david berenstein- supersymmetry: a string theory point of view

8/3/2019 David Berenstein- Supersymmetry: a string theory point of view

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Supersymmetry: a string theory point of view

David Berenstein

Department of Physics, University of California at Santa Barbara, Santa Barbara, CA 93106,

U.S.A.

Abstract. In this article I describe recent advances that show the relationship between stringtheory and supersymmetry, and how string theory ideas have revolutionized our understanding ofsupersymmetric field theories

1. INTRODUCTION

These lecture notes were presented at the XXXV Latin American School of Physics,

whose topic was on Supersymmetries and its physical applications. The course lasted

for five one hour sessions. The original title that I was given to work around was " Super-

symmetry in particle physics, string theory and cosmology" . I was clearly overwhelmed

by the wealth on information that is available on each one of the three topics. So I de-

cided to go with just one of the three, the one I am most familiar with, string theory, andtry to keep in mind the other two as the lectures progressed.

I decided that my purpose for the lectures was to explore some modern aspects

of four dimensional field theories and how these results are inspired and related to

geometrical ideas in string theory. I also wanted to give a flavor of what ingredients are

being used to understand the problem of the cosmological constant and supersymmetrybreaking, as well as the string theory ideas that are inspiring new trends in the study of

phenomenological models for particle physics beyond the standard model.

The choice of topics is by no means complete and reflects some personal choice on

what I believe are very interesting phenomena that have been found in string theory,mostly from familiarity with subjects themselves, together with constraints given by

time limits. I also tried perhaps to no avail to refrain from being too technical onmost parts, but not at the expense of ruining the understanding of ideas. Because of this,

on some sections these lectures are very technical. Hopefully the ideas that are meant to

be illustrated with the technical sections are put in a better light with these sections.

The choice of references is also for the purpose of illustrating ideas and not meant to

be exhaustive. I hope I have not missed giving credit where appropriate. Many facts can

be learned on review papers, of which I have cited a few.

The lectures are organized as follows.

On lecture one I give an introduction to supersymmetry, and supersymmetric string

theory in flat space. I discuss how worldsheet supersymmetry on the string givesfermionic states in spacetime, and I discuss briefly the superstrings with N 2 and

N 1 SUSY in ten dimensions. I also introduce the notion of D-branes, whose physicsoccupies much of the rest of the talks. The second and third lecture will deal with max-


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imally supersymmetric Yang Mills theory in four dimensions and the AdS/CFT corre-spondence. I will also introduce the notion of plane wave limits and I will discuss the

integrable aspects of the planar diagram expansion. In lecture four I will present recent

results in theories with N 1 SUSY in four dimensions which can be realized by plac-

ing D-branes on various geometries. Then I will describe how the vacuum structure of

these theories is captured by studying random matrix models. Finally I will have some

comments on recent attempts to understand the cosmological constant, supersymmetry

breaking and the problem of finding metastable vacua in string theory.

2. LECTURE 1

2.1. What is supersymmetry and why we like it

Supersymmetry is a symmetry that relates fermions to bosons. In the simplest case, the

supersymmetry charge commutes with the Hamiltonian, with the result that boson and

fermion states are degenerate in energy. The supersymmetry charges are to be interpreted

as infinitesimal generators of a symmetry, similar to Lie algebra generators.

In these lectures we will be concerned with supersymmetry in relativistic quantum

field theories. In a theory of a Lorentzian spacetime, it is a non-trivial extension of theLorentz group, and because of spin-statistics, the charge that generates the supersymme-

try transformation must be some form of spinor. For renormalizable theories of particle

physics in dimension 4, we can only have spin helicities from 1 to 1. The supersym-

metry charge is of spin 1

2.The supersymmetry charge can be represented by a Weyl spinor

Q

Q

(1)

By unitarity (complex conjugation), there is another doublet of charges

Q

Q

(2)

The non-triviality of the extension of the Lorentz group is captured by the commutator

Q

Q

P

P (3)

and

Q

Q

Q

Q

0

(4)

From these commutators it follows that the generator of time translations H, the Hamil-

tonian, is given by a positive definite operator

H

QQ

(5)

so that if there is a supersymmetric ground state, the energy of the ground state has to

vanish. (A ground state is supersymmetric if Q 0 Q 0 0). This fact has been

argued as a natural mechanism to cancel the cosmological constant.


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One can have more than one supersymmetry charge, in which case we add an indexto the supersymmetry charge to keep the distinction between the charges. The spacetime

algebra is then

QI QJ

IJP (6)

and we can also deform the

Q

Q

commutator

QI QJ

Z

IJ

(7)

where Z is called a central charge. This means Z commutes with the elements of theLorentz group.

Supersymmetries can be counted either by the total number of generators of SUSY, or

by the total number of spinor charges. The first counting is independent of dimension,

while the second one depends on dimension. The convention is that N 1 SUSY in fourdimensions corresponds to having four supersymmetries.

Supersymmetry predicts that for every boson, there is a fermion with the same quan-

tum numbers (except spin) and vice versa. In particular these are degenerate in mass.

This is not seen in experiments. Indeed, a lot of familiar physics would be different if

this were the case: electron orbits in the atom would not be stable. The electrons would

decay to a lower orbit by emitting superpartners of the photon and becoming bosons.

How can we reconcile this fact with low energy physics? The assumption underlying

the equality of bosons and fermions is that supersymmetry is a symmetry of the ground

state. However, it can be spontaneously broken.

Other than being a non-trivial extension of the Lorentz group, we like supersymmetry

for various technical properties of supersymmetric field theories.

If we consider loop diagrams, the fermions in the loop contribute with opposite sign tothe bosons, providing natural cancelations of certain divergences and making the theory

more finite. For example, a generic boson in a four dimensional renormalizable QFT hasquadratic divergent corrections to its mass. However, a fermion only has logarithmic

corrections to its mass. By the Bose-Fermi degeneracy, the boson mass can not receive

larger corrections than the fermion. This means that these quadratic divergences cancel,

and the mass gets small corrections with respect to its tree level value. This is the reason

why Supersymmetry is a technically natural solution to the hierarchy problem. If attree level the Higgs mass is small compared to some other scale (quantum gravity for

example), then perturbative corrections can not make it large.

Another reason for supersymmetry is that it fits the data for grand unification in the

simplest models better than the corresponding theories without supersymmetry. Finally,a reason to be interested in SUSY is that it is predicted by many string theory models as

part of the UV structure of the theory which makes the string theory consistent.

2.2. Basic string theory: bosonic string theory

The basic assumption of string theory is that elementary particles are not point-like,but that they arise as elementary excitations of an extended object of dimension one (the

fundamental string). The time evolution of the string spans a two dimensional surface

embedded in spacetime.


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FIGURE 1. Worldsheet of a string

The minimal assumption about the dynamics of the string is that the action of the

string is given by the induced volume of this surface in spacetime. We suggest reading[1, 2, 3, 4] as an introduction to strings and superstrings.

By doing dimensional analysis we find that the volume of this spacetime has engi-

neering dimension equal to 2 ( we are counting powers of momenta), while a natural

action has dimension zero. To compensate for units, we need to introduce a scale in the

system. This is the string tension

T

1

ls 2

(8)

The action just described is the Nambu-Goto action and it is given by

S

1

d

2

!

det gind

(9)

A second equivalent way to obtain the same classical physics is by introducing an

auxiliary worldsheet metric , so that the action becomes

S

d2"

gx

x

(10)

Notice that in the above action does not appear with derivatives, so that when we takeits equation of motion we get constraints that let us solve for almost uniquely, and ifwe substitute the values of we get back the Nambu-Goto action.

The action has a symmetry under local rescalings of # exp f . This classical

symmetry is the conformal invariance of the worldsheet action. This second version ofthe string action is due to Polyakov and it is the version we usually consider for quantiz-

ing the string. Strings where we have gravity on the worldsheet are called critical strings.

