witten, e. - duality, spacetime and quantum mechanics

8/4/2019 Witten, E. - Duality, Spacetime and Quantum Mechanics

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DUALITY, SPACETIME ANDQUANTUM MECHANICS

The purpose of this articleis to describe somethemes in theoretical physicsthat developed indepen-dently for many years, insome cases for decades, andthen converged rather sud-denly beginning around1994-95. The convergenceproduced an upheaval some-times called "the second su-perstring revolution." It isas significant in its own wayas "the first superstring revo-lution," the period around 1984-85 when the potential ofstring theory to give a unified description of natural lawwas first widely appreciated.Some of the themes whose convergence we explorehere are depicted in figure 1. Their diversity is part ofthe charm of the second superstring revolution, in whicha major new perspective on the quest for superunificationof the forces of nature has developed from the interplayof esoteric ideas about physics at ultrahigh energies withdown-to-earth investigations of the physical properties ofgauge theories in four dimensions.

netic monopoles).More fundamentally, thesymmetry seems to be vio-lated when we derive themagnetic field from a vectorpotential A, with B=VxA,while representing the elec-tric field (in a static situ-ation) as the gradient of ascalar.But the vector potentialis not just a convenience insolving Maxwell's equations.It is needed in 20th-centuryphysics for three very goodpurposes:

[> To write a Schrodinger equation for an electron in amagnetic field.[> To make it possible to derive Maxwell's equations froma Lagrangian; this is a necessary step in treating electro-magnetism quantum mechanically.[> To write anything at all for non-Abelian gauge theory,which-in our modern understanding of elementary par-ticle physics-is the starting point in describing the strong,weak and electromagnetic interactions.Though it thus seems impossible to have symmetrybetween electric and magnetic charges in quantum fieldtheory, there definitely are field theories with both electricand magnetic charges, as we know from the work ofGerard 't Hooftand Alexander Polyakov in the 1970s. Forweak coupling-the only region where one traditionallycan understand how quantum field theories behave-theelectric and magnetic charges appear in completely differ-ent ways. We recall that in the case ofelectromagnetism,weak coupling means that the fine structure constant

Widely disparate themes fromseveral decades of theoretical physicshave recently converged to becomeparts of a single story. The result isa still-mysterious 'M-theory' that mayrevise our understanding of the roleof quantum mechanics.

Edward Witten

Electric-magnetic dualityWe begin with a piece of late-19th-century physics. Thevacuum Maxwell equations for the electric and magneticfields E and B,

a EVxB=-a t 'aBVxE=-- a t '

have a symmetry underE~B,B~-E

that has been known for nearly as long as the Maxwellequations themselves. This symmetry is known as duality.The symmetry still holds in the presence of chargesand currents if one adds both electric and magneticcharges and currents. In nature, such symmetry seemsto be spoiled by the fact that we observe electric chargesbut not magnetic charges (which are usually called mag-

28 MAY 1997 PHYSICS TODAY

is small. Here, e is the basic unit of electric charge andit is also the couplingconstant ofquantum electrodynamics.For weak coupling, electric charges appear as elemen-tary quanta, obtained by quantizing fields. The electriccharge q of any particle is of the form q = ne with someinteger n.By contrast, magnetic monopoles arise for weak cou-pling as collective excitations of the elementary particles.Such collective excitations appear, in the weak couplinglimit, as solitons-extended solutions ofnonlinear classicalfield equations. The magnetic charge m of any particlethat is obtained by quantizing such a soliton is an integermultiple of a fundamental quantum of magnetic charge,which (according to a celebrated analysis by Dirac) equals

1997 American Institute of Physics, S,0031,9228,9705,020,5


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Electric-magnetic dualityin four dimensions

Dualities ofstring theory

FIGURE 1. THE SECOND SUPERSTRINGREVOLUTION has resulted from theconvergence of four main themes. Clockwisearound the circle, starting from the top, theyare electric-magnetic duality in fourdimensions; the strange symmetries ofsupergravity; the nonclassical symmetries ofstring theory that violate the classical conceptsof space and time; and investigations of gaugetheory dynamics in four dimensions.

