JHEP12(2002)052 t a to at .pdf are y 2002 2002 b w 2, effects taking ho 18, ariance en t space v constan er there er directions By in a , . emb /jhep122002052 e brok ´ c SISSA/ISAS emb 52 c is compact 20 discuss De differen De or acua e 200 o f compact V oincar 12 w w in t ep P compact. jh directions the Generically s/ is non-compact er the in that, ccepted: Received: pap erstring A e/ discuss ersymmetry the iv e space Publishing ch Sup w After compact where sup ar On planes theories. t/ ch erse .i the n. ssa when space, dels). tifold Physics On cer I transv motion. interactions mo I examples D-, of orien e of the yp http://jhep.si t compact some and a rane in Strings, cancelled. when in consequences Institute el equations not D-b analyse (cosmological lev ra by planes the D-branes e Frederic.Zamora@ ysical harges are , c e W h Heterotic ph disk Zamo .c tifold and oles solv en. and the among direction to some electric orien Published at tadp brok Switzerland ws of rederic oles time is F el. oles 23, allo CERN set erstrings lev global teraction a analyse the and/or and y eve ` in e tadp of that Sup tadp 2003 W vit ariance an ´ Gen v del when Raul.Rabadan@cern along Division, in ears finite ct: mo e ergra ´ a en ords: D-branes ory y Rabad app sup to earing of ´ ul The CH-1211 E-mail: brok SISSA/ISAS oincar c Abstra set global Ra the Dilaton spaces obtain Keyw P is ° the app term JHEP12(2002)052 1 8 8 5 1 3 7 14 16 12 11 10 10 18 19 11 20 16 22 21 for are the tifold of tials jects solutions orien y ob uncacelled. oten ]. vit p [4 and e in these Ramond-Ramond redefinition ergra as remain the sup oles effectiv D- can When ], of tadp [5 that ]. Some ]. t oles set 2 theories en a disk , ]–[9 y [1 etc. tadp consequences: b [7 the brok accoun in en is oid v to a erstring ysical in brok NS-NS spaces e ph found and sup – considered e ariance amplitudes, of the tak 1 b v systems in – in een e ´ b can trary string series oles compact e should ersymmetry v ersymmetry a metrics circle oles con D-brane in ha oincar a sup sup spaces tadp electrostatics one ed P in has higher the in tadp in in arp with oles On that case erators breaking theories y ]. oles a NS-NS global dels compact harges of op [3 tadp c (bi)w w , del y metric in of e mo a ergences mo tadp for string consequences implications equation h these ed equation y div these I ativ of teractions I of ossibilit some suc ], arp duction in p of e field compact ersymmetric [6 cancelled example: in ysical ysical tro a yp the deriv tensor t ears, duction duction effect Ph Propagator Sup Dilaton Ph (Bi)w RR T-dualit Sugimoto In An are y is ts duli on t in presence terpretation tro tro otal mo ten presence oles 5.3 5.2 4.1 3.1 3.4 3.2 3.3 4.3 4.2 5.1 2.1 Ricci T Conclusions Examples D-brane In The In There kground In the recen 1 . . cated ...... B A 6 4 5 3 2 planes 1. In Con 1 lo tadp The in some bac JHEP12(2002)052 t e e ´ a is w of on w On are our the the the the one this ho sum case to that with with wing some fields space harge of ected, global space. of of In ergen c for NS-NS jellium system oincar ariance ed ose in uniform the 3 v allo rom P w a ordinates div exp F in ones), the to condensate O-planes co inclusion equation e ´ of these sho system t. viour shifting supp these As energy rigid equations global distance equations to zero, a e compact y a of solution small the eha w breaks b del This a y the to electrostatic b the kground. they the ortan b If oincar with expanded no consequences. and/or mo in dilaton short from P ortional compact non-compact es w space. is bac These that er, regularization with with ]. imp similar y go at in No but deal the usual to Ramond-Ramond the [12 is tial, presence prop When pap non-compact ysical this is, arises to deal cancelled term there is in replaced simplified the ph the term and e ecome computing w the oten D-branes equation fields. motion the to case. directions. b This condensate. first t b suitable p compact e y are ho that the b of to w a on same instance, manifold the tegrating anish, that tak the can ho e ed in the v or propagator in ariance ortan for these term manifold. e to brane w hanges oisson dilaton solid v neutralise due similarly lik ysics c In The w in P propagator dilaton a in wn By not D-brane the see, sho imp ph for they the than a limit, e it, ´ the kground. in ] endence es o sho the that space this images (and w equations compact [5 on term the the w jellium electrostatic the bac do t them. – to as a dep ho of of ions compact with see to of electrons oincar w 2 the e y erse define the similar. the ximation in ducing example, P manifold. sources – teractions This the it the factor is ving the to and with e to to will in ts similar ariance tro arra pass shap harges’ non-linear) v ha ]. some of e er, the appro in from c transv alence e Susskind ds oin that in 11 v solutions W the dic y w p , viour breaks e ´ del equations b pap global The There, disks) e neutralising ciated compact the h [10 eing and i.e.: olume, mo solution erio a e eha find solution b olume if v anishing. (highly illustrativ p metho b re-visit b decompactification a while v v y some a the asso oincar deriv b to whic hler to ysics. is ’dilatonic P generated, analogy space. second at form, the that corrections will the Einstein jellium not ph ws uniform es find can is other ysical e directions. Fisc the y harge ed eliminated a term infinite is c that e b tend the get the the guiding to the ph explanation, allo comes of D-branes e e tak w e they in e cated ergences concerns 2 a to arp an b ers, In e w with e term lik lo The solutions. are the complicated y matter preserv compact w w div the b t sum e harge term order As h pap case bi-w ositiv can c space ole er detailed divided p en. effect a ears compactification see, electrostatic the in kground of the duction the solv more space, distances, terms. of arian whic our when with kground. harges in pap v tro tadp bac non-compact app c has coincide the to will total the brok more propagator in created jellium in and or pair condense and a a e other e ole e bac (in hand, harges. to harges on, e field ´ for b w a large e this ergences c an compact c in lik the tial rom the lik as oles or at In In The F result ect tadp div one compact metric F See, kground del ariance the electric the 3 2 v ould oincar oten anishes harges, tadp the of handles the acts additional suitable global c when the should mo in propagator resp P of system. the case, bac But redefined v but, turned considerably w p the compact dilaton the in JHEP12(2002)052 . ] e ]. h p in er at v on all [1 8 ws for the v the the o ∆ , The elop zero o that (2.1) of order name [7 + ab string whic allo del jellium c out dilaton dev features in h compact compact i.e.: t although distance, extended ∆ sum mo the calised the the define a the the term. consistency lo = oin ). will teraction laplacian higher the the of whic are p of see, the consider condition of A main e in tensions ∆ a ole e and in y w to under the times b definition, is w propagating these picture, fact will the t the with this jects sum previously a e sources Sugimoto tadp of endix y correctly an of the a ob w of w disk harge er this presence oses, call Then c factors tegration D-brane larger the in b app e endence: the to deca string As with is defined in One w t D-branes ed in . in , equations (the um to The A for will prop dep should the en n analysed the ears ] mix. t harged the one e e arp . the analogous: global c [7 op absence y an w osed W due y a een app to a not One results shortly the t b than space space. is w oundary equation the the . the 0. from e Einstein b infinite these ) do v x. i explained due of irrelev see = in term y ha ergen with an een Mourad illustrativ decomp ysical i the faster singularit some − understand there y q of w (as case oisson e oles ph will div parallel y comes e e et if b P in for ( transformations. vit compact ws P parado and e b δ – without is i yp y is ecomes also from w if q t en the obtain that 3 tadp and gro them b the the ers can windings t ev – ergra dimensions of in As of is X in φ coinciding can case can terpretation Dudas space-lik er only Olb e spaces analogy er sup = in v e term condition endence space a o of T-dualit jellium diagram w ev φ tial one in most and Our solv obtained y the differen w compact of A the dep electrostatics parallel ∆ a to This a equation, ering in Ho oten sum : een binations v the p in dilaton to has taking b finite compact solutions and propagator co M in of dimensions. e and y analogy the cylinder general, solutions space. need regularised. com in v b solutions captures the the zero. ening electrostatic e e the oisson has manifold. presence oles the In ha erse it space P w ole split to ]. to term in extra from and the e naiv that the condition. linear happ b systems. others del, t [15 the tadp manifold tum meaning amplitude tadp add is transv is compact correctly to zero mo The of compact can equation the is jellium images the the electrostatic equation sphere). cosmological not the some the to coming taking Rules conditions presen solutions electrostatics it what on the consider momen y e in the the es is this do D-brane on b the the the w space anishing ts , to space. on find go of effect Sum basic del: solutions of non-compact cancellation the solutions also when oin making simplified to Similar mo p ers rom in than a ole construct us b Non-v F Since The deduce metric branes case revisit whole y ws understand The de, ergence viously find these some Brane is e to o um can of conditions field. and W of coupling term compact div n similarly propagator term allo mo T equation to the images 2. the space. the Let in tadp Ob the a some it JHEP12(2002)052 e e h ), in es no the the W that then (2.6) (2.5) (2.4) (2.7) (2.3) (2.2) (2.8) (2.2 term, whic es in Stok has system created of effectiv can a giv existence e the tial propagator, the term yp for space, the equation kground. propagator equation that ) equation solution and in the oten ysics. bac the p to the equation (2.1 , no ph in 5 jellium reflects ) jellium-t is from oisson term harges the compact eq. c ) P a a of state (2.3 oisson equation configuration: the P (2.3 the as the . there section in duce ) that without the i . neutralising in considering solid in y een . obtained consider The . ¶ ears t tro y of i w oisson harges. e that ) . − φ q b c i in solution to b P p i M et see equation finds y y q , φ . app V ( b ( term from equation M P to j δ φ , V + the i and i q the equation P can M the it a 1 the i q c will ) − q one equation constan V q case, energy j φ of ) + e has The i x i a solution. y , − w name X φ X = i obtained the X ) ν e the ariance. x using − as using e oisson φ – y v − = one as ( v teraction ( theory = e, space, b sources y P φ∂ the p y 4 induces δ G in consider ( ha = µ in p i By b φ δ re-defined q – ositions ∂ e ´ φ i p 2 = p to q the p the to can part, φ e ole prob ) µν ∆ obtain manifold. ij the ) ∆ b t wing g y is ∇ X a ( + non-compact X µ not een the M satisfies, terpreted reduced c function, G space. = M oincar compact tadp V and do 2 1 w (2.6 c orro φ = is Z in P c on b one: A harges et obtained ∆ the c enden e duce e c Z = b ∆ φ the should the b c oles in delta compact = E tro = dep end solv could een ∆ the on computing in S b φ as term, dified compact can to the tadp , satisfies equation ∆ dep of to propagator one een usual It space, mo of h dimensional y try the had w teraction kground the a the parallel will et w in to the and the jellium b a This whic the equation what bac k just ws a to olume equation the e v In but er, bac and w satisfy allo compact the this energy w, solutions. the ev h branes If space. term the another w expression ole. the as turn part kno teraction is that is that The find Ho t that addition e whic in this in us dimensions M , w other solution, tadp to in V result. tegrating M second there call So, Let The As no finds compact enden in harges ws the written /V the signals c 1 e y same parallel allo of where It dep b By will But the b solution. where action: − one of The has theorem. JHEP12(2002)052 ), of to On y the the e osite that (2.9) viour (2.5 L (2.10) (2.11) (2.12) t created distance opp circle eha from limit ected. o escap a b w 2 the propagator tial t q exp in constan on can of 2 equation as the oten y a p of system: tial the (flux teraction and . the plus in t | endence the 1 oten endence sum is y p . . nL of transformation ) dep 1 φ ts dep y solving + the q ( presen zero. 2 M decompactification y oin G y what correct y Z b b is uous p teraction i no M − energy linear q in the Z the tin M en y is | the 2 V quadratic e can, P electrostatic ) Z the . obtains: i gets giv con a images ∈ the q ) at harges + ole M X n tak i (y) c a es is , V ) the duces one 2 x φ i 2 P e one one q , the y ( q is finds w giv y ( ) tadp ( + ted tial of φ + duces i | if – these G repro q j i d one and (2.9 5 q anish, q the ) nL of there v oten i 1 j – satisfied q i p X to repro that x + and X , represen in 1 tegrated − i not metho are y y sum system x that e ) = in ( = v where c − L do G 2 φ the ha i circle the y the φ harges, | Notice q system (2.10 e y c that a if Z ∇ cancelled b of w oles 2 ∈ ij o cases, X n X q M the in w suggests 1 electrostatic tial conditions t Z not oles. − q side find tadp of it of = = ole = the are energy to oten ) satisfied E tadp So p E the y ole. hand finds ( tadp harges the oles substituting c V are tries energy harges, hand, system left tadp As c one 1 non-compact global tadp a where anish. that Just oles q one the v the the the , the the the see other y to circle. adp ely case b On electrostatic a T harges, that to to c on example: side . terms the 1: harges. on consider the t naiv an L c y). ole easy An On In If, ortional us ends righ (y) these means the is φ ultiplied y harges It compact infinit 2.1 Let tadp Figure c dep It m length prop of the obtain b JHEP12(2002)052 , e y y of 0 b b are the the φ See non also y arra + ed , (2.13) (2.16) (2.14) (2.17) (2.15) . viour. b It can harged 2 0 for c y that dic. φ there 2 cancelled eha distances, q of b y absence + a energy + erio 2 y e describ infinite are w . y viour p 1 that 2 . 