Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. Scn ato h oreSprymtyadEtaDimensio Extra and Supersymmetry course the of part (Second 05Pr I etrso Extra on Lectures III Part 2015 Dimensions ennoQuevedo Fernando swt e Allanach) Ben with ns Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. 2 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. re vriwo compactifications of overview brief A 3 uesmer nhge dimensions higher in Supersymmetry 2 xr dimensions Extra 1 Contents . ia eak ...... Remarks . . . . Final ...... 3.4 . . . Compactifications Calabi-Yau ...... 3.3 . . Compactifications . . Freund-Rubin . . . 3.2 . Compactifications Toroidal . . 3.1 ...... Reduction Dimensional 2.3 . uesmer ler ...... algebra Supersymmetry 2.2 . pnr nhge iesos...... dimensions . higher . in . Spinors . . 2.1 ...... scenario world brane The 1.3 . ooi il hoisi xr iesos...... dimensions extra in theories Field Bosonic 1.2 . aiso ihrDmninlTere ...... Theories Dimensional Higher of Basics 1.1 .. uesmer ler and Algebra d higher Supersymmetry in algebra supersymmetry 2.2.2 of Representations lectu 2.2.1 in covered not material (extra spinors Majorana 2.1.3 .. ag xr iesos...... dimensions . . odd in . . representations . . Spinor dimensions . . even . . in 2.1.2 representations . . Spinor . . . . 2.1.1 ...... problem . . hierarchy the . . and . . scenarios world . . Brane . . . . 1.3.3 . compactifications Warped . dimensions 1.3.2 extra Large 1.3.1 theory Klein Kaluza dimensions: extra in Gravitation 1.2.4 .. clrfil n5dmnin ...... and . Duality . . fields, tensor higher . Antisymmetric and . dimensions 5 . in 1.2.3 . fields Vector dimensions 5 in 1.2.2 field Scalar 1.2.1 .. h Consider Why 1.1.1 .. re itr ...... History Brief 1.1.2 D ≥ iesos 5 ...... Dimensions? 4 p Bae 26 ...... -Branes 3 p D D bae 10 ...... -branes 2 = 2 = n n 22 ...... 1 + 21 ...... e)...... 23 ...... res) mnin 24 ...... imensions 7 ...... 12 ...... 5 ...... 19 ...... 32 . 31 ...... 30 ...... 21 ...... 8 ...... 27 ...... 23 ...... 15 ...... 17 ...... 7 ...... 16 ...... 34 ...... 6 . . . . . 29 21 5 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. 4 CONTENTS Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. .. h Consider Why Theories Dimensional 1.1.1 Higher of Basics 1.1 dimensions Extra 1 Chapter • • • as’lwipisfrteeeti field electric the for implies law Gauss’ gravity). nuhntt aebe bevd( observed been space-like have extra to many not as enough be may there but complicate causaliy) are affecting dimensions time Extra dimensions. extra the However lntr ytm.Eecs:Poeit. Prove Exercise: systems. planetary (Φ n ruetagainst argument One nte ruetagainst argument Another oin So nlge ruet odfrgaiainlfilsadtheir and fields gravitational for hold arguments Analogues − foedmnini opcie (radius compactified is dimension one If Φ 1 ∝ /g ∝ 1 2 D /R 1 R /R xeietlyw observe we Experimentally . d pctm dimensions spacetime o itnebtenbodies between distance for ) S S I I D D 2 3 xF − E E ~ ~ 3 .Ol oetasΦ potentials Only ). MN · · d d S S ~ ~ F MN = = D + D Q Q ≥ · · · ≥ D h rvttoa oc ewe w bodies two between force gravitational The . 4 = = ic [ Since . k 4 ≥ k ⇒ k ⇒ E ~ Dimensions? .Ol n4dmnin ag opig r dimensionless are couplings gauge dimensions 4 in Only 4. ∝ k k E ~ ∝ r ∝ k E E ~ ~ D R . ∝ k ∝ k 1 A E ~ /R R r n4dmnin n ngnrldimension general in and 4-dimensions in M .Btw nwti nypt on ntesz of size the on bound a puts only this know we But 4. = 10 iein like ) n t oeta fapitcharge point a of Φ potential its and D 1 1 = ] − aesal ris ool n4 hr a estable be can there 4D in only So orbits. stable have −      18 2 5 R R o ogaiainlpyisand physics nongravitational for m 1 1 , R R 3 2 1 1 , 2 3 [ , , M F MN 4 : : Φ × R r < R Φ Φ ,s [ so 2, = ] S ∝ ≫ 1 then , ∝ ∝ uulyipycoe ielk curves time-like closed imply (usually d potentials. r R iesosa oga hyaesmall are they as long as dimensions D 1 R R . 1 − 1 2 3 g . (4 = ] dimensions 5 dimensions 4 − F D Q ∝ ) : / D .S properly So 2. 1 r /R , . F 2 ∝ 10 (potential 1 − /R 6 for m S D − = 2 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. .. re History Brief 1.1.2 6 • • • • • ∼ • • • eew ilsattedsuso fpyisi xr dimensi extra in physics of discussion the start will we Here • • • • fe 96 eea epedvlpdteK da ( ideas KK the developed people several 1926, After 1926 n17’ n 90s uesrnsrequired Superstrings 1980’s: and 1970’s In 9hcentury 19th u ob h aiu ubro iesos( dimensions of number maximum the be S to for out dimensions of numbers maximum possible and dimensions 1960’s: ae on based D n1914 In ehia simplifications Technical drsigtepolm fteSadr Model. Standard the rol of important problems an the play addressing may that extr symmetries and spacetime supersymmetry tro of that sions other lecture address first the to in also mentioned We and solves supersymmetry that lems symmetries spacetime Enhancing ic noddmnin,teei oaaou fthe of analogue no is there dimensions, odd in since implied also which magnetism. ab etargument best Maybe oeta ato udmna theories? fundamental of part Potential ro to prior nte curiosity Another rpgtn ere ffedmi oe dimensions). lower in freedom of degrees propagating iesos hoyicuigbt rvt n ag inter gauge and gravity both di including 4 theory in a dimensions, only exist fields gauge D of theories field quantum defined D -iesosi osuypyisi rirr ubro dime of number arbitrary in physics study to is 4-dimensions 5 AD 150 26. = ≥ ≥ r o-eomlsbe.Hwvrkoigta rvt is gravity that knowing However non-renormalisable). are 4 4. Klein eWitt de Einstein Nordstrom Weyl Ptolemy yidi nvrewt t ieso fsalradius small of dimension 5th with universe cylindric : Nordstrom aly M Cayley, sideas. ’s rvt s‘o-rva’for ‘non-trivial’ is Gravity . . bann iesoa agMlstere n4 from 4d in theories Mills Yang dimensional 4 obtaining Kaluza D h o?Atraltebs a oadesteqeto fwhy of question the address to way best the all After not? Why . O dimensionality” ”On n 99-1921 - 1919 and 1t etemnmm 1dmnin,hwvr ontamtch admit not do however, dimensions, 11 minimum. the be to 11 = dim( fe uesmerctere n4 r airt en sta define to easier are 4D in theories supersymmetric Often . bu,Riemann obius, ¨ a tepiga nucsflter fgaiyi em of terms in gravity of theory unsuccessful an attempting was G/H G/H sdgnrlrltvt xeddt v iesos i con His dimensions. five to extended relativity general used 8+3+1) + 3 + (8 = ) ti motn olo o lentv ast drs h p the address to ways alternative for look to important is It . = tigMter ossetin consistent String/M-theory . Kaluza SU SU (3) N (2) D dmninlgoer,... geometry, -dimensional × D 0 eeomnsi uegaiyrqie extra required supergravity in Developments 10. = × Nahm SU neednl re ouiygaiyadelectro- and gravity unify to tried independently U ≥ (1) Einstein (2) i h es httegaio edhsno has field graviton the that sense the (in 4 γ − 5 ). × arxi ordimensions. four in matrix × 3+1+1 7 = 1) + 1 + (3 Witten U ons. U HPE .ETADIMENSIONS EXTRA 1. CHAPTER (1) (1) cin saraynnrenormalisable. non already is actions nsions. , norudrtnigo aueand nature of understanding our in e S eediscussed: were USY iesosaetentrlexten- natural the are dimensions a Jordan besoso h tnadModel. Standard the of spots uble o-eomlsbearayi four in already non-renormalisable esos(ag edtere in theories field (gauge mensions xmndtecoset the examined R , D Pauli > D 10 = .As tig with strings Also 5. , , 11. Ehrenfest D clrfields, scalar 1turned 11 = eobserve we tn from rting et were cepts ,...) irality rob- Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. x Exercise: n opr h eutwt h tnadoedmninlsi one-dimensional standard the with result the compare and eoddmnini iceo radius or circle a is dimension second nnt ubro)4 clrfils h qain fmto f motion of equations The fields. equations wave scalar stan massive) 4D the of of) functions number are infinite coefficients Fourier the that Notice dimensional iue1.1: Figure DIMENSIONS EXTRA IN THEORIES FIELD BOSONIC 1.2. e h xr dimension extra the Set iesos olwdb etrfilsadohrbsncfield bosonic other and fields vector by star followed will dimensions, we gravity, dimensional dimensions higher 5 discussing Before in field dimensions Scalar extra in 1.2.1 theories Field Bosonic 1.2 u pctm snow is spacetime Our osdramsls Dsaa field scalar 5D massless a Consider fields. fermionic dimensions. include higher be to and will five We in terms. theories simple gravitational in reach dimensions extra having of effects ∈ • (0 90s20’:Sprtig revived Superstrings 1990’s-2000’s: sions). a , ) and M xml fafiedmninlspacetime dimensional five a of Example osdrteSh¨dne qainfrapril oigin moving particle a for Schr¨odinger equation the Consider 4 AdS . V ∂ M = D ∂ C T /CF ∞ M ϕ M tews) eieteeeg eesfrtetwo-dimension the for levels energy the Derive otherwise). 4 x D = 0 = 4 × − = 1 S 1 dualities... y eidct in Periodicity . ϕ enn iceo radius of circle a defining ( x ⇒ µ ϕ y , ( S x = ) 5D M n r. = X ) D ∞ −∞ M , h oeta orsod oasur well square a to corresponds potential The = 1( hoy.Baewrdseai lreetadimen- extra (large scenario world Brane theory). (M 11 = n  = y M X ∞ Z 0 = ∂ −∞ ieto mle iceeFuirexpansion Fourier discrete implies direction 4 µ × d ∂ , 5 µ ϕ S 1 ∂ x n 1 ..., , − ( where x M µ ihtesmlrcsso clrfilsi extra in fields scalar of cases simpler the with t ihaction with 4 n r r exp ) ∂ ϕ 2 2 ntenx hpe eetn h discussion the extend we chapter next the In with uto ntelimit the in tuation  S fhelicity of s ad4 oriae n hrfr r (an are therefore and coordinates 4D dard M 1 ϕ uligu ntelvlo opeiyto complexity of level the on up building . ϕ  sacrua xr ieso nadto ofour to addition in dimension extra circular a is n y ( iny rteFuirmdsae(ngeneral (in are modes Fourier the or x r ≡ µ exp )  y 2 + . λ w dimensions two  πr ≤ iny . r .Ti ililsrt the illustrate will This 1.  r lSchr¨odinger equation al ≪ 0 = a. ( V x ( x and 0 = ) y. The for 7 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. ahcmoethsadsrt ore expansion Fourier discrete a has component Each edi D ecnsltamsls etrfield fie vector vector massless abelian a an split of can We case 4D. simpler in next field higher the to and move dimensions now us 5 Let in fields Vector 1.2.2 included. are modes Fourier the as long ihtefit ieso’ aenumber wave dimension’s fifth the with fmsiesaa cin n4.I eaeol neetdi e in interested only are mas we a If for 4D. action 4D in one actions to scalar reduces massive action of 5D the that means This nodrt banteeetv cini Dfralteepart these all for action, 4D 5D in original action the effective into the obtain to order In compactification ocnrt nyo h oeato.I ersrc u atte our restrict of we speak If case action. mode then 0 did), the on only concentrate iue1.2: Figure fo equations Gordon Klein of mode number Fourier each infinite an then are These 8 a iulz h ttsa nifiietwro asv state massive of tower infinite an as states the visualize can hsi called is This states ic hycm rmtemmnu nteetadimension: extra the in momentum the from come they since , S ϕ 5D h auaKentwro asv ttsdet nextra an to due states massive of tower Klein Kaluza The ( x auaKentower Klein Kaluza iesoa reduction dimensional M = 2 = enn htteetadmnini opc n t existen its and compact is dimension extra the that meaning , = ) A ϕ Z n µ r π ϕ d sa4 atcewt mass with particle 4D a is ( 4 x = Z x µ .Ti ol eeuvln ojs rnaigaltemassiv the all truncating just to equivalent be would This ). Z A d n 4 M d = X x = ∞ ∂ y −∞ ⇒ ∂ = M µ n h asv tts( states massive the and A n ϕ ϕ∂ µ n ∈ 0 oegnrly fw epaltemsiemdsw akabou talk we modes massive the all keep we if generally, More .    ( ∂ exp Z M x µ . µ A A ∂ ϕ ) µ µ 4  ∂ = ϕ =: µ iny n ϕ r ( Z 0 x ρ (  µ A x d ) m µ 4 M vco n4dimensions) 4 in (vector ρ , saa n4dimensions) 4 in (scalar x ) − n 2 ∗ ( n x = = X + M ∞ n r −∞ 2 2 n r into ) 2 2 ... ϕ nytezr oe( mode zero the Only . n 