Somewhat analogous phenomena werefamiliar in the mathematical physics of1+ 1 dimensions (one space and one timedimension), and they have sometimes haduseful applications in two-dimensional con-densed matter physics such as the quantumHall system (see PHYSICS TODAY, February1997, page 17). But similar physics in3 + 1 dimensions was scarcely expected.Testing Montonen-Olive duality wasout of reach in the 1970s and 1980s, primar-ily because the duality exchanges a with1 / a. It is impossible for a and 1/ a to bothbe small, so-if one's knowledge ofquantum

field theory is limited to weak coupling-the duality ap-pears impossible to test.But before the subjectwent into eclipse, a fewimportantinsights were obtained. One was that Montonen-Oliveduality makes more sense with supersymmetry.

Ztthc] e, Thus,2 7 T h cm=n--e

with some integer n',In short, there seem to be very deep reasons for alack of symmetry between electricity and magnetism in

modern physics. Yet,in 1977, Claus Montonenand DavidOlivenoted a surprising symmetry between electricity andmagnetism in the classical limit of a certain four-dimen-sional field theory- (The model in question was a limitingcase of a then-current model of weak interactions.) Mon-tonen and Olive saw that in this model the mass M ofany particle of electric and magnetic charges q and m wasgiven by a beautiful symmetric formula,

M = ( 1 - v q2 +m2 ,where (l is a constant that measures the gauge symmetrybreaking. They conjectured that the theory had an exactsymmetry that exchanges q and m.

A symmetry that exchanges electric and magneticcharges must exchange the quantum ofelectric charge witha multiple of the quantum of magnetic charge. For in-stance, in the Montonen-Olive case the transformation is

47Tnce H e

and hence1a Ha

Such a symmetry will exchange electric and magneticfields, so, to a classical observer, it will looklike the dualityof Maxwell's equations. Finally, such a symmetry mustexchange elementary quanta with collective excitationssince, for weak coupling, electric charges arise as elemen-tary quanta and magnetic charges arise as collectiveexcitations.

SupersymmetrySupersymmetry is a conjectured symmetry between fer-mions and bosons.f Itis an inherently quantum mechani-cal symmetry, since the very concept of fermions is quan-tum mechanical. Bosonic quantities can be described byordinary (commuting) numbers or by operators obeyingcommutation relations. Fermionic quantities involve an-ticommuting numbers or operators.Two-dimensional supersymmetry emerged historicallyfrom Pierre Ramond's discovery in 1970 of how to incor-porate fermions into string theory. Supersymmetry wasformulated as a four-dimensional symmetry by Julius Wessand Bruno Zumino in 1974. It also was conceived inde-pendently byYuri Gol'fand and Eugeny Likhtman in 1971.

Supersymmetry is an updating of special relativity toinclude fermionic as well as bosonic symmetries ofspacetime. In developing relativity, Einstein assumedthat the spacetime coordinates were bosonic; fermions hadnot yet been discovered! In supersymmetry the structureof spacetime is enriched by the presence of fermionic aswell as bosonic coordinates.

If true, supersymmetry explains why fermions existin nature. Supersymmetry demands their existence.From experiment, we have some hints (especially fromthe observed values of the strong, weak and electromag-netic coupling constants) that nature may be supersym-metric. Determining whether this is actually so is one ofthe main goals of present and future elementary particleexperiments. Experimental discovery of supersymmetrywould be the beginning of probing the quantum structureof space and time.

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FIGURE 2. QUARK CONFINEMENT AND THE MEISSNEREFFECT. a: The strong-force field lines (green) between a

widely separated quark and antiquark (red) form a thin fluxtube (white) in the vacuum (blue). The energy of the tube

grows linearly with its length, making it impossible to isolatea lone quark. b: The analogous Abrikosov-Gorkov flux tube

between a magnetic monopole and an antimonopole in asuperconductor. The magnetic field lines are confined to a

thin nonsuperconducting tube (white) by the Meissner effect,which excludes magnetic flux from a superconductor (gray).

