Jhep09(2010)087

JHEP09(2010)087 UV Springer July 30, 2010 : August 31, 2010 : f the Klebanov- September 24, 2010 : S-CFT Correspon- Received 10.1007/JHEP09(2010)087 Accepted astable Klebanov-Strassler ng theory constructions of Published m for the force on a probe doi: tured metastable vacuum in y anti-D3 branes must have a m field strengths whose energy rved by a stack of anti-D3 branes not admissible, then our analysis , lution describes the backreaction of Published for SISSA by [email protected] , of the deformed conifold. We find that the only solution whose 3 0912.3519 S Flux compactifications, Gauge-gravity correspondence, Ad We solve for the complete space of linearized deformations o [email protected] CEA Saclay, CNRS-URA 2306, 91191E-mail: Gif sur Yvette, France Institut de Physique Th´eorique, [email protected] On the existence ofKlebanov-Strassler meta-stable vacua in Open Access Abstract: Iosif Bena, Mariana and Gra˜na Nick Halmagyi Keywords: dence, Superstring Vacua ArXiv ePrint: the Klebanov-Strassler field theory.strongly If this suggests singularity that is anti-D3vacua, branes which do would have notde dramatic give Sitter consequences rise space. for toD3-brane some met Key in stri to our ansatz. this result is a simple, universal for physics is consistent with thatsingularity in of the a infrared, perturbation coming from produceddensity NS diverges. b and RR If three-for this singularitythe is anti-D3 admissible, branes, our so and is thus likely to be dual to the conjec smeared on the Strassler background consistent with the symmetries prese JHEP09(2010)087 6 8 9 2 7 19 20 21 24 25 26 27 28 30 32 37 38 10 11 11 12 16 17 19 24 27 33 36 37 13 16 – 1 – = 0 1 X ˜ ξ 6= 0, UV and IR expansions ˜ ξ equations is non zero equations for 1 i 1 i X ξ φ X ˜ ξ IR, UV → → Equations equations i i 6.1.1 IR singularities 4.3.1 The4.3.2 space of solutions IR4.3.3 behavior UV behavior ξ φ B.1 IR behaviorB.2 of UV behavior of 6.2 IR 5.3 Anti-BPS 3-branes in Klebanov-Strassler 6.1 UV 4.4 The zero energy condition and gauge freedom 5.1 BPS three-branes 5.2 Anti-BPS 3-branes in anti-Klebanov-Strassler 4.1 Solving the 4.2 Solution for 4.3 Solving the 2.6 Imaginary (anti)-self duality of the three form flux 2.1 The first2.2 order formalism Papadopoulos-Tseytlin ansatz2.3 for the perturbation The zero-th2.4 order solution: Klebanov-Strassler 2.5 1.1 Strategy and results B Behavior of 7 Discussion A Conventions 6 Constructing the anti-D3 brane solution 5 Boundary conditions for three-branes 3 The force on a4 probe D3 brane The space of solutions 2 Perturbations around a supersymmetric solution Contents 1 Introduction JHEP09(2010)087 = 0 = 1 s g N stants in QCD brane persymmetric ity (AdS-CFT) ]. Another obvious 2 tions do not have the etric brane configura- tric branes, and hence nes, which source fields : essentially all MQCD le vacua in SQCD theo- menon does not happen. ound for massive l building, as these vacua stable vacuum. At , it sources an NS5 world- ry (where supersymmetry CD[ e theory. d source fields of different ertained by computing the onfiguration describing the persymmetric field theories -fashioned dynamical super- he NS5 brane to bend loga- UV, which is responsible for IB supergravity backgrounds uge theories engineered using arithmically, the tiny infrared cuum of a nonsupersymmetric nfigurations whose low-energy istic of four-dimensional gauge e D-branes and NS5 branes in 4 branes ending on NS5 branes; 6= 0 these non-supersymmetric configura- s g ], and soon thereafter in quite a large number 6= 0 this probability is zero because the branes 1 – 2 – = 0 this probability is zero because the branes s g s = 2 SQCD, this bending encodes the logarithmic g N at infinity. This bending encodes the running of coupling con /r ]. The D4 branes in the deep infrared of the susy and non-susy M ] that are dual to some of the SQCD theories via the gauge-grav 2 ]. The fact that the putative MQCD non-supersymmetric and su 3 4 ) SQCD in the free-magnetic phase [ It is also interesting to note that for three-dimensional ga For MQCD theories one can construct putative non-supersymm An obvious question posed by the ubiquitousness of metastab The key reason for which the susy and non-susy brane configura Nc Over the past few yearshas the begun analysis to of play metastablecan an vacua bypass important of many role su of in thesymmetry phenomenological familiar breaking mode problems (DSB). associated Such with old meta-stable vacua were first f 1 Introduction the susy an non-susy configurations not being vacua ofD3 the branes sam and NS5The branes D3 in branes type end IIBthat in string decay codimension-three like theory, 1 this sources pheno on the NS5 bra configurations end on thekinds. NS5 Since branes these fields in dodifference different not is decay fashions, at transposed infinity, an but into grow a log log-growing difference in the configurations are not vacua oftunneling the probability same between theory them: can for differ also by be an asc infinite amount, while for question is whether such metastablesuch vacua as exist [ in thecorrespondence. type I tions whose low-energy physics matches that of the SQCD meta of similar theories. ries is whether suchphysics vacua also yields exist these in theories.type the string-theory IIA co These string constructions theory, involv and are generically referred to as MQ SU( theory [ these brane configurations haveMQCD the supersymmetric same asymptotics vacua. as However,tions the for do c not have thedo same not asymptotics describe as a theis metastable MQCD broken vacuum dynamically), supersymme of but a rather a supersymmetric non-supersymmetric theo va rithmically. For the MQCDrunning dual