the theory formerly known as strings

At a time when certain pundits are predicting the End of Sci-

ence on the grounds that allthe important discoveries have alreadybeen made, it is worth emphasizing thatthe two main pillars of 20th-centuryphysics, quantum mechanics and Ein-stein’s general theory of relativity, aremutually incompatible. General relativ-ity fails to comply with the quantumrules that govern the behavior of ele-mentary particles, whereas on the op-posite scale, black holes are challengingthe very foundations of quantum me-chanics. Something big has to give. Thispredicament augurs less the bleak fu-ture of diminishing returns predicted bythe millennial Jeremiahs and more an-other scientific revolution.

Until recently, the best hope for a the-ory that would unite gravity with quan-tum mechanics and describe all physi-cal phenomena was based on strings:one-dimensional objects whose modesof vibration represent the elementaryparticles. In the past two years, however,strings have been subsumed by M-theo-ry. In the words of the guru of stringtheory (and according to Life magazine,the sixth most influential American babyboomer), Edward Witten of the Insti-tute for Advanced Study in Princeton,N.J., “M stands for Magic, Mystery orMembrane, according to taste.” Newevidence in favor of this theory is ap-pearing daily, representing the most ex-citing development since strings firstswept onto the scene.

M-theory, like string theory, reliescrucially on the idea of supersymmetry.Physicists divide particles into two class-es, according to their inherent angular

momentum, or “spin.” Supersymmetryrequires that for each known particlehaving integer spin—0, 1, 2 and so on,measured in quantum units—there is aparticle with the same mass but half-in-teger spin (1/2, 3/2, 5/2 and so on), andvice versa.

Unfortunately, no such superpartnerhas yet been found. The symmetry, if itexists at all, must be broken, so that thepostulated particles do not have thesame mass as known ones but insteadare too heavy to be seen in current ac-celerators. Even so, theorists have re-tained belief in supersymmetry primari-ly because it provides a frameworkwithin which the weak, electromagnet-ic and strong forces may be united withthe most elusive force of all: gravity.

Supersymmetry transforms the coor-dinates of space and time such that thelaws of physics are the same for all ob-servers. Einstein’s general theory of rel-ativity derives from this condition, andso supersymmetry implies gravity. Infact, supersymmetry predicts “super-gravity,” in which a particle with a spinof 2—the graviton—transmits gravita-

tional interactions and has as a partnera gravitino, with a spin of 3/2.

Conventional gravity does not placeany limits on the possible dimensions ofspace-time: its equations can, in princi-ple, be formulated in any dimension.Not so with supergravity, which placesan upper limit of 11 on the dimensionsof space-time. The familiar universe, ofcourse, has three dimensions of space:height, length and breadth, while timemakes up the fourth dimension of space-time. But in the early 1920s Polish phys-icist Theodore Kaluza and Swedish phys-icist Oskar Klein suggested that space-time may have a hidden fifth dimension.This extra dimension would not be in-finite, like the others; instead it wouldclose in on itself, forming a circle.Around that circle could reside quantumwaves, fitting neatly into a loop. Onlyinteger numbers of waves can fit aroundthe circle; each of these would corre-

The Theory Formerly Known as Strings64 Scientific American February 1998

The Theory Formerly Known as StringsThe 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

Copyright 1998 Scientific American, Inc.

spond to a particle with a different en-ergy. So the energies would be “quan-tized,” or discrete.

An observer living in the other fourdimensions, however, would see a set ofparticles with discrete charges, ratherthan energies. The quantum, or unit, ofcharge would depend on the circle’s ra-dius. In the real world as well, electricalcharge is quantized, in units of e, thecharge on the electron. To get the rightvalue for e, the circle would have to betiny, about 10–33 centimeter in radius.

The unseen dimension’s small size ex-plains why humans, or even atoms, areunaware of it. Even so, it would yieldelectromagnetism, and gravity, alreadypresent in the four-dimensional world,would be united with that force.

