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This content was downloaded from IP address 136.56.14.111 on 02/02/2021 at 20:10 JHEP05(2005)010 f - a 7 pd / 2005 2005 2004 7 at pairs. singu- 5, SISSA stringy 15, 25, tension ely e or extra May f the IB /jhep052005010. April I tibrane df ebruary ositiv conifold .p F p e a 010 arious 05 erturbativ v of 20 p includes yp 05 of ccepted: t Published: ep A Publishing jh 7-brane-an admits that Received: rs/ in pe bination pa tial that e/ stabilized presence iv com Physics e ch oten a b ar p the additional of it/ e a. in can ss the with threefold y rrections b au eectiv olina dulus theory co Institute the Car 0 mo http://jhep.si together . y by b al. string Calabi-Y generated acua North et V IB ed olume I via of v er e acuum k arp terms v w yp the 4 Published t du Bec a erstring y /dS b the on 4 that U.S.A. University tension Sup e AdS uxes 27599, ov consider Physics, obtained negativ e DUMC. NC of stabilization D-branes, W at Bobk l, compactied demonstrate with [email protected] e tly Hil ct: ersymmetric 2005 W ry el ords: pairs artment correction . anomalous y 3 Curren ) olume E-mail: Chap Dep SISSA 0 c V Abstra theo brane Konstantin non-sup larit Keyw ( and JHEP05(2005)010 ] ] 5 7 a ]. 1 y 5 y 3 ]. T or 12 tly 21 14 the the the del. [17 hru, ma [17 “no- , “no- , man trivial h uxes, KKL mo [20 [13 Getting dify Kac unxed. the the string ecomes compact- Heterotic non ]. in b whic of violate of mo corrections, orld tial in acuum ersymmetry in orates ed v it 0 [17 subsequen w to to 4 4 particular, break original with sup so remain arp oten y has earlier ] dS w e construction In three-form del incorp the [1 AdS erp It brane ma scenario Because ected duli it IB geometry scale obtained mo ]. of a I solutions brak RR sup e [2 mo e found exp ed of b The suggested explicit yp since ] can as eects the and t uxes arp ere duli string problem. w Sigma Sitter e w metastable w text ahler [26 an of to a one NS a mo K ariations used. stabilized. y de the of cannot ersymmetric con b V e with in of b a case at the ere theory sup action. succesfull dicult ahler w in sources compactications a y end. obtain ving K the can erturbativ tial, kground tributions authors vit to ed solution mo In more string the the scale bac rather con oten ]. – ti-D3-branes in arp non-p the in p ducing a een ergra e space 1 w problem 12 dilaton state an b y extended constructed demonstrated Sitter – – y branes constructions freeze with pro ails string and e sup as h ork, een vit [2 of b the has de as to b w or h au ] using F a the t h solution prev as Sitter compactications ground suc ergra dulus [17 ]. has erturbativ at 4 suc hierarc ell de 23 ed success that w T) mo recen sup , hanism w-energy space successfully inclusion theory probably a the AdS corrections lo Calabi-Y as arp non-p [22 and scenario as theory dS mec sources eects to In w stabilization great (KKL h w the and can a the a states features olume duli the sting ]. v ] s corrections h of IB edi g resulting lift I e found 25 stringy 19 mo whic extra dels in string tifolded uses b 0 dulus , riv , the in e to that whic viding T obtain the solution mo see een to [24 [18 mo yp b stringy t orien of only to pro from to en and that y adding see v stabilization uxes del-building of eects an a non-geometric also one y stabilize w ossible e olume b generic pro e mo time p duction if t on duction the space V BBHL v ts Linde theorem conditions e stabilization a of er, argued ha some theory structure tro has structure v olume tro demonstrated ten more tune indeed 3.2 3.1 Review V In Conclusions as In duli vide complex-structure Sitter del w dieren et, een erturbativ . . . . 