McMahon D. String Theory Demystified

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256

String Theory Demystiﬁ ed

A Statement of the Holographic Principle

The holographic principle was ﬁ rst proposed by Gerard t’Hooft in 1993 and has

been worked on extensively by Leonard Susskind. It can be asserted using two

postulates:

• The total information content in a volume of space is equivalent to a theory

that lives only on the surface area that encloses the region.

• The boundary of a region of space-time contains at most a single degree of

freedom per Planck area.

The holographic principle really applies to gravity and we have already seen it in

action when talking about black holes. Information content, which is another way

of saying entropy, is about counting the number of states in a system and so is

proportional to area. We have already seen that in the case of a black hole that

entropy is proportional to the area of the event horizon:

where G is Newton’s gravitational constant. The area A is measured in Planck

units.

This is a surprising result because we would intuitively expect that the number of

states is proportional to the volume of the enclosed region. Following Susskind, we

illustrate that this is in fact the case when gravity is not involved. Imagine that a

volume V contains a set of spins on a lattice. We take the lattice spacing to be a, and

imagine that the lattice ﬁ lls the entire volume. Then the total number of spins

contained in V is

# spins =

The total number of states the system can have is

= 2

Using thermodynamics, we arrive at a relationship between the number of states

and entropy S:

NS∝ exp

CHAPTER 15 The Holographic Principle

257

Hence we ﬁ nd that 2

exp= , or taking the logarithm of both sides:

==ln( ) ln

We’ve found what we intuitively expect—the entropy (and by extension the

amount of information) in the region is proportional to the volume. After all we

started off assuming we had a lattice of spins that ﬁ lled the volume—so what else

could we get?

For black holes we found something very different. In that case, the entropy is

directly proportional to the area of the even horizon. So in some sense, gravity must

be different from other interactions. It turns out that the case of a black hole provides

the maximum entropy that a gravitational system can have.

A Qualitative Description of

AdS/CFT Correspondence

The framework of the holographic principle which comes out of string/M-theory is

known as AdS/CFT (anti-de Sitter/conformal ﬁ eld theory) correspondence. We can

quantitatively describe the space-time using AdS space in ﬁ ve dimensions. The

ﬁ ve-dimensional AdS model has a boundary with four dimensions that looks like

ﬂ at space with three spatial directions and one time dimension.

The AdS/CFT correspondence involves a duality, something we’re already

familiar with from our studies of superstring theories. This duality is between two

types of theories:

• Five-dimensional gravity

• Super Yang-Mills theory deﬁ ned on the boundary

By “super” Yang-Mills theory we mean theory of particle interactions with

supersymmetry. The holographic principle comes out of the correspondence between

these two theories because Yang-Mills theory, which is happening on the boundary, is

equivalent to the gravitational physics happening in the ﬁ ve-dimensional AdS geometry.

So the Yang-Mills theory can be colloquially thought of as a hologram on the boundary

of the real ﬁ ve-dimensional space where the ﬁ ve-dimensional gravitational physics is

taking place.

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String Theory Demystiﬁ ed

The Holographic Principle and M-Theory

Now let’s make our description more quantitative. In the ﬁ nal chapter of the book

we discuss stringy cosmology. There we will encounter a model of space-time that

has sprung out of string/M-theory that might in fact describe our actual universe.

That same model has a nice application in the topic of this chapter as well. The

model is a ﬁ ve-dimensional AdS space. It can be described as follows.

We start with a ﬁ ve-dimensional AdS space. In a nutshell, this is a four-

dimensional spatial ball and an inﬁ nite time axis. The radius of the ball is

01≤<r

The radius of curvature is denoted by R, and we lump the remaining spatial

dimensions together into a unit three-sphere denoted by Ω. The metric which

describes the AdS is written as

rdt dr rd

22 2 2 2

144=

−

+−−

()

[( ) ]Ω

Note that there are different, equivalent ways to write this metric which you might

encounter elsewhere. AdS space has negative curvature and acts like a cavity of size

R with reﬂ ecting walls. Light or objects and reﬂ ect off the boundary and return to the

center (see “The Illusion of Gravity” by Juan Maldacena in Scientiﬁ c American,

November 2005, for a nice popular level description of AdS).

