Contents Introduction and motivations 1 1 Space-time supersymmetry in string theory 6 1.1 Why...

Universita degli Studi di Perugia

Facolta di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Specialistica in Fisica

Tesi di Laurea in Fisica Teorica

Type IIB superstring theory

on rotated pp-wave backgrounds

Candidato:

Andrea Marini

Relatori:

Prof. Gianluca Grignani

Dott.ssa Marta Orselli

Febbraio 2009

Contents

Introduction and motivations 1

1 Space-time supersymmetry in string theory 6

1.1 Why superstring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Introduction to supersymmetry . . . . . . . . . . . . . . . . . . . . 7

1.3 The classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 The superparticle . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.2 The supersymmetric string action . . . . . . . . . . . . . . . 15

1.3.3 The local fermionic symmetry . . . . . . . . . . . . . . . . . 16

1.3.4 Types of superstrings . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Light-cone gauge . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.2 Lorentz invariance . . . . . . . . . . . . . . . . . . . . . . . 26

2 Superstrings on pp-wave backgrounds 28

2.1 Pp-wave metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Penrose limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Penrose limit of AdS5 × S5 spaces . . . . . . . . . . . . . . . 32

2.3 Spectrum of type IIB superstrings on the pp-wave background . . . 33

2.3.1 Bosonic sector . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.2 Fermionic sector . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 The AdS/CFT correspondence 40

3.1 ’t Hooft’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 The Maldacena conjecture . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 The plane-wave/Super Yang-Mills duality . . . . . . . . . . . . . . 47

3.3.1 Decomposition of N = 4 fields into D, J eigenstates . . . . 48

3.3.2 Stating the BMN proposal . . . . . . . . . . . . . . . . . . . 49

i

4 New Penrose limit 52

4.1 Pp-wave backgrounds without flat directions . . . . . . . . . . . . . 53

4.2 Pp-wave backgrounds with one flat direction . . . . . . . . . . . . . 54

4.3 Pp-wave backgrounds with two flat directions . . . . . . . . . . . . 55

5 Rotated pp-wave backgrounds 57

5.1 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Spectra of type IIB superstrings on rotated pp-wave backgrounds . 59

5.2.1 Bosonic sector . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2.2 Fermionic sector . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Decoupling limits of N = 4 SYM gauge theory . . . . . . . . . . . . 67

5.4 Matching of spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Conclusions 73

A Conventions for N = 4, D = 4 Yang-Mills gauge theory 75

B Gamma matrices and spinors 78

C Finding the new Penrose limit 83

Bibliography 91

ii

Introduction and motivations

A theory of strings was first proposed in the late 1960’s as a model for the strong in-

teractions describing the dynamics of hadrons. In those years Nambu [1], Nielsen [2]

and Susskind [3] independently understood that the dual resonance model devised

by Veneziano [4] could be interpreted in terms of quantum fluctuations of a one-

dimensional object, namely a string.

The discovery of asymptotically free Yang-Mills quantum field theories [5, 6, 7, 8],

predicting slight deviation from Bjorken scaling, led to the triumph of Quantum

Chromo-Dynamics (QCD) as a candidate to describe strong interactions and ruled

out string theory. In fact, among the massless string states there is one of spin 2

which could not be interpreted at all in the spectrum of nuclear resonances.

Although quantum field theory has been proved to be extremely successful in pro-

viding a description of elementary particles and their interactions, it does not work

so well for gravity. If one naively tries to quantize general relativity, which is a

classical field theory, using the methods of quantum field theory, divergences that

cannot be removed with the conventional renormalization techniques arise.

In 1974, when it seemed that string theory would be bound to be definitely aban-

doned, it was shown that the massless spin 2 particle, which appears in the string

spectrum, interacts like a graviton [9], so the theory actually includes general rel-

ativity. Furthermore a consistent quantum string theory does not suffer from any

ultraviolet divergences, thus it was promoted from an effective theory of strong

dynamics to a theory of fundamental strings and put forward as a candidate for a

quantum theory of gravity.

In conventional quantum field theory the elementary particles are mathematical

points, whereas in string theory the fundamental objects are one-dimensional: open

or closed lines, of finite length, that while moving in the space-time describe two-

dimensional surfaces, called world-sheets.

Strings have a characteristic length scale, which can be estimated by dimensional

analysis. Since string theory is a relativistic quantum theory that includes gravity


it must involve the fundamental constants c (the speed of light), ~ (Planck’s con-

stant divided by 2π), and GN (Newton’s gravitational constant). From these one

can form a length, known as the “Planck length”

lP =~

MP c= 1.616 · 10−33 cm ,

where

MP =

√~cGN

= 1.22 · 1019 GeV/c2

is the “Planck mass”. These indicate the length and mass scale at which the

physics should begin to depend on the fundamental one-dimensional structure.

Of course these values are extremely far from the ones that we are able to reach

experimentally nowadays. Experiments at energies far below the Planck energy

cannot resolve distances as short as the Planck length. Thus, at such energies,

strings can be accurately approximated by point particles. From the viewpoint of

string theory, this explains why quantum field theory has been so successful.

In 1974 Gerard ’t Hooft [10] observed a property of SU(N) gauge theories which

was very suggestive of a correspondence between the gauge dynamics and string

theory. The main point consists in treating the rank of the gauge group N as

a parameter of the theory, and considering the limit N → ∞, while keeping the

product λ = g2Y MN fixed (g2

Y M being the Yang-Mills coupling). λ is known as the

’t Hooft coupling and it is the true expansion parameter of such a theory. ’t Hooft

noted that, in addition to the expansion in powers of λ, one may also classify

the Feynman graphs appearing in the correlation function of generic gauge theory

operators in powers of 1/N2. The latter is a topological expansion and, in fact,

1/N2 turns out to correspond to the genus of the surface onto which the Feynman

diagrams can be mapped without overlap.

The double expansion in λ and in powers of 1/N2 looks remarkably similar to the

perturbative expansion for a string theory with coupling constant gs ∼ 1/N and

with the expansion in powers of λ playing the role of the α′ expansion1 around

classical string solutions.

Despite this fascinating interpretation, before 1997, the observation of ’t Hooft had

not been realized in the context of string theory. In other words the ’t Hooft strings

and the “fundamental” strings seemed to be different objects. Amazingly in 1997

Maldacena, moving from symmetry considerations and from the study of a stack

1The constant α′, the Regge slope, is equal to the square of the string scale and is related tothe string tension Ts according to the relation Ts = (2πα′)−1.


of D3-branes [11] conjectured that strings of type IIB superstring theory on the

AdS5×S5 background are the ’t Hooft strings of an N = 4, D = 4 supersymmetric

Yang-Mills theory.

According to this conjecture any physical object or process in the type IIB theory

on AdS5 × S5 background can be equivalently described by N = 4, D = 4 super

Yang-Mills (SYM) theory. In particular, the ’t Hooft coupling is related to the

AdS radius R as (R

ls

)4

= λ ,

where ls =√

α′ is the string scale.

On the string theory side of the duality, ls/R appears as the world-sheet coupling;

hence when the gauge theory is weakly coupled the two dimensional world-sheet

theory is strongly coupled and non-perturbative, and vice versa. In this sense

the AdS/CFT duality is a weak/strong duality. This is in a way the power of

this correspondence because it allows to obtain information on the strong coupling

behavior of one of the two theories by studying the weak coupling regime of the

other. On the other hand it is also the main reason why it is very hard to check

the Maldacena conjecture.

Due to the difficulties of solving the world-sheet theory on the AdS5 × S5 back-

ground, our understanding of the duality on the string theory side has been mainly

limited to the low energy supergravity limit, and in order for the supergravity ex-

pansion about the AdS background to be trustworthy, we generally need to keep

the AdS radius large. At the same time we must also ensure the suppression of

string loops. On the gauge theory side these requests correspond in limiting to the

regime of large ’t Hooft coupling and of N →∞.

However, there is a specific limit, known as Penrose limit, that can be used to

reduce the AdS5×S5 background to a background of plane-fronted wave with par-

allel propagation (pp-wave background) [12, 13, 14, 15, 16, 17] on which the string

theory σ-model becomes solvable [18, 19]. Moving from this evidence and from

the AdS/CFT correspondence, in 2002 Berenstein, Maldacena and Nastase [20]

answered to the open question on how this specific limit translates to the gauge

theory side. They formulated a definite proposal, now called the BMN conjecture,

for mapping the operators of the gauge theory to string states. The pp-wave/BMN

correspondence represent thus a concrete framework where to test the AdS/CFT

duality.

Few months later it was found a Penrose limit that leads to a different pp-wave


background with a flat direction, i.e. which possesses a space-like isometry [21].

Beginning from 2006, a series of decoupling limits of the N = 4 super Yang-Mills

theory in four dimensions with gauge group SU(N) was discovered [22, 23, 24, 25].

These decoupling limits lead to decoupled theories that are much simpler than the

full N = 4 SYM, but still contain many of its interesting features. Furthermore it

was found how, according to the the AdS/CFT duality, the particular limit under

which the theory decouples to the SU(2) sector of the gauge theory translates in

the string theory side. Then it was shown that the spectrum of the gauge theory in

this sector matches that of the string on the pp-wave background with a space-like

isometry, after taking the corresponding decoupling limit for string theory [23].

It is natural at this point to ask whether it is possible to find other Penrose limits

that allow us to get different pp-wave backgrounds, for instance, with more than

one space-like isometry. This would be very interesting for various reasons. First

it is not trivial at all to discover a new background achievable through a limiting

procedure from AdS5×S5 in which the superstring theory is exactly solvable. Then

we could try to extend the matching of spectra found for the SU(2) sector of the

N = 4 SYM and the strings on a pp-wave background with a flat direction, to the

other sectors of the gauge theory. It would be really amazing to succeed in finding

which pp-wave background in the string theory side corresponds to each sector of

the gauge theory. This would be a way to shed light upon another important area

of the AdS/CFT correspondence. These are exactly the motivations justifying the

work presented in this thesis.

The content of the thesis is as follows:

Chapter 1 contains an introduction to the superstring theory. It is shown how the

type IIB theory can be quantized on the flat Minkowski space-time, and how the

spectrum can be obtained.

In chapter 2 the pp-wave space-time is introduced. This play a remarkable role

in superstring theory, being, obviously along with the flat background, the only

background on which the theory is exactly solvable. Then the Penrose limit pro-

cedure is explained. This allows to get a pp-wave background as a limit of a given

space-time. In particular it will be demonstrated that pp-waves arise as Penrose

limit of AdS5 × S5.

Chapter 3 is dedicated to the celebrated AdS/CFT correspondence. First the

’t Hooft model is presented, and it is shown why this suggest a duality between

gauge theories and string theories. Subsequently the Maldacena conjecture is in-


troduced and it is discuss how the procedure of taking the Penrose limit on the

string theory side translates on the gauge theory, according to the AdS/CFT cor-

respondence.

Chapters 4 and 5 contain the original results of the thesis.

In chapter 4 first the already known Penrose limits that lead to pp-wave back-

grounds with zero and one flat direction are generalized, by introducing some arbi-

trary parameters. Then it is presented a new Penrose limit that I found, through

which one can obtain a pp-wave background with two space-like isometries, start-

ing form the AdS5×S5 background. This is one of the original result of the thesis.

In chapter 5 it is shown how to derive, through a coordinate transformation (in

practice a set of four rotations) depending on some parameters, a pp-wave metric

that, varying the values of the parameters, covers all the backgrounds studied in

the previous chapter. In this way the theory can be directly quantized on this

“rotated” background, and then the spectra of each case can be retrieved fixing

opportunely the parameters. This is a very important result within the context of

the AdS/CFT correspondence. As a matter of fact, once one has all these spectra

it is possible to compare them with the ones of the various sectors in which the

gauge theory decouples [25]. From this comparison it is straightforward to match

the results coming from the two sides of duality by relating the relevant parameters

introduced in the rotation of coordinates (that are only four) with the parameters

that distinguish each decoupled sector of the gauge theory.

Appendices A and B contain the convention used for the N = 4 SYM theory in

four dimensions and for the ten dimensional gamma matrices and Majorana-Weyl

spinors.

In appendix C there are all the calculations that lead to the definition of the new

Penrose limit discussed in chapter 4.

Chapter 1

Space-time supersymmetry in

string theory

First we shall explain what are the main motivations that lead to the introduction of

the supersymmetry in string theory. Then we will discuss the fundamental notions

about supersymmetry in a four-dimensional space-time, in order to understand

more easily its generalization to generic dimensions and to a new type of object,

like string. Exploiting the analogy to the case of the supersymmetric relativistic

point particle, we will be able to derive the superstring dynamics. Finally we will

quantize the theory in a particularly convenient gauge, the light-cone gauge.

Throughout this thesis we shall use units in which ~ = c = 1, and unless otherwise

specified the mostly plus metric.

1.1 Why superstring

The bosonic string theory, despite all its beautiful features, has a number of short-

comings. The most obvious of these are the presence of tachyons and the absence

of fermions. It is conceivable that the former feature merely indicates that the

vacuum has been incorrectly identified, and that (as in a Higgs theory) there is

some other stable vacuum that does not give rise to tachyons. The absence of

fermions in the theory is instead a more serious problem, since all the common

matter is constituted by this type of particles. For this reason the bosonic string

theory cannot describe any real phenomenon. The only way to make up for these

problems is to try to formulate another string theory.

There are two different ways to introduce supersymmetry in string theory. One is

based on the introduction of a two-dimensional world-sheet supersymmetry that

relates the space-time coordinates Xµ(τ, σ) to fermionic partners ψµ(τ, σ), which

are two-component world-sheet spinors. In this formalism it is possible to write an

action that is invariant under supersymmetric transformation, with N = 1 internal

Chapter 1. Space-time supersymmetry in string theory 7

degrees of freedom. The application of the action principle gives rise to a consis-

tent string theory with critical dimension D = 10. However the resulting spectrum

shows the presence of tachyons. Thus the spectrum must be truncated in a very

specific manner first proposed by Gliozzi, Scherk and Olive (GSO) in 1977. This

method, called “GSO projection”, through the use of the operator (−)Nf , not only

eliminates tachyons from the spectrum but gives also a ten-dimensional space-time

supersymmetry to the theory.

Note that this formulation was a real amazing discovery, since it proves that string

theory predicts supersymmetry; if string theory is correct so it is supersymmetry.

The other formalism (the one we shall follow) leads to the same theory in a way that

makes supersymmetry manifest. In fact, in this approach, one starts directly with

a covariant world-sheet action with space-time supersymmetry. While it seems to

be difficult to quantize this action covariantly, it can be quantized in light-cone

gauge. Though the resulting formalism is not manifestly Lorentz invariant, it can

be shown to be Lorentz invariant in D = 10.

The GSO conditions are automatically built in from the outset, without having to

make any truncations; bosonic and fermionic strings are unified in a single Fock

space.

1.2 Introduction to supersymmetry

The Coleman-Mandula theorem states that the supersymmetry algebra is the only

graded Lie algebra of symmetries of the S-matrix consistent with relativistic quan-

tum field theory.

The most general relations that define the supersymmetry algebra are

Q Aα , QβB+ = 2σ µ

αβPµδ

BA ,

Q Aα , Q B

β + = QαA, QβB+ = 0 ,[Pµ, Q

Aα

]− =

[Pµ, QαA

]− = 0 ,

[Pµ, Pν ] = 0 ,

(1.1)

where we denote with Qα the supersymmetry generators and with Pµ the energy-

momentum vector.

The indices α, β, . . . and α, β, . . . run from one to two and denote two-component

Weyl spinors (i.e. with defined chirality). The indices (µ, ν, . . . ) = 0, 1, 2, 3 identify

Lorentz four-vectors in a four-dimensional space-time. The capital indices A,B, . . .


refer to the internal degrees of freedom of the physical system (the supersymmetry

degrees of freedom); they run from 1 to some number N > 1. The algebra with

N = 1 is called the supersymmetry algebra, while those with N > 1 are called

extended supersymmetry algebras.

Let us introduce two antisymmetric tensors εαβ and εαβ, that are invariant under

Lorentz transformations:

ε21 = ε12 = 1, ε12 = ε21 = −1, ε11 = ε22 = 0 , (1.2)

Analogous relations hold for ε-tensor with dotted indices. Spinors with upper and

lower indices are related through the ε-tensor:

ψα = εαβψβ, ψα = εαβψβ . (1.3)

The σ-matrices form a basis of the space of 2 × 2 complex matrices and have the

following index structure σ µαα ; σ0 is the identity matrix I2 and σi, i = 1, 2, 3 are

the Pauli matrices (see (B.1) in appendix B). The ε-tensor may also be used to

raise the indices the σ-matrices

σµαα = εαβεαβσ µ

ββ. (1.4)

The σ-matrices satisfy the relations:

(σµσν + σν σµ) βα = −2ηµνδ β

α ,

(σµσν + σνσµ)αβ = −2ηµνδα

β,

Tr (σµσν) = −2ηµν ,

σµαασββ

µ = −2δ βα δ β

α .

(1.5)

The main feature of a supersymmetric theory is that it contains an equal number

of bosonic and fermionic states. It is not hard to demonstrate this claim. Let us

define the “Fermion number” operator Nf , such that (−1)Nf has eigenvalue +1

when it acts on a bosonic state and -1 when it acts on a fermionic state. Thus

(−)NF Qα = −Qα(−)NF . (1.6)

For any finite dimensional representation of the algebra, we find

Tr[(−)NF Q A

α QβB]

= Tr[(−)NF (Q A

α QβB + QβBQ Aα )

]

= Tr[−Q A

α (−)NF QβB + Q Aα (−)NF QβB

]

= 0 .

(1.7)


Using the first relation of (1.1) we conclude

Tr[(−)NF Q A

α , Q Bβ]

= 2σµ

αβδA

BTr[(−)NF Pµ

]= 0 . (1.8)

For any fixed non-zero momentum Pµ this reduce to

Tr(−)NF = 0 . (1.9)

The trace is independent of the choice of the basis. Thus, choosing a basis in which

the operator (−)Nf is represented by a diagonal matrix, from (1.9) we immediately

see that this operator has as many positive as negative eigenvalues. In other words,

the number of bosonic eigenstates is equal to that of fermionic ones.

Now let us consider representations of the supersymmetry algebra, which corre-

spond to massive one-particle states, so that P 2 = −M2; in the rest frame the

anticommutation relations becomeQ A

α , Q Bβ

=

QαA, QβB

,

Q A

α , QβB

= 2MδαβδA

B .(1.10)

Then rescaling the generators as follows

a Aα =

1√2M

Q Aα ,

(a A

α

)†=

1√2M

Q Aα , (1.11)

we see that the algebra (1.10) is isomorphic to the algebra of 2N fermionic creation

and annihilation operators(a A

α

)†and a A

α :

a A

α , a Bβ

=

(a A

α

)†,(a B

β

)†= 0 ,

a A

α ,(a B

β

)†= δαβδAB .

(1.12)

The representation of this algebra are well known, and they are constructed from

a “Clifford vaccum” Ω, defined by the condition

a Aα Ω = 0 . (1.13)

Ω has the remarkable feature to be eigenstate of P 2 with eigenvalue M2:

P 2Ω = −M2Ω (1.14)

Any other state that belongs to the same representation can be achieved from the

Clifford vacuum through the application of creation operators

Ω(n)α1...αn

A1...An=

1√n!

(a A1

α1

)†. . .

(a An

αn

)†Ω . (1.15)


Since the operators a Aα anticommute, the state Ω(n) is antisymmetric under the

exchange of two pairs of indices αi Ai ↔ αj Aj.

Note that all the states of the representation Ω(n) have the same mass. In fact

the operator P 2 commutes with Q Aα , and obviously also with

(a A

α

)†; thus it is

straightforward to see that Ω(n) is eigenstate of P 2 with M2 as eigenvalue.

In order to formulate a supersymmetric field theory we must first represent the al-

gebra (1.1) in terms of fields. Thus let us introduce the anticommuting parameters

ξα along with their complex conjugate ones ξα:

ξα, ξβ = ξα, Qβ = [P µ, ξα] = 0 , (1.16)

with analogous relation that hold for ξα.