If we want to quantize this action, we want to have the conformal symmetry to survive


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the quantization. This leads to constraints from asking for the quantum corrections to theconformal symmetry to vanish. For flat g this amounts to the dimension of spacetime

being fixed at 26. Fortunately, this is higher than four dimensions, so we can consider

string theory on geometries where four of the 26 dimensions are non-compact and the

other 22 are compact. For a more general metric, we get a quantum field theory in two di-

mensions. The conformal invariance contains invariance under rescalings. This implies

that the -model action must be invariant under the renormalization group equations.To first order in the loop expansion on the worldsheet about a free field theory the beta

functions ofg give rise to the vanishing of the Ricci tensor, so that consistency of thestring theory predicts Einsteins equations.

This is not assumed a priori, and it is one of the reasons why string theory is so

exciting to study. We will now give a quick sketch of the quantization of the string.

Notice that any Riemann surface has a conformally flat metric, where is constant,and we pick a complex set of coordinates z

z (lightcone in Lorentz signature) so that

%

%

1 and other components vanish.

Substituting this metric in the action and varying with respect to the target space

coordinates X we get very simple equations on the worldsheet: namely, the target spacecoordinates X satisfy free wave equations. These are given by X being holomorphic

X Xz

&X

z

(11)

equivalently, in Lorentzian signature, we get

X

X

%

&X

(12)

where

(

)

(13)

The periodicity in #

&

2 means we get an oscillator expansion for the solutionsof the equations of motion.

X

p& %

n

expin

%

& n

expin

(14)

and since we have gauge fixed the metric, we still need to impose its equations of motion.This gives us a set of local constraints in the space of solutions (the Virasoro con-

straint). It is quadratic in the fields X, and takes the form

pp Oscillators

(15)

The variables p are interpreted as momenta in spacetime. Each positive integer n

produces one left moving quantum and one right moving quantum of energy n, for each. These quantize the zero modes for translations. It follows that p2 gives essentially aset ofintegers as the allowed masses for particles.

We also need to know the ground state energy of the system. This is a system whereall oscillators are at zero occupation number. The ground state has a mass given by

m2 2

(16)


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which is negative, and signals a tachyonic instability in the system, so at best the bosonicstring describes a consistent theory expanded about an unstable saddle point. We also

get states with m2 0 and spin 2: these are identified as gravitons. These are one left

oscillator and one right oscillator with n 1. This means critical string dynamics will

generically include gravity.

Another thing to notice is that the bosonic string we have described above does not

describe fermions.

2.3. Superstrings

As we have seen, when we considered the bosonic string theory, we obtained world-

sheet quantum fields for the coordinates of spacetime, and a very large symmetry alge-bra (the conformal group). We can speculate that increasing the worldsheet symmetries

might lead to a more interesting theory.

This is how one can conceive of the type II string theories. The main idea is to

have supersymmetry on the worldsheet of the string, so that instead of just having the

conformal group, we get its supersymmetric extension. Now, on top of gravity on the

worldsheet, we have its superpartner (the gravitino), which also appears as an auxiliary

field. Also, for every X

we get a pair of fermions, one left moving, one right moving, . Fixing the superconformal invariance we get 10 dimensions for flat spacetime,as opposed to 26. This number is still bigger than four, and much closer to four. From

the point of view of topology it is a great improvement to have only 6 extra dimensions

instead of 22.

Again, we consider the quantization in superconformal gauge. So the

will havean oscillator expansion as well and they can have zero modes. The quantization of spin

zero modes leads to representations of Clifford algebras. Here the zero modes of the

become like ten-dimensional gamma matrices , so the degeneracy of the ground stateimplies that the ground state has spin degrees of freedom. The energy of this state is

zero (the boundary conditions are supersymmetric) and leads to massless particles. We

get one set of gamma matrices from the left movers and one from the right movers. Thisground state is bosonic.

Fermion fields are not usually observables, but their currents are (e.g. the charge of a

fermion state, but not the sign of its wave function). This means we can also take anti-

periodic boundary conditions (the boundary conditions are not supersymmetric). If we

can do this independently on the left and right movers, then these new ground states with

mixed boundary conditions will be spinors in 10D, because we only get one set of zero

modes.

One can check that these ground states are still massless, because half of the super-

symmetries on the worldsheet are unbroken. One of these states is a spin 3

2 particle

which is massless in ten dimensions! The only possibility of such a thing happening is

if the ten dimensional theory in target space is supersymmetric itself.

From here, worldsheet SUSY can imply spacetime SUSY, so the massless degrees offreedom of the theory lead to supergravity in ten dimensions.

Consistency of the theory requires that the tachyon state is not physical, so this theory


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is perturbatively stable. The projection onto physical states is called the GSO projection.The details are too technical for the scope of this paper. We will just notice that there

are two ways to do this consistently: type IIA and type IIB strings, which differ on the

chirality of the spacetime supersymmetries.

In general one wants to compactify these theories to obtain a four dimensional world.

If we require that the compactification include only a non-trivial metric and that super-

symmetry survives in four flat dimensions then the compactification manifold should be

a Calabi-Yau geometry [5].

A problem with these theories and phenomenology is that they have too many super-symmetries: on compactifying to 4D the spectrum is non-chiral, and we would have avery hard time reconciling this theory with the standard model.

2.3.1. Heterotic strings

The heterotic strings were invented by noticing that the left and right moving degrees

of freedom of the string theory could be different. Worldsheet supersymmetry can have

chirality on the worldsheet too, so one can get away with imposing supersymmetry only

for the left moving degrees of freedom, and not have any supersymmetry on the right

movers. From this point of view, the left movers live in 10 dimensions, and the rightmovers on 26. The sixteen extra dimensions are to be considered as an internal symmetry

space of the theory.

The worldsheet supersymmetry is used to generate only one set of gravitinos in ten

dimensions instead of two.

It turns out that the theory is very constraining. The sixteen extra dimensions have to

be realized as current algebras either for E80

E8 symmetry, or SO 32 symmetry. These

lead to massless vector particles in 10D. So we get on top of N 1 supergravity in 10D

a particular gauge group of interactions. This theory is chiral in 10D, and the chiralitycan descend to 4D on compactifications. One of the great successes of string theory is

that the theories that are found from string consistency conditions give rise to all of the

solutions to anomaly free supergravity theories in 10D. The gauge groups that appearare big enough to contain the standard model, the SSM and many different GUTs.

2.4. D-branes

So far I have argued that classical solutions of the heterotic string are the best candi-

dates for 4D-particle physics, and that type II classical solutions are not.

However, we can consider solitons on the type II theory which preserve some amountof SUSY and which do not involve only the metric. We can explicitly write black

hole/brane solutions of type II supergravity which preserve some amount of supersym-metry, so it is important to understand if these have a nice description in terms of the

type II strings. The answer to this question is yes, and the objects that are described in

this simple form are called D-branes [6], see also [7]. The idea is to have a topological

defect in spacetime which is the geometric locus where strings can end.


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FIGURE 2. A D-brane with a string attached

From the string theory point of view we are quantizing strings with open ends. The

open ends are required to lie on the D-brane itself. These objects are physical branes:

they have tension and they deform the geometry. The geometry they produce coincides

with the black brane geometry in many cases.A very similar analysis for the closed type II strings can be done for the open strings

ending on D-branes. It can be shown that a flat dimension kD-brane preserves half of the

supersymmetries in type IIA if k

0

2

4

, and for the type IIb if k

1

1

3

5

.In particular, we can use them to break the total number of supersymmetries in 4D

from 32 to 16. The quantization of modes on these D-branes produces massless particles

with spin 1: gauge bosons!. This gives us another route towards getting gauge field

theories in 4D. If we place M branes on top of each other, we get M2 open strings

stretching between them. The interactions of the massless particles are governed by the

dimensional reduction ofU M , N 1, 10D gauge theory to k & 1 dimensions.

For k 3, this gives us N 4 SUSY in four dimensions. We can also considerconfigurations of D-branes at angles that reduce the supersymmetry further [8]. In this

way, one is able to obtain a chiral spectrum of particles. These can in principle realizethe standard model of particle physics, and the construction does not necessarily need a

GUT embedding of the model into higher dimensions. Most of the modern developmentsin string theory in the last few years have been tied to understanding D-branes.

We will explore some of the physics associated to D-branes and its relation to super-

symmetry in the rest of the lectures.