0 y the is jumps. yp a large q e obtained 1 φ t . is equation ¶ arra the q b eha e t | oles h + in e b + L A + 12 tial of v 2 acuum to 2 nL dic o v suc 2 y ativ y ) + y tadp ) − a 2 ) ab 2 oten in q 2 (also C erio L q y e ected case. 2 L p q | 2 L p µ q deriv 2 + laplacian the dified N + 2 difference the + giv propagator 1 duction L ∈ . L 1 + exp a q ) 1 X n ): q ( 2 1 mo q L ( to tro q q ( first + The and − solving is the | ≤ only in − (2.9 endence + where − − infinite | y equation: from ¶ 1 ) ¶ y ) the b ¶ nL , q | 2 ) tial 2 . ) the ( dep but 2 an y substracted, 2 q 1 dimensional 1 + L q electrostatic case the q y + using duces solution − y b − | + oten − for ected + y y ¶ 1 , p N 1 hand, harge ) ( one b the the 1 q 2 cancelled. then q c ∈ δ y ( q ) ving C ( figure X 2 ( n – ( The 2 exp propagator repro 2 finds: L a q In δ so 2 y L 2 + L 6 our + earing not teraction in and . and L y + as quadratic + | = is − 2 other – ysical 1 + L in the b − y ) in 2 y ) one 1 2 | 2 a 1 2 y app y are y ph , y tial y 2 y ( ( seen y 2 = the − origin. 2 q is the propagator q computed G | − q | size, there − c uit the as figure 2 + y e viour oles y | + oten | + On ∆ ( b 2 1 nL the propagator endence p 1 δ 1 tin y y in y = obtained 1 1 y eha + ergence 0. there affect 1 q regularized 1 at q − ) tadp b e q y dep can q lines, the con the duces where y | µ 1 = b ted div = µ ( µ y Z 2 | not tly y c of L in G ∈ ts the y q L y L φ µ X n and c the also 2 = repro do + linear non-compact = source oin = q linear = y ∆ if ) 1 olume p system 1 ) enien compact ) ) q y v term q a y v ( represen y y can tial the ( where c oles ( the ( − is c instead The compactification c dicit φ is the is the φ con φ = div : ts onder : oten : 2 of e large ) i.e.: put G case tadp In erio ds. w the unless 1 y L p 2 b tial p y y t e space , there ≤ ≤ v for to 1 ≤ the quadratic y y The y can oten ha segmen y ( erges finite space, metho the p migh propagator energy in lines. ≤ ≤ e osing no E is ≤ that . that 1 2 w t div 2 obtained: is olic 0 y y ens oles). h imp The The The One • • • images happ Notice parab straigh where whic and Notice Because compact easily there comparable tadp figure images of resulting previous JHEP12(2002)052 to on 12. des, rom tion is case: some F made space. L/ (2.19) (2.20) (2.18) action, mo − tibranes compact atten idea D-branes. = motion an the what y obtain the C of is pa the ourier The of dify F to . L with case, will electrostatic non-compact in mo branes, ) e view ) in compact. of there. W the of e inclusion is they t equations (2.14 lik to parallel (2.19 its compact theories oin from is the p space space. system in ed space, , , electrostatic in a the ]) 2 ) viour our e equation oles , 1) /L the [10 deriv y eha analogous 2 ( tak directions rom − L term b 12 n Therefore, that in erstring in the propagator F compact a tadp π t us e − are 2 a part (2 the the e 2 of lik 1 8 t sup the L let y in extra 6=0 non-compact systems. − – X n ts − accoun in solving conditions | 7 2 and, along 1 ergen y 24 y to case, π L the oin | – effects b redefined. 4 p in distances = these t div express substracted. = oles t − e on ) ) are e space in a necessary y = consequences tak w the ( ariance ) a short − v y if tadp to t reg ( n differen in action obtained ( as A differen consistency G 0 e reg o e ysical ´ t. electrostatic at w X analogy n> G solutions that compact t dified ph the completelly ears e an the circle. mo discarding ergen the oincar are NS-NS a d een effectiv app e and The cated P teraction o b b in div lo also of in on and observ kground ole ) has there the to has bac break should de reduced tadp (2.17 planes equation understo the G(x) equation space. mo e oles term the conditions. the duces v Propagator of curious (regulating the tifold ha that zero distance t tadp 2: e is dilaton repro terpretation w oin jellium the h orien Summarising: It obtains compact propagator p using get In large the tegrate e t consistency dilaton in dimensions to Once the 3. the The the and meaning whic i.e.: Figure one A By w JHEP12(2002)052 a is M are the the the the and in sum (3.1) (3.2) (3.4) (3.3) e × dilaton , +1 tak compact jects osed p 0 solutions, break equations. conditions obtain (D-branes, brane the -brane R ob us equation = p e to and t + w of expanding for + ! ) Let the i 2 the ´ decomp | has ology 2)! φ x,y . of e ( sources +2 Einstein N b on φ + 2 p name differen | ∂ top . part. action 4) 2)! p F dilaton / | -branes field consistency N ( ¢ ersymmetric a side +2 the p 2 h can 3) + g p M | the . the some − D ) p F the − The j p . | ( sup non-compact y metric ) ( (( √ g GG to these dy ). with e y φ φ hand i for ) ( − y − ∂ i the compact dilaton the ( 2) of | t ij y / where dilaton under dy √ √ φ ) B h ) e φ p ³ ) y x, − + x oth consider the 2) − ( the righ e lik ( ) / the considering M b induced ) e x, B x nc ((3 ∂ coupled ( p in + y ( e g e W in manifold − c ij equation b 2 g ϕ the | g − ts i.e.: + . the ositions − that ) ((3 binations p ) if = p cases, ) x e scenarios + p i ( x the i 0, oin ) µν − branes. ( y ν q ϕ p – ˆ y g M ed 9 com dilaton ∂ 6= the factors Z µν case, − | dx i 8 that x, g orld µ the X Ω p ( y ) φ – arp obtained w ( ed y − φ is µ the 3 4 p are − w some dx of e e ∂ Ω( ) − ¡ linear i 4 dimensional b 3 − e 9 y arp y tegrating . at δ for G p + w i = , B x, in q − brane ( ten can that dimension ) usual = y √ dimensions, i . nc µν a in of X ´ endence: g electrostatic and with cated and µν x, Z φ ˆ form: g form: the ( consider η directions of lo ν = Finally ten A 2 − N ∂ dep consequence the 1 κ equation = N require M 4 the in the p can µν ) the in and M φ G x case metric − metric G of applied ( conditions to e 4) the G endix manifold / G has = one µν ysical is includes condition: lik 3) − g the the ) S along − een has . app tensions. metric ph √ p . b metric (( ³ for general case e and frame, e µ its more, ordinates the frame the one v non-compact ∂ à ed 0 compact equation co a less en p is in ha M this ariance are a = consider, consistency consider a v 4 Z planes,. − arp ) Ev N p ansatz is e consistency in x 3 and − us M ( ]. immediate can 7 s e ´ l Einstein that + ideas G tak Einstein B M s e this anishing, [15 g tifold (Bi)w Dilaton Let An Similar W wing us the explained the ∼ ordinates, cated. oincar i q where non-compact lo With metric where Let 3.2 follo non-v P Notice compact These in orien co where In 3.1 as rules JHEP12(2002)052 the the the and (3.8) (3.9) (3.6) (3.5) (3.7) equa- (3.10) (3.11) in in on − − dilaton t Einstein ) ! i equations 2 y | the term the − +2 factor ordinates of p y differen e ( in equation F co | v ed δ i of h q Einstein ha olume √ arp v e i 2 set | w . , X w the 2)Ω endence a ) dilaton / dilaton , +2 ) E 4 p E and compact 2)Ω the to equations. M F / dep . +1 0 − | and ) . V − p in the the ) ) y ) y T (( 6= x . the +1 T y ( − p h ) ( ( to term one φ y ϕ B ) these φ 2)Ω( √ on i osed µ µ i 3) 2) and relates +(( / y ) / ∂ ∂ ∂ φ of ) 4 ole obtain 2)Ω( x − 2)Ω p 4) / )Ω( 3) 7) +1 / ) / fields − p 7 ) p ) 1 is ( 1 3 − − − (( − factor the ((3 p − can tadp − p − ( p p e p previous decomp = ) p / ( ( e (( RR y (( 8 of (( ed ( e e − equation e w )) e − c φ 0. = = b the 3 the e g – x Assuming 2) à ) i ) ( the = arp / q 4 9 y √ x 3 − ) ϕ of w ( ). terpretation 3 manifold B ν This can – p of 4 i y µ − Ω( − M B ∂ in ( i from ∂ p X Z µ p − ). ∂ φ the (( µν ∂ ) = B ) g − = p manifold, = ) p e x T the h ( een − factorisation where M ysical − )) B compact w √ V y equation 3, e coupling analogues ( ph ( et endix 2(3 obtained the and φ M = 2(7 µ b j e the Z ∂ p ∂ the b the the app compact M ij = in for V h w from to dilaton E can are the (see 2)Ω ello relation / ) the b form: ordinates ordinates: due M on V +1 comes co co is p case: equation discussion the (( linear indices a equation and of he ansatz a is discuss term ) √ T is ( x i tegrating ( relation E dilaton ∂ this will ϕ in mixed there similar e w for equation. This non-compact the frame: W The non-compact where electrostatic and A The There By With Belo 2. 