In 1978 Eugene Cremmer, BernardJulia and Joel Scherk of the École Nor-male Supérieure in Paris realized thatsupergravity not only permits up to sev-en extra dimensions but is most elegantwhen existing in a space-time of 11 di-mensions (10 of space and one of time).The kind of real, four-dimensional worldthe theory ultimately predicts dependson how the extra dimensions are rolledup, à la Kaluza and Klein. The severalcurled dimensions could conceivably al-low physicists to derive, in addition toelectromagnetism, the strong and weaknuclear forces. For these reasons, manyphysicists began to look to supergravity

in 11 dimensions in the hope that itmight be the unified theory.

In 1984, however, 11-dimensionalsupergravity was rudely knocked off itspedestal. An important feature of thereal world is that nature distinguishesbetween right and left: the laws govern-ing the weak nuclear force operate dif-ferently when viewed through a mirror.(For instance, neutrinos always haveleft-handed spin.) But as Witten andothers emphasized, such “handedness”cannot readily be derived by reducingspace-time from 11 dimensions downto four.

P-Branes

Supergravity’s position was usurpedby superstring theory in 10 dimen-

sions. Five competing theories held sway,designated by their mathematical char-acteristics as the E8 × E8 heterotic, theSO(32) heterotic, the SO(32) Type I,the Type IIA and Type IIB strings. (TheType I is an “open” string consisting ofjust a segment, whereas the others are“closed” strings that form loops.) Onestring in particular, the E8× E8, seemed—at least in principle—capable of explain-ing the known elementary particles andforces, including their handedness. Andunlike supergravity, strings seemed toprovide a theory of gravity consistentwith quantum effects. All these virtues

enabled string theory tosweep physicists off theirfeet and 11-dimensionalsupergravity into the dog-house. Murray Gell-Mann

of the California Institute ofTechnology encapsulated the

mood of the times by declar-ing at a meeting: “Eleven-di-

mensional supergravity—ugh!’’After the initial euphoria over

strings, however, nagging doubts be-gan to creep in. First, many important

questions—especially how to confrontthe theory with experiment—seemed in-capable of being answered by tradition-al methods of calculation. They calledfor radically new techniques. Second,why were there five different string the-ories? If one is looking for a uniqueTheory of Everything, surely this is anembarrassment of riches. Third, if super-symmetry permits 11 dimensions, whydo superstrings stop at 10? Finally, if weare going to conceive of pointlike parti-cles as strings, why not as membranesor more generally as p-dimensional ob-jects—inevitably dubbed p-branes?

Consequently, while most theoristswere tucking into super-spaghetti, asmall but enthusiastic group were de-veloping an appetite for super-ravioli. Aparticle, which has zero dimensions,sweeps out a one-dimensional trace, or“worldline,” as it evolves in space-time[see illustration on next page]. Similarlya string—having one dimension, length—sweeps out a two-dimensional “world-sheet,” and a membrane—having twodimensions, length and breadth—sweepsout a three-dimensional “worldvolume.”In general, a p-brane sweeps out a world-volume of p + 1 dimensions. (Of course,there must be enough room for the p-brane to move about in space-time, sop + 1 must not exceed the number ofspace-time dimensions.)

As early as 1962, Paul A. M. Dirac,one of the fathers of quantum mechan-ics, had constructed an imaginative mod-el based on a membrane. He postulatedthat the electron, instead of resemblinga point, was in reality a minute bubble,a membrane closed in on itself. Its oscil-lations, Dirac suggested, might generateparticles such as the muon, a heavier

The Theory Formerly Known as Strings Scientific American February 1998 65

DU

SAN

PET

RIC

IC

LIFE, THE UNIVERSE ANDEVERYTHING may arisefrom the interplay of strings,bubbles and sheets in higherdimensions of space-time.


version of the electron. Although his at-tempt failed, the equations that Diracpostulated for the membrane are essen-tially the ones we use today. The mem-brane may take the form of a bubble,or it may be stretched out in two direc-tions like a sheet of rubber.

Supersymmetry severely restricts thepossible dimensions of a p-brane. In thespace-time of 11 dimensions floats amembrane, discovered mathematicallyby Eric Bergshoeff of the University ofGroningen, Ergin Sezgin, now at TexasA&M University, and Paul K. Town-send of the University of Cambridge. Ithas only two spatial dimensions andlooks like a sheet. Paul S. Howe of King’sCollege London, Takeo Inami of KyotoUniversity, Kellogg Stelle of ImperialCollege, London, and I were able toshow that if one of the 11 dimensions isa circle, we can wrap the membranearound it once, pasting the edges togeth-er to form a tube. If the radius of the cir-cle becomes sufficiently small, the rolled-up membrane ends up looking like astring in 10 dimensions; in fact, it yieldsprecisely the Type IIA superstring.