The supersymmetry algebra (1.1) becomes

[ξQ, ξQ] = [ξQ, ξQ] = 0 ,

[P µ, ξQ] = [P µ, ξQ] = 0 ,

[ξQ, ξQ] = 2ξσµξPµ ,

(1.17)

where we use the summation convention:

ξQ = ξαQα , ξQ = ξαQα . (1.18)

A component multiplet is a set of fields (A,ψ, . . . ) on which we define the infinites-

imal transformation δξ:

δξA = (ξQ + ξQ)× A ,

δξψ = (ξQ + ξQ)× ψ .(1.19)

The transformation δξ satisfies

(δηδξ − δξδη)A = 2(ησµξ − ξσµη)PµA

= −2i(ησµξ − ξσµη)∂µA .(1.20)

in accord with (1.17). This supersymmetry transformation maps tensor fields into

spinor fields and vice versa. From the algebra (1.1) we see that Q has mass dimen-

sion 1/2. Therefore, fields of dimension l transform into fields of dimension l +1/2

or into derivatives of fields of lower dimension.

Starting with the scalar field A, we define the spinor ψ as the field into which A

transforms:

δξA =√

2 ξαψα . (1.21)


The field ψ transforms into a tensor field of higher dimension F and into the

derivative of A itself:

δξψα = i

√2 σµξα∂µA +

√2ξβFαβ . (1.22)

The coefficient of ∂µA is chosen to guarantee that the commutator of

δηδξA = 2iξσµη∂µA + 2ξηF (1.23)

closes in the sense of (1.20). The same commutator acting on the field ψ closes if

δξF = i√

2 ξσµ∂µψ . (1.24)

If we had been willing to use the field equations, −iσµ∂µψ = mψ, equation (1.24)

could have been satisfied by F = −mA∗. In this case we would have said that the

transformations (1.21) and (1.22) close through the field equations. In extended

supersymmetry we are sometimes forced to close the commutators through the field

equations because we do not yet know the full multiplet structure of the theory.

The component multiplet which we have constructed is called the “scalar multi-

plet”:

δξA =√

2 ξψ ,

δξψ = i√

2 σµξ∂µA +√

2ξF ,

δξF =√

2 ξσµ∂µψ .

(1.25)

These fields form a linear representation of the supersymmetry algebra (1.1). If A

has dimension 1, then ψ has dimension 3/2, while F has dimension 2 and must

assume the role of auxiliary field. The relations (1.25) are called “supersymmetry

tranformations”.

To construct an invariant action it is sufficient to find combinations of fields which

transform into space derivatives. Such combinations are given by

L0 = i∂µψσµψ + A∗¤A + F ∗F , (1.26)

and

Lm = AF + A∗F ∗ − 1

2ψψ − 1

2ψψ (1.27)

Therefore the complete Lagrangian is

L = L0 + mLm , (1.28)


from which we get the equations of motion

iσµ∂µψ + mψ = 0

F + mA∗ = 0

¤A + mF ∗ = 0

(1.29)

They describe a Weyl spinor ψ and a complex scalar A, both of mass m.

The Lagrangian (1.28) has the property to be automatically normal-ordered. This

simply reflects the fact that supersymmetric theories must contain an equal num-

ber of bosonic and fermionic degrees of freedom for a given mass.

Now let us introduce an explicit expression for the generators Q of the supersym-

metry algebra:

QαA =∂

∂θαA− iσ µ

αβθβ

A∂µ ,

QαA =

∂

∂θ Aα

− iσ µ

αβεβαθα∂µ .

(1.30)

The new anticommuting coordinates θ and θ are as many as the number of super-

symmetries (A = 1, . . . , N) of the theory; the indices α and α refer to two different

types of two components Weyl spinors, θ and θ, corresponding to the two repre-

sentations with opposite chirality of the Lorentz group.

The main point is the introduction of the two anticommuting coordinates θ and θ,

which, together with the usual space-time coordinates xµ, form a new space called

“superspace”, whose elements are z = (x, θ, θ).

This formalism, that describes a supersymmetric theory on a superspace, makes the

supersymmetry manifest. The operator ξQ+ ξQ is the generator of the superspace

coordinates transformations:

δθA = [ξQ + ξQ, θA] = [ξQ, θA] = ξA ,

δθA = [ξQ + ξQ, θA] = [ξQ, θA] = ξA ,

δxµ = [ξQ + ξQ, xµ] = iθσµξ − iξσµθ .

(1.31)

In this way the supersymmetry is realized as a “geometric” transformation of the

superspace.

1.3 The classical theory

In this section we present the classical theory of the manifestly supersymmetric

superstring action. Its symmetries are subtle and need to be understood before


attempting quantization. An important ingredient is a new type of local fermionic

symmetry on the world-sheet. It is not ordinary supersymmetry, although it is

certainly related to it. In order to illustrate how it works we first give a brief

description of the super particle, which involves fewer complications than the string

and is therefore easier to analyze in detail.

1.3.1 The superparticle

The action of a relativistic point particle of mass m is

S =1

2

∫dτ

(e−1xµxµ − em2

), (1.32)

where e(τ) is an auxiliary coordinate, that can be identified as the square root of

a one-dimensional metric. An important virtue of this formula is the existence of

the limit m → 0. In generalizing (1.32) to the superparticle we set m = 0, since

the mass term is not relevant for the introduction of supersymmetry.

In addiction to the local reparametrization symmetry τ → f(τ), the action (1.32)

is also invariant under global space-time Poincare transformations generated by

δxµ = aµ + bµνx

ν

δe = 0.(1.33)

In order to achieve space-time supersymmetry we generalize Minkowski space, with

its bosonic coordinates xµ, to a superspace with fermionic coordinates as well as

bosonic ones. If there are to be N supersymmetries, we introduce N anticom-

muting spinor coordinates θAa(τ), with A = 1, 2, . . . , N . The index a is that

of a space-time spinor appropriate to D dimensions. For a general Dirac spinor

a = 1, 2, . . . , 2[D/2]. The superspace we are dealing with is therefore that with

coordinates z = (x, θ, θ), where the spinors θAa and θAa belong to representations

with opposite chirality of the Lorentz group.

In the superspace the supersymmetry transformations are simply variations of the

coordinates of the superspace itself. Introducing N infinitesimal Grassmann pa-

rameters εA, spinors of the same type as the corresponding θA, the transformation

formulas are

δθA = εA δxµ = i εAΓµθA

δθA = εA δe = 0.(1.34)


We consider the spinors that we have just introduced as Majorana spinors, i.e.

with real components. Therefore also the Dirac matrices Γµ turn out to be in the

Majorana representation (purely imaginary).

We now generalize the bosonic point particle, which propagates in Minkowski space,

to a supersymmetric point particle propagating in superspace. Many supersym-

metric actions can be written, since xµ− iθAΓµθA and θA are both invariant under

supersymmetry. Many Lorentz invariant Lagrangians can be built from out of

them. The simplest and most straightforward generalization of (1.32) is

S =1

2

∫dτ e−1

(xµ − iθAΓµθA

)2

. (1.35)

This obviously is Lorentz invariant and supersymmetric and so has the full super-

Poincare symmetry. It gives the equations of motion

p2 = 0, pµ = 0, Γ · p θ = 0, (1.36)

where we define

pµ = xµ − iθAΓµθA. (1.37)

Since (Γ · p)2 = −p2 = 0, the matrix Γ · p has half of its eigenvalues equal to zero.

Furthermore θ always appears multiplied by Γ·p. As a result, half of its components

are actually decoupled from the theory. This is a consequence of a far from obvious

additional symmetry of (1.35). The new symmetry is a local fermionic symmetry,

called κ symmetry, whose transformation law is

δκθA = iΓ · p κA,

δκxµ = iθAΓµδθA,

δκe = 4e ˙θAκA.

(1.38)

where κAa(τ) denote N infinitesimal Grassmann spinor parameters.

The κ transformation is not ordinary supersymmetry either on the world-line or in

space-time. In fact, the action contains no world-line spinors at all. To see what

it is, let us consider the algebra obtained by commuting two κ transformations.

Let δ1 and δ2 represent κ variations with parameters κ1 and κ2, respectively. Then

[δ1, δ2] θA =

(2iΓµκ

A2

˙θBΓ · p ΓµκB1 + 4iΓ · p κA

2˙θBκB

1

)− (1 ↔ 2) . (1.39)

We are dealing here with a symmetry for which the action is lacking the auxiliary


coordinates required for off-shell closure of the algebra. Therefore we must use the

equations of motion to demonstrate closure of the algebra. The equation Γ ·p θ = 0

eliminates the first term in the last expression (1.39). This leaves

[δ1, δ2] θA ∼ iΓ · p κA (1.40)

for the choice

κA = 4κA2

˙θBκB1 − (1 ↔ 2) . (1.41)

Thus the commutator of two κ transformations is a κ transformation. This result

is only possible because there is no on-shell conserved charge associated with κ.

1.3.2 The supersymmetric string action

We start from the bosonic string action [26]

Sbos = − 1

4πα′

∫dτdσ

√−hhαβ∂aX

µ∂bXµ, (1.42)

and using the analogy with the superparticle, an obvious guess is to replace the

derivative term ∂αXµ with the term Πµα defined as

Πµα = ∂αXµ − θAΓµ∂αθA, (1.43)

in this way the supersymmetric superstring action results

S1 = − 1

4πα′

∫dτdσ

√−hhαβ Πµ

α Πνβ. (1.44)

This obviously possesses local reparametrization invariance and N global super-

symmetries. It is not the action we want, however. The local κ symmetry of the

superparticle action is lost in this generalization. As a result θ describes twice as

many degrees of freedom as it should. Also, the equations of motion constitute a

complicated nonlinear system that is quite intractable. Fortunately it is possible

to add a second term S2 so that the total action S = S1 + S2 does have local κ

symmetry. As a result, half the components of θ are again decoupled, and the

equations of motion can be completely solved, at least in a particular gauge.

The construction that restores the local κ symmetry does not work for arbitrary

N , in contrast to the superparticle case. We need to take N ≤ 2 so that there are

at most two supersymmetries. We will present formulas in terms of the coordinates

θ1 and θ2 as appropriate to the N = 2 case. The N = 1 or N = 0 cases can then


be obtained by setting one or both of the θ equal to zero. The extra term in the

action that completes the description of the supersymmetric superstring action is

S2 =1

π

∫dτdσ

[−iεαβ∂αXµ(θ1Γµ∂βθ1 − θ2Γµ∂βθ2

)+ εαβ θ1Γµ∂αθ1θ2Γµ∂βθ2

]

(1.45)

The alternating tensor density εαβ (1.2) explains why there is no factor of√

h. In

fact, the term S2 is completely independent of hαβ. Therefore it does not contribute

to the energy-momentum tensor Tαβ.

The term (1.45) clearly has local reparametrization symmetry and global Lorentz

symmetry, but it is not trivial to see that it also has global N = 2 supersymmetry.

In particular S2, and consequently the total action S, is supersymmetric only under

certain conditions that regard the number of dimensions of the space-time and the

type of spinors θ that we are dealing with. These conditions are verified only in

four cases:

• D = 3 and θ is Majorana;

• D = 4 and θ is Majorana or Weyl;

• D = 6 and θ is Weyl;

• D = 10 and θ is Majorana-Weyl.

Thus, even at the classical level, the superstring theory exists only in four cases.

This is the counterpart of the statement that the classical bosonic string theory

exists for any dimension. This result should not be a surprise, since supersymmetry

is well-known to restrict the possible values of D even at the classical level. The

quantization of the theory will single out the D = 10 case as special.

1.3.3 The local fermionic symmetry

We now wish to show that the sum S1 + S2 has a local fermionic symmetry, that

is not possessed by either term separately, and that is the analogous of the κ

symmetry for the superparticle. The infinitesimal parameters κ now carry three

indices κAαa. The index A = 1, 2 corresponds to the label on θA. The index

α = 0, 1 is a world-sheet vector index and a is a D-dimensional space-time index

corresponding to one of the four possible types of spinors. The spinor index a is

suppressed in most formulas.

From the two-dimensional world-sheet point of view we have a theory without any


spinors at all that has local fermionic symmetries that transform as vectors. The

various fermionic quantities that appear are spinors in the D-dimensional sense, of

course.

In two dimension a vector representation, as that of κAαa in the world-sheet, is

reducible. In particular a decomposition of a vector into what might be called

“self-dual” and “anti-self-dual” pieces is conveniently achieved using the projection

tensor

Pαβ± =

1

2

(hαβ ± εαβ

√h

), (1.46)

which satisfy the projection conditions

Pαβ± hβγ P γδ

± = Pαδ±

Pαβ± hβγ P γδ

∓ = 0.(1.47)

For A = 1 k is restricted to be anti-self-dual:

k1α = Pαβ− k1

β; (1.48)

for A = 2 instead it has to be self-dual:

k2α = Pαβ+ k2

β. (1.49)

It will tum out that A = 1 describes right-moving modes and symmetries whereas

A = 2 describes left-moving modes and symmetries.

Bearing in mind that in the superstring case pµ is replaced by Πµα, we can guess,

in analogy with the superparticle case, that

δθA = 2iΓ · ΠαkAα

δXµ = iθΓµδθA,(1.50)

with δhαβ still to be determined. Using (1.50), the variation of the Lagrangian L1

(that corresponds to the action S1) gives

δL1 = −√

hhαβΠα · δΠβ − 1

2δ(√

hhαβ)

Πα · Πβ, (1.51)

where

δΠµα = 2i∂αθAΓµδθA. (1.52)

The Lagrangian L2 can be rewritten in the form

L2 = −iεαβΠµα

(θ1Γµ∂βθ1 − θ2Γµ∂βθ2

)+ θ4 terms. (1.53)


Varying the θ’s in the Π θ2 piece of L2 and combining the first term in the variation

of L1 gives a term that can be precisely canceled by the second term in δL1 for the

choice

δ(√

hhαβ)

= −16√

h(Pαγ− k1β∂γθ

1 + Pαγ+ k2β∂γθ

2). (1.54)

Since√

hhαβ is unimodular (det = 1) and symmetric, it is important that the

right-hand side of (1.54) be symmetric and traceless. The self-duality properties

of κ1 and κ2 as well as identities such as

Pαγ+ P βδ

+ = P βγ+ Pαδ

+ (1.55)

ensure that this is the case.

In addition to the fermionic symmetries there is a further local bosonic symmetry

of action S1 + S2. One way of discovering it is by considering the algebra of κ

transformations. The closure requires further local bosonic transformations with

infinitesimal parameters λα:

δθ1 =√

hP αβ− ∂βθ1λα

δθ2 =√

hP αβ+ ∂βθ2λα

δXµ = iθAΓµδθA

δ(√

hhαβ)

= 0.

(1.56)

From the supersymmetric superstring action, through the application of the prin-

ciple of least action, we get the equations of motion

Πα · Πβ =1

2hαβhγδΠγ · Πδ

Γ · ΠαPαβ− ∂βθ1 = 0

Γ · ΠαPαβ+ ∂βθ2 = 0

∂α

[√h

(hαβ∂βXµ − 2iPαβ

− θ1Γµ∂βθ1 −2iPαβ+ θ2Γµ∂βθ2

)]= 0

(1.57)

The first of these corresponds to Tαβ = 0; these are complicated nonlinear equa-

tions, but we will show later that they collapse to simple free theory equations in

the light-cone gauge.

In the point-particle case, the supersymmetric action with N θ coordinates pos-

sesses a manifest global SO(N) symmetry corresponding to a rotation of these

variables into one another. In the superstring case we were only able to emulate

the superparticle construction for N = 0, 1 or 2. However, there is no global ro-

tation symmetry in any of these cases. In the N = 2 case the SO(2) symmetry is

explicitly broken by S2 component of the total action.


1.3.4 Types of superstrings

The critical value of space-time dimensions for superstring theory is D = 10; we

will prove this by requiring that the quantized theory is Lorentz invariant. In ten

dimensions we have seen that the coordinates in the superstring action must be

chosen to be Majorana-Weyl spinors, i.e. θA (with A = 1, 2) must be real and

have a definite handedness. The overall meaning of left and right is a matter of

convention, but there are two physically distinct possibilities: either θ1 or θ2 are

chosen to have the same handedness or to have the opposite handedness. In the

case of closed strings the only boundary conditions that are imposed are periodicity

in σ. This is possible in either case, since it does not relate θ1 and θ2. For open

strings, on the other hand θ1 and θ2 must be equated at the ends of the strings.

Since a left-handed spinor cannot equal a right-handed one, this is only possible

in the case when θ1 and θ2 have the same handedness. As a consequence of these

observations it is possible to classify the following different superstring theory:

• Type I superstring theory: superstring theory based on open superstring; the

boundary conditions reduce the space-time supersymmetry to N = 1 only, which

is part of the motivation for the name “type I”.

• Type IIA superstring theory: it is based on closed superstrings in which

θ1 and θ2 have opposite handedness so that the resulting theory necessarily in-

volves oriented strings, since θ1 describes modes that propagate one way around

the string, while θ2 describes modes that propagate in the opposite direction. This

theory has two conserved D = 10 supersymmetries of opposite handedness.

• Type IIB superstring theory: a theory of closed superstrings with two coor-

dinates θ of the same handedness. In this case one has the option of symmetrizing

the left- and right-moving modes to define a theory of unoriented closed strings or

to not do so leaving a theory of oriented closed strings. It is obviously a left-right

asymmetric (chiral) theory.

In this work we will consider only this last kind of theories. There is yet another

possibility for constructing a consistent supersymmetric string theory:

• Heterotic string theory: it is based on using only one θ coordinate rather

than two. In this case we choose to consider bosonic strings in D = 26 that describe

the left-moving modes and superstrings in D = 10 that describes the right-moving

modes.


1.4 Quantization

The structure of phase-space constraints in the supersyinmetric superstring action

makes covariant quantization very difficult. Fortunately, quantization works very

nicely in the light-cone gauge, so we concentrate on its description.

1.4.1 Light-cone gauge

In order to be specific, the following discussion assumes that D = 10 and that the

spinors θ1 and θ2 are Majorana-Weyl. As we will see, the D = 10 case is the only

one in which the Lorentz algebra commutator [J i−, J j−] vanish; none of the other

possible cases D = 3, 4 or 6 has this propriety.

The local reparametrization and Weyl invariances allow us to take

hαβ = ηαβ ,

where ηαβ is the flat metric, and leave a residual symmetry (corresponding to the

conformal invariance) that can be used to impose the condition

X+(τ, σ) = x+ + 2α′p+τ, (1.58)

where X± are the light-cone coordinates defined as

X± =X0 ±XD−1

√2

. (1.59)

In order to understand deeply what this particular choice implies, let us analyze

the consequence of (1.58) on the bosonic degrees of freedom. Thus let us consider

the Polyakov action [27]

Sboslc = − 1

2πα′

∫dτdσ∂τX

− +1

4πα′

∫dτdσ

(∂τX

i∂τXi − ∂σX

i∂σXi), (1.60)

then defining x− as

x−(τ) =1

π

∫ π

0

dσX−(τ, σ)

we can write the Lagrangian corresponding to (1.60):

L = − 1

2α′∂τx

−(τ) +1

4πα′

∫dσ

(∂τX

i∂τXi − ∂σX

i∂σXi). (1.61)

The momentum conjugate to x−(τ) is a constant:

p− = −p+ =δL

δ (∂τx−)= − 1

2α′, (1.62)


whereas the momenta conjugate to the transverse fields, X i with i = 1, . . . , D− 2,

are

Πi =δL

δ (∂τX i)=

1

2πα′∂τX

i =p+

π∂τX

i. (1.63)

The Hamiltonian takes the form

H = p−∂x− +

∫ π

0

dσΠi∂X i −L =1

2

∫ π

0

dσ

(2πα′ΠiΠi +

1

2πα′∂σX

i∂σXi

),

(1.64)

and so the Hamilton equations turn out to be

∂τx− =

∂H

∂p−=

H

p+

∂τp− = −∂τp+ = − ∂H

∂x−= 0

∂τXi =

∂H

∂Πi= πα′Πi

∂τΠi − ∂H

∂X i=

1

4πα′∂2

σXi.

