M(atrix) theory: matrix quantum mechanics as fundamental theory

43 Pages·2001·0.387 MB·English

by  Taylor W.

REVIEWSOFMODERNPHYSICS,VOLUME73,APRIL2001
M(atrix) theory: matrix quantum mechanics as a fundamental theory
Washington Taylor*
InstituteforTheoreticalPhysics,UniversityofCalifornia,SantaBarbara,California93106
(Published 8June2001)
ThisarticlereviewsthematrixmodelofMtheory.Mtheoryisan11-dimensionalquantumtheoryof
gravitythatisbelievedtounderlieallsuperstringtheories.Mtheoryiscurrentlythemostplausible
candidateforatheoryoffundamentalphysicswhichreconcilesgravityandquantumfieldtheoryina
realistic fashion. Evidence for M theory is still only circumstantial—no complete
background-independentformulationofthetheoryexistsasyet.Matrixtheorywasfirstdevelopedas
a regularized theory of a supersymmetric quantum membrane. More recently, it has appeared in a
differentguiseasthediscretelight-conequantizationofMtheoryinflatspace.Thesetwoapproaches
tomatrixtheoryaredescribedindetailandcompared.Itisshownthatmatrixtheoryisawell-defined
quantum theory that reduces to a supersymmetric theory of gravity at low energies. Although its
fundamental degrees of freedom are essentially pointlike, higher-dimensional fluctuating objects
(branes)arisethroughthenon-Abelianstructureofthematrixdegreesoffreedom.Theproblemof
formulating matrix theory in a general space-time background is discussed, and the connections
betweenmatrixtheoryandotherrelatedmodelsarereviewed.
CONTENTS C. Membranes 448
1. Compactmembranes 448
2. Infinitemembranes 449
I. Introduction 419 3. Wrappedmembranesasmatrixstrings 450
A. Supergravity,strings,andmembranes 421 D. 5-branes 450
B. DualityandD-branes 422 VI. MatrixTheoryinaGeneralBackground 452
C. M(atrix)theory 424 A. Matrixtheoryontori 452
II. MatrixTheoryfromtheQuantizedSupermembrane 425 B. Matrixtheoryincurvedbackgrounds 453
A. Thebosonicmembranetheory 425 VII. RelatedModels 454
B. Thelight-frontbosonicmembrane 426 A. TheIKKTmatrixmodelofIIBstringtheory 454
C. Matrixregularization 426 B. Thematrixmodeloflight-frontIIAstring
D. Thebosonicmembraneinageneralbackground 429 theory 454
E. Thesupermembrane 429 C. TheAdS/CFTcorrespondence 455
F. Covariantmembranequantization 430 VIII. Conclusions 456
III. TheMatrixModelofMTheory 430 Acknowledgments 457
A. Membrane‘‘instability’’ 431 References 457
B. TheconjectureofBanks,Fischler,Shenker,and
Susskind 431 I. INTRODUCTION
C. Matrixtheoryasasecond-quantizedtheory 432
D. Matrixtheoryanddiscretelight-cone In the last two decades, a remarkable structure has
quantization 433
emerged as a candidate for the fundamental theory of
IV. InteractionsinMatrixTheory 435
nature. Until recently, this structure was known prima-
A. Two-bodyinteractions 436
rily under the rubric ‘‘string theory,’’ as it was believed
1. Thebackgroundfieldformalism 436
that the fundamental theory should be most effectively
2. Two-gravitoninteractionsatleadingorder 436
described in terms of quantized fundamental stringlike
3. Generaltwo-bodysystemsandlinearized
supergravityatleadingorder 437 degrees of freedom. Since 1995, however, several new
4. Generaltwo-bodyinteractions 439 developments have drastically modified our perspective.
5. Generaltwo-gravitoninteractions 441 An increased understanding of nonperturbative aspects
B. TheN-bodyproblem 443 of string theory has led to the realization that all the
C. Longitudinalmomentumtransfer 444 knownconsistentstringtheoriesseembespeciallimiting
D. Summaryandoutlookforthecorrespondenceof cases of a more fundamental underlying theory, which
matrixtheoryandsupergravity 444
has been dubbed ‘‘M theory.’’ While the consistent su-
V. M-theoryObjectsfromMatrixTheory 445
perstring theories give microscopic models for quantum
A. Supergravitons 446
gravity in ten dimensions, M theory seems to be most
1. Classicalsupergravitons 446
naturallydescribedinelevendimensions.Wedonotyet
2. Quantumsupergravitons 446
B. Extendedobjectsfrommatrices 447 have a truly fundamental definition of M theory. It may
bethatinitsmostnaturalformulation,thedimensional-
ity of space-time emerges in a smooth approximation to
*Electronic address: [email protected]; Permanent address: a nongeometrical mathematical system.
Center for Theoretical Physics, MIT, Bldg. 6-306, Cambridge, Atthesametimethatstringtheoryhasbeenreplaced
MA02139. by M theory as the most natural candidate for a funda-
0034-6861/2001/73(2)/419(43)/$28.60 419 ©2001TheAmericanPhysicalSociety
420 WashingtonTaylor: M(atrix)theory
mentaldescriptionoftheworld,thestringitselfhasalso shown for all linearized gravitational interactions and a
lostitspositionasthemaincandidateforafundamental subset of nonlinear interactions. This is the first time
degree of freedom. Both M theory and string theory that it has been possible to show explicitly that a well-
contain dynamical objects of several different dimen- defined microscopic quantum-mechanical theory agrees
sionalities. In addition to one-dimensional string excita- with classical gravity at long distances, including some
tions(1-branes),stringtheoriescontainpointlikeobjects nonlinear corrections from general relativity. Under-
(0-branes), membranes (2-branes), three-dimensional standing the correspondence between matrix quantum
extended objects (3-branes), and objects of all dimen- mechanicsandclassicalsupergravityindetailgivessome
sions up to eight or nine. Eleven-dimensional M theory, important new insights into the connections between
on the other hand, seems to contain dynamical mem- quantum-mechanical systems with matrix degrees of
branes and 5-branes. Amongst all these degrees of free- freedom and gravity theories.
dom, there is no obvious reason why the ‘‘string’’ of OneremarkableaspectofthematrixdescriptionofM
string theory is any more fundamental than, say, the theory is the fact that it describes classical gravitational
pointlike or 3-brane excitations of string theory, or the interactions through quantum-mechanical effects. In
membrane of M theory. While the perturbative string classical matrix theory separated objects experience no
expansion makes sense in a regime of the theory where
interactions.Performingaone-loopcalculationinmatrix
thestringcouplingissmall,therearealsolimitsinwhich
quantum mechanics gives classical Newtonian (linear-
the theory is described by the low-energy dynamics of a
ized) gravitational interactions. Higher-order general
system of higher- or lower-dimensional branes. It seems
relativistic corrections to the linearized gravity theory
that by considering the dynamics of any of these sets of
arise from higher-loop calculations in matrix theory.
