The Theory Formerly Known as Strings

The Theory of Everything is emerging as one in
which not only
strings but also membranes and black holes play
a role

by Michael J. Duff

At a time when certain pundits are predicting the
End of Science on the grounds that all the
important discoveries have already been made, it is
worth emphasizing that the two main pillars of
20th-century physics, quantum mechanics and
EinsteinÕs general theory of relativity, are
mutually incompatible. General relativity fails to
comply with the quantum rules that govern the
behavior of elementary particles, whereas on the
opposite scale, black holes are challenging the
very foundations of quantum mechanics. Something
big has to give. This predicament augurs less the
bleak future of diminishing returns predicted by
the millennial Jeremiahs and more another
scientific revolution.

Until recently, the best hope for a theory that
would unite
gravity with quantum mechanics and describe all
physical phenomena was based on strings:
one-dimensional objects whose modes of vibration
represent the elementary particles. In the past two
years, however, strings have been subsumed by
M-theory. In the words of the guru of string theory
(and according to Life magazine, the sixth most
influential American baby boomer), Edward Witten of
the Institute for Advanced Study in Princeton,
N.J., "M stands for Magic, Mystery or Membrane,
according to taste." New evidence in favor of this
theory is appearing daily, representing the most
exciting development since strings first swept onto
the scene.

M-theory, like string theory, relies crucially on
the idea of
supersymmetry. Physicists divide particles into two
classes, according to their inherent angular
momentum, or "spin." Supersymmetry requires that
for each known particle having integer spin--0, 1,
2 and so on, measured in quantum units--there is a
particle with the same mass but half-integer spin
(1/2, 3/2, 5/2 and so on), and vice versa.

Unfortunately, no such superpartner has yet been
found. The
symmetry, if it exists at all, must be broken, so
that the postulated particles do not have the same
mass as known ones but instead are too heavy to be
seen in current accelerators. Even so, theorists
have retained belief in supersymmetry primarily
because it provides a framework within which the
weak, electromagnetic and strong forces may be
united with the most elusive force of all: gravity.

Supersymmetry transforms the coordinates of space
and time
such that the laws of physics are the same for all
observers. EinsteinÕs general theory of relativity
derives from this condition, and so supersymmetry
implies gravity. In fact, supersymmetry predicts
"supergravity," in which a particle with a spin of
2--the graviton--transmits gravitational
interactions and has as a partner a gravitino, with
a spin of 3/2.

Conventional gravity does not place any limits on
the
possible dimensions of space-time: its equations
can, in principle, be formulated in any dimension.
Not so with supergravity, which places an upper
limit of 11 on the dimensions of space-time. The
familiar universe, of course, has three dimensions
of space: height, length and breadth, while time
makes up the fourth dimension of space-time. But in
the early 1920s Polish physicist Theodore Kaluza
and Swedish physicist Oskar Klein suggested that
space-time may have a hidden fifth dimension. This
extra dimension would not be infinite, like the
others; instead it would close in on itself,
forming a circle. Around that circle could reside
quantum waves, fitting neatly into a loop. Only
integer numbers of waves can fit around the circle;
each of these would correspond to a particle with a
different energy. So the energies would be
"quantized," or discrete.

An observer living in the other four dimensions,
however,
would see a set of particles with discrete charges,
rather than energies. The quantum, or unit, of
charge would depend on the circleÕs radius. In the
real world as well, electrical charge is quantized,
in units of e, the charge on the electron. To get
the right value for e, the circle would have to be
tiny, about 10-33 centimeter in radius.

The unseen dimensionÕs small size explains why
humans, or even atoms, are unaware of it. Even so,
it would yield electromagnetism, and gravity,
already present in the four-dimensional world,
would be united with that force. In 1978 Eugene
Cremmer, Bernard Julia and Joel Scherk of the Čcole
Normale Supˇrieure in Paris realized that
supergravity not only permits up to seven extra
dimensions but is most elegant when existing in a
space-time of 11 dimensions (10 of space and one of
time). The kind of real, four-dimensional world the
theory ultimately predicts depends on how the extra
dimensions are rolled up, ą la Kaluza and Klein.
The several curled dimensions could conceivably
allow physicists to derive, in addition to
electromagnetism, the strong and weak nuclear
forces. For these reasons, many physicists began to
look to supergravity in 11 dimensions in the hope
that it might be the unified theory.