of of the coupling constanttheories with [ the energy, character same asymptotics comes frombrane the descriptions phenomenon of of four-dimensionalsince theories brane the contain bending end D of D4volume brane field is that codimension is 2 inside logarithmic the at NS5 infinity, brane and that causes t have infinite mass. JHEP09(2010)087 ], is 10 this ] so- 11 M = P er (KS) solu- ng to the KS ] but in fact break 7 2 nts of these theories ], who argued that a , 5 6 able gauge theory vac- t work in gravity duals rently in the literature. , acua in four-dimensional ] and furthermore, both the 4 8 rmalizable modes. If this d the non-supersymmetric e-dimensional MQCD does on-supersymmetric theory. -BPS solution would differ logarithmic modes that are here exist non-backreacted ne configurations decays at urce-free KS solution. The e dynamical but explicit. a metastable DSB vacuum of nfigurations. As these modes It is possible that a putative garithmic modes in different le vacuum of a supersymmet- rangian of the supersymmetric cua [ the Klebanov-Tseytlin [ hey use the fact that the force y. Since the difference between acted solution would allow one l gauge theories, it is only four- 1 en in [ non-supersymmetric vacuum of . s must contain fields that depend ime of parameters. ] such a field is given by the integral of the NS B-field ]) as well as other duals of KPV-like configurations [ 3 9 – 3 – that the anti-D3 branes source only a normalizable a-priori ] (analyzed in [ 5 ] once backreaction is taken into account. 4 ] was in the probe approximation and it important to extend th 5 ] assumes D3 branes at the bottom of the throat. In particular, if 12 P − anti-D3 branes placed at the bottom of the Klebanov-Strassl M units of 3-form flux can decay into a BPS solution correspondi P M = 1 four-dimensional gauge theories. As the coupling consta In this paper we will analyze whether a similar mechanism is a Thus, the mechanism that ensures the absence of metastable v Note that this question is in no way settled by the results cur The best-known candidate for the IIB gravity dual of a metast In the well-known Klebanov-Strassler solution [ Worldsheet evidence in support of this MQCD analysis was giv N 1 2 brane configurations that are candidates for meta-stable va to a backreacted supergravity solution.to establish Finding whether the this backre ric configuration theory, is or indeed it aThis is metastab rather necessity a is non-supersymmetric particularly vacuum evident, of given a that n in MQCD t tion with solution with argument presented by [ collection of anti-D3 branes could decay into the smooth, so run logarithmically with the energy,logarithmically all their on gravity the dual AdS radius, at least asymptotically sourced differently in the infrared forare the present BPS in and non-BPS four-dimensional co butdimensional not MQCD in which three does dimensiona nothave have metastable such vacua, vacua. thre of the fields of theinfinity, non-supersymmetric the and two do supersymmetric correspond bra configuration to is vacua really of a the 3D same MQCD theory, metastable an vacuum. MQCD appears to rely on just one ingredient: the existence of three dimensions, which is linear in the inverse of the energ on the 2-cycle of the conifold. MQCD dual of the configuration ofsuffer [ from the same problems in the UV, at least in certain reg supersymmetry explicitly [ For example the ultraviolet perturbationlution expansion presented in around [ uum has been proposedcollection by of Kachru, Pearson and Verlinde (KPV) [ indeed happens, then the non-BPS solutionthe will supersymmetric not field be dual theory. to Instead,a it non-supersymmetric theory will obtained describe by a perturbingtheory. the Lag As a consequence, supersymmetry breaking will not b metastable configuration in themanner infrared than the couples supersymmetric toin configuration; these the hence lo ultraviolet the from non the BPS solution by one or more non-no mode, and uses the consistency of the result (in particular t JHEP09(2010)087 ], ], × 5 14 of the normal- SU(2) 3 -invariant S × 2 by explicit Z × 6.1 , and thus could on the 4 ], who reduced the n a six-dimensional /r SU(2) 16 × section ds that are sourced by rassler solution, as well ]) to argue that such an de, then the backreacted d that could amplify the neral solution preserving n [ ldacena and [ N˘astase heories also have MQCD 13 ty as 1 her zero or has the precise ns into a non-normalizable lebanov-Strassler geometry d on three or four variables. able mode, and hence would ate arena for studying these ources in the internal space, decay at least as strongly as n [ echnology. reacted supergravity solution ponding to anti-D3 branes in nti-D3 brane. In addition, we ar in the inverse of the energy, ]) into D5 or NS5 branes [ er solution, and to see whether differential equations for eight modes. To do this in generality es whose coupling constants run ar) Klebanov-Tseytlin geometry , all SU(2) eakest of assumptions about the es that behave asymptotically as 15 olution, whether normalizable or ion preserves the SU(2) 3 gauge theory, for which none of the arguments – 4 – of the KS solution. τ 2+1 dimensional ]. Hence this force calculation alone will not determine the 13 . If the anti-D3 brane in the deep infrared couples to such a mo . Furthermore, as we have argued above, both the Klebanov-St Our purpose in this paper is to establish the nature of the fiel One way to simplify the problem is to smear the anti-D3 branes One can argue that because an anti-D3 brane is a point source i It is also interesting to note that the only known fully-back 3 r symmetry of the original solution. The ansatz for the most ge 2 /r Z this