To them, I said,

the truth would be literally nothing

but the shadows of the images.

*-Plato, The Republic (Book VII)*

the truth would be literally nothing

but the shadows of the images.

Plato, the great Greek philosopher, wrote a series of `Dialogues' which summarized many of the things which he had learned from his teacher, who was the philosopher Socrates. One of the most famous of these Dialogues is the `Allegory of the Cave'. In this allegory, people are chained in a cave so that they can only see the shadows which are cast on the walls of the cave by a fire. To these people, the shadows represent the totality of their existence - it is impossible for them to imagine a reality which consists of anything other than the fuzzy shadows on the wall.

However, some prisoners may escape from the cave; they may go out into the light of the sun and behold true reality. When they try to go back into the cave and tell the other captives the truth, they are mocked as madmen.

Of course, to Plato this story was just meant to symbolize
mankind's struggle to reach enlightenment and understanding
through reasoning and open-mindedness. We are *all*
initially prisoners and the tangible world is our cave.
Just as some prisoners may escape out into the sun, so may
some people amass knowledge and ascend into the light of
true reality.

What is equally interesting is the *literal*
interpretation of Plato's tale: The idea that reality could
be represented completely as `shadows' on the walls.

In 1993 the famous Dutch theoretical physicist G. 't Hooft
put forward a bold proposal which is reminiscent
of Plato's Allegory of the Cave. This proposal, which
is known as the *Holographic Principle*, consists of
two basic assertions:

**Assertion 1**
The first assertion of the Holographic Principle is that
all of the information contained in some region of space
can be represented as a `Hologram' - a theory which `lives'
on the boundary of that region. For example, if the region
of space in question is the DAMTP Tearoom, then the holographic
principle asserts that all of the physics which takes place
in the DAMTP Tearoom can be represented by a theory which
is defined on the walls of the Tearoom.

**Assertion 2** The second assertion of the
Holographic Principle is that the theory on the boundary of
the region of space in question should contain at *most*
one degree of freedom per Planck area.

A *Planck area* is the area enclosed by a little
square which has side length equal to the Planck length,
a basic unit of length which is usually denoted *L*_{p}.
The Planck length is a fundamental unit of length, because
it is the parameter with the dimensions of length which can
be constructed out of the basic constants *G* (Newton's
constant for the strength of gravitational interactions),
(Planck's constant from quantum mechanics),
and *c* (the speed of light). A quick calculation reveals
that *L*_{p} is very small indeed:

To many people, the Holographic Principle seems strange and counterintuitive: How could all of the physics which takes place in a given room be equivalent to some physics defined on the walls of the room? Could all of the information contained in your body actually be represented by your `shadow'?

In fact, the way in which the Holographic Principle appears in M-theory
is much more subtle. In M-theory we *are* the shadows on the wall.
The `room' is some larger, five-dimensional spacetime and our four-dimensional
world is just the boundary of this larger space. If we try to move away from
the wall, we are moving into an extra dimension of space - a fifth dimension.
In fact, people have recently been trying to think of ways in which we might
actually experimentally `probe' this fifth dimension.

At the heart of many of these exciting ideas is a version of the Holographic Principle known as the adS/CFT correspondence.

The adS/CFT correspondence is a type of duality, which states that two
apparently distinct physical theories are actually equivalent. On one side of
this duality is the physics of gravity in a spacetime known as
*anti-de Sitter* space (adS). Five-dimensional anti-de Sitter space
has a boundary which is four-dimensional, and in a certain limit looks like
flat spacetime with one time and three space directions. The adS/CFT correspondence
states that the physics of gravity in five-dimensional anti-de Sitter space,
is equivalent to a certain supersymmetric Yang-Mills theory which is defined
on the boundary of adS. This Yang-Mills theory is thus a `hologram' of the
physics which is happening in five dimensions.
The Yang-Mills theory has gauge group *SU*(*N*),
where *N* is very large, and it is said to be `supersymmetric' because it has
a symmetry which allows you to exchange bosons and fermions.
The hope is that this theory will eventually teach
us something about QCD (quantum chromodynamics), which is a
gauge theory with gauge group *SU*(3). QCD describes interactions between
quarks. However, QCD has much less symmetry than the theory defined on the
boundary of adS; for example, QCD has no supersymmetry.
Furthermore, we still don't know how to incorporate a crucial property of
QCD, known as *asymptotic freedom*.

Here in DAMTP, we have been working to see if the adS/CFT correspondence can be generalized. Working with collaborators in such far-flung places as the United States, Canada, and Durham, we have managed to show that the duality is still true even when you replace adS with more complicated five-dimensional spacetimes. In particular, we have calculated what happens when you put electric charge in adS, or rotation in adS, or even what happens when you put a certain exotic charge known as `NUT-charge' into adS.