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FIGURE 3. A set of parallel D-branes with a string ending on both

3. LECTURE 2

3.1. A stack of D-branes

Let us look at type IIB string theory in flat ten dimensions and let us look for the

simplest configuration of D-branes which leads to a four dimensional gauge theory ontheir worldvolume. This is, let us consider a stack of N parallel D3-branes located on

top of each other, or separated just a little bit, but at a distance much smaller than the

string scale, let us call it r.

We are interested in understanding the degrees of freedom of the D-brane stack which

are very light. This is, we are looking for degrees of freedom whose energy is much

smaller than the string scale. A generic string state will have a mass comparable to

m

l

1s (17)

However, the ground state of the string stretching between two of these D-branes will

have a mass of order m

l

1s l

ls 1 l 1

s (18)

where l is the typical separation of the branes. This can be made arbitrarily small

compared to the string scale by moving the D-branes sufficiently close to each other.


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FIGURE 4. A stack of D-branes

These strings are essentially massless. The D3 brane is a supersymmetric D-brane (BPS)

for type IIB string theory, so the degrees of freedom on the D-branes are supersymmetric.

The D3 brane preserves 16 supersymmetries. Also the ground states for an open string

state have spin less than or equal to one.

We get that the low energy dynamics we want to capture is given exactly by N 4

SYM theory in four dimensions (the worldvolume coordinates of the D-brane). Thetheory is not free: the coupling constant in four dimensions does not have a scale. So it

can have a finite value at very low energies. Now we will describe some of the physicsassociated to this theory.

The stack ofN branes leads to a UN

gauge symmetry. The field content is summa-

rized by a vector particle, four weyl spinors and six scalars, all in the adjoint of UN

,

A, 1

3 3 34, 1 3 3 3 6. The theory has an SO

6

SU

4

R-symmetry that rotates the four

chiral spinors into each other as a 4, and rotates the six scalar fields as a 6 of SU4

.

Non-renormalization theorems for sypersymmetry imply that this theory is finite. The

field theory has a superconformal symmetry and leads to an enhancement in the infrared

to 32 supersymmetries when all D-branes are exactly on top of each other.

The superconformal symmetry is an extension of the Lorentz group, which has addi-tional generators: dilatations and special conformal transformations (this is like a dou-

bling of the translations). There is a similar doubling of supersymmetries, by addingwhat are called special supersymmetry transformations. The R-symmetry is part of the


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superconformal algebra too. The generators are named as follows

P

L

K

Qi Qi S

i S

i Ri j (19)

The Ri j are global internal symmetries of the quantum field theory, is the generator

of dilatations. L is the set of Lorentz transformations, P and Q are the translations and

supersymmetries, while K

S are the special conformal transformations and the special

supersymmetries respectively.

The bosonic part of this superconformal group is given by

SO 4

2 0

SO 6

(20)

where the first factor is the extension of the Lorentz group to the conformal group, while

the second factor is due to the R-symmetry. We will come back to explore the theory

further. For the time being, this is all we need to keep in mind.

3.2. Another route to the infrared (IR)

Let us again consider the same stack of D-branes. However, we will look at them

as sources for supergravity fields and we will look at the background geometry which

includes the back-reaction of the D-branes. These are solutions of supergravity theory

with 16 supersymmetries. One can show explicitly that two parallel D-branes which

respect the same supersymmetry do not attract or repel each other. They attract each

other with gravitational interactions. This means there is a compensating force that

cancels the gravitational attraction. This force has to be communicated by massles fields

[6].It turns out that the supergravity theory to which these branes couple have some form

of generalized electromagnetism. The branes carry this charge. It acts like the central

charge of extended supersymmetries in four dimensions.

The D3 branes are sources for flux for a 4-form potential

A (21)

where this is the generalized potential associated to the extended charge that the D-

branes carry. The solution for the stack of D3 branes is given explicitly by the following

background [9]

ds2 f 1 6 2dx27 7 & f1 6 2 dr2 & r2d25

F5

d5 & S

D

(22)

where

f 1 & 4gsN

2

r4

(23)

or more generally, f is a solution of the Laplace equation in the six directions transverse

to the D-brane, with sources given by the positions of the parallel D-branes. The high


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amount of supersymmetry implies that the background does not get corrected. Wecan consider this geometry as a black hole/black brane solution of the supergravity

equations.

A standard property of black holes is that there is a redshift factor between being close

to the horizon and asymptotic infinity. This is seen by the ratio

8

g00 x

g00 f

1 6 4r

r9 0

#0

(24)

Therefore if we add a particle which has proper energy E at a radius r, it has a globalenergy Eg which is reduced compared to E by the fact that clocks run at different

speeds at r and at infinity. States with very low global energy Eg 1 l 1

s , do not need

to have small proper energy. However, massive stringy states with this property becomelocalized near r

0. This suggests that we can look for the infrared physics in the region

of the geometry where there is a very large redshift [10].

In the metric we discussed above, this is the region where we can ignore the first term

in f and take r very small. We do this as a limit procedure. The end result is called the

near-horizon geometry of the D3-brane system.

The geometry we obtain in this limit procedure is given by a very simple metric. The

spacetime geometry is given by a round AdS5 0 S5 geometry with Nunits of flux through

the S5 . In may ways this is like a five dimensional theory (because S5 is compact). Thecosmological constant on this effective theory is negative.

The isometry group of this geometry is exactly given by SO 4

2 0

SO 6 , and the

supergravity solution of the near horizon geometry of the stack of D3-branes also has 32

supersymmetries. Notice that this is the same result we obtained when we discussed the

symmetries of the N 4 SYM theory.This result, together with some other calculations suggest that the two descriptions

of the low energy dynamics (type IIB superstrings and N 4 SYM ) are the same[10]. The idea that these two systems might be describing the same physics is known

as the AdS/CFT correspondence. The correspondence goes beyond the supergravity

approximation. As discussed previously, we can have stringy states whose proper energy

is of order ls, but with very small global energy.The question we need to answer is the following: What is the nature of this correspon-

dence?

This is a very non-trivial question and other than direct verification of the correspon-

dence, one would want to have some set of tools which give some intuition with regards

to this issue.The ideas that suggest that this is possible are related to the entropy of gravitational

systems, namely that one can not put more energy in a region of spacetime than the

energy required to build a black hole which fills the region under consideration. Thus

the entropy of the energy inside a region of spacetime will grow at most like the area of

the region under consideration [11, 12].

This suggests that one can map all of the degrees of freedom inside a region of space-time to the boundary. This map is called a "hologram". The AdS/CFT correspondence

is a holographic correspondence: the CFT is the theory living on the boundary of AdS

space that encodes all of the gravitational degrees of freedom of the bulk.


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The notion of boundary here is due to Penrose: one has to consider the conformalcompactification ofAdS

5. This is a manifold with topology S3

0

R in Lorentzian signa-

ture, and whose conformal structure is equivalent to a product of a round S3 and a time

direction. The S5 shrinks to zero size on the boundary.

Even though we do not get a full metric on the boundary, we can still write the N 4SYM on the corresponding conformal geometry. This is because the N 4 SYM theory

is a conformal field theory, so it only needs to be coupled to a conformal structure to be

well defined.

3.3. The AdS/CFT correspondence

We will now describe the AdS

CF T correspondence for this particular background.

The following table shows the parametric correspondence between the two theories

SYM AdS5 A

S5

Symmetry SUB 2 C 2 D 4 E SUB 2 C 2 D 4 E

Rank UB NE FS5

F G N

Coupling constant g2YM

gs

Radius g2YM

N R4

In the table above we have described the relations between the gauge group, coupling

constant and t Hooft coupling [13] to the flux, string coupling and radius of curvature ofAdS5 0 S

5 in string units. The correspondence between gYM gopen and gs is standard.The radius can be read directly from the metric in string units, which we wrote as a limit

of equation Eq. (22).Now, we want to write tests of the above correspondence. For this we want to insist

on calculable properties of the field theory and the gravitational background. Lacking a

complete solution for either of the two problems, we want to see what is calculable as aperturbation expansion around a simple configuration.

For type IIB string theory, we need large radius in string units and small coupling

constant, so that we can perturb around a classical metric

RH

1

gs 1 1

(25)

This implies that we need gsN very large and gs very small. Both of these together imply

that we need to take N very large.