Notwithstanding such results, themembrane enterprise was largely ig-nored by the orthodox string communi-ty. Fortunately, the situation was aboutto change because of progress in an ap-parently unrelated field.

In 1917 German mathematician Ama-lie Emmy Noether had shown that themass, charge and other attributes of el-

ementary particles are generally con-served because of symmetries. For in-stance, conservation of energy followsif one assumes that the laws of physicsremain unchanged with time, or aresymmetric as time passes. And conser-vation of electrical charge follows from asymmetry of a particle’s wave function.

Sometimes, however, attributes maybe maintained because of deformationsin fields. Such conservation laws arecalled topological, because topology isthat branch of mathematics that con-cerns itself with the shape of things.Thus, it may happen that a knot in a setof field lines, called a soliton, cannot besmoothed out. As a result, the soliton isprevented from dissipating and behavesmuch like a particle. A classic exampleis a magnetic monopole—the isolatedpole of a bar magnet—which has notbeen found in nature but shows up astwisted configurations in some fieldtheories.

In the traditional view, then, particlessuch as electrons and quarks (which car-ry Noether charges) are seen as funda-mental, whereas particles such as mag-netic monopoles (which carry topologi-cal charge) are derivative. In 1977,however, Claus Montonen, now at theHelsinki Institute of Physics, and DavidI. Olive, now at the University of Walesat Swansea, made a bold conjecture.Might there exist an alternative formu-lation of physics in which the roles ofNoether charges (like electrical charge)

and topological charges (like magneticcharge) are reversed? In such a “dual”picture, the magnetic monopoles wouldbe the elementary objects, whereas thefamiliar particles—quarks, electronsand so on—would arise as solitons.

More precisely, a fundamental parti-cle with charge e would be equivalentto a solitonic particle with charge 1/

e

.

Because its charge is a measure of howstrongly a particle interacts, a mono-pole would interact weakly when theoriginal particle interacts strongly (thatis, when e is large), and vice versa.

The conjecture, if true, would lead toa profound mathematical simplification.In the theory of quarks, for instance,physicists can make hardly any calcula-tions when the quarks interact strongly.But any monopoles in the theory mustthen interact weakly. One could imag-ine doing calculations with a dual theo-ry based on monopoles and automati-cally getting all the answers for quarks,because the dual theory would yield thesame final results.

Unfortunately, the idea remained onthe back burner. It was a chicken-and-egg problem. Once proved, the Mon-tonen-Olive conjecture could leap be-yond conventional calculational tech-niques, but it would need to be provedby some other method in the first place.

As it turns out, p-branes can also beviewed as solitons. In 1990 AndrewStrominger of the Institute for Theoret-ical Physics in Santa Barbara found thata 10-dimensional string can yield a soli-ton that is a five-brane. Reviving an ear-lier conjecture of mine, Strominger sug-gested that a strongly interacting stringis the dual equivalent of weakly inter-acting five-branes.

There were two major impedimentsto this duality. First, the duality pro-posed by Montonen and Olive—be-tween electricity and magnetism in or-dinary four dimensions—was still un-proved, so duality between strings and

The Theory Formerly Known as Strings

SIMULTANEOUS SHRINKING of a membraneand a dimension of space-time can result in a string.As the underlying space, shown here as a two-dimen-sional sheet, curls into a cylinder, the membranewraps around it. The curled dimension becomes acircle so small that the two-dimensional space endsup looking one-dimensional, like a line. The tightlywrapped membrane then resembles a string.

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

DU

SAN

PET

RIC

ICD

USA

N P

ETRI

CIC


five-branes in 10 dimensions was evenmore tenuous. Second, there were allkinds of issues about how to find thequantum properties of five-branes andhence how to prove the new duality.