4. 1. 3. 4 tions: JHEP12(2002)052 , e do ) re- ole the the the RR can this n b from is y static (3.13) (3.12) (3.14) When to to to to ansatz. − for tadp dilaton. dilaton. and solution function example e y That ordinates ( branes v this e due p ed. o co the comes ). the the The − y In 9 ab gauge, the that: Then, δ to of ordinates ortional solution NS-NS Ω( co a RR shift) us n 0. the preserv q harmonic t the alue prop = illustrativ order. endence v compact images. is n order, tell 7. couple ) X construct is t the with x ´ < ( ) the endence dep the the ) anishes y not when this p er. B x compact v can appropriate if constan ( in µ higher planes. t all Ω( dep g ∂ do a space A eak for t an This the − ole ecause constan at case As one equations: w b that equations to symmetry a I p of in get is e ) e tifold . v . e e x ´ e factor v factor tadp ) o ( p i o form: w ose cated, accoun (up B case yp y tial, ab t ed ( ) ed 2) dx lo ab compact orien means 1 ho ecomes the / · to ) is c D3-branes coupling. the endence. · olution − b to oincar 7 in arp · coupling. arp the oten (3.5 or − P ti are H ev the the tribution 0 of these w space p p i w and dep – ersymmetric y This q can (( 7 an a field is dels dx e the in on con ) i string ³ < 10 from y the sup X taking ( mo of one form p has and RR – where = coupling sign. infinit C y = 1 factor zero the < ´ b equation the D-branes D-brane compact = action T + 3 to the case, C d . in to that ed j function in p space cylinder ∂ same +1 cosmological es T-dual found the el string the rom p for the the us ij endence arp cally F e go C with in RR w the ears lev the b the hh lo metho for equal D3-branes on implications the e tell dep a √ ), is ) ) the lik where app ϕ can disk compact harmonic ecial. motion with e since T + olume has ts since (3.5 φ reducing ) v -branes (3.7 ersymmetric ysical sp of harges ). x also p tak T y the c the ( 2)( oin function this smaller, endence dilaton is ph / b D cancelled ϕ p ) that sup the of p µ to 3 (3.6 them, of ole of can ected, motion k RR ∂ − if dep cancelled V and not = the e , the up equation some to of 2 hec w exp p equation the tadp endence c equation equation are implications at getting in are directly as harmonic among 2)Ω+((3 rom system / olume dep w are the ole the ) olume the function lated are dilaton equation F As can case oles v a p v orking field seen the +1 extract section – – in w p RR n ysical e or y e harges q (( teract The Also F The c b That the in equations One b tadp T has case, That ximation, w − RR Ph Then, in are e • • • • w ³ e i ∂ 3.3 W where not appro 3.4 No JHEP12(2002)052 e e e e y = In v b b an w w the the o will cos- erse erse case with 5 with RR these n e y harge t. ysical ab (3.15) string on a in q b can w ≥ -branes (T-dual n of ph univ p Then p this or ordinates as the the transv P conditions D F RR-c if ends co planes presen the In of case Then particular only no of the equations The oles. dep ensated ]. time This In are cally explained y sign. cancel, [7 are obtain brane tifold binations in The lo y vit tadp as to . b to only comp oles cancel images. torus. the com osite orien no there a brane. has of to T-dualities ergra tly cally larger p expansion t tadp harges the ersymmetric − on opp lo equations. found c are 9 the but ole ] sup 2 e linear . of the solution v manifold enien sup [1 , order on v RR conditions d enden taking ha the tadp there also In theory ecoming y dilaton ) con const to I b b Einstein and e string e RR cancelled theory = indep are cancelled metho consider b (3.5 ) familiar compact is since yp the is x t units. solutions the ( to where ) not g t the can to branes ole the ersymmetric – − consistency tension There y solutions more ole, has are b on (3.13 p taking del result, of sup Sugimoto 11 ) One arian dimensions -brane tadp ) equation y the x v the µν p that ( – erturbation the b mo tadp ]. g oles B in p D set the h ) the examples torus. 7 RR e the ´ [15 (3.13 finite in 2 out − in of p p suc tadp a ( 5 in cancel el − equation dimensional the finding 9 ), − uous − expressed review T-dual compact metric p NS-NS e some x es lev T es where oincar Op-plane. ( 2 e er a e in tin Rules the can P b a w w B h carried giv the that the to the of ativ lo equation solution e of NS-NS con disk of h b eac y of k One Sum can a b duli ts string) the First side with the suc on consider h lac case with equal can terested -deriv I mo en images oin case x at t. where in e p illustration the e giv Brane whic will hand dels obtain the yp b e the the t the needed. an e brane, . case b as that, of -branes to solution mo w cosmological tension presen A to tegrating left as analysis 3, some all p to e decreases. will are v in the D brane the wn will e < er at the y 5 the ha and w v b to function − some p o e trary kno endix p the terms ersymmetric w 2 similar section , or consider -branes cated Con A F Since coupling condition are mological seen app 0. x on expansion harge p Sup consider re-visit disk c lo us • • • • Examples this D will summing 16 solution. 4.1 4. In the harmonic particular, RR directions construct putting will Let consequence are configurations JHEP12(2002)052 e e e ´ es to v en for ere are the o tial but can fact The par- with w (4.1) (4.2) close op giv tifold ab cancel all affects action. images dilaton defined oincar jects. oten the tation. ected isotropic tial, analysed P the e p to there space aub-NUT tation that the ob orien plane to and T erturbativ not exp ere future. the frame): terms oten tifold t, and rom problem w together is p the directions metric, F the order due disk, of from effectiv a are oin e represen tifold dilaton non-p p All orien represen tension In is the lik these en. compact y a The the They e the b infinite a the to orien hange is, Einstein metric the of c in effects brok in ergence tessence just if ersymmetries 5 an along breaking the There tifold. el (in and . due un is the div idea a cured 7 negativ y homogeneous sup quin at (32 ). lev constructed gauginos. − term e anishes). tisymmetric in p symmetric conditions. effects kground. duce left | These orien ole a v b space: form and e the sources The L an I (4.1 the is space the i ole the e tro bac zero to the e ~ ]. m in the kground lik tadp symmetry at the for term [1 one. to wn yp consider + from sign, singularit solution tadp t e ´ these t of i bac in do reflected e y eeps ~ the y and a go del | ected is a compact k one funcion , its hin But tiD9-branes is p the will the erturbativ w compact 2 is the IB cancellation − a with mo similar 9 an e I oincar exp – φ/ tial the Z 3 from of e P ersymmetry ect constan W break ∈ the oles but X op. dimensional ole − ature is i far tifold 12 in e ~ ery non-p space-lik hange yp lo m ah-Hitc there oten t c v sup – Λ i the resp a ects p ten q tadp curv tiy harmonic tadp suffer orien Z e H no That where one A transforming i viour. (so planes oks tisymmetric equations Sugimoto has the X from resp correct dynamical with tibranes the notation. lo y not an 6-plane the e function the + eha from in the that case an ond b scalar O the b 16 zero as 1 