(1.65)

These contain the string dynamics and show that the only relevant degrees of

freedom are the transverse ones; as a matter of fact these are sufficient to describe

the whole system, since x− is univocally determined by the coordinates X i.

In the fermionic sector the κ symmetry can be used to enforce the condition

Γ+θ1 = Γ+θ2 = 0, (1.66)

where Γ+ and Γ− are the two light-cone components of the ten-dimensional Dirac

matrices

Γ± =1√2

(Γ0 ± Γ9

). (1.67)

The relations (1.58) and (1.66) constitute the light-cone gauge conditions.

The matrices Γ+ and Γ− are nilpotent, i.e

(Γ+

)2=

(Γ−

)2= 0, (1.68)

but their sum is nonsingular. It follows that exactly half the eigenvalues of each

must be zero. Therefore the gauge choice (1.66) amounts to setting half the com-

ponents of θ equal to zero. This is the number of components that are decoupled

as a consequence of the local κ symmetries.

A generic spinor in ten dimensions has 32 components. The Majorana condition

makes them real and the Weyl condition sets half of them equal to zero, leaving


16 real components. The light-cone gauge condition (1.66) reduces the count by

another factor of 2 leaving just 8 real components (just as the number of transverse

bosonic fields).

The only manifest symmetry in the light-cone gauge is the rotational invariance

of the eight transverse dimensions. Thus the eight surviving components of each

θ can be regarded as forming an eight-dimensional spinor representation of the

transverse SO(8) group. There are three irreducible representations of the SO(8)

group: one is the fundamental vector representation 8v and the other two are the

spinor representations 8s and 8c, which correspond to the two different chiralities.

We use the letters i, j, k, . . . for 8v labels (e.g. X i), a, b, c, . . . for 8s labels, and

a, b, c . . . for 8c labels.

For the type IIB theory θ1 and θ2 belong to the same representation, that, as a

convention, we choose to be 8s.

Using the symbol S for the eight surviving components of θ in the light-cone gauge,

we have

S1a =√

p+θ1a, (1.69)

S2a =√

p+θ2a. (1.70)

The important point to realize is that the condition Γ+θ = 0 implies that θΓµ∂αθ

vanishes unless µ = −. Thus the equations of motion (1.57) in light-cone gauge

become:(

∂2

∂σ2− ∂2

∂τ 2

)X i(τ, σ) = 0, (1.71)

(∂

∂τ+

∂

∂σ

)S1a(τ, σ) = 0, (1.72)

(∂

∂τ− ∂

∂σ

)S2a(τ, σ) = 0. (1.73)

These equations of motion can be obtained from the following superstring action

Sl.c. = −1

2

∫dτdσ

(T∂αX i∂αX i − i

πSaρα∂αSa

)(1.74)

with i = 1, . . . , 8 and a = 1, . . . , 8. The ρα matrices are two-dimensional Dirac

matrices, that satisfy the algebra

ρα, ρβ = −2ηαβ. (1.75)


In the action (1.74) S1a and S2a have been combined into a two-component Majo-

rana world-sheet spinor Sa

Sa =

(S1a

S2a

). (1.76)

S1a ed S2a can be regarded separately as one component Majorana-Weyl world-

sheet spinors, describing right- and left-moving degrees of freedom, respectively.

In the covariant action S = S1 + S2 the variables θAa transform as world-sheet

scalars. Yet, by the time the light-cone gauge has been fixed, the remaining nonzero

components belong as the components of a two-dimensional spinor.

Choosing the Dirac matrices as follows

ρ0 =

(0 −i

i 0

), ρ1 =

(0 i

i 0

). (1.77)

and exploiting that Sa is simply S†ρ0 and that S1a and S2a are real components of

Majorana spinors, we can write the Lagrangian density of the superstring as

L =T

2X iX i − T

2X ′iX ′i i

2π

(S1aS1a + S2aS2a

)+

i

2π

(S1aS ′1a + S2aS ′2a

). (1.78)

From the Lagrangian density we can get the momenta conjugate to the bosonic

variables

P i =δL

δX i=

p+

πX i, (1.79)

and that conjugate to the fermionic ones

Π1a =δL

δS1a=

i

2πS1a, Π2a =

δL

δS2a=

i

2πS2a. (1.80)

Thus the Hamiltonian turns out to be

H =1

4α′p+

∫ π

0

dσ

(2πα′P iP i +

1

2πα′X ′iX ′i − i

πS1aS ′1a +

i

πS2aS ′2a

). (1.81)

Bearing in mind that we are dealing only with type IIB superstrings, that are

closed strings, we must impose both for the bosonic coordinates X i(τ, σ) and for the

fermionic ones SAa(τ, σ), periodicity boundary conditions over the σ’s coordinate.

X i(τ, σ) = X i(τ, σ + π),

SAa(τ, σ) = SAa(τ, σ + π).(1.82)


The mode expansion of the solution of the equations of motion (1.71), (1.72), (1.73)

are

X i(τ, σ) = xi + 2α′p+τ +i

2

√2α′

∑

n 6=0

1

nαi

ne−2in(τ−σ) +i

2

√2α′

∑

n 6=0

1

nαi

ne−2in(τ+σ),

(1.83)

S1a(τ, σ) =+∞∑

n=−∞Sa

ne−2in(τ−σ), (1.84)

S2a(τ, σ) =+∞∑

n=−∞Sa

ne−2in(τ+σ). (1.85)

Reality of these coordinates implies that

Sa−m = (Sa

m)† . (1.86)

In the type IIB superstring case, S1 and S2 belong to the same spinorial represen-

tation with defined chirality.

The quantization of the theory takes the coordinates to obey to precise commuta-

tion and anticommutation rules. In particular the bosonic fields X i must obey[P i

τ (τ, σ), Xj(τ, σ′)]

= −iδ (σ − σ′) δij

[X i(τ, σ), Xj(τ, σ′)

]=

[P i

τ (τ, σ), P jτ (τ, σ′)

]= 0,

(1.87)

that, written in terms of the oscillators α and α, take the form[xi, pj

]= iδij,

[αi

m, αjn

]= mδm+nδ

ij,[αi

m, αjn

]= 0,

[αi

m, αjn

]= mδm+nδ

ij.

(1.88)

The SAa coordinates have instead canonical anticommutation relations

SAa(τ, σ), SBb(τ, σ′)

= πδabδABδ(σ − σ′), (1.89)

that, expressed in terms of the expansion coefficients, become

Sa

m, Sbn

= δabδm+n,

Sa

m, Sbn

= δabδm+n. (1.90)

Replacing the fields in (1.81) with their mode expansions we get the expression of

the Hamiltonian,

H =1

2p+

(pipi +

2

α′(N + N)

), (1.91)


where N and N are the number operators, in which there are both the bosonic and

fermionic oscillators:

N =∞∑

n=1

(αi−nαi

n + nSa−nS

an

)=

∞∑n=1

n(ai†

n ain + Sa†

n San

),

N =∞∑

n=1

(αi−nαi

n + nSa−nS

an

)=

∞∑n=1

n(ai†

n ain + Sa†

n San

),

(1.92)

with i = 1, . . . , 8 and a = 1, . . . , 8.

Note that in the Hamiltonian (1.91) does not appear any normal ordering constant,

because the zero-point energies cancel as a consequence of the application of a

commutation algebra for the bosonic modes and of an anticommutation algebra

for the fermionic modes.

From the covariant action S1 +S2 we can get the equation of motion for the metric

hαβ. This is the analogue of the Virasoro constraints that one meets in the bosonic

string theory [26]. through the light-cone gauge we can solve these constraints and

obtain the Hamiltonian in terms of the oscillators. So the Virasoro constraints play

no longer the crucial role of selecting the physical states in the spectrum, but they

furnish only informations that we already know. As a matter of fact, solving these

constraints, we get directly α−n , that in the closed string case turns out to be:

α−n =1√

2α′p+

+∞∑m=−∞

(αi

n−mαim +

(m− n

2

)Sa

n−mSam

),

α−n =1√

2α′p+

+∞∑m=−∞

(αi

n−mαim +

(m− n

2

)Sa

n−mSam

).

(1.93)

For closed strings α−0 = α−0 =√

2α′2

p−, thus

p− =1√2α′

(α−0 + α−0

). (1.94)

Using (1.93) we can see that p− is equal to the Hamiltonian:

p− =1

2α′p+

+∞∑m=−∞

[αi

n−mαim + αi

n−mαim + m

(Sa−mSa

m + Sa−mSa

m

)]= H. (1.95)

Replacing p− with H (1.91) in the mass-shell condition M2 = 2p+p− − pipi, we

obtain the superstring spectrum

M2 =2

α′

(N + N

). (1.96)


The level-matching condition

N = N (1.97)

prove that M2 can’t be negative. This means that the superstring spectrum is

tachyon-free.

1.4.2 Lorentz invariance

We have seen that the choice of light-cone gauge is extremely convenient in order

to quantize the superstring theory. However in this way the resulting formalism

is not manifestly Lorentz invariant. Of course we want the theory to be Lorentz

invariant, so we must show that the light-cone Lorentz generators satisfy the Lie

algebra of the Lorentz group

[Jµν , Jρλ

]= −iηνρJµλ + iηµρJνλ + iηνλJµρ − iηµλJνρ, (1.98)

where ηµν is the Minkowski flat metric with signature (−, +, +, . . . , +).

Let us express the angular momentum operators Jµν in the following way

Jµν = lµν + Eµν + Kµν , (1.99)

where

lµν = xµpν − xνpµ, (1.100)

Eµν = −i

∞∑n=1

1

n

(αµ−nαν

n − αν−nαµ

n

), (1.101)

and

Kµ+ = 0, K i− =1

2α′p+

+∞∑n=−∞

K ij−nαj

n,

Kij = Kij0 , K ij

n = − i

4

+∞∑m=−∞

San−mγij

abSbm.

(1.102)

It is easy to verify that Lorentz generators satisfy the Lorentz algebra, except for

the case [J i−, J j−], that, as results from (1.98), is supposed to vanish. In order to

prove this we see that terms quartic in oscillators are guaranteed to cancel as a

consequence of the Lorentz invariance of the classical theory. Only terms quadratic

in oscillators need to be evaluated carefully. Using the identities

[K ij

m, α−n]

= mKijm+n, (1.103)


[K ij

m, Kkln

]=− i

(K il

m+nδjk −Kjlm+nδ

ik −K ikm+nδ

jl + Kjkm+nδil

)

+ m(δikδjl − δilδjk

)δm+n,

(1.104)

after some manipulation, we can see that [J i−, J j−] = 0 only if the number of

space-time dimensions is D = 10.

The main point is that imposing the Lorentz invariance of the theory leads to fix

the dimensionality of the space-time and select the case D = 10 as the only possible

case in which the theory is consistent.

Chapter 2

Superstrings on pp-wave

backgrounds

Supergravity was formulated in the 70’s as a quantum field theory that would

contain supersymmetry, gravity and compactified extra dimensions.

There are four different kinds of supergravity: three of these models correspond to

the low energy limit of the IIA, IIB and heterotic E string theory. The maximally

supersymmetric solutions of supergravity in 10 dimensions of type IIB are the

Minkowski space and AdS5×S5, where AdS5 is a 5-dimensional space with negative

curvature, whose metric solves the Einstein’s equations with cosmological constant

(5-dimensional Anti-de Sitter space) and S5 is a 5-dimensional sphere.

The σ-model for strings on AdS5 × S5 is difficult to solve. However, there is a

specific limit in which AdS5×S5 reduces to a plane-wave [12,13,14,15,16,17], and

in this limit the string theory σ-model becomes solvable [18, 19]. In this special

limit we then know the string spectrum, at least for non-interacting strings.

2.1 Pp-wave metric

Plane-fronted gravitational waves with parallel rays, pp-waves, are a general class

of space-times which support a covariantly constant null Killing vector field vµ ,

∇µvν = 0 , vµvµ = 0 . (2.1)

In the most general form, they have metrics which can be written as

ds2 = −4dudv − F (u, xI)du2 + 2AJ(u, xI)dudxJ + gJK(u, xI)dxJdxK , (2.2)

where gJK(u, xI) is the metric on the space transverse to a pair of light-cone di-

rections given by u, v and the coefficients F (u, xI), AJ(u, xI) and gJK(u, xI) are

constrained by (super-)gravity equations of motion. The pp-wave metric (2.2) has

Chapter 2. Superstrings on pp-wave backgrounds 29

a null Killing vector given by ∂∂v

which is in fact covariantly constant by virtue of

the vanishing of the Γvvu component of the Christoffel symbol.

The pp-waves generally considered in the literature have AJ = 0 and are flat in

the transverse directions, i.e. gIJ = δIJ , for which the metric becomes

ds2 = −4dudv − F (u, xI)du2 + δIJdxIdxJ . (2.3)

A more restricted class of pp-waves, plane-waves, are those admitting a globally

defined covariantly constant null Killing vector field. One can show that for plane-

waves F (u, xI) is quadratic in the xI coordinates of the transverse space, but still

can depend on the coordinate u, F (u, xI) = fIJ(u)xIxJ , so that the metric takes

the form

ds2 = −4dudv − fIJ(u)xIxJdu2 + δIJdxIdxJ . (2.4)

Here fIJ is symmetric and by virtue of the only non-trivial condition coming form

the equations of motion, its trace is related to the other field strengths present.

For the case of vacuum Einstein equations, it is traceless.

There is yet a more restricted class of plane-waves, homogeneous plane-waves, for

which fIJ(u) is a constant, hence their metric is of the form

ds2 = −4dudv − µ2IJxIxJdu2 + dxIdxI , (2.5)

with µ2IJ being a constant.

Now we shall focus on a very special plane-wave solution of ten dimensional type

IIB supergravity which admits 32 supersymmetries. In particular we consider a

special case of (2.5) with µ2IJ = µ2δIJ . This metric, however, is not a solution to

source-free type IIB supergravity equations of motion and we need to add a constant

self-dual RR five-form flux; moreover, the dilaton should also be a constant. The

(bosonic) part of this plane-wave solution is then

ds2 = −4dx+dx− − µ2(xixi+xaxa)(dx+)2+ dxidxi + dxadxa, (2.6a)

F+ijkl =2

gs

µ εijkl , F+abcd =2

gs

µ εabcd , (2.6b)

eφ = gs = constant , i, j = 1, 2, 3, 4 , a, b = 5, 6, 7, 8 , (2.6c)

In the above, µ is an auxiliary but convenient parameter, and can be easily removed

by taking x+ → x+/µ and x− → µx− (which is in fact a light-cone boost).

Let us now consider some of the isometries of the background (2.6). The solution


is invariant under translations in the x+ and x− directions. These translations can

be thought of as two (non-compact) U(1)’s with the generators

i∂

∂x+≡ P+ = −P− , i

∂

∂x−≡ P− = −P+ . (2.7)

Due to the presence of the (dx+)2 term, a boost in the (x+, x−) plane is not a

symmetry of the metric. However, the combined boost and µ scaling:

x− →√

1− v

1 + vx− , x+ →

√1 + v

1− vx+ , µ →

√1− v

1 + vµ , (2.8)

is still a symmetry.

Obviously, the solution is also invariant under two SO(4)’s which act on the xi

and xa directions. The generators of these SO(4)’s will be denoted by Jij and Jab

where

Jij = −i

(xi

∂

∂xj− xj

∂

∂xi

), Jab = −i

(xa

∂

∂xb− xb

∂

∂xa

). (2.9)

Note that although the metric possesses SO(8) symmetry, because of the five-form

flux this symmetry is broken to SO(4)×SO(4). There is also a Z2 symmetry which

exchanges these two SO(4)’s, acting as

xi Z2←→xa . (2.10)

So far we have identified 14 isometries which are generators of a U(1) × U(1) ×SO(4)× SO(4)× Z2 symmetry group. One can easily see that translations along

the xI = (xi, xa) directions are not symmetries of the metric. However, we can

show that if along with translation in xI we also shift x− appropriately, i.e.

xI → xI + εI

1 cos µx+

x− → x− − εI1 µxI sin µx+

,

xI → xI + εI

2 sin µx+

x− → x− + εI2 µxI cos µx+

, (2.11)

where εI1 and εI

2 are arbitrary but small parameters, the metric and the five-form

remain unchanged.

2.2 Penrose limit

In this section we discuss a general limiting procedure, known as Penrose limit

which generates a plane-wave geometry out of any given space-time. This pro-

cedure has also been extended to supergravity by Gueven [12], hence applied to


supergravity this limit is usually called Penrose-Gueven limit [14, 13]. Although

the Penrose limit can be applied to any space-time, if we start with solutions of

Einstein’s equations (or more generally the supergravity equations of motion) we

end up with a plane-wave which is still a (super)gravity solution. In other words

Penrose-Gueven limit is a tool to generate new supergravity solutions out of any

given solution.

The procedure of taking the Penrose limit can be summarized as follows:

i) Find a light-like (null) geodesic in the given space-time metric.

ii) Choose the proper coordinate system so that the metric looks like

ds2 = R2[−4dudv + dv

(dv + AI(u, v, xI)dxI

)+ gJK(u, v, xI)dxJdxK

]. (2.12)

In the above R is a constant introduced to facilitate the limiting procedure, the

null geodesic is parametrized by the affine parameter u, v determines the distance

between such null geodesics and xI parametrize the rest of coordinates. Note that

any given metric can be brought to the form (2.12).

iii) Take R →∞ limit together with the scalings

v =v

R2, xI =

xI

R; u, v, xI = fixed. (2.13)

In this limit AI term drops out and gIJ(u, dv, xI) now becomes only a function of

u, therefore

ds2 = −4dudv + gIJ(u)dxIdxJ . (2.14)

This metric is a plane-wave, though in the Rosen coordinates. Under the coordinate

transformation

xI → hIJ(u)xJ , v → v +1

2gIJh′IKhJLxKxL ,

with hIKgIJhJL = δKL and h′IJ = ddu

hIJ the metric takes the more standard form of

(2.4), the Brinkmann coordinates. The only non-zero component of the Riemann

curvature of plane-wave (2.4) is RuIuJ = fIJ(u) and the Weyl tensor of any plane-

wave is either null or vanishes.

The above steps can be understood more intuitively. Let us start with an observer

which boosts up to the speed of light. Typically such a limit in the (general)

relativity is singular, however, these singularities may be avoided by “zooming”

onto a region infinitesimally close to the light-like geodesic the observer is moving

on, in the particular way given in (2.13), so that at the end of the day from the


original space-time point of view we remove all parts, except a very narrow strip

close to the geodesic. And then scale up the strip to fill the whole space-time, which

is nothing but a plane-wave. The covariantly constant null Killing vector field of

plane-waves correspond to the null direction of the original space-time along which

the observer has boosted.

2.2.1 Penrose limit of AdS5 × S5 spaces

We showed that a plane-wave space-time can be obtained through a Penrose limit

from any space-time. Let us now apply this procedure to AdS5 × S5:

ds2 = R2[− cosh2 ρ dτ 2 + dρ2 + sinh2 ρ dΩ2

3

+(cos2 θdφ2 + dθ2 + sin2 θ dΩ′32] (2.15)

We then boost along a circle of radius R in S5 directions, i.e. we choose the light-

like geodesic along τ − φ direction at ρ = θ = 0. Next, we send R →∞ and scale

the coordinates as

x+ =1

2(τ + φ) , x− =

R2

2(τ − φ) , (2.16a)

ρ =x

R, θ =

y

R, (2.16b)

keeping x+, x−, x, y and all the other coordinates fixed. Inserting (2.16) into (2.15)

and dropping O( 1R2 ) terms we obtain

ds2 = −4dx+dx− − (xixi + yaya)(dx+)2+ dxidxi + dyadya, (2.17)

where i = 1, 2, · · · , 4 and a = 1, 2, · · · , 4.