isometry was written down by Papadopoulous and Tseytli first-order perturbations around thenon-normalizable, Klebanov-Strassler s give a forcebehavior on predicted a in [ probe D3 brane that is eit one would need tothe find warped the deformed fully-backreacted conifold. solutionand corres Since since these moreover branes are they point polarize s (by the Myers effect [ tiny infrared difference betweenUV the mode. BPS and non-BPS solutio anti-D3 branes placed at the bottomthese of branes the give Klebanov-Strassl rise to normalizable or non-normalizable presented above applies.metastable DSB As vacua, we and their haveand coupling discussed hence constants do are above, not line correspond these to t bulk fields that grow in the UV an logarithmically with the energy, contain supergravitylog mod as other backgrounds dual to four-dimensional gauge theori on a probe D3 brane is not inconsistent with that calculated i supergravity equations of motionscalar to functions second-order of ordinary the “radial” variable the fields of such aFinding non-supersymmetric such a solution solution would is depen clearly beyond hope with current t describes a vacuum of a calculations, this intuition is notbackreacted correct solution even sourced under by the anti-D31 w branes, which must will point out that subleadingdo perturbations not of in the (singul factand correspond as to such the perturbationsissues. Klebanov-Tseytlin of geometry the is (regular) an inappropri K space, the fields sourced by such a brane should decay at infini assumption is correct. However as we will show in section izability or non-normalizabilty of the modes sourced by an a deformed conifold, so that the resulting backreacted solut not correspond to non-normalizable modes. As we will show in corresponding to a DSB metastable vacuum, constructed by Ma solution would differ from thenot BPS correspond solution by to a a non-normaliz metastable DSB vacuum. JHEP09(2010)087 ]. These 13 fields than the nother one must less d of the Klebanov- ation. This conden- atures of the anti-D3 arametrically large, and f eight coupled nonlinear t second-order equations ane that drives inflation. re to say whether it will -normalizable modes, the ominant compared to the ions to give a solution to es cannot remain smeared sibility that the localized . Furthermore, because of tring phenomenology and he potential existence of a the AdS vacuum giving rise give rise to consistent super- discussed before, when these hies. he tip. Hence, one can study om of the Klebanov-Strassler ts by polarization, and hence oblem tractable one must use und the KS solution, factorize throat to the ambient Calabi- for the localized and polarized -Strassler throat that is glued ed system. Note that this fac- es couple to ent in the KKLT construction e branes bring is parametrically ion around the supersymmetric e smeared branes do not source nstants. In fact, as we will show tentially-dangerous mode that is utions that we construct depends – 5 – ], the second-order equations satisfied by the per- 17 ] and in the KKLMMT model for string theory inflation [ 18 , one of these integration constants is a gauge artifact and a 4.4 , but will eventually condense into a single-center configur 3 S Fortunately, as shown in detail in [ Even after smearing the anti-D3 branes, solving the system o A non-normalizable mode would “climb” up all the way to the en Before beginning it is worth discussing the implication of t One might object that this smearing washes down important fe the fact that thefields fields of sourced the background, by at thethese least fields anti-D3 in by the branes performing region are a farKS subd first-order away solution. from perturbation t expans localized branes will very likelynon-normalizable do the modes, same; this however, ifpolarized does th branes not will source exclude them. at all thedifferential equations pos is no easy task. To make the general pr localized polarized ones, theirnon-normalizable potential in for the sourcing ultraviolet a willanti-D3 po be branes. much lower than Hence, if the smeared branes give rise to non Furthermore, one generally assumes that the extraredshifted energy in th the KS throat, and can give riseStrassler to throat, mass and hierarc willYau. probably affect However, the as gluing of this this gluing is not understood, it is prematu solution. Suchstring anti-branes cosmology are constructions: partof they of de are the Sitter a space “staple crucialconstructions [ diet” involve ingredi placing of anti-D3 s to branes an in ambient a Calabi-Yau. Klebanov to The anti-branes de-Sitter lift vacua the and energy can of create a potential for a probe D3 br be fixed to zeroEinstein’s in equations. order Hence, the foron class the fourteen of solutions physically-relevant perturbative integration sol to constants! the sixteen equat non-normalizable mode sourced by anti-D3 branes at the bott turbations of the eight scalarinto functions sixteen of first-order the equations PTtorization of ansatz aro which does eight not form throwand a away the clos any sixteen of first-order the onesin have modes: section sixteen integration both co the eigh branes that give the putativeanti-D3 metastable branes vacuum. are Indeed, together as theycouple acquire to extra more dipole supergravity momen the fields broken than supersymmetry the one smeared canon D3 argue the branes that the anti-D3 bran sation however takes place overhence a does not time affect that the cangravity fact solution. the be that Furthermore, tuned smeared as to anti-D3 the branes be smeared anti-D3 p bran JHEP09(2010)087 × anti-D3 SU(2) × N to solve these ]. We would like 18 then identify which ontribution of these