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Home , String duality, Supergravity, Supermembranes

PROGRES M-THEORYN SI *

M . DUFFJ . * Michigan Center for Theoretical Randall Laboratory, Department of Physics, University of Michigan Ann Arbor, MI 48109-1120, USA

After reviewing how M- subsumes theory, we report on some new and interesting developments, focusin e "-world"th n go .

1. M-theory and dualities widelwas it Noty lon believeso gago d that there were five different superstring each competing for the title of "," that all-embracing theory that describes all physical phenomena. See Table 1. Moreover, on the (d, D) "branescan" of supersymmetric extended objects with d worldvolume movin spacetima n gi dimensionsD f eo l thesal , e theories occupied the same (d = 2, D — 10) slot. See table 2. The orthodox wisdom was that while (d = 2, D = 10) was the Theory of Everything, the other on the scan were Theorie Nothingf changedo knosw w no no w thal s e thaAl .ha W t. t there ear t fivno e different theorie l butal t , a stogethe r wit —hsupergravityD 1 1 , they form merely six different corners of a deeper, unique and more profound theory called "M- theory" wher stane M Magicr dfo , Myster r Membraneyo . M-theory involvef o l sal the other branes on the branescan, in particular the eleven-dimensional membrane (d — 3, D — 11) and eleven-dimensional fivebrane (d — 6,D — 11), thus resolving the mystery of why strings stop at ten dimensions while super allows eleven2. Although we can glimpse various corners of M-theory, the big picture still eludes . Uncompactifieus d M-theor dimensionleso n s yha s parameters, whic goohs i d from the uniqueness point of view but makes ordinary impossible since smal o thern e l ear couplin g constant provido st expansioe eth n parametersA . low energy, £?, expansio possibls ni powern ei E/Mp,f so Plance th wit p khM , supergravitfamilia e 1 leadd 1 th an o = t s D r y plus correction highef o s r powern si curvaturee th . Figurin t whagou t governs these corrections wouln i lona y o dgg wa pinning down what M-theory really is. Why, therefore, do we place so much trust in a theory we cannot even define? Firs knoe tw w tha equations tit s (thoug generan i t hvacuas no it l ) hav maximae eth l number of 32 charges. This is a powerful constraint and provides many "What else can it be?" arguments in guessing what the theory looks like when

* Research supporte GranE DO parn ti d y DE-FG02-95ER40899b t . t [email protected]

CP624, Cosmology and Elementary , edited by B. N. Kursunoglu et al. © 2002 American Institut Physicf eo s 0-7354-0073-3/027$ 19.00 245 Gauge Group Chiral? Supersymmetry charges Type I 50(32) yes 16 Type IIA U(l) no 32 Type IIB - yes 32

Heterotic x Es yes 16 Heterotic 50(32) yes 16

Table 1: The Five Superstring Theories

compactifie dimensions1 1 < D r exampleo dt Fo . , when M-theor compactifies yi d on circla e 5radiu1of s J?n Typlead it , the string eIIA sto , with string coupling