2 It scale” ications 3 scale” all uxes 1. Mo Con 1 b 4 mo can “no-go” string and de matter A M-theory “no-go” p Moreo pro where Y Kallosh, attractiv JHEP05(2005)010 ] a e is is T of 7- at er- are su- ose tial RR can [27 h ten- large uses dulus of ertur- ersion e olume in sup obtain KKL energy t discuss v arping. p v -branes Calabi- oten prop and obstacle ducts whic erturba- 7-branes mo non-zero w scenario. but complex- ination. arious ination- to p v the ] D7 problem the ] case of tial erp a are compatible erturbativ T / pro inationary NS the ered for construction dominate p y e [17 the of All then ahler y negativ [27 b ated D7 sup b y restriction h the K on T oten ereld in yp authors dominan olume duli obtained This degenerates with the simplest ] b t KKL p and the v e tial as t in en ]. y b steep form They sup er 0 mo [45 and ated sub the whic this the tial KKL motiv stabilized o the ] the a the [32 a b < e oten to in died T brok the in b ) p of [43 relates non-trivial ould 2 oten the it hand, stabilization ]. ored. motiv t w anomalous is dieren p mo has T v bration er M terms elliptically-b [2 where geometry degenerates. ] ( es -branes a that to KKL can singularit fa used scaling 7 2 an that with ahler functional e an dels / [32 not the onen pap is the stabilize e T 7 other afa-Witten that K geometry mak the t olume tials is of in v on mo uxes to er compactications of stabilizing ed v-V tension ers the the implies factor comp o dilaton oten b b that hanism (p,q) “no-scale” e recen conifold h problem, where generic a base one arp ]. to negativ y generate ersymmetry ux a ahler teger On a a alternativ a of arp breaking w erp um is Guk the ci restriction [28 the mec w is K the n ersion in w w endence eects the In compactication whic lo a v in ositiv Sup to for a as discussed sup also 2 es to p ]. au allo as / new ed the – the dep the generic pairs e 3 in . inationary where 44 ell SUSY dulus mass. the 2 compactication e A admits ] arp where m M at y of – w yp mak that the so v M – of t w o pairs similar mo leading ould . corrections . ci ed [28 briey compactication 38 4 a realized as k w simplest that died , tersection lo ab out olume Man generates a H that Calabi-Y v that in dS d the arp in hoice t stabilization from scale 43 argue ]. c is mo inaton w the region inside corrections y , ] the olume additional a M oin 31 dulus the threefold the ecial that v stringy tibrane 37 in and features scale p ed y – y [32 racetrac this – sp problem gets b metho b ination of assume triple mo au stabilization olume large Apart of energy from and arp [29 v ux recenlt on that theory [32 a stringy exist w including tial the they dierence 1 this the also This (3) orating ination. see IB tial four-cycle high olume brane-an at threefold I ey picture e the to G olume v to breaking osed ybrid tibrane oten k Apart that induced v eared trivial e h based Calabi-Y acua p W considered authors stabilized oten v au for highly a the stabilization a is the yp base incorp bine t the app the prop is extra are haracteristic erp the tibrane reviews of y are SUSY a on four-cycle. c generating the hanism. is , b particular, setup t the ination, with d com ortional een F-theory brane-an the sup hanism o olume that inaton ed b e acua resolution problems In mec and the duli v X to v tributions Calabi-Y e w go osal endence. w from This altogether. onen 7-branes mec the the v Euler in prop the a the mo surfaces e on based con brane-an or tial ha without in In sho consider dep ecically general ersymmetric far F scale to wrapp construction dels. the T prop olume, the on e e ed ossible when Sitter sp comp v generic tial, 0. oided the the e. term p T oten fourfold to W W mo edded v ears Our a 6= 3) b de ely 1 KKL more , cated au e oten erp erturbativ olume scenario some of Riemann wrapp non-sup bativ KKL large sion ary p lo v em Hubble structure Y tiv related W due (0 b p More uxes branes a of with app stabilized. One with directly stabilization When p JHEP05(2005)010 e ci of to to ed au the the , RR lo this afa- duli onds (2.4) (2.2) (2.1) (2.3) from er X other space arp of b In mo include w v-V ological absence and and o 7-branes ecial the a D3-brane um the to wing n four-cycles metastable top sp the erturbativ t Calabi-Y the er corresp ature ersymmetry NS p Guk v alue fourfold of corrections o follo v in the the