In 1984, however, 11-dimensional supergravity was
rudely
knocked off its pedestal. An important feature of
the real world is that nature distinguishes between
right and left: the laws governing the weak nuclear
force operate differently when viewed through a
mirror. (For instance, neutrinos always have
left-handed spin.) But as Witten and others
emphasized, such "handedness" cannot readily be
derived by reducing space-time from 11 dimensions
down to four.

P-Branes

Supergravity's position was usurped by superstring
theory in
10 dimensions. Five competing theories held sway,
designated by their mathematical characteristics as
the E8 X E8 heterotic, the SO(32) heterotic, the
SO(32) Type I, the Type IIA and Type IIB strings.
(The Type I is an "open" string consisting of just
a segment, whereas the others are "closed" strings
that form loops.) One string in particular, the E8
X E8, seemed--at least in principle--capable of
explaining the known elementary particles and
forces, including their handedness. And unlike
supergravity, strings seemed to provide a theory of
gravity consistent with quantum effects. All these
virtues enabled string theory to sweep physicists
off their feet and 11-dimensional supergravity into
the doghouse. Murray Gell-Mann of the California
Institute of Technology encapsulated the mood of
the times by declaring at a meeting:

"Eleven-dimensional supergravity--ugh!"

After the initial euphoria over strings, however,
nagging
doubts began to creep in. First, many important
questions--especially how to confront the theory
with experiment--seemed incapable of being answered
by traditional methods of calculation. They called
for radically new techniques. Second, why were
there five different string theories? If one is
looking for a unique Theory of Everything, surely
this is an embarrassment of riches. Third, if
supersymmetry permits 11 dimensions, why do
superstrings stop at 10? Finally, if we are going
to conceive of pointlike particles as strings, why
not as membranes or more generally as p-dimensional
objects--inevitably dubbed p-branes?

Consequently, while most theorists were tucking
into
super-spaghetti, a small but enthusiastic group
were developing an appetite for super-ravioli. A
particle, which has zero dimensions, sweeps out a
one-dimensional trace, or "worldline," as it
evolves in space-time [see illustration]. Similarly
a string--having one dimension, length--sweeps out
a two-dimensional "worldsheet," and a
membrane--having two dimensions, length and
breadth--sweeps out a three-dimensional
"worldvolume." In general, a p-brane sweeps out a
worldvolume of p + 1 dimensions. (Of course, there
must be enough room for the p-brane to move about
in space-time, so p + 1 must not exceed the number
of space-time dimensions.)

As early as 1962, Paul A. M. Dirac, one of the
fathers of
quantum mechanics, had constructed an imaginative
model based on a membrane. He postulated that the
electron, instead of resembling a point, was in
reality a minute bubble, a membrane closed in on
itself. Its oscillations, Dirac suggested, might
generate particles such as the muon, a heavier
version of the electron. Although his attempt
failed, the equations that Dirac postulated for the
membrane are essentially the ones we use today. The
membrane may take the form of a bubble, or it may
be stretched out in two directions like a sheet of
rubber.

Supersymmetry severely restricts the possible
dimensions of a
p-brane. In the space-time of 11 dimensions floats
a membrane, discovered mathematically by Eric
Bergshoeff of the University of Groningen, Ergin
Sezgin, now at Texas A&M University, and Paul K.
Townsend of the University of Cambridge. It has
only two spatial dimensions and looks like a sheet.
Paul S. Howe of KingÕs College London, Takeo Inami
of Kyoto University, Kellogg Stelle of Imperial
College, London, and I were able to show that if
one of the 11 dimensions is a circle, we can wrap
the membrane around it once, pasting the edges
together to form a tube. If the radius of the
circle becomes sufficiently small, the rolled-up
membrane ends up looking like a string in 10
dimensions; in fact, it yields precisely the Type
IIA superstring.