For SYM, we need to be in a perturbative regime. The effective perturbation expansion

is in the t Hooft couplingg2YMN (26)


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from summing over all colors in Feynman diagrams. This needs to be small to havecalculable physics. Indeed, the ideas of t Hooft suggest that we get a double expansion

in g2YMN and 1

N. The first one is characterized by planar diagrams, while the second

one gives non-planar diagrams and which topologically suggests Riemann surfaces of

higher genus. This should become the

and gs expansion in the type IIB string theory

on AdS5 0 S5. From this (perturbative) point of view, the AdS/CFT correspondence is a

strong/weak coupling duality for the t Hooft coupling constant.

We now come to the main technical problem of the AdS/CFT correspondence: how to

set up reliable calculations at weak coupling that are valid at strong coupling. We havealready seen that both systems have the same number of free parameters and symmetries.Can we test more?

This is the point where supersymmetry comes to the rescue. The supersymmetry alge-

bra should be realized unitarily, which places some unitarity constraints on the represen-

tations. Certain of these representations are short (they preserve some supersymmetry),

and they saturate a BPS bound (this is a consequence of unitarity).

These can not get corrected because the bound is an inequality between the and R

charge quantum numbers. However, the R charge is related to a non-abelian symmetry,which means that the charge is quantized and can not receive corrections, while in principle can. It turns out that all supergravity single particle states preserve some

amount of SUSY, essentially because these are massless in ten dimensions and arenaturally short (they lack the longitudinal components to make full massive multiplets).So it is natural to calculate their spectrum. The most convenient choice is to write the

AdS5 0

S5 metric in global coordinates

dt2 cosh2&

d2&

sinh2d23 & d2 cos2

&

d2&

sin2d

23 (27)

Here t, , 3 are the coordinates of AdS5, where t is associated to a timelike-killingvector and , 3 are spherical coordinates around an initial point. We have also chosen

to write the S5 in similar coordinates, where is a coordinate related to a particularkilling vector, and ,

are spherical coordinates with respect to the circle represented

by the angle .Notice that the gravitational potential term as read from the metric gtt implies thatthe particles are confined near

0. If we want to solve for wave propagation in

this geometry, this fact means that the "Schrdinger operator" will have a discrete

spectrum. This time slicing gives a killing vector which is identified with the generator

of dilatations in the superconformal group.

The problem of studying the full supergravity modes of this geometry was studied a

long time ago [14, 15], and a complete classification of states was obtained as represen-

tations of the superconformal group SU 2

2 4 .

The list of representations are all in short multiplets of the symmetry algebra (half

BPS). The lowest lying state of each multiplet is given by a totally symmetric tracelesstensor of SO 6 , which is a scalar with respect to the rotations of the superconformal

group which commute with the dilatations. This state is annihilated by all of the K andthe S generators, and half of the Qs.

We will now reproduce these results using the N 4 SYM theory.


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3.3.1. The operator state correspondence.

Because N 4 SYM theory is conformally invariant, the theory on different metrics

related by a conformal transformation are equivalent.

In particular, all fields are massless so we want to introduce an infrared regulator.

There are two ways to do this: do a Wick rotation to Euclidean BR4 and consider

correlation functions of gauge invariant operators inserted at various positions. The

distances between the operators acts as the IR regulator. We can also study the theory on

a box with a particular geometry. It turns out these two approaches are equivalent, if thebox is a round sphere. This is seen as follows.

The metric of flat space is given in radial coordinates by

ds2I 4 r2

d23 & dr

2

r2

(28)

This metric of flat P 4 , removing the origin is conformally equivalent to the metric of

P

0

S3 , they differ by the (local) scale factor r2. So if we insert an operator at the origin,

the scaling dimension of the operator is the eigenvalue under dilatations. This is the same

as translations in the Euclidean time t log r . So local operators inserted at the origin

in P 4 are equivalent to states of the theory on the round sphere. I will go back and forth

between these two descriptions. This property of conformal field theories in arbitrarydimensions is called the operator-state correspondence.

3.4. Matching supergravity and SYM

We have seen that the natural time in AdS is related to the generator in the supercon-

formal group. This is equivalent to using radial quantization in the dual SYM Euclidean

field theory, namely, we need to study the evolution along t in the metric defined by

Eq. (28).

Super-primary states correspond to local operators O such that

Q

K

OR

Q

S

OR

0

(29)

These are chiral if it also holds thatQ

Q

OR

0 for some of the supersymmetries.

States obtained from a chiral primary by application of Q

P are called descendants.

The conformal dimension of the operator O is the eigenvalue of the energy operator of

the stateO

on S3.

Unitarity implies that for any primary

S J

(30)

and equality is saturated for chiral primaries (BPS states). J is a generator of an SO 2

subgroup ofSO

6

, such that there is an SO

4

which commutes with JT

J 0

0 0 U (31)


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Highest weight states of the representation algebra are the ones that saturate this bound.In the free field theory, there is only one complex scalar field such that its canonical

dimension and J coincide. Let us call the field Z. The half BPS states with respect to

J are descendants of objects that can only be built out of Z. Remember that Z carries

gauge symmetry indices Zij in the adjoint (they are N0

N matrices). The operators that

correspond to the BPS chiral primaries need to be gauge invariant. This means that

all gauge indices need to be contracted. This can be accomplished by using a matrix

notation. The list of operators are given by (multi)-traces of Z.

n1 n2 nk

tr Zn1

tr Znk

(32)

How does this relate to the description of gravitons?

Each trace is considered as a single graviton state [16, 17]. Multi-trace objects are

considered as multi-graviton states. One can prove that in the large N limit (for finite ni),

different traces are approximately orthogonal. This correspondence receives correctionsin 1

N2, which can be interpreted as the result of interactions. The effective gravitational

constant in five dimensional AdS5 is exactly 1

N2 in units where the radius of AdS5 is

one. Do we match states with SUGRA? Yes. The state above is considered as the highest

weight state of an SO 6 representation. This representation is totally symmetric andtraceless, matching exactly the supergravity result.

3.5. Beyond supergravity in the BPS sector

Are there other interesting half BPS states in AdS50

S5 that are not gravitons?Consider the following scenario: fix N large, but do not take the infinite N limit. Then

it turns out that traces are algebraically independent up to order N. In particular, one can

show that tr ZN% 1

is a polynomial in tr Zk for k W N. This means that the description

in terms of quanta for gravitons breaks down for very large angular momentum. This

upper limit on the gravitons is called a stringy exclusion principle.An old calculation by Susskind showed that strings with a lot of energy grow trans-

versely to their direction of motion. Also holographic arguments suggest that this isnecessary to avoid making a black hole. Therefore, we could look for D-brane solution

which represent this growth.

Place a D3-brane wrapping a round S3 in S5 and give it a lot of angular momentum on

the S5. One can find dynamically stable solutions of this kind. Moreover, one can showthat they preserve half of the supersymmetries. The size of the S3 inside the S5 that the

D-branes span has a maximum size. The name for these branes is Giant gravitons [18].

These are additional objects that have the same quantum numbers as ordinary gravitons.

Their associated dimension is of order N, and these are candidate objects that capture

the stringy exclusion principle, however, are they traces?

Well, it turns out that there is another giant graviton! This one wraps an S3 inside

AdS5 and has a lot of angular momentum along the S5. It is also dynamically stable andit also preserves half of the SUSYs [19, 20].

Neither of these two giant gravitons is a trace. There are conjectures for what bothof these states are [21, 22]. They pass a remarkable set of consistency conditions (for


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S giant

p

EF

Closed strings

AdS giant

FIGURE 5. The quantum hall droplet analog of the giant gravitons

example, we understand their low lying fluctuations both from the DBI action and from

the SYM theory [23, 24]). There is still work to be done verifying these conjectures.The dynamics describing these BPS states is very simple from the gauge theory point

of view. It corresponds to a gauged matrix quantum mechanics of a single N0

N matrix.