re-definition vit NS-NS es effectiv cancelled, h tifold start t ey tial the to y is is to = R-R) b that do b the in fermions term ) to the suc ected solution solution The symplify y ergra not The ecause oten functions. duces ( and These orien see for to b p to ole and ]. extreme exp H el. ected harmonic sup differen tro with this harges The are as [7 e harge c the t). in lev tifold, c consider can string, due w y exp the function tadp the torus osons dimensional a I the This ery e e to del b that the v is is w as e planes w propagator one determined one to . disk (NS-NS the orien a harmonic this 9 ten harges yp mo all viour accoun c from t T e of Mourad close are squared oles the of the solutions. same gauge of 32 has negativ eha to notice t the tifold branes that y string b , an gauge at in and b harges jection function = example e the flat c solution tadp field sum effects NS-NS the negativ and orien with en ortan ed Λ in tak pro (32) a e configuration the to first RR Sugimoto The Notice As Finally closed tak solution R-R massless osite W imp Dudas constructed the ariance 5 USp v e e y that the b non-trivial and spatially in energy a This effects with to the already opp harmonic the where construction b are metric The b string the 4.2 preserv allel JHEP12(2002)052 a y is is of the tial. , as (4.3) (4.4) zero, erges 2 || alue related dilaton to acceler- v solution dx div case oten acts is p 8) is solution solution / singularit 0 the ) the c 2 The deccelerates. ) that 0 ature that the t erse c and The Λ. the In − t decrease bang) erse dilaton alues √ Λ term. µν curv of v / univ that η √ to ) 0 the ((( zero. univ (big c dilaton e the alue 9 strings. of − v / to other scalar 1 kinetic the ]: the 3 starts the ) metric / es 0 [7 um, of c it free the also of The (2 (for go flat − is zero, to = Λ t t demonstrate um form and c tial . 0 dilaton Λ t to maxim Then en cation to √ ole = 16 . ( at lo es 3 viour. 0 oten the its zero / − wing driv solution. + p φ 1 go maxim constan 2 in tadp 2 8 easy − eha . to t is / = e b alue 2 time Λ dt gets the 3 t is follo v the 0 es / – 9 the the t 8) √ the 252 2 / , of It / te go ) in c the 0 3) zero. 2 13 + the t energy ) c φ energy / 72 0 4) 4 – e c system / − t to Dilaton (2 at ) when , when − dilaton 2 0 t has y tial ) ecomes 81 c = φ es is Λ 3: 0 t maximal b e trated the c − 0 some √ t a the − go coupling. 3 ): t = oten is / = y 4 Λ determine ((9( 2 c p parameters to b frame ) and e √ φ R tial solution 0 scalar 2 t deceleration. e y 3( concen c / obtained metric Figure singularit ( 0 − a where string e is 0 that there φ zero (( , 0 the 3 φ figure oten b e c t e the − p t 3 the Ricci dilaton e , of / the um t t 1 y 2 = coupled see constan Einstein ) can − A from of φ 0 suffers ) the t; singularit ecause oin the c e basically 0 t. b p c the the ws of ature − is infinit dilaton maxim eakly limit − t alen the . in parameters w alue t 3 gro are times Λ e ´ to v case um o of φ curv Λ times, 0 √ the t The w ( φ e es √ t constan ery equiv critical 0 ( . energy v at φ long 1 go that − e to figure oincar − y and ) maxim dilaton solution P scalar the at larger normalization 0 in osition = = are in 0 up t The p constan c to φ 2 E or − e ysically a the erse The The The The F The flat dilaton t ds seen the ( ph where Strings as gets not to Close the as is unique singularit univ ating. cosmological and JHEP12(2002)052 are the the (4.5) (4.9) (4.7) (4.8) (4.6) tifold tifold + ternal erform on calised 2 || in p also ˆ x lo direction: d orien orien profile 8) the can / ) een of 2 e one the ) w in where 0 w c et of , expression − b k t T-dualities dilaton h x Λ t √ the space eac the , ((( . e 2 of 9-direction 4 ) 9 / has 0 / further 2 , c relation . 1 and ) singularities ) 2 k − 2 0 solution. t ternal the top ) 0 c compactifying ordinates 0 Λ c − c dx in the t no frame: ) √ co − on ( Λ t − t the metric ( ature is t √ Λ 4) ( / the has Λ √ − on ( (3 erforming e √ e 8) curv 2 string in 9 ( p 3 / b / Consider spatial / 4 Einstein (5 2 + ) + original − − 0 general 2 ) e c 2 Dudas-Mourad the ceed calised 0 – 9 dt solution the c / w scalar distribution dt − lo 7 our ) 8) 2 9 ) t on the t / 14 pro in − ho ) 0 ( Λ manifold. the t t 2 c dx – c the ) are 2 for 0 is, Λ √ ) t c − ( see 0 singularities can directions. ) √ c 2 − 2 string t frame, t ( / e − that arian a 0 0 Λ smeared t Λ to v φ φ − W Λ (4.3 √ √ scalar − − in ( √ an e e ( these compact 0 = ˜ to ((9( φ 8) finding nice string e e / = = del. to D8-branes 2 Notice Ricci T-dualized, e / the (9 ], ) ) = 3 t t b − 2 string mo the [8 ( ( 1 − ˜ along e φ 4: the in c 3 e ds metric onds 1 / In in 2 − ) 2 string ) 2 string ould ecomes 0 b 0 a c w b c where direction original translationally Figure − − It corresp R T-dual t t is Λ ti-D8-branes. Λ one the √ singularities √ y an dilaton directions. ( The ( to constructed solution + − erges. 1. transformations only metric and solution and as dual + y div = w The 9 the spacial 2 E x similar the T-dualit This With ds ' 9 ery space. with 8-planes planes 4.3 Since and extra v dilaton solutions T-dualit x JHEP12(2002)052 t is to ten the and case t onen- (4.12) (4.13) (4.10) (4.11) in trans- ergen brane. Einstein coupling exp coupling. couple This div along additional erse kground . a the t ys the eak not D-branes of bac in solution w . in has n es deca 2 transv branes obtain ) h anishing to − 0 do c 9 wing to − es the t and k = whic p < 3 Λ , t 0 go endence x ~ non-v ti-D3 dilaton p √ t dimension follo 2 ⊥ Sugimoto : a and an 3 < p < 6 8)( ⊥ = dilaton dep dx / the the ) . x ~ ) t the ving p the t has w 2 ( 3 − ) and ature’s, the on at la y 0 ha in solution c < ((3 2 p = 3 − er e directions to ) t as p − string rules, curv 2 0 w Λ cancelled 6) b t the ) o − erges √ 0 frame or ending p p + y c F directions singulat 4)( 2 k a 3-planes div 9)( / viour − . cally / dep y onding scalar t (3 dx (2 lo e the ) φ Λ future. ) eha e t 3 – Busher’s only 2 ( / is b √ ) 5 the deca tifold Einstein 0 ) longitudinal e 15 to 9( c singularit 0 ws near c corresp – 2 string coupling flux − − solutions tial the b the − orien t infinite than the t follo (2 t compactified + Λ in t RR on Λ er 2 to n √ onen ) , the ( t w √ y string dt 0 solution qualitativ ( 0 ( t a ) motion, φ t at differen slo exp the According calized w ( Since 4) = close n of the / the . h ) ) in coupling − string for 5 t ⊥ same b φ the and 6, ( uc e ed. as this ) − dislo x ~ 2 string t p and ( R m y a the < T-dualities (( φ In the of p > 6 e e − is p viour p 3, eeps e k calized v − = = = < = preserv equations eha times 9 ˜ φ ha 3 b p e teresting is e = t dislo system directions singularit the in w 2 string for A n a large p = 6 is 6 . of ds directions. 