n = ∂ )aecalled are 0) 6= ( wt nraigms rprinlto proportional mass increasing (with s x µ = ce,ltu lgtemd xaso of expansion mode the plug us let icles, µ ege mle hnthe than smaller nergies ϕ S n 0 = ) 1 n = X HPE .ETADIMENSIONS EXTRA 1. CHAPTER ∞ ( to otezr oe(like mode zero the to ntion di D iia oa electromagnetic an to similar 5D, in ld −∞ ieso.Masses dimension. asv Dfils hsmasthat means This fields. 4D massive r S x 4D ls clrfil lsa nnt sum infinite an plus field scalar sless µ ) ρ ∂ n + µ exp ϕ . n . . ... ( auaKlein- Kaluza x  . µ ei ae noacutas account into taken is ce iny ) r ∗ n −  m )i ases One massless. is 0) = n . n r = 2 2 | 1 r | ed.I this In fields. e ϕ n | n cl,w may we scale, or /r | 2  momentum rwlinearly grow Kaluza n ). ϕ t Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. neaanpu hsit h Daction: 5D the 5D into in this state plug massless again each once for states massive of tower infinite hnti biul eoe qiaett h clrfil ca field scalar the to equivalent becomes obviously this then h oet ler sjs iei iesos replace dimensions, 4 in like just is algebra Lorentz The hoeague ..Lorenz e.g. gauge, a Choose In implying ihfil strength field with where osdrteaction the Consider DIMENSIONS EXTRA IN THEORIES FIELD BOSONIC 1.2. s bu h eeaiaino hs eut oahge dim dimension higher 4 a in to algebra c results Lorentz and these Recall one of spin generalization has the dimensions 4 about in ask particle gauge a that counting know We freedom of degree and spin on Comments interesti further have fields electromagnetic dimensional fmsievco n clrfils oieta h ag cou gauge the that Notice fields. of scalar (massl (coefficients particle and gauge vector a of massive theory 4D of a with up end we Therefore M o nt iesoa asesrpeettosin representations massless dimensional finite For fdgeso reo n4dmnin s2( 2 is dimensions 4 in freedom of degrees of oaiain n ( and polarizations 45 D and pctm iesos hsgnrlssto generalises this dimensions, spacetime V n M stevlm fthe of volume the is 67 S o xml.Dfietesi ob h aiu ievleo an of eigenvalue maximum the be to spin the Define example. for F 5D MN h M D →S 7→ F µν − MN M , )in 2) 4D and ρσ D i = ∂ F dimension, M µν n = A J Z F iesoa opc pc eg an (e.g. space compact dimensional i ∂ S M µν M d 5D i F 4 = P r eae by related are ) MN ∂ η x = 0 = M µ µσ  = A ǫ M ijk A ( = 2 g N := πr 5D 2 g M g νρ 1 1 Z A 4 2 4 M 2 − 7→ µ F d ,E, E E, jk + ∂ ⇒ = 5 (0) = ∂ 7→ D M A x J , M η i A µν dimensions, g νρ ∂ A where V 2 5D N 2 1 N ∂ g g πr D i F M 0 | 5 2 gise htw ast discuss. to pass we that issues ng M D A 2 − O (0) with − F . , ... , µ M ∂ µσ 4 ( . MN D , {z µν M ∂ ninlguefield. gauge ensional i ∝ nodrt n h Deetv cinwe action effective 4D the find to order In . − ν e(o ahcomponent each (for se A N − 1 = . by ... , 2) 0 = i + F N A s) asessaa n nnt towers infinite and scalar massless a ess), 0 } M 2 = MN η lnso -ad5dmninlactions dimensional 5 and 4- of plings M ) O , ristodgeso reo.W may We freedom. of degrees two arries νσ 2 g 2 23 0 = ( πr 5D 2 .,D ..., , D . , M . )crepnigt h photon 2 the to corresponding 3) M − ∂ µρ µ , )i itegroup: little is 2) , ρ n N − 0 − peeo radius of sphere ∂ .,so ..., , 2. µ η ρ µρ 0 y M + M νσ M ... i A (
23 i  +1) M omtswith commutes niaigan indicating ) , h number The . r .Higher ). 9 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. .. niymti esrfils ult and Duality fields, tensor Antisymmetric 1.2.3 10 ewl e o hti xr iesosteeaefrhrfiel further are there dimensions extra in helicity that now see will We lya motn oei tigter o instance). example for for theory Consider string in role important an play as known symmetry a by fields ofrw osdrdsaa-advco fields: vector and scalar- considered we far So n a eeal tr iha niymti ( antisymmetric an with start generally can One • • • eetepotentials the Here E bet htaentda osaaa rvectors. or scalaras to dual not are that objects NEW D ese ob e ido hsclfields. physical of kind new Ho a be fields. to electromagnetic seen and be scalars as such objects known ofrin far So exchange The iial n5dmnin,oecuddfiei analogy in define could one dimensions, 5 in Similarly identities. Bianchi and hnMxelseutosi au read: vacuo in equations Maxwell’s then asesatsmerctensor antisymmetric Massless n4dmnin,dfieada edsrnt oteFrdyten Faraday the to strength field dual a define dimensions, 4 In w-ne asesatsmerctensor antisymmetric massless two-index a h action The 1 /g λ 6 = 2 ≤ R .Teeaeatsmerctno ed,wihi Daejust are 4D in which fields, tensor antisymmetric are These 1. d 4 xH D S 4 = F µνρ MNP a esont eeuvln to equivalent be to shown be can F , H niymti esr fhge akhv ensont ee be to shown been have rank higher of tensors antisymmetric 5 ↔ µνρ F B ˜ H = ∂ ∂ NP µνρ (the D > D µ µ F F ∂ ˜ 4 = duality [ ↔ µν µν M lcrmgei duality electromagnetic = 4 B B ˜ H NP Baciidentities) (Bianchi 0 = fil equations) (field 0 = RS ϕ ϕ clrvco ne range - index vector scalar ∂ M B u netadmnin hs ilb e ye fparticles of types new be will these dimensions extra in But . ( [ ] ( µ 1 x µν x r ftesm ye otayt the to Contrary type. same the of are ...M B M F µ ˜ νρ ) = ) in MNP F ˜ p ⇒ ] +2 µν D A A M = µ ihfil strength field with 4 = = F ⇒ B := ˜ ( ( = QRS x x µν p µ M S ∂ ) )-tensor - 1) + ) ssi ob ult asesscalar massless a to dual be to said is ǫ [ 2 1 F M ∝ ˜ N QR MNP σ orsodn to corresponding ) ǫ 1 = M g µνρσ A 2 M = R 0 = µ ǫ 2 ...M d N QRS MNP F 0 = 4 ǫ F ρσ , ∂ x screpnigt ooi atce of particles bosonic to corresponding ds σµνρ HPE .ETADIMENSIONS EXTRA 1. CHAPTER QR 1 p +2 p .,D ..., , , , B µ 1 -branes ee in wever ] , a∂ M . H 2 sor 1 , µ µνρ ...M F 3 a − MNP F seeapeset.Therefore sheet). example (see H p qiaett clr rvector or scalars to equivalent 1 µν E +1 ~ µνρ = D ↔ via n eieafil strength field a derive and ∂ n ihrte can they higher and 6 = D [ ∂ = B µ ~ σ B a 4 = wp edequations field swaps νρ ∂ [ ] uvln da)to (dual) quivalent Q , ihaction with ae hs are these cases 5 B ˜ RS ] a . (that S = Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. osetenme fdgeso reo,cnie h iteg little the dimensions consider transverse freedom, the of in degrees of number the see To hs are These u nta fitgaigoe eitgaeb at h La the parts by integrate we Λ over integrating of instead But q action: ag edi uegueadcnstteguechoice gauge the set can and gauge pure a is field gauge ∂ osligteeutoso oinaetesm sintegratin as same the are motion of equations the solving so Exercise: couplings weak to coupling strong maps theo actually Yang-Mills duality supersymmetric the as such cases, general more aslrerdu osalradius coordinates small to the radius of large one maps with theory string of worldsheet tensor ntneltu aetesmls ae esro ak0(sca 0 rank of tensor a a theory case: dual simplest the the obtain take to us integration sets let of that instance order glo constrain the the multplier Change Gauge Lagrange a 3. 1. Introduce then: a are 2. of dualisation field, existence to the steps for The condition sufficient symmetry. a is That symmetry. n t ul(with dual its and DIMENSIONS EXTRA IN THEORIES FIELD BOSONIC 1.2. R n xtegauge the fix and oeta ne ult,couplings duality, under that Note nteesml ae the cases simple these In ov h qaino oinfrteLgag multiplier Lagrange the for motion of equation the Solve abRmn field Kalb-Ramond o h rpgtn xo field axion propagating the for µ aktno.Altesesaetesm saoe oietha Notice above. as same the are steps the All tensor. rank 1 + 2 X R niymti esr ar pn1o less, or 1 spin carry tensors Antisymmetric h rcdr etoe ntepeiu xriecnb gene be can exercise previous the in mentioned procedure The d + 2 B σ∂ A S M µ µ 1 hnw e h edsrnt ozr yadn arnemul Lagrange a adding by zero to strength field the set we Then . = X∂ ··· D p osdrtefloigLagrangian following the Consider M +1 − R µ q 2 d X
h lblsmer is symmetry global the 2 σ h lblsmer is symmetry global the , needn components. independent X D B ( L D B µν − ed oteda action dual the to leads 0 = µ MN . XD ( = lentvl,sleteeuto fmto o h field the for motion of equation the solve Alternatively, p B S g )indices) 2) + M 2 µ g = 1 1 X atr a npicpeb bobdi h eento fth of redefinition the in absorbed be principle in can factors = 2 ...M H ˜ a. ( + M ∂        p Z [ +1 hthpest h coupling the to happens What ǫ 1 M ...M µν R g B B B 1 d B Λ 4 µν µ 56 r apdt mlilso)terinverses, their of) (multiples to mapped are → 7→ D x M 5 F − µν 2 B , B 1 p  ...M − / M nertn vrΛsets Λ over integrating ) B g 2 2 1 µ X 2 R 1 i p 1 6 ··· +2 ...i H = M n scalled is and → ] e.g. µνρ ) p aktotno n4dimensions 4 in tensor two rank : q +1 2 X ǫ → M H n6dimensions: 6 in i , S ˜ + 1 ↔ B µνρ ...M (2 = M c etr n4dimensions 4 in vectors 2 : agn tmasw change we means it Gauging . 1 a D ··· X + A g k clri dimensions 4 in scalar : ooti nato o rpgtn massless propagating a for action an obtain to H M 2 R µ T iigo iceo radius of circle a on living ( q M ) ǫ a is(rdsrt ae ieteIigmodel) Ising the like cases discrete (or ries out. g [3]. 1 = ∂ opadcuttenme fcomponents of number the count and roup n hngtbc h rgnlaction. original the back get then and 0 = − + − D [ h orsodn edsrnt ozero, to strength field corresponding the M 2 µνρσ − rnemlile emadslefor solve and term multiplier grange g c ulter steeitneo global a of existence the is theory dual R duality 1 a)i -iesos h action The 2-dimensions. in lar) p M aie oaysse hthsaglobal a has that system any to ralised trfiiga prpit ag.For gauge. appropriate an fixing fter − a ymtyb nrdcn gauge a introducing by symmetry bal B ne hstransformation? this under ˜ ..., , d 1 h ahitgasaegusa and gaussian are integrals path the t 1 M 2 ...M ∂ ··· F σ∂ µ 2 M µν ...M ( H D D µ q o h eea antisymmetric general the For . νρσ Λ = n h ag edwudb a be would field gauge the and D . − ∂ − µ  ∂ 2) ( .Ti stecs o the for case the is This Λ. p µ +2) A . . H ν ile osritt the to constraint tiplier ] ) νρσ 2 − . , ∂ ooti naction an obtain to ν A ∂ R µ µ ed u for but fields e hsduality This . X hnthe then 0 = → D µ X S A 11 = = µ Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. eaenwrayt osdrtegraviton the consider to ready now are we niymti esr fdffrn ak nrdc e kind new introduce ranks different of tensors Antisymmetric p 12 hymv in move they hc r xeddojcswith objects extended are which dimensions where hscnb ena olw:teeetoantcfil couple field electromagnetic the follows: as seen R be can This fprilsvia particles of that h urn a ewitna nitga vrtewrdln o line world the over integral an as written be can current the ..ne tig(r1bae ih2dmninlworldshee dimensional 2 with 1-brane) (or string a need i.e. ihtoidcs h nlgeis analogue the indices, two with bet orsodn opitlk antcmnpls i monopoles, (magnetic) magnetic point-like to corresponding objects hslast h ocp fa of ra concept higher the of to tensors leads antisymmetric This that see can we Therefore ecnetn hsie o ihrdmninltere with theories dimensional higher for idea this extend can We fe icsigsaa- etr n niymti tenso theory antisymmetric Klein and vector- Kaluza scalar-, discussing dimensions: After extra in Gravitation 1.2.4 esr frank of tensors akatsmerctensor antisymmetric rank osre urn.I h aesense, same the In current. conserved d branes 4 e srcl h iuto ihpitprilsin particles point with situation the recall us Let atcecriscag ne etrfil,sc selectr as such field, vector a under charge carries particle A xA R ,ν µ, J µ 0 J d µ 3 0 = D x wt ia current Dirac (with = p > D − , 1 p Z Z q p , 1. + 2 n oteculn becomes coupling the so and − B B , 3. rnsta opet h ulfields dual the to couple that branes 4 MNP M dimensions. 1 + 1 ...M niymti tensor antisymmetric d p +1 x Z M i 1 d ...i ∧ x p J M p d µ p rn sagnrlsto fapril htculst antis to couples that particle a of generalisation a as brane vector 1 x = = scalar pta iesossann a spanning dimensions spatial N ∧ G ψγ R ... ∧ ¯ MN d p d ∧ µ D A Z x ψ ϕ rnscrya carry branes ∼ S d − P M G x B 1 o neeto edfrisac) o atceo charge of particle a For instance). for field electron an for = MN M J x A MN p M R +1        0 1 i d fKlz li hoy e ssataanwith again start us Let theory. Klein Kaluza of ...M 1 d Z 4 ...i x G G G D xJ M A p p µν 44 µ +1 ntesm a htin that way same the In . .Eetoantcfilscul oteworldline the to couple fields Electromagnetic 4. = 4 µ µ ∧ A d fields r d µ x graviton rirr dimensions arbitrary n new 0 0 vectors x pndg ffreedom of deg. spin scalar µ = B ˜ N ocul.Frhrgnrlstosare generalisations Further couple. to t 1 + 1 0 , , oacnevdcreti iesosas dimensions 4 in current conserved a to s M , ihrrn esr.Frapotential a For tensors. rank higher q 1 1 fetne bet nw as known objects extended of s , 1 kcul aual oetne objects. extended to naturally couple nk h particle the f R ...M ido hrewt epc oahigher a to respect with charge of kind HPE .ETADIMENSIONS EXTRA 1. CHAPTER omagnetism d mmrn r2-brane) or (membrane ξ p D µ − iesoa olvlm while worldvolume dimensional 1 + A p D µ − D p . 3 +1 . − − 2 2