From this point of view one finds a remarkable analogy with a Quantum hall droplet

that helps describe all of these states: the strings are interpreted as edge excitations of

the quantum hall droplet. The giant along AdS5 is interpreted as an electron and the giant

growing along S5

is interpreted as a hole state [25]. From here, we see that already theBPS content shows that there are objects beyond supergravity which are captured by the

SYM theory. This is shown pictorially in Fig. 5.In the next lecture we will go beyond supergravity states and produce a full string

spectrum out of SYM theory, with many non-BPS particles.

4. LECTURE 3

4.1. Plane wave limits

Before we describe plane wave geometries, let us begin with a few observations aboutgeometries that one uses for general relativity.

First, any geometry in the large radius limit becomes essentially flat. For a particle

moving in a geometry, this characterization is correct if we can localize the particle to a


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very small region of space compared to the scale at which the geometry is varying: thisis a limit where the curvature does not matter. This is a high energy limit in the sense

that to localize the particle one needs a relatively high amount of kinetic energy (this

follows from the uncertainty principle). From here it follows that in principle the flat

spacetime spectrum of strings is encoded in the AdS/CFT, but we need to think of it as

a high energy limit of the spectrum of states.

One can then ask the following question: is the ultra-relativistic limit of particles in

a weakly curved background interesting? This question is really addressing whether in

the high energy limit we get something other than a flat spacetime. The answer dependson the background.

There are various ways to look at this problem. The backgrounds we are dealing with

have geometric curvature and fluxes. These objects are tensorial.

We can ask how important they are by looking at the numerical value of their compo-

nents in a frame were the relativistic particle is at rest in its own frame. The background

fields in this frame are related to the original coordinate system where we describe the

background fields by a boost transformation. The boost transformation is an element of

a non-compact SO n

1 group, and this being non-compact, a large boost results in largecomponents for the transformation matrix . In particular, an otherwise small compo-

nent of a tensor field, can get to be very large in a different coordinate system related by

a boost. This is a feature of geometries where we have time. On Euclidean geometriesthe group of local rotations is compact and does not lead to large components of tensors.To define the ultra-relativistic limit, we can then take a double scaling limit where

we take the background to zero in some particular coordinate system, and we make a

compensating boost at the same time so that some components in the boosted system

are kept finite.

In a more invariant language, we have a particle with momentum p, and possibly

some polarization tensors (like spin ). The background field is characterized by some

tensor T.

We take the limit T # 0 , so that some contraction between T with p, and perhaps

stays finite. This defines the scaling of p in the limit. Notice that since the mass of the

particle should roughly be fixed p2 m2, the fact that the energy goes to infinity while

the proper mass stays finite means that the particle trajectory is well approximated by anull curve.

If we consider a point particle coupled to gravity, the natural trajectories for this limitare null geodesics. If we adapt the coordinate system to the null geodesic so that p0 is

finite, the new geometry is a non-trivial limit of the original geometrical system. The

typical scalar one can keep finite is Rpp. This limit is called the Penrose limit .

The geometry one obtains this way is of the plane wave type, and it is a considerable

simplification of the geometrical system.

4.2. Plane wave limit ofAdS5 X

S5

We will now explicitly take the plane wave limit of the AdS5 0 S5 geometry [26]. We

will keep a lot of the technical details in the following derivation, but the main ideas


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were already presented above. The main idea is to look at a null geodesic where we stayat a fixed position in AdS

5(varying global time), and consider the motion at the speed

of light along one great circle in S5.

First, we write the metric along geodesics of AdS5 0 S5, parametrized by one time

coordinate on AdS and one circle ofS

dt2 cosh2 & d2 & sinh2d23 & d

2 cos2 & d2 & sin2d

23 (33)

and we want to consider a particle at the origin

0,

0. We will focus on thegeometry near this trajectory. We can do this systematically by introducing coordinates

x

t

2 and x%

t. We also perform the rescaling

x%

x%

x

R2 x

r

R

y

R

R#

(34)

In this limit the metric becomes

ds2

4dx%

dx a

r 2& a

y 2

dx%

2&

da

y 2&

dar 2

(35)

which is the plane wave background. After an extra rescaling of x(

we can introduce a

scale .

ds2

4dx%

dx

2az 2dx

% 2&

da

z 2

F% 1234

F% 5678

const0

(36)

The # 0 limit of this geometry is flat space.What makes this limit interesting is that the free string theory can be exactly quantized

in this background.

4.3. Quantization of strings in the RR plane wave

Notice that in Eq. (36) the x(

coordinates are a light cone for this geometry. If

we choose the light cone gauge for the Green-Schwarz string, the action simplifies

dramatically. This is, we choose time on the string worldsheet so that it coincides withx

%

, x%

. From here we can see that the bosonic worldsheet action simplifies, because

the only terms that are not quadratic in the target space fields,

a

z 2dx

%

ddx

%

d

(37)

become quadratic due to x%

, so that its derivatives on the worldsheet are constant.

From here

Sbos

a

z 2&

a

z

a

z

(38)

It is a free massive field theory with 8 bosons.


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Some of the supersymmetries of spacetime are realized linearly in this background,and they commute with the Hamiltonian. So the fermions are also massive. The mass

comes from the flux. The exact details can be found in the paper of Metsaev [27].

Quantizing the string is straightforward: it is a free field theory on the worldsheet, and

we get an oscillator expansion. The spectrum of particles is given by

p

Hlc

%

n

g

Nn 2 &4n2

p%

2 (39)

In addition we have the condition that the total momentum on the string vanishes

P

n

g

nNn 0

(40)

Here p

is the lightcone Hamiltonian, and p%

is a conserved quantum number associated

to the quantization of the zero modes of translations on x

(the lightcone momentum).

4.4. What operators in the SYM survive the limit?

We need to consider what states correspond to p(

1 in the limit we are taking.

From the definition ofx%

and x

, we can understand what the quantum numbers related

to their translations are in the geometry, and translate these generators of symmetry into

the dual SYM

2p

p

%

ix

h

ix

h

it &

J

2p%

p

p

R2

1

R2ix

i

J

R2

(41)

where J is the rotation in the direction, one of the R-charges of the theory, exactly the

one characterized by the metric Eq. (27).We therefore need operators with J finite,

J

R2

" N, and remember alsothat J is the BPS condition. From these considerations it follows that we need to

consider states which are almost BPS and of very large conformal dimension.

Rewriting the spectrum of the string in the field theory parameters we find that the

contribution to the energy from each oscillator is roughly given by

J n wn

p 1&

4gNn2

J2

(42)

and we find that it looks like an expansion in the tHooft coupling gN, that we can

reproduce by doing perturbation theory! [28].

Notice that we are making a non-trivial statement. The large radius limit is taken firstmaking gN large, and the quantum numbers are taken large to compensate for the large

radius limit (this is the ultra-relativistic limit we were considering). We are trying to

compare these results to a perturbation expansion around gN 0. What saves the day is


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that there is a new effective expansion parameter which reduces the value of the t Hooftcoupling parametrically with the quantum numbers of the particle, so that the limits are

not incompatible.

4.5. Discretized worldsheets

We have now found a set of quantum numbers which we need in order to reproduce

the string spectrum. Considering that these states have large quantum numbers one hasto wonder whether planar diagrams dominate or not in the limit. It is a non-trivial

calculation to show that one is still safe. Just like for the supergravity fields, each string

state should be well approximated by single trace operators. These are words in the fields

(the ordering inside the operator matters), but we have an identification between them

due to the cyclic property of the trace.

As argued before, the operators are almost BPS with respect to J. Again, of the six

real scalars of the SYM theory, only one complex scalar Z has charge 1 and dimension

1. From here we conclude that the operator is made mostly out of Z. For example

r

tr

Z

ZrZ

Z

(43)

where there is a finite number of r. One can interpret these as a small number ofexcitations scattered in the operator which is mostly Z.

Planarity of the diagrams makes the position of the defects important. States with

defects in different places are orthogonal in the planar theory in the free limit. All of

these operators have the same conformal dimension in the free field limit. However,

when we introduce interactions we need to find the proper basis of operators, and we

end up with a problem in degenerate perturbation theory.The interaction term

g2YMtr

Q

Z

jR

Q

Z

jR

(44)

will make the defects hop to the left or right in the operator.Indeed, we can not get jumping by more than one site at first order perturbation theory

because each change of position would count as one extra non-planarity. To cancel it,

we need to add an interaction each time we hop positions. The idea now is to interpretthe position of the defects in between the Z as a lattice. The number of sites is of order

J

" N, so in the large N limit we obtain a continuum theory, as we are also taking the

interaction terms gsN # .