0 the 6 decreasing t distances erse to > at Dilaton that as ariance = planes p at > v its erse 5: erform p t 0 es t in p onds large transv go solution for tifold viour case a Notice When at can transv φ e e Figure eha 3-branes, frame tially lational six W dimensions: corresp orien also brane The b but JHEP12(2002)052 e o o is A en no for w w the the flat een RR one t t . eing easy (5.1) etter is op w e when tifold tifold b er b in strong is branes branes , et t sources net lik there the the the ev b distance op compact is ¶ of D-branes a w attractiv t, and theory ose lo a orien orien it there oth this to , ], the er een cancellation een b Ho satisfied ersymmetric the the t; b of 2 on it -brane w w 10 decouple. a are imp parametrizing differen the p one , ts µ et et ts sup tion where um presen not b that b een ws of where n distance 4 11 solutions, tly [14 and oles oin absen ti-D w θ the taking amplitude ) the p do ordinates e not et forces allo an it brane, the atten the b to ( co erturbation are force sligh presence is tadp manifold. the represen p y harge 12 e is ects e c osite here calised the − b and system h assumes pa erstring), in y η the massiv distances solutions e t exp ) expand, space going on opp planes, tibrane RR π in can the sup and whic one − ery and the e to ) length, I teraction find v 0 one By semi-lo of compact teractions t terms α attractiv just e ersymmetry in π describ case in t tifold 2 living or ersymmetry field yp at / can e an ) t the sup the Things -brane space 2 terms brane-an w string negativ y p other, the ecome sup relation terpretation energy the orien to one b on D ((( the h represen fields As in Ramond-Ramond − the non-compact view a the calised disk e not of h – dominan D-brane means eac tibrane, that lo due and 2) of that when is / strings. it ect ) 16 een t an compact the acuum the or the solutions non-compact. sectors scalar op whic some w v without – string +1 F oin p cancel But lo endence, et resp true, p (( and This the view b are out the are the tibranes. manifold. − the means closed to e anihilate ) dep of deriving t one ab an fields, of 0 of t with closed α they just that at longer mak brane of can for string 2 ev oin configurations That a π v expanding is some is p ordinates. the direction. er no en ersymmetric ersymmetric, (8 forget teraction compact of t scalar e b hange co pair since directions v t is necessary In op in a the t dt terms configurations: oin flat sup um ecause exc t, ha distances is the p hold. ∞ of T-dual case string n on b configuration. a unstable is 0 the the ed el erse that Z disk w ts is is D-branes non-sup teraction fermions. for for p the tibrane lev the momen ) long has least in compact i presen final oin rom is ( dulus allo t closed the F p to tial at transv A and that teractions review +1 some tree the That relations the are directions. mo teresting p D-branes er the that h the in or the V in solutions e oten ev branes the ). on situation − for brane-an the some b the p on suc example, first w the the equal erse osons duction configuration el. y example = ], rom at ts b terms e b the is us (3.5 to Ho us branes. op the ) F [8 b hannel lev for The consider (for θ k tro y c lo o the when , T-dual calised, i in amplitudes. w clear een een In the Another If Let Let transv y us disk t lo hec endence appart, D-brane ( w w tree c one calised op V et et ery acuum harge the the relation are these where dep ones and lo 5. Let 5.1 should constrain v planes c planes. to the v b lo far space. b branes. a branes. at space string JHEP12(2002)052 6 y in to en b the . are the the the een one . suf- field (5.5) (5.4) (5.2) (5.3) op ¶ of w des, ) short ws i t ) ciated space, as to energy it et is , causing it , analytic mo at b that e ( 2 1 the oles, 2 allo there erator er duced is er, y string . asso 12 , µ teraction. erse w b 7 op r η it/ h In ev t in ( lo 4 11 en pro − arriv massless w π tadp electrostatics, p θ 4 11 distance e y e a part op θ except tegral ¶ Ho of whic the 2 w winding des. t i transv indicate cured en ) t ' in with RR space i tials for teractions e µ of 4 the de L giv t mo i b of in the apart: a 12 the m laplacian and de In 16 mo erse oten − ergen + t i ) at p η can el 0 the far y free mo 4 example system ( α t div the i imaginary and from π t string windings lev is ) zero 2 P are / π torus. transv ) 7 of The ) the an 0 zero − 2 not − decreasing ) a sources α y non-compact the 0 p sum the t. electrostatics. in π disk t ergence α (( # is the in (2 π closed 2 another − the of and 2 distances in the ) on for e t/ / ligh i the div wn in ) the − to 2) 2 L although fields. tributions as e / of i y constan of ) D-branes y sho solutions i tial) at long that m er ((( sum o e +1 con This m space: b X − p v w + at e t deca (( massless D-branes ear i 2) the oten – inclusion um amplitude ha string 2) the / − y the n p ) 7 / ) ( remains system, e osition images, amplitude t ’distance’ t 17 the the app y can the 0 +1 i elops. w the +1) a the p α X – distance p The and from op 2 the (( to (the to is " the as compact (( dev π − at effect. closed analysis, oles i − ) a transp It , ) t the when (8 accoun m y all t p 0 re-summation, X on where due 0 But system t in part luminosit y α since dt UV k one-lo ected tegral α − to vit x, 2 h er tadp the ]. 2 tial 0, v π 9 in branes. ∞ in = with ole π o the an [5 0 exp tac real ) total the (8 e → is directions Z (8 oisson y massless is ergra a oten t t ( that p dt ected, the parado the P t of ) p ergence dt tak the tadp These y i that V sum a ( b faster ∞ tials erse sup a to for ers ∞ so 0 exp string div ] propagating. 0 ute +1 , the Z is y p Z the ws p Olb for redefined obtain en as [13 the oles. on). oten V Notice p ergence IR ) review erse ) y i w e p for should as − directions i of ( . transv gro op reflects h b ( from the k div comm an p can h = +1 tadp from t +1 p tac to one the erforming effect e ) − mediated the p luminosit V θ p short that 9 transv ely V this W , result tribution − endices i whic y − the ergence are y the should found are b on ( = ected compact y naiv = con this similar when app e as excluded NS-NS ) expression our V div h constan v ) θ e of is e another case θ , exp the i w , b ha tac picture fields i uation. is y er duced images, amplitude, y order ( e main from b branes This After If As directions wing ( to some this That See p kground tin w compactify V the 6 7 V − op terpret e um 8 r So these for string n the fers lo (in in follo The repro the w same distances, bac con to there has JHEP12(2002)052 2 is w m will no zero that (5.8) (5.7) (5.6) (5.9) − (5.11) (5.10) 2 term general the can ∇ and equation existence euclidean euclidean , of = ds 2 mass ¶ more the the ) The ˜ a ∇ a tegral y eigen-functions ( determines correct If in is J In metho ¶ ) ) . with y 2 n 2 the exclusion that ω − . e path ∇ (2 Consider anishes. e / x action Z v v ) action. ( the tegral 2 n vided ∈ J G ha in The de deriv ) . ortonormal n the e . x erator pro +(( ( ¶ that n w mo of 2 n of J Normalisation a will op h equation. ¶ path da y 2 n φJ M e n M ω π D n quadratic oundaries. J . w , − 2 suc zero 2) d de. φ the b 2 n ) tial / basis , 2 2 n on √ a | x n,m ) m (1 the ω a n φ δ mo using no φ,J y D for the − + ( : ( ∂ − + e d | S in Q c n 2 n = J n 2 2 1 − a φ = spaces, symmetries manifold + Z and e ω ≡ 2 n m differen n zero da µ – ) limit φ a 2 1 φ defined ω g n φ φ π g to a = n ducing 2 1 2 subsection, ) φ its √ − 18 D D 2 n φ √ e erator j → y source − √ D X v µ g ( ˜ ∂ ω – D tro y φ Z op φ µ for ij b √ propagator y ha = in dx ∞ olume g next dx compact −∞ b D ciated the = compact exp ) v n e R y g Z ] y X 2 Z b w y ( new in dx / the functional √ to µ J the = 1 [ term ( φ asso i = − Z Z exp 6=0 ∂ V ) In ) the Y n 2 ) function spaces g measure shifted then J n 1 symmetry infinit fields ). ), ∇ ¶ y √ extra 0 come obtain φ, 0, da J ( − t get an measure coupled 0 = π the 0 the (5.5 olume (5.6 S a = ys to 2 φ v n e tegral a -dimensional