Since AdS5 and S5 are not Ricci flat, AdS5 × S5 geometry can be supergravity

solution only if it is accompanied with a self-dual five-form flux of type IIB. Taking

the Penrose limit we find that

F(5) =2

gs

dx+ ∧ (dx1 ∧ dx2 ∧ dx3 ∧ dx4 + dy1 ∧ dy2 ∧ dy3 ∧ dy4). (2.18)

Finally the metric can be brought to the form (2.6) through the coordinate trans-

formation

x+ → µx+ , x− → 1

µx−. (2.19)

We would like to note that as we see from the analysis presented here, for the

AdS5×S5 case the xi come from the AdS5 and ya from the S5 directions. However,


after the Penrose limit there is no distinction between the xi or ya directions. This

leads to the Z2 symmetry of the plane-wave.

Starting with a maximally supersymmetric solution, e.g. AdS5 × S5, after the

Penrose limit we end up with another maximally supersymmetric solution, the

plane-wave. In fact that is a general statement that under the Penrose limit we

never lose any supersymmetries.

2.3 Spectrum of type IIB superstrings

on the pp-wave background

In this section we shall explicitly show that the type IIB superstring theory on the

pp-wave background (2.6) is exactly solvable and then we shall derive the string

spectrum for such a theory. The procedure that we shall follow is analogous to that

seen in the previous chapter, where we considered the easiest case (superstrings on

a flat space-time).

2.3.1 Bosonic sector

The bosonic string σ-model action in the background (2.6), which has metric Gµν ,

is

S = − 1

4πα′

∫d2σ

√−g gabGµν∂aXµ∂bX

ν

= − 1

4πα′

∫d2σ

√−g gab(−2∂aX

+∂bX− + ∂aX

I∂bXI − µ2X2

I ∂aX+∂bX

+),

(2.20)

where gab is the world-sheet metric, σa = (τ, σ) are the world-sheet coordinates and

I = 1, 2, · · · , 8. Note that the RR background fluxes do not appear in the bosonic

action. We first need to fix the two dimensional gauge symmetry, a part of which

is done by choosing

√−g gab = ηab , −ηττ = ησσ = 1 . (2.21)

To fix the residual world-sheet diffeomorphism invariance, we note that the equa-

tion of motion for X+, (∂2τ − ∂2

σ)X+ = 0, has a general solution of the form

f(τ + σ) + g(τ − σ). Choosing the light-cone gauge we set f(x) = g(x) = 12α′p+x,

i.e.

X+ = α′p+τ , p+ > 0 . (2.22)


The choices (2.21) and (2.22) completely fix the gauge symmetry. As we know, in

this gauge X+ and X− are not dynamical variables anymore and are completely

determined by XI ’s through the constraints resulting from (2.21) [26]

δLδgτσ

= 0 ,δLδgττ

=δL

δgσσ

= 0 .

Using the solution (2.22) for X+ and setting −gττ = gσσ = 1, these constraints

become

∂σX− =

1

α′p+∂σX

I∂τXI , (2.23)

∂τX− =

1

2α′p+

(∂τX

I∂τXI + ∂σX

I∂σXI − (µα′p+)2XIXI

). (2.24)

We can now drop the first term in (2.20) and replace X+ with its light-cone solution.

After rescaling τ and σ by α′p+, we obtain the light-cone action

Sbos.l.c. =

1

4πα′

∫dτ

∫ 2πα′p+

0

dσ[∂τX

I∂τXI − ∂σX

I∂σXI − µ2X2

I

]. (2.25)

This action is quadratic in XI ’s and hence it is solvable. The equations of motion

for XI , (∂2

τ − ∂2σ − µ2

)XI = 0 , (2.26)

should be solved together with the closed string boundary conditions

XI(σ + 2πα′p+) = XI(σ) . (2.27)

In fact X± should also satisfy the same boundary condition. From (2.22) it is

evident that X+ satisfies this boundary condition. We will come back to the

boundary condition on X− at the end of this section. The solutions to these

equations are

XI = xI0 cos µτ +

pI0

µp+sin µτ +

√α′

2

∞∑n=1

1√ωn

[αI

n e−i

α′p+ (ωnτ+nσ)

+ αIn e

−iα′p+ (ωnτ−nσ)

+ αI†n e

+iα′p+ (ωnτ+nσ)

+ αI†n e

+iα′p+ (ωnτ−nσ)

],

(2.28)

where

ωn =√

n2 + (α′µp+)2 , n ≥ 0 , (2.29)

and α and α correspond to the right and left moving modes. The case of n = 0

has been included for later convenience. The canonical quantization conditions

[XI(σ, τ), P J(σ′, τ)] = iδIJδ(σ − σ′) , (2.30)


where P I = 12πα′∂τX

I , yield

[xI0, p

J0 ] = iδIJ , [αI

n, αJ†m ] = [αI

n, αJ†m ] = δIJδmn . (2.31)

Next, using the light-cone action we work out the light-cone Hamiltonian

Hbos.l.c. =

1

4πα′

∫ 2πα′p+

0

dσ[(2πα′)2P 2

I + (∂σXI)2 + µ2X2

I

]. (2.32)

As we expect, the light-cone Hamiltonian density is the momentum conjugate to

light-cone time X+, P− = 2α′p+ (∂τX

− + µ2X2I ). Plugging the mode expansion

(2.28) into (2.32) we obtain

Hbos.l.c. =

1

α′p+

[α′µp+αI†

0 αI0 +

∞∑n=1

ωn(αI†n αI

n + αI†n αI

n)

]

+8

α′p+

(1

2α′µp+ +

∞∑n=1

ωn

),

(2.33)

where the last term is the zero point energies of bosonic oscillators (after normal

ordering) and we have defined

αI0 ≡ αI

0 =1√

2µp+pI

0 − i

√µp+

2xI

0 . (2.34)

It is easy to check that [αI0, α

J†0 ] = δIJ . In the next section we will see that this

zero point energy is canceled by the zero point energy of the fermionic modes, a

sign of supersymmetry.

Now let us check whether X− also satisfies the closed string boundary condition

X−(σ + 2πα′p+) = X−(σ). From (2.23) we learn that

X−(σ + 2πα′p+)−X−(σ) =

∫ 2πα′p+

0

dσ∂σXI∂τX

I

=∞∑

n=1

n(αI†n αI

n − αI†n αI

n) = 0 ,

(2.35)

where we have used the mode expansion (2.28). Equation (2.35) is the level match-

ing condition, which is in fact a constraint on the physical excitations of a closed

string [27].

The vacuum of the light-cone string theory, |0, p+〉 is defined as a state satisfying

αIn|0, p+〉 = αI

n|0, p+〉 = 0 , n ≥ 0 . (2.36)


Note that this vacuum is specified with the light-cone momentum p+, i.e. for

different values of p+ we have a different string theory vacuum state and hence

a different Fock space built from it. As we see from (2.33), in the plane-wave

background all the string modes, including the zero modes, are massive. In other

words all the supergravity modes (created by α†0 and α†0 ) are also massive.

2.3.2 Fermionic sector

The fermionic sector of the Green-Schwarz superstring action for type IIB strings

is [26]

SF =i

4πα′

∫d2σ (θα)> (βab)αρ∂aX

µΓµ (Db)ρβθβ +O(θ3) . (2.37)

In the above θα, α = 1, 2, are Weyl-Majorana spinors,

(βab)αρ =√−g gabδαρ − εab(σ3)αρ , (2.38)

and (Db)ρβ is the supercovariant derivative

(Db)ρβ = δρ

β∂b + ∂bXν (Ων)

ρβ , (2.39)

where Ων has components given by

Ω− = 0 , (ΩI)αβ =

iµ

4Γ+(Π + Π′)ΓI(σ2)α

β ,

(Ω+)αβ = −1

2µ2xIΓ+I δα

β +iµ

4(Π + Π′)Γ+Γ+ (σ2)α

β ,(2.40)

with I = i, a = 1, 2, · · · 8, Π = Γ1234 and Π′ = Γ5678.

In order to have space-time supersymmetry for the on-shell string modes κ-symme-

try is a necessary fermionic symmetry. In fact by fixing the κ-symmetry we remove

half of the fermionic degrees of freedom (the unphysical ones), so that after gauge-

fixing we are left with 16 physical fermions, describing on-shell space-time fermionic

modes. This number of fermionic degrees of freedom is exactly equal to the number

of physical bosonic degrees of freedom coming from the XI modes after fixing the

light-cone gauge (note that there are left and right modes).

It has been shown that the action (2.37) for the plane-wave background possesses

the necessary k-symmetry [18], and to obtain the physical fermionic modes we need

to gauge fix it, which can be achieved by choosing

Γ+θα = 0, α = 1, 2. (2.41)


Similar to the flat space case, the above suffices to fix the full κ-symmetry of the

plane-wave background [18]. By imposing (2.41) we can reduce the ten dimen-

sional fermions to SO(8) representations, and since the two θα have the same ten

dimensional chiralities, both of them end up to be in the same SO(8) fermionic

representation, which we have chosen to be 8s.

To simplify the action we note that (2.41) implies

(θα)>ΓIθβ = 0 ∀α, β , (ΩI)αβθβ = 0 .

From the ∂aXµΓµ term in the action only ∂aX

+Γ+, and from the Ωµ terms only

Ω+ survive and hence

Sferl.c. =

i

4πα′

∫dτ

∫ 2πα′p+

0

dσ[(θα)> (βab)αρ(∂aX

+Γ+) (δρβ∂b + ∂bX

+ (Ω+)ρβ)θβ

].

Next, we use (2.40) and (2.22) to further simplify the action; after some straight-

forward algebra we obtain

Sferl.c. =

−i

4πα′

∫dτ

∫ 2πα′p+

0

dσ[θ†∂τθ + θ∂τθ

† + θ∂σθ + θ†∂σθ† − 2iµθ†Πθ

].

(2.42)

Note that in the above we have replaced θ1 and θ2 which are now eight component

8s fermions with their complexified version

θa =1√2(θ1

a + iθ2a) , θ†a =

1√2(θ1

a − iθ2a) . (2.43)

The last term in the action is a mass term resulting from the RR five-form flux of

the background. As we see the spin connection does not contribute to the action

after fixing the κ-symmetry.

The above action takes a particularly nice and simple form if we adopt SO(4) ×SO(4) representations for fermions. In that case θ and θ† are replaced with θαβ, θαβ

and their complex conjugates, where α and α are Weyl indices of either of the

SO(4)’s 1. In this notation the action reads

SFerlc =

−i

4πα′

∫dτ

∫ 2πα′p+

0

dσ

[θ†αβ∂τθ

αβ + θαβ∂τθ†αβ + θαβ∂σθ

αβ

+ θ†αβ∂σθ†αβ − 2iµθ†αβθαβ + θ†

αβ∂τθ

αβ + θαβ∂τθ†αβ

+ θαβ∂σθαβ + θ†αβ∂σθ

†αβ− 2iµθ†

αβθαβ

].

(2.44)

1We should remember that SO(4) ' SU(2)× SU(2).


As we see θαβ and θαβ decouple from each other. The coupled equations of motion

for the fermions are

(∂τ + ∂σ)(θαβ + θ†αβ)− iµ(θαβ − θ†αβ) = 0 ,

(∂τ − ∂σ)(θαβ − θ†αβ)− iµ(θαβ + θ†αβ) = 0 .(2.45)

The solution to the above is

θ =1√p+

β0eiµτ

+1√2p+

∞∑n=1

c−n

[(1− ρ−n)βn e

−iα′p+ (ωnτ+nσ)

+ (1 + ρ−n)β†n e+i

α′p+ (ωnτ+nσ)]

+cn

[(1− ρn)βn e

−iα′p+ (ωnτ−nσ)

+ (1 + ρn)β†n e+i

α′p+ (ωnτ−nσ)]

(2.46)

where ωn is defined in (2.29) and

ρ±n =ωn ± n

α′µp+, c±n =

1√1 + ρ2±n

. (2.47)

In the above, since there was no confusion, we have dropped the fermionic indices.

θαβ’s also satisfy a similar equation, with similar solutions.

Imposing the canonical quantization conditions

θαβ(σ, τ), θ†ρλ(σ′, τ) = 2πα′δα

ρδβλδ(σ − σ′) , (2.48)

leads to

β0, β†0 = 1, βn, β

†m = βn, β

†m = δmn , (2.49)

where again we have suppressed the fermionic indices.

Using the light-cone action and the mode expansion (2.46), we work out the light-

cone Hamiltonian:

Hfer.l.c. =

1

α′p+

[α′µp+β†0β0 +

∞∑n=1

ωn(β†nβn + β†nβn)

]

− 8

α′p+

(1

2α′µp+ +

∞∑n=1

ωn

),

(2.50)

where in the above we have used β†nβn as a shorthand for β†nαββαβn + β†

nαββαβ

n for

n ≥ 0.

In the full light-cone Hamiltonian, which is a sum of bosonic and fermionic contri-

butions, the zero point energies cancel and

H(2)l.c. =

1

α′p+

[α′µp+(αI†

0 αI0 + β†0β0) +

∞∑n=1

ωn(αI†n αI

n + αI†n αI

n + β†nβn + β†nβn)

].

(2.51)


Having worked out the Hamiltonian and the mode expansions we are now ready to

summarize and list the low lying string states in the plane-wave background. First,

we note that the level matching condition (2.35) also receives contributions from

fermionic modes. Again using the fact thatδ(Lb+Lf )

δgτσ= 0 we find that a term like

θ†θ should be added to the right-hand-side of (2.23) and hence the improved level-

matching condition in which the fermionic modes have been taken into account

is ∞∑n=1

n(αI†n αI

n + β†nβn − αI†n αI

n − β†nβn)|Ψ〉 = 0 , (2.52)

with |Ψ〉 a generic physical closed string state.

As usual the free string theory Fock space, H, is [27]

H = |vacuum〉 ∞⊕m=1

Hm , (2.53)

where Hm, the m-string Hilbert space, is nothing but m-copies of (or the direct

product of m) single-string Hilbert spaces H1.

The string theory vacuum state in the sector with light-cone momentum p+, which

will be denoted by |v〉, is the state that is annihilated by all αn and βn:

αn|v〉 = αn|v〉 = 0 , βn|v〉 = βn|v〉 = 0 , ∀n ≥ 0 . (2.54)

This state is clearly invariant under SO(4)×SO(4) symmetry and has zero energy.

However, it is possible to define some other “vacuum” states which are invariant

under the full SO(8). These states all necessarily have higher energies.

|0〉 ≡ β†011β†012β

†021β

†022|v〉 , or |0〉 ≡ β†011β

†012β

†021β

†022|v〉 . (2.55)

It is evident that both |0〉 and |0〉 have energy equal to 4µ. The interesting and

important property of |0〉 and |0〉 is that they are SO(8) invariant and hence it is

natural to assign them with positive Z2 eigenvalues2. On the other hand it is not

hard to check that under Z2

β0 12 ←→ β0 21 and β0 12 ←→ β0 21 .

Therefore |v〉 and |0〉 should have opposite Z2 charges; with the positive assignment

for |0〉, |v〉 should have negative Z2 eigenvalue.

2Note that as discussed in section 2.1 Z2 is a specific SO(8) rotation.

Chapter 3

The AdS/CFT correspondence

The first evidence of the existence of a correspondence between gauge and string

theories arose in 1974 when ’t Hooft [10,28] noted the analogy in the perturbation

expansion of a SU(N) gauge theory, in the large N limit, and of a string theory.

However, the observation of ’t Hooft had not been realized in the context of string

theory until 1997, when Maldacena, exploiting considerations on symmetries and

studying the geometry of a stack of D3-branes, made the brilliant hypothesis that

type IIB string theory on AdS5 × S5 is dual to N = 4 super Yang-Mills (SYM)

theory in D = 4 dimensions.

Unfortunately the Maldacena conjecture is difficult to check, because it establishes

a duality between theories in opposite regimes. Our ability does not go beyond

the large N limit which corresponds to the supergravity limit on the string theory

side, except for quantities which are protected by supersymmetry.

In the previous chapter we have shown that the superstring σ-model is solvable on

the pp-wave space-time, which can be obtained the from AdS5 × S5 background

through the procedure of Penrose limit. This result suggests a way to go beyond

the supergravity limit: first we have to understand how the procedure of taking

the Penrose limit translates to the gauge theory side. We then need a definite

proposal for mapping the operators of the gauge theory to (single) string states.

This proposal, following the work of Berenstein, Maldacena and Nastase [20], is

known as the BMN conjecture.

3.1 ’t Hooft’s model

Attempts at understanding strong dynamics in gauge theories led ’t Hooft to intro-

duce a remarkable expansion for such theories with large gauge groups (with rank

∼ N). He suggested treating the rank of the gauge group as a parameter of the

theory, and expanding in 1/N2, which turns out to correspond to the genus of the

surface onto which the Feynman diagrams can be mapped without overlap. In this

Chapter 3. The AdS/CFT correspondence 41

way one obtains a topological expansion analogous to the genus expansion in string

theory, with the gauge theory Feynman graphs viewed as “string theory” world-

sheets. In this correspondence, the planar (non-planar) Feynman graphs may be

thought of as tree (loop) diagrams of the corresponding “string theory”.

Asymptotically free theories, like SU(N) gauge theory with sufficiently few matter

fields, exhibit dimensional transmutation, in which the scale dependent coupling

gives rise to a fundamental scale in the theory. For QCD, this is the confinement

scale ΛQCD. Since this is a scale associated with physical effects, it is natural to

keep this scale fixed in any expansion. This scale appears as a constant of integra-

tion when solving the β function equation, and it can be held fixed for large N if

we also keep fixed the product g2Y MN while taking N →∞. This defines the new

expansion parameter of the theory, the ’t Hooft coupling constant1

λ ≡ g2Y MN. (3.1)

To see how the expansion works in practice, we can consider the action for a

gauge theory, for example the N = 4 Super Yang-Mills theory2 written down in

component form

L =1

g2Y M

Tr(− 1

2FµνF

µν +θI

16π2FµνF

µν +6∑

i=1

DµφiDµφi

+4∑

A=1

iΨAΓµDµΨA +1

2

6∑i,j=1

[φi, φj]2 +6∑

i=1

ΨAΓi[φi, ΨA])

.

(3.2)

In this action we have a single vector, four Weyl fermions and six real scalars, all

in the adjoint representation of the gauge group; the covariant derivative is defined

as Dµχ = ∂µ − i[Aµ, χ] (for a more detailed discussion see appendix A). We have

scaled our fields so that an overall factor of 1/g2Y M appears in front. We write this

in terms of N and the ’t Hooft coupling λ, using 1/g2Y M = N/λ. The perturbation

series for this theory can be constructed in terms of Feynman diagrams built from

propagators and vertices in the usual way. With our normalization, each propagator

contributes a factor of λ/N , and each vertex a factor of N/λ. Loops in diagrams

appear with group theory factors coming from summing over the group indices of

1Note the difference with a conformally invariant theory such as N = 4 SYM, in which the β

function vanishes for all values of the coupling gY M . There is no natural scale in this theory thatshould be held fixed. This makes limits different from the ’t Hooft limit possible

2Supersymmetry is not consequential to this discussion and we ignore it for now.


the adjoint generators. These give rise to an extra factor of N for each loop. A

typical Feynman diagram will be associated with a factor

λP−V NV−P+(L+1) (3.3)

if the diagram contains V vertices, P propagators and L loops. These diagrams

can be drawn them using the ’t Hooft double line notation. For U(N), the group

index structure of adjoint fields is that of a direct product of a fundamental and

an anti-fundamental. The propagators are represented with two lines showing the

flow of each index, and the arrows point in opposite directions (see Figure 3.1).

The vertices are drawn in a similar way, with directions of arrows indicating the

Scalar propagator

Gluon propagator

Fermion propagator

Planar diagram Non-planar diagram

Typical interaction vertices

Figure 3.1: Typical Feynman rules for adjoint fields and sample planar andnon-planar diagrams.

fundamental or anti-fundamental indices of the generators.

In this diagrammatic presentation, the propagators form the edges and the insides

of loops are considered the faces. The one point compactification of the plane

then means that the diagrams give rise to closed, compact and orientable surfaces,

with Euler characteristic χ = V − P + F = 2 − 2h, where h is the genus of the

surface. The number of faces is one more than the number of loops, since the

group theory always gives rise to an extra factor of N for the last trace. With the


one-point compactification the outside of the diagram becomes another face and

can be interpreted as the last trace.