The first of these impediments wasremoved, however, when Ashoke Sen ofthe Tata Institute of Fundamental Re-search in Bombay established that su-persymmetric theories would require theexistence of certain solitons with bothelectrical and magnetic charges. Theseobjects had been predicted by the Mon-tonen-Olive conjecture. This seeminglyinconspicuous result converted manyskeptics and unleashed a flood of papers.In particular, it inspired Nathan Seibergof Rutgers University and Edward Wit-ten to look for duality in more realistic(though still supersymmetric) versions ofquark theories. They provided a wealthof information on quantum fields, of akind unthinkable just a few years ago.

Duality of Dualities

In 1990 several theorists generalizedthe idea of Montonen-Olive duality

to four-dimensional superstrings, inwhose realm the idea becomes evenmore natural. This duality, which none-theless remained speculative, goes bythe name of S-duality.

In fact, string theorists had alreadybecome used to a totally different kindof duality called T-duality. T-duality re-lates two kinds of particles that arisewhen a string loops around a compactdimension. One kind (call them “vi-brating” particles) is analogous to thosepredicted by Kaluza and Klein andcomes from vibrations of the loop ofstring [see illustration on next page].Such particles are more energetic if thecircle is small. In addition, the stringcan wind many times around the circle,

like a rubberband on a wrist; its

energy becomes higher themore times it wraps around and the

larger the circle. Moreover, each energylevel represents a new particle (callthem “winding” particles).

T-duality states that the winding par-ticles for a circle of radius R are thesame as the “vibration” particles for acircle of radius 1/

R

, and vice versa. To aphysicist, the two sets of particles areindistinguishable: a fat, compact dimen-sion may yield apparently the same par-ticles as a thin one.

This duality has a profound implica-tion. For decades, physicists have beenstruggling to understand nature at theextremely small scales near the Plancklength of 10–33 centimeter. We have al-ways supposed that laws of nature, aswe know them, break down at smallerdistances. What T-duality suggests, how-ever, is that at these scales, the universelooks just the same as it does at largescales. One may even imagine that ifthe universe were to shrink to less thanthe Planck length, it would transforminto a dual universe that grows biggeras the original one collapses.

Duality between strings and five-branes still remained conjectural, how-ever, because of the problem of quantiz-ing five-branes. Starting in 1991, a teamat Texas A&M, involving Jianxin Lu,Ruben Minasian, Ramzi Khuri and my-self, solved the problem by sidesteppingit. If four of the 10 dimensions curl upand the five-brane wraps around these,the latter ends up as a one-dimensionalobject—a (solitonic) string in six-di-mensional space-time. In addition, afundamental string in 10 dimensionsremains fundamental even in six di-

mensions. So the concept of duality be-tween strings and five-branes gave wayto another conjecture, duality betweena solitonic and a fundamental string.

The advantage is that we do knowhow to quantize a string. Hence, the pre-dictions of string-string duality couldbe put to the test. One can show, for in-stance, that the strength with which thesolitonic strings interact is given by theinverse of the fundamental string’s in-teraction strength, in complete agree-ment with the conjecture.

In 1994 Christopher M. Hull ofQueen Mary and Westfield College,along with Townsend, suggested that aweakly interacting heterotic string caneven be the dual of a strongly interact-ing Type IIA string, if both are in six di-mensions. The barriers between the dif-ferent string theories were beginning tocrumble.

It occurred to me that string-stringduality has another unexpected payoff.If we reduce the six-dimensional space-time to four dimensions, by curling uptwo dimensions, the fundamental stringand the solitonic string each acquire aT-duality. But here is the miracle: the T-duality of the solitonic string is just theS-duality of the fundamental string, andvice versa. This phenomenon—in whichthe interchange of charges in one pic-ture is just the inversion of length in thedual picture—is called the Duality ofDualities. It places the previously spec-ulative S-duality on just as firm a foot-ing as the well-established T-duality. Inaddition, it predicts that the strengthwith which objects interact—their charg-es—is related to the size of the invisibledimensions. What is charge in one uni-verse may be size in another.

In a landmark talk at the Universityof Southern California in 1995, Wittensuddenly drew together all the work onT-duality, S-duality and string-string du-ality under the umbrella of M-theory in11 dimensions. In the following months,literally hundreds of papers appeared

The Theory Formerly Known as Strings

“BRANE” SCAN lists the membranesthat arise in space-times of different di-mensions. A p-brane of dimension 0 is aparticle, that of dimension 1 is a stringand that of dimension 2 is a sheet or bub-ble. Some branes have no spin (red), butDirichlet-branes have spin of 1 (blue).