At any rate, it became clear at an early stage ofresearch into supersymmetry that its special propertiesmade Montonen-Olive duality more plausible in the su-persymmetric case. But evidence for Montonen and Ol-ive's conjecture still appeared meager because, even in thesupersymmetric case, one's understanding of field theorywas limited to weak coupling.Quark confinementAnother important development in the 1970s (pioneered,among others, by YoichiroNambu, Stanley Mandelstamand 't Hooft)was the realization that duality of some sortbetween electricity and magnetism could be relevant tounderstanding the surprising phenomenon of quark con-finement. Accordingto quantum chromodynamics (QeD),strongly interacting particles such as protons are made ofquarks and gluons. But experimentally, we never observean isolated quark. Experiment and computer simulationseem to show that if one creates a quark-antiquark pairand separates them by a distance r, the energy growslinearly with r because of a mysterious "non-Abelian fluxtube" that forms between them (see figure 2a). This lineargrowth in the energy makes it impossible to separate thequark and antiquark to infinity and observea single quarkin isolation.Quark confinement in QeD is a difficult strong cou-pling problem, but a somewhat similar phenomenon innature is much better understood. The Meissner effect isthe fundamental observation that a superconductor expelsmagnetic flux. Suppose that magnetic monopoles becomeavailable for study and that we insert a monopole-anti-monopole pair into a superconductor, separated by a largedistance r (see figure 2b). What will happen? Amonopoleis inescapably a source ofmagnetic flux, but magnetic fluxis expelled from a superconductor. The optimal solutionto this problem, energetically, is that a thin, nonsupercon-ducting tube forms between the monopole and the anti-monopole. The magnetic flux is confined to this region,which is known as an Abrikosov-Gorkov flux tube (or aNielsen-Olesen flux tube in the context ofrelativistic fieldtheory). The flux tube has a certain nonzero energy perunit length, so the energy required to separate the mo-nopole and antimonopole by a distance r grows linearlyin r, for large r. (In practice, monopoles are not availablefor study, but flux tubes in superconductors can be createdand studied by, for instance, applying external magneticfields in an appropriate way.)As a non-Abelian gauge theory, QeD has fields rathersimilar to ordinary electric and magnetic fieldsbut obeyinga nonlinear version of Maxwell's equations. Quarks areparticles that carry the QeD analog of electric charge,and are confined in vacuum just as ordinary magneticcharges would be in a superconductor. This analogy ledin the 1970s to the idea that the QeD vacuum is to asuperconductor as electricity is to magnetism. This is animportant idea, but developing it concretely was out ofreach in that period.30 MAY 1997 PHYSICS TODAY

a

SupergravityMeanwhile, under development was "supergravity," theextension of supersymmetry to include gravity. Super-gravity is an enrichment of ordinary general relativity inwhich the spacetime coordinates are fermionic as well asbosonic. It is an updating of general relativity to includefermions just as supersymmetry is an updating of specialrelativity to take into account that fermions exist.When supergravity theories were constructed, start-ing in 1976, they proved to be remarkably rich. Theirexistence always looks miraculous. The conditions thatmust be satisfied to construct these theories are highlyoverdetermined, but the theories do exist; they delicatelyhang together, using every trick in the classical field theorybook. This situation is probably unfamiliar to most read-ers as it does not have a good analog in physical theoriesthat were discovered prior to supergravity.In constructing supergravity theories, physicists ranextensively into curious symmetries-technically they arenoncompact global symmetries-that acted by duality onmassless spin-1 fields such as the electromagnetic field.These mysterious symmetries were intensively studied inthe late 1970s.3In the late 1980s and early 1990s, some physicistsbegan to take seriously the idea of interpreting thoseduality symmetries in quantum theory. Since (as in theMontonen-Olive case) quantum duality symmetries al-ways exchange ordinary particles with solitons, this idealed to a thorough study of solitonic solutions of varioussupergravity theories. A rich variety of solutions wasfound, describing extended objects of various kinds."

Such ideas, however, could not be pressed too far inthe context of supergravity. Supergravity has in commonwith ordinary general relativity that it apparently doesnot work as a quantum theory, because the nonlinearitiesrequired by Einstein's principle of equivalence are toosevere-with or without supersymmetry. Physicists havelong puzzled over how to reconcile general relativity withquantum mechanics-that is, with the rest of physics.String theoryPrecisely this problem is overcome in string theory, inwhich elementary particles are understood as vibratingstrings, and the structure of spacetime is coded in the