n-normalizable mode ] it was assumed that all by lengthening the 18 the ambient Calabi-Yau. m of the KS throat, but ll on the presence of extra D3 branes to the bottom 3 brane sources. t are relatively simpler to l contain both a redshifted bution to the energy could on-redshifted contribution, is involves solving the eight ot to overrun the quantum behavior. This amounts to tially small by placing them do not. Our first strategy is heless, if a non-normalizable g theory [ or the addition of tter vacua in string theory. ry contains fields that diverge meared in an SU(2) rmalizable modes, this energy This without doubt negatively . In [ ties that come from explicit D- branes source non-normalizable r equations factorize into sixteen t govern the metric and fluxes of s a perturbation of the Klebanov- , exactly as needed in the KKLMMT 5 ) these singularities do not appear to , the force felt by a probe D3 brane in 3 /r 6.1.1 – 6 – ]. We simplify the problem by solving this system ]. 16 13 units of additional Maxwell charge, that there is a non-zero N − ]. One of the main achievements of our paper is that we are able 17 It would be interesting to explore whether this extra contri On the other hand, as we will show in section As we will discuss in detail in subsection ( Having set up this rather heavy computational machinery, we -invariant way at the tip of the warped deformed conifold. Th 2 Z the Papadopoulous-Tseytlin ansatz [ second-order coupled nonlinear differential equations tha invalidate the KKLT construction of de Sitter vacua in strin mode is present,rather its up energy the will throat,Hence, not the at be extra the energy localized junction broughtcontribution, about at coming between from by the this the an bottom anti-D3 botto throatfrom of brane and the the wil KS junction throat, region. and a n be negatively affected by the non-normalizable mode. Nevert have a distinct physical origin, as opposed to say singulari branes in the KSrequiring background and that investigate there the are resulting IR of equations perturbatively, treating the anti-D3Strassler branes solution, a and using thefirst-order fact linear that the ones second-orde outsolve [ of which form a closed system, tha 1.1 Strategy andOur results goal is to identify the modes sourced by anti-D3 branes, s impacts the existence of large parts of the landscapethe of background sourced de by Si anti-D3non-normalizable branes modes. does It not is depend always at of a the form 1 the contribution of these anti-D3in branes an can arbitrarily-long be made KSmodes, exponen throat. the Nevertheless, if energy anti-D3 throat. they Hence, bring in some cannot circumstancesmay be be and too for large made certain to non-no allow exponentially moduli the K¨ahler to sm be stabilized. to remind theof reader a that KS in throatanti-D3 this in branes construction to order one the to totalcorrections adds lift energy that anti- the needs are to AdS needed be vacuum small to to in stabilize de order the Sitter. n moduli K¨ahler The c force on a probewhich D3-brane we allow in for. the We findin UV that a the and resulting subtle the infrared manner geomet charge which is seems the to be only incompatible no with anti-D model for inflation in string theory [ equations exactly in terms of integrals. parameters correspond to non-normalizable modesto and enforce which the correct UV boundary conditions appropriate f JHEP09(2010)087 n A first n first order to 2 upergravity n τ . Appendix hen perturbing 7 erturbation theory ttempt to construct ditions in the infrared . For this procedure to a Whether this singularity ξ ranes in KS and since the upersymmetric linearized equence we argue that in other that it is of utmost ], who reduce the set of 3 n 4 we solve for the modes l, variable. viously always be done, the ions. If the singularity is to e KS boundary conditions. 17 d anti-D3 branes in the KS ter, as their divergent energy deemed inadmissible then we annot match that of the KS linear second order equations s singularity in the discussion d singularities must normally sults in section nse. n the KS boundary conditions iscuss the boundary conditions n und a supersymmetric solution, amine the consequences for the y problem are sufficiently strong tten as a system of 2 of the singularity which we have s finite is the negative-mass Schwarzschild – 7 – shows the asymptotic expansion of the auxiliary B depending on a single (radial) variable a φ fields n and their “canonical conjugate variables” a φ ] allows eight of these equations to form a closed subsystem w 17 ]. In section 3 we compute the force on a probe D3-brane in our s 17 This paper is organized as follows: In section 2 we set up the p There can of course only be one correct solution for anti-D3 b Our second strategy is to enforce anti-D3 brane boundary con Since we are perturbing around a known solution, the The best example of an unphysical singularity whose action i 3 solution. presents our conventions and appendix density integrates to a finitebe excluded action. in order However, for even the such gauge/gravity mil duality to make se brane sources. On the other hand, they are milder than the lat second order equations for 2 Perturbations around a supersymmetricTo solve solution the IIB supergravitywe equations will perturbatively use aro the method developed by Borokhov and Gubser [ for D3 andthe anti-D3 backreacted branes solution in correspondinggeometry. our to We conclude backgrounds. and a discuss stack the In implications of of section smeare our re 6fields. we a section. following [ ansatz and show thatand is present has their a perturbative expansions. simple universal In section form. 5 In we d sectio this case the correct UV