The perturbative expansion of the vacuum correlation function takes the form of

a double expansion∞∑

h=0

N2−2hPh(λ) (3.4)

with h the genus and Ph some polynomial in λ, which itself admits a power series

expansion

Ph(λ) =∞∑

n=0

Ch,nλn (3.5)

The simple idea is that all the diagrams generated for the vacuum correlation

function can be grouped in classes based on their genera, and all the diagrams in

each class will have varying dependences on the ’t Hooft coupling λ. Collecting

together all the diagrams in a given class again into groups sharing the same

dependence on λ, we can extract the h and n dependent constant Ch,n. It is

clear from (3.4) that for large N , the dominant contributions come from diagrams

of the lowest genus, the planar (or spherical) diagrams.

The double expansion (3.4) and (3.5) looks remarkably similar to the perturbative

expansion for a string theory with coupling constant 1/N and with the expansion

in powers of λ playing the role of the world-sheet expansion. The analogy extends

to the genus expansion, with the Feynman diagrams loosely forming a sort of

discretized string world-sheet. At large N , such a string theory would be weakly

coupled. The string coupling measures the difference in the Euler character for

world-sheet diagrams of different topology. This has long suggested the existence

of a duality between gauge and string theory.

So far we have considered only the vacuum diagrams, though the same arguments

go through when considering correlation functions with insertions of the fields.

The action appearing in the generating functional of connected diagrams must be

supplemented with terms coupling the fundamental fields to currents, and these

terms will enter with a factor of N . The planar3 (leading) contributions to such

correlation functions with j insertions of the fields will be suppressed by an extra

factor of N−j relative to the vacuum diagrams. The one particle irreducible three

and four point functions then come with factors of 1/N and 1/N2 relative to

the propagator, suggesting that 1/N is the correct expansion parameter. The

3With the point at infinity identified, planar diagrams become spheres, and higher genusdiagrams spheres with handles.


expansion (3.4) for these more general correlation functions still holds if we account

for the extra factors of N coming from the insertions of the fields. The extra factor

depends on the number of fields in the correlation function, but is fixed for the

perturbative expansion of a given correlator.

3.2 The Maldacena conjecture

The most important realization of ’t Hooft’s observation is the celebrated AdS/CFT

correspondence. The duality is suggested by the two viewpoints presented by D-

branes. The low energy effective action of a stack of N coincident D3-branes is

given by N = 4, D = 4 super Yang-Mills theory with gauge group U(N). For large

N the stack of D-branes will modify the geometry seen by the type IIB strings. So

the stack of D-brane can be analyzed either by the gauge theory viewpoint or by

the supergravity viewpoint. The ability to take these different viewpoints is the

essence of the AdS/CFT duality.

Let us consider a stack of parallel Dp-branes. These realize p+1 dimensional U(N)

SYM theories. The metric in type II supergravity have been known since the early

90’s [29] and may be expressed in the following simple form:

ds2 = H−1/2(r)

[−f(r)dt2 +

p∑i=1

(dxi)2

]+ H1/2(r)

[f−1(r)dr2 + r2dΩ2

8−p

], (3.6)

where

H(r) = 1 +R7−p

r7−p, f(r) = 1− r7−p

0

r7−p, (3.7)

and dΩ28−p is the metric of a unit 8− p dimensional sphere. The horizon is located

at r = r0 and the extremality is achieved in the limit r0 → 0.

Among the solutions (3.6) p = 3 has a special status: in the extremal limit r0 → 0

the 3-brane solution

ds2 =

(1 +

R4

r4

)−1/2 (−dt2 + dx21 + dx2

2 + dx23

)+

(1 +

R4

r4

)1/2 (dr2 + r2dΩ2

5

)

(3.8)

is perfectly non-singular.

The limiting form of the extremal metric as r → 0 is

ds2 =R2

z2

(−dt2 + d~x2 + dz2)

+ R2dΩ25 , (3.9)


Figure 3.2: Spacetime around D3-branes.

where z = R2

r. This describes the direct product of 5-dimensional Anti-de Sitter

space, AdS5, and the 5-dimensional sphere, S5, with equal radii of curvature R.

Since both factors of the AdS5×S5 space (3.9) are maximally symmetric, we have

Rabcd = − 1

R2[gacgbd − gadgbc] (3.10)

for the AdS5 directions, and

Rijkl =1

R2[gikgjl − gilgjk] (3.11)

for the S5 directions. This shows that near r = 0, the extremal 3-brane geometry

(3.8) is non-singular and, in fact, all appropriately measured curvature components

become small for large R. Roughly speaking, this geometry may be viewed as a

semi-infinite throat of radius R, which for r À R opens up into flat 9+1 dimensional

space (see Figure 3.2). Thus, for R much larger than the string scale√

α′, the entire

3-brane geometry has small curvatures everywhere and is appropriately described

by the supergravity approximation to type IIB string theory. Let us see how the

requirement R À √α′ translates into the language of U(N) SYM theory on N

coincident D3-branes. To this end it is convenient to equate the ADM tension of

the extremal 3-brane classical solution to N times the tension of a single D3-brane.


In this fashion we find the relation

2

κ2R4Ω5 = N

√π

κ, (3.12)

where Ω5 = π3 is the volume of a unit 5-sphere, and κ =√

8πG is the 10-

dimensional gravitational constant. It follows that

R4 =κ

2π5/2N . (3.13)

Since κ = 8π7/2gstα′2, (3.13) gives R4 = 4πNgstα

′2. In turn, gst determines the

Yang-Mills coupling on the D3-branes through g2YM = 4πgst. Thus, we have

R4 = g2YMNα′2 , (3.14)

i.e. the size of the throat in string units is measured by the ‘t Hooft coupling. This

remarkable emergence of the ‘t Hooft coupling from gravitational considerations

is at the heart of the success of the AdS/CFT correspondence. Moreover, the

requirement R À √α′ translates into g2

YMN À 1: the gravitational approach is

valid when the ‘t Hooft coupling is very strong and the traditional field theoretic

methods are not applicable.

The duality is also supported by considerations on the symmetry groups of the

two theories. The isometry group of AdS5, SO(2, 4), coincides with the conformal

group in 3 + 1 dimensions. The isometry of the sphere S5, SO(6), corresponds to

SU(4) ∼ SO(6) R-symmetry group of N = 4 SYM theory.

These motivations led Maldacena to conjecture that

Type IIB superstring theory on the AdS5×S5 background is dual to N = 4, D = 4

supersymmetric Yang-Mills theory.

From (3.14) we see that when the gauge theory is weakly coupled the two dimen-

sional world-sheet theory is strongly coupled and non-perturbative, and vice-versa.

In this sense the AdS/CFT duality is a weak/strong duality.

The specific prescription for the correspondence is suggested by the matching of

the global symmetry groups and their representations on the two sides of the du-

ality. The matching extends to the partition function of the N = 4 SYM on the

boundary of AdS5 (R × S3) and the partition function of IIB string theory on

AdS5 × S5 ⟨e

∫d4x φ0(x)O(x)

⟩CFT

= Zstring

[φ|boundary = φ0(x)

], (3.15)


where the left-hand-side is the generating function of correlation functions of gauge

invariant operators O in the gauge theory (such correlation functions are obtained

by taking derivatives with respect to φ0 and setting φ0 = 0) and the right-hand-side

is the full partition function of (type IIB) string theory on the AdS5×S5background

with the boundary condition that the field φ = φ0 on the AdS boundary. The

dimensions of the operators O (i.e. the charge associated with the behavior of the

operator under rigid coordinate scalings) correspond to the free-field masses of the

bulk excitations.

3.3 The plane-wave/Super Yang-Mills

duality

The technical difficulties of solving the σ-model for strings on the AdS5×S5 back-

ground have made our understanding of the string theory side of the duality mainly

limited to the low energy supergravity limit. In order for the supergravity expan-

sion about the AdS background to be trustworthy, we generally need to keep the

AdS radius large. At the same time we must also ensure the suppression of string

loops. As a result, most of the development and checks of the duality from the

string theory side have been limited to the regime of large ’t Hooft coupling and

the N →∞ limit on the gauge theory side.

However, one might wonder if it is possible to go beyond the supergravity limit,

which corresponds to restricting to some particular sector of the gauge theory. In

this way real string theory calculations could be performed from the gauge theory

side. We would then need to have similar results from the string theory side to

compare with.

In section 2.2 we demonstrated the fact that plane-waves may generically arise as

Penrose limits of given geometries and, in particular, the maximally supersymmet-

ric plane-wave appears as the Penrose limit of AdS5 × S5 geometry. On the other

hand the Maldacena’s conjecture states that type IIB string theory on AdS5 × S5

background is dual to the N = 4, D = 4 (super-conformal) gauge theory. Here

we would like to show the latter duality can be revived for type IIB strings on the

maximally supersymmetric plane-wave. Thus we need to understand how this spe-

cific limit translates to the gauge theory side. We then need a definite proposal for

mapping the operators of the gauge theory to (single) string states. This proposal,

following the work of Berenstein-Maldacena-Nastase [20], is known as the BMN


conjecture.

The basic idea of the BMN proposal is to start with the usual AdS/CFT duality

and find what parallels the procedure of taking the Penrose limit in the dual gauge

theory side.

As we argued in section 2.2 the process of taking the Penrose limit consists of

finding a light-like geodesic and rescaling the other light-like direction, as well as

all the other transverse directions, in the appropriate way given in (2.13). For

the case of AdS5 × S5 the geodesic was chosen as a combination of a direction

in S5 and the global time (2.16). The generator of translation along this light-

like geodesic, P−, is then a combination of translation along the global time and

rotation along the S1 inside S5 (2.16a). According to the AdS/CFT duality, how-

ever, translation along global time corresponds to the dilatation operator of the

N = 4 gauge theory on R4 while the rotation in the S1 direction corresponds to a

U(1) of the R-symmetry. Explicitly the dilatation operator D is the generator of

U(1)D ∈ SU(2, 2) ' SO(4, 2) (the conformal group in four dimensions) and J is

the generator of U(1)J ∈ SU(4) ' SO(6) R-symmetry.

3.3.1 Decomposition of N = 4 fields into D, J eigenstates

The matter content of the N = 4 gauge multiplet naturally falls into the represen-

tations of SO(4, 2) × SO(6). However, in order to trace the Penrose limit in the

gauge theory and state the BMN proposal we need to study their representations

in the SO(4)× SO(4)× U(1)× U(1) subgroup of SO(4, 2)× SO(6).

The N = 4 gauge multiplet contains six real scalars, φI , I = 1, · · · , 6, four gauge

fields Aa, a = 1, 2, 3, 4, and eight complex Weyl fermions, ψAα , α = 1, 2 and

A = 1, 2, 3, 4 (see appendix A). Here we are only interested in U(N) gauge theories

where scalars and fermions are both in the adjoint representation of the U(N),

so they are N × N hermitian matrices. Aa are not in the adjoint representation

however (but they do transform in the adjoint for global transformations), and as

in any gauge theory one might consider the covariant derivative of the gauge theory

Da = ∂a + iAa (3.16)

which is in the adjoint of the local U(N). In all our arguments we will consider

Euclidean gauge theory on R4 so the a index of Da is an O(4) index. We might,

however, switch between field theories on R4 and its conformal map, R× S3.

The eigenvalues of J will be denoted by J . Since J is the generator of a U(1)


subgroup of U(4) R-symmetry group, the gauge fields are trivial under it. That is,

[J , Da] = 0 ; (3.17)

in other words Da has charge J = 0. The scalars, however, decompose into two

sets. We choose J to make rotations in the φ5 and φ6 plane, i.e.

Z =1√2(φ5 + iφ6) ; [J , Z] = +Z , (3.18)

and hence [J , Z†] = −Z†. Therefore Z has J = 1 (and Z†, J = −1). The other

four scalars, which will be denoted by φi, i = 1, 2, 3, 4 commute with J and have

J = 0. The 16 fermionic fields also decompose into two sets of eight with J = ±12.

The eigenvalue of D will be denoted by ∆. For fields in the N = 4 gauge multiplet

at free field theory level, ∆ = 1 for scalars and Da and ∆ = 32

for fermions.

Hereafter we will use ∆0 to denote the dimension of operators at free field level

(the engineering dimensions) and ∆ for the full interacting theory. More explicitly,

[D, Z(0)] = (1 + O(g2Y M))Z(0) , [D, Z†(0)] =

(1 + O(g2

Y M))Z†(0)

[D, φi(0)] = (1 + O(g2Y M))φi(0) , [D, Da(0)] =

(1 + O(g2

Y M))Da(0)

[D, ψAα(0)] =

(3

2+ O(g2

Y M)

)ψA

α(0) , [D, ψAα(0)] =

(3

2+ O(g2

Y M)

)ψA

α(0) .

(3.19)

After taking out the two U(1) factors (D, J ) of the SO(4, 2)×SO(6) (or SU(2, 2)×SU(4)), the bosonic part of four dimensional superconformal group, we remain

with an SO(4) × SO(4) (one SO(4) ∈ SO(4, 2) and the other SO(4) ∈ SO(6))

subgroup. We also need to find the SO(4) × SO(4) representation of the fields.

We shall omit all the details for the sake of brevity, and we shall only report the

SO(4) × SO(4) × U(1) × U(1) representations of all fields of the N = 4 gauge

multiplet in Table 3.1.

3.3.2 Stating the BMN proposal

Having worked out the SO(4) × SO(4) × U(1)D × U(1)J representation of the

N = 4 fields, we are ready to take the BMN limit, restricting to the operators

with parameterically large R-charge J , but finite ∆0−J . In fact, starting with the

AdS/CFT correspondence, the BMN limit on the gauge theory side parallels the


Field ∆0 − J ∆0 + J SO(4)× SO(4)

Z 0 2 (1,1)

Z† 2 0 (1,1)

φi 1 1 (1,4)

Da 1 1 (4,1)

ψαβ 1 2 ((2,1), (2,1))

ψαβ 1 2 ((1,2), (1,2))

ψαβ 2 1 ((2,1), (1,2))

ψαβ 2 1 ((1,2), (2,1))

Table 3.1: SO(4) × SO(4) × U(1) × U(1) representations of all fields of theN = 4 gauge multiplet. The dimensions are those of the free theory. For the J

charge of fermions note that ψαβ and ψαβ are related by CPT and hence haveopposite J charge; similarly for the other two fermionic modes.

Penrose limit on the gravity side, according which

−i∂

∂φ←→ J (3.20a)

i∂

∂τ←→ D (3.20b)

Then, (2.16) or (3.14) imply that

iµ∂

∂x−=

iα′

2R2

(∂

∂τ− ∂

∂φ

)←→ 1

2√

g2Y MN

(D + J ) , (3.21a)

i

µ

∂

∂x+= i

(∂

∂τ+

∂

∂φ

)←→ D −J . (3.21b)

On the gravity (string theory) side i ∂∂x− and i ∂

∂x+ are the light-conemomentum and

the light-coneHamiltonian, respectively.

Taking the Penrose limit (2.16) is then equivalent to taking g2Y MN and J to infinity

while keeping J2

g2Y MN

fixed. According to (3.21a) the value of J2

g2Y MN

is equal to the

string light-conemomentum (squared) on the string theory side.

In summary, part one of the plane-wave/SYM duality can be stated as

The light-cone string field theory Hamiltonian in the plane-wave background is equal

to the difference between the dilatation operator D and the R-charge operator J :

1

µHSFT = D − J , (3.22)


in the sector of the gauge theory consisting of gauge invariant operators with para-

metrically large R-charge, the BMN sector.

Chapter 4

New Penrose limit

In section 2.2 we showed how the Penrose limit allows us to reduce the AdS5 × S5

space-time to a plane-wave background.

From the Penrose limit of AdS5×S5 we can get different plane-wave backgrounds.

This is a consequence of the fact that one can choose any light-like geodesic of

AdS5×S5 in the procedure of taking the Penrose limit. Of course different choices

for the geodesic, i.e. for the light-cone coordinates x+ and x−, lead to different

pp-wave backgrounds.

In subsection 2.2.1 we have considered only a particular choice of the geodesic,

through which we were able to obtain the plane-wave metric (2.6).

In 2002 it was found a “new” Penrose limit [21] that leads to a background which

is again of pp-wave type, but it results slightly different from the one that had been

known till then. The peculiarity of this background is that it possesses a space-like

isometry, since the metric does not depend on one of the coordinates. The direction

determined by this particular coordinate is usually called flat direction.

It is important to emphasize that, correctly speaking, in order to find the new

pp-wave background, it has not been introduced a “different” Penrose limit. In

fact the process of taking the Penrose limit is unique and is the one explained in

section 2.2. The difference, as we said before, relies instead in the definition of

coordinates made within the procedure. Thus the expression “new Penrose limit”

must be understood in this way. In the next chapter we shall show that all the

pp-wave backgrounds achievable through the Penrose limit are tied by a time-

dependent coordinate transformation. This proves that mathematically they are

all equivalent. The same statement is not true from the physical point of view, since

the transformation involves time. Thus what changes from a background to another

is what we call time, and consequently what we call Hamiltonian. Therefore the

physics is different when we consider the theory on different pp-wave backgrounds,

and for this reason it is important to find as much of these backgrounds as possible.

The main aim of this chapter is to find how we can reach a pp-wave background

Chapter 4. New Penrose limit 53

with two space-time isometries, by applying the Penrose limit on AdS5× S5. This

is an interesting case that has not been considered in literature yet.

Before doing so, we will retrieve the plane-wave backgrounds with zero and one flat

direction introducing some arbitrary parameters, in order to find a whole class of

such a backgrounds. In reality, the role played by these parameter is non trivial,

and it will become clear in the next chapter.

4.1 Pp-wave backgrounds without flat

directions

The first case that we consider is that of pp-wave backgrounds without flat direc-

tions.

We start from the AdS5 × S5 metric

ds2 = R2[− cosh2 ρ dt2 + dρ2 + sinh2 ρ (dΩ′

3)2

+ dθ2 + sin2 θ dα2 + cos2 θ (dΩ3)2] ,

(4.1)

in which

(dΩ′3)

2= dβ2 + sin2 β dγ2 + cos2 β dλ2, (4.2)

(dΩ3)2 = dψ2 + sin2 ψ dφ2 + cos2 ψ dχ2. (4.3)

The procedure that we shall follow here to implement the Penrose limit is a little

different from the one presented in subsection 2.2.1. In fact we shall define the

coordinate in a different way, that has the advantage of being more easily general-

izable to the other other cases that we shall present in this chapter.

Let us introduce the new coordinates ϕ0, ϕ1, ϕ2, ϕ3, ϕ4, that are connected to the

original ones through the relations:

χ = ϕ0 ,

φ = ϕ1 + η1ϕ0 ,

α = ϕ2 + η2ϕ0 ,

γ = ϕ3 + η3ϕ0 ,

λ = ϕ4 + η4ϕ0 .(4.4)

η1, η2, η3, η4 are exactly the parameters that we introduce to generalize the case

analyzed in chapter 2. Now we need to make another transformation of variable in

which we will introduce the radius R, which plays the main role in the procedure


of taking the Penrose limit.

The transformation (4.4) leads to a set of coordinates which we will call x+, x−, xi

(i = 0, 1 . . . , 8):

x+ =t + ϕ0

2µ, x− =

µR2

2(t− ϕ0) ,

x1 + ix2 = r1eiϕ1 , x3 + ix4 = r2e

iϕ2 ,

x5 + ix6 = r3eiϕ3 , x7 + ix8 = r4e

iϕ4 ,

(4.5)

where r1, r2, r3, r4 are defined as follows

r1 = Rψ , r2 = Rθ ,

r3 = Rρ sin β , r4 = Rρ cos β .(4.6)

Now, expressing the metric (4.1) in these new coordinates and taking the Penrose

limit (R →∞ while keeping all coordinates fixed), we get

ds2 =− 4dx+dx− + dxidxi − µ2

4∑

k=1

(1− η2

k

) [(x2k−1

)2+

(x2k

)2] (

dx+)2

+ 2µ4∑

k=1

ηk

[x2k−1dx2k − x2kdx2k−1

]dx+.