constant gs given by 9s = #ii3/2 (1) recovee W e weath r k coupling regime only when RU , —»whic0 • h explain e earsth - lier illusion that the theory is defined in D = 10. Similarly, if we compactify on a 1 line segment (known technicall Ss ya /Zrecovee heterotiS 2w ) E E e x sth r c string. Moreover, althoug cornere hth M-theorf so understane yw d best correspone th o dt weakly coupled, perturbative, regimes where the theory can be approximated by a dualitiesf o , theb relatee anothee ywe ar on a , o dy somt b r whicf eo h are rigorously established and some of which are still conjectural but eminently plausible. For example, if we further compactify Type IIA string on a circle of radiu shon sca w R,e rigorouslw y equivalenthas i t i tTyp e th B strin eo II t g com- pactified on a circle or radius I/R. If we do the same thing for the ES x ES heterotic string we recover the 5O(32) heterotic string. These well-established relationships which remain within the weak coupling regimes are called T-dualities. The name S-dualities e lesreferth s o well-establishet s d strong/weak coupling relationships. For example 5O(32the , ) heterotic strin believegis 5-dua5O(32be the dto to l ) Typ Type / string th B strin self-5-duale eII d b an o ,g t compactif e w f I . y more dimensions, other dualities can appear. For example, the heterotic string compact- six-dimensionaa ifie n do l torus Talss i 6 o believe self-5-duale b o dt . There alss i o e phenomenoth f dualityno f dualitieso whicy b e T-dualithth theore e on th f ys o yi 5-duality of another. When M-theory is compactified on Tn, these 5 and T duali- ties are combined into what are termed {/-dualities. All the consistency checks we have been able to think of (and after 5 years there dozens of them) have worked and convince thas du l thes al t e dualitie facn i te validsar compactifcoursn f O ca . e ew y M-theor morn yo e complicated manifold sfour-dimensionae sucth s ha e l th # r 3o six-dimensional Calabi-Yau space thesd an s e bewilderina lea o dt g arra othef yo r dualities exampler Fo . heterotie th : c strin Typ e T n duas th g4i o / e/ o strint l n go heterotiK3]e th Type c th strin / e/ e strin Tn duas th i g6o Calabi-Yau n o gt o l ; e Typ strinth A Calabi-Yaa eII n go u manifol e Typ duas th strindi B eo II t l n go mirroe th r Calabi-Yau manifold. These more complicated compactifications leao dt many more parameter theorye th n si , knowmathematiciane th o nt ,s sa t bu

246 T S . 1 1 s V V V V V S/ V V V V S/ V . 10 V S/V V V V 9 s s 8 s T 5 . 7 s 6 . V S/V V S/V V V 5 s s 4 V S/V S/V V 3 S/V S/V V 2 s 1 ...... 0 . 01 2 3 45 6 789 10 11 d-*

Table 2: The branescan, where 5, V and T denote scalar, vector and antisymmetric tensor multiplets.

in physical uncompactified spacetime have the interpretation as expectation val- f scalao s rue fields. Within string perturbation theory, these scalar fields have flat potential theid san r expectation value arbitrarye ar s decidino S . g which topology Nature actually chooses and the values of the moduli within that topology is known as the vacuum degeneracy problem.

2. Branes In the previous section we outlined how M-theory makes contact with and relates the previously known super string theories, but as its name suggest, M-theory also relies heavily on membranes or more generally p-branes, extended objects with —d spatia 1 = p l particla o (s 0-branea s s ei strina , a 1-brane s gi a , membrane is a 2-brane and so on). In D = 4, a charged 0-brane couples naturally Maxweln a o t l vector potential A^ with strength carrie d F^an electri n sa c charge Q~ f *F (2) Js* 2 magnetiand c charge

P ~ j^ F2 (3)

2 where F2 is the Maxwell 2-form field strength, *F2 is its 2-form dual and 5 is a 2-sphere surroundin chargee gth . Thi generalizee b s idey ama p-braneo dt D n si

dimensions p-branA . e couple (p-Io st - l)-form potential A^lp>2^^p+l with (p-f 2)- form field strength F^^.,^ carried +2an "electricn sa " charg r uniepe t p-volume

Q f *^D-p-2 (4) JSD-p-2

247 "magneticd an " charg r uniepe t p-volume

P~ [ Fp+i (*)

wher e(p-f-2)-fore Fth p+2s i m field strength, *FD- Ps (D-p-2)-for-2it m duad an l Sn is an n-sphere surrounding the brane. A special role is played in M-theory by the EPS (Bogomolnyi-Prasad-Sommerfield) branes whose mass per unit p-volume, or tensio charge equas i th , r unino T t l e pe t p-volume

T ~ Q (6)