behavior is a largetwo deformation conclusions of we th have offeredimportance here to differ rigorously so settleuncovered. the wildly We issue from present of each various arguments admissibility for and against thi argue that there is noin solution the which UV can and interpolate anti-D3 betwee brane boundary conditions inwhile the disallowing IR. the aforementionedultraviolet. singularities and We ex showsolution that with the no ultraviolet non-normalizable of modes this turned solution on. c As a cons is to be allowedbe or allowed, not then gives we risesolution argue to for that two anti-D3 we very branes have distinct in found conclus KS. the If correct the non-s singularity is to be from linearizing Einstein’s equation can of course be rewri equations without losing anyformalism information. of [ While this can ob work, it is essential thatso the as symmetries of to the just supergravit allow a dependance of the fields on a single, radia order equations for JHEP09(2010)087 (2.7) (2.3) (2.8) (2.4) (2.6) (2.5) (2.2) (which ), while a 0 φ 2.6 is measured . a 1 in (  , φ c a ξ ∂τ ∂φ ∂W ∂W 2 1 bc G − otion reduce to a set of is linear in them. The  be written in terms of a  a 1 ]) d c 3 φ erturbation ∂ ∂φ ∂φ ∂W 2 1 eory working directly with the ion [ ac . , ) (2.1) ≡ ) b G φ 2 ξ . ( d , 2 1 b b ab X V , ( − G ∂φ ∂W = 0 − O a a b = b dτ , hence they are of the same order as the dφ ) = 0 dτ b 1 ) + dφ ∂φ ∂W ∂φ X 0 ∂W ,M φ a φ ) X ab ab ( ( 0    ]. dτ a a 1 G c dφ φ d 1 G b – 8 – ( φ 8 1 17 φ 2 1 b ab ) ∂φ ∂W a M + 0 b G ]. − φ ξ ac a 0 1 2 M ) = ( ): a φ 19 G d φ a + b − − ( dτ 2 1 dφ = a satisfies the gradient flow equations , φ a 1 V M = a a − dτ a 0 dξ dτ ξ φ L − dφ φ a b 1 , given by dτ a = dφ dτ ξ dφ a  φ ) are just a rephrasing of the definition of the  vanish the deformation is supersymmetric. ab ) 0 a G 2.8 ξ φ 1 2 ( − ab are expanded around their supersymmetric background value G = a φ ≡ L a ξ represents the set of perturbation parameters, and ) imply the equations of motion [ are linear in the expansion parameters X a 2.7 ξ The main point of this construction is that the equations of m The fields , and when all the a 1 first order linear equations for ( φ supersymmetric solution Note that equations ( where will correspond in our case to the Klebanov-Strassler solut The second order equations appeared in [ around a supersymmetric solution. Related perturbation th eqs. ( 2.1 The first orderThe formalism starting point is the one dimensional Lagrangian which we require tosuperpotential have the simplifying property that it can by the conjugate functions while the deviation from the gradient flow equations for the p where JHEP09(2010)087 ).  2 ′ 2.6 (2.9) ( Φ (2.11) (2.10) (2.12) 1 4 a , ξ , transform 2 5 )+ or the PT ) g 2 ′ b 4 x a g F − , . p , M ) ∧ 6 ) of the 2Φ 4 5 2 − e g g g e kF ∧ ∧ +2 + 3 4 symmetry which are . 2 ′ ) + 3 g g ) + ′ 2 A 4 g 2 k rding the choice of def- . So we are looking for k g y ∧ ] F Z 0 ∧ 2 3 have the liberty to smear e + + C 1 16 definition − g × g 2 2 3 + ndix ( ansatz consistent with these ∧ g g P 2 otion that the anti-D3 branes ′ ( he nate after having derived the oes not alter the definition of 2 ∧ ∧ y f (2 g ally the other terms just get a y − 1 . f 2 x ∧ SU(2) g dτ ( ′ ′ , and a prime denotes a derivative − e in the ten-dimensional ansatz, or 1 x e τ f and the connection × F g 2 Φ) ( 8, in the following order ( τ x − ) + + )) 2 ∧ p ab 2 2 ) is 5 − F g G +4 g Φ dτ ,..., A − − 2.1 + ∧ 4 e symmetry. We are interested in a solution e 2 1 4 P 4 1 = 1 ) + g 2 1 g 2 . The fluxes and dilaton are 4 ( (2 a + g y τ Z ∧ + f – 9 – 2 , + ′ 3 ∧ a x y × g + A e φ 2 ) 6 are functions of g x,y,p,A,f,k,F, + F symmetry (which exchanges the two copies of SU(2)). − kF F + cosh 2 2 ), this will lead to a significant alteration in the equa- 2 − 3 . If one redefines p ′ . Of course both such redefinitions are completely le- = ( SU(2) ) in which case the only transformation comes from the Z g 1 dτ p τ = ( a P x +4 2.1 − and × ∧ 5 φ ) 2.8 A − ) since the metric +6 1 4 p F 2 6 e g ′ . + (2 ( f, k − 2.8 y 5 + e ˜ τ/∂τ 1 2 ∧ g x = 0 ∂ 5 + 2 , ) and ( + ∧ 0 g 3 2 − 5 ) , ′ 2 p 2 1 2.7 f x 2 g ) and ( ∗F −  ,C ds ∧ − ) A x + . There are several different conventions in the literature f A 1 4 2.7 τ k − τ e ( 5 p +4 p 2 1 F F g 4 +2 e ) = A = = = 2 φ = e ( 3 5 3 Φ = Φ( b is a constant while F F ′ H = W φ P a ′ . 2 10 a φ We will denote the set of functions At this point it is worth emphasizing an important point rega ξ ds ab G The metric in the one dimensional Lagrangian ( gitimate, however they arethe not equivalent since the latter d nontrivially. Moreover this also leads to an alteration in t continuously connected to the KSsymmetries solution. was The proposed most by general Papadopoulos and Tseytlin (PT) [ The KS background has SU(2) 2.2 Papadopoulos-Tseytlin ansatz for the perturbation equations of motion ( derivative with respect tomultiplicative the factor radial of coordinate ( so essenti tions of motion ( Alternatively one might choose to redefine the radial coordi and the superpotential is a family of non-supersymmetric solutions with SU(2) Furthermore one can see from theand supergravity equations the of fields m they source do not create a source for the axion inition for the radial coordinate for the backreaction ofthese a branes smeared without set breaking the of anti-D3 branes and we where all functions depend on the variable equivalently in the Lagrangian ( where with respect to ansatz, we compare our conventions to those of others in appe JHEP09(2010)087 ] 11 ue to (2.14) (2.13) 4 ) (2.15) π becomes zero. The (2 ] found a particularly dt . / lin (KT) solution [ h 3 eformations of the KT 20 / 1 the KS solution with the τ , = 3 of radial variable can lead  2 t d we will employ the same ion, we tabulate the form of . − is must be kept in mind when where , , , a 2 ) sinh t  , P s 1) 3 4 r / 3 on it is not reasonable to compare / 2 / + 1) − 2 ]) which have subleading terms are ) 1 of the KS solution. 