(4.7)

This metric obviously falls within the class of metrics described by (2.2). Thus we

have just obtained a pp-wave background, that, as we can immediately see, does

not have any spacial isometry.

Note that fixing all the ηk equal to 1 gives back the metric (2.6).

4.2 Pp-wave backgrounds with one flat

direction

Now we are going to repeat an analogous procedure, in order to obtain a general-

ization of the limit derived in [21].

Following the steps of the previous section, let us define the new variables in term

of which we describe the AdS5 × S5 metric (4.1):

χ = ϕ0 − ϕ1 ,

φ = ϕ1 + ϕ0 ,

α = ϕ2 + η2ϕ0 ,

γ = ϕ3 + η3ϕ0 ,

λ = ϕ4 + η4ϕ0 .(4.8)


Note that these relations are slightly different from (4.4): in particular the main

differences are the definitions of the variable χ, which is not anymore given just by

ϕ0, and φ, where we fixed the parameter η1 = 1. So this time our generalization

contains only three parameters η2, η3, η4.

The definition of the x’s coordinates remains unchanged, except for x1 and x2:

x+ =t + ϕ0

2µ, x− =

µR2

2(t− ϕ0) ,

x1 = Rϕ1 , x2 = R(π

4− ψ

),

x3 + ix4 = r2eiϕ2 , x5 + ix6 = r3e

iϕ3 ,

x7 + ix8 = r4eiϕ4 ,

(4.9)

where r2, r3, r4 are the same as before:

r2 = Rθ , r3 = Rρ sin β , r4 = Rρ cos β . (4.10)

Tacking the limit R →∞, the metric reduces to

ds2 =− 4dx+dx− + dxidxi − µ2

4∑

k=2

(1− η2

k

) [(x2k−1

)2+

(x2k

)2] (

dx+)2

+ 2µ4∑

k=2

ηk


]dx+ − 4µx2dx+dx1.

(4.11)

This describes plane-wave backgrounds with one flat direction, since x1 does not

appear in the metric and consequently there is a space-like isometry along this

direction. Of course the background obtained in [21] is included in (4.11), and in

particular corresponds to the choice η2 = η3 = η4 = 0.

4.3 Pp-wave backgrounds with two flat

directions

Now we show how to choose the definitions of the x’s coordinates in order to arrive

to a pp-wave background with two flat direction, after taking the Penrose limit.

In the same manner as in the two previous cases we introduce the variables ϕ0, ϕ1,


ϕ2, ϕ3, ϕ4, that this time we now are defined as

χ = ϕ0 −√

2ϕ1 − ϕ2 ,

φ = ϕ0 +√

2ϕ1 − ϕ2 ,

α = ϕ0 + ϕ2 ,

γ = ϕ3 + η3ϕ0 ,

λ = ϕ4 + η4ϕ0.(4.12)

In appendix C we explain why this is exactly the transformation that is suitable

for our goal and we show explicitly the procedure followed to achieve it.

The definitions of the x’s coordinates are quite obvious if we remember how these

coordinates were defined in the previous cases:

x+ =t + ϕ0

2µ, x− =

µR2

2(t− ϕ0) ,

x1 = Rϕ1 , x2 =R√2

(π

4− ψ

),

x3 = Rϕ2 , x4 = R(π

4− θ

),

x5 + ix6 = r3eiϕ3 , x7 + ix8 = r4e

iϕ4 ,

(4.13)

and again r3 and r4 are

r3 = Rρ sin β , r4 = Rρ cos β . (4.14)

Substituting the new coordinates in the metric (4.1) and taking the Penrose limit

we get

ds2 = −4dx+dx− + dxidxi − µ2

4∑

k=1

(1− η2

k

) [(x2k−1

)2+

(x2k

)2] (

dx+)2

+ 2µ∑

k=3,4

ηk


]dx+ − 4µ

(x2dx1 + x4dx3

)dx+.

(4.15)

Note that this is the metric that we are looking for; we can immediately see that

the coordinates x1 and x3 do not enter in the metric and so the two corresponding

directions are flat.

Let us emphasize that this is the first original result of the thesis and it is an

highly non-trivial result, since until now only two pp-wave backgrounds have been

obtained from a Penrose limit of AdS5 × S5. What we have shown in this chapter

concludes this kind of investigations. In fact, as we shall see in the next chapter,

we have just found all the interesting pp-wave backgrounds achievable through the

procedure of taking the Penrose limit, in view of the AdS/CFT correspondence.

Chapter 5

Rotated pp-wave backgrounds

In this chapter we obtain a pp-wave metric, which depends on some parameters

introduced through a time-dependent coordinate transformation of the maximally

supersymmetric background (2.6). For this reason, in practice, this metric de-

scribes a set of infinite pp-wave backgrounds (one for each point of the parameter

space). We refer to them as to rotated pp-wave backgrounds.

More precisely we generalize the procedure explained in [24] applying a transfor-

mation that introduces eight parameters, instead of just two.

Note that the backgrounds obtained in this way do not have any particular physical

meaning, within the AdS/CFT correspondence. This is quite obvious, since these

backgrounds do not come from a Penrose limit of AdS5 × S5.

Despite this, the procedure that we are going to show results to be very useful

because allows us to obtain a general formula in which are contained also the

physically interesting pp-wave backgrounds. Let us explain more clearly this fun-

damental concept: the rotated metric itself does not have any physical meaning,

because of the way we obtain it, but, for certain values of the parameters, it

describes exactly the backgrounds studied in the previous chapter, the ones we

achieved through Penrose limits of AdS5 × S5 background.

We can then proceed in finding the spectra on these generic rotated backgrounds.

The final goal is to exploit these spectra in order to reproduce the ones found in [25]

for the nine decoupled sectors of N = 4 SYM which contain scalars.

5.1 Coordinate transformation

In order to reach our goal we start from the pp-wave background metric (2.6), that

we rewrite as

ds2 = −4dx+dx− − µ2xixi(dx+

)2+ dxidxi , (5.1)

Chapter 5. Rotated pp-wave backgrounds 58

where i = 0, 1, . . . , 8 and five-form field strength

F(5) = 2µdx+(dx1dx2dx3dx4 + dx5dx6dx7dx8

). (5.2)

We consider the following coordinate transformation

x− = z− +µ

2

(C1z

1z2 + C2z3z4 + C3z

5z6 + C4z7z8

),

x2k−1

x2k

=

cos(ηkµz+) − sin(ηkµz+)

sin(ηkµz+) cos(ηkµz+)

z2k−1

z2k

,

(5.3)

where k = 1, 2, 3, 4 and C1, C2, C3, C4 and η1, η2, η3, η4 are the parameters.

Note that the transformations for the transverse coordinates are rotations whose

angles depends on the η’s, hence the name “rotated pp-wave background”. After

the transformation the differentials are changed in this way

dx+ = dz+ ,

dx− = dz− +µ

2

4∑

k=1

[Ckz

2k−1dz2k + Ckz2kdz2k−1

],

dx2k−1 = ηkµ[sin(ηkµz+)z2k−1 + cos(ηkµz+)z2k

]dz+

+ cos(ηkµz+)dz2k−1 − sin(ηkµz+)dz2k ,

dx2k = ηkµ[cos(ηkµz+)z2k−1 sin(ηkµz+)z2k

]dz+

+ sin(ηkµz+)dz2k−1 + cos(ηkµz+)dz2k .

(5.4)

The metric (5.1) becomes

ds2 =− 4dz+dz− + dzidzi − µ2

4∑

k=1

(1− η2

k

) [(z2k−1

)2+

(z2k

)2] (

dz+)2

− 2µ4∑

k=1

[(Ck − ηk)z

2k−1dz2k + (Ck + ηk)z2kdz2k−1

]dz+ ,

(5.5)

while the five-form field (5.2) is invariant under the coordinate transformation

F(5) = 2µdz+(dz1dz2dz3dz4 + dz5dz6dz7dz8

). (5.6)

It is straightforward to check that the metric (5.5) contains all the backgrounds

obtained in chapter 4: setting all Ck’s equal to zero we retrieve the metric without

flat directions (4.7); the choice C1 = η1 = 1 and C2 = C3 = C4 = 0 leads to

the metric with one flat direction (4.11); if we set C1 = η1 = C2 = η2 = 1 and

C3 = C4 = 0 we obtain exactly the metric with two flat directions (4.15).


5.2 Spectra of type IIB superstrings on

rotated pp-wave backgrounds

As we have already argued, equation (5.5), depending on eight parameters, de-

scribes a whole set of backgrounds of a pp-wave type.

Now we shall proceed in finding the superstring spectra on these rotated back-

grounds.

5.2.1 Bosonic sector

The Lagrangian density of the bosonic σ-model is

L B = − 1

4πl4sp+

gµν ∂αZµ∂αZν , (5.7)

where gµν is the metric tensor of the background. In the light-cone gauge, Z+ =

l2sp+τ the light-cone Lagrangian density is

L Blc =− 1

4πl4sp+

(∂αZi∂αZi + f 2

4∑

k=1

(1− η2

k

) [(Z2k−1

)2+

(Z2k

)2]

+ 2f4∑

k=1

[(Ck − ηk)Z

2k−1Z2k + (Ck + ηk)Z2kZ2k−1

]),

(5.8)

where we have defined f = µl2sp+. The conjugate momenta are

Π2k−1 =Z2k−1 − f (Ck + ηk) Z2k

2πl2s,

Π2k =Z2k − f (Ck − ηk) Z2k−1

2πl2s,

(5.9)

so that the classical bosonic Hamiltonian HBlc =

∫dσ(ΠiZ

i − LBlc) is given by

HBlc = − 1

4πl4sp+

∫ 2π

0

dσ

[ZiZi + (Zi)′(Zi)′

+ f 2

4∑

k=1

(1− η2

k

) [(Z2k−1

)2+

(Z2k

)2] ]

.

(5.10)

From the Lagrangian density (5.8) we get the equations of motion

∂α∂αZ2k−1 + 2fηkZ2k − f 2

(1− η2

k

)Z2k−1 = 0 , (5.11a)

∂α∂αZ2k − 2fηkZ2k−1 − f 2

(1− η2

k

)Z2k = 0 , (5.11b)


In order to solve these equations, it is useful to decouple them by introducing four

complex fields

Xk = Z2k−1 + iZ2k , (5.12)

in terms of which the above equations read as follows

∂α∂αXk − 2ifηkXk − f 2

(1− η2

k

)Xk = 0, (5.13a)

∂α∂αXk + 2ifηk˙Xk − f 2

(1− η2

k

)Xk = 0 . (5.13b)

One can see that a solution of the form

Xk = e−ifηkτY k (5.14)

solves (5.13a) if Y k satisfy the equation

∂α∂αY k − f 2Y k = 0 . (5.15)

Therefore for Y k and its conjugate Y k we have the following mode expansions

Y k = ils

+∞∑n=−∞

1√ωn

(ak

ne−i(ωnτ−nσ) − (

akn

)†ei(ωnτ−nσ)

), (5.16a)

Y k = ils

+∞∑n=−∞

1√ωn

(ak

ne−i(ωnτ−nσ) − (

akn

)†ei(ωnτ−nσ)

). (5.16b)

We can rewrite the bosonic Hamiltonian in terms of the complex fields

HBlc = − 1

4πl4sp+

∫ 2π

0

dσ

4∑

k=1

(˙XkXk + (Xk)′(Xk)′ + f 2

(1− η2

k

)XkXk

). (5.17)

Then we quantize in the canonical way imposing the following equal time commu-

tation relations [Zi (τ, σ) , Πj (τ, σ′)

]= iδijδ (σ − σ′) , (5.18)

which written in terms of the oscillators become[ak

n, ak′m

]= 0 ,

[ak

n, (ak′m)†

]=

[ak

n, (ak′m)†

]= δkk′δnm . (5.19)

Substituting the mode expansion (5.16) in the Hamiltonian (5.17) and using the

commutation relations (5.19) we get the bosonic spectrum in this background

HBlc =

1

l2sp+

+∞∑n=−∞

∑

k=1,2

[(ωn + ηkf) M (k)

n + (ωn − ηkf) M (k)n

]

+∑

k=3,4

[(ωn + ηkf) N (k)

n + (ωn − ηkf) N (k)n

],

(5.20)


where

ωn =√

n2 + f 2, for all n ∈ Z , (5.21)

and M, M, N, N are the number operators, defined as

M (k)n = ak†

n akn , M (k)

n = ak†n ak

n for k = 1, 2 ,

N (k)n = ak†

n akn , N (k)

n = ak†n ak

n for k = 3, 4 .(5.22)

We have decided to distinguish the cases k = 1, 2 and k = 3, 4 because the former

refers to the number operators relating to the S5 coordinates (z1, z2, z3, z4) and the

latter to the ones relating to the AdS5 coordinates (z5, z6, z7, z8). This convention

might look weird, but it will result useful for our final aim.

5.2.2 Fermionic sector

Starting with θA as a Majorana-Weyl spinor with 32 components and A = 1, 2, we

choose the light-cone gauge

Z+ = l2sp+τ, Γ+θA . (5.23)

The Green-Schwarz fermionic action is then given by [19]

SFlc =

i

4πl4sp+

∫dτdσ

[(ηαβδAB − εαβ (σ3)AB

)∂αZ+θAΓ+ (Dβθ)B

], (5.24)

and the generalized covariant derivative takes the form

Dα = ∂α +1

4∂αZ+

(ω+ρσΓρσ − 1

2 · 5!FλνρσκΓ

λνρσκiσ2Γ+

), (5.25)

where ω is the spin connection of the metric, and σk’s are the Pauli matrices (see

appendix B).

iσ2 = ε =

(1 0

0 −1

), σ3 =

(0 1

−1 0

). (5.26)

In order to get the spin connection of the metric (5.5), we need to calculate the

vierbeins ea. Each of the ten vierbeins is a 1-form

ea = eaµdxµ , (5.27)

where the latin letter (the first index) is a flat index and the greek letter (the

second index) is a curved index. The vierbeins should obey

gµν = ηabeaµe

bν , (5.28)


or equivalently

ds2 = ηabeaeb , (5.29)

where ηab is the flat Minkowski metric. The spin connection is defined by

dea + ωabe

b = 0 , (5.30)

with ωab being a 1-form

ωab = ωµabdxµ . (5.31)

The components of the spin connection are given by

ωµab =1

2ec

µ (Ωcab + Ωbac + Ωbca) , (5.32)

where Ωabc is given by

Ωabc = e µa e ν

b (∂µecν − ∂νecµ) . (5.33)

For commodity we define

A2 =4∑

k=1

(1− η2

k

) [(z2k−1

)2+

(z2k

)2]

,

Fk = (Ck − ηk)z2k−1,

Gk = (Ck + ηk)z2k ,

(5.34)

in this way we can rewrite the metric (5.5) as follows

ds2 = −4dz+dz−+dzidzi−µ2A2(dz+

)2−2µ4∑

k=1

[Fkdz2k + Gkdz2k−1

]dz+ . (5.35)

Using (5.29) we can get the vierbeins for this metric

e− =µA

4dz+ +

dz−µA

+Fk

2Adz2k +

Gk

2Adz2k−1 ,

e+ = µAdz+ ,

ei = dzi with i = 1, . . . , 8 .

(5.36)

So the components of eaµ are

e−+ =µA

4, e−− =

1

µA, e−2k−1 =

Gk

2A, e−2k =

Fk

2A,

e++ = µA ,

eii = 1 with i = 1, . . . , 8 .

(5.37)


For our purposes here it is enough to find the components of the spin connection

where the curved index is +. It turns out that the only components that don’t

vanish are

ω+,2k−1,2k = −ω+,2k,2k−1 = −µηk ,

ω+,+,2k−1 = −ω+,2k−1,+ = −µz2k−1 (1− η2k)

A,

ω+,+,2k = −ω+,2k,+ = −µz2k (1− η2k)

A.

(5.38)

The only relevant components are ω+,2k−1,2k and ω+,2k,2k−1 because all the other

non-vanishing components are contracted with Γ+ in the covariant derivative and

thus are killed in the light-cone gauge.

The five-form F(5) (5.6) has the components F+1234 = F+5678 = 2µ turned on and

obviously all the other components which can be obtained from these by permuting

the indices. Thus the Green-Schwarz action (5.24) can be written as

SFlc =

i

4πl2s

∫dτdσ

θAΓ+

[(δAB∂τ + (σ3)AB ∂σ

)θB

− 1

2l2sp

+µ

(4∑

k=1

ηkΓ2k−1,2kθA +

1

2

(Γ+1234 + Γ+5678

)i (σ2)AC Γ+θC

)].

(5.39)

Let us make some manipulations on this formula. First let us consider

(Γ+1234 + Γ+5678

)Γ+ = Γ+Γ+

(Γ1234 + Γ5678

),

here we use Γ+ = −2Γ− and then write

Γ+Γ− =Γ+, Γ−

− Γ−Γ+ = −I32 − Γ−Γ+ , (5.40)

with the term Γ−Γ+ that vanish when it acts on θC for the light-cone gauge con-

ditions.

In appendix B we show that, θA being Majorana-Weyl spinors, only 16 of their

total 32 components are non-zero

θA =

(ψA

0

). (5.41)


Using the relation θAΓ+ =(2(ψA)T 0

)(see (B.27)) we get

SFlc =

i

2πl2s

∫dτdσ

(ψ1

)T

[∂+ − f

2

4∑

k=1

ηkγ2k−1,2k

]ψ1

+(ψ2

)T

[∂− − f

2

4∑

k=1

ηkγ2k−1,2k

]ψ2 − f

2

(ψ1

)T (γ1234 + γ5678

)ψ2

+f

2

(ψ2

)T (γ1234 + γ5678

)ψ1 ,

(5.42)

Expressing the spinors ψ according to (B.23)

ψA =

(SA

0

), (5.43)

which is a consequence of the light-cone gauge condition Γ−θA = 0, we can write

the action as follows

SFlc =

i

2πl2sp+

∫dτdσ

(S1

)T

[∂+ − f

2

4∑

k=1

ηkγ2k−1,2k

]S1

+(S2

)T

[∂− − f

2

4∑

k=1

ηkγ2k−1,2k

]S2 − 2f

(S1

)TΠS2

.

(5.44)

where Π = γ1234, ∂± = ∂τ ± ∂σ and we recall that f = l2sp+µ. From the action

(5.44) we can derive the equations of motion(

∂+ − f

2

4∑

k=1

ηkγ2k−1,2k

)S1 − fΠS2 = 0 , (5.45a)

(∂− − f

2

4∑

k=1

ηkγ2k−1,2k

)S2 + fΠS1 = 0 . (5.45b)

It’s useful to observe that a field of the form

SA = e

f

2

4∑

k=1

ηkγ2k−1,2kτ

ΣA (5.46)

satisfies the above equations if the fields ΣA obey the equations of motion of the

fermionic fields in the usual pp-wave background [19]:

∂+Σ1 − fΠΣ2 = 0 , (5.47a)

∂−Σ2 + fΠΣ1 = 0 , (5.47b)


whose solutions are

Σ1 = c0 e−ifτS0 −∑n>0

cne−iωnτ

(Sne

inσ +ωn − n

fS−ne−inσ

)+ h.c. , (5.48a)

Σ2 = −c0 e−ifτ iΠS0 − iΠ∑n>0

cne−iωnτ

(S−ne−inσ − ωn − n

fSne

inσ

)+ h.c. ,

(5.48b)

where, for all values of n, ωn is defined as in equation (5.21) as ωn =√

n2 + f 2,

while cn = 1√2[1 + (ωn−n

f)2]−1/2.