The theory on the lattice is translation invariant, mainly from the cyclic property of

the trace. Therefore it is convenient to rewrite the states in a momentum basis, so we

apply position dependentphases for inserting the same defect at different positions.

For example, a state with one excitation of momentum n will roughly be given by

a

4

n

0

p

%

1

" J

J

l g 1

1

" JNJ6 2%

1 6 2 tr

Q

Z

l

4

Z

J

lR

e

2inl 6 J

(45)

Obviously for n s 0 the above vanishes by cyclicity of the trace. In the string theory this

state has angular momentum along the string and it is not a physical state. However, we


22/38

FIGURE 6. Hopping interactions from planar diagrams: The dashed line indicates the defect. We see

one crossing in the diagram

can create states with two oscillators of the string excited:

a 4n a 3

n 0 p % t l3

c3 v

#

1

" J

J

l

g 1

1

NJ6 2% 1

trQ

3Zl4ZJ

lRe2inl 6 J

(46)

where we have used cyclicity to keep one of the operators at zero. It is the relative phases

that count.

The eight bosonic oscillators of the string are supplied by the four neutral scalarsunder J and by the four directions in which we can take covariant derivatives. The

fermions of the string are made by putting fermionic operators on the trace.

4.5.1. The perturbative calculation

We want to calculate 2p

J for these states. It is clear that bosonic states with

n 0 are protected by SUSY. For these states

J ndefects (47)

The ones with ns

0 acquire an anomalous dimension: this is the quantum contribution

to , J is an angular momentum so it is always quantized and fixed by the free field

theory value. It is also clear that the planar perturbation theory will not notice defectsthat are far away from each other, so the contribution to the dimension of the operatoris a sum over individual defects (dilute gas approximation). The dimension is usually

split into the free field theory part, plus the anomalous dimension . What we need to


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calculate is the anomalous dimension of these operators. This is a standard field theoryproblem.

Concentrating on one defect we have

osc gN cos 2n

J

a

gN

n2

J2

(48)

and it has to vanish for n 0 because that operator is protected. This helps to fix

a 1, and one finds agreement with the first term in the expansion of the light cone

Hamiltonian of the string; we reproduce the full set of oscillator energies to first order inthe t Hooft coupling. All of this follows because we have already identified (roughly)

the states that diagonalize the perturbation [28]. A fairly extensive review is found in

[29].

4.6. Spin chains and integrability.

We will now generalize some of the ideas presented previously to more general

operators. It is easy to realize that the full list of single trace operators made out of

only scalars can be represented by the degrees of freedom of a spin chain. The idea is

that we can at first ignore the cyclic property of the trace and consider instead a wordmade out of the scalar fields by using matrix multiplication with free indices at the end.

The idea is that for each possible field i we can consider it as a possible spin state ofthe spin chain at the position along the chain that the letter i arises.

This is, we make a correspondence

i

i

(49)

letter by letter along the chain. We map for example

1312

1312

(50)

In the free field theory level any two words made out of the same letters in differentorder are degenerate, and correspond to essentially orthogonal states (up to the cyclic

property of the trace). This is a property of the large N limit of free field theory. The

states132

and

123

are different.

The spectrum of anomalous dimensions is a problem in degenerate perturbation

theory. At first order this leaves the length of the trace invariant because states with

different length have different scaling dimension.

The fact that we restrict to planar diagrams means that to first order the only thing that

can happen is that we get a nearest neighbor interaction that can change the values of

the spin labels at each point. Again, this is because going further away from the initial

configuration costs us in non-planarity, which needs to be compensated by interactions.

The spectrum of anomalous dimensions gives rise to a nearest neighbor Hamiltonianfor the spin chain. This spin chain for scalars has SO 6 symmetry, which simplifies

the Hamiltonian considerably. The space of two nearest neighbor pairs decomposes into


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three different irreps ofSO 6 SU 4

1

(51)

The spin chain hamiltonian assigns a different value to each irrep. The last one can be

seen to be zero, as the BPS states belong to it. The computation can be done explicitlyand it was found that the spin chain Hamiltonian in question is integrable [30]. One can

also have spin chains with SO6

symmetry of the above form which are non-integrable,

so this is a non-trivial fact.

The integrability of the spin chain means the eigenvalues can be found analyticallyin implicit form by using the Bethe ansatz technique. Cases with just two impurities

in the scalar sector can be solved exactly for any length of the chain. The result has

been generalized to all possible single trace operators and it was found that there is an

integrable spin chain with SU2

24

symmetry describing the full one loop calculation

[31].

From the strings on AdS5 0 S5 one also finds that the sigma model action is classically

integrable BPR.It has been argued that both of these integrability structures are associated to the

same type of symmetry (a Yangian symmetry). It is conjectured that one can connect

weak coupling perturbation theory with the strong coupling string theory by a family of

integrable systems, so that string theory on AdS5 0 S5 should be solvable!: one should

be able to resum the planar perturbation expansion of SYM to obtain the string sigma

model [33]. This is very hard.

4.6.1. More recent results

Instead of proving integrability to all orders, one can try to match relatively simplestates on both descriptions, and hope that the simplicity is carried along from weak to

strong coupling.

On the string sigma model one can look for semiclassical strings with large quantum

numbers, similar to the large quantum numbers on the plane wave limit. From the

spin chain, one conjectures that these states can be described by the thermodynamic

Bethe ansatz. The agreement to various orders in perturbation theory is spectacular and

very non-trivial. A recent review of these results can be found in [34]. There are also

conjectures as to how the resummation looks like to all orders in perturbation theory forsome sectors of the theory [35].

This ends the description of the AdS/CFT correspondence for these lectures. In thenext lecture, we will move to systems with less supersymmetry.


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FIGURE 7. A D-brane wrapping a cycle

5. LECTURE 4

So far we have dealt with flat D3-branes in flat ten dimensional space and their lowenergy dynamics. We have seen that this system is very rich and shows us how to obtain

string dynamics from gauge theory and gives new insight into what a theory of quantum

gravity could look like. However, the amount of supersymmetry is too large, so it is also

important to understand less supersymmetric systems that can arise from branes. Part of

the motivation is to make contact with phenomenology of particle physics.In order to do that we need to consider higher dimensional branes (let us say D5

branes) which are wrapping some compact geometrical cycle C or set of cycles, so that

their low energy effective field theory is a four dimensional supersymmetric field theory.

We need to wrap a compact cycle because the effective four dimensional coupling

constant is given by a dimensional reduction formula on the world volume of the brane

1

g24YM

1

gs

CdVol

(52)

For what we will be studying, supersymmetric cycles can be characterized by holo-

morphic curves or surfaces on a Calabi-Yau (CY) geometry. We will consider type IIBstrings compactified on a CY manifold, and branes wrapping some of its holomorphic

cycles and spacetime in the transverse directions to the brane.

Since this system has less supersymmetry, there are fewer constraints on the form of

the Lagrangian from the four dimensional point of view. Moreover, the system is not

as protected from receiving quantum corrections to the effective action. In practicalterms this means that it is a lot harder to make exact statements about the system.

However, there are some quantities whose origin is essentially topological which canstill be calculated exactly because there is still some amount of non-renormalization due

to supersymmetry.


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To study these effects, we do not need to know the specific metric and geometry tounderstand some details of the low energy supersymmetric dynamics. For example, the

number of massless degrees of freedom are controlled by topological aspects of the

geometry, a set of N branes wrapping a cycle will still lead to a UN

supersymmetric

gauge theory.

If the cycles are rigid (one can not deform them and keep the cycle as a holomorphic

cycle), then there are no transverse motions of the brane in the higher dimensions which

cost no energy. This means that in the low energy effective field theory all we would get

is pure SYM theory.Considering more cycles and branes wrapping them one can get more general theo-

ries. Also, if there are some transverse motions that are allowed, these lead to massless

matter fields in the adjoint representation of the group, one for every holomorphic de-

formation.

Our interest will be in studying the low energy dynamics of these classes of theories.