2373 . τ τ τ ,  a 0 12 sinh(2 aP τ 3 − φ / 1 2 = 18 1 −  ) τ τ  ) + ) ]. The authors of [ τ τ 0 dτ τ 1 3 1)(cosh 1)(cosh / 20 − τ t 1 − , sinh sinh ) r/r 4 for more details). ) − − 2 2 ) t τ r τ τ τ τ – 10 – sinh(2 4 A ln( − , sinh(2 ) with the 2 1 τ ) 2 − sinh sinh ]. The KT solution has a warp factor ] and [ 1 2 τ coth 2) 3  τ 2.8 h coth coth t  aP 17 sinh(2 h τ/ 2 τ τ 0 τ sinh 2 1 / ( ( 9 + x 0 1 sinh(2 2 Z 1 (sinh h · 1 P P 2 +2 N 2 0 ( 24 ( 1 3 4  ) and ( − − P p 1 P 4 π − τ τ − ======0 = tanh( = 2 32 2.7 0 27 0 0 0 0 0 0 0 0 A − y p sinh 4 x coth f k A F Φ 6 e e 0 τ 6 − e ) related to the warp factor is given by the integral ) = e e h but the subleading behavior of the KT solution is ambiguous d r ∞ τ 0 ( ( 3 R = = h 2 τ/ h e KT P h = = 32 r 0 h There is an important difference between the Klebanov-Tseyt This is an important consideration since a judicious choice We have 4 not in general solutions of ( and the Klebanov-Strassler (KS) one [ Since we are expandingthe around zeroth-order the functions Klebanov-Strassler here: solut to a significant simplification incomparing the the equations radial of coordinates motion. innatural Th [ choice such thatchoice the in the equations current simplify work somewhat (see appendix an 2.3 The zero-th order solution: Klebanov-Strassler where the function leading UV behavior ofidentification the KT solution does in fact agree with which has a naked singularity at some finite distance ability to absorb a shiftthe into the subleading singularity. behavior For this of reas solution the (in KT particular the and deformation KS proposed solution in and [ in fact d JHEP09(2010)087 ) 5 8 ˜ φ , 7 ˜ φ (2.19) (2.20) (2.26) (2.25) (2.21) (2.24) (2.16) (2.22) (2.17) (2.18) (2.23) , 6 ˜ φ . , 5 7 ) ˜ ξ φ 8 ) , ] this system ) 0 , ξ ”  4 k 8 7 ˜ 6 20 φ ˜ ξ ˜ ξ , 6 , ξ − ) equations, which 7 + 6 ˜ 0 ξ 0 1 ξ j , 1 F f ˜ . ) φ ˜ φ flow). This solution ) ξ ( 6 − ˜ − 0 ξ
2 1 olve them) ∗ ( Φ) y 5 ) 1 3 + 0 ] this system was solved = 0, which remarkably , + , ξ 5 k ) ˜ ξ 6 = 0 4 6 4 17 ( rposes. In [ ξ ˜ ˜ rturbations. ξ ξ ll of these constraints and φ this work is to connect the − 1 2 ) 5 1 , N sinh(2 0 0 + ˜ ) ξ = f d redefinition − 6 F ) 5 ˜ ( − ξ ) . In [ 0 3 n generated by a mass term for 0 3 i A, f, k, F, 0 ˜ 0 , ξ k φ + ξ ξ F 2 y y 4 5 2 ξ indices down. − ˜ ξ + 3 − − 2 = ( 0 1 2 1 0 P e p f 1 ˜ , − ) for φ . ( 2 ξ 4 0 τ 3 ˜ ξ cosh(2 (2 P f + ( x ξ − 2.7 1 2 ) 15 − 5 − (2 4 ˜ − , + ξ 1 0 ˜ ξ 0 ˜ 2 e ) ξ x 1 y ˜ p, x 0 φ 2 6 1 − ξ 2 , + 1 e , F ) 3 ˜ ξ 3 ) 3 4 1 + ˜ ξ ˜ 0 + 3 ξ ˜ P e − − ˜ φ – 11 – ξ )) 6 y , 0 + ˜ 0 ξ 4 + p )) ) 1 k ξ + ( + 5 6 0 ˜ ξ 1 0 1 ˜ 3 , 5 k − ξ ˜ y ξ − − ˜ ˜ 2 0 ξ φ ) ˜ ξ ) x A, y, x 0 (2 3 0 − , cosh(2 2 0 0 ξ 5 f ) F 0 + 2 ( 4 − y F ˜ − f ξ 0 ) e 1 − -equations in general, it is convenient to pass to the basis ( − 0 ˜ 5 0 φ F i + 2 cosh(2 0 + p y y 2 ξ 1 3 2 2 P F sinh − ˜ − − ξ ξ ( 8 ( + 0 3 0 ˜ 5 3 1 − − ξ 1 5 + P e x 1 ˜ 3 ) ξ ˜ ( − ˜ 2 0 ξ ξ 0 1 ) x ˜ , + ξ p 0 0 P f  0 − sinh(2 = ( 2 p x y e 0 + (3 6 P f 2 (2 0 a 0 y = ( − 3 1 1 3 , ξ − − +4 0 φ A 4 (2 0 0 “ a x ξ y x 4( 0 ˜ − 2 φ − 2 2 P e = x − − 7 a + 2 cosh − sinh(2 3 1 ˜ e a ξ e φ e ξ − 2 3 P e 4 ˜ − − e − − ξ ξ − equations are solved, one can insert the solutions in the − = = = = ( = 3 = = = = i 1 ′ ′ ′ ′ ′ ′ ′ ′ 6 7 1 8 3 4 5 2 ξ ξ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ′ 8 ξ ξ ξ ξ ξ ξ ξ ξ Equations equations ˜ φ (3 i i ξ φ ≡ Then the system of equations is (in the order in which we will s We find that to solve the a The inverse is The inverse is ˜ ξ 5 6 we write here explicitly. We start by performing a linear fiel The first step is to solve the system of equations ( 2.4 where all the fields on the right hand side are the first order pe Once the where a prime indicates derivative with respect to 2.5 perturbatively in the UVUV and and IR IR seperately. asymptotics, Since thatwas our analysis solved goal will in analytically not suffice with for the our pu assumption that The equations in the order in which we solve them, are and to avoid confusion we abuse notation at from now on put all the gaugino in the dual field theory (often referred to as the corresponds exactly to the non-supersymmetric deformatio is still not generalconsider the enough most for general our solution purposes, space. we need to lift a JHEP09(2010)087 ry he (2.36) (2.35) (2.37) (2.34) (2.32) (2.33) (2.27) ]), it is 5 23 [ ˜ φ ) 5 0 ) (2.30) 6 F ˜ ξ AdS − + P 5 ) (2.28) ˜ ξ 4 ( (2 ) (2.31) ˜ ξ ) 0 6 0 x ˜ ξ y 2 + + 2 − − 0 e 3 1 x 5 ˜ ˜ plex three-form flux of the ξ 2 1 ξ e ˜ ξ y is ) rmations of ) (2.29) ( warped Calabi-Yau solutions 0 . It is important to note that . 