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laws by which the strings propagate.String theory makes three general predictions:I > General relativity.[> Supersymmetry.[> Non-Abelian gauge theory.I have often thought that if physics has been devel-oped on thousands of planets throughout our universe,then those planets can be sorted into eight classes accord-ing to which of the three great ideas-general relativity,supersymmetry and Yang-Mills theory-predate stringtheory and which are regarded as consequences of it. Wehappen to live on a planet of type + - +. (In other words,on our planet, general relativity and Yang-Mills theorypredate string theory, but supersymmetry was discoveredat least partly because of its role in string theory.) If welived, for instance, on a planet of type - + +, discussionsof the relation of string theory to general relativity wouldhave a different flavor!String theory also leads in a strikingly elegant wayto models of particle physics with the qualitative proper-ties of the real world (such as the existence of quarks withelectric charge e/ 3 and the V - A structure of weak inter-actions). Most hopes for a proof that string theory de-scribes nature depend on eventually improving the deri-vations of particle physics from string theory and makingcontact with experiment, perhaps on the heels of anexperimental discovery of supersymmetry. Some of thepossible scenarios are reviewed in Gordon Kane's articlein PHYSICS TODAY (February 1997, page 40).String theory, if correct, entails a radical change inour concepts ofspacetime. That is what one would expectof a theory that reconciles general relativity with quan-tum mechanics. Much of the story is still out of reach;string theorists spent much of the late 1980s and early1990s studying that part of the story that was accessiblewith the techniques of the time. The answer involvedduality again.A vibrating string is described by an auxiliary two-dimensional field theory, whose Lagrangian is roughly

Here, X(T,U) is the position of the string at proper timeT, at a coordinate U along the string. In string theory,this auxiliary two-dimensional field theory plays a morefundamental role than spacetime, and spacetime exists onlyto the extent that it can be reconstructed from the two-di-mensional field theory. (For more detail about this, see myprevious article in PHYSICS TODAY,April 1996,page 24.)Duality symmetries of the two-dimensional field the-

FIGURE 3. DUALITY of one model can imply a seeminglyunrelated wonder in another: Model A is dual to model B.A perturbation 8 turns model A into model N. The dualitymight map 8 to a perturbation 8' that turns model B intomodel B'. This implies that model N is dual to model B'.Often, model B' is "easy" and model N is "hard" (or viceversa) in which case the duality between them reveals some ofthe secrets of model A'. The secrets in question may havehad, to begin with, no obvious relation to duality.

ory put a basic restriction on the validity of classicalnotions of spacetime. The basic duality is.o : aXa : ; Hau'

and is just analogous to the more familiar electromagneticduality E HB. In each case the duality exchanges aregime where familiar ideas in physics are adequate withone where they are not. In the case of electric-magneticduality, the "easy" region is weak coupling and the "hard"region is strong coupling. In the case of the two-dimen-sional string theory dualities, the "easy" situation is thatof large distances and the "hard" region is that in whichsome distances become very small.5How many string theories are there? Compared toordinary field theory, which has innumerable models,string theory is relatively unique. There are five consis-tent relativistic string theories-type I, type IIA, type lIB,and the Es x Es and SO(32)heterotic superstrings. (Eachof these five theories involves ten spacetime dimensions,some of which can be "compactified," or rolled up intounobservably small manifolds. Each theory consequentlyhas various classical solutions and quantum states, andthus might be manifested in nature in different ways.)In addition to the ten-dimensional string theories, thereis a wild card, eleven-dimensional supergravity. Thoughnot related to any known quantum theory, it has been-atleast to some-too intriguing to ignore. Efforts in the lastgeneration to understand the unification of natural lawby means of supertheories focused mainly on these sixtheories.

Most familiar theories in physics have one or moredimensionless parameters that can be adjusted (for in-stance, the fine structure constant a in the case of quan-tum electrodynamics). Bycontrast, in string theory-thatis, in each of the traditionally known string theories-there is no adjustable dimensionless parameter. Insteadthere is a field, the "dilaton" field , whose expectationvalue determines the fine structure constant by a formulathat is roughly a=e=. If nature is kind and fortunesmiles and we are one day able to calculate from firstprinciples the vacuum expectation value of the dilaton,then we could predict the value of the fine structureconstant. Traditionally, our understanding of string the-ory is limited to situations where is large and theeffective coupling is small; this is usually called the regionofweak coupling.Dynamics of gauge theoriesThe last major influence leading to the second superstringrevolution was the renewed investigation of four-dimen-sional quantum gauge theories in recent years, focusingon the supersymmetric case."Asymptotic freedom" offour-dimensional gauge theo-ries enables one to understand what happens at highenergies in terms ofweak coupling, but it also means thatthe structure of the vacuum, which determines what theparticles actually are, is governed by strong coupling andhence appears out of reach.