This formulgeneralizee b y casee ama th so dt wher branee eth s carry several electric and magnetic charges. The supersymmetric branes shown on the branescan are al- ways EPS, although the converse is not true. M-theory also makes use of non-EPS non-supersymmetrid an c brane t branescane showth no s e supersym n th no t bu , - metric ones do play a special role because they are guaranteed to be stable. The letters S, V, and T on the branescan refer to scalar, vector and antisym- metric tensor supermultiplets of fields that propagate on the worldvolume of the brane. Historically, these branescae pointth n so n were discovere thren di e differ- ent ways. The S branes were classified by writing down spacetime supersymmet- ric worldvolume actions that generalize the Green-Schwarz actions on the super- string braneT d contrasty an 5sB . werV e eth , show ariso nt solitos ea n solutions^bf the underlying theories6. However, the solitonic V branes found this way were bound by p < 7. The 8-brane and 9-brane slots were included on the scan only because they were allowed by supersymmetry6. Subsequently, all the V-branes were given a new interpretation as Dirichlet p-branes, called D- branes, surface dimensiof so whicn o np h opewhicd an n d hstring carren R n ysR- ca (Ramond-Ramond) charge IIe A9Th . theor D-branes yha s wit 0,2,4,6,= hp d 8an the IIB theory has D-branes with p = 1,3,5,7,9. They are related to one another

by T-duality. In terms of how their tensions depend on the string coupling gs, the D-branes are mid-way between the fundamental (F) strings and the solitonic (S) fivebranes:

> rrt r > f2 ^^ r np r *____ fj~> ° ( 7\ J.Fi ^ ''"S 5 *~'•*•Dp ^S5 ?o 9s \) 9s2 Since the EPSe yar , ther no-forca s ei e condition betwee branee nth s that allows su to have many branes of the same charge parallel to one another. The gauge group on singla e D-bran abelian a s ei n stac e U(l). anotherw e suckf N I on f ho branep , to n so e gaugth e grounon-abeliae th s pi separatne w U(N). s A e them this decomposes into its subgroups, so in fact there is a at work whereby the vacuum expectation values of the Higgs fields are related to the separation of the branes r examplFo . theore eth y tha tbrane 3 livestaca TypD N n f so B ks o si e II four-dimensionaa supe4 l U(N)= rn Yang-Mill s theorylimie th largf o tn I .e N °The 3-brane solito Typf no supergravit B eII earln a s y candidatywa 'brane-world'a r efo , firstly because of its dimensionality7'8 and secondly because gauge fields propagate on its worldvolume8.

248 the geometry of this configuration tends to the product of five dimensional anti-de Sitter space and a five dimensional sphere, AdS$ x 55. In D — 11, M-theory has two BPS branes, an electric 2-brane and its magnetic dual which is a 5-brane. Their tensions are related to each other and the Planck mass by

T| ~ T5 ~ Af£ (8)

if we stack N such branes on top of one another, the M2-brane geometry tends

4 in the large N limit to AdS± x S 7 and the M5-brane geometry to AdS-j x 5. In additio otheno thertw re objectear —D plan11e n si th , Kaluza-Kleie wavth d ean n monopole, which though not branes are still BPS. When spacetime is compactified a p-brane may remain a p-brane or else become a (p — &)-brane if it wraps around A; of the compactified directions. For example, the Type IIA fundamental string emerge wrappiny sb M2-brane gth e around shrinkin d 51an radius git zeroo st d an , Type th e IIA 4-brane emerge similaa n fros i y M5-branee mrth wa .

3. -off f M-theoro s y What do we now know with M-theory that we did not know with old-fashioned string theory examplesw ?fe Hera e ear , reference foune whicb o st Refy n di . h2 ma . 1) Electric-magnetic (strong/weak coupling) duality in D = 4 is a consequence of string/strin gwhic6 dualit— tur n hconsequenci a D s nyn i i membrane/fivebranf eo e 11dualit= . D n yi 2) Exact electric-magnetic duality, first propose e maximallth r dfo y supersym- metric conformally invariant n = 4 super Yang-Mills theory, has been extended o effectivet dualit Seibery yb d Wittegan non-conformao nt theories2 — e n l th : so-called Seiberg-Witten theory. This has been very successful in providing the first proofs of confinement (albeit in the as-yet-unphysical super QCD) and in generating new pure mathematics on the topology of four-manifolds. Seiberg- Witten theory and other n — 1 dualities of Seiberg may, in their turn, be derived from M-theory. 3) Indeed t seemi , s likely tha l supersymmetrial t c quantum field theories with any gauge group, and their spontaneous symmetry breaking, admit a geometrical interpretation within M-theor e worldvolumth s ya e fields that propagate th n eo common intersection of stacks of p-branes wrapped around various cycles of the compactified dimensions, with the Higgs expectation values given by the brane separations. 4) In perturbative string theory, the vacuum degeneracy problems arises be- cause there are billions of Calabi-Yau vacua which are distinct according to classi- cal topology. Like higher-dimensional Swiss cheeses, each can have different num- ber of p-dimensional holes. This results in many different kinds of four-dimensional gauge theories with different gauge groups, numbers of families and different choices of quark and representations. Moreover, M-theory introduces new non- perturbative effects which allow many more possibilities, making the degeneracy problem apparently even worse. However t ,all f thesmosno o ) f (i et manifolde ar s