degrees of freedom, we can access at least some part of
This connection between a classical theory of gravity
the full physics of M theory.
and a quantum system with matrix degrees of freedom
This review article concerns itself with a remarkably
was the first example found of what now seems to be a
simple theory that is believed to be equivalent to M
verygeneralfamilyofcorrespondences.Theconjectured
theory in a particular reference frame. The theory in
equivalence between strings (ten-dimensional quantum
question is a simple quantum mechanical system matrix
gravity) propagating on an anti–de Sitter background
degrees of freedom. The quantum-mechanical degrees
offreedomareafinitesetofbosonicN3N matricesand and a conformal quantum field theory (the celebrated
fermionic partners, which combine to form a system AdS/CFT correspondence) gives another wide class of
with a high degree of supersymmetry. It is believed that examples of this type of correspondence. We discuss
this matrix quantum mechanics theory provides a other examples of such connections in Sec. VII.
second-quantized description of M theory around a flat Anotherremarkableaspectofmatrixtheoryistheap-
space-time background and in a light-front coordinate pearance of the extended objects of M theory (the su-
system.ThefiniteintegerN servesasaregulatorforthe permembrane and M5-brane) in terms of apparently
theory, and the exact correspondence with M theory in pointlike fundamental degrees of freedom. There is a
flat space-time emerges only in the large-N limit. Since rich mathematical structure governing the way in which
this system has a finite number of degrees of freedom objects of higher dimension can be encoded in noncom-
foranyvalueofN, itismanifestlyawell-definedtheory. muting matrices. This structure may eventually lead us
Since it is a quantum mechanics theory rather than a to crucial new insights into the way in which all the
quantum field theory, it does not even exhibit the stan-
many-dimensional excitations of M theory and string
dard problems of renormalization and other subtleties
theory arise in terms of fundamental degrees of free-
that afflict any but the simplest quantum field theories.
dom.
It may seem incredible that a simple matrix quantum
Thisreviewfocusesprimarilyonsomebasicaspectsof
mechanics model can capture most of the physics of M
matrix theory: the definitions of the theory through
theory, and thus perhaps of the real world. This would
regularization of the supermembrane and through light-
imply that matrix theory provides a calculational frame-
frontcompactification,theappearanceofclassicalsuper-
work in which, at least in principle, questions of quan-
gravity interactions through quantum effects, and the
tumeffectsingravityandPlanckscalecorrectionstothe
construction of the objects of M theory in terms of ma-
standard model could be determined to an arbitrarily
trix degrees of freedom. There are many other interest-
high degree of accuracy by a large enough computer.
ingdirectionsinwhichprogresshasbeenmade.Reviews
Unfortunately, however, although it is only a quantum
mechanics theory, matrix theory is a remarkably tricky ofmatrixtheoryandrelatedworkthatemphasizediffer-
modelinwhichtoperformdetailedcalculationsrelevant ent aspects of the subject include those of Bigatti and
tounderstandingquantumcorrectionstogeneralrelativ- Susskind(1997),Banks(1998,1999),NicolaiandHelling
ity, even at very small values of N. (1998), Taylor (1998, 2000), Bilal (1999), de Wit (1999),
Although it is technically difficult to study detailed Obers and Pioline (1999), and Konechny and Schwarz
aspects of quantum gravity using the matrix-theory ap- (2000).
proach, it is possible to demonstrate analytically that In the remainder of this section, we give a brief over-
classical 11-dimensional gravitational interactions are view of a number of ideas that form the background for
produced by matrix quantum mechanics. This has been the discussion in the remainder of the review. This sec-
Rev.Mod.Phys.,Vol.73,No.2,April2001
WashingtonTaylor: M(atrix)theory 421
tion is intended to be a useful introduction to matrix theories in lower dimensions can be derived from the
theory and M theory for the nonspecialist. In Sec. I.A 11D theory by compactifying some subset of the dimen-
we review some basic aspects of classical supergravity sions (or by considering a dual limit of a compactifica-
theories and the appearance of strings and membranes tion, as in the ten-dimensional type-IIB supergravity
in these theories. In Sec. I.B we discuss the two major theory, which we shall discuss momentarily). We recall
developments of the second superstring revolution: du- heresomebasicfeaturesofeleven-andten-dimensional
alityandD-branes.Wefocusinparticularontheduality supergravity theories. For more details the reader may
relating M theory to a strongly coupled limit of string consultGreen,Schwarz,andWitten(1987)orTownsend
theory. Section I.C gives a brief introduction to matrix (1996b).
theory in the context of the developments summarized By examining the structure of the supersymmetry
in Secs. I.A and I.B. The material in this section is es- multiplet containing the graviton, one may determine
the set of classical fields that appear in any supergravity
sentially an overview of the remainder of the review.
theory. In 11D supergravity, there are the following
A. Supergravity,strings,andmembranes propagating fields:1
ea: vielbein field (bosonic, with 44 components)
I
The principal outstanding problem of theoretical A : 3-formpotential(bosonic,with84components)
IJK
physics at the close of the 20th century is to find a the- c: Majorana fermion gravitino (fermionic, with 128
I
oreticalframeworkthatcombinestheclassicaltheoryof components).
general relativity at large distance scales with the stan- Thevielbeinea isanalternativedescriptionofthespace-
I
dardmodelofquantumparticlephysicsatshortdistance timemetrictensorg . The3-formfieldA isantisym-
IJ IJK
scales. At the phenomenological and experimental lev- metric in its indices and plays a role very similar to the
els, the next major challenge is to extend the standard vector potential Amof classical electromagnetism.
model of particle physics to describe physics at and In ten dimensions there are two supergravity theories
above the TeV scale. For both of these endeavors, a with 32 SUSY generators. These are N52 theories,
potentially key structure is the idea of a ‘‘supersymme- since the supersymmetry generators comprise two 16-
try,’’ which relates bosonic and fermionic fields through component spinors. In type-IIA supergravity these
a symmetry group with anticommuting (Grassmann) spinors have opposite chirality, while in type-IIB super-
generators Qa, where ais a spinor index. For an intro- gravity the spinors have the same chirality. In addition
ductiontosupersymmetry,seeWessandBagger(1992). tothemetrictensor/vielbeinfield,bothtype-IIAandIIB
Inasupersymmetrictheoryinflatspace,theanticom- supergravity have several other propagating bosonic
mutator of a pair of supersymmetry (SUSY) generators fields. The IIA and IIB theories both have a scalar field
Qais a translation generator or linear combination of f(the dilaton) and an antisymmetric 2-form field Bmn.