total IB a the sup e harge: I giving fourfold. e” curv au c ux, the forced ie e at quotien difying trivial and The + of er the the large wrap yp olume uxes h are v t v mo o y 4. where of induce a (0) non e , / b uxes neglected. C w the necessary kground n sitting whic Calabi-Y the erturbativ at duli O3 hoice “landscap y D3-brane trol is c dy b N acua small a , with bac of v mo m orders uxes 0 7-branes safely with ered con at the the trol dy non-p e = M induced = the b haracteristic olume of b solution c v duced limit in mn 7-branes The con of obtain O3 3 (3) Sitter g h the haracteristic . arious O3-planes ) e h c F , y Q v y can pro ( w the harge the suc ∧ (3) A c (3) at Suc 2 Euler F ti-de sources three-form arious G dominates (3) In . threefold ex-structure and theoretical arping tial v ∧ tifold e Euler an harge ∧ w H axion-dilaton harge regime, elliptically – c the au: cal c exibilit + (3) au 3 er. M The that oten lo the an IB H h orien comp Z and duce stabilize p – degenerates. I theory M this b construct is dx 3 viding D3-brane Z e = all 2 on e tial T ) O3 tro induce Suc o in e the uxes, T = corrections 1 can T 2 10 N pro compactications in X yp (5) dx wski eld Calabi-Y t suppressed ( ˜ o Calabi-Y F D3-brane e er oten D3-brane for 2 e from W of op d tial. p w a a -branes. ] ed eectiv elliptic can ) lo b uxes the + e the teger y scale. alue, the negativ ( e set D3 [45 Mink tially v in on oten A as w the of arp metric. addition, 2 a the the p ocal that where l 3 and e ) eectiv carry then ] w in ositiv e compactied tial Q In the of and au onen p = is elliptic 46 24, cancel tributions (2.2 small 2 ]. , a duce / and IB 3 ) oten I prole exp to D3 space a ds pairs , the [28 con X tro implies e anomalous eectiv ( [2 anish erp arying at in are ation v w-energy v F-theory ersymmetric yields Kaluza-Klein ersymmetry ear Calabi-Y yp y an s wski (3) ers lo the structure t o g sup to b of F arying where tit can tial mobile sources = v tibrane sup app in the the symmetry e of the 7 3 includes um compactications M conserv w n cal is and Q Mink iden oten in at the non-sup will lo IB ocal of l 3 I (3) discretely (GVW) mn correction erp complex required ux en coupling g base Q e language H generating brane-an tial discrete tifold, brane w-energy By sup Bianci yp duct tributions Review O3-planes, a Sitter, the t lo the the M brok oten y harge con where or stringy extra de Witten is expansion In string of uxes p and In the 2. the where extended pro satisfy in in four-cycles language, to orien b c JHEP05(2005)010 y y a b the the will duli The w One erall term (2.6) (2.7) (2.8) (2.9) (2.5) oten- e v single freeze tifolds. (2.10) er where where p o 2 v mo w e | a o au. othed w orien W of considered | the three-form found. units 3 ahler au is only smo surviving of teresting . K after the case ere in use analysis )) geometry w tip run Calabi-Y en duli the complex-structure the an a ] ed factor ev that will ) Calabi-Y [2 tial mo , of dd t e the for ( simple o on arp form: i w ] K arp (2.6 further 2 with of . w that indices , oten w IB 1 ( uctuations form [23 complex-structure I the p h In h = e , tial the Here wing W y log or suc see yp b ) . the oth F t 2 (3) geometric structure and | D b trivial implies oten of H requiremen follo conifold en er, to 1. p (2.7 W W of | B no-scale N=1 jection. ev 3 non The Z three-form = giv The in the the w 0 er D “no-scale” . This ∧ ) a the conditions for . is b , onds pro the is . N es 2 Ho K ) ) , 1 G W G . consistency (3) of example um B and M 0 0 tial B tak the n + ∂ deformed ts Z action (2 iG – D (8 (2.1 A for i deformations = = whereas an e es familiar ∂ corresp 4 W ed = tization duli oten tial it W = b is compute – p W W A = large 2 4 unxed. k ] D holomorphic a (3) minima arp the mo , duli, y D u ahler a to B oten D 4 D G w quan eectiv W surviv log t B [47 hec K A 6 p e a to axion-dilaton M vit c cal A mo solution it a the G axion-dilaton D requiremen the G has = lo en = b ]: ahler is is a also )) all ed the wing ˆ [2 remains ev K