For us, we are interested in superstring theory. The number of space-time

dimensions in superstring theory is D = 10. So the complete space is

AdS S⊗

where S

is a unit ﬁ ve-sphere containing the remaining dimensions from string theory.

If we denote the extra ﬁ ve coordinates by y

they are incorporated into our metric by

adding a term

Rdy

. We can imagine compactifying these dimensions to a very small

size so that they can be effectively ignored. So the universe can be effectively treated

as the ﬁ ve-dimensional “bulk” which is the interior of the sphere and the boundary

which is the surface. The surface has three spatial dimensions and time.

In the M-theory picture, the world we know is in essence a “shadow” or hologram

living on the boundary of a larger dimensional universe. The physics is divided as

follows:

• The boundary conformal theory lives on the surface of the sphere at x = 1.

These are the particles and interactions of the standard model, plus any

supersymmetric extension of it.

• Gravity is everywhere.

CHAPTER 15 The Holographic Principle

259

• But, gravity can propagate into the bulk. In the bulk, which is the interior

volume of the AdS sphere, gravity is the only interaction. Inside the ball of the

AdS geometry, the theory is supergravity. We won’t get into supergravity in

this book but you can look it up on the arXiv if interested in learning about it.

The conformal theory that describes particles and their interactions is

supersymmetric and is called super Yang-Mills theory or SYM for short. The

gauge group for SYM is SU(N). So the AdS/CFT correspondence can be framed

as follows:

• There is a super Yang-Mills theory with SU(N) on the surface of the ball.

• There is bulk supergravity in the interior of the ball.

In string theory, the number of degrees of freedom for the SYM is constrained by

three factors:

• The fundamental string length

• The string coupling

• The curvature of AdS space

The number of degrees of freedom for SYM is ∼ N

since the gauge group is

SU(N) and it has a gauge coupling g

. The constraint on N is quantiﬁ ed in the

following relationship:

RgN

=  ()

/14

The gauge-coupling is related to the string-coupling constant as:

YM s

Now we would like to introduce a cutoff in the bulk. We divide up the sphere into

little cells such that the total number of cells in the sphere is

∼

−3

for some cell d.

That is,

• We cut off the information storage capacity by replacing the continuum of

space by cells of size d.

• There is a single degree of freedom in each cell.

With the total number of degrees of freedom for the SYM theory proportional to

, we ﬁ nd that the total number of degrees of freedom with the cutoff is

dof

260

String Theory Demystiﬁ ed

Now since

RgN

=  ()

/14

we can write

dof



In ﬁ ve-dimensions, the Newton gravitational constant is



Hence we ﬁ nd that

dof

This agrees with the holographic principle, and is the same as the result obtained

for black holes with the exception of the factor of 1/4.

More Correspondence

In this section we describe connections between the supergravity theory of the bulk

and the SYM of the boundary. We can convert between bulk variables and SYM

variables as follows. Let E

SYM

be energy on the boundary and M be the energy in the

bulk. They are related as

ERM

SYM

Temperature is related in the same way:

TRT

SYM

where T is the temperature in the bulk. Now consider a thermal Yang-Mills state

with temperature T

SYM

. The entropy is

SNT

SYM

()

A thermal state of temperature T

SYM

corresponds to an AdS Schwarzschild black

hole at the center of the AdS ball.

CHAPTER 15 The Holographic Principle

261

Using

SYM

and

RgN

=  ()

/14

we obtain

()TR



Now if we take S = A/4G then we ﬁ nd

SYM

Now we regulate the SYM so that the maximum T

SYM

is 1/d. Then we ﬁ nd the

maximum area to be

max

Regulation of the super Yang-Mills theory on the boundary gives a holographic

description with one bit per Planck area.

An interesting result derived by Susskind and Witten is the IR-UV connection.