generators: $Q,Q%;Pm. If supersymmetry plays any Each of the type-II theories also has a set of antisym-
role in describing physics in the real world, it must be metric ‘‘Ramond-Ramond’’ p-form fields C(p) . For
m m
1fl p
necessary to incorporate local supersymmetry into Ein- for the type-IIA theory, pP$1,3% is even, and for the
stein’s theory of gravity. The supersymmetry generators type-IIB theory pP$0,2,4% is odd.
cannot simply describe a global symmetry of the funda-
Like the 3-form field A of 11D supergravity, the
IJK
mentaltheory,sinceingeneralrelativitythemomentum
antisymmetric 2-form field Bmn and the Ramond-
generator that appears as an anticommutator of two
Ramond p-form fields of the type-II supergravity theo-
SUSY generators becomes a local vector field generat-
riesarecloselyanalogoustothevectorpotentialofelec-
ingadiffeomorphismofspace-time.Inatheorycombin-
tromagnetism. In both the type-IIA and IIB
ing general relativity with supersymmetry, supersymme-
supergravity theories there are classical stringlike ex-
trygeneratorsbecomespinor-valuedfieldsonthespace-
tremal black-hole solutions of the field equations which
time manifold.
are charged under the 2-form field (Dabholkar etal.,
It is possible to classify supersymmetric theories of
1990),aswellashigher-dimensionalbranesolutionsthat
gravity (supergravity theories) by constructing super-
couple to the p-form fields (for a review, see Duff,
symmetry algebras with multiplets containing particles
Khuri, and Lu, 1995). The dynamics of these stringlike
of spin 2 (gravitons). In any dimension greater than
and branelike solutions can be described through an ef-
eleven, supersymmetry multiplets automatically contain
fectiveactionlivingontheworldvolumeofthestringor
particles of spin higher than 2, so that the maximal di-
higher-dimensional brane. Just as the electromagnetic
mension for a supergravity theory is eleven. Indeed,
vector potential Am couples to an electrically charged
there is a unique such classical theory in eleven dimen-
particle through a term of the form
sions with local supersymmetry (Cremmer, Julia, and E
Scherk, 1978). This theory has N51 supersymmetry, m
AmdX (1)
meaning that the supersymmetry generators live in a L
single 32-component spinor representation of the 11D
Lorentzgroup.ThegeneratorsQaextendtheusual11D
Poincare´ algebra into a super-Poincare´ algebra. Eleven- 1We denote space-time indices in 11 dimensions by capital
dimensionalsupergravityisinanaturalsensetheparent roman letters I,J,K,...P$0,1,...,8,9,11%, and indices in 10 di-
of all other supergravity theories, since all supergravity mensionsbyGreeklettersm,n,...P$0,1,...,9%.
Rev.Mod.Phys.,Vol.73,No.2,April2001
422 WashingtonTaylor: M(atrix)theory
whereListhetrajectoryoftheparticle,the2-formfield sight into the theory, although the work of Sen (1999)
of type-II supergravity couples to the two-dimensional and others has recently generated a new wave of devel-
string world sheet through a term of the form opment in this direction.
E
To summarize our discussion of string theory, it has
Bmneab~]aXm!~]bXn!, (2) been found that a natural approach to finding a micro-
S scopicquantumtheoryofgravitywhoselow-energylimit
where Xm are the embedding functions of the string isten-dimensionalsupergravityistoquantizethestring-
world sheet S in ten dimensions and a,bP$0,1% are likedegreesoffreedomthatcoupletotheantisymmetric
world-sheet indices. 2-form field Bmn.
The tension of the string is given by T 51/2pa8, Because 11-dimensional supergravity seems to be in
s
where l 5Aa8 is the fundamental string length. The some sense more fundamental than the ten-dimensional
s
theory, it is natural to want to find an analogous con-
starting point for perturbative string theory is the quan-
struction of a microscopic quantum theory of gravity in
tization of the world-sheet action on a string, treating
m 11 dimensions. Unlike the ten-dimensional theories,
the space-time coordinates X as bosonic fields on the
however, in 11-dimensional supergravity there is no
string world sheet. The remarkable consequence of this
stringlikeblack-holesolution;indeed,thereisno2-form
quantization is that quanta of all the fields in the super-
for it to couple to. There is, however, a ‘‘black-
gravitymultipletariseasmasslessexcitationsofthefun-
membrane’’ solution in 11 dimensions, which has a
damental string. It has been shown that there are five
sourceextendedinfinitelyintwospatialdimensions.Just
consistent quantum superstring theories which can be
as the black string couples to the 2-form field through
constructed by choosing different sets of fields on the
Eq.(2),theblack-membranesolutionof11Dsupergrav-
string world sheet. These are the type-I, IIA, IIB, and
heterotic E 3E and SO(32) theories. In each of these ity couples to the 3-form field through
8 8 E
cases, string theory seems to give a consistent micro-
scopic description of interactions between gravitational AIJKeabc~]aXI!~]bXJ!~]cXK!, (3)
S
quanta. In addition to massless fields, there is also an
infinitetoweroffieldsineachtheorywithmassesonthe where now a,b,cP$0,1,2% are indices of coordinates on
orderof1/l . Inprinciple,anyscatteringprocessinvolv- the three-dimensional membrane world volume.
s
ingafinitenumberofmasslesssupergravityparticlescan It is tempting to imagine that a microscopic descrip-
be systematically calculated as a perturbative expansion tion of 11D supergravity might be found by quantizing
in string theory. The strength of string interactions is thesupermembrane,justasamicroscopicdescriptionof
encoded in the dilaton field through the string coupling 10Dsupergravityisfoundbyquantizingthesuperstring.
g5ef. The perturbative string expansion makes sense This idea was explored extensively in the 1980s, when it
when g is small. was first realized that a consistent classical theory of a
We shall not discuss string theory in any detail in this supermembrane could be realized in 11 dimensions. At
review; for a comprehensive introduction to superstring thattime,whilenosatisfactorycovariantquantizationof
theory, the reader should consult the excellent text- the membrane theory was found, it was shown that the
books by Green, Schwarz, and Witten (1987) and by supermembrane could be quantized in light-front coor-
Polchinski (1998). We would like, however, to empha- dinates. As we shall discuss in greater detail in the fol-
lowing sections, this construction leads to precisely the
size the following points:
matrix quantum mechanics theory that is the subject of
(i) The world-sheet approach to superstring quanti-
this article. Not only does this theory provide a micro-
zation yields a first-quantized theory of gravity
scopic description of quantum gravity in 11 dimensions,
fromthepointofviewofthetargetspace—thatis,
but, as more recent work has demonstrated, it also by-
astateinthestringHilbertspacecorrespondstoa
passes the difficulties mentioned above for string theory
single-particle state in the target space consisting
by directly providing a nonperturbative definition of a
of a single string.
theory that is second quantized in target space.