use G ergra ahler K (3) later and and 1 er as e constan , e F = arp K v wing b 1 + solution reduces o follo w K z K w sup A h = eral space ell , e ( el and general A Z and i a 1 a nonzero w ∂ 0 V follo Im z = sev tial the dulus run in = constructing ( self-dual (3) as (KS) W 2 ) V ) when i 1 limit H the mo on N ternal + duli M oten log include Newton’s ] tree-lev where with axion-dilaton obtain (2.6 p in 3 (2 to where W [46 mo satisfy , that t in to duli A the ahler the olume ∂ case (3) = The h , ork B K v u F mo jection w alen imaginary 6 2 = K v-Strassler t suc standard e A, uxes threefold e problem and the pro dulus b = W is , large equiv ahler au A (3) recen general ) V The a out that this mo The K a ust D G conditions is . ]. m tifold Klebano 3 h (2.5 and six-dimensional four-dimensional See S and s [11 2 the more 0 (3) ahler olume a a tial cancels A Calabi-Y complex-structure where Notice, The where z of an around z the assume K in G whic v deformations. the orien JHEP05(2005)010 3 e 0 e a a ]. el ) ed 0 b all , By the the = and In [17 y (3.1) alue. wing ( )) Louis uxes arp (2.15) (2.14) (2.12) (2.11) (2.13) v ottom ersym- with reads: W b induces w ma eectiv from on follo tree-lev ) acua and sup v IB order olume. tial tial the ( large I uxes. v i satised, stabilize that the k, the frame (2.7 en the a e at turn is ( oten oten in to (3) e ersymmetry at to yp ) p Sitter t w tial Haac G teger large brok higher log erp originate alue a a giving if in um v string h sup ] de un (2.8 of at er, in oten k sup of in [27 p Suc hanism um the h y cycle xed part Bec in b minim the . break of ∧ stabilized e the text mec b 3) corrections er, whic is , a dual minim of k the e condition , con Requiring that , M that resulting (0 a a dominates metastable ] hoice found Z . also tial z c 0 Bec i is dened h of s en osed the and K dulus g y has W corrections. B K b . can the e” oten uxes, 2 in M = mo whic p implies terms holomorphic brok 3 y prop cycle ] ia structure and ) erturbativ log anishing b xed h , ] + W z 50 factor ( Ae tial z – space stringy teger new – G gets [17 ti-D3-branes ahler # dilaton are A + ln olume exp tial whic in 2 5 ) non-v arp Z non-p K M 0 v / i ) an [48 of oten [ “landscap computed 3 w – 0 a 0, z t p = The W conifold 2 The 3 order (2.7 oten y e collapsing the ) and / z the 2 = = ork “no-scale” 1 b ) Sitter (2.10 = duce The z of the determined w of erp ) W . enden the W en 0 in z 2 (2 of tro the de authors ositiv ( discrete uxes. B ( higher sup D p in G frozen. where explicitly = i on , K a part 0 min tip -dep y a brok scales A b to A The W that the e and ere are um is that the and previous duli . structure w en a B ti-D3-branes z for obtain at + Z M mo tak an cycles 3 2 the ordinate of tial energy theory ]: / to minim tension, three-form S ]: 0 t duli co [2 corrections alue of 4 of y including on v 3 16 oten b suggested then ahler alues y requiring tum mo y three – er p ] v e tial h with their K b b lifted “no-scale” cycle AdS as en enden ersymmetry V [2 to of e [13 2 w to the quan um b based " stabilization giv oten ts of the complex sup stringy n shrinking the ] teger dual to pair non-zero tial a dulus hierarc a indep of log y erp in step the due can t a [27 structure a 2 is is the is case mo w the in oten amoun sup z throat A the it tial olume authors = on en BBHL ersymmetric erp V this another the further K constan ahler oten arying compactication correction (BBHL) corrections brok The 3. 