The fermionic conjugate momenta can be computed from the action (5.44)

λA =i

2πSA , (5.49)

so we can obtain the fermionic part of the classical Hamiltonian

HFlc =

1

l2sp+

∫ 2π

0

dσ[(

λ1)T

S1 +(λ2

)TS2 −L F

lc

]. (5.50)

Substituting the fermionic Lagrangian density LFlc this becomes

HFlc =

−i

2πl2sp+

∫ 2π

0

dσ

[(S1

)T (S1

)′ − (S2

)T (S2

)′

− f

2

((S1

)T4∑

k=1

ηkγ2k−1,2kS1 +

(S2

)T4∑

k=1

ηkγ2k−1,2kS2

)− 2f

(S1

)TΠS2

],

(5.51)

and making use of the equations of motion (5.45)

HFlc =

i

2πl2sp+

∫ 2π

0

dσ((

S1)T

S1 +(S2

)TS2

). (5.52)

Now we quantize the theory imposing the canonical equal time anticommutation

relations SAa (τ, σ) , SBb (τ, σ′)

=

1

2δABδabδ (σ − σ′) , (5.53)

which imply for the oscillators San

San,

(Sb

m

)†= δabδnm . (5.54)

After some algebra one gets for the fermionic part of the Hamiltonian

HFlc =

1

l2sp+

+∞∑n=−∞

S†n

(ωn + i

f

2

4∑

k=1

ηkγ2k−1,2k

)Sn . (5.55)


The 8×8 matrices γ2k−1,2k can be easily obtained from (B.3) and (B.14), and result

to be:

γ12 = ε× σ3 × I2 , γ34 = −σ1 × σ1 × ε ,

γ56 = −σ1 × ε× σ1 , γ78 = ε× I2 × σ3 .(5.56)

(If the notation used to write these matrices is not clear, in appendix B equation

(B.4), it is explained the way in which these have to be read).

All the matrices i γ2k−1,2k have eigenvalues ±1, each with multiplicity four. Since

they commutes we can find a set of common eigenvectors. Choosing this set as

basis we can write the fermionic spectrum as

HFlc =

1

l2sp+

+∞∑n=−∞

[(ωn +

f

2(−η1 − η2 + η3 + η4)

)F (1)

n

+

(ωn +

f

2(−η1 − η2 − η3 − η4)

)F (2)

n +

(ωn +

f

2(η1 + η2 + η3 + η4)

)F (3)

n

+

(ωn +

f

2(η1 + η2 − η3 − η4)

)F (4)

n +

(ωn +

f

2(−η1 + η2 + η3 − η4)

)F (5)

n

+

(ωn +

f

2(η1 − η2 + η3 − η4)

)F (6)

n +

(ωn +

f

2(η1 − η2 − η3 + η4)

)F (7)

n

+

(ωn +

f

2(−η1 + η2 − η3 + η4)

)F (8)

n

],

(5.57)

where F(b)n are the fermionic number operators defined by the relation

F (b)n =

(Sb

n

)†Sb

n . (5.58)

The full Hamiltonian Hlc of quantized strings on rotated pp-wave backgrounds is

given by the sum of (5.20) and (5.57)

Hlc =HBlc + HF

lc =1

l2sp+

+∞∑n=−∞

∑

k=1,2

[(ωn + ηkf) M (k)

n + (ωn − ηkf) M (k)n

]

+∑

k=3,4

[(ωn + ηkf) N (k)

n + (ωn − ηkf) N (k)n

]+

8∑

b=1

(ωn +

f

2db

)F (b)

n

,

(5.59)

and the level matching condition is

+∞∑n=−∞

[ ∑

k=1,2

(M (k)

n + M (k)n

)+

∑

k=3,4

(N (k)

n + N (k)n

)+

8∑

b=1

F (b)n

]= 0 . (5.60)


In equation (5.59) we rewrote the fermionic part of the Hamiltonian in a more

compact form, by introducing the coefficients db of the fermionic number operators,

that from the comparison with (5.57), result to be

d1 = −η1 − η2 + η3 + η4 , d5 = −η1 + η2 + η3 − η4 ,

d2 = −η1 − η2 − η3 − η4 , d6 = η1 − η2 + η3 − η4 ,

d3 = η1 + η2 + η3 + η4 , d7 = η1 − η2 − η3 + η4 ,

d4 = η1 + η2 − η3 − η4 , d8 = −η1 + η2 − η3 + η4 .

(5.61)

Note that the spectrum does not depend on the four parameters Ck, but only on

the η’s, that for this reason are the only relevant ones.

5.3 Decoupling limits of N = 4 SYM

gauge theory

Now let us briefly review the main results found in papers [22, 23, 24, 25], in order

to compare them with what we have just get for strings.

The idea developed in these papers is to consider decoupling limits of N = 4

SYM on R × S3 with gauge group SU(N). By taking such a decoupling limit,

the remaining decoupled theory is significantly simpler than the full N = 4 SYM

theory and this makes it possible to take a strong coupling limit of the decoupled

theory in a controllable manner.

The decoupling limits can be taken in the microcanonical ensemble.

Let n = (n1, n2, n3, n4, n5) be given such that J can be written as

J ≡ n1S1 + n2S2 + n3J1 + n4J2 + n5J3 . (5.62)

Then the decoupling limit of SU(N) N = 4 SYM on R× S3 is given as

λ → 0, H ≡ D − J

λfixed, J À 1, (5.63)

where λ is the ’t Hooft coupling, defined as usual by λ = g2Y MN . D is the dilation

operator, which, for small λ, is expanded as D = D0 + λD2 + O(λ3/2), with D0

being the bare scaling dimension and D2 the one-loop contribution.

We see that the limit (5.63) indeed is in the microcanonical ensemble since H and

J are linear combinations of the Cartan generators of psu(2, 2|4).

In order for the decoupling limit (5.63) to be translatable to the string theory side


we also need to demand that |D − J | ¿ λ ¿ 1; this requirement means that only

states with D0 = J can be present. Since D ' D0 + λD2 for small λ, from (5.63)

we get that H = D2 for D2 acting on the surviving states.

A particular limit of the above kind was found and studied in [22,23] with n given

by (n1, n2, n3, n4, n5) = (0, 0, 1, 1, 0). In the limit (5.63) all the states decouple

except for those in the SU(2) sector. According to the AdS/CFT correspondence

this particular decoupling limit is dual to the corresponding decoupling limit of

string theory on AdS5 × S5 [23]. By employing a certain Penrose limit [21], it was

found [23] the spectrum and matched this to the spectrum found on the gauge

theory side, for large J = J1 + J2. Therefore, the decoupling limit (5.63) provides

us with a precise way to match gauge theory with string theory.

In paper [25] all the decoupling limits of the form (5.63) that correspond to different

choice of n = (n1, n2, n3, n4, n5) were found. There are a total of fourteen such

decoupling limits of N = 4 SYM on R×S3, which correspond to fourteen different

subgroups of the total symmetry group PSU(2, 2|4) of N = 4 SYM.

Sectors (n1, n2, n3, n4, n5)

U(1)bos (0,0,1,0,0)

U(1)fer

(35,−3

5, 3

5, 3

5, 3

5

)

SU(2) (0,0,1,1,0)

SU(1, 1)bos (1,0,1,0,0)

SU(1, 1)fer

(1, 0, 2

3, 2

3, 2

3

)

SU(1|1)(

23, 0, 1, 2

3, 2

3

)

SU(1|2)(

12, 0, 1, 1, 1

2

)

Sectors (n1, n2, n3, n4, n5)

SU(2|3) (0,0,1,1,1)

SU(1, 1|1)(1, 0, 1, 1

2, 1

2

)

SU(1, 1|2) (1,0,1,1,0)

SU(1, 2) (1,1,0,0,0)

SU(1, 2|1)(1, 1, 1

2, 1

2, 0

)

SU(1, 2|2) (1,1,1,0,0)

SU(1, 2|3) (1,1,1,1,1)

Table 5.1: The table shows all the fourteen decoupling limits found in [25]:in the left column are listed the sectors that survive the decoupling limit forthe corresponding choice of n = (n1, n2, n3, n4, n5) reported in the right column.The abbreviations “bos” and “fer” that appear in some sectors indicate that weare referring to the bosonic or fermionic theory respectively.

The list of all the fourteen decoupling limits is reported in Table 5.1; among them

two give rise to trivial decoupled theories (the bosonic and the fermionic U(1)

limits). The remaining twelve non-trivial decoupled theories are divided into nine

theories with scalars and three without scalars. The theories with at least one scalar

are those for which n3 is equal to 1. Thus the three theories without scalar results


to be that corresponding to the fermionic SU(1, 1) limit, the SU(1, 2) limit and the

SU(1, 2|1) limit. One of the theories with scalars has a SU(1, 2|3) symmetry, and

all the other decoupled theories can be seen to be just a subsector of this theory.

We are interested only in the nine decoupled theories that contain at least one

scalar. Their spectra can all be described in the same way using the number

operators Mn, Nn and Fn. Depending on their letter content, the decoupled theories

have different number of these operators appearing and in Table 5.2 we list how

many there are of each of the three possible types.

SU(·) (2) (1, 1)bos (1|1) (1|2) (2|3) (1, 1|1) (1, 1|2) (1, 2|2) (1, 2|3)

a 1 0 0 1 2 0 1 0 2

b 0 1 0 0 0 1 1 2 2

c 0 0 1 1 2 1 2 2 4

Table 5.2: The table shows how many number operators we have of each type(a for scalars Mn, b for derivatives Nn, and c for fermions Fn) in each of the ninetheories that contain at least one scalar. SU(1, 1)bos corresponds to the bosonicSU(1, 1) theory.

As shown in [25] the spectra for these nine different theories all take the form

D − J =λ

J2

∑

n∈Zn2

(a∑

i=1

M (i)n +

b∑j=1

N (j)n +

c∑α=1

F (α)n

), (5.64)

with the cyclicity (zero momentum) constraint

P ≡∑

n∈Zn

(a∑

i=1

M (i)n +

b∑j=1

N (j)n +

c∑α=1

F (α)n

)= 0 . (5.65)

Note that F(α)n ∈ 0, 1 while M

(i)n , N

(j)n ∈ 0, 1, 2, .... The numbers a, b and c are

given in Table 5.2.

5.4 Matching of spectra

Equation (5.64), along with Table 5.2, gives us the spectrum for each of the nine

decoupled gauge theories that contain at least one scalar. Now we want to compare

these spectra with the ones that we have obtained for the type IIB strings on the


rotated pp-wave backgrounds. Before doing that we must apply to the string theory

the decoupling limit that correspond to that just seen for the N = 4 SYM theory.

It has been shown [23] that the decoupling limit for string results to be µ →∞, or

equivalently f →∞, and that, after taking this limit, according to the AdS/CFT

duality, the string Hamiltonian divided by µ should be dual to D− J ∼ D2, where

D2 is the one-loop dilatation operator

Hlc

µ←→ D − J . (5.66)

So let us first apply the large f limit to the string spectra (5.59). Remembering

the definition of ωn, its expansion for f →∞ takes the form

ωn =√

f 2 + n2 = f

√1 +

n2

f 2' f

(1 +

n2

2f 2

)= f +

n2

2f, (5.67)

where we neglect all the terms O(f−2). In order for the spectra to be finite, the

divergent term contained in the expansion of ωn should cancel.

In the bosonic part of the Hamiltonian (5.59) we deal with terms of the kind

(ωn + ηkf) M (k)n '

[f (1 + ηk) +

n2

2f

]M (k)

n , (5.68a)

(ωn − ηkf) M (k)n '

[f (1− ηk) +

n2

2f

]M (k)

n , (5.68b)

for k = 1, 2, and the analogous ones for N(k)n and N

(k)n when k = 3, 4.

Instead in the fermionic part of the Hamiltonian (5.59) we have(

ωn +f

2db

)F (b)

n '[f

(1 +

db

2

)+

n2

2f

]F (b)

n . (5.69)

The only terms that survive the limit f → ∞ are those for which the coefficient

of the linear part in f vanish. All the other terms are divergent and thus decou-

ples in the large f limit. The bosonic number operators will survive only if the

corresponding ηk results to be ±1 and the fermionic number operators only if the

corresponding db results to be −2. It is not hard to see that a suitable choice to

match the η’s parameters with the parameters n1, n2, n4, n5 is the following

η1 = n4 ,

η2 = n5 ,

η3 = −n1 ,

η4 = n2 .

(5.70)


Remember that we are dealing only with theories that contains at least one scalar,

so as we mentioned earlier n3 is fixed and must be equals to 1.

According to the guess (5.70) we can obtain the spectra calculated from the string

theory side, that should correspond to each decoupling limit of the N = 4 SYM.

Let us begin trying to match the spectra of the decoupled theories that contain

only one scalar, i.e. those for which n3 = 1 and n4 6= 1 6= n5:

• The bosonic SU(1, 1) limit: n = (1, 0, 1, 0, 0) ⇒ η = (0, 0,−1, 0)

In this case we have that the only non-vanishing ηk is η3 which results to be −1.

As a consequence, after the limit f → ∞, only one bosonic number operator

survive (N(3)n , which is the one corresponding to η3, of course); whereas none of the

fermionic number operators survive, since db can never be equal to −2. Thus the

spectrum (5.59), after the limit, becomes

Hlc =1

2l2s p+ f

∑

n∈Zn2 N (3)

n . (5.71)

Since f of strings is related to the angular momentum J of the gauge theory

according to the relation (for a detailed discussion see [23])

f 2 =(µl2sp

+)2

=J2

λ, (5.72)

we can write Hlc/µ, i.e. the term that should reproduce D2, in this way

Hlc

µ=

λ

2J2

∑

n∈Zn2 N (3)

n . (5.73)

• The SU(1|1) limit: n =(

23, 0, 1, 2

3, 2

3

) ⇒ η =(

23, 2

3,−2

3, 0

)

In this limit none of the η’s is ±1 and d1 = −η1 − η2 + η3 + η4 is the only db equal

to −2. So we have that the the only number operator that survive is F(1)n :

Hlc

µ=

λ

2J2

∑

n∈Zn2 F (1)

n . (5.74)

The procedure that we have to follow for all the other cases is analogous, so, starting

from now, for the sake of brevity we shall leave out the detailed discussion, and we

shall report only the final results for the spectra.

• The SU(1, 1|1) limit: n =(1, 0, 1, 1

2, 1

2

) ⇒ η =(

12, 1

2,−1, 0

)

Hlc

µ=

λ

2J2

∑

n∈Zn2

(N (3)

n + F (1)n

). (5.75)


• The SU(1, 2|2) limit: n = (1, 1, 1, 0, 0) ⇒ η = (0, 0,−1, 1)

Hlc

µ=

λ

2J2

∑

n∈Zn2

(N (3)

n + N (4)n + F (5)

n + F (6)n

). (5.76)

Now we shall take into consideration the decoupled theories with two scalars, for

which n3 = n4 = 1 and n5 6= 1:

• The SU(2) limit: n = (0, 0, 1, 1, 0) ⇒ η = (1, 0, 0, 0)

Hlc

µ=

λ

2J2

∑

n∈Zn2M (1)

n . (5.77)

• The SU(1|2) limit: n =(

12, 0, 1, 1, 1

2

) ⇒ η =(1, 1

2, 1

2, 0

)

Hlc

µ=

λ

2J2

∑

n∈Zn2

(M (1)

n + F (2)n

). (5.78)

• The SU(1, 1|2) limit: n = (1, 0, 1, 1, 0) ⇒ η = (1, 0,−1, 0)

Hlc

µ=

λ

2J2

∑

n∈Zn2

(M (1)

n + N (3)n + F (1)

n + F (5)n

). (5.79)

In the end let us consider the decoupled theories that contains three scalars, i.e.

the ones for which n3 = n4 = n5 = 1:

• The SU(2|3) limit: n = (0, 0, 1, 1, 1) ⇒ η = (1, 1, 0, 0)

Hlc

µ=

λ

2J2

∑

n∈Zn2

(M (1)

n + M (2)n + F (1)

n + F (2)n

). (5.80)

• The SU(1, 2|3) limit: n = (1, 1, 1, 1, 1) ⇒ η = (1, 1,−1, 1)

Hlc

µ=

λ

2J2

∑

n∈Zn2

(M (1)

n + M (2)n + N (3)

n + N (4)n + F (1)

n + F (2)n + F (5)

n + F (6)n

).

(5.81)

We immediately see that these results are fully compatible with that presented in

the previous section in (5.64), along with Table 5.2.

Note that there is a relation between the numbers of scalars contained in the gauge

theory and the number of flat directions of the pp-wave background on which lies

the string theory. We can summarize this relation as follow: the decoupled theories

with m scalars correspond to pp-wave backgrounds with m− 1 flat directions.

Conclusions

The first evidences of the existence of certain limits, that allow to reduce AdS5×S5

to another background on which the type IIB supersymmetric string theory is ex-

actly solvable, arose in 2002. In that year it was first discovered that, through a

Penrose limit, AdS5 × S5 reduces to a pp-wave background [13, 14, 15], and that

in this limit the string theory σ-model becomes solvable [18, 19]. For this reason,

studying type IIB superstring theory on a pp-wave background, it has been proved

to be a very useful way to explore the AdS/CFT correspondence, that otherwise

results extremely hard to verify.

The conjecture proposed by Berenstein, Maldacena and Nastase moved in this di-

rection, finding how the procedure of taking the Penrose limit translates in the

gauge theory side of the AdS/CFT duality.

Until now only two Penrose limits have been discovered, from which it has been

possible to obtain a pp-wave background with zero or one flat direction. In chap-

ter 4 first we have found a way to generalize the procedure, by introducing some

parameters, that allows to obtain a set of such backgrounds. Then we have found

a new Penrose limit of AdS5×S5 that leads to a pp-wave background with two flat

directions. This is a remarkable result since it concludes this kind of investigations;

we have just found all possible pp-wave backgrounds achievable through a Penrose

limit of the AdS5×S5 metric, which are important in the context of the AdS/CFT

correspondence.

In chapter 5, by applying a time-dependent coordinate transformation (5.3) to

the maximally supersymmetric background (2.6), we have obtained a set of back-

grounds of the pp-wave type (5.5). This result has been reached through the

introduction of some arbitrary parameters (ηk and Ck, with k = 1, 2, 3, 4) on the

transformation law. We have called the backgrounds obtained in this way rotated

pp-wave backgrounds, since the transformation for the transverse coordinates con-

sists in a set of rotations.

Although these backgrounds do not have any particular physical meaning within

Conclusions 74

the AdS/CFT correspondence, as a consequence of the way in which we have got

them, we have shown that among them are contained all the backgrounds we have

achieved from a Penrose limit of AdS5 × S5 in chapter 4. Being related one to

another by a coordinate transformation, all these backgrounds result to be math-

ematically equivalent. The same statement is not true from the physical point

of view, since the transformation involves time. Thus what changes from a back-

ground to another is what we call time, and consequently what we call Hamiltonian.

For this reason the physics is different when we consider the theory on different

pp-wave backgrounds.

In 2006 it was shown in [22,23,24,25] that taking certain decoupling limits on the

N = 4 D = 4 super Yang-Mills theory with gauge group SU(N) leads to decoupled

theories significantly simpler than the full N = 4 SYM theory. It was also found

how these decoupling limits translate on the string theory side according to the

AdS/CFT duality.

In chapter 5 we have calculated the string spectra on the rotated pp-wave back-

grounds with the final aim of reproducing the spectra of the decoupled gauge

theories [25]. First we have taken the decoupling limit on the string spectra and by

comparing the spectra in both the string and gauge theory sides we have been able

to relate the η’s parameters, which distinguish the various string theory spectra,

to the numbers n1, n2, n3, n4, n5, which instead distinguish the various decoupled

gauge theories (Eq. (5.70)). In this way we have succeeded in reproducing from

the string theory side all the spectra of the decoupled gauge theories.

It is worth mentioning that the various pp-wave backgrounds that we have con-

sidered in this thesis, namely with zero, one and two flat directions, represent all

possible backgrounds giving rise to string spectra which, after taking appropriate

limits, precisely match the corresponding gauge theory spectra. It is in fact with

the aim of achieving this matching that we engaged in the derivation of the pp-

wave background with two flat direction starting from type IIB string theory on

AdS5 × S5. This is one of our original results and the final piece of information

needed to successfully compare the string theory and gauge theory spectra in the

limit discussed in this thesis.