249 in fact smoothly connected in M-theory by shrinking the p-branes that can wrap aroun p-dimensionae dth l manifole holeth n si whicd dan h appea blacs a r k holen si spacetime. As the wrapped-brane volume shrinks to zero, the black holes become massles effecd san smoota t h transition froCalabi-Yae mon u manifol anothero dt . Although this does not yet cure the vacuum degeneracy problem, it puts it in a different light topologe e questio on live Th .longen w o ei n y ys ni rathewh r r than another but why we live in one particular corner of the unique topology. This may well hav edynamicaa l explanation. 5) Ever since the 1970's, when Hawking used macroscopic arguments to predict that black holes hav entropn ea y quarte e equaon aree o lt rtheith f ao r event horizon, a microscopic explanation has been lacking. But treating black holes as wrapped p- branes, together with the realization that Type II branes have a dual interpretation as Dirichlet branes, allows the first microscopic prediction in complete agreement with Hawking. The fact that M-theory is clearing up some long standing problems in quantum gives us confidence that we are on the right track. 6) It is known that the strengths of the four change with energy. In supersymmetric extension standare th f so d model finde on , s tha fine th te structure constants a3,0:2,0:1 associated wit S77(3e hth 577(2x ) {7(1x ) l meeal ) t aboua t t 1016 GeV, entirely consistent with the idea of grand unification. The strength of the dimensionless number ac — GE2, where G is Newton's constant and E is the energy, also almost meet othee sth t rquite threeno t . bu ,Thi s near mis bees sha n a source of great interest, but also frustration. However, in a of the kind envisioned by Witten, spacetime is approximately a narrow five dimensional layer bounded by four-dimensional walls. The particles of the live on the wall t gravitbu s ye five-dimensiona liveth n i s l resulta bulk s s possibli A t .i , o et choose the size of this fifth dimension so that all four forces meet at this common scale. Note that thi mucs si h less tha Plance nth k scal 10f eo 19 GeV, gravitationao s l effects may be much closer in energy than we previously thought; a result that would hav l kindeal cosmologicaf so l consequences. So what is M-theory? There is still no definitive answer to this question, although several different proposals have been made. By far the most popular is M(atrix) theory10. The matrix models of M-theory are U(N) supersymmetric gauge quantum mechanical models with 16 . Such models are also interpretable as the effective action of N coincident Dirichlet 0-branes. theore Th y begin compactifyiny sb elevente gth h dimensio circla n nradiuf o e o s R, so that the longitudinal momentum is quantized in units of l/R with total PL N/R wit hN —> e theor ooTh . s holographicyi than i t i containt s only degrees of freedom which carr e smallesyth t uni f longitudinao t l momentum, other states being composites of these fundamental states. This is, of course entirely consistent with their identification wit Kaluza-Kleie hth n modes conveniens i t I . describo t t e thes degreeeN matricesN f freedox o s N s ma . When these matrices commute, their simultaneous eigenvalues are the positions of the 0-branes in the conventional