J Q µ ( p = = D brane) D q R h ulojcsare objects dual the R d hr r dual are there 4 = d 3 xJ ξ µ δ 0 4 ( with x − ymmetric p ξ J branes B D such ) µ [ MN 5 = the q , ] Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. ymtiso h akrudand background the of symmetries ..temti so the of is metric the i.e. osdrtepyia xiain otebcgon metric w background it the where to later excitations it physical consider the will Consider but constant a to factor warp where n lgtezr oepr noteEnti ibr action: Hilbert Einstein the into part mode zero the plug and η Exercise: unifie a obtained fields! have scalar we Notice scale. Planck 4-dimensional dimensional 5 the in appears metric background The DIMENSIONS EXTRA IN THEORIES FIELD BOSONIC 1.2. hr nodrt bobtecntn nteMxeltr eha we term Maxwell the in constant the absorb to order in Where where ecnwrite can we h lnkmass Planck The Comment space. compact the of ordmninlPac as) n osbeslto st is solution possible One mass!). Planck four-dimensional Here hrfr ecnadjust can we therefore r (5) oa hslevel this at So . MN R G = (+ = M MN κ M (4) ∗ S sacntn ob xd efrigteFuirexpansion Fourier the Performing fixed. be to constant a is 3 4 stefnaetlms cl ftehg iesoa theory dimensional high the of scale mass fundamental the is R , D × − − = hwta h ateuto olw rmapr gravitationa pure a from follows equation last the that Show S , − 2 = 1 e , φ | × − − (0) M σ M − g , S ∇ µν − − pl 2 1 Z ∗ 2 1 3 ,aohroei hto iesoa ikwk spacetime Minkowski dimensional 4 of that is one another ), e = a nyb eemndatrterdu ftecrl sfixed. is circle the of radius the after determined be only can seulyvld nti setting, this In valid. equally is =   σ d M M 4 − M X −∞ G S x ∞ 4 ∗ MN ∗ 4 1 g 3 × p and e µν (0) g · = 2 S µν | n 2 σ g πr 1 F | e − r = µν ( iny/r type − − ogv h ih eut u hr sn te osritt fi to constraint other no is there But result. right the give to sadrvdqatt.W nweprmnal that experimentally know We quantity. derived a is auaKenansatz Klein Kaluza d M M φ κ κ F y s 2 φ 2 µν ∗ pl 3 (4) 2 srsrce oteitra [0 interval the to restricted is φ (0) − A , Z (0) 1 3 where = A d R {z   A ν (0) 5 x µ (0) W + p µ g G ( A µν y = 4 1 | 55 ν (0) ) G φ η X −∞ | − (0) =
∞ − µν (5) A φ κ e κ isenHletaction Hilbert Einstein d F A , R − 2 2 x µν σ (0) µ n φ κ µ A φ . e W − eaetegueculn othe to coupling gauge the Relate iny/r d ν F (0) φ x e5dmninlMnosimetric Minkowski dimensional 5 he ( µ (0) y ν (0) A sawr atrta salwdb the by allowed is that factor warp a is ) l lya motn role. important an play ill hoyo rvt,eetoants and electromagnetism gravity, of theory d A (5) µν − ν φ , µ (0)
R d + MN   y , − } 2 2 M A φ κ eset ve πr − + 6 = hoyi v-iesos using five-dimensions, in theory l pl 2 φ 0 = .Frsmlct ewl e the set will we simplicity For ]. ∞ X −∞ ∞ ntt ecnue ihthe with confused be to (not ∂ µ µ oe fmsiemodes massive of tower κ   φ φ n edequations field and ( . − (0) n φ 1 e (0) iny/r ∂ = ) µ M 2 φ M (0) 4 pl ie circle a times M + pl = U ≈ (1) ... p ) 10 x hc/G isometry G 19 M MN ∗ GeV, and S the 13 = 1 , Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. The aiod oeta h lnkms o eae like behaves now mass Planck corresponding the group that gauge Note with manifold. dimensions 4 in Mills Yang to eeaiaint oedimensions more to Generalization rmhr ecnepiil e h yere ftelwenerg low the of symetries the see explicitly can we here From rmteKlz-li nazw a rt h ieeeetas element line the write can we ansatz Kaluza-Klein the From Symmetries 14 • • • K y eea iesoa oriaetransformations coordinate dimensional 4 general nodrt ev d leave to order In vrl scaling overall rbe o hs hois tloslike looks It theories: these for problem in,epann h auaKenpormeo nfigal unifying of programme dimensions. Klein Kaluza the coord explaining general from sions, derived be can symmetries gauge standard aietto ftepolmta h hoycno prefer cannot pro theory the not over that does radius) problem theory the the of and manifestation good a equally are dimensions extra hc r ag rnfrainframsls field massless a for transformation gauge are which usi nyoeaogahg ubro them). of number huge a (each among solutions one only string is of ours ”landscape” so-called leading the dimension, is extra the This of shape and volume devel the Recent of value problem. this share theories String theories. h t ieso sarbitrary. is dimension 5th the φ i m (0) transformation F r iln etr fa nenlmanifold internal an of vectors Killing are samassless a is ( x µ y y , = ) 7→ M G A , y λ 4 M MN × y pl ouu field modulus 2 s S d 2 + s 1 x nain,need invariant, 2 = µ = (or f = ( x M 7→ φ µ (0)   µ M (0) = ) ∗ D 3 − − x g × 7→ a ieto nteptnil so potential, the in direction flat a , 2 3 1 ′ µν µ φ V  ⇒ S ( D (0) x g 1 + A λ − ν µν (0) × ) 4 y scle rahn oe ainor radion mode, breathing called is γ κ dx d κ g , S µ (0) y 2 ∼ 1 ′ µ 7→ mn γ all t..Ti sthe is This etc.). , dx φ , mn d = M K ν h auso h ais(rvlm ngnrl fthe of general) in volume (or radius the of values the µν K (0) y − i ∗ ′ m D i m M − φ (graviton) A y (0) K = 2 (0) ν i D j r n + D A −  F 7→ − 4 dy µ i ∂x A ( 4 ∂f A ihmetric with x µ (0) ffcieato in action effective y + µ µ ν j pet nsrn hoyalw ofi the fix to allows theory string in opments oteioer fteetadimensional extra the of isometry the to = λ y , 1 HPE .ETADIMENSIONS EXTRA 1. CHAPTER
hsi h a oudrtn that understand to way the is This ! kaA A , d n eciigadffrn nvreand universe different a describing one 2 ) x oalrebtdsrt e fsolutions. of set discrete but large a to nt rnfrain netadimen- extra in transformations inate φ ouiproblem moduli µ h neatosb en fextra of means by interactions the l M γ κ (0) a DMnosisae(infinite space Minkowski 5D flat a µ (0) A , ∗ 2 mn ieawyt xti ie tis It size. this fix to way a vide dx ( µ (0) γ M mn h = µ K γ φ ∗  mn ⇒ (vector) (0) i n r 2 µ ′ ) (0)  A h hoycorresponds theory The . D i µ i dilaton − n hrfr h ieof size the therefore and d 4 s   = . D 2 fetadimensional extra of 4. = A 7→ hsi major a is This . µ (0) λ − 3 2 d κ 1 s 2 ∂x ∂f µ Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. rvt ae lc nthe in place takes gravity ol ouecodntso a of coordinates volume world hrfr ehv odsigihthe distinguish to have we Therefore aual elzdi tigter hr atrtnst li to tends matter where theory string in realized naturally i which colliders ar at they experimentally because probed dimensions be extra can the that scale seen theo not Klein have Kaluza We standard the sional. discussion been have we scenario far So world brane The 1.3 reo ( freedom sfollows: as iesos oieta nyin only that Notice dimensions. lsdsrnscnlaei h ulhge iesoa ( dimensional higher full att the are in strings leave open can of strings closed ends where surfaces to corresponding h oa cincnb rte as: therefore written and be spacetime can bulk action full total the The d on higher lives a hand inside other brane) the (the (quarks surface particles spatial Model dimensional Standard 3 a the all follows: as is this with ewl nrdc o ieetadmr eea ihrdime higher general a more is and universe different our a now introduce will We where rn u npicpei ol eayohrdimensionality other any be could it principle in but brane h rnvredimensions. transverse the orsodn otenme fcmoet fataeessym traceless a of components of number the to Corresponding r far so explored energies highest the that know we general In SCENARIO WORLD BRANE THE 1.3. esnt xetalrevleo h oueadi a enusu been has it and volume the of experim value any large in a seen expect been to has reason dimensions extra of signature no and ial ecnas on h ubro ere ffedmo a of freedom of degrees of number the count also can we Finally γ µν N steidcdmti ntebae hc o ipiiyw ar we simplicity for which brane, the on metric induced the is dof 2). = p rn,o ufc nieahge dimensional higher a inside surface a or brane, D ukdmnin.Ti cnrosesvr dhca rtsigh first at hoc ad very seems scenario This dimensions. bulk N dof p D rn.Mte ie nthe in lives Matter brane. h rvtnadtepoo aetesm ubro degrees of number same the have photon the and graviton the 4 = S = S bulk brane D gravity matter ( D iesoa uksae(akrudsaeie rmte( the from spacetime) (background space bulk dimensional = − = S − 2)( Z = M 2 ←→ ←→ D ∗ S D d bulk 4 − − x 2 p 1) Z + pnstrings open strings closed | γ − d S | D D ( brane = 1 L x etn u loguefils r rpe on trapped are fields) gauge also but leptons , 0 pctm.Te h orsodneis correspondence the Then spacetime. 10) = eo rns( atclrcasof class particular (a branes D on ve p ( p matter 10 s mninlsaeie(h uk.Gaiyon Gravity bulk). (the spacetime imensional | equire ≤ G ce o.Weesgaiy oigfrom coming gravity, Whereas to). ached d nygaiypoe h xr dimensions. extra the probes gravity only yi hc u nvrei ihrdimen- higher is universe our which in ry n.I auaKentere hr sno is there theories Klein Kaluza In ent. =4 iesoso h rn,whereas brane, the of dimensions 4) (= | − D ( D eysal(mle hntesmallest the than (smaller small very e lyasmdthat assumed ally D soa cnro h dahr sthat is here idea The scenario. nsional 16 erctno nthe in tensor metric ( − ) )) D ukspacetime bulk rvtnin graviton R M cm). 2 1. − ∗ > 3) osdrn tt ea be to it considering e e and TeV 1 D iesosuigagain using dimensions yia xml of example typical A . M < r ∗ D ≈ 10 − M − pl transverse 2 16 . u tis it but t msince cm p branes p p 1) + 3 = 15 of Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. nKlz li hois rmteEnti ibr action, Hilbert Einstein the From theories. Klein Kaluza in hshsa motn mlcto loa otevleof value the to as also implication important an has This cnroi spsil ohv xr iesosa ag s0 as large as dimensions extra have to possible is it scenario sdffiutt eteprmnal n ofrtebs experim best the far so fo and constraints ≈ experimentally the test t use to in to However, difficult have is we probe. dimensions, can extra accelerators the that feels of scale TeV the to osrie h ieo h xr dimensions extra the of size to the up constrained probe can which observations experimental contradict udmna scale fundamental eakdta h udmna ihrdmninlscale dimensional higher fundamental the that remarked with e sfis osdra napdcmatfiain hti c a is that compactification, unwarped an consider first us Let dimensions extra Large 1.3.1 differ two distinguish can we purposes phenomenological For spacetime. bulk full the probing strings closed by incorporated iue1.3: Figure 16 0 . m hsi uhlre hnte10 the than larger much is This mm. 1 V D − 4 ∼ rn ol cnrowt atrcrepnigt pnsrnswihsta which strings open to corresponding matter with scenario world Brane r D − 4 M eoigtevlm fteetadmnin.Btnww a ha can we now But dimensions. extra the of volume the denoting ∗ fw lo h ouet elreeog.W a vntyt hav to try even may We enough. large be to volume the allow we if M − pl 2 16 r mo h tnadMdl hrfr,i h rn world brane the in Therefore, Model. Standard the of cm obe to = M < r ∗ D − 2 M 10 M V . ∗ D mwtotcnrdcigayexperiment! any contradicting without mm 1 − ∗ h lnkmass Planck the hteeg.B h aeagmn ehave we argument same the By energy. that n lse fbaewrdscenarios. world brane of classes ent − slmtdt be to limited is 16 rvt ny ic rvt ss ek it weak, so is gravity Since only. gravity r wihi sal ae ob forder of be to taken usually is (which 4 nscnol eti osae agrthan larger scales to it test only can ents mbcueti stelnt associated length the is this because cm ebaewrdseai,i nygravity only if scenario, world brane he HPE .ETADIMENSIONS EXTRA 1. CHAPTER ntn apfactor warp onstant tadedo h rn n gravity and brane the on end and rt M M ∗ pl ≥ ssilgvnby given still is e nodrt not to order in TeV 1 eamc smaller much a ve h fundamental the e W ( y .W have We ). M pl ) Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. xoetal ag ie fteetadimensions. extra the of sizes large exponentially cl ob forder of be to scale SCENARIO WORLD BRANE THE 1.3. it tmyhl ocnie h integral the consider to help may It Hint: M order pctm eget we spacetime used oieta hswl eur xoetal ag xr dim extra how large of exponentially scale question require dynamical will this a into that it Notice turns it because problem, e n rvt xeiet,suyn eitoso h squ the accele of the deviations both, studying in experiments, experiments gravity of and round TeV next the by tested be rvttoa xeiet swl sSadr oe tests. Model of Standard as well as experiments gravitational rbe o scagdt xli h h ieo h xr dim extra the 10 of size of the scale why explain to changed is now problem nepeso o h lcrc(n rvttoa)potenti gravitational) (and electric the for expression an hsi h so-called the is This compactifications Warped 1.3.2 dimension. fifth the of size the oapitpril nfiedmnin eue othe to reduces dimensions five in particle point a to ihtefloigpoete.Isedo h xr dimensi extra the of Instead properties. following the with wihcnb enda an as defined be can (which ete aewre emtiswt a with geometries warped have then We egh2 length iia oabae en iesoa ufcsisd di 5 factor warp a the inside that surfaces is dimensional ingredient 3 being brane, a to similar ew r oieta hsstu hne h aueo h irrh pr hierarchy the of nature the changes up set this that Notice xrie eosrt httevlm fa of volume the that Demonstrate Exercise: n ilas ecnitn.Teitrsigtigaotthe about thing interesting The consistent. be also will and ≈ r M r = M ≈ ∗ πr .Teheacypolmte eoe h rbe ffidn a finding of problem the becomes then problem hierarchy The ). ∗ M 10 h interval the , 18 ≈ pl 8 2 e trigfo ml scale small a from starting GeV e)i udmna hra h ag lnksaei deri a is scale Planck large the whereas fundamental is TeV) 1 /M mi re ohv lnkms fteosre value observed the of mass Planck a have to order in km ∗ 3 r .Ti sceryrldotb xeiet.Hwvr starti However, experiments. by out ruled clearly is This ). 2 M = adl udu scenario Sundrum Randall ∗ M ∼ I ubro xr dimensions extra of Number pl 2 e.I v iesos hswl eur ieo h extra the of size a require will this dimensions, five In TeV. 1 ilhv afta size, that half have will /M orbifold ∗ 4 hc gives which , d s 2 W ( exp = y 1 2 6 3 of sdtrie ysligEnti’ qain nti back this in equations Einstein’s solving by determined is ) S y V 1 I N eedn apfco exp factor warp dependent N − yietfigtepoints the identifying by r W 1 = ≈ = M ( h ipetcs saana5dmninlter but theory dimensional 5 a again is case simplest The . N R y 0 πr ∗ ) Γ( . d − m for 1mm 2
4 N ≈ π .Tesrae tec n fteitra lyarole a play interval the of end each at surfaces The ). η N/ dmninlptnila itne uhlre than larger much distances at potential -dimensional 1 N/ xe µν e.Ti hne h aueo h hierarchy the of nature the changes This TeV. 1 peeo radius of sphere ieof Size 2) 2 − d lin al ρ x r 2 µ N nin i nt fteivrefundamental inverse the of units (in ensions M ihrdmnin ol iesalrvalues smaller give would dimensions Higher with nbigacircle a being on d esoa pctm.Tescn important second The spacetime. mensional − rdlwa clssalrta 0 than smaller scales at law ared x D ∗ r 1 ν ao xeiet rbn clso order of scales probing experiments rator iesoa aei hti spsil to possible is it that is case dimensional 6 10 e.Ti ste ossetwt all with consistent then is This TeV. 1 = nin ss ag ognrt h Planck the generate to large so is ensions for 10 10 iesos hwta h oeta due potential the that Show dimensions. 10 ofi h ieo h xr dimensions. extra the of size the fix to ρ − be eas o h ml cl (i.e. scale small the now because oblem − 2 − − 11 M 12 = 9 4 d m m m m y ∗ P y 2 W 1TeV = r M − ≡ . i N ( is =1 pl y ) ehns htgvsrs to rise gives that mechanism x ≈
e uniy h hierarchy The quantity. ved y n5dimensions 5 in , i 2 S fteoiia icehad circle original the if , s hsrsl oderive to result this Use . 10 1 gwt dimensional 6 a with ng ti o ninterval an now is it , 18 e weew have we (where GeV ieso ob of be to dimension . 1mm. ground. (1.1) 17 I Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. osdrte h w rnsoeat branes,one two the then Consider h nrysae r esitds htw a aePac cl ntelf b left the on scale Planck have can we that so redshifted are scales energy The isenseutosimply equations Einstein’s iue1.4: Figure volume The 18 asheacybtentePac rn at brane Planck the between hierarchy mass clscag with change scales rn”,tettlato a otiuin rmtetobra two the from contributions has action total the brane”), hne from changes sln sw oeaa rmti n fteitra,alteen the all interval, the of end this from factor away move we as long as clsi prpit o h irrh rbe.I h fund the If problem. hierarchy the for appropriate is scales ohv tension have to h lnklength Planck the opcie na interval an on compactified h hsc ilb oendb hssaebtat but scale this by governed be will physics the ofidamcaimt xtevleof value the fix to mechanism a find to atclrw a aeteeetoekscale electroweak the have can we particular Exercise: r that case unwarped the (unlike experiment with compatible udmna cl n5D in scale fundamental e −| ky | rn ofiuaini h adl-udu cnro h apdgeo warped The scenario: Randall-Sundrum the in configuration Brane V osdrafiedmninlgaiyter ihangtv co negative a with theory gravity dimensional five a Consider ni erahteohredo h nevli which in interval the of end other the reach we until D y − to 0 = ± 4 r Λ a factor a has y /k ≥ ftefnaetlsaeis scale fundamental the If . 50 epciey Here respectively. y M ℓ = pl W hsi oeeeatwyt sle h irrh problem. hierarchy the ”solve’ to way elegant more a is This . (0 and πr ( π , y ) via d ) Λ ∝ s . . 2 ahedo h nevlcrepnst 3bae hc ec we which ’3-brane’ a to corresponds interval the of end Each e η eiyta h apdmetric warped the that Verify S y −| V µν = y D ”h lnkbae)adoeat one and brane”) Planck (”the 0 = ky n h tnadMdlbaeat brane Model Standard the and 0 = − = 7−→ r | 4 e with fodr50 order of k − S 2 ∼ exp( M sacmo cl ob eemndltri em fthe of terms in later determined be to scale common a is y W =0 ( ew k − θ Z ) + − π M osat(e 5 n xml he ) otemetric the so 4), sheet example and [5] (see constant a + η y ≈ π kπr µν d ∗ = M y S ≈ d ℓ y ) r pl exp pl x = η M µ µν ewl aea xoetal mle cl.In scale. smaller exponentially an have will we πr oieta nti cnro5dmnin are dimensions 5 scenario this in that Notice ! · pl e d hsmasta l h eghadenergy and length the all that means This . − x W the , + e n h ukitself: bulk the and nes πkr mna cl stePac cl,at scale, Planck the is scale amental ν qie aismn ioeeslarge). kilometers many radius a equired ( y − S ) ≈ HPE .ETADIMENSIONS EXTRA 1. CHAPTER bulk
y rysae ilb rdsitd ythe by shifted” ”red be will scales ergy r e if TeV 1 . y rn are hsc at physics carries brane 0 = aeadteTVsaeo h ih brane. right the on scale TeV the and rane 2 d = θ 2 πr er nthe in metry y = hsepnnilcagsof changes exponential This . y r πr mlgclconstant smological = sol lgtybge than bigger slightly only is oieti sol cartoon. a only is this Notice . πr ”h tnadModel Standard (”the y ieto ie iet a to rise gives direction eol need only We M Λ pl y hoose but , < 0 = 0 , Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. soeo h otatv ra frsac ihnsrn theo string within research of areas active most the determ of that one moduli is the fix to mechanisms Studying theories). ehnsshv enfudt eraie nsrn hoy(p theory are string they in otherwise realised since be to extra found the of been size have the mechanisms fixing of problem the scenarios into turned both been in has that Notice straightforward. not is brane napdcase. unwarped ov for Solve atbet stabilise to factible n h -i scale 5-dim the and hssle h irrh rbe hog apn sln as long as warping through radius the problem hierarchy the solves This ig aste sgvnby suc given field is Higgs then the mass normalised Higgs canonically we step last the In bv (Randall-Sundrum). above scale 5D the from differ Does scale scale? Planck Planck the the from scale electroweak the reproduce aifisEnti’ qain.Here equations. Einstein’s satisfies SCENARIO WORLD BRANE THE 1.3. idtevleo h constant the of value the Find ntelreetadmnin cnroteheacyproble hierarchy volumes. the large very scenario with dimensions compactifications extra large super the to alternatives In problem are hierarchy dimensions extra the warped and and Large scenarios world Brane 1.3.3 h lnksae ooti ig asa h eksaew ne we scale weak the at mass Higgs a obtain To scale. Planck the aoia omadso httems spootoa otef the to proportional is mass the that show and form canonical h nevl o a s htEnti’ qain euet reduce equations Einstein’s that use can You interval. the uligcnrt oesta nld h tnadmdlan model standard the include that models concrete Building h ig arninis Lagrangian Higgs The ntewre aete‘ouin smr lgn sw a see can we as elegant more is ‘solution’ the case warped the In W r S ( vis oterqie au.Teavnaeoe h ag xr dim extra large the over advantage The value. required the to θ ) n s h apfco oso htteeffective the that show to factor warp the use and g vis He 6 r = M W r krπ ovalues to e ∼ ∼ = 2 − ′ 2 r eecmaal as comparable here are → 2 krπ adhoc M H g = pl 2 4 m πkr H Z Z Z = − cnro ae nhge iesoa (nonrenormalisable dimensional higher on based scenarios k. = d d d 2 osdrteHgsLgaga ntebaeat brane the on Lagrangian Higgs the Consider 4 4 4 M ∼ M Λ e e x x x − − 0ta h irrhclylrevle eddfrtevolum the for needed values large hierarchically the than 50 3 3 p p p krπ 2 r W , | | | Z g g g ( v M θ 4 4 vis − m 0 ) | | π π ? h e n eed ntewr atr h aua cl for scale natural The factor. warp the on depends and H stewr atrand factor warp the is | d − g 3 h 4 µν e θ 4 g W r e krπ vis ∼ µν 2 − D − ′′ 2 D kr 2 µ h W e H g µ − 4 µν = stiny. is H † krπ D † e = 2 D ν 2 krπ M h rbe fsligteheacyproblem hierarchy the solving of problem the , H yadhge iesoa hoisi general. in theories dimensional higher and ry o ν M pl M ymtyt drs h irrh problem. hierarchy the address to symmetry H hssleteheacypolm o does How problem? hierarchy the solve this httekntctrsaecnncl The canonical. are terms kinetic the that h Λ iesos ti ot eakn htboth that remarking worth is It dimensions. secagdt h rbe ffinding of problem the to exchanged is m k D actor nstesz n hp fetadimensions extra of shape and size the ines 3 − tigte nfimrtertclgrounds theoretical firmer on them utting 3 kr µ − ntesimple the on H ehns a efudt stabilise to found be can mechanism a λ 1 4 ed drsigisohrpolm na on problems other its addressing d  † λ ( D e δ r | D − H ( − ( | sacntn esrn h ieof size the measuring constant a is θ πkr kπr ν H lnksaei now is scale Planck | 2 H − e | − 2 − . π 2 ∼ o ag can large How − kr − ) e nin sta tlosmore looks it that is ensions 0 h -i lnkscale Planck 4-dim The 50. −
− v λ 2 D 0 . 2 krπ ( δ ) | ( 2 H xml mentioned example 5 = θ i v ) | 0  2 2 θ ) . 2 − = i v π, r 0 2 gravitational ) ) ei re to order in be 2 rn tinto it bring i nthe in e v 0 19 is Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. 20 HPE .ETADIMENSIONS EXTRA 1. CHAPTER Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. Define Let o hoyo emosi oeta ordmnin,ne so need dimensions, four than γ more in fermions of dimensions theory a higher For in Spinors 2.1 9]. di 8, extra [6, in d are fields extra aspects fermionic in about fields What bosonic metrics). possible and tensors the discussed have we far So dimensions higher in Supersymmetry 2 Chapter in e of set a .. pnrrpeettosi vndimensions even in representations Spinor 2.1.1 hr h Σ the where hs emtct rprisaedet (Γ to due are properties hermiticity whose fttlnumber total of arcs ..rpeettoso h lffr algebra Clifford the of representations i.e. matrices, η MN | 0 i n eoetevcu uhthat such vacuum the denote a a n diag(+1 = 0 j ar flde operators ladder of pairs emoi oscillators fermionic := := MN h Σ MN r eeaosof generators are 2 2 i i number states n + 1 , a Γ Γ , − i 2 0 Σ 1 j , n ..., , Q P Γ + n ( − Γ a M j + i | ) i 0 1 1 † Γ i , −
o   2 = j Γ )sgaue tflosta the that follows it signature, 1) +1 ( N n 2 = a i o