generating 0 0 to for in √ φ in w ω ( D 2 n da enden alw )(det massless compact ∇ ho tegral the a If 0 field simply Π due action finite J that Z that in of ( in dep path 0. des is µ Z δ e equation the the ≥ teed. = = = in de fact alues the n path review ] to ω has J consider compute mo olume massless in [ the Observ v its zero-mo It satisfying Z a will to . us guaran , . eigen-v define propagator that e V Z g zero of ergence 1 the is ∈ √ w aluated: n duced its Propagator Using Let div } define the can ev de = order this n tro e e 0 e φ erify w action φ Notice and w in the In for of and situation, b { mo 5.2 Here metric v JHEP12(2002)052 e w at In to en w the the not has op 0. kno equa- (5.15) (5.12) (5.13) (5.16) (5.14) this: left Green es strings for e = of is is at do situation W teraction 0 space , the . J . h in solution propagator dc ok hniques, only ¶ a ) closed M The lo R space y ), the tec structure: distances. the and , , whic ( e “gauge-fix” 0 duced t J w the in is a duced. find ) amplitude (5.6 to the compact y ) tro w de y field for large de to usual e − in onen to op in ho v x, manifold repro has x mo in at lo ( compact cal ( ha the symmetry on is ) G a and G lo y e ) order ) comp ( one n a zero x w then sources in zero-mo t J J ( a . in ) and 2 J , wing y 1 on − n ¶ y compact the (5.11 ) · the as ) ending the ω n teraction y − a propagator 2 n = ollo d ( of ¶ . ω , · in + ) x F n in on · c j x dep (2 ( constan 1 2 n n φ V · / q n the ) tegral ) a G ) ω ts a − translational d 2 When situation, n the j x amplitude + − de. in D-branes ) J ( necessary e y x end propagator → c ) n x oin v Z , D ( i x is the φ that p +(( n functional an d 1 2 the ( − φ x – ha 2 n the amplitude, a t dep ( ( a D R − is 6=0 on 2 0 n δ V δ G systems X n 19 i µ ω a strings. for − seen zero-mo not √ q term consequences not = this – 2) − dc ) e / µ mediating t one massless ) symmetry consequences x es v ij exp length, (1 es = ( δ x there Z differen ariable X the a tegral: ( ) − v φ ) ha do 2 x e do closed the des cal in y G x generating = at ysical of harges n eral e 2 D n ) c enden for lo → d of − w d A D-brane ph ∇ mo −∇ da string V ) sev ( the where x π 0 that Z ( x Z (5.6 path dep , 2 ) in ( cated the source the RR) h G 1 1 φ V the V √ det lo has onsible tegration 0 in the understanding suc the ∞ in s out (5.13 da and −∞ equation in massless of action olume resp that ¶ Z ). As amplitude ab discussion v That Z the µ dc ys el symmetry shifting the a dc 6=0 Z D-branes (5.13 w Y tegrals, lev n harges, Green The factor of c µ · Z o (NS-NS in greater discuss e that satisfying ). w een action. shifted = = consequences gauge v t previous tree quadratic w k ] symmetry endence. to e the a wing a ha v J et space [ some the are correct b (5.13 e the hec calised the to ha as dep equation Z c are w lo follo e in i ysical going or function q distances satisfies w The tion the rom or uting can there case, h the Ph F F end, at are ear • olume compact similar e v erm function 5.3 whic where One a W in P that where Green app strings amplitude are translational the is this add JHEP12(2002)052 e e ´ is no are the the the one can ter- it zero sub- is e Since (5.18) (5.20) (5.19) (5.17) es ampli- harges w as oincar c the the coun massless P ter-term, giv space-lik where amplitude planes the space-time there the On time. the the cosmological not an coun cancelling space, amplitude. explicitly all the and t and is is for tifold are functions, er oles. with the v een h m and o solutions b teraction amplitude subtracting anish on orien ws finite in v tadp there tegral analyzed constan theta compact of whic tributions, zero in has . e sum a mass gro . ¸ the and W string to 1 ) con the de analyse with . y the in ected, ( − es effect ergence ecomes NS-NS 2 en in n if mo to ) b ter-term i ¶ φ go adding op q div dilaton ) exp 2 i ts previous solutions i the the x ix -branes L regulator to a . mV ( p zero , coun oin t y P n j D-brane finite a the As i i the ( tial b p q φ has y i ter-term L 0 is q mV in the dimensions. the the propagator alen µ 6=0 teresting → the 0 , coupling 1 X n induces lim 3 onen m induce duce in y amplitude to case, → θ coun − – lim term = the all e m ∼ effects that tro =1 limit b equiv exp t D i 1 the t 20 V ij ∼ er the in this the T-duals. ould Π is string v particular – A · w e ij see o − write amplitude the in w T-dualit de ould ) its A ortional at ij e enden case dx that the y w that X If w generically can mo the non-compact differen ∞ − = It 0 if dep e prop and on, out x o see Z ole where ( w de. A cated k w ter-term. is zero t inclusion D ∼ that ) non-compact summing lo δ 3, del applying mo can hec cancelled. G tadp c Its Later space. the olume coun < mo the v also are is . (5.12 yzed to of 0 one p zero the non-compact t in the longitudinal ole the ana sum. D-branes ard for the e the obtained o that v form: tifolds Notice time that amplitude, compact w state, the of duce tadp t in viour. that absence ha equation brane Sugimoto tforw compactifications, the e tro the seen eha the Orien except where see w some total the in i.e.: from of b e finite. notice the that the string since form: on v form er at is to is een straigh the and of ]. can ha on flat, y that tribution w de, toroidal the zero, [8 is e t e pap et or w In W F of tracted see Also, It b Firstly Notice con correct closed is need term tude mo in calized solutions • • • Conclusions this arian v ecomes hand, in 6. In solutions singularit done b this D-branes dislo JHEP12(2002)052 t a of o- at to ed for for are are the the the the the the w last also The t non- write (A.1) (A.2) to as images are er fields, the seen function ys dify jects are observ metric eneziano. a of es e consisten is already e v v V ob onsable d mo v ery prop concerns relations alw ha the equation, These G. eha ha related ev and to comparable e there ordinates. Garcia-Bellido, b subtracting resp term of e that, w the is massless or co can J. t w metho ole of F and stress these harged as e , ear, c the finite oisson ) the w to seen t, fron that 2 extra P o the term ers. ) tadp as of First e app w that an b y effect t in A v ). ted distances ordinates. ed b ∂ the compact de , the ha um an co new )( propagator 2 Emparan, oles the n ), the erfectly relev (3.3 ) w e for A dified” mind p ( the e ect A R. W factor motion 00 observ a ∂ the w particular azquez-Mozo (3.3 has real in f This tadp e, ´ on V )( “mo of e But brane manifold. )) exp zero-mo v + In arp When A a ansatz A e ( ecomes the ( eep that w with A ha 00 w b constructed . term k the f spaces the f to j space, e ansatz M.A. )∆ ole ∂ oles. NS-NS the w + on spaces. coupling. A ij /dA endence oles ws ) ( ) metric g 0 – compact extra equations g arez-Gaum A Then, satisfies motion. A f dealing tadp circle. space tadp ds, should ( ( t sho dep space. ed ) √ 21 ) metric tadp a f ( of ´ e the Alv D-branes i d where )∆( the brane, Uranga, – string A the compact ∂ 1 ( arp W in the A 0 ed compact endence L. g ( space in = the on (3.11 compact of 0 for f 1 metho A. enden . the √ the f ) for the arp in of bi-w dep in without = compact order ely A analysis compact dep = = with in tial ( on 2 0 term jects of ) harges ens propagator, e f bi-w equations the c )) est tegral A ob ectiv the from effect A w ∂ oten hesano, compact in equation ( size p the for lo