Notwithstanding such results, the membrane
enterprise was
largely ignored by the orthodox string community.
Fortunately, the situation was about to change
because of progress in an apparently unrelated
field.

In 1917 German mathematician Amalie Emmy Noether
had shown
that the mass, charge and other attributes of
elementary particles are generally conserved
because of symmetries. For instance, conservation
of energy follows if one assumes that the laws of
physics remain unchanged with time, or are
symmetric as time passes. And conservation of
electrical charge follows from a symmetry of a
particleÕs wave function.

Sometimes, however, attributes may be maintained
because of
deformations in fields. Such conservation laws are
called topological, because topology is that branch
of mathematics that concerns itself with the shape
of things. Thus, it may happen that a knot in a set
of field lines, called a soliton, cannot be
smoothed out. As a result, the soliton is prevented
from dissipating and behaves much like a particle.
A classic example is a magnetic monopole--the
isolated pole of a bar magnet--which has not been
found in nature but shows up as twisted
configurations in some field theories.

In the traditional view, then, particles such as
electrons
and quarks (which carry Noether charges) are seen
as fundamental, whereas particles such as magnetic
monopoles (which carry topological charge) are
derivative. In 1977, however, Claus Montonen, now
at the Helsinki Institute of Physics, and David I.
Olive, now at the University of Wales at Swansea,
made a bold conjecture. Might there exist an
alternative formulation of physics in which the
roles of Noether charges (like electrical charge)
and topological charges (like magnetic charge) are
reversed? In such a "dual" picture, the magnetic
monopoles would be the elementary objects, whereas
the familiar particles--quarks, electrons and so
on--would arise as solitons.

More precisely, a fundamental particle with charge
e would be equivalent to a solitonic particle with
charge 1/e. Because its charge is a measure of how
strongly a particle interacts, a monopole would
interact weakly when the original particle
interacts strongly (that is, when e is large), and
vice versa.

The conjecture, if true, would lead to a profound
mathematical simplification. In the theory of
quarks, for instance, physicists can make hardly
any calculations when the quarks interact strongly.
But any monopoles in the theory must then interact
weakly. One could imagine doing calculations with a
dual theory based on monopoles and automatically
getting all the answers for quarks, because the
dual theory would yield the same final results.

Unfortunately, the idea remained on the back
burner. It was a
chicken-and-egg problem. Once proved, the
Montonen-Olive conjecture could leap beyond
conventional calculational techniques, but it would
need to be proved by some other method in the first
place.

As it turns out, p-branes can also be viewed as
solitons. In
1990 Andrew Strominger of the Institute for
Theoretical Physics in Santa Barbara found that a
10-dimensional string can yield a soliton that is a
five-brane. Reviving an earlier conjecture of mine,
Strominger suggested that a strongly interacting
string is the dual equivalent of weakly interacting
five-branes.

There were two major impediments to this duality.
First, the duality proposed by Montonen and
Olive--between electricity and magnetism in
ordinary four dimensions--was still unproved, so
duality between strings and five-branes in 10
dimensions was even more tenuous. Second, there
were all kinds of issues about how to find the
quantum properties of five-branes and hence how to
prove the new duality.

The first of these impediments was removed,
however, when
Ashoke Sen of the Tata Institute of Fundamental
Research in Bombay established that supersymmetric
theories would require the existence of certain
solitons with both electrical and magnetic charges.
These objects had been predicted by the
Montonen-Olive conjecture. This seemingly
inconspicuous result converted many skeptics and
unleashed a flood of papers. In particular, it
inspired Nathan Seiberg of Rutgers University and
Edward Witten to look for duality in more realistic
(though still supersymmetric) versions of quark
theories. They provided a wealth of information on
quantum fields, of a kind unthinkable just a few
years ago.