SO i )   n † Σ δ | a (1 2 = MQ = 0 = ij i + i ⇒ | ⇒ D , 0 , i η ... ( 0 ,te hr r states are there then 0, = NP − η a ) MN i † ( ( = 1 + ) )sbett h oet algebra Lorentz the to subject 1) a a +Γ = † n 0 j ( Σ + ( a ) ) n a 2 † † i , j ) a , ) 21 † = 0 = NP | 0 j n (Γ and o Σ · · · i k X η 2 2 i MN i =0 n MQ · · · =   − Γ M esos odrfrne o h technical the for references Good mensions? 2 Γ j − = n n mnin saas etr,antisymmetric vectors, (scalars, imensions k 6=0 0 eaaou ftefu iesoa Dirac dimensional four the of analogue me ( a ( + a a Γ + j Σ )   n † i 4 MP (where i ) ) i = † † D Γ h ( 2 = , 1 2 Γ a − j
η 2 = n M ( +1 NQ Γ a − M j
1 , ) n j ) 1 6=0 † n j , † Γ o 0 = − ... N rmteCiodalgebra Clifford the From . 2 = i Σ ( , 0 = a , 1 NQ 1 .,n ..., , ) 1 = D † 2 η | 0 . . MP .,n ..., , i −
o)furnish now) 1 . − 1 , Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. ersnaini h aea for as same the is representation satisfies tflosfrom follows It ersnain nee iesosaerdcbe ic th since reducible, are dimensions even in Representations spinor. component 4 a form hs of those utadd Just oeta h eeaosΣ generators the that Note { aea” a have D .. pnrrpeettosi d dimensions odd in representations Spinor 2.1.2 as to referred are chirality definite of spinors and ic hr sn ute arxwt hc Γ which with matrix Γ further no is there Since althose Call by defined are representations spinor the in States 22 ntesnethat sense the in ihegnau 1free ubr of numbers even for +1 eigenvalue with ngnrl define general, In htalthe all that n n the find and Γ 2 2 = n +1 n , ihetne metric extended with 1 + Γ γ SO M 5 i | ,teei ociaiy h pnrrpeettos dimen representations’ spinor The chirality. no is there ”, Γ s } | (1 2 0 s | n .,s ..., , n (Γ and 0 = s 0 , n +1 0 .,s ..., , 2 .,s ..., , Γ n 2 = 2 N n − n n +1 − n D )aegvnby given are 1) i n S 1 − S n − i ogv h ubro pnrcomponents: spinor of number the give to 0 1 , i Γ 1 i ia spinors Dirac S 0 Γ i | 2 r iesae oΓ to eigenstates are Γ 1 := M s n endaoet etesmlaeu iesae of eigenstates simultaneous the be to above defined ( ( 1 . S ... 0 a a +1 (2 o . s , ... , ... 0 j i ) ) N )(2 ) † †    Γ 2 n D a 0 = a − 2 i Γ ,i efcl xed h lffr ler in algebra Clifford the extends perfectly it 1, = +1) 0 j n S 1 ( ( 2 D n − a a η n i − − − := 1 i 0 +1 µν | Γ 1 2 = s 2 = = ) uulycmue ow ignlz l of all diagonalize we So commute. mutually ) In . 2 i i † 2 , 0 oteΓ the to † HPE .SPRYMTYI IHRDIMENSIONS HIGHER IN SUPERSYMMETRY 2. CHAPTER n Γ 1 2 2 1 | . s , ... , (+1 = a a s +1    M i n 0 0 i =( := s n . s , ... , u o reuil.The irreducible. now but , D n Γ i = + = − − − 2 2 h 2 = 1 4 1 iesoswith dimensions 4 = n n := Γ n n Γ , 2 +1 − 2 − n a − 2 = 2 = 2 1  2 4 1 0 M i n 1 +1 n 0 +1 Γ 1 + 2 1 i 1 4 ) − h with i ..., , † 1 Σ = Γ arcsof matrices n = 1 D D ( and 4 2 h i elspinors Weyl ... − i s 2 , = 2 − Γ  n 2 0 j 1 1 n + 0 Σ Γ − n = Γ , +1 − = M − 2 1 2 MN , − 1). 0 (2 s n Γ ) i 1 : : D/ i Σ Γ o d ns hspoet scalled is property This ones. odd for 1 − Γ i 2 ... ol epie oafurther a to paired be could 0 = | )(2 j 1 h ±| 1 1 01 s D D i +1 eeaiainof generalization e i Γ 0 unu numbers quantum 2 ( ... i a . s , ... , s 0 +1) 2 = 2 = i , Γ = n 0 = 0 Γ D 1 , − . s , ... , 2 ..., , . 1 Γ n n n = Σ = ) 2 = − n 1 † : : n odd 1 + even i 2 1 ( D − 2 ,frisac,testates the instance, for 2, = , n s ... i i − n , n n 1 +1 ini 2 is sion n SO 1 = 0 = − − i (2 i 2 = h − 1 Σ 1 . Γ iesos rmisproperties its From dimensions. + j . i )(2 01 (1 .Sneoddmnin onot do dimensions odd Since 1. 2 (Γ .,n ..., , 2 1 n n ) , − j 2 2 +1) | n 2 0 1 + n +1 D i γ , eeaosi diinto addition in generators ) 2 − − . 5 ) Γ 1 s , 2 1 i . 2 D n = − = 1 2 = ± i 2 1 1 a n i . iesosto dimensions prtr the operator, |± chirality 2 1 , ± 1 2 i , Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. ojgt fteciaiymatrix chirality the of conjugate aini ojgt oisl.Wy pnr xs nee dim even in exist spinors Weyl itself. to conjugate is tation ttrsotta hrecnuainol rsre h spin the preserves only conjugation charge that out turns it ..in i.e. oMjrn elsiosaepsil ndimensions in possible are con spinors Weyl Majorana Majorana or so Weyl the impose either D can one and other, each where fpriua eeac stegenerator the is relevance ”spins” particular several have Of i can seen we have Then we representation. as spinorial diagonalized simultaneously be thus can and where between relations that guaranteed not is it eainbetween relation A transformations. rnfrain,spinors transformations, where ial h uesioilpart: spinorial pure the Finally aoaacniincnb moe naWy pnrif spinor Weyl a on imposed be can condition Majorana condition. A Majorana a admit not do e etr ftePicreagbai htaltegenerat the all that is Poincar´e algebra dimension the 4 of in feature those new to A identical are proof the for arguments .. aoaasios(xr aeilntcvrdi lectu in of covered notion the not introduce now material us (extra Let spinors Majorana 2.1.3 ALGEBRA SUPERSYMMETRY 2.2. rn hre.Ti sthe is This charges. brane dimensions tnadPicreagbai ihrdimensions, higher in Poincar´e algebra standard D D in algebra SUSY The algebra Supersymmetry 2.2 ) h ler a h aesrcuea n4dmnin,wi dimensions, 4 in as structure same the has algebra The 4). = o ,teWy ersnain r efcnuaeadcom and conjugate self are representations Weyl the 8, mod 2 = iesos(ihindex (with dimensions hr Σ where O  C D Ω and sthe is αβ AB 4 = D MN  , O M 0 = 8 ∗ , hrecnuainmatrix conjugation charge sdfie bv ersn h oet rnfraini the in transformation Lorentz the represent above defined as 12 r ieso-eedn osat n h eta charges central the and constants dimension-dependent are aetesm weight. same the have , 1 ... , ψ , 2 ↔ , D iesos h w nqiaetWy ersnain are representations Weyl inequivalent two the dimensions, 3 , ψ ψ o .I te od,aogtepyial esbedimens sensible physically the among words, other In 8. mod 4 iesoscnit fgenerators of consists dimensions ∗ A rnfr into transform > D srfre oa the as to referred is onigtenme fsprymtisa nteetne SU extended the in as supersymmetries of number the counting (Γ n 2 oea adl-o L eeaiaino the of generalization HLS or Mandula- Coleman 4 n Q +1 reality α A ) [ ∗ Q , M M h MN ( = o pnr nMnosisaeie ne nntsmlLor infinitesimal Under spacetime. Minkowski in spinors for β B M ψ ossec eurs( requires Consistency . 01 o ′ 01 ψ Q , hsi sdt en a define to used is This . = − n t ope conjugate complex its and = , α A ψ 1) aoaacondition Majorana O = ] Q + n i +1 α A iω Ω − αβ AB rnfriga pnr imply: spinors as transforming C MN = D − Σ
and 2 = 1 MN M Σ − Γ ensions MN D iw 0 − enda h ievle fteeoperators. these of eigenvalues the as defined P M r’ciaiyi ( if chirality ors’
1 n ewl kpte here. them skip will we and s M α β 0 = O MN Γ ors h icsino h ihrdimensional higher the of discussion the n ψ 2 Q ic h Σ the Since . n + D , α A +1 , D M 1 ail ihteMjrn condition, Majorana the with patible C iin u o oh ndimensions In both! not but dition, thst eo h form the of be to has It . P , 10. = Z htebsncgnrtr enn a defining generators bosonic the th Γ 2 2 = (2 Γ M αβ AB , 0 0 j weight 3 , )(2 ) , C , n Q ∗ o n h elrepresen- Weyl the and 8 mod 4 ψ j n yaayigtecomplex the analyzing by and , α A +1) C ∗ Γ ato hc r pnr in spinors are which of last r ossetwt Lorentz with consistent are − MN w Z 0 omt ihec other each with commute 1) αβ AB hc spsil in possible is which 1 = fa operator an of pnra representation. spinorial n r ngnrlcomplex, general in are +1 res) ope ojgt to conjugate complex o a loinclude also can now D 1 If +1. = ler.The algebra. 4 = ions, ψ Ycs in case SY D ∗ n O = 5 = seven, is by C , entz Γ 6 0 23 , ψ 7 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. helicity eeae yatn on acting by generated iesoa ersnain ersrc otesalrlit smaller the to restrict we representations dimensional gi h itegop( group little the again d.Te uns nagbao h form the of algebra an furnish They 4d). hsipista uesmercmliltcnb constru be can multiplet supersymmetric a that implies This s with h aiu ievleof eigenvalue maximum the oceto-adanhlto prtr.T e hs consi this, see To operators. annihilation and creation- to hslast afo h ubro opnnsof components of number the of half to leads This s massless the consider d we case higher 4-dimensional in the algebra to Similar supersymmetry of Representations 2.2.1 24 oi esatwt rirr spinors arbitrary with start we if So utemr,w a eaaethe separate can we Furthermore, hrfr h egtof weight the Therefore sn h ue ona´ ler,w a loso hsexpr this show also can P we Poincar´e algebra, super the Using -iesoa aei hc tcnb rudta hr sn hsclevi si physical argument clear no understanding. less is better a needs there issue is that This this observed. argued dimensions been higher be in can However it representations. which in case 4-dimensional so particle eurn egt+ weight requiring ersnain,teweight the representations, { j ( ,Q Q, α 2 1 Q ... ) oieta etitn ool nt iesoa ersnain sa is representations dimensional finite only to restricting that Notice + α P N } D s a ob fteform the of be to has D h j ( nwihboth which in − λ P β M opnns(ealthat (recall components 1 ) 1 niiae ythe by annihilated (2 aet aihleaving vanish to have hc r l eoi u case our in zero all are which − j )(2 P 0 j +1) n that and 0 = 1 2 Q Q , Q en ht( saseilcs f[ of case special a as (, that means α Q α + ISO Q

h shv weight have ’s

( w Q M w α P + =+ =+ α Q hrfr hywl eo h form the of be will they Therefore . 1 01 ( β w M D ± 2 1 1 2 ) Q = Q , i MN P − = − h = = M N 0 α = ) a nnt iesoa ersnain.I elmtto limit we If representations. dimensional infinite has 2)) ± | { Q operators, − i D Q is 01 nterpeetto.Ntc htfrtemmnu famass a of momentum the for that Notice representation. the in sinto ’s | a eecue n eol edt osdrcmiain of combinations consider to need only we and excluded be can 1 | Q α + 2 = + − 1 2 + w ↓ P , HPE .SPRYMTYI IHRDIMENSIONS HIGHER IN SUPERSYMMETRY 2. CHAPTER = Q , α , w Q 1 2 2 1 ± = P fteform the of D ( + = β − 2 , 1 , α − { = µ P 2 1 Q ± + o vnand even for Q ± ± S ↓ α , Σ ( = µ + Q j } + 2 1 .A h ” the As 1. 01 ± 2 1 1 2 P , Q ( = and Q , − . steol ozr anticommutators. nonzero only the as , , E , E 2 1 Q 0 ± β | , i ± λ ) + α , E , E 2 1 i · · · Q 1 2 } Q , − ,adters ftesae ntemliltare multiplet the in states the of rest the and 0, = = , α − = = · · · · · · , S ntemsls ae namely case, massless the in 0 codn oegnausof eigenvalues to according { M ± N j ∓ Q , . , ... , l group tle − Q , 0 e h commutator the der D i 2 1 MN − − ± iS i ± ( . , ... , combination ” Q , α 2 = P ae endb momenta by defined tates 1 2 α , 0 1 2 tdsatn rma”aum state ”vacuum” a from starting cted 0) Q , Q i 1 i Q − sint ealna obnto fthe of combination linear a be to ession α , β togasmto.I sls utfidta the than justified less is It assumption. strong D α , ± } α α ) osqety l h combinations the all Consequently, 0). ) 2 − = ] and 0 = 1 ec o h nnt iesoa massless dimensional infinite the for dence O P = = c xr iesostesle aenot have themselves dimensions extra nce o d iesoaiyrespectively), dimensionality odd for 1 = ! ( 0 − D