(ii) The world-sheet approach to superstrings is per-
turbativeinthestringcouplingg. Asweshalldis- B. DualityandD-branes
cussinthefollowingsection,therearemanynon-
perturbative objects that should appear in a Although 11-dimensional supergravity and the quan-
consistent quantum theory of 10D supergravity. tum supermembrane theory were originally discovered
In order to have a definition of string theory that cor- at around the same time as the five consistent super-
responds to a true quantum theory of gravity in space- string theories, much more attention was given to string
time, it is necessary to overcome these obstacles by de- theory in the decade 1985–1995 than to the 11-
velopingasecond-quantizedtheoryofstrings.Workhas dimensional theory. There were several reasons for this
been done towards developing such a string field theory lack of attention to 11D supergravity and membrane
(see,forexample,Zwiebach,1993;GaberdielandZwie- theory by (much of) the high-energy community. For
bach, 1997a, 1997b). It is currently difficult to use this one thing, heterotic string theory looked like a much
formalism to do practical calculations or gain new in- more promising framework in which to make contact
Rev.Mod.Phys.,Vol.73,No.2,April2001
WashingtonTaylor: M(atrix)theory 423
with standard-model phenomenology. In order to con- however,berelevanttothisarticle.TheDp-braneswith
nect a ten- or 11-dimensional theory with four- p<3 coupletotheRamond-Ramond(p11)-formfields
dimensionalphysics,itisnecessarytocompactifyallbut of supergravity through expressions analogous to Eqs.
fourdimensionsofspace-time[or,ashasbeensuggested (2)and(3).Thesebranesarereferredtoasbeing‘‘elec-
more recently by Randall and Sundrum (1999) and oth- trically coupled’’ to the relevant Ramond-Ramond
ers, to consider our 4D space-time as a brane living in fields. The Dp-branes with p>4 have ‘‘magnetic’’ cou-
the higher-dimensional space-time]. There is no way to plings to the (72p)-form Ramond-Ramond fields,
compactify 11D supergravity on a smooth 7-manifold in which can be described in terms of electric couplings to
such a way as to give rise to chiral fermions in the re- the dual fields C˜(p11) defined through dC˜(p11)
sulting 4D theory (Witten, 1981). This fact made 11D 5*dC(72p). D-branesarenonperturbativestructuresin
supergravity for some time a very unattractive possibil-
first-quantizedstringtheory,butplayafundamentalrole
ity for a fundamental theory; more recently, however,
in many aspects of quantum gravity. In recent years
singular (orbifold) compactifications of 11D M theory
D-branes have been used to construct stringy black
have been considered (Horˇava and Witten, 1996) which
holes and to explore connections between string theory
lead to realistic models of phenomenology with chiral
and quantum field theory. Basic aspects of D-brane
fermions(seeforexample,Donagietal.,2000).Another
physics are reviewed by Polchinski (1996) and Taylor
reason for which the quantum supermembrane was
(1998); applications of D-branes to black holes are re-
dropped from the mainstream of research was the ap-
viewed by Skenderis (1999), Mohaupt (2000), and Peet
pearance of an apparent instability in the membrane
(2000); a recent comprehensive review of D-brane con-
theory(deWit,Lu¨scher,andNicolai,1989).Asweshall
structions of supersymmetric field theories is given by
discuss in Sec. III, rather than being a problem this ap-
Giveon and Kutasov (1999).
parent instability is an indication of the second-
CombiningtheideasofdualityandD-branes,wehave
quantized nature of the membrane theory.
a new picture of fundamental physics as having an as-
As was briefly discussed in the introduction, in 1995
yet-unknown microscopic structure, which reduces in
two remarkable new ideas caused a substantial change
certain limits to perturbative string theory and to 11D
in the dominant picture of superstring theory. The first
supergravity. In the 10D and 11D limits, there are a va-
ofthesewastherealizationthatallfivesuperstringtheo-
riety of dynamical extended objects of various dimen-
ries, as well as 11-dimensional supergravity, seem to be
sionsappearingaseffectiveexcitations.Thereisnoclear
related to one another by duality transformations that
reason for strings to be any more fundamental in this
exchange the degrees of freedom of one theory for the
structure than the membrane in 11-dimensions, or even
degrees of freedom of another theory (Hull and
than D0-branes or D3-branes in type-IIA or IIB super-
Townsend, 1995; Witten, 1995). It is now generally be-
string theory. At this point, in fact, it seems likely that
lieved that all six of these theories are realized as par-
these objects should all be thought of as equally impor-
ticular limits of some more fundamental underlying
tant pieces of the theory. On one hand, the strings and
theory, which may be describable as a quantum theory
branes can all be thought of as effective excitations of
in 11 dimensions. This 11-dimensional quantum theory
someas-yet-unknownsetofdegreesoffreedom.Onthe
of gravity, for which no rigorous definition has yet been
other hand, by quantizing any of these objects to what-
given, is often referred to as ‘‘M theory’’ (Horˇava and
ever extent is technically possible for an object of the
Witten, 1996).2
relevant dimension, it is possible to study particular as-
The second new idea in 1995 was the realization by
pectsofeachofthetheoriesincertainlimits.Thisequal-
Polchinski (1995) that black p-brane solutions that are
itybetweenbranesisoftenreferredtoas‘‘branedemoc-
charged under the Ramond-Ramond fields of string
racy.’’