Duality of Dualities

In 1990 several theorists generalized the idea of
Montonen-Olive duality to four-dimensional
superstrings, in whose realm the idea becomes even
more natural. This duality, which nonetheless
remained speculative, goes by the name of
S-duality.

In fact, string theorists had already become used
to a
totally different kind of duality called T-duality.
T-duality relates two kinds of particles that arise
when a string loops around a compact dimension. One
kind (call them "vibrating" particles) is analogous
to those predicted by Kaluza and Klein and comes
from vibrations of the loop of string [see
illustration]. Such particles are more energetic if
the circle is small. In addition, the string can
wind many times around the circle, like a rubber
band on a wrist; its energy becomes higher the more
times it wraps around and the larger the circle.
Moreover, each energy level represents a new
particle (call them "winding" particles).

T-duality states that the winding particles for a
circle of
radius R are the same as the "vibration" particles
for a circle of radius 1/R, and vice versa. To a
physicist, the two sets of particles are
indistinguishable: a fat, compact dimension may
yield apparently the same particles as a thin one.

This duality has a profound implication. For
decades,
physicists have been struggling to understand
nature at the extremely small scales near the
Planck length of 10-33 centimeter. We have always
supposed that laws of nature, as we know them,
break down at smaller distances. What T-duality
suggests, however, is that at these scales, the
universe looks just the same as it does at large
scales. One may even imagine that if the universe
were to shrink to less than the Planck length, it
would transform into a dual universe that grows
bigger as the original one collapses.

Duality between strings and five-branes still
remained
conjectural, however, because of the problem of
quantizing five-branes. Starting in 1991, a team at
Texas A&M, involving Jianxin Lu, Ruben Minasian,
Ramzi Khuri and myself, solved the problem by
sidestepping it. If four of the 10 dimensions curl
up and the five-brane wraps around these, the
latter ends up as a one-dimensional object--a
(solitonic) string in six-dimensional space-time.
In addition, a fundamental string in 10 dimensions
remains fundamental even in six dimensions. So the
concept of duality between strings and five-branes
gave way to another conjecture, duality between a
solitonic and a fundamental string.

The advantage is that we do know how to quantize a
string.
Hence, the predictions of string-string duality
could be put to the test. One can show, for
instance, that the strength with which the
solitonic strings interact is given by the inverse
of the fundamental stringÕs interaction strength,
in complete agreement with the conjecture.

In 1994 Christopher M. Hull of Queen Mary and
Westfield
College, along with Townsend, suggested that a
weakly interacting heterotic string can even be the
dual of a strongly interacting Type IIA string, if
both are in six dimensions. The barriers between
the different string theories were beginning to
crumble.

It occurred to me that string-string duality has
another
unexpected payoff. If we reduce the six-dimensional
space-time to four dimensions, by curling up two
dimensions, the fundamental string and the
solitonic string each acquire a T-duality. But here
is the miracle: the T-duality of the solitonic
string is just the S-duality of the fundamental
string, and vice versa. This phenomenon--in which
the interchange of charges in one picture is just
the inversion of length in the dual picture--is
called the Duality of Dualities. It places the
previously speculative S-duality on just as firm a
footing as the well-established T-duality. In
addition, it predicts that the strength with which
objects interact--their charges--is related to the
size of the invisible dimensions. What is charge in
one universe may be size in another.

In a landmark talk at the University of Southern
California
in 1995, Witten suddenly drew together all the work
on T-duality, S-duality and string-string duality
under the umbrella of M-theory in 11 dimensions. In
the following months, literally hundreds of papers
appeared on the Internet confirming that whatever
M-theory may be, it certainly involves membranes in
an important way.