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FIGURE 4. THE MYSTERIOUSQUANTUMWORLD OF MTHEORY, with some of the

previously known theories as different classicallimits (weak coupling). Via a web of dualities,each of these weakly coupled theories can be

interpreted as infinite-coupling limits of others.

One of the main problems is to understand quarkconfinement. As discussed above, some sort ofrelation ofconfinement to duality was conjectured in the 1970s.There are many other mysteries about the vacuum struc-ture of gauge theories. In general, one would like to beable to determine what symmetries are broken, and whatparticles have lowmass, in a given four-dimensional gaugetheory, such as QCD.These problems were a prime focus ofparticle physicsin the mid-to-late 1970s. Many qualitative results wereobtained by a variety ofmethods, including lattice strong-coupling expansions, computer simulations, 1/N expan-sions and matching relations between short-distance andlong-distance calculations. In the 1980s and 1990s, com-puter simulations of QCD improved steadily. But thestudy ofmore general four-dimensional gauge theories wasclearly in need of new ideas.The new idea that was brought to bear in the lastfewyears was, in the first instance, simply to study thesequestions in the supersymmetric case. Supersymmetricgauge theories (for instance, the supersymmetric extensionof QCD) exhibit many of the phenomena that can occurwithout supersymmetry, but supersymmetry brings muchsimplification and enables one to settle questions thatotherwise are out of reach.Investigations along these lines in the last few yearsgave many new results of a sort that physicists hadspeculated about in the 1970swithout being able to exhibitthem in concrete models." (See PHYSICS TODAY, March1995, page 17.) Examples included strange patterns ofsymmetry breaking and the appearance ofexoticmasslessbound states. The results were elegant and surprising;they also seemed disparate and impossible to unify.ConvergenceIn the last three years, it has become clear that thesethings are all part of one story.The new gauge theory results should be derived fromnon-Abelian duality, generalizing that of Montonen andOlive. Such duality is often to be derived from stringtheory. Certain ofthe supergravity symmetries carry overto string theory, where they generalize the dualities thatspelled the doom of spacetime as a fundamental notion.The result is a very new perspective on field theory andstring theory.In fieldtheory,wenowunderstand (assketchedin figure3) that not just quark confinement but the whole range ofsurprises ofstrongly coupledfield theory should be derivedfrom duality, at least in the supersymmetric case.

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Type I

For string theory the change in viewpoint is perhapseven wider and includes the discovery that there is onlyone theory.For weak coupling the five string theories-and thewild card, eleven-dimensional supergravity-are all differ-ent. That is why they have been traditionally understoodas different theories. Understanding them as differentlimits ofone theory requires understanding what happensfor strong coupling.The novelty of the last couple of years, in a nutshell,is that we have learned that the strong-coupling behaviorof supersymmetric string theories and field theories isgoverned by a web ofdualities relating different theories.When one description breaks down because a couplingparameter becomes large, another description takes over.For instance, in uncompactified ten-dimensionalMinkowski space, the strong-coupling limit of the type Isuperstring is the weakly coupled heterotic SO(32) super-string; the strong-coupling limit ofthe type lIAsuperstringis related to eleven-dimensional supergravity; the strong-coupling limit oftype lIB superstring theory is equivalentto the same theory at weak coupling; and the strong-cou-pling limit of the Es x Es heterotic string involves eleven-dimensional supergravity again.From this list, and additional items that appear aftercompactifying some dimensions, we learn that the differ-ent theories are all one. The different supertheoriesstudied in different ways in the last generation are dif-ferent manifestations of one underlying, and still myste-rious, theory, sometimes called M-theory,where M standsfor magic,mystery ormembrane, according to taste." Thistheory is the candidate for superunification of the forcesofnature. It has eleven-dimensional supergravity and allthe traditionally studied string theories among its possiblelow-energymanifestations.The relations between the different string theoriesoften look at low energies like the electric-magnetic du-ality ofMaxwell's equations. Knowledgeof string theorydualities has shed much light on field theory dualities,and vice versa.To understand these dualities, we have had to learnabout new degrees of freedom in string theory, such asD-branes (quantum versions of objects that were firstfound as solitonic solutions of supergravity)." As an illus-tration of the power of the new insight, it has becomepossible for the first time-with the aid of the new vari-ables-to count the quantum states of a black hole (incertain cases), thereby settling a longstanding problem."(See PHYSICS TODAY, March 1997, page 19.)