EXTRA DIMENSION curled into atube offers insights into the fabric ofspace-time.

DU

SAN

PET

RIC

IC

DU

SAN

PET

RIC

IC


on the Internet confirming that whateverM-theory may be, it certainly involvesmembranes in an important way.

Even the E8× E8 string, whose hand-edness was thought impossible to derivefrom 11 dimensions, acquired an originin M-theory. Witten, along with PetrHorava of Princeton University, showedhow to shrink the extra dimension ofM-theory into a segment of a line. Theresulting picture has two 10-dimension-al universes (each at an end of the line)connected by a space-time of 11 dimen-sions. Particles—and strings—exist onlyin the parallel universes at the ends,which can communicate with each oth-er only via gravity. (One can speculatethat all visible matter in our universe lieson one wall, whereas the “dark matter,”believed to account for the invisible massin the universe, resides in a parallel uni-verse on the other wall.)

This scenario may have importantconsequences for confronting M-theorywith experiment. For example, physi-

cists know that the intrinsic strengths ofall the forces change with the energy ofthe relevant particles. In supersymmet-ric theories, one finds that the strengthsof the strong, weak and electromagnet-ic forces all converge at an energy E of1016 giga electron volts. Further, the in-teraction strengths almost equal—butnot quite—the value of the dimension-less number GE2, where G is Newton’sgravitational constant. This near miss,most likely not a coincidence, seems tocall for an explanation; it has been asource of great frustration for physicists.

But in the bizarre space-time envi-sioned by Horava and Witten, one can

choose the size of the 11th dimensionso that all four forces meet at this com-mon scale. It is far less than the Planckenergy of 1019 giga electron volts, atwhich gravity was formerly expected tobecome strong. (High energy is connect-ed to small distance via quantum me-chanics. So Planck energy is simplyPlanck length expressed as energy.) Soquantum-gravitational effects may be

68 Scientific American February 1998

THREE FORCES CONVERGE to thesame strength when particles are as ener-getic as 1016 giga electron volts. Untilnow, gravity was believed to miss thismeeting point. But calculations includingthe 11th dimension of M-theory suggestthat gravity as well may converge to thesame point.

T-duality connects the physics of large space-timeswith that of small ones. Visualize a curled space-time

as a cylinder. A string looped around it has two kinds of en-ergy states. One set arises from the waves in the string thatfit around the cylinder; call these the “vibration” modes. Ifthe cylinder is fat, the vibrations tend to have long wave-lengths and less energy. So the energies corresponding

to different numbers of waves around the cylinder areseparated by small amounts—that is, they are “closelyspaced.”

The string can, however, also loop around thecylinder like a stretched rubber band. If the cylinderis fat, the string needs to stretch more, requiring

more energy. So the energies of the states corre-sponding to different numbers of loops—callthese the “winding” modes—are widely spaced.

Now look at the energy levels for a thin cylin-der. The waves fitting around it are small andso have high energy. As a result, the vibrationstates are widely spaced. But the loops re-

quire less energy, and so the winding modesare closely spaced.

To an outside observer, however, thedifferent physical origins of the vibrationand winding states are not apparent.Both the thin and the fat tube yield ulti-

mately the same energy levels, whichphysicists interpret as particles. Thus,the minute scales of the thin space-time may yield exactly the same

physics as the large scales of ouruniverse. — M.J.D.

Duality between Large and Small

DU

SAN

PET

RIC

ICD

USA

N P

ETRI

CIC


far closer in energy to everyday eventsthan physicists previously believed, aresult that would have all kinds of cos-mological consequences.

Recently Joseph Polchinski of the In-stitute for Theoretical Physics at SantaBarbara realized that some p-branes re-semble a surface discovered by 19th-century German mathematician PeterG. L. Dirichlet. On occasion these branescan be interpreted as black holes or,rather, black-branes—objects from whichnothing, not even light, can escape.

Open strings, for instance, may be re-garded as closed strings, part of whichare hidden behind the black-branes.Such breakthroughs have led to a newinterpretation of black holes as inter-secting black-branes wrapped aroundseven curled dimensions. As a result,there are strong hints that M-theory mayeven clear up the paradoxes of blackholes raised by Stephen W. Hawking ofthe University of Cambridge.