The point we will take in these lectures is that the geometry of the embedding can give

us a lot of information about the low energy dynamics, and in the end a lot of the vacuum

structure can be obtained from understanding the geometry better.To separate the problem of gravity from the field theory, we can consider geometries

where the corresponding cycles we wrap are small, and the volume of the CY is very

large. Under these circumstances most of the CY geometry does not matter, and onlythe geometry very close to the brane is important. We can thus take a large volume CY,even infinite.

This second option is done by writing a simple non-compact CY geometry and

doing the analysis there. A second advantage is that the four dimensional Planck

scale is sent to infinity in the process, so that gravitational interactions are decoupled

and can be ignored. Thus the low energy degrees of freedom are those of an ordinary

supersymmetric field theory.

Instead of treating the most general case, the best thing to do is to start with an

example, where we know what the low energy field theory on the D-branes is. A

particularly simple case is to begin with an orbifold space 2

2, so that the other

directions form a six-dimensional flat geometry. This space is the identification of flat

2 under x

y #

x

y

The characterization of this geometry can be done as anequation in three complex coordinates which are the holomorphic invariants of the

2action

uv

w2 0

(53)

Here u x2, v y2, w xy. This produces a singular space, where the singularity is at

u v w 0 (the fixed point of the

2 action), which is resolved by blowing-up the

singularity: replace the singular points by a P1 S2.

If we wrap a D5-brane on the sphere, we obtain a theory which in six dimensions

would have N 1 SUSY, so that in four dimensions it has N 2 supersymmetries.From here we know its Lagrangian at weak coupling.

For M branes it consists of a U

M

gauge theory (vector multiplet) with additionalscalars in the adjoint (one chiral superfield), we call this field . represents the positionof the D5-branes in the transverse directions to

2

2 and the worldvolume of the D5-

branes themselves, and there is also no superpotential.


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The vacuum structure of this theory is exactly solvable due to the fact that we haveN 2 SUSY. The explicit solution was found in the work of Seiberg and Witten [36].

The theory is solved by understanding the geometry of a Riemann surface associated to

the gauge theory.

Now take the N 2 theory and deform it slightly to N 1 SUSY, by adding a

superpotential term to . This can be done as a perturbation of the N 2 system.The superpotential will be polynomial and given explicitly by

W

tr

k%

1i

g1

aiii

(54)

Classically, for vacuum configurations to survive, all the eigenvalues of have to sit onthe zeroes of the polynomial

W

0 P

(55)

so now the D5-branes have some potential on the transverse directions to the orbifold.

This deformation is associated to a change of the complex structure of the transverse

directions to the D5-brane. The new geometry is given by [37]

uv w2

Pz

2

(56)

The D5-branes can only be placed at the singularities of the above geometry: the

zeroes of P

z, otherwise they are not supersymmetric. The reason for this is that we

have to remember that the true geometry results from blowing up the original singular

locus. It is only at the singularities of the geometry above that the blow-up gives

rise to a holomorphic S2. Generically the field is massive, unless we fine-tune thesuperpotential. The structure of the theory in the deep IR reduces to a product of pure

SYM theory, with a gauge group U Ni . In this setup there are Ni eigenvalues ofat each root ofP, each eigenvalue is interpreted as a D5-brane. We have good reason to

believe that pure N 1 SYM theory with group U Ni has Ni vacua [38].

The order parameter characterizing these vacua is the vacuum expectation value

(v.e.v.) of the gaugino condensate

trWW

3

(57)

which is another element of the chiral ring. This information is holomorphic, so it is

natural that is should be measurable in the geometry. In practical terms this means the

equation should be deformed [37]

uv w2

Pz

2

fz

(58)

Because we can vary the coefficients ofP by varying the superpotential deformation, thef associated to this process should be of lesser degree than P. One can see that P has

degree k, so f has degree k 1, and it is characterized by kcoefficients (one for each rootof P). These should be measuring the partial gaugino condensates for each singularity.

In practice, there is one complex parameter per gauge group. Roughly, this takes into

account the possible phases of the gaugino condensate associated to each singularity.


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The effective superpotential of the gaugino condensate is (for an orientable theory)

Weff

Ni

0

Si

(61)

and this is a function of the partial gaugino condensates. One then takes

W

S 0

(62)

to get to the vacuum structure, where one also includes a term NiSi log Si for each partialgaugino condensate.

5.2. Two proofs of the Dijkgraaf-Vafa conjecture

There are two proofs of the above result which arise from a completely different point

of view. In the first proof [41], one does a perturbative calculation in the presence of a

classical gaugino background.

After some superspace manipulations, one can integrate out the antichiral superfields,

and one finds the that the propagator for chiral fields looks as follows

d4pd2m

p2&

m2&

W

(63)

where m is the mass of the particle, and p

are superspace momenta, while W isthe covariant chiral super-field strength for the gauge field and gaugino fields. This

propagator requires m s 0, and does not apply for a chiral theory where there are no

bare mass terms allowed in the Lagrangian.

After using all the delta functions of the vertices on an L loop integral, and countingpowers of, one gets a total measure d2L

% 2. A local term in the action will have ameasure which is d2, so we need 2L powers of to get a non zero result. This translatesinto 2L powers ofW. Also there can not be more powers ofW than two per color loop

(numbers of faces) for the result to be chiral. This is a result of some classical superfieldmanipulations.If we use the t Hooft double line notation (fat graphs), the number of faces of the

double line graph is bounded from below F

L, which singles out only planar diagrams

(for orientable surfaces).

Some work with Schwinger parameters shows that the momentum dependence disap-

pears from the diagram, and one is left with the planar amplitude of the matrix model:

the difference is that instead of N in the loops, one has S tr W2 from the gaugino

insertions. There is also one loop which has no W. This loop contributes the factor ofNi from tr 1 . After some combinatorics, one can see that the end result is given by

summing derivatives of the prepotential.

There is a second proof of the result which is more technical, and was done in the work[42]. They study the chiral ring of the supersymmetric field theory in a lot of detail. They

found that the chiral ring is generated algebraically by

tr k

tr kW

tr kW2

(64)


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The correlation functions of chiral fields in supersymmetric backgrounds are indepen-dent of position. Together with the cluster decomposition principle this implies that the

correlation functions are just the product of v.e.v.s of the individual chiral fields on the

vacuum. This is why these form a ring: multiplication of two chiral fields produces a

chiral field.

The idea now is that if there is some relation to matrix models, then the observables of

the matrix model should be correlated to the chiral superfields, and one can try to imitate

the proof of the loop equations for the matrix model in the supersymmetric field theory.

The loop equations of the matrix model can then be interpreted as relations in the chiralring of the field theory.

The proof goes by studying a generalized Konishi Anomaly for the (Virasoro con-

straint) variations

n

nWW

(65)

The idea is that a variation of the fields is generated by a supercurrent, so one has theconservation law for the associated current

D2

J

W

& gauge anomaly

(66)

There are two sources for chiral violation of the supercurrent conservation. The first is

the fact that the superpotential depends on the fields , so the current is not a symmetryof the theory, and there is explicit violation from the action. The second term is due to

a mixed anomaly with the gauge superfields. On taking v.e.v.s on a supersymmetric

background the left hand side vanishes, and the right hand side produces a set of

polynomial relations in the chiral ring.

Some of these coincide exactly with the planar loop equations of the matrix model

associated to the superpotential W. Integrating these equations one obtains the effective

superpotential again. The advantage of this second proof is that one also obtains a fulldictionary between matrix model observables and the chiral ring. This permits one to

also write down formal operators which correspond to the partial gaugino condensates,so these fields which arise from low energy considerations after a Higgs mechanism hasbeen taken into account can be given a gauge invariant realization in the UV theory.

5.3. Brane-flux transitions

Now, let us return to the problem we started this lecture with. As we saw, the geomet-

ric realization of gaugino condensation led to a deformation of the singularities of the

complex structure to obtain a new smooth geometry. One of the things one can notice is

that once we accept that the singularities of the curve (they are also singularities of the

higher dimensional geometry) are deformed away the topology of the CY geometry haschanged. Each of these singularities we have seen (double roots) is of the conifold type.