theory can be described in the language of perturbative
Asweshallseeintheremainderofthisreview,matrix
strings as ‘‘Dirichlet-branes,’’ or ‘‘D-branes,’’ that is, as
theory can be thought of alternatively as a quantum
hypersurfaces on which open strings may have end-
theory of membranes in 11 dimensions or as a quantum
points. Type-IIA string theory contains Dp-branes with
theory of pointlike D0-branes in ten dimensions. In or-
p50, 2, 4, and 6, while type-IIB string theory contains dertorelatethesecomplementaryapproachestomatrix
Dp-branes with p521, 1, 3, 5, and 7. Dp-branes with theory, it will be helpful at this point to briefly review
p>8 also appear in certain situations; they will not, oneofthesimplestlinksinthenetworkofdualitiescon-
necting the string theories with M theory. This is the
duality that relates M theory to type-IIA string theory
2The term ‘‘M theory’’ is usually used to refer to an 11- (Townsend, 1995; Witten, 1995). The connection be-
dimensional quantum theory of gravity that reduces to N51 tween these theories essentially follows from the fact
supergravity at low energies. It is possible that a more funda-
thattype-IIAsupergravitycanbeconstructedfrom11D
mental description of this 11-dimensional theory and string
supergravity by performing ‘‘dimensional reduction’’
theory can be given by a model in terms of which the dimen-
along a single dimension. To implement this procedure
sionalityofspace-timeiseithergreaterthan11orisanemer-
we assume that 11D supergravity is defined on a space-
gentaspectofthedynamicsofthesystem.Generallytheterm
time with geometry M103S1 where M10 is an arbitrary
M theory does not refer to such models, but usage varies. In
thisarticlewemeanbyMtheoryaconsistentquantumtheory 10D manifold and S1 is a circle of radius R. When R is
ofgravityin11dimensions. small we can systematically neglect the dependence of
Rev.Mod.Phys.,Vol.73,No.2,April2001
424 WashingtonTaylor: M(atrix)theory
all fields in the 11D theory along the 11th (compact) we describe in detail in the following sections. As dis-
direction, giving an effective low-energy 10D theory, cussed above, it seems that a natural way to try to con-
which turns out to be type-IIA supergravity. In this di- struct a microscopic model for M theory would be to
mensional reduction, the different components of the quantize the supermembrane that couples to the 3-form
fields of the 11D theory decompose into the various
field of 11D supergravity. In general, quantizing any
fields of the 10D theory. The metric tensor g in 11
IJ fluctuating geometrical object of higher dimensionality
dimensions has components gmn, 0<m,n<9, which be-
thanthestringisaproblematicenterprise,andforsome
bcoemcoemteheth1e0DRmametorincd-tRenasmoro,ncdomvpeoctnoerntsfiegldm11C, (w1)hicihn time it was believed that membranes and higher-
m
dimensional objects could not be described in a sensible
the 10D theory, and a single component g , which
11 11
becomes the 10D dilaton field f. Similarly, the 3-form fashion by quantum field theory. Almost two decades
fieldofthe11Dtheorydecomposesintothe2-formfield ago,however,Goldstone(1982)andHoppe(1982,1987)
Bmnand the Ramond-Ramond 3-form field Cm(3n)l in ten foundaverycleverwayofregularizingthetheoryofthe
dimensions.Justasthefieldsof11Dsupergravityreduce classical membrane. They replaced the infinite number
to the fields of the type-IIA theory under dimensional of degrees of freedom representing the embedding of
reduction, the extended objects of M theory reduce to the membrane in space-time by a finite number of de-
branes of various kinds in type-IIA string theory. The grees of freedom contained in N3N matrices. This ap-
membrane of M theory can be ‘‘wrapped’’ around the
proach, which we describe in detail in the next section,
compactdirectionofradiusR tobecomethefundamen-
wasgeneralizedbydeWit,Hoppe,andNicolai(1988)to
tal string of the type-IIA theory. The unwrapped
the supermembrane. The resulting theory is a simple
M-theory membrane corresponds to the Dirichlet
quantum-mechanicaltheorywithmatrixdegreesoffree-
2-brane(D2-brane)intypeIIA.Inadditiontothemem-
dom. The Hamiltonian of this theory is given by
brane, M theory has an M5-brane (with six-dimensional S D
world volume), which couples magnetically to the 1 1 1
3-form field A . Wrapped M5-branes become D4- H5Tr X˙ iX˙i2 @Xi,Xj#@Xi,Xj#1 uTg@Xi,u# . (5)
IJK 2 4 2 i
branesintypeIIA,whileunwrappedM5-branesbecome
solitonic (NS) 5-branes in type IIA, which are magneti- In this expression, Xi are nine N3N matrices, uis a
callychargedobjectsundertheNS-NS2-formfieldBmn. 16-component matrix-valued spinor of SO(9), and g
i
Throughthedimensionalreductionof11Dsupergrav-
are the SO(9) gamma matrices in the 16-dimensional
ity to type-IIA supergravity, the string coupling g and
representation. Even before its discovery as a regular-
string length l in the 10D theory can be related to the
s izedversionofthesupermembranetheory,thisquantum
11D Planck length l and the compactification radiusR
11 mechanics theory had been studied as a particularly el-
through
S D egant example of a quantum system with a high degree
R 3/2 l3 of supersymmetry (Baake, Reinicke, and Rittenberg,
g5 , l25 11. (4)
l s R 1985; Claudson and Halpern, 1985; Flume, 1985).
11
Although the Hamiltonian (5) describing matrix
From these relations we see that in the strong-coupling
theory and its connection with the supermembrane has
limit g!‘, type-IIA string theory ‘‘grows’’ an extra di-
been known for some time, this theory was previously
mension R!‘ and should be identified with M theory
believed to suffer from insurmountable instability prob-
in flat space. This motivates a definition of M theory as
lems. It was pointed out several years ago by Townsend
the strong-coupling limit of the type-IIA string theory
(1996a) and by Banks, Fischler, Shenker, and Susskind
(Witten,1995);becausethereisnononperturbativedefi-
(1997) that Eq. (5) can also be seen as arising from a
nitionoftype-IIAstringtheory,however,thisdefinition
systemofN Dirichletparticlesintype-IIAstringtheory.
is not completely satisfactory.
Using the duality relationship between M theory and
WhenthecompactificationradiusR usedtoreduceM
type-IIAstringtheorydescribedabove,Banks,Fischler,
theory to type IIA is small, momentum modes in the
11th direction of the massless fields associated with the Shenker, and Susskind (henceforth ‘‘BFSS’’) made the
11D graviton multiplet become massive Kaluza-Klein bold conjecture that in the large-N limit the system de-
particles in the 10D type-IIA theory. These particles finedbyEq.(5)shouldgiveacompletedescriptionofM
couple to the components gm11 of the 11D metric, and theoryinthelight-front(infinite-momentum)coordinate
thereforetoC(1) intendimensions.Thustheseparticles frame. This picture cleared up the apparent instability
m
problems of the theory in a very satisfactory fashion by
can be identified as the Dirichlet 0-branes of type-IIA
making it clear that matrix theory should describe a
stringtheory.Thisconnectionbetweenmomentumin11
second-quantized theory in target space, rather than a
dimensions and Dirichlet particles, first emphasized by
first-quantized theory as had previously been imagined.