Appendix A

Conventions for N = 4, D = 4

Yang-Mills gauge theory

There are various formulations of N = 4 supersymmetric Yang-Mills theory. We

present the two realizations which are most commonly encountered in the literature,

one realized by writing the Lagrangian in terms of N = 1 superspace gauge theory

coupled to a set of chiral-multiplets, the other based on dimensional reduction of

the ten dimensional component formulation of SYM.

N = 4 SYM Lagrangian in N = 1 superfield language

Let us fix our conventions for the N = 4, D = 4 gauge theory action in terms of

N = 1 gauge theory in superspace.

An N = 4 vector multiplet decomposes into one N = 1 vector and three chiral

multiplets. We coordinatize N = 1 superspace as z = (x, θ).

The generators of supertranslation on superspace, written as chiral and anti-chiral

superderivatives, are

Dα =∂

∂θα+

i

2θασµ

αα∂µ ,

Dα =∂

∂θα+

i

2θασµ

αα∂µ .

(A.1)

Squares of fields and derivatives are defined with a customary factor of 1/2, and

with the index conventions as in

D2 =1

2DαDα , D2 =

1

2DαDα , (A.2)

and likewise for the fields. The superderivatives satisfy theN = 1 anticommutation

relations

Dα, Dβ = 0 , Dα, Dα = iσµαα∂µ . (A.3)

Grassmann Delta functions are given by

δ4(θ − θ′) = (θ − θ)2 (θ − θ′)2 (A.4)

Appendix A. Conventions for N = 4, D = 4 Yang-Mills gauge theory 76

Some useful identities are

[Dα, D2] = iσµαα∂µD

α , D2θ2 = D2θ2 = −1 ,

D2D2D2 =¤D2 , [Dα, Dα]σµαα∂µ =

1

2DαDασµ

αα∂µ + i¤ .(A.5)

The non-AbelianN = 4 supersymmetric Yang-Mills action cast inN = 1 superfield

form is

S =2

g2Y M

Tr( ∫

d8z e−V ΦieV Φi +1

2

∫d6z W αWα +

1

2

∫d6z WαW α

+i√

2

3!

∫d6z εijk [Φi, Φj] Φk +

i√

2

3!

∫d6 zεijk

[Φi, Φj

]Φk

),

(A.6)

with the field strength given by

Wα = i D2(e−V DαeV

). (A.7)

Here, Φi (i = 1, 2, 3) are chiral superfields and all superfields take values in the Lie

algebra whose generators obey

[tA, tB

]= i fABC tC . (A.8)

The superspace measures are defined as d8z = d4z d2θ d2θ, d6z = d4x d2θ, and

d6z = d4x d2θ.

N = 4 SYM Lagrangian from dimensional reduction

The component formulation is more useful when actually computing Feynman di-

agrams and studying the combinatorics which lead to the double expansion char-

acteristic of the double scaling limit proposed by BMN.

We use the mostly minus metric convention, gµν = diag(+,−,−,−). The La-

grangian (and field content) of the N = 4 Super Yang-Mills theory can be deduced

by dimensionally reducing the ten-dimensional N = 1 SYM theory (with 16 su-

percharges) on T 6 (which preserves all supersymmetries). There is a single vector,

four Weyl fermions and six real scalars, all in the adjoint representation of the

gauge group. The reduced Lagrangian, in component form, is

L =1

g2Y M

Tr(− 1

2FµνF

µν +θI

16π2FµνF

µν +6∑

i=1

DµφiDµφi

+4∑

A=1

iΨAΓµDµΨA +1

2

6∑i,j=1

[φi, φj]2 +6∑

i=1

ΨAΓi[φi, ΨA])

.

(A.9)

Appendix A. Conventions for N = 4, D = 4 Yang-Mills gauge theory 77

Decomposing the ten dimensional Dirac matrices yields four (Γµ) and six (Γi)

dimensional ones. This Lagrangian is manifestly invariant under a U(N) gauge

symmetry. The generators of U(N) are chosen with the (non-standard) normaliza-

tion

Tr(tAtB) = δAB ,

(A, B = 1, ..., N2), and satisfy the appropriate completeness relation

δAB(tA)ab (t

B)cd = δa

dδcb ,

a, b = 1, ..., N , since these are the generators in the adjoint representation. The

fields take values in the U(N) algebra

χ(x) = χA(x)tA ,

with χ any of the fields in the N = 4 multiplet. The sums above are taken

over the N2 − 1 generators of SU(N) and the single generator of the U(1) factor

in U(N). The covariant derivative is defined as Dµχ = ∂µ − i[Aµ, χ]. When

diagrams are computed, Feynman gauge is chosen to simplify calculations, taking

advantage of the similarity between scalar and vector propagators in this gauge.

There is also a global SU(4) ∼ SO(6) R-symmetry, under which the scalars φi

transform in the fundamental of SO(6), and the fermions ΨA in the fundamental of

SU(4) = Spin(6). The vectors are singlets of the R-symmetry. The θ term counts

contributions from non-trivial instanton backgrounds, which is ignored when one

assumes the trivial vacuum.

Appendix B

Gamma matrices and spinors

We briefly review our conventions for the representations of Dirac matrices in ten

dimensions and for Majorana-Weyl spinors. As usual, we shall use the mostly plus

metric.

Gamma matrices

Let In denote the n× n unit matrix, σ1, σ2, σ3 the 2× 2 Pauli matrices

σ1 =

(0 1

1 0

), σ2 =

(0 −i

i 0

), σ3 =

(1 0

0 −1

), (B.1)

and ε the antisymmetric tensor of rank two

ε = iσ2 =

(0 1

−1 0

). (B.2)

We can define the real 8× 8 matrices γ1, ..., γ8 as

γ1 = ε× ε× ε , γ5 = σ3 × ε× I2 ,

γ2 = I2 × σ1 × ε , γ6 = ε× I2 × σ1 ,

γ3 = I2 × σ3 × ε , γ7 = ε× I2 × σ3 ,

γ4 = σ1 × ε× I2 , γ8 = I2 × I2 × I2 .

(B.3)

This should be read as

γ7 = ε× I2 × σ3 =

(0 I2 × σ3

−I2 × σ3 0

), I2 × σ3 =

(σ3 0

0 σ3

), (B.4)

and so on. It is easy to verify that the matrices γ1, ..., γ8 obey the following relations

γiγTj + γjγ

Ti = γT

i γj + γTj γi = 2δijI8 , i, j = 1, ..., 8

γ1γT2 γ3γ

T4 γ5γ

T6 γ7γ

T8 = I8 , γT

1 γ2γT3 γ4γ

T5 γ6γ

T7 γ8 = −I8 .

(B.5)

Appendix B. Gamma matrices and spinors 79

Now we introduce the 16× 16 matrices γ1, ..., γ9 defined as

γi =

(0 γi

γTi 0

), i, j = 1, ..., 8

γ9 = σ3 × I8 =

(I8 0

0 −I8

).

(B.6)

The matrices γ1, ..., γ9 are symmetric and real, and they obey

γi, γj = 2δijI16 , i, j = 1, ..., 9

γ9 = γ1γ2 · · · γ8 .(B.7)

At this point we are ready to define the Dirac matrices in ten dimensions, which

are the following 32× 32 matrices:

Γ0 = −ε× I16 =

(0 −I16

I16 0

),

Γi = σ1 × γ =

(0 γi

γi 0

), i = 1, ..., 9

Γ11 = σ3 × I16 =

(I16 0

0 −I16

).

(B.8)

We see that these matrices are real and satisfy the relations

Γa, Γb = 2ηabI32 , a, b = 0, 1, ..., 9, 11

Γ11 = Γ0Γ1 · · ·Γ9 .(B.9)

It is convenient to introduce the light-cone Dirac matrices Γ±, given by

Γ± =Γ0 ± Γ9 ,

Γ± = −1

2Γ∓ =

1

2(Γ0 ± Γ9) .

(B.10)

The raising and lowering of these indices are done according to a flat space metric

with η+− = −2.

We then define

Γa1a2···an = Γ[a1Γa2 · · ·Γan] , (B.11)

and analogously the 16× 16 matrices

γi1···in = γ[i1 γi2 · · · γin] , (B.12)


with il = 1, ..., 8. Since γi is symmetric we have that

γTijkl = γijkl , (B.13)

i.e. that γijkl is also symmetric.

Furthermore we define the 8× 8 matrices

γi1···i2k= γ[i1γ

Ti2· · · γT

i2k] , γi1i2···i2k+1= γT

[i1γi2 · · · γT

i2k+1]. (B.14)

with il = 1, ..., 8. In particular we call Π the matrix

Π ≡ γ1234 = γ1γT2 γ3γ

T4 , (B.15)

which has the following proprieties

Π2 = I8 , ΠT = Π , Π = γ5678 . (B.16)

The last equation follows from (B.5). Finally it is possible to show that Π satisfies

the relations

Πγij = γijΠ = −εijklγkl , Πγi′j′ = γi′j′Π = −εi′j′k′l′γ

k′l′ , (B.17)

with i, j = 1, 2, 3, 4 and i′, j′ = 5, 6, 7, 8.

Spinors

The spinors θA are 32-component Majorana-Weyl spinors. The Majorana condition

imposes that the 32 components of θA are real. The Weyl condition is

Γ11θA = θA , (B.18)

for both A = 1, 2. Note here that we choose the two spinors to have the same

chirality since we are considering type IIB string theory. Using (B.9) we see that

the Weyl condition means that only the first 16 components of θA are non-zero,

whereas the last 16 components are zero. We write therefore

θA =

(ψA

0

), (B.19)

where ψA, A = 1, 2, are two real 16 component spinors. We impose the light-cone

gauge

Γ−θA = 0 . (B.20)


Since

Γ−

(ψA

0

)= (Γ0 − Γ9)

(ψA

0

)

=

(0 −I16 − γ9

I16 − γ9 0

)(ψA

0

)=

(0

ψA − γ9ψA

),

(B.21)

we immediately see that the light-cone gauge becomes equivalent to

γ9ψA = ψA , (B.22)

which resembles a Weyl condition for the transverse directions. Indeed, using (B.6),

we see that the last 8 components of ψA are zero. Thus, we write

ψA =

(SA

0

), (B.23)

where SA, A = 1, 2, are two real 8 component spinors.

Now let us see how the gamma matrices act on the spinors:

ΓiθA =

(0

γiψA

), γiψ

A =

(0

γTi SA

), (B.24)

for i = 1, ..., 8. Furthermore we have

ΓiΓjθA =

(γiγjψ

A

0

), γiγjψ

A =

(γiγ

Tj SA

0

),

ΓijθA =

(γijψ

A

0

), γijψ

A =

(γijS

A

0

),

ΓijklθA =

(γijklψ

A

0

), γijklψ

A =

(γijklS

A

0

).

(B.25)

In particular for Γ1234 and Γ5678 these relations become

Γ1234θA =

(γ1234ψ

A

0

), Γ5678θ

A =

(γ5678ψ

A

0

)=

(γ1234ψ

A

0

). (B.26)

For our calculations of the fermionic spectra of the type IIB superstring, the fol-

lowing relations turn out to be very useful:

θA = (θA)T Γ0 = (0, (ψA)T ) ,

θAΓ+ = (2(ψA)T , 0) , θAΓ− = 0 ,(B.27)


and

Γ−Γ+θA = −4θA , Γ+Γ−θA = −θA ,

θAΓ+Γ− = −4θA , θAΓ−Γ+ = −θA .(B.28)

Using these relations along with Γ−+ = 12(Γ−Γ+ − Γ+Γ−) we obtain that

Γ−+θAθBΓ−+ = Γ09θAθBΓ09 = θAθB . (B.29)

We can also write the following more general relation

ΓabθAθBΓab = Γijθ

AθBΓij + 2θAθB , (B.30)

where i, j = 1, ..., 8 and a, b is either 0, 1, ..., 9 or +,−, 1, ..., 8.

Appendix C

Finding the new Penrose limit

In this appendix we shall show how to get the transformations of variables, that

in section 4.3 allowed us obtain the pp-wave background with two flat directions

from a Penrose limit of the AdS5 × S5 geometry. Thus we start exactly from the

AdS5 × S5 metric (4.1)

ds2 = R2[− cosh2 ρ dt2 + dρ2 + sinh2 ρ (dΩ′

3)2

+ dθ2 + sin2 θ dα2 + cos2 θ (dΩ3)2] ,

(C.1)

where

(dΩ′3)

2= dβ2 + sin2 β dγ2 + cos2 β dλ2, (C.2)

(dΩ3)2 = dψ2 + sin2 ψ dφ2 + cos2 ψ dχ2. (C.3)

We expect that the transformation of variables suitable for our aim, in its more

general form, must be:

χ = uϕ0 + aϕ1 + bϕ2 ,

φ = v ϕ1 + c ϕ0 + dϕ2 ,

α = w ϕ2 + e ϕ0 + fϕ1 ,

γ = ϕ3 ,

λ = ϕ4 ,(C.4)

being u, v, w, a, b, c, d, e, f parameters to be determined. Note that we have

omitted the η’s parameters, which in this treatment are irrelevant.

Let us make the following guess for the definition of the coordinates x+, x−, xi

(i = 0, 1 . . . , 8), that is quite obvious if we remember how we defined these variables

in the cases teated in sections 4.1 and 4.2:

x+ =t + ϕ0

2µ, x− =

µR2

2(t− ϕ0) ,

x1 = Rϕ1 , x2 =R√2

(π

4− ψ

),

x3 = Rϕ2 , x4 = R(π

4− θ

),

x5 + ix6 = r3eiϕ3 , x7 + ix8 = r4e

iϕ4 ,

(C.5)

Appendix C. Finding the new Penrose limit 84

in which r3 and r4 are given by

r3 = Rρ sin β , r4 = Rρ cos β . (C.6)

Now we are going to show explicitly all the calculations necessary in order to

express the metric in the new coordinates. Therefore, using (C.4) and (C.5), from

which we find out

t = µx+ +x−

µR2, ϕ0 = µx+ − x−

µR2, (C.7)

and defining r as

r2 = r23 + r2

4 = R2ρ2 ⇒ ρ =r

R, (C.8)

we can write the square of the infinitesimal space-time length as follows

ds2 = R2

[− cosh2 r

R

(µ

(dx+

)2+

2

R2dx+dx− +

(dx−)2

µ2R4

)

+(dr)2

R2+ sinh2 r

R(dΩ′

3)2+

(dx4)2

R2

+1

2

(cos

x4

R− sin

x4

R

)2 [w

dx3

R+ f

dx1

R+ e

(µdx+ − dx−

µR2

)]2

+1

2

(cos

x4

R+ sin

x4

R

)2

(dΩ3)2

].

(C.9)

The metric of the three-sphere (dΩ3)2 becomes

(dΩ3)2 = 2

(dx2)2

R2+

1

2

(1− sin

2√

2x2

R

)[vdx1

R+ c

(µdx+ − dx−

µR2

)+ d

dx3

R

]2

+1

2

(1 + sin

2√

2x2

R

)[u

(µdx+ − dx−

µR2

)+ a

dx1

R+ b

dx3

R

]2

.

(C.10)

Bearing in mind that there is an overall factor R, that multiply all the terms in

ds2, we can neglect all the terms O(R−3) in (dΩ3)2 since these does not survive the


Penrose limit R →∞:

(dΩ3)2 ' 2

(dx2)2

R2+

(1

2−√

2x2

R

)[v2 (dx1)

2

R2+ c2µ2

(dx+

)2 − 2c2

R2dx+dx−

+d2 (dx3)2

R2+ 2

v d

R2dx1dx3 + 2µ

c v

Rdx+dx1 + 2µ

c d

Rdx+dx3

]

+

(1

2+

√2x2

R

) [u2µ2

(dx+

)2 − 2u2

R2dx+dx− + a

(dx1)2

R2+ b2 (dx3)

2

R2

+2µau

Rdx+dx1 + 2µ

u b

Rdx+dx3 + 2

a b

R2dx1dx3

].

(C.11)

For the same reason we can neglect all the terms O(R−1) in (C.9):

ds2 ' −µ2R2(dx+

)2 − µ2r2(dx+

)2 − 2dx+dx− + (dr)2 + r2 (dΩ′3)

2

+(dx4

)2+

1

2

[w2

(dx3

)2+ f 2

(dx1

)2+ 2f wdx1dx3 − 2e2dx+dx−

+2µR(w e dx+dx3 + e f dx1dx+

)+ R2µ2e2

(dx+

)2]

− 2µx4(w e dx3dx+ + e f dx1dx+

)−Rµ2e2x4(dx+

)2

+ R2

(1

2+

x4

R

)(dΩ3)

2 ,

(C.12)

where the last term is given by

R2

(1

2+

x4

R

)(dΩ3)

2 ' (dx2

)2+

1

4

[(v2 + a2

) (dx1

)2+

(b2 + d2

) (dx3

)2

+ (v d + a b) dx1dx3 − 2(c2 + u2

)dx+dx−

]

+µR

2

[(c v + a u)dx+dx1 + (c d + b u)dx+dx3

]+

R2

4µ

(c2 + u2

) (dx+

)2

+√

2µx2[(a u + c v)dx+dx1 + (b u− c d)dx+dx3

]

+ R

√2

2µ2x2

(u2 − c2

) (dx+

)2+ µx4

[(c v + a u)dx+dx1(c d + b u)dx+dx3

]

+ Rµ2

2x2

(u2 + c2

) (dx+

)2+√

2µx2x4(u2 − c2

) (dx+

)2.

(C.13)

Now we should take the limit R →∞ (while keeping all the x’s coordinates fixed,

of course), but in this way clearly the metric diverges. In order to avoid that it

happens we must fix the parameters such that the coefficients of the divergent


terms vanish. Let’s do that now. The first condition we have to impose is that the

coefficient of R2µ2 (dx+)2

should be equal to zero:

e2

2+

c2

4+

u2

4− 1 = 0 ; (C.14)

the same thing should happen for the coefficients of Rµdx+dx3

2w e + c d + b u = 0 , (C.15)

that of Rµdx+dx1

2e f + c v + a u = 0 (C.16)

that of Rµ2x4 (dx+)2

− e2 +1

2

(u2 + c2

)= 0 , (C.17)

and in the end that of Rµ2x2 (dx+)2

u2 − c2 = 0 . (C.18)

These constraints make the procedure of Penrose limit possible and meaningful.

But these by themselves are not sufficient in order to reach our goal. In fact we

are searching for a pp-wave metric that would have two space-like isometries, and

in practice we already know how this metric should be like. For this reason there

are other constraints are required. First we want all the coefficients of (dxi)2

to be

one; this is automatically true in all cases, except for (dx1)2

and (dx3)2, for which

this request implies that

f 2 +v2 + a2

2= 2 , (C.19)

and

w2 +b2 + d2

2= 2 . (C.20)

We see that in ds2 there is a term ∼ dx1dx3, that we do not want in the metric so

its coefficient must vanish:

f w +v d + a b

2= 0 ; (C.21)

Analogously the demand that terms of the type x2dx+dx3 and x4dx+dx1 would not

appear in ds2, implies respectively

b u− c d = 0 , (C.22)


− 2e f + c v + a u = 0 . (C.23)

Finally the last two constraints that have to be imposed is that the coefficients of

the terms x2dx+dx1 and x4dx+dx3 would be −4µ:

a u− c v = −2√

2 , (C.24)

− 2w e + c d + b u = −4 . (C.25)

Once all these conditions are verified the metric (C.12), (C.13) becomes

ds2 = −4dx+dx− + dxidxi − µ2

4∑

k=1

(1− η2

k

) [(x2k−1

)2+

(x2k

)2] (

dx+)2

+ 2µ∑

k=3,4

ηk


]dx+ − 4µ

(x2dx1 + x4dx3

)dx+ ,

(C.26)

that is just the metric (4.15) that we are looking for.

Solving the system of all the twelve constraints equations (C.14-C.25) we get the

values of the parameters that bring to this metric. A possible choice is to fix these

parameters in the following way

u = 1 , a = −√2 , b = −1 ,

v =√

2 , c = 1 , d = −1 ,

w = 1 , e = 1 , f = 0 .

(C.27)

This is exactly the choice that we made in section 4.3.

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