PROGRES M-THEORYN SI * M . DUFFJ . * Michigan Center for Theoretical Physics Randall Laboratory, Department of Physics, University of Michigan Ann Arbor, MI 48109-1120, USA After reviewing how M-theory subsumes string theory, we report on some new and interesting developments, focusin e "brane-world"th n go . 1. M-theory and dualities widelwas it Noty lon believeso gago d that there were five different superstring theories each competing for the title of "Theory of everything," 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 dimensions 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 branes 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 symmetry 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 perturbation theory 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 mass, 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 supersymmetry 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 Particle Physics, 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 E8 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 string theory 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 t Type 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 bewilderinleaa 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 moduli,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 dimension 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 field 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 worldsheet braneT d contrasty an 5sB .