Townsend(1996a),isacrucialingredientinunderstand-
Following the BFSS conjecture, there was a flurry of
ingtheconnectionbetweenthetwoperspectivesonma-
activity for several years centered on the matrix model
trix theory that we develop in this review.
defined by Eq. (5). In this period of time much progress
C. M(atrix)theory was made in understanding both the structure and the
limitsofthisapproachtostudyingMtheory.Ithasbeen
In this section we briefly summarize the development shownthatmatrixtheorycanindeedbeconstructedina
ofmatrixtheory,givinganoverviewofthematerialthat fairly rigorous way as a light-front quantization of M
Rev.Mod.Phys.,Vol.73,No.2,April2001
WashingtonTaylor: M(atrix)theory 425
theory by taking a limit of spatial compactifications. A tothetheoryofaclassicalrelativisticbosonicstring.We
fairly complete picture has been formed of how the ob- do not assume familiarity with string theory, and give a
jects of M theory (the graviton, membrane, and M5- self-containeddescriptionofthemembranetheoryhere;
brane) can be constructed from matrix degrees of free- readers unfamiliar with the somewhat simpler classical
dom. It has also been shown that all linearized bosonic string may wish to look at the texts by Green,
supergravitational interactions between these objects Schwarz,andWitten(1987)andbyPolchinski(1998)for
and some nonlinear general relativistic corrections can comparison with the discussion here.
bederivedfromquantumeffectsinmatrixtheory.Some
Just as a particle sweeps out a trajectory described by
simple compactifications of M theory have been con-
aone-dimensionalworldlineasitmovesthroughspace-
structed in the matrix-theory formalism, leading to new time,adynamicalmembranemovinginD21 spatialdi-
insight into connections between certain quantum field
mensions sweeps out a three-dimensional world volume
theories and quantum theories of gravity. The goal of
in D-dimensional space-time. We can think of the mo-
this article is to review these developments in some de-
tion of the membrane in space-time as being described
tailandtosummarizeourcurrentunderstandingofboth
byamapX:V!RD21,1 takingathree-dimensionalmani-
the successes and the limitations of the matrix-model
fold V (the membrane world volume) into flat
approach to M theory.
D-dimensionalMinkowskispace.Wecanlocallychoose
In the following sections, we develop the structure of
a set of three coordinates sa,aP$0,1,2%, on the world
matrix theory in more detail. Section II reviews the
original description of matrix theory in terms of a regu- volume of the membrane, analogous to the coordinatet
larization of the quantum supermembrane theory. In used to parametrize the world line of a particle moving
Sec. III we describe the theory in the language of light- in space-time. We shall sometimes use the notation t
front quantized M theory and discuss the second- 5s and we shall use indices a,b, ... to describe ‘‘spa-
0
quantizednatureoftheresultingspace-timetheory.The tial’’coordinatessaP$1,2% onthemembraneworldvol-
connectionbetweenclassicalsupergravityinteractionsin ume. In such a coordinate system, the motion of the
space-time and quantum loop effects in matrix theory is membranethroughspace-timeisdescribedbyasetofD
presented in Sec. IV. In Sec. V we show how the ex- functions Xm(s0,s1,s2).
tendedobjectsofMtheorycanbedescribedintermsof Thenaturalclassicalactionforamembranemovingin
matrix degrees of freedom. In Sec. VI we discuss exten- flat space-time is given by the integrated proper volume
sions of the basic matrix-theory conjecture to other swept out by the membrane. This action takes the
space-time backgrounds, and in Sec. VII we briefly re-
Nambu-Goto form
view the connection between matrix theory and several E
other related models. Section VIII contains concluding A
S52T d3s 2dethab, (6)
remarks.
where T is a constant that can be interpreted as the
membrane tension T51/(2p)2l3, while
II. MATRIX THEORY FROM THE QUANTIZED p
SUPERMEMBRANE hab5]aXm]bXm (7)
In this section we describe in some detail how matrix is the pullback of the flat space-time metric (with signa-
theory arises from the quantization of the supermem- ture 211 1) to the three-dimensional membrane
fl
brane.InSec.II.Awedescribethetheoryoftherelativ- world volume.
istic bosonic membrane in flat space. The light-front de- Because of the square root, it is cumbersome to ana-
scription of this theory is discussed in Sec. II.B, and the lyze the membrane theory directly using this action.
matrix regularization of the theory is described in Sec. There is a convenient reformulation of the membrane
II.C. In Sec. II.D we briefly describe the bosonic mem- theory that leads to the same classical equations of mo-
braneinageneralbackgroundgeometry.InSec.II.Ewe tion using a polynomial action. This is the analog of the
extend the discussion to the supermembrane. The prob- Polyakov action for the bosonic string. In order to de-
lemoffindingacovariantmembranequantizationisdis- scribethemembraneusingthisapproach,wemustintro-
cussed in Sec. II.F. duce an auxiliary metric gab on the membrane world
The material in this section roughly follows the origi- volume. We then take the action to be
nal papers by Hoppe (1982, 1987) and de Wit, Hoppe, E
and Nicolai (1988). Note, however, that the original S52T d3sA2g~gab]aXm]bXm21!. (8)
2
derivation of the matrix quantum mechanics theory was
done in the Nambu-Goto-type membrane formalism, The final constant term 21 inside the parentheses does
while we use here a Polyakov-type approach. not appear in the analogous string theory action. This
additional‘‘cosmological’’termisneededduetotheab-
A. Thebosonicmembranetheory
sence of scale invariance in the theory.
Computing the equations of motion from Eq. (8) by
Inthissectionwereviewthetheoryofaclassicalrela-
tivisticbosonicmembranemovinginflatD-dimensional varying gab, we get
Minkowski space. This analysis is very similar in flavor gab5hab5]aXm]bXm. (9)
Rev.Mod.Phys.,Vol.73,No.2,April2001
426 WashingtonTaylor: M(atrix)theory
Replacing this in Eq. (8) again gives Eq. (6), so we see theoryisstillcompletelycovariant.Itisdifficulttoquan-
that the two forms of the action are actually equivalent. tize,however,becauseoftheconstraintsandthenonlin-
The equation of motion that arises from varying Xmin earity of the equations of motion. The direct quantiza-
Eq. (8) is ]a(A2ggab]bXm)50. tion of this covariant theory will be discussed further in
Tosimplifytheanalysis,wewouldnowliketousethe Sec. II.F.
symmetries of the theory to gauge-fix the metric gab.