250 sense. That they wil generan i l e non-commutingb l , however, suggests thao t t properly understand M-theory muse w , t entertai e idea fuzz nth f ao y spacetime in which spacetime coordinates are described by non-commuting matrices. In any event, this matrix approac succesd ha s reproducinn si hha g expectee manth f yo d properties of M-theory such as D = 11 Lorentz covariance, D = 11 supergravity as the low-energy limit, and the existence of membranes and fivebranes. It was further proposed that when compactified on Td-1, the quantum mechan- ical model shoul replacee db d-dimensionan a y db l U(N) Yang-Mills field theory definee duath ln o torud s fd~l. Another tes f thio t s M(atrix) approach, then, is that it should explain the {/-dualities. For d = 4, for example, this group is SL(3,Z) x 5L(2,Z). The SL(3,Z) just comes from the modular group of T3 wherea e electric/magneti th e 5L(2 s th i s ) ,Z c duality grou f four-dimensionapo l n =Yang-Mills 4 however, 4 > d r , thisFo . picture looks suspicious becaus core eth - responding becomes non-renormalizabl fule th l [/-dualitd ean y group has still escaped explanation. There have been speculations on what compacti- fied M-theory might be, including a revival of the old proposal that it is really M(embrane)theory. In other words, perhaps D = 11 supergravity together with configurationsS BP s it : plane wave, membrane, fivebrane monopolK e K , th d ean embeddin1 1 D = Typ e th f ego IIA eightbrane l therM-theoro al t s e ei ar , d yan tha neee w t d loodegree w furtheo kn ne r freedomf nonsw o fo r ne - t a onl r bu , yfo perturbative quantization schemetime th f writin eo t A . g thi stils si l being hotly debated. What seems certain, however, is that M-theory is not a string theory. It can be approximated by a string theory only in certain peculiar corners of parameter space "strino S . g phenomenology" will becom oxymoron ea n unless,t somr ye fo s ea unknown reason r universou , e happen occupo thesf st o e e yon corners .

4. AdS/CFT and the brane-world The year 1998 marked a renaissance in anti de-Sitter space (AdS) brought about by Maldacena's conjectured duality between physics in the bulk of AdS and a conformal field theors boundaryit n yo r example11Fo . , M-theor duas i o 7 AdS^n t lyS o x a non-abelian (n = 8, d = 3) super conformal theory, Type IIB string theory on AdS*> x 55 is dual to a (n = 4, d = 4) U(N) super Yang-Mills theory and M-theory on AdS? Sduax s non-abeliaa i 4 o t l n ((n+,ra_) = (2,0), ) conforma6 d= l theory. particularn I s beeha ns a ,spelle U(N)t 4 mos dou = t d clearlYang-Mille th n yi s case, ther correspondenca e sees eb i o nt e betwee Kaluza-Kleie nth n mass spectrum e conformath e bulid nth kan l dimensio f operatorno e boundaryth n o s 12e '13W . note that, by choosing Poincare coordinates on AdSs, the metric may be written as

ds2=e-2y/L(dx»)2+dy2, (9)

where #M, 0,1 (IJL— e four-dimensiona , th 2,3) e ar , l brane coordinates thin I . s case e superconformath l Yang-Mills theor takes yi resido ne boundart th t ea y—y >• — oo.

251 lengtS Ad he scal gives Th i eL y nb

4 2 L 47ra'= (^MAT) (10)

strine Th g couplin Yang-Mille th gd gan s s coupling QYM are relatey db

9YM9= s 2 (11)

The full quantum string theory on this spacetime is difficult to deal with, but we can approximate it by classical Type IIB supergravity provided

L2»a' (12)

so that stringy correction to supergravity are small, and that gs « I or

TV ->• oo (13)

so that loop correction neglectede b n sca . overwhelminTherw no s ei g evidencn ei favor of this correspondence and it allows us to calculate previously uncalculable strong coupling e gaugeffectth n ei s theory starting from classical supergravity. Models of this kind, where a bulk theory with gravity is equivalent to a boundary theory without gravity, have also been advocate t SusskindHooft' y b y db d 1an 4 15 who call them holographic theories. Many theorists are understandably excited abou AdSe th t / CFT correspondence becaus teacn f wha eca o s abou ht u i t t non- perturbative QCD. In my opinion, however, this is, in a sense, a diversion from the really fundamental question: What is M-theory? So my hope is that this will be a two-way process and that superconformal field theories will also teach us more about M-theory. The Randall-Sundrum mechanism16 also involves AdS but was originally moti- decoupline th a vatedvi t gravitf go no , y from D3-branes t rathe possiblbu a , s a r e mechanis evadinr mfo g Kaluza-Klein compactificatio localiziny nb g gravite th n yi presenc uncompactifien a f eo d extra dimension. Thi accomplishes swa insertiny db g a positive tension 3-brane (representing our spacetime) into AdSs • In terms of the Poincare patch of AdSs given above, this corresponds to removing the region y < 0, and either joining on a second partial copy of AdSs > °r leaving the brane at the end osingla f e patc f AdSsho . In either cas resultine eth g Randall-Sundrum metris ci given by + dy2, (14) wher (-00,00G ey y G r [0,oo o )'two-sideda r fo ) r 'one-sidedo ' ' Randall-Sundrum brane respectively. e similaritTh f these notio yo scenarioo th tw eo t n d thale s tfacn i the te yar closely tied together. To make this connection clear, consider the one-sided Randall- Sundrum brane. By introducing a boundary in AdSs at y = 0, this model is conjectured to be dual to a cutoff CFT coupled to gravity, with y = 0, the location of the Randall-Sundrum brane, providing the UV cutoff. This extended version of