4 ) s, it is worthwhile considering p )+2( ) ˜ 0 ξ + 5 0 . y ) + + 7 sing the perturbed metric. f spect to the background metric. p ˜ 1 0 3 4 ξ − ˜ ξ + 0 ˜ A 0 φ − + 3 0 H x , x 5 4( (9 A 1 k 3 ) ˜ ξ − ( 0 4( − )+2 ( e H p Φ ) 0 )+2( = + − 3 1 3 0 0 p + , , . − e − ˜ p φ p 3 0 e + 2 ′ ′ 2 + + 0 A F − 1 2 = f k 0 0 A − 4( 7 = 0 ∗ − 3 A A ∓ ∓ ∓ ˜ 4( − 7 φ ) = 0 ) = 0 7 1 F Φ 4( 4( ) e ˜ ˜ − φ 6 6 ξ ˜ e 0 φ = = = − ˜ ˜ − ∗ e 0 1 6 ξ ξ f e e y ′ 2 y y – 12 – 2 Φ 2 1 6 − + 2 )(2 2 − e e F − 2 − 0 e ˜ 5 5 − ξ 2 − ) 0 ˜ ˜ + ) ξ ξ F ˜  φ 0 k ) ( ( 0 F 2 8 ) + ( p ISD : F e 7 0 k ˜ ˜ 0 8 φ φ ˜ + y φ Φ − x ) ˜ 0 0 φ − e 2 0 y A ISD : − f − ) + P + 4( ) 0 e sinh 2 8 (2 − − 1 2 F anti ˜ ˜ φ φ e sinh Φ 0 − e 2 − k + 3 − − ) and the fact that the zeroth order three-form flux is imagina )(2 6 ˜ ]) and one might be tempted to consider the possibility that t φ − P 2 3 ˜ 0 φ P + ( ˜ ˜ 2 φ φ x 22 0 2 (2 ˜ 6 0 2 φ 2.26 0 , F (2 x ˜ φ 0 − f 0 − 2 0 0 ( y x 21 p − 0 )–( F 0 2 − 6 0 ( x F − p 5 0 2 − ( 6 e y ˜ φ e 0 − 2 − cosh 2.30 2 1 y 3  e e − 2 1 1 2 5 e − e − + ======2 ′ 7 6 ′ ′ 4 ′ 2 ′ 5 ′ 3 ′ 1 ˜ ˜ ˜ ˜ ˜ ˜ ˜ φ φ φ φ φ φ φ In our conventions, the condition for imaginary self dualit Using eqs. ( In terms of the PT ansatz, these conditions lead to and the condition for imaginary anti self duality is self dual, the conditions that the first order flux be ISD becom flux turned on by ansince anti-D3 brane the is metric imaginary is anti-self deformednot dual consistent at to first consider order the self (asInstead, duality opposed one of to should the flux defo analyze with the re condition for self duality u of IIB supergravity [ 2.6 Imaginary (anti)-self dualityBefore of moving on the to solving three thethe form first conditions flux order system for of self equation KS duality solution of is the well known three to form be flux. imaginary self The dual (as com are all with the upper sign for the ISD condition. JHEP09(2010)087 τ ]. It (3.1) (2.38) (2.39) 24 d [ symmetry in 2 Z ) to be constant. . 8 ]. ˜ ξ ,  , ) 26  4 0 7 ˜ , ] does not have purely ξ are of course more in- f ˜ φ probe D3 brane in the 25 f motion, most terms in 17 orce in the unperturbed − ). [ 0 s we will find in the next terms to be proportional to ) + der for the first-order per- , be equal to k 8 ) have ISD three-form fluxes ed in the perturbed solution,  ( ˜  2.37 on a probe D3 brane is quite φ 7 8 t both the Dirac-Born-Infeld ure of the presence of anti-D3 . Allowing for a delta-function ˜ ˜ in also contains even powers of tions [ φ φ 2.8 + ... 2 − ), ( ˜ + ) + φ ) 33 8 5 g ˜ 3 ˜ φ φ 2.7 )(2 22 τ = 0 and force ( 315 , g − P − x 1 2 2 2 11 6 ˜ ξ − ˜ g ˜ − φ φ − p ( τ 00 (2 0 15 2  g 0 +4 0 F ’s vanish. If one relaxes the x ( F A − + ξ 2 4   – 13 – √ e − 1 τ 0 0 ) ]. One can also see this by examining the small- x x 0  2 2 = = p 2 24 − − + ) ) C 0 0 0 p p A + DBI ) implies certain conditions on the integration constants + + 4( 0 1 0 e V A A C 1 2 4( 4( 2.38 e − e = φ = ), which are actually weaker than ( ) = ) = 7 7 ˜ 6 6 ξ ˜ ˜ ξ ξ X multiples the free Green’s function of the deformed conifol 2 + − = C 5 5 6 ˜ ˜ ξ ξ ( ( X − = 5 X = 0, The DBI action for the probe D3 brane is It is interesting to note that all BPS solutions of ( The conditions for the first order deformation to be anti-ISD 1 . Setting those to zero and demanding the coefficient of the odd X the PT ansatz then there exist BPS solutions with non-ISD flux Hence, the generic solution to the system of first-order equa simple. perturbed solutions. AsKlebanov-Strassler is solution. well known, It(DBI) and such appears the Wess-Zumino a at (WZ) actions brane first arehowever rather feels we glance complicat show no tha below f the that total by action using cancel the and first the order final equations expression o for the force branes: the force on a probe D3 brane. 3 The force on aBefore probe solving D3 the brane above equations, we compute the force on a One can try arguingturbed that solution these to conditionssection correspond are there to necessary exists anti-D3 in a much or branes. simpler and However, more physical a signat It is easy to see that these immediately lead to volved: those in the Green’s function( ( since for supersymmetric solutions all the where the constant turns out that the fullτ expansion of the dilaton near the orig ISD flux, contrary to recentexpansion claims of in the dilaton. [ ISDsource fluxes do at not the source the origin, dilaton the dilaton in an ISD background should JHEP09(2010)087 : 1 ˜ ξ (3.2) (3.3) (3.7) (3.6) (3.5) (3.4) ) ′ 1 x 2 . − is 1 ′ 1 ˜ ξ τ p 0 is action with x  )] 2 ) 1 4 + 4 − x ˜ e φ ′ 1 4 . ating the RR field of the RR potential 3 2 A 10 )] − , 0 ) 1 − F ′ 4 + 4 p )] + 3 ˜ φ , ′ 0 0 ˜ − φ 2 ) x . F 2 + 4 butions to the force cancel, , he term proportional to ′ 0 ) 2 x ) cancel against correspond- P ) − 4 ′ 4 1 x − 1 − ˜ ) 2 − − φ (2 A x ′ 0 1 p 3.5 1 2 P . ′ 0 ˜ x φ f − on with respect to 1 p − +4 2 ˜ (2 ξ − ′ 0 we get ))(4 1 1 0 0 A ) p 0 4 x p x − f ′ 0 ) + ′ 4 ))(2 + 4 2 4 e F 0 0 ′ 0 ˜ x φ ′ 0 − + − f p )) + 4 2 F 0 e − ) + 1 A p ′ 0 F 3 0 2 − − − A f P A + 4 +4 0 ′ 0 )(4 − 0 = k P ′ 0 p 1 (2 (4 − ( A 0 0 x P ). A 4 1 (2 0 f x 2 e 0 k F 2 = 10( (2 + 4 WZ (4 f ( )) + − 4 − f 1 4 ′ 0 – 14 – F + 0 0 2.25 ˜ φ 0 ˜ + 1 p φ F A 0 F + 2 p F + 0 WZ τ F 0 +4 10 + (4 − in ( F − dτ 1 0 k 4 ( 0 0 k kF 0 ˜ A + 4 4 − φ [( [ dV k ( (5) P F 4 x 0123 ˜ DBI 2 ( φ 1 3 0 0 2 1 e ). Hence to first order in perturbation theory, the only − − F ˜ x (2 x 5 1 φ k F − − A 4 4 [ − 0 ( 0 2  f p 0 − − 0 = = = 3.6 = 0 0 0 x x = − x p p 4 2 + +4 4 , we have that 1 F 0 − − i 0 ˜ − +4 +4 0 0 (1 + 4 φ WZ A ˜ φ 0 0 0 p p F 0 4 2 DBI p 0 F 0 x A A e 2 4 4 k +4 +4 F +4 0 e ( e 0 − − 0 0 A A − − − A p 4 4 4 e e e +4 = = + − 0 DBI DBI 0 dτ dτ A 4 = = 2 dV dV e WZ WZ 1 0 . It is however easier to compute instead the derivative of th is the DBI action in the unperturbed solution, − − − F F , which gives the force exerted on the D3 brane by the RR field: (5) DBI τ 1 = = = F F DBI 0 V DBI along the worldvolume of the brane, which we obtain by integr To compute the form of the WZ action one needs to know the value Using this, as well as the equation of motion for F (4) respect to where and in the first-order expansion, the derivative of this acti and we have used the definition of ing terms from thecontribution DBI to force the ( force felt by a probe D3 brane comes from t C strength thus the first and the second term of the Wess-Zumino force ( From the definition of the The zero-th order and first order WZ forces are as expected for a BPS D3 brane in the KS background. It is not hard to check that the zeroth-order WZ and DBI contri JHEP09(2010)087 5 ). /r (3.9) 3.12 (3.10) (3.13) (3.12) (3.11) : τ ) proportional i force predicted φ 5 − r the wn to scale like 1 rane in the first-order he -dependence regardless on’s third law that the r must be proportional to 3 branes at the bottom. 