Unfortunately, unlike the case of the classical string, in
B. Thelight-frontbosonicmembrane
which there are three components of the metric and
three continuous symmetries (two diffeomorphism sym-
We now consider the membrane theory in light-front
metries and a scale symmetry), for the membrane we
coordinates
havesixindependentmetriccomponentsandonlythree
diffeomorphism symmetries. We can use these symme- X65~X06XD21!/&. (17)
tries to fix the components g0aof the metric to be The constraints (14) and (15) can be explicitly solved in
4 4 light-front gauge,
g 50, g 52 ¯h[2 deth , (10)
0a 00 n2 n2 ab X1~t,s ,s !5t. (18)
1 2
where n is an arbitrary constant whose normalization We have
has been chosen to make the later matrix interpretation
1 1
transparent.Oncewehavechosenthisgauge,nofurther X˙ 25 X˙ iX˙ i1 $Xi,Xj%$Xi,Xj%, ]X25X˙ i]Xi.
components of the metric g can be fixed. This gauge 2 n2 a a
ab (19)
canonlybechosenwhenthemembraneworldvolumeis
oftheformS3R, whereSisaRiemannsurfaceoffixed We can go to a Hamiltonian formalism by computing
topology. The membrane action becomes in this gauge, the canonically conjugate momentum densities. The to-
1
using Eq. (9) to eliminate g, tal momentum in the direction P is then
S D E
E
S5T4n d3s X˙ mX˙ m2n42¯h . (11) p15 d2sP152pnT, (20)
and the Hamiltonian of the theory is given by
Itisnaturaltorewritethisactionintermsofacanoni- S D
E
cal Poisson bracket on the membrane where at constant nT 2
t, $f,g%[eab]af]bg with e1251. We shall assume that H5 4 d2s X˙ iX˙ i1n2$Xi,Xj%$Xi,Xj% . (21)
thecoordinatessarechosensothat,withrespecttothe
symplectic form associated with this canonical Poisson The only remaining constraint that the transverse de-
bracket, the volume of the Riemann surface S is *d2s grees of freedom must satisfy is
54p. In terms of the Poisson bracket, the membrane
$X˙ i,Xi%50. (22)
action becomes
S D
E This theory has a residual invariance under time-
Tn 2
S5 4 d3s X˙ mX˙ m2n2$Xm,Xn%$Xm,Xn% . (12) ifneodmepoernpdheisnmtasrdeao-pnroetsecrhvainnggeditfhfeeomsyomrpplheicstmics.fSourmchadnifd-
The equations of motion for the fields Xmare thus manifestly leave the Hamiltonian (21) invariant.
We now have a Hamiltonian formalism for the light-
X¨ m5n42]a~¯hhab]bXm!5n42(cid:136)$Xm,Xn%,Xn(cid:137). (13) fsrtoilnltramtheemrbdrifafinceultthteoorqyu.anUtnizfeo.rtUunnlaitkeelys,trtihnigstthheeoorryy, iins
which the equations of motion are linear in the analo-
The auxiliary constraints on the system arising from
gous formalism, for the membrane the equations of mo-
combining Eqs. (9) and (10) are
tion (13) are nonlinear and difficult to solve.
4 2
X˙ mX˙ m52n2¯h52n2$Xm,Xn%$Xm,Xn% (14)
C. Matrixregularization
and
X˙ m]aXm50. (15) A remarkably clever regularization of the light-front
membrane theory was found by Goldstone (1982) and
It follows directly from Eq. (15) that Hoppe (1982) for the case in which the membrane sur-
$X˙ m,Xm%50. (16) face S is a sphere S2. According to this regularization
procedure, functions on the membrane surface are
We have thus expressed the classical bosonic mem- mapped to finite-sized matrices. Just as in the quantiza-
brane theory as a constrained dynamical system. The tion of a classical mechanical system defined in terms of
m
degreesoffreedomofthissystemareD functionsX on a Poisson bracket, the Poisson bracket appearing in the
the three-dimensional world volume of a membrane membrane theory is replaced in the matrix regulariza-
withtopologyS3R, whereSisaRiemannsurface.This tion of the theory by a matrix commutator.
Rev.Mod.Phys.,Vol.73,No.2,April2001
WashingtonTaylor: M(atrix)theory 427
Itshouldbeemphasizedthatthisprocedureofreplac- trix entries is precisely equal to the number of indepen-
ing functions by matrices is a completely classical ma- dent spherical harmonic coefficients that can be
nipulation. Although the mathematical construction determined for fixed N,
usedissimilartothoseusedingeometricquantizationof N21
(
classical systems, after regularizing the continuous clas- N25 ~2l11!. (27)
sical membrane theory the resulting theory is a system l50
thathasafinitenumberofdegreesoffreedombutisstill The matrix approximations (26) of the spherical har-
classical. After this regularization procedure has been monics can be used to construct matrix approximations
carried out, we can quantize the system just like any to an arbitrary function of the form (24)
other classical system with a finite number of degrees of
(
freedom. f~j ,j ,j !!F5 c Y . (28)
1 2 3 lm lm
l,N,m
The matrix regularization of the theory can be gener-
alized to membranes of arbitrary topology, but is per- The Poisson bracket in the membrane theory is re-
haps most easily understood by considering the case placed in the matrix-regularized theory with the matrix
commutator according to the prescription
originallydiscussedbyHoppe(1982),inwhichthemem-
brane has the topology of a sphere S2. In this case the 2iN
world sheet of the membrane surface at fixed time can $f,g%! @F,G#. (29)
2
be described by a unit sphere with an SO(3) invariant
Similarly, an integral over the membrane at fixed tis
canonical symplectic form. Functions on this membrane
replaced by a matrix trace through
can be described in terms of functions of the three Car-
E
tesian coordinates j ,j ,j on the unit sphere satisfying 1 1
j21j21j251. The 1Poi2sso3n brackets of these functions 4p d2sf!NTrF. (30)
1 2 3
are given by $j ,j %5e j . This is the same alge-
A B ABC C The Poisson bracket of a pair of spherical harmonics
braic structure as that defined by the commutation rela-
takes the form
tions of the generators of SU(2). It is therefore natural
9 9
to associate these coordinate functions on S2 with the $Ylm,Yl8m8%5gllmm,l8m8Yl9m9. (31)
matrices generating SU(2) in the N-dimensional repre-
Thecommutatorofapairofmatrixsphericalharmonics
sentation.Intermsoftheconventionsweareusinghere,
(26) can be written
when the normalization constant nis integral, the cor-
9 9
rect correspondence is @Ylm,Yl8m8#5Gllmm,l8m8Yl9m9. (32)
2 It can be verified that in the large-N limit the structure
j ! J , (23)
A N A constants of these algebras agree:
whereJ1,J2,J3 aregeneratorsoftheN-dimensionalrep- lim 2iNGl9m9 !gl9m9 . (33)
resentation of SU(2) with N5n, satisfying the commu- 2 lm,l8m8 lm,l8m8
N!‘
tation relations 2i@J ,J #5e J .