Even the E8 X E8 string, whose handedness was
thought
impossible to derive from 11 dimensions, acquired
an origin in M-theory. Witten, along with Petr
Horava of Princeton University, showed how to
shrink the extra dimension of M-theory into a
segment of a line. The resulting picture has two
10-dimensional universes (each at an end of the
line) connected by a space-time of 11 dimensions.
Particles--and strings--exist only in the parallel
universes at the ends, which can communicate with
each other only via gravity. (One can speculate
that all visible matter in our universe lies on one
wall, whereas the "dark matter," believed to
account for the invisible mass in the universe,
resides in a parallel universe on the other wall.)

This scenario may have important consequences for
confronting
M-theory with experiment. For example, physicists
know that the intrinsic strengths of all the forces
change with the energy of the relevant particles.
In supersymmetric theories, one finds that the
strengths of the strong, weak and electromagnetic
forces all converge at an energy E of 1016 giga
electron volts. Further, the interaction strengths
almost equal--but not quite--the value of the
dimensionless number GE2, where G is NewtonÕs
gravitational constant. This near miss, most likely
not a coincidence, seems to call for an
explanation; it has been a source of great
frustration for physicists.

But in the bizarre space-time envisioned by Horava
and
Witten, one can choose the size of the 11th
dimension so that all four forces meet at this
common scale. It is far less than the Planck energy
of 1019 giga electron volts, at which gravity was
formerly expected to become strong. (High energy is
connected to small distance via quantum mechanics.
So Planck energy is simply Planck length expressed
as energy.) So quantum-gravitational effects may be
far closer in energy to everyday events than
physicists previously believed, a result that would
have all kinds of cosmological consequences.

Recently Joseph Polchinski of the Institute for
Theoretical
Physics at Santa Barbara realized that some
p-branes resemble a surface discovered by
19th-century German mathematician Peter G. L.
Dirichlet. On occasion these branes can be
interpreted as black holes or, rather,
black-branes--objects from which nothing, not even
light, can escape.

Open strings, for instance, may be regarded as
closed
strings, part of which are hidden behind the
black-branes. Such breakthroughs have led to a new
interpretation of black holes as intersecting
black-branes wrapped around seven curled
dimensions. As a result, there are strong hints
that M-theory may even clear up the paradoxes of
black holes raised by Stephen W. Hawking of the
University of Cambridge.

In 1974 Hawking showed that black holes are in
fact not
entirely black but may radiate energy. In that
case, black holes must possess entropy, which
measures the disorder of a system by accounting for
the number of quantum states available. Yet the
microscopic origin of these quantum states stayed a
mystery. The technology of Dirichlet-branes has now
enabled Strominger and Cumrun Vafa of Harvard
University to count the number of quantum states in
black-branes. They find an entropy that agrees
perfectly with HawkingÕs prediction, placing
another feather in the cap of M-theory.

Black-branes also promise to solve one of the
biggest
problems of string theory: there seem to be
billions of different ways of crunching 10
dimensions down to four. So there are many
competing predictions of how the real world
works--in other words, no prediction at all. It
turns out, however, that the mass of a black-brane
can vanish as a hole it wraps around shrinks. This
feature miraculously affects the space-time itself,
allowing one space-time with a certain number of
internal holes (resembling a Gruy¸re cheese) to
change to another with a different number of holes,
violating the laws of classical topology.

If all the space-times are thus related, finding
the right
one becomes a more tractable problem. The string
may ultimately choose the space-time with, say, the
lowest energy and inhabit it. Its undulations would
then give rise to the elementary particles and
forces as we know them--that is, the real world.

10 to 11: Not Too Late

Despite all these successes, physicists are
glimpsing only
small corners of M-theory; the big picture is still
lacking. Recently Thomas Banks and Stephen H.
Shenker of Rutgers University, together with Willy
Fischler of the University of Texas and Leonard
Susskind of Stanford University, have proposed a
rigorous definition of M-theory. Their "matrix"
theory is based on an infinite number of
zero-branes (particles, that is). The coordinates,
or positions, of these particles, instead of being
ordinary numbers, are matrices that do not
commute--that is, xy does not equal yx. In this
picture space-time itself is a fuzzy concept in
which the coordinates cannot be defined as the
usual numbers but instead as matrices.