252 the Maldacena conjecture17 then reduces to the standard AdS/CFT duality as the boundary is pushed off to y -» —oo, whereupon the cutoff is removed and gravity becomes completely decoupled. Not particulan ei r that this connection involvesa

e boundarth t a singla singl f T yo eeCF patc e casbrana th f f AdS eho o r e Fo 5. sitting betwee patcheo ntw f AdSsso woule on , d instead requir copieo e th etw f so patchese eacr th CFTfo f ho e cruciaA on ., l tes thif o t s assumed complementarity Maldacene oth f Randall-Sundrud aan m picture thas si t both should yiel same dth e corrections to Newton's law 18. A third developmen brane-worle th n i t bees idee dha nth a tha extre tth a dimen- sions are compact but much larger than the conventional Planck sized dimensions traditionan i l Kaluza-Klein theories19. Thi possibls si standare th f ei d model fields are confined to the d — 4 brane with only gravity propagating in the d > 4 bulk19. We shall not pursue this possibility here.

References

1. J. H. Schwarz, Superstrings. The first fifteen years of super string theory, (World Scien- tific, 1985). 2. M. J. Duff, The world in eleven dimensions: supergravity, and M- theory, (IOP Publishing, 1999). 3. J. H. Schwarz, Recent progress in , hep-th/0007130. . SchwarzH . 4J . , Does superstring theory have a conformally invariant limit?, hep- th/0008009. 5. A. Achucarro, J. Evans, P. Townsend and D. Wiltshire, Super p-branes, Phys. Lett. B198 (1987)1 44 , . , StringLu . X solitons, . . KhurJ . DuffR J d . . R 6an M i,. Phys. Rep. 259 (1995)3 ,21 , [hep-th/9412184]. 7. G. T. Horowitz and A. Strominger, Black strings and p-branes, Nucl. Phys. B360 197 (1991). 8. M. J. Duff and J. X. Lu, The self-dual Type IIB superthreebrane, Phys. Lett. B273 409 (1991). 9. J. Polchinski, Dirichlet-branes and Ramond-Ramond charges, Phys. Rev. Lett. 75, 4724 (1995). 10. T. Banks, W. Fischler, S. H. Shenker and L. Susskind, M-theory as a matrix model: a conjecture, Phys. Rev. D55, 5112 (1997). 11. J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998) [hep-th/9711200]. . GubserS . . Polyakov . KlebanoS M R . . . I ,A 12 d ,van Gauge theory correlators from non- critical string theory, Phys. Lett. B428, 105 (1998) [hep-th/9802109]. 13. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998) [hep-th/9802150]. 14. G. 't Hooft, in , gr-qc/9310026. . SusskindL . world15 e hologram,a Th ,s a . Math.J Phys. , 63736 7 (1995) [hep-th/9409089]. 16. L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83, 4690 (1999) [hep-th/9906064]. . Susskin L . Witten E . d 17 e Holographican dTh , Bound n Anti-dei Sitter Space, hep- th/9805114. 18. M. J. Duff and James T. Liu, Complementarity of the Maldacena and Randall-Sundrum

253 pictures, Phys. Rev. Lett. , 20585 2 (2000) [hep-th/0003237]. 19. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, The universe's unseen dimensions, Scientific American, August 2000. 62 ,

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