A B ABC C As a result, it can be shown that for any smooth func-
In general, any function on the membrane can be ex-
tions f,g on the membrane defined in terms of conver-
panded as a sum of spherical harmonics,
gent sums of spherical harmonics, with Poisson bracket
( $f,g%5j we have, defining F, G, and J in terms of f, g,
f~j1,j2,j3!5 clmYlm~j1,j2,j3!. (24) and J through Eq. (28),
l,m E
1 1
The spherical harmonics can in turn be written as sums lim TrF5 d2sf (34)
N 4p
of monomials in the coordinate functions: N!‘
and
Y ~j ,j ,j !5( t(lm) j j , (25) FS D G
lm 1 2 3 k A1flAl A1fl Al lim 2iN @F,G#2J 50. (35)
where the coefficients t(lm) are symmetric and trace- N!‘ 2
A A
1fl l
less (because j j 51). Using the correspondence (23), This last relation is really shorthand for the statement
A A
we can construct matrix approximations Y to each of that
lm HFS D G J
the spherical harmonicSs wDith l,N through 1 2iN
lim Tr @F,G#2J K 50, (36)
2 l( N 2
Y ~j ,j ,j !!Y 5 t(lm) J J . (26) N!‘
lm 1 2 3 lm N A1flAl A1fl Al
where K is the matrix approximation to any smooth
For a fixed value of N only spherical harmonics with l function k on the sphere.
,N canbeconstructedbecausehigher-ordermonomials We now have a dictionary for transforming between
in the generators J do not generate linearly indepen- continuumandmatrix-regularizedquantities.Thecorre-
A
dentmatrices.Notethatthenumberofindependentma- spondence is given by
Rev.Mod.Phys.,Vol.73,No.2,April2001
428 WashingtonTaylor: M(atrix)tSheory D
E
2 2iN 1 1 1
j $ J , $ , %$ @ , #, d2s$ Tr.
A N A (cid:149) (cid:149) 2 (cid:149) (cid:149) 4p N 1
(37)
V5 , (42)
(cid:29)
Thematrix-regularizedmembraneHamiltonianisthere-
1
fore given by
S D 1
1 1
H5~2pl3!Tr PiPi 2 where
p 2 ~2pl3!
S D p q5e2pi/N. (43)
1
3Tr @Xi,Xj#@Xi,Xj# The matrices U,V satisfy
4
S D UV5q21VU. (44)
1 1 1
5 Tr X˙ iX˙i2 @Xi,Xj#@Xi,Xj# . (38) In terms of these matrices we can define
~2pl3! 2 4
p Y 5qnm/2UnVm5q2nm/2VmUn. (45)
nm
This Hamiltonian gives rise to the matrix equations of
The matrix approximation to an arbitrary function on
motion
the torus is then given by
X¨i1 @Xi,Xj#,Xj 50, (39) ( (
(cid:134) (cid:135) f~h,h!5 c Y ~h,h!!F5 c Y .
1 2 nm nm 1 2 nm nm
n,m n,m
which must be supplemented with the Gauss constraint
(46)
@X˙i,Xi#50. (40) Just as in the case of the sphere, the structure constants
ofthePoissonbracketalgebraoftheFouriermodes(41)
Thisisaclassicaltheorywithafinitenumberofdegrees arereproducedbythecommutatorsofthematrices(45)
of freedom. The quantization of such a system is in the large-N limit, where the symplectic form on the
straightforward, although solving the quantum theory torus is taken to be proportional to e . Combining Eq.
ij
can in practice be quite tricky. (46)withEqs.(29)and(30)thengivesaconsistentregu-
We have now described, following Goldstone and larization of the membrane theory on the torus, which
Hoppe, a well-defined quantum theory arising from the again leads to the matrix Hamiltonian (5).
matrix regularization of the relativistic membrane The fact that the regularization of the membrane
theory in light-front coordinates. This model has N3N theoryonaRiemannsurfaceofanygenusgivesrisetoa
matrixdegreesoffreedom,andasymmetrygroupU(N) family of theories with U(N) symmetry can be related
with respect to which the matrices Xi are in the adjoint to the fact that the symmetry group of area-preserving
representation.Themodeljustdescribedarosefromthe diffeomorphisms on the membrane can be approxi-
regularization of a membrane with world-volume topol- mated by U(N) for a surface of any genus. This was
ogy S23R. A similar regularization procedure can be emphasized in the case of the sphere by Floratos, Il-
followed for an arbitrary genus Riemann surface. Re- iopoulos, and Tiktopoulos (1989), and discussed for ar-
markably, the same U(N) matrix theory arises as the bitrary genus by Bordemann, Meinrenken, and Schli-
regularization of the theory describing a membrane of chenmaier (1994). How this connection should be
any genus (Bordemann, Meinrenken, and Schlichen- understood in the large-N limit is, however, a subtle is-
maier, 1994). While this result has been demonstrated sue.Itispossibletoconstruct,forexample,sequencesof
implicitly only for Riemann surfaces of genus greater matricesinU(N) thatcorrespondinthelarge-N limitto
than one, the toroidal case was described explicitly by singular area-preserving diffeomorphisms of the mem-
Fairlie, Fletcher, and Zachos (1989) and Floratos (1989; brane surface. These singular maps may have the effect
see also Fairlie and Zachos, 1989). In this case a natural of essentially changing the membrane topology by add-
basis of functions on the torus parametrized by h,h ing or removing handles. Thus it probably does not
1 2
P$@0,2p#% is given by the Fourier modes make sense to think of the matrix membrane theory as
being associated with membranes of a particular topol-
Ynm~h1,h2!5einh11imh2. (41) ogy. Indeed, as we shall emphasize in Sec. III, matrix
configurations with large values of N can approximate
To descrSibe the matrix approximDations for these func- any system of multiple membranes with arbitrary to-
tions we use the ’t Hooft matrices
pologies. Thus in some sense the matrix regularization
of the membrane theory contains more structure than
1
the smooth theory it is supposed to be approximating.
q This additional structure may be precisely what is
U5 q2 , neededtomakesenseofMtheoryasaquantizedtheory
of membranes.
(cid:29) Another way to describe mathematically the matrix
qN21 regularization of a theory on the membrane is in the
Rev.Mod.Phys.,Vol.73,No.2,April2001