Physicists have long suspected that unifying
gravity--in
other words, the geometry of space-time--with
quantum physics will lead to space-time becoming
similarly ill defined--at least until a new
definition is discovered. The matrix approach has
generated great excitement but does not yet seem to
be the last word. Over the next few years, we hope
to discover what M-theory really is.

Witten is fond of imagining how physics might
develop on
another planet, where major discoveries such as
general relativity, quantum mechanics and
supersymmetry are made in a different order than on
Earth. In a similar vein, I would like to suggest
that on planets more logical than ours, 11
dimensions would have been the starting point from
which 10-dimensional string theory was subsequently
derived. Indeed, future terrestrial historians may
judge the late 20th century as a time when
theorists were like children playing on the
seashore, diverting themselves with the smoother
pebbles or prettier shells of superstrings while
the great ocean of M-theory lay undiscovered before
them.

Further Reading

THE MEMBRANE AT THE END OF THE UNIVERSE. Michael
Duff and
Christine Sutton in New Scientist, Vol. 118, No.
1619, pages 67--71; June 30, 1988.

UNITY FROM DUALITY. Paul Townsend in Physics
World, Vol. 8,
No. 9, pages 1--6; September 1995.

EXPLAINING EVERYTHING. Madhusree Mukerjee in
Scientific
American, Vol. 274, No. 1, pages 88--94; January
1996.

REFLECTIONS ON THE FATE OF SPACETIME. Edward
Witten in
Physics Today, Vol. 49, No. 4, pages 24--30; April
1996.

DUALITY, SPACETIME AND QUANTUM MECHANICS. Edward
Witten in
Physics Today, Vol. 50, No. 5, pages 28--33; May
1997.

The Author

MICHAEL J. DUFF conducts research on unified
theories of
elementary particles, quantum gravity,
supergravity, superstrings, supermembranes and
M-theory. He earned his Ph.D. in theoretical
physics in 1972 at Imperial College, London, and
joined the faculty there in 1980. He moved to the
U.S. in 1988 and has been a Distinguished Professor
at Texas A&M University since 1992. Duff has acted
as a spokesperson for British Scientists Abroad, a
group of expatriate scientists concerned about the
underfunding of British science and the consequent
brain drain. He is a Fellow of the American
Physical Society.

LIFE, THE UNIVERSE AND EVERYTHING may arise from the interplay
of strings, bubbles and sheets in higher dimensions of
space-time.

TRAJECTORY of a particle in space-time traces a worldline.
Similarly, that
of a string or a membrane sweeps out a worldsheet or
worldvolume, respectively.

SIMULTANEOUS SHRINKING of a membrane and a dimension of
space-time can
result in a string. As the underlying space, shown here as a
two-dimensional sheet, curls into a cylinder, the membrane wraps
around it. The curled dimension becomes a circle so small that
the two-dimensional space ends up looking one-dimensional, like
a line. The tightly wrapped membrane then resembles a string.

EXTRA DIMENSION curled into a tube offers insights into the
fabric of
space-time.

"BRANE" SCAN lists the membranes that arise in space-times of
different
dimensions. A p-brane of dimension 0 is a particle, that of
dimension 1 is a string and that of dimension 2 is a sheet or
bubble. Some branes have no spin (red), but Dirichlet-branes
have spin of 1 (blue).

THREE FORCES CONVERGE to the same strength when particles are
as energetic
as 1016 giga electron volts. Until now, gravity was believed to
miss this meeting point. But calculations including the 11th
dimension of M-theory suggest that gravity as well may converge
to the same point.

M-THEORY in 11 dimensions gives rise to the five string theories
in 10 dimensions. When the extra dimension curls into a circle,
M-theory yields the Type IIA superstring, which is further
related by duality to the Type IIB string. If, however, the
extra dimension shrinks to a line segment, M-theory becomes the
physically plausible E8£E8 heterotic string. The latter is
connected to the SO(32) string theories by dualities.