1 = Σ − − 1 = . .,N ..., , − MN ( 2 i s 2) ..., , { j ( P ..., , Q α Q 1 Q 1 ) edfietesi obe to spin the define We . α D + α N − + Q , N ,) , 2 D 4 P D s M − 0 imensions . j ( 6 } β N 23 szr nmassless in zero is ) 2 D corresponding 0 = ) sadr pnin spin (standard Q . ( α Q β nyfinite only ) . | λ i less of Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. ie oestate some Given e stk lsrlo ttesetu of spectrum the at look closer a take us Let h uesmerctere in theories supersymmetric the iesoaiy hsipisamxmmnme fspacetime of number maximum a implies this dimensionality, imposing be will number total the and ALGEBRA SUPERSYMMETRY 2.2. tflosfrterneo occurring of range the for follows It oietesmlrt fti ruetwt h rvospro was previous dimensions the 4 with in argument metries this of similarity the Notice eteeoeoti h olighlcte yapplication by helicities follwing the obtain therefore We • • fteetere switni h table. the in written is theories these of yeIA yeIBsrn hois( theories string IIB type IIA, type hsalw w different two allows This 10 = D Only 11 = D ooi ere ffedm8 44. + 84 freedom of degrees bosonic ohidcs,adtesbrcino h 2 the of subtraction the comp and vector indices), the times both components spinor the of product the is In nain esr(uha h ercadtehge dimensio higher Ψ the and trace” metric the as (such tensor invariant etrsio Ψ spinor vector n a a httespin the that say can one nodrt on h o hl)dgeso reo o ahfie each for freedom of group degrees little shell) the on (on based the count to order In h utato ftetaefrtegaio.I em of terms In graviton. the for trace the of subtraction the reo,i h aeof case the in freedom, o h rvtn spinor gravitino the For in tensor symmetric a to corresponding the D D | λ 1 eoti 9 obtain we 11, = N ≤ | M 1cs.A niymti esro rank of tensor antisymmetric An case. 11 = UYi osbe h nymliltcnit of consists multiplet only The possible. is SUSY 1 = M α 23 hsrequires thus 2 Γ Q | αβ M λ α + i | fhelicity of a ozr,te twudb oe reuil representa irreducible lower a be would it then nonzero, was λ M α i nyfrihsa reuil oet ersnaini co if representation Lorentz irreducible an furnishes only = = graviton λ N A | g ·  h ψ max N O MN {z 2 M MNP N 4 λ α µ .W ilseltrta precisely that later see will We 8. = λ D 4 D | ( λ ehv 2 have we , D N (i.e. } . 23 1 2 − i hoisadone and theories 2 = − 0and 10 = − D − otiuino h ih adsd sdsadd oegener More discarded. is side hand right the on contribution , 2 ψ , Q , λ ) h rvtnin graviton The 2). 4 with 1 2 ≤ λ M sthat ’s 2 opnnsfrtegaiiowihmthstenme o number the matches which gravitino the for components 128 =  min α + 23 2 u eebrn that remembering But 32. N | i Q gravitino λ | p λ | |{z} α + λ − )adtp rhtrtc( heterotic or I type and 2) = D ,ti s( is this 3, = 1 + i = D i M D α 2 − | = λ D 2 1 3 + i 11. = i 1and 11 = · λ λ ( − D . , ... , | D A , Q λ 2 − iesosmnstetae hc s45 is which trace, the minus dimensions 2 − i 3 − α + ,teato fany of action the ), ere ffedmo spin a of freedom of degrees 2) M niymti esr(non-chiral) tensor antisymmetric λ N − 23 D − D 2 fthe of orsodn otemsls pcrmof spectrum massless the to corresponding 1 = | iesoscarries dimensions p D λ | λ 10: = N 2 − i in 1 + 3 9 fta h aiu ubro supersym- of number maximum the that of 8 su 3 − D 84. = ) dimensions needn opnns h rtfactor first The components: independent =
2 ersnain (1) representations (2) 1 2 Q | · a arcs aih fte”gamma the If vanish. matrices) Γ nal α + = MNP nnso h rvtn snei carries it (since gravitino the of onents N − {z

D N 4 w D N Σ dw aet efr h analysis the perform to have we ld =+ i iesoshas dimensions 23 } uegaiyi bandfrom obtained is supergravity 8 = N D Q 8 . D 2 1 α + N Q 2 = D α + illwrthe lower will , ) h pcrmfreach for spectrum The 1). = 10 = | λ ( D 2 D i , − 2 + 2)( , D 11. 2 2 1 ino t w right. own its on tion 2 − D Q λ tatoswt any with ntractions atcei iia to similar is particle 1 − ⊗ 1) o vnadodd and even for α + D − p M 1 2 | +1 − λ components, 1
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− eigenvalue: ere of degrees 4in 44 = 1 2 3
ly a ally, ⊕ 1 2 25
f , Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. hre fpbae r e xmlso eta hre nth in charges central of examples new are p-branes of Charges hrfr h UYagbacnb rte as written be can algebra SUSY the Therefore Where 1sprrvt r -rnsad5bae.Tecorrespondi The 5-branes. and 2-branes are supergravity 11 26 of ftesiosin spinors the of ultno orsod to corresponds tensor dual hsagbai hto h H hr r 32 are there LHS the on that is algebra this ealta bu eea niymti tensors antisymmetric general about that Recall .. uesmer ler and Algebra Supersymmetry 2.2.2 each. freedom of degrees 8 + 8 with multiplets ntp I aetesm hrlt s h hoyi chiral). is theory the (so field chirality 4-index (s same chirality the opposite have have IIB case type IIA in type in fermions the Here ffedmi ohIAadIBcssad pt 2.I h yeI type the In 128. to up adds cases IIB and N IIA both in freedom of h aea h ubro opnnso h H.S ed o exp not do we So RHS. the of components of algebra. number the as same the 11 h opigi eedn fteojc’ charges object’s the of dependent is coupling The × utpes h nqegaiainloewt 4+6 degr 64 + 64 with one gravitational unique the multiplets, 1 = A o ntnein instance For • • • • 10 MNP B B B B / C M MN MNP M 5from 55 = 2 1 hc ope ote(magnetic) the to couples which ) stecag ojgto arx ett on h ubro number the count to test A matrix. conjugation charge the is ...M ope oaparticle a to couples ope oastring a to couples A oammrn ... membrane a to p +1 n N Q MNP † Q D oa to α 1i 2 hra nteLS h rttr a 1cmoet f components 11 has term first the LHS, the in whereas 32) is 11 = IIA Q , IIB D Z I MN p β ssl ul ti ayt hc httenme fbsncand bosonic of number the that check to easy is It dual. self is 1tepeec fthe of presence the 11 = o brane ( g g = g n h atoe( one last the and MN MN MN A C N QRS MNP Γ R R n M 2 2 B B Q , Q B
× × MN M αβ MN p particle ψ ψ object d HPE .SPRYMTYI IHRDIMENSIONS HIGHER IN SUPERSYMMETRY 2. CHAPTER P string brane M M α α issvnidxfil teghi ult h -ne edstr field 4-index the to dual is strength field index seven (its x d M o M x M + hr d where , 2 ∝ 11 × × 5 ∧ B C p 6 ogvn oa o h H f1+542=528 = 11+55+462 of LHS the for total a giving so 462 = ) Z MN d 33 B ψ φ Γ hreculsto couples charge P a M x rns otentrletne bet in objects extended natural the So branes. 5 = MN MN Z / N B 1 p 2 needn opnns(ic h dimension the (since components independent 528 = 2 A q M A Z ...M M -Branes wrdsheet) (world x MNP + i
1 1 M αβ ...M ...i p 2 b eest h ol line world the to refers × p M Z M A φ α p B = MN +1 1 esripiste opet -rns The 2-branes. to couple they implies tensor A φ gaiy ( (gravity) ) ...M M B R loi h I aetefil tegho the of strength field the case IIB the in Also 1 fsi r1 eknow: we 1, or 0 spin of h hoyi o hrl n h fermions the and chiral) not is theory the o ...M MN + d gcagsaethen are charges ng p UYalgebra: SUSY e M D Z N Q MNP † MNP − p M e ffedmadtecia rmatter or chiral the and freedom of ees C +1 1 1 Γ J ...M aeteeaetodffrn ye of types different two are there case N QR MNP 0 i 1 ...i p c oesrrsst d othe to add to surprises more ect 2 p A A : × needn opnnsof components independent f M M
λ αβ Z λ N QR MNP (chiral) ) 2 Z × MN rom emoi degrees fermionic λ and P M h second the Z N QR MNP ength D = . Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. iesoscudb elcdby replaced be could dimensions 5 h xml fasaa n5dimensions 5 in scalar a of example the e srve h eea rcdr frdcn n ubrof number any reducing of procedure general the review us Let Reduction Dimensional 2.3 su original the of part preserve branes that implies usual as h uegaiywt h iglrt o en point-like being not hol singularity black the as with supergravity appear the They equations. field bran supergravity corresponding the the of of charges opposite but tension same P odto sstigtercagseult hi tensio their in to objects equal pointlike charges correspondin their the setting be is can condition p-branes BPS The states. BPS the defining osdreg h euto of reduction the e.g. Consider 2 emoi ere ffedm iesoa euto to reduction Dimensional freedom. of degrees fermionic 128 h rn ftetnoilcharges tensorial the of brane the REDUCTION DIMENSIONAL 2.3. gi,tesbrcini netasio ereo freedom. of degree spinor extra A an is subtraction the Again, hnteFuirmode Fourier the then o iesoa euto,ol epthe keep only reduction, dimensional For ealhr htatreidxatsmerctensor antisymmetric index three a that here Recall n httoidxatsmerctensors antisymmetric index two that and 9 · 2 MNP 10 h eta hre si h aeo xeddsupersymmetry extended of case the in as charges central The − 4dgeso reo n h gravitino the and freedom of degrees 44 = 1 htcris( carries that A MNP g MN ψ M α D A 3 9 7→ 7→ 7→ 4dgeso reo.Nt htw ae18bsncdegree bosonic 128 have we that Note freedom. of degrees 84 = ) M ) sal h hreta per nteBScniini t is condition BPS the in appears that charge the Usually 4). = ϕ ϕ B ( ( n x x graviton A |{z} |{z} 32 ∂ MN 4 ψ 2 ihrsett the to respect with M M |{z} ψ M g µνρ n =8 µ α µν ) ) ∂ M ψ , D A , 7→ 7→ 7→ 7→ ϕ g , 1to 11 = Z ∞ h ini hscag ensbae essatbae tha antibranes versus branes defines charge this if sign The . = 0 = B ϕ A 7 1 4 tensors 7 · 2 | =6fermions 8=56 ( ayfilsin fields many µ µν vectors 7 n x ( µνm {z M µ |{z} x 4D |{z} |{z} B , ) µm µ n 5 } ψ d ) − m α A mode, 0 = spinors A , :Tefnaetlfilsaegraviton are fields fundamental The 4: = A , µνm = ⇒ S g , vectors |{z} 1 M µn ieso a asof mass a has dimension r ult clr.Tesetu stesm sthe as same the is spectrum The scalars. to dual are | 1vectors 21 ∂ . scalars 4 ψ m | µ 7 × d {z A M B , 2 α ∂ · ( µmn {z 8 x µ µνρ .If 4. = 2 clr (symmetry!) scalars =28 S µ ϕ ih9 with } 1 } u fhge ieso ( dimension higher of but ) n tels fwihhsradius has which of last (the lk h ascag eaini h aeof case the in relation mass/charge the (like n m , n4dmnin are odgeso freedom of degrees no carries dimensions 4 in scalars | A , persymmetry. − .pbae loapa ssltncsolutions solitonic as appear also p-branes e. {z d ido ouin nw sbakbae of black-branes as known solutions of kind e mn |{z} ϕ h nlfil sa niymti tensor antisymmetric an is field final The ed to: leads 4 = mn } iesosbge hn4to 4 than bigger dimensions · R P bet.Tegnrlsto fthe of generalisation The objects. BPS g n 2 smassless, is 2 2 7 1 9 · · − 2 6 2 ϕ 1 · · 5 3 4 = n 3 clr (antisymmetry!) scalars =35 − in 2 0 = D 9 .,D ..., , − 2 1 R n lya motn role important an play 4 = | 8 = . mnp {z , − } · p 6=18components. 128 = 16 1 .TeBScondition BPS The ). g R ffedmand freedom of s epoeto to projection he MN hr edin field where ) d htcarries that .Recall 4. = carry t 27 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. iesoa euto of reduction dimensional euto to reduction I uegaiy losatn rmthe from starting Also supergravity. di IIA extra in theory simpler the to going c by explicit done the originally Actually, was reduction. dimensional di by 4 in related supersymmetries supersym of number of maximum dimensionality the that maximum statement the why of statement the rnsi 0dmninlsrn hoycnpoiechiral provide can theory string dimensional 10 a in branes o4hsa ffc of effect an has 4 to hr sater of theory a is There N 28 uegaiyi dimensions: 4 in supergravity 8 = te neetn iesoa eutosae from are: reductions dimensional interesting Other D number ie iet h pcrmof spectrum the to rise gives 4 = 56 28 70 1 8 N N 1 = uegaiybsdo the on based supergravity 8 = D helicity →N 7→ 10, = 2 0 1 2 3 2 1 .This 8. = N atcetype particle uegaiygvsrs to rise gives supergravity 1 = gravitino graviton fermion HPE .SPRYMTYI IHRDIMENSIONS HIGHER IN SUPERSYMMETRY 2. CHAPTER vector scalar N N atrmliltin multiplet matter 1 = oe snncia,btohrcmatfiain and compactifications other but non-chiral, is model 8 = 8 nseldgeso reo in freedom of degrees shell on ·  N g D 1 2 7+21) + (7 MN · 4 etrmlpe in multplet vector 4 = − 1to 11 = 2  2 56 ntuto fetne uegaiytheories supergravity extended of onstruction (4 N and · − · (4 8+7+3 70 = 35 + 7 + 28 oescoet h SM oiethat Notice MSSM. the to close models 1 = 2)(4 2 esosaddmninlyrdc it. reduce dimensionally and mensions 2 · − 4 esosis mensions A − (4 2 D − erctere s1 sietclt the to identical is 11 is theories metric 2 MNP 2) 1) − 56 = 0i ie rcsl h pcrmof spectrum the precisely gives it 10 = D − D − )=28 = 2) 4, = 2 euigtedmninfo 11 from dimension the Reducing . 0adpromn dimensional performing and 10 = 1 4  · − 2 112 = 2 2 1 = N  N · 8 = supergravity. 4 = 56 = 2 ic ohtere are theories both since 8 = · 2 = 2 · D d 16 = 2 4 = n ngeneral in and 4 = p Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. ymtisadte ssal ne unu orcin.Si corrections. quantum under stable is then and symmetries n that manifold euclidean compact usually volume, finite a is edter sdtrie yteaction: the by determined is theory field iesos h eeatfilsaetebsncfilsi h m the in fields metric bosonic the the are t fields relevant dimensions The four dimensions. in symmetry maximal preserving s solutions touch only will we here and broad very compactifi is circle geometries simplest general the through reduction was dimensional dimensions on extra concentrated mostly have we far So compactifications of overview brief A 3 Chapter h functions niymti esrtr satal u vrtedifferen the over inclu sum terms a actually derivatives is higher term as tensor antisymmetric such terms, bosonic order itrdpnigi h au ftevcu nryvnse o vanishes energy vacuum the of value the if depending Sitter M fatsmerctnosadvr e scalars. few very and tensors antisymmetric of xlctykoncs ycs,ta aihwith vanish that case, by case known explicitly qain o ausof values for equations D 4 M oiefis httesmls ouino hs qain is equations these of solution simplest the that first Notice h yia trigpiti uegaiyter nhg d high in theory supergravity a is point starting typical The hr the where h edeutostk h eea ceai form: schematic general the take equations field The where ersnsmxmlysmercsaeiein spacetime symmetric maximally represents φ i hc stefull the is which 0 = f T ( G MN φ MN S aeawl enddpnec ntefields the on dependence defined well a have ) · · · = stecrepnigsrs nrytno and tensor energy stress corresponding the is niymti esr fdffrn ranks different of tensors antisymmetric , tn o em nldn emoswihaentrlvn fo relevant not are which fermions including terms for stand − Z d h D G x MN p D | i G , | iesoa ikwk pc.Ti ouinuulypreser usually solution This space. Minkowski dimensional h H D M P i ∗ D and − fH 2 D M P h R φ i + 1 ...M R uhta h emtyi ftetype the of is geometry the that such ∂ MN  M H q φ