The local topology is a cone over S20

S3. The blow-up process fills the S3, so that the

topology is globally S20

P

4 , while the deformation process fills the S2 to give P 30

S3.


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We have seen that the deformation process follows just from field theory alone, anddoes not require us to think in terms of strings in a CY geometry. This is why I spent

some time describing how the proofs of the Dijkgraaf-Vafa were argued. However, the

geometrical picture can help us to explain some interesting features of the field theory.

The geometrical picture of gaugino condensation process is that we have to make

a change of topology. In topological gravity this was stated earlier in [43], and was

shown with a supersymmetric field theory calculation in [44]. In the new topology there

is no compact holomorphic cycle in which to wrap the D-branes, so one does not see

the sources of the generalized fluxes (these are the D-branes). However, the change intopology allows one to have the flux without a source.

From the point of view of looking at the system from asymptotic infinity both of these

pictures lead to the same boundary conditions, but the topological structure inside has

changed. This operation can be called a brane-flux transition.

It replaces D-branes by fluxes. The flux through the corresponding three cycle S3 is

equal to the number of original branes we placed in the geometry. This is very similar

to our discussion of the AdS5 0 S5 geometry, where the remnant of the fact that we have

D-branes is that the flux through the S5 is quantized.

Another important thing to notice is that since we have lost the D-branes, we do not

have a place for open strings to end anymore. This is interpreted as confinement in the

gauge theory, since we do not see the charges at the end of the strings anywhere. One alsosees that in the deformed geometry all of the degrees of freedom have become closed

strings. These include the gauge invariant "composites" of the D-brane themselves, these

states are like glueballs in QCD. This is also similar to the description of AdS5 0 S5.

However, there is a lot less data on the spectrum of these strings.

5.4. Moduli stabilization and racetracks

So far we have discussedN 1 geometries where there are no moduli fields. However,

there are other phenomena in supersymmetric field theories that one would want to

understand. In particular, there can be non-generic situations which lead to families ofsupersymmetric vacua in the IR field theory. These are not lifted by perturbation theory

and lead to trouble for understanding certain cosmological issues.

Systems like these can be obtained by considering a slightly more singular situation:a curve of singularities rather conifold points, or point-like D-branes in a CY. These two

situations are different. In the case of the curve of singularities one can also blow-up

these singularities, so that a brane wrapping some compact cycle can now also move

freely along the curve. This means that there is a degenerate set of vacua. In SUSY

theories this is called the moduli space of vacua, and the massless modes that connect

the vacua are moduli fields.

Now let us assume that this curve of singularities has some singularities itself. At

these places there are more than one different type of 2-cycle that a D-brane can wrap.

Placing branes at these extra cycles produces some amount of confinement, and as we

argued above, leads to a change of the topology and geometry.

The branes that can move along the curve of singularities feel the effects of this


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confinement and one usually generates an effective superpotential for these branes,see for example [45]. The upshot is that gaugino condensation produces and effective

superpotential for some flat directions that lifts the degeneracy of vacua, giving only

a discrete set of solutions. Geometrically, the effect of the gaugino condensation is to

deform the geometry, and this usually lifts the curve of singularities to something less

singular. This is, one finds that instead of the full curve of singularities one only has a

finite number of conifold points left over.

In the simplest cases, one obtains runaway behavior: the saddle point of the effective

superpotential happens at infinity [46]. Usually one needs more than one extra singular-ity on the curve. Thus one makes two different gaugino condensation effects competeagainst each other. This is called a racetrack mechanism. This fixes some of the moduli

problems. For branes in the bulk, this does not seem to happen [47], and that one gets

instead is just the deformed geometry.

From the flux-geometry point of view, the fluxes produce effective potentials for

geometrical moduli that describe the shape of the local Calabi-Yau geometry. Thus the

shape of the CY can be fixed by the fluxes.

6. LECTURE 5

This is the last part of the lecture notes, where I try to discuss advances in N 0 SUSY

(supersymmetry breaking or no supersymmetry at all). Most of this discussion is on amuch less rigorous footing than then previous lectures, but we will use extensively the

intuition we have gotten from the more supersymmetric cases.

6.1. Warping

We want to begin again with AdS5 0 S5 as a geometric template for four dimensional

field theories. The first description I gave in terms of flat D-branes shows that we have

Lorentz invariance along the four longitudinal directions to the brane, with a metric ofthe formds2 dx27 7 f

16

2r

&dr2

&

(67)

The fact that de coefficient of dx27 7 varies with r is called a warp factor. The topology is

still a product, but the metric is not a sum of terms for each factor independent of the

other one. The observation that this theory can be interpreted as ordinary field theory

without gravity when f is 1

r4 motivated the construction of the Randall-Sundrum

scenario for solving the hierarchy problem [48].

The reason in the above geometry there is no four dimensional Newton constant is

because such a constant mode deformation of the metric is non-normalizable. If we

add a little geometric (UV) cutoff to the story, we can recover the constant mode for

gravity, but it is weakly coupled. This would produce a hierarchy between the four andfive dimensional gravitational constant from warping. We also have this redshift factor

which can reduce the mass of objects from the gravitational point of view. How strongly

you feel 4D gravity depends on where you are.


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This ingredient shows up very often in string compactifications with fluxes [49],although in general we do not know how to write the metrics with very much detail

at all.

6.2. The most general SUSY background

We have seen that there is a lot of interesting physics to be obtained from placing

branes in a geometry. We believe this is a very general construction, and might accountfor most supersymmetric vacua in string compactifications. The emphasis on most is

because this is the largest class of backgrounds that we know of where we believe

we can control the physics. Via string dualities we can connect these backgrounds to

other constructions in heterotic strings, so we dont seem to be losing generality byconsidering these backgrounds instead of others.

As we argued from the Dijkgraaf-Vafa approach, strong coupling effects produce

brane-flux transitions, so that a good way too understand the low energy effective field

theory is to study compactifications with fluxes. The fluxes can in principle fix all of the

moduli, and usually have a non-zero value for the total superpotential of the theory. This

implies when coupling to gravity that the cosmological constant will be negative.

The geometry also captures all of the order parameters for the gaugino condensatephases and values. If we consider very generic theories, we can expect that there are a

lot of solutions of the equations which correspond to these different phases of the field

theory.

A generic CY geometry can have a very large number of cycles, lets say 200. Placing

a modest number of branes on each such cycle and counting vacua can lead easily

to numbers of the order of 10200 just by multiplying a factor of 10 for each cycle.

Estimates based on physical considerations can be found in the work of Douglas [50, 51].

Moreover, there are over 10

000 CY three-fold geometries, and it has not been proventhat there is not an infinitude of these. Add to that the ability to change the quantization

of fluxes through the cycles, and the numbers become a lot larger.

From the point of view of string theory this is a huge number of supersymmetricvacua with negative cosmological constant. Usually the bigger the fluxes, the smaller

the cosmological constant in four dimensions. This is just like AdS5 0

S5. This means

that we can get very small numbers for the absolute value of the cosmological constant,

but the wrong sign. Current observations indicate that we are accelerating away from

each other.

This can only be accomplished with SUSY breaking, because one can have a super-

symmetry algebra for negative cosmological constant, but not for a positive one. So the

question is how do we do this and retain control of the theory?

We already have most of the ingredients in place. We just need some way to break

supersymmetry:

Start with a SUSY model: branes on a CY geometry. Do brane-flux transitions (these capture non-perturbative effects) and one can con-

struct models which solve the moduli problem. There can still be some branes in

the bulk and that lead to the SSM or something similar.


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Assume that the regime one is studying is geometrical, so that the solution cantolerate geometric reasoning. This produces one value of the cosmological constant

of the wrong sign.

6.2.1. The KKLT scenario

There is one ingredient missing which I have not discussed so far.

Branes are sources for fluxes, a generalized form of electromagnetism. On a compactgeometry the total charge for these fluxes to be consistent must be zero. There are

contributions from topology, D-branes and the fluxes themselves. In the discussion above

we might have not cancelled the D3-brane charge, and it can be the case that we are

missing an anti-brane.

We now add it by hand to fix this problem and see what happens. The description ofthe above scenario is due to Kachru, Kallosh, Lind

david berenstein- supersymmetry: a string theory point of view

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