In 1974 Hawking showed that blackholes are in fact not entirely black butmay radiate energy. In that case, blackholes must possess entropy, which mea-sures the disorder of a system by ac-counting for the number of quantumstates available. Yet the microscopic ori-gin of these quantum states stayed a mys-tery. The technology of Dirichlet-braneshas now enabled Strominger and Cum-run Vafa of Harvard University to countthe number of quantum states in black-branes. They find an entropy that agreesperfectly with Hawking’s prediction,placing another feather in the cap ofM-theory.

Black-branes also promise to solveone of the biggest problems of stringtheory: there seem to be billions of dif-ferent ways of crunching 10 dimensionsdown to four. So there are many com-peting predictions of how the real worldworks—in other words, no prediction atall. It turns out, however, that the mass

of a black-brane can vanish as a hole itwraps around shrinks. This feature mi-raculously affects the space-time itself,allowing one space-time with a certainnumber of internal holes (resembling aGruyère cheese) to change to anotherwith a different number of holes, vio-lating the laws of classical topology.

If all the space-times are thus related,finding the right one becomes a moretractable problem. The string may ulti-mately choose the space-time with, say,the lowest energy and inhabit it. Its un-dulations would then give rise to the el-ementary particles and forces as weknow them—that is, the real world.

10 to 11: Not Too Late

Despite all these successes, physicistsare glimpsing only small corners of

M-theory; the big picture is still lacking.Recently Thomas Banks and Stephen H.Shenker of Rutgers University, togetherwith Willy Fischler of the University ofTexas and Leonard Susskind of Stan-ford University, have proposed a rigor-ous definition of M-theory. Their “ma-trix” theory is based on an infinite num-ber of zero-branes (particles, that is).The coordinates, or positions, of theseparticles, instead of being ordinary num-bers, are matrices that do not com-

mute—that is, xy does not equal yx. Inthis picture space-time itself is a fuzzyconcept in which the coordinates can-not be defined as the usual numbers butinstead as matrices.

Physicists have long suspected thatunifying gravity—in other words, thegeometry of space-time—with quantumphysics will lead to space-time becom-ing similarly ill defined—at least until anew definition is discovered. The ma-trix approach has generated great ex-citement but does not yet seem to be thelast word. Over the next few years, wehope to discover what M-theory really is.

Witten is fond of imagining howphysics might develop on another plan-et, where major discoveries such as gen-eral relativity, quantum mechanics andsupersymmetry are made in a differentorder than on Earth. In a similar vein, Iwould like to suggest that on planetsmore logical than ours, 11 dimensionswould have been the starting point fromwhich 10-dimensional string theory wassubsequently derived. Indeed, future ter-restrial historians may judge the late 20thcentury as a time when theorists werelike children playing on the seashore,diverting themselves with the smootherpebbles or prettier shells of superstringswhile the great ocean of M-theory layundiscovered before them.

The Theory Formerly Known as Strings Scientific American February 1998 69

The Author

MICHAEL J. DUFF conducts research on unified theoriesof elementary particles, quantum gravity, supergravity, su-perstrings, supermembranes and M-theory. He earned hisPh.D. in theoretical physics in 1972 at Imperial College,London, and joined the faculty there in 1980. He moved tothe U.S. in 1988 and has been a Distinguished Professor atTexas A&M University since 1992. Duff has acted as aspokesperson for British Scientists Abroad, a group of expa-triate scientists concerned about the underfunding of Britishscience and the consequent brain drain. He is a Fellow of theAmerican Physical Society.

Further Reading

The Membrane at the End of the Universe. Michael Duff and Chris-tine 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 To-

day, Vol. 49, No. 4, pages 24–30; April 1996.Duality, Spacetime and Quantum Mechanics. Edward Witten inPhysics Today, Vol. 50, No. 5, pages 28–33; May 1997.

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 dimen-sion shrinks to a line segment, M-theory becomes the physically plausible E8× E8 het-erotic string. The latter is connected to the SO(32) string theories by dualities.

SA

DU

SAN

PET

RIC

IC


the theory formerly known as strings

Documents