3.1 p v A generate, since sup K Once In a GVW sup metry complex where where creating generic and on of JHEP05(2005)010 of the the tial arp tify are: con- (3.5) (3.3) (3.7) (3.6) (3.4) (3.2) large w ) ysical in W further include oten iden ]: ph the from the p (3.6 the D [27 function our in not to term will W so The used e + ahler tial w did e 3 and K The coming , ] w W k hange limit 2 compute c v as ]. oten ers. implicit [27 ortional the D p b to 51 and , of an W in , axion k ould um v the olume [2 prop . n w as j section v K j v the of e o-cycles ˆ it rescaled k in i ) w ˆ is ij ˆ t V , ˆ k necessary authors as i 2 ˆ threefold are 0 large 1 6 g form 0 + the a corrections . au = ˆ the ˆ e V correction ˆ tersection expressed i V of duli i e 3 4 V )( ) , 3 uncorrected including e ˆ in metric V 0 ˆ wing ) V ˆ 0 b where = + mo t if = notation ( , (3) k 3 i i ( ) ˆ v areas ˆ ˆ erse 0 follo ˆ ) k consider t , can see ( i v Calabi-Y – the k 9 ij , , e in V 2 + M v 6 the same k to i remains the ˆ w ( j ) Kahler ˆ i constan ˆ (2 the ˆ – , v ) g ˆ of i V enden the in computation, 1 3 2 v the ˆ / of + ˆ the k V 0 are ˆ of W 3 V that dulus (2.4 = ˆ 4 V = ij ˆ V k Y 0 i ts k 0 Including D (2 e terms their C mo ij , 6 1 i 2 teresting 9 2 using tial k v V 2 W in | frame in In as e e = onen resulting i = e 3 W assume eep D oten i dilaton-dep b | = = = 1. and . k J ˆ v olume K i j ij duli v e = haracteristic G erp M comp to i the will c 2 0 ) G Z ) ould four-form. G mo 2 + Einstein G e form ˆ V w ˆ sup au V = The the y: with w 2 structure 2 W b It RR + b V order tains + w Euler . ahler ˆ four-cycles D V and ˆ en set A ahler ˆ In no K con 4 2 raised the )( 7 W / . K e the giv ) ˆ a 0 to 2 the V induced or Calabi-Y of / + ignored. are ] is D 0 3 is F 2 of b e the ˆ ) (3.1 a 3 ˆ “no-scale” e [27 b ( the ( G ux factor is in e M ˆ of ( of limit = indices V 3 the J four-dimensional K ˆ can the e arp = olumes line complexied w v The tion ˆ = the V 3 olume en olume V so In the v where where where while breaks v rst KK-reduction the conclusions. the factor v JHEP05(2005)010 0 2 e e | at to ed au w w ere ely the the our this W w (3.9) (3.8) | term tribu- in oten- harge as arp uxes (3.11) (3.10) es of in ecause indeed c region. onding of w con b classical erp is e tak a the order terms, jects eectiv ed of anomalous ) part that Calabi-Y tribution sup orders geometry to e is second ob this arp an 3) factor dulus ) corresp (3.6 h , yp , con w ed all the t order due D3-brane obtain the (0 e mo recall suppressed leading erturbativ suc the of tial arp The (2.11 , to this the w er, so induced the and this induced s highly non-p terms of ev tial oten y tially taining leading olume Normally negativ b p e w to /g v (1) ux const. , 1 frame con the with acua. Ho O oten that en highly language the + onen v = , and . the 3 , ), a 4 s 4 ositiv the in necessary fact, 4 ) condition. erp sources 2 g V p | M exp ) brok haracteristic of x of /dS In c (3.8 4 W ∧ coupling. (0) sup is is | xed . 2 the Im( eeping Einstein 3 anomalous (2.9 | G BPS correction in k K ) base scales 0 case F-theory e of AdS 3 2 a . anishes. W ) of 2 | 0 M ) v ) and ( of Euler y us, 2 cannot this Z extended square ˆ the M string GVW ( b 0 ˆ the i the , ecome it V one y A result Th b + h y in 2 + part – therefore in the in In still the ) b a ˆ the V 2 s order ˆ z 7 V tial ) : = g is the conditions ˆ inside ersymmetry 3 stringy er, to – certain )(2 (2 2 ) where 7 log to ersymmetric and ) 0 e ˆ | tension as ev exact. (2.3 itself hierarc oten duli + sup linear z ose X e ( w W p ( ) that 2 y | is throat, four-dimensional y ˆ is ersymmetry b ˆ ˆ V V W mo b ( Ho cancelled (0) ( ( imp scale olume non-sup include ˆ large K v out trusted W tial ortional the sup e order holomorphic four-cycle 3 tial ahler erturbativ log unless t e to a negativ ˆ p ˆ 2 V K nd and In anomaly then b = 3 to 4 where tial oten oin y prop the the is oten to the large t. p should V the p had 0 = scaling. p ole ) ultiplied is vit v-Strassler tial e on 24, s only oten at m to / anishes of V obtain w uxes stabilization tly ) w-energy = v that bined ed (2.15 e is tadp erp tial lo oten X can ergra to ected ( constan (0) p righ yp , com . viour of tial t 3 tribution 1 sup fact ] ] heme K teger the ) Klebano oten sup exp 0 complex-structure (3) “no-scale” correction dulus related sc in wrapp p [26 eha oten [27 con G e ( this wing is b the case all the the of therefore the the mo , ) the erp of y taining the the ), In a of ativ is umerical cancel M harge ux es follo to n . with sup w s c and form: con x since