This relates IR divergences in the bulk to UV divergences on the boundary. Consider

a string in the bulk that ends on the boundary. The ends of the string correspond to

a point charge in the Yang-Mills theory. Now, just thinking back to the self-energy

of an electron, you will realize that a point charge in the Yang-Mills theory has a

divergent inﬁ nite self-energy. This is an UV divergence. The divergence of the bulk

string is proportional to 1/d, while d plays the role of a short distance regulator for

UV divergence in SYM theory.

The energy of the string is linearly divergent at the boundaries. Since this

divergence is softer, we say that it is an IR divergence. The propagator for a particle

of mass m in the bulk is given by

∆=

−

where we have relgulated the area using

AR≈

and



. Super Yang-Mills

theory is a conformal ﬁ eld theory. Remember Chap. 5? We learned how to calculate

operator product expansions there. For super Yang-Mills theory:

YX YX X X

()()

12 12

=−

−

You can see that you can transform between these two expressions. What this

means is that a propagator for a particle of mass m in the bulk can be transformed

into a power law in the conformal ﬁ eld theory on the boundary.

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String Theory Demystiﬁ ed

Summary

In this chapter we provided a brief and heuristic introduction to two interesting

ideas that have sprung from string theory: the holographic principle and the AdS/

CFT correspondence. These two ideas are related. The holographic principle tells

us that for an enclosed volume, the informational content of the volume can be

described by an equivalent theory that lives on the bounding surface area. This

notion is codiﬁ ed in black hole mechanics where the entropy of the black hole is

proportional to the area of the horizon, not the volume it encloses. The AdS/CFT

correspondence describes a ﬁ ve-dimensional universe where ﬁ ve-dimensional

supergravity in the bulk is equivalent to a super Yang-Mills conformal ﬁ eld theory

on the boundary.

Quiz

A solution of supergravity gives the metric for a D-brane as:

ds F z dt dx F z dz

22212

=−−

−

()( ) ()

where

ag N

()

⎛

⎝

⎜

⎞

⎠

⎟

−

1. Find an expression for F(z) in the limit

ag N

1

2. Using your answer to Prob. 1, ﬁ nd a new expression for the metric.

3. The holographic principle can be best described by

(a) The informational content of a region is encoded in its volume.

(b) The informational content of a region can be described entirely by the

surface area.

bounding surface.

4. In AdS/CFT correspondence, the number of degrees of freedom available

to the super Yang-Mills theory on the boundary is

(a) Independent of the AdS geometry.

(b) Related to the string coupling strength only.

(d) Is related to the fundamental string length only.

CHAPTER 15 The Holographic Principle

263

5. In AdS/CFT correspondence, the number of degrees of freedom is

(a) Proportional to the area of the bounding surface and to Newton’s

gravitational constant.

(b) Proportional to the area of the bounding surface and inversely

proportional to Newton’s gravitational constant.

string length.

(d) Proportional to the area of the bounding surface and the string coupling

constant.

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String Theory and

Cosmology

Conventional cosmology, which grew out of general relativity, astrophysics, and

quantum ﬁ eld theory, proposes that the universe began with a “big bang” at a ﬁ nite

time in the past with an inﬂ ationary rush, and will expand forever until the universe

dies with a whimper, as a result of increasing entropy eventually sapping the useful

life out of it. Proposals which originated in string/M-theory have led to different

cosmological models. These models have the unexpected and shocking ability to

describe the universe before the big bang. Based on a brane-world-type universe,

they involve the collision of two branes which get rid of the “singularity” of big-

bang theory and replace it with an eternal universe, which could be described as

“cyclic.” In this chapter, we give an overview of some of the cosmological models

that have arisen from string/M-theory. Unfortunately, the details of these models

using string/M-theory are well beyond the scope of this book, so our description

will be more of a qualitative nature. The motivated reader is urged to consult the

references for details. Cosmology is sure to be an active area of research in the

coming years with many new and possibly unexpected developments.

CHAPTER 16