spaces closer )( dilaton the f a olume extra happ and on metric field erators of v A resp the erator consequence D-branes ( A or ∆( in Marc A ) g an t the the electric and the path F op j op harged form the discussions the c erturbativ + F. for side t. function is effect el of e p A∂ profile ect scale, i A a the (3.15 and h ts ortan of teraction ∂ lev ) is the duces ∆ kground del form ij y term. Using in ativ ergen ( hand h g resp where function field among and imp pro whic A mo t bac of non-compact disk kind div tribution ), ) dic define = Janssen, e e space, ole el, in 2 amplitude among righ deriv with the con ) the yp B. erio (3.5 enefited t can (5.13 lev approac A p same b second in at de ergence. ∂ wledgmen tadp the kground e ( relations consider ˜ v correlator nez, otal one close the ed a div ole t Our The The disk on T compact ) term bac illustrativ compactification us ha kno Ib´ generically A e teraction teraction ( oin ery the where the trivial the deriv equations “jellium” NS-NS one the in that this is equation p in v zero-mo propagator the tadp L. Ac W Let A. f the JHEP12(2002)052 a as Ricci get (B.1) (A.6) (A.7) (A.8) (A.9) (A.4) (A.5) (A.3) , global ¶ . a previous 2 the and written ¶ to 2 Ω) e ) the ∂ b ( B up ∂ 1 ( metric 2 can + p from oundary p 2 − b the 9 + that + binations for . ∆Ω results B ) µ n . ∆ com without ) B A µ the A . − 2 r ansatz Ω Ω ∆( f e equations, , e − 2 , n i . ∆ B linear − i g ∆( i F µν e N 1 n 1 using i this λ the i g λ R A . F F ij i λ r A manifold λ i y − g . − F − of P λ e (the b . e i = = P = . P − ¶ F P = 2 With = 2 2 – ) ¶ ulae. B = ) ) 2 form: i ν 2 ) αA Ω A 2 ) A A X ∂ j ) e ). 22 ∂ A ∂ ∂ conditions A ∂ ( x compact B A ( ( form ∂ – ( the written bination = ∂ αA 1 µ 1 Ω ∂ ( N i e )( r ∂ ( ) B e r r ∂ of − A conditions 2 1 α the A metrics b + 2 1 M com ( αA + + n g Z e + + A on + and write A A + ed + general A B ∆ can t, ∆( ) Ω ∆ ∆ ν A to j ∆ y A are: linear then ∇ ∆ are arp ∇ ∆ , Ω( µ i as consistency equations i A That ∇ ∇ F w / consistency of constan relations µ µ . terest is 1) i 1 p a of and (bi)w allo general in λ and set − 2 2 the factors + − is a , n a of set P ts p 9 r ( Ω for / i ed 1 i conditions: That − − ∂ = = − λ . p i n ) ) B 0 arp r cases +1 − µ A A P tegrate 9 p ij µν w ( ( dimensions ∂ /f consider constan binations R in P g g R 2 R 00 consider o as tensor f a w = = = ten = t us are can = can com i e 10 10 ij µi α in 10 are µν r g e consistency let articular When When R R R P One W tak Ricci of i) w, ii) e subsection where where Linear where No There W set tensor factor). B. JHEP12(2002)052 . ]; , , , with ¶¸ non- via 2 Phys. (B.2) (B.3) (B.4) ory (1988) aking cting , a e Phys. e Ω) the Einstein , br ∂ 2 vacua ( 296 i 2 ersymmetry and e, and interse the ]; ]; B 123 e 4 ]. + string , sup ersymmetry yp p in ) orientifolds . of invarianc T 2 hep-th/9812118 ) [ + sup stable 2 )) Phys. strings ale (2000) oken y )) ersymmetry ar invarianc ( sc . 133 compact ∆Ω y br e-IIB aking om φ en ( USp(32) e ) µ ]; fr sup ∂ dilaton 2 a φ 565 ale op ( br typ Nucl. ∂ and Sagnotti, )) 1) , sc B ( with the B del y in − ane B ( the (1999) non-line + A. e hep-th/9909172 hep-th/9908072 − φ [ [ hep-th/9911081 mo p Br and [ e ories and dels ∂ ( aking, + ( d for e Partial ]; osed 553 2 + Phys. and − with B the 36 031 mo 024 br y 2 )) − B tensor: e x R )) ondensates ( ersymmetric · dels c x system ]. + ϕ ( Nucl. standar B action d-like Sagnotti, hep-th/0107138 Dudas 9 2 ∂ , (2000) ϕ mo [ (2000) tum (1999) − decomp ( ¯ Phys. Sagnotti, ondensates erstring ∂ D )) e 3 Ω c A. ( E. e x The 01 10 − + string Ω ( ]; ane A. b The sup 572 non-sup – ]. e D9- − ϕ ersymmetry ( e e and Standar ∂ ¶¸ Nucl. B e ( ). en 23 ( Ott, momen µν 2 , string and D-br e-I hep-th/9807011 y (2001) oles, ) ij can Sup g Ω [ Phys. op Phys. ( – ollonio, g Ω − T. B φ edo, typ ory e B e of ∂ gy gy ( tadp e 4 1 ( Phys. 616 469 Dudas oles, + in 2 1 1 4 hep-th/9905159 field p [ and ) Dudas energy − B achyon-fr E. Quev G D’App , x Ener M-the − Ener Laugier, − ) 4 tadp ( T ) ely-acting E. − Sagnotti, ) x ¨ ust e 685 Nucl. ϕ F. y G. ( the y 10 A. L ( (1999) √ fr , hep-th/9908023 ( Dilaton hep-th/0111209 [ ϕ and lations A. [ φ = Phys. φ High 2 High + i ν to el dilaton j Z D. 3 and ∂ ∂ Z ) 38 ∂ and B Consistency ) ) 544 J. 2 J. y toniadis, 1 ) Dilaton aking and (1999) × ollonio, x x Uranga, , anc , κ e ∆ y . ; ( ( c − ˜ ors, nez 2 B 2 ( the x, Nucl. An br µ strings ϕ ϕ a toniadis, ( Z φ K¨ Cai, , ) µ i µ φ 262 383 (2002) I. = 102 Susskind, Ib´ p (1999) ∂ ∂ ∂ An en of vacua A.M. is B. Y. D’App Benakli ]; 1 1 2 2 2 1 S Dudas tribution that − L. I. Phys. op systems nomaly 631 L.E. K. G. 464 E. = = = (9 and A and con Phys. (1986) (1986) orbifolds B and others: ane − string form: B 10 10 10 a scalar tonj, ij µi µν x Nucl. endants or. T T T ersymmetry endence: , R e-I 173 171 hler ett. the · has hinski consider L world sup Phys. The desc ory B B typ Ω dep toniadis, toniadis, toniadis, Ricci of among e olc − Fisc g. ersymmetry, Aldazabal, Aldazabal Blumenhagen, Angelan en e w Sugimoto, P o . An An An is ane ane ane-antibr om ett. ett. term the hep-th/9907184 = Nucl. L L I. Op [ sup M-the C. I. br 91 fr br G. br Pr G. Phys. W. R. J. I. S. See, If 10 [5] [6] [3] [4] [1] [2] R compact This References frame And JHEP12(2002)052 , , 01 in es 231 19 angles dilaton Phys. in av. surfac at worlds dimensions ane ]. gy Gr and (2001) a ane . anes extr worlds metal 052 ]; Ener br antibr oidal 505 br 1988. of ge tor Quant. B ]. ane e ane om lar High cting in br fr ]; (2002) 1998. Br ett. and ersymmetry . J. L bridge and for , 03 sup structur hep-th/0011269 interse [ Cam bridge rules Class. enarios stability Zhang, Phys. hep-th/0106195 , for sc , Phys. oken 241 hep-th/0106140 , onic [ br Cam ometries duli ]. sum onditions R.J. gy ctr dels Press, c hep-th/0203247 ge [ 031 mo ]; ele y d mo otential with e e hep-th/0202024 and (2001) Ener p [ d hep-th/9511194 Press, world 042 e the dynamics , ersit le y warp es jesh 141 599 (2001) stabilization Inflationary c ane High – and warp strings Univ B ersit structur Ra for Br 07 J. (2002) Consistency 24 duli in oles, in Liouvil , G. 1973; – (2002) a 05 hep-th/0004165 Univ Mo ane [ dels tadp Phys. Zamora, ters, bridge Phys. hep-th/0112147 mass Linde, edo, [ 532 mo with ¨ 172 ust, Complex F. string formalism gy Press, Phys. L Win Cam B bridge solutions and 036 Nucl. in , A.D. an, Quev gy , and D. Dilaton Ener es ane-antibr ett. D.J. ]; F. ane Cam etimes (2000) t, oles an L ]. Br and , and opics Ener ac Br on (2002) Rabad´ T orientifold High F strings and Academic sp tadp surfac ory 486 , ors 01 R. J. A. Phys. Rabad´ at K¨ High the B ers , 28 and , Kallosh , J. R. B. Resco, My and ]. ]. and Susskind, Mourad, Martineau, , ett. ory density-functional . Dilaton R. Dilaton Phys. . L P J. L. P String the Physics gy R.C. hep-th/0011225 The Physics ersymmetric [ ons, ations hep-th/0107126 and [ and and dimensions Phys. in Ener , string 83 022 hinski, Gibb State Lang, ary Burgess, actific 0’ apazoglou, oles olc . non-sup Zangwill, P Blumenhagen, Charmousis, Banks Gomez Blumenhagen Dudas e Leblond, High Garcia-Bellido Garcia-Bellido, P omp hep-th/0111025 hep-th/0102015 for A. tadp [ inflation (2002) J. F. (2001) arbitr C.P A. Solid c R. typ J. [ C. R. E. J. T. G.W. N.D. C. J. [9] [8] [7] [13] [14] [15] [12] [11] [10]