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D-branes have a very strange property: Their "posi-tions" are described,in general, bynoncommuting matrices.Whenthe matrices commute,their simultaneous eigenvaluesare the D-brane positions inthe traditional sense. This hassuggested that, to properly understand M-theory, thespacetime coordinates must be reinterpreted as noncom-muting matrices, roughly as happened to the coordinates ofclassical phase space when quantum mechanics emerged.This intuitive idea has been implemented with partial suc-cess in a "matrix model"ofM-theory.lOStatus of quantum mechanicsFinally, we are perhaps beginning to see a change in thelogical role of quantum mechanics. Since the 1920s, wehave had quantum systems that were obtained by quan-tizing classical systems. In a sense, that has been thefoundation stone of physics for almost 70 years.Now, in the quest to unify the forces of nature, weare dealing with one mysterious quantum theory that hasthe previously known theories as different classical limits,as sketched in figure 4. The competing classical limitsare equally significant; no one of them is distinguished.The different classical limits are related by dualitiesthat generalize the Montonen-Olive duality

47TliceH'--.eAs we see from the appearance of Ii in this formula, suchdualities are symmetries that exist only in the quantumworld.Thus, in the search for superunification, quantummechanics mikes possible fundamental new symmetriesjust as gravity does in the light of general relativity. Theawareness that gravity makes possible a new symmetryprinciple-general covariance or the principle of equiva-lence-changed forever our understanding of the role ofgravity in the scheme of things. As we approach the 21stcentury, it seems that a similar process may be beginningfor quantum mechanics.References1. C.Montonen, D. Olive, Phys. Lett. B 72, 117 (1977).2. J. Wess, J. Bagger, Supersymmetry and Supergrauity, 2nd ed.,

Princeton U.P.,Princeton (1992). S.J. Gates Jr., M.T.Grisaru,M. Rocek, W. Siegel, Superspace: One Thousand and OneLessons in Supersymmetry, Benjamin/Cummings, Reading,Mass. (1983).

3. For example, see B. Julia, in: Superspace and Supergrauity,S. W. Hawking, M. Rocek, eds., Cambridge U. P. , Cambridge,UK (1981), p. 331.

4. M. J. Duff, R.R. Khuri, J. X.Lu, Phys. Rep. 259, 213 (1995).5. A. Giveon, M.Porrati, E. Rabinovici, Phys. Rep. 244,77 (1994).6. K. Intriligator, N. Seiberg, Nucl. Phys. B Proc. Suppl. 45B,C,

1 (1996). M. E. Peskin, preprint hep-thl9702094 (on the LosAlamos server, http://xxx.lanl.gov/).7. J. H. Schwarz, preprint hep-thl9607201. A. Sen, preprint

hep-thl9609176. P.K. Townsend, preprint hep-thl9612121, toappear in proceedings ofthe Summer School onHigh EnergyPhysics and Cosmology,held at ICTp, Trieste, June 1996.8. J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995). J. Polchinski,S. Chaudhuri, C.V . Johnson, preprint hep-thl9602052. M. R.Douglas, preprint hep-thl9610041, to appear in proceedings ofthe Les Houches session on Quantum Symmetries held inAugust 1995. C. Bachas,preprint hep-thl9701019, to appearin proceedings of the Workshop on Gauge Theories, AppliedSupersymmetry and Quantum Gravity, held at Imperial Col-lege, London, July 1996.

9. A. Strominger, C.Vafa,Phys. Lett. B379,99 (1996). Reviewedin G.Horowitz, preprint gr-qc/9604051, to appear in proceed-ings of the Pacific Conference on Gravitation and Cosmologyheld in Seoul, South Korea, February 1996.

10. T. Banks, W. Fischler, S. H. Shenker, L. Susskind, preprinthep-thl9610043.


witten, e. - duality, spacetime and quantum mechanics

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