φ∂ 29 and D M = = = φ htcnb ikwk,d itro nide anti or Sitter de Minkowski, be can that 4 = φ + h su st n xlctsltoso these of solutions explicit find to is issue The . T B A MN f ( ( φ ,H φ H, G, ,H φ H, G, B ( ain.Tesuyo opcictosfor compactifications of study The cations. ahsprrvt hoyhsa spectrum an has theory supergravity Each . spstv rngtv respectively. negative or positive is r φ est edetermined. be to eeds mssm opc aiodfrteextra the for manifold compact some imes sls pcrmo h hoyta include that theory the of spectrum assless ausof values t M igpwr ftecraue t. The etc.. curvature, the of powers ding n h nydsuso ftegoer of geometry the of discussion only the and ( c tde o lal ecieorworld our describe clearly not does it nce ) ,B A, m ftemi points. main the of ome ,H φ H, G, H 1 ...M ohave to M mnin o hc ewl erhfor search will we which for imensions (3.3) ) 1 (3.4) ) q ipefntoso hi arguments, their of functions simple ...M n oescalars some and (3.2) ) q +1 q h G H rsn ntesetu.The spectrum. the in present MN M 1 u upssadhigher and purposes our r ...M i = q +1 η M MN + 4 φ ... M ⊗ h effective The . and
e l super- all ves D h − H 4 MN M where (3.1) D i − = 4 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. n w-ne esri h w xr iesosare dimensions extra two the in tensor tw0-index and a and compactifications in role important an play fields These h ieadsaeo h xr dimensions. extra the of shape and size the r asessaa ed nfu iesoswt a poten flat a with dimensions four in tensors, fields scalar massless are ie by mined hr safurther a is there T pc eeaie to generalises space hrlwa neatos h xr-iesoa componen extra-dimensional The u interactions. therefore weak chiral and supers non-chirality all Implying preserves dimensions. also four It in case. dimensional five the in was opcictosteei ute ymtyecagn th exchanging symmetry a further is there compactifications with irrsymmetry mirror eet h hf ymtyfratsmti tensors, antisymetric for symmetry shift the reflects stegnrlsto of generalisation the is hc sjs aietto fteivrac ne deform under invariance the of manifestation a just is which aiod hnigteK¨ahler modulus the Changing manifold. SL h eodsmls aei h aewhere case the is case simplest second The Compactifications Toroidal 3.1 o an than dimensions more flat all). be almost at will (if four, there them with that universe idea our the of with geometry live to have will we 30 h standard the hmprmtieapaedfie ytecstspace coset the by defined plane a parametrise them ftetorus. the of mle h standard the implies → (2 ecnclettefu needn opnnsi em ftw of terms in components independent four the collect can We e sdsustesmls aeo w xr iesos The dimensions. extra two of case simplest the discuss us Let hr here Where , T ( R aT D ) − e.g. /O 4 + √ ( = (2) b G o aktotno h components the tensor two rank a for ) ope structure complex / .Teeaetpclfilsta loapa nmr eea co general more in appear also that fields typical are These ). S T ( ⊗ 1 cT G ) oeo hs eut a egnrlsd o a For generalised. be can results these of Some . SL scle the called is D SL = + − O (2 4 SL (2 d G ( the hc nldste‘ag’t sal iedaiy( duality size ‘small’ to ‘large’ the includes which ) , ,d, d, , 11 R R G (2 Z ) G mn naineo h pcrmadpriinfnto associa function partition and spectrum the of invariance ) → /O , D 22 R Z − emti invariance geometric ) ) 1 2 = (2) U − /O /R = iesoa ou.Ti savldslto steone-dime the as solution valid a is This torus. dimensional 4 K¨ahler structure fatodmninltrs hnigtevleof value the Changing torus. two-dimensional a of → G ( d ult o iceadi called is and circle a for duality 12 2   HPE .ABIFOEVE FCOMPACTIFICATIONS OF OVERVIEW BRIEF A 3. CHAPTER ) aU cU 2 O . stedtriato h w-iesoa erc h field The metric. two-dimensional the of determinant the is G G (2 + + 11 12 , 2 d b U T , R G G T M ) /O 12 22 ≡ ≡ hne h ieo h ou snetevlm sdeter- is volume the (since torus the of size the changes ,b ,d c, b, a, 4 (2)   s4dmninlMnosisaeand space Minkowski 4-dimensional is ouu ic h w-iesoa ou saK¨ahler a is torus two-dimensional the since modulus B G G B 2 12 11 Since . + mn B + ∈ i B √ → ilb asessaa ed nfu dimensions. four in fields scalar massless be will SL Z mn i G raitcsetu odsrb u ol with world our describe to spectrum nrealistic so h metric the of ts G √ B mere n hrfr a xeddSUSY extended has therefore and ymmetries (2 22 G U , + e = R ecalled re toso h torus the of ations ad ml xr iesos sol n of one only is dimensions, extra small d ileeg.Smlryfrantisymmetric for Similarly energy. tial stecmlxsrcueo h ou it torus the of structure complex the is U ) ,k k, /O needn opnnso h metric the of components independent − e rbbymn,sltosadthe and solutions many, probably ne,   T and ope clrfilsa follows: as fields scalar complex o 2.Tefull The (2). T bc ∈ duality − d 0 (3.6) 1 = B Z ou for torus T ouifields moduli utemr nsrn theory string in Furthermore . G fields a B 0 mn The . =   d pcictos ahof Each mpactifications. with U 0 = d c = T ouispace moduli ↔ U 2 b , 0 = ,n m, ic hymeasure they since nsrn theory string In . D hne h shape the changes e othe to ted T = − a , M hc scalled is which 1 = − ,temoduli the 4, = soa circle nsional D c − , ) This 1). = d · · · 4 case 1 = = sthen is T D , T (3.5) U field D − − is 4 4 Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. hsptnilhsamnmmat minimum a has potential This h rttr oigfo the from coming term first The non-t a has that field only the case toroidal the in that Notice Compactifications Freund-Rubin 3.2 This stabilised. moduli all have dimensions. an which experiments equations by field out the ruled experiment of are strong fields very are moduli there with which dimensions for forces int new long-range of mediate type will Furthermore particles dimensions. massless four these than imply s know more very observed also only not We that have know we we mass. since Planck experiments the with the as incompatible of such size the quantities knowing physical before seen the do have we we that as means But arbitrarily changed dimensions. be can they that fact The COMPACTIFICATIONS FREUND-RUBIN 3.2. niymti esr a ennvnsigi eea n s and general in non-vanishing be In can dimensions. tensors di four antisymmetric extra in in invariance fields Lorentz bosonic breaking other without many usually are there know we are h awl field Maxwell The atmxmlsmer nfu iesosw a rt h bac the write can we dimensions four in symmetry maximal want rat-eSitter), anti-de or f a ob unie rmteDrcqatsto condition: quantisation t Dirac actually the is from sphere quantised a be on to a has field space Maxwell compact the the of with value solutions nontrivial give equations field the in isenHlettr nfu iesos ie ieapote a rise gives dimensions) transfo Weyl four a in doing term (and Einstein-Hilbert equations field the in this Plugging nabtaycntn and constant arbitrary an ouifilsgv iet euiu ahmtc.Hwvrth However mathematics. beautiful to rise give fields Moduli e scnie h ipetsc ae trigwt i di six with Starting case. a such simplest the consider us Let Here F mn ,n m, , g µν stemti famxmlysmercfu-iesoa spa four-dimensional symmetric maximally a of metric the is 4 = , F g ic ozr ausfor values nonzero since 5 mn MN h G mn h erco aial ymti opc w dimensional two compact symmetric maximally a of metric the a ennvnsigbtteol opnnsta a ediffe be can that components only the but non-vanishing be may i ǫ = mn F S   R h eiCvt esri w iesos lgigteeex these Plugging dimensions. two in tensor Levi-Civita the MN = ∼ g − µν 0 F N Z MN Z n hrfr xstesz fteetadmnin!Tevalu The dimensions! extra the of size the fixes therefore and S d g V 2 mn 6 0 emo h cinadtescn emfo h curvature. the from term second the and action the of term x F ( p R F =   µν ) | G ∼ ,N N, ,ν µ, , , |  N R (6) 6 2 R h 0 = − F + MN rcin htwudgv iet fhforce- fifth a to rise give would that eractions R ta o h radius the for ntial , 1 F 1 4 ∈ tigte ozr svr arbitrary. very is zero to them etting iilbcgon stemetric the is background rivial atclrtesaasadfil tegh of strengths field and scalars the particular , i stemjrcalnefrtere fextra of theories for challenge major the is MN mto ftemti ohv h standard the have to metric the of rmation esm samgei oooeflxthat flux monopole magnetic a as same he 2 Z lcntans hrfr hoiso extra of theories Therefore constraints. al w-iesoa peeo radius of sphere two-dimensional alvlmsaealwdt xli why explain to allowed are volumes mall , esosta a aenntiilvalues trivial non take can that mensions hrfr ene olo o solutions for look to need we therefore d o nwtesz n hp fteextra the of shape and size the know not ol ra oet naine fwe If invariance. Lorentz break would 3 xr iesosi rca odetermine to crucial is dimensions extra = esoa rvt-awl theory: gravity-Maxwell mensional aigsaa ed ihfltpotentials flat with fields scalar having F yaepolmtcfrsvrlreasons. several for problematic are ey gon ed as: fields kground MN htms auso h oueare volume the of values most that f    0 0 0 eie Mnosi eSitter de (Minkowski, cetimes ǫ mn R :   pc (sphere), space etfo zero from rent h G MN pressions R i The . But . (3.9) (3.7) (3.8) 31 e Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. stribttedfiigpoet sta opreserve to that is property defining the but torii as with t ntndmnin hthsarn-v edstrength field rank-five a has that dimensions ten in ity ntentr ftecmatsxdmninlmnfls The manifolds. six-dimensional compact the T of spacetime. nature four-dimensional the flat on a having supersymme also of and properties the problem from benefit still and preserve theories that compactifications for search can srltdt h rgnloeb a by one group original corr the the to on related trajectory is closed a around vector a transporting trigfo (chiral) from Starting Compactifications Calabi-Yau 3.3 four-d a to equivalent be theory. to claimed is compactification this aial ymti pc,sc saseven-sphere a as such space, symmetric maximally as atrsto fteeee-iesoa pctm noei into spacetime eleven-dimensional the of factorisation utbefrteKlz-li elsto fguesymmetr gauge of realisation manif Kaluza-Klein the the particular for in since suitable known not are metrics explicit strength in.I atclr trigfrom starting particular, In tions. trce uhatninoe h years. the over attention chir much is attracted spectrum the solution compactifications Their toroidal 1984. to in contrary Sezgin compa and for Salam case by the found dimensions is This spacetime. Minkowski dimensional atrsto ftesxdmninlsaebtenafu di four a between case space this dimensional six the of factorisation lostepsiiiyo iiu ihvnsigvcu en vacuum vanishing with minimum a of constan possibility cosmological and positive the Freund a allows as by such introduced action, the was to and terms spaces) broke dimensional space two six-dimensional full and of symmetries the the gauge be in could breaking symmetry spontaneous the with analogy lota h atta h niymti edsrnt tens strength field antisymmetric the that to fact therefore the and solution that different also a val to the rise but gives quantised value are each fluxes the usual As dimension. extra Minkows not is space dimensional four the Therefore energy. fteptnila h iiu snegative is minimum the at potential the of 32 AdS h nweg fClb-a aiod slmtdsnei is it since limited is manifolds Calabi-Yau of knowledge The eeaiain fteFen-ui nazt ihrdimen higher to ansatz Freund-Rubin the of Generalisations oieta ue fteeetoantcfil rvddtek the provided field electromagnetic the of fluxes that Notice SU SO 5 3 oooygop ogl paigtehlnm group holonomy the speaking Roughly group. holonomy (3) × AdS F 6.Bigholonomy Being (6). N Q MNP S 5 4 hc a entesatn on ftecelebrated the of point starting the been has which × suigmxmlsmercsae hsalw for allows this spaces symmetric maximal assuming , S 2 hstp fcmatfiaini ald‘pnaeu compa ‘spontaneous called is compactification of type This . N 1, = SU D 0sprrvt tp )mtvtdb tigcompactificat string by motivated I) (type supergravity 10 = 3 enstemnfl ob aaiYumanifold. Calabi-Yau a be to manifold the defines (3) HPE .ABIFOEVE FCOMPACTIFICATIONS OF OVERVIEW BRIEF A 3. CHAPTER G D rnfrain nsxdimensions six In transformation. 