the opposite chirality in

type II A theory. The states are

• The dilatino, the spin-1/2 superpartner of the dilaton, with 28 states.

• The gravitino, the spin-3/2 fermion which is the superpartner of the

graviton. For type II A theory the 2 gravitinos have opposite chirality.

THE

R〉 ⊕

R〉 SECTOR

These are boson states formed from the tensor product of two Majorana-Weyl

spinors. In type II A theory the left and right states have opposite chirality,

while in type II B theory they have the same chirality. For type II A theory, there

are two states:

• A vector gauge ﬁ eld

• A “3-form” gauge ﬁ eld (see Relativity Demystiﬁ ed for a description of

forms)

In type II B theory the states are

• A scalar ﬁ eld

• A two-form gauge ﬁ eld

• A four-form gauge ﬁ eld

Summary

In this chapter, we have summarized the states of type II A and type II B string

theories, which are based on the R and NS sectors first introduced in Chap. 7.

Type II A theory consists of states that are tensor products of states with

opposite chiralities, while type II B theory consists of states that are tensor

products of states with the same chiralities. This results in different “particle”

states. These theories predict the existence of superpartners that have not yet

been observed in nature.

CHAPTER 11 Type II String Theories

205

Quiz

1. Type II string theories:

(a) Include only open strings

(b) Include open and closed strings

(d) Include open strings only if they are oriented

2. States in the NS sector can be described in space-time as:

(a) NS sector states must be combined with R sector states to give

space-time fermions, otherwise they do not describe states in

space-time.

(b) Space-time bosons.

(d) Space-time bosons for oriented strings, space-time fermions for

unoriented strings.

3. The spin ﬁ eld operator

Se e

ai i

±±

φφ

22//

is what type of spinor? How

many components does it have in general?

(a) SO(10), 16 components

(b) SO(10), 32 components

(d) SO(10), 10 components

4. The superpartner states in the

NS R R NS⊗⊗,

sectors include:

(a) The dilatino and the gravitino. For type II A theory the 2 gravitinos

have opposite chirality.

(b) The dilatino and the gravitino. For type II A theory the 2 gravitinos

have the same chirality.

have opposite chirality.

(d) The dilatino and the gravitino. For type II A theory the 2 dilatinos

have the same chirality.