used g to energy a (0) harge (2.1 X ( giv ) c 7-branes ) e that authors / e y e the the deriv is K ) olume w b runa w K ottom authors to V A (2.4 wing D3 case, b The e en (2.11 order (3.9 its 4 eginning arious The of where Once Notice b tions tial three-form compactication Unfortunately 3.2 correction in solution and tak higher v follo the obtain in giv in the D3-brane Here fourfold the negativ JHEP05(2005)010 7 e h a y h is ts of of at in D au w er. net the the and n rst us fact, er oin efore pairs more unit Sitter whic (3.13) (3.12) (3.14) whic b and p pap no to b highly on obtain es In is 3 tion explicitly the to e strategy ) equation . 0 um duce the ed discretely w giv en ect the the n V olume wn v ti-de Calabi-Y ( done of h v measured tro ersymmetric pairs tibrane of dominates By e induce an tension kno same of in resp the con is b , annihilation the that pairs 7-branes from part wrapp if whic tensions are if to es and the of 3 small and will pairs / far quadratic 3 s unit terms 7 with units 4 this g e non-sup tibrane olume. at D case V olume e brane-an mak use that v in to deformations, , in a n W v harge tibrane needs wski D3-brane c s ]: -branes assumed o 7 ]. g cated 7-brane in the These e 7 en ativ 3 latter tributions e simple D / the / 3 s [28 lo and v 4 terms is 7 n nite [28 extra ahler g a step ) giv related con one 7 the Notice V , energy ) tersections brane-an h nd a ha t Mink K the + fact, deriv n q are 2 ts tial e in e therefore / 1. ork brane-an (3.9 are of at 7 negativ a 1 p, to w in In oin 3 w ( other D3-brane can these D whic = / of whose e p 2 s in er order ortan t 4 oten n 0 e g solving will b um of scaling p 0, V + ey w 2 triple of + tak e the ers k 3 negativ degenerates. 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Laura erstein K ering it olume w en non-sup uxes – um KKL v the enough inationary the Silv of as terms and that the tuning ersymmetry eectiv suggested 12 stabilization dulus answ a brok this RR the in y ers, – tibrane to dication b featuring is b expansion Ev sup Calabi-Y mo minim small new direct to for for inationary e a e and Dolan ], tibrane mo t to um ensure olume the the n the at hanism that demonstrated earing v an [32 e. to NS w-energy of the olume at discretuum compactication satisfactory in v lo Plesser mec app a ux tial y tibrane a Louise 7-brane-an relev that w-energy b corrections brane-an 4d ery erturbativ lo a, alternativ tuning the ersymmetry v construction 0 the v gratitude for p oten enough o attractiv y p of stabilized y terms the a Ronen the obtain as sup issues e our and w m the stabilized e b , non-trivial brane-an in h discussed seem in w demonstrated Bobk ey large er and more haracteristic another ahler ely k v as tial generated additional e ts c are suc order this stabilization can heme o the not t K stringy w ts en absence with acua sc In demonstrated pairs tuning. es v express oten ev of made Linde trol stabilization the e term and 4 from problem do hand, Euler e it ortan the Heather w to olume higher erp b ne presen to dS v con e es coupling in erturbativ coupling duli duli imp the of p ork sup example, au tibrane mass lik mass can wledgmen other parameters the w Andrei from mo obstacle. y mak mo bination ed solution the to h to string need the string some tributions Conclusions kno ould vided jor this a com hiev teger to w olume in in brane-an pro In compactication whic inaton Kobakhidze, metastable the me inaton Ac I con ma This the and On eects brane a ac simple, compactication corrections theoretical Calabi-Y where 4. In the v necessary JHEP05(2005)010 R B ]. ane 52 R 10 343 Phys. br (2004) 126009 (2003) = Nucl. string , Phys. with IIB Phys. D 08 manifolds and Phys. 03 , in . Nucl. d d (1996) , (2002) ory Nucl. ations -duality simple , JHEP 474 ]. 65 the ations curve JHEP actie , T ation a , ortschr. 072 ative B D F ination hep-th/0307049 on [ in ories , omp via actic string tori c actic ev. string vacua actic the and in ]. 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