1sprrvt hthsatrefr edwt akfu fi rank-four with field three-form a has that supergravity 11 = h V N − ∼ i uesmer in supersymmetry 1 = 1 S N /N 7 nte eeaiaino o I supergrav- IIB for os generalisation Another . uesmer hyhv ob manifolds be to have they supersymmetry 1 = ther 4 li ordmnin.Frteeraosi has it reasons these For dimensions. four in al F dsetatrst h clrptniland potential scalar the to terms extra adds t hc en eaievleo h vacuum the of value negative a means which oteidpnetsmere ftefour the of symmetries independent the to n e.Teoii fguesmere nfour in symmetries gauge of origin The ies. ieetfu-iesoa nvre Notice universe. four-dimensional different a M l a oioere.Ti ae hmnot them makes This isometries. no has old mninlnngaiainlcnomlfield conformal non-gravitational imensional sodn aiodadtersligvector resulting the and manifold esponding rctere,a o drsigtehierarchy the addressing for as theories, tric esoa n w iesoa pc,in space, dimensional two a and mensional ibtat-eSitter. anti-de but ki or eo h integer the of ue 1 AdS ··· tfiaino uegaiyter nsix in theory supergravity a of ctification re i hscs h ymtybreaking symmetry the case this (in ories eerqieet u togconstraints strong put requirements hese lopreserves also ry htis that ergy, aet eRciflt( flat Ricci be to have y nw hyamtRcifltmtisbut metrics flat Ricci admit they known M F yigein osaiietesz fthe of size the stabilise to ingredient ey in rvd neetn compactifica- interesting provide sions MN 5 7 ui nte18’.Adn extra Adding 1980’s. the in Rubin htgvsrs obcgonssuch backgrounds to rise gives that d/FT AdS/CF × G a w nie loe natural a allowed indices two has S 4 G stegopdfie yparallel by defined group the is F D or µνρσ sasbru fterotation the of subgroup a is M nodrt aechiral have to order in 4 = N AdS N 4 orsodnei which in correspondence uesmer and supersymmetry 1 = ∝ × tfiain omk the make to ctification’ sntflydetermined. fully not is 4 S ǫ × µνρσ 2 X with 7 R ie iet a to rise gives where mn M 4 )just 0) = h four the X oswe ions 7 sa is eld Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. e eeatpoete a emnind h two-dimensi The mentioned. two-torus be the can properties relevant few a ehiuso leri topology. manifo algebraic these of about techniques accumulated been has knowledge existi of already amount symmetries gauge from comes then dimensions COMPACTIFICATIONS CALABI-YAU 3.3. yls(ult ahohr.BigsxdmninlCalabi- 2 six-dimensional dimension Being of cycles other). each to (dual cycles r o o-rva r5cce) h ieo h he cycle three the of size The cycles). 5 or 1 non-trivial not are a h yia ausaeo order of are values typical aiod sntee nw ob nt.Frec aaiYut Calabi-Yau each For finite. mani like be these on to that known even equations not field is the manifolds of solutions of number the 1 ahClb-a aiodetmtdt eo re 10 order of be to estimated manifold Calabi-Yau each novdisebtfruaeyfrti aeeprmna s experimental case this problem for hierarchy fortunately the but issue probably unsolved that p argued this be neglect may to it justified However is it ju then well question not environmental is problem constant any cosmological problem, the constant neglecting cosmological For the problem. solves hierarchy what the know to not solution the is energies low at ruet lot1 er eoetedsoeyo h accele the of discovery the before of years value 10 observed almost current arguments the predicted Weinberg explanati that anthrop proper sense the for named searching physicists (mis) theoretical usually for is b what to happens within constant) universe, cosmological own ’so the major unusual a of was value an it the is thought is we This (what what that one. va indicates observed the only It the of value physics. of right order the the with of vacuum qua energy a all be after will obtained there is computed, constant cosmologic cosmological the the of of values value different cosmologic the with address solutions to of way density vacu concrete the a of as value proposed different been a has with come will solutions these of , 11 1 = · · · ti eodtesoeo hs oe odsusfrhrtede the further discuss to notes these of scope the beyond is It o ahtreidxatsmerctno edstrength field tensor antisymmetric three-index each for o h ups fti orei a ei ntesneta t that sense the in merit a has it course this of purpose the For r h og ubr ftemnfl.Te eemn h Eul the determine They manifold. the of numbers Hodge the are h , · · · 12 hsi h ansuc fwa skona h tiglandsca string the as known is what of source main the is This . h , 12 h ieo h -o -ylsaethe are 4-cycles or 2- the of size The . T 2 ehv icse.W nwthat know We discussed. have we , 3 , .Te2ad4cce r ult ahohradtotpso 3- of types two and other each to dual are 4-cycles 2-and The 4. ≃ 10 3 − 10 4 o h nw aaiYumnfls hsgvsu nie of idea an us gives This manifolds. Calabi-Yau known the for Z γ i H 3 K¨ahler moduli = T 500 2 n fntr.I a oesinicmrti the in merit scientific some has It nature. of ons N a w ye ftplgclynntiilone- non-trivial topologically of types two has i ace ntena uuemygv sa idea. an us give may future near the in earches rmr.We l ouiaesaiie each stabilised are moduli all When more. or tfid ftecsooia osati nyan only is constant cosmological the If stified. lcntn shg nuhs htwhatever that so enough high is constant al olmi drsigteheacyproblem. hierarchy the addressing in roblem a aiodhv oooial non-trivial topologically have manifold Yau tmcretost h aumeeg are energy vacuum the to corrections ntum hc h anciiimi htsnew do we since that is criticism main the which aino h universe. the of ration gi h xr iesoa hoy great A theory. dimensional extra the in ng meeg o omlgclcntn) This constant). cosmological (or energy um r the are s H nya niomna atrltdt our to related fact environmental an only e h omlgclcntn yteekn of kind these by constant cosmological the rpslt drs h irrh problem hierarchy the address to proposal usint easee yfis principles first by answered be to question lcntn rbe.Teie sta the that is idea The problem. constant al nlvrino aaiYumnfl is manifold Calabi-Yau a of version onal od rvd:tenme fCalabi-Yau of number the provide: folds mnp cpicpe hsmyb disappointing be may This principle. ic eeaemn ue htcnb turned be can that fluxes many are here d otyo h oooia ie using side, topological the on mostly lds ol lob niomna.Ti san is This environmental. be also could uin ooeo h anpolm of problems main the of one to lution’ al fClb-a aiod.However manifolds. Calabi-Yau of tails umeeg ogv h oa vacuum total the give to energy cuum emi oiainfrsupersymmetry for motivation main he T n ahnntiil3-cycle non-trivial each and i rnumber er ope structure complex with ewt ubro ouin for solutions of number a with pe i 1 = χ , · · · 2( = h , h 11 12 moduli Here . − yls(there cycles h 11 .Their ). U h γ 12 a i (3.10) , with and i 33 = Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. oeaon ffietnn a enee n hrfr h whol the therefore and needed be may tuning question. fine of amount Some ot and dimensions extra (supersymmetry, problem hierarchy hois.I ste opligt td h hsclimpli physical the study to compelling dimensions. then ingre is important also It are They theories). cases. unwarped and warped the hyhv xaddorudrtnigo hscltere w Inde theories physical patient. of study. remain understanding to our need expanded have we re they be future, may near they the theory but in study, string available further of deserve ingredients such as basic and both gravity) are they that Notice o uesmer ecnsyta ti eyeeataduniq and elegant very a is it that say can we instance. they for supersymmetry For but problem hierarchy properties, the formal as beautiful such of questions set standa a of super have symmetries both both spacetime that They the seen extend have to We way lectures. natural these the of end the is This Remarks Final 3.4 34 • • ohsprymtyadetadmnin a esbett be to subject be may dimensions extra and supersymmetry Both oee h ako vdneo e hsc rmLCarayp already LHC from physics new of evidence of lack the However • xr iesosaei oesnecmeigpooast ad to proposals competing sense some in are dimensions Extra ti oeflto oudrtn Fs seilynon-per especially QFTs, understand to tool powerful a is It tmyb nesniligein ffnaetlter Mth (M theory fundamental of ingredient essential an be may It tmyb elzda o nris h nryo UYbreakin SUSY of energy matter) the dark energies, unification, (hierarchy, low reach at realized be may It AdS/CFT). HPE .ABIFOEVE FCOMPACTIFICATIONS OF OVERVIEW BRIEF A 3. CHAPTER e lentvs ntninwt experiments. with tension in alternatives) her ain fsprymti hoisi extra in theories supersymmetric of cations ihi odagmn ocniu their continue to argument good a is hich dfil theories. field rd ta drse h rbe fquantum of problem the addresses (that loadesipratusle physical unsolved important address also edn fayeprmna verification experimental any of pendent eatol thge nrista those than energies higher at only levant inso udmna hois(string/M theories fundamental of dients ymtyadetadmnin provide dimensions extra and symmetry eetnino pctm symmetry: spacetime of extension ue ubtvl Sdaiy Seiberg-Witten, (S-duality, turbatively rs h irrh rbe ohfrom both problem hierarchy the dress oy strings). eory, t l h rpsl oadesthe address to proposals the all uts ute etdso nexperiments. in soon tested further auans su a eunder be may issue naturalness e f1TVi ihnexperimental within is TeV 1 of g Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission. Bibliography 1 .Qeeo .KipnofadO Schlotterer, O. and Krippendorf S. Quevedo, F. [1] 2 .Weinberg, S. [2] 4 .Akn-ae,S ioolsadG .Dvali, R. G. and Polchinski, Dimopoulos S. Arkani-Hamed, J. N. [4] see: dualities of discussion recent a For [3] 7 .Weinberg, S. [7] 8 .Polchinski, J. [8] 5 .RnaladR Sundrum, R. and Randall L. [5] 6 .C West, C. P. [6] 9 .A Strathdee, A. J. [9] xr Dimensions, Extra ri:4250 [hep-th]. arXiv:1412.5704 ftere ihsbilmtrdmnin n e cl qua scale TeV and dimensions submillimeter with theories of 19)060 [arXiv:hep-ph/9807344]. 086004 (1999) Lett. 83 19)37 [arXiv:hep-ph/9905221]. 3370 (1999) uegaiy rn yaisadsrn duality, string and dynamics brane Supergravity, h unu hoyo ed,Vlm Foundations, I Volume fields, of theory quantum The h unu hoyo ed,Vlm I Foundations, III Volume fields, of theory quantum The uesrn hoyadbyn,Vlm II, Volume beyond, and Theory Superstring xeddPoincar´e Supersymmetry, Extended ri:0119 [hep-th]. arXiv:1011.1491 ag asheacyfo ml xr dimension, extra small a from hierarchy mass large A 35 abig etrso uesmer and Supersymmetry on Lectures Cambridge hnmnlg,atohsc n cosmology and astrophysics Phenomenology, n.J o.Py.A Phys. Mod. J. Int. U 2005. CUP ulte fFed n Strings, and Fields of Dualities arXiv:hep-th/9811101. tmgravity, ntum U (1995). CUP U (1998). CUP 2 hs e.D Rev. Phys. 18)273. (1987) hs Rev. Phys. 59