1 e, the cancelation of terms ) for large values of ) (3.8) nsider its physical implica- X e; it may be zero in certain 3 ame ibility that an anti-D3 brane , s perturb the KS solution and 4.1 τ/ ) D3 branes in the infrared, one . τ 10 ) 2 − 3 ]. In that paper, the force felt by − e e τ/ ( ( 13  -preserving perturbation parameters, 10 O O 2 . − 11 1 Z e + r + 1 ( 4 , 3 ×  r X 3 O 1) τ/ , in fact this is the most convergent of the 4 O 8 τ/ + + − e − − 3 + e r τ C ) 8 = 1 SU(2) τ/ – 15 – 5 τ , it is rather straightforward to compute the UV (4 4 r ∼ r 4 4 X 2 × − e as well P − V 1 ∼ 3 0 r X x 2 = 3(1 F = 0) but when non-zero it has a universal form ( 1 ∼ − 0 1 e x τ X 2 X F − is universal. = e is related to the canonical radial coordinate ], although the mode itself was not identified. However, the by expanding the integrand in ( 4 1 ˜ ξ τ 1 12 , and furthermore that the constant /r ˜ ξ 1 ˜ ξ ], where 20 is a normalizable mode decaying as ] was discussed in [ have a nontrivial 1 As we will show in detail in section Second, this result agrees with that obtained in [ We would also like to note that in the ultraviolet the mode (in 13 X Recalling that in the UV, of whether non-normalizable modes arein turned the on force or up not. to order Henc 1 Having obtained this simpletions. universal First, result, the force weon is pause whether largely to these independent modes co ofsolutions which are (such mode normalizable as or [ non-normalizabl and thus the potential goes like force analysis alone insources no a way non-normalizable supports mode. or The disfavors force the has poss exactly the s Our result implies thereforemust that in order to describe anti- a probe anti-D3 braneand in to KS be with linear D3same in branes force the at should D3 the be bottom charge. felt was by It sho a was probe also D3-brane argued in using KS Newt with anti-D Out of the 14 physically-relevant SU(2) asymptotic expansion of we can see thatperturbed the UV solution expansion is of always the force felt by a probe D3 b only one enters in the force. it follows that By expanding in the ultraviolet the number of anti-D3 branes. to by [ 14 modes. The necessity for having such a mode in order to find t JHEP09(2010)087 d (4.5) (4.4) (4.6) (4.3) (4.7) (4.1) . τ analytically. . Having set i 1 ξ ˜ ξ modes. On the i ξ 3)) sinh . 5 ), which depends on − xteen IR integration X  τ )] = 2.33 0 or the full solution with 6 k e immediately found that )–( X − . ion including the second order equation for 6 (cosh 2 − 0 ˜ ξ 7 physical). As we will see, the , having argued above that the f itly but these integrals can be 2.17 ) its one to relate numerically the ( = 0 X 0 y 7 F + X . 5 , − 5 ) 0 X τ X τ cosh(2 (2 = 0 (4.2) P f 2 = ) are constant 1 ) − − ˜ 5 [2 ξ 7 5 ˜ ξ 5 ˜ ˜ ξ ξ 0 τ , , sinh(2 x X 4 ] one sets ⇒ 2 6 ˜ ) 7 ) one easily determines the solution ξ , 2 sinh ˜ ξ − 0 4 20 – 16 – = 0 X e y ′ 4.5 X 1 = 0 ) cosh + 7 dτ 1 X = 7 τ , X 4 0 τ 7 ˜ τX ξ sinh(2 to zero, finding an analytic expression for all the other ˜ ξ Z τX 2 2 1 , but one should keep in mind that they both appear in − −  7 − X − 6 = = X 2 sinh 6 exp ′ ′ X 7 6 ) is solved by ˜ ˜ X 1 ξ ξ ) and using ( + and X + 5 4.4 6 is the only mode which contributes to the force on a D3 brane an 5 2.17 = X 1 equations for X ) X 1 = 0, we can fully solve the system of equations for the ) ˜ τ ξ i equations for the choice X τ 1 ξ i ξ X is nonzero there is no analytic expression for any of the ((1 + 2 (1 + 2 1 . = = X 1 7 6 . To compare to the solution in [ ˜ ˜ X ξ ξ 7 ˜ ξ The first equation ( have been denoted and = 0, we then immediately read off that ( , and then building on top of this, the integral expressions f i 6 6 1 ˜ ˜ ˜ We proceed by initially setting ξ Doing the derivative of ( ξ nonzero 4.1 Solving the Here we solve the We now have a pair of coupled o.d.e’s for ( is thus the signatureas of long the as anti-D3 brane background, we hav To ease notation the two integration constants appearing in ξ and this integral cannot beintegration constant done explicitly. In a cruel twist In this section we find the generic solution to the system ( 4 The space of solutions UV and IR integration constants. sixteen integration constants (althoughfull only solution fourteen involves are integralsexpanded that both cannot in be theconstants. done UV Having explic the and expression IR, for the giving full rise solution perm to sixteen UV and si ξ other hand, when and later we will find integral expressions for the full solut JHEP09(2010)087 , 1 8 ˜ ξ ˜ ξ (4.8) (4.9) τ (4.13) (4.14) (4.10) (4.12) τ )) sinh 7  X τ ) τ ) and therefore all )) cosh 1 7
˜ n, but the integrals ξ , one first finds the 7 ) sinh τ 4 3 + 5 τX 2 X τX − ). Then one constructs . 4 . − − ation of the homogeneous i e will keep the 8 constants + ( − b 6 2 i 6 6 = 0 j nd then it is easy to see that X X st these cs. ) ) (4.11) ( stem of coupled o.d.e’s, which X 7 X , . . . , n 1 8 τ 2 ), cannot be found explicitly. − h g ,X − PX ) + ( = 1 4.1 7 + 2 + 2(6 . ) τ ) 7 Z τ τ j 7 + 9 ,X 5 X i 6 = 3 b τX − + j X . A simple way to understand this extra 2 (1 + 4 + ,X , 5 1 4 equations (except for the equation for 5 (1 ) to find 5 . Given the system which are modified by j −  5 )) cosh X X φ g ˜ ξ 7 X −
7 ) , 6 j ( X ,X τ 6 i 2 5 ) and solve the o.d.e. to find , ( 4 2.23 X τ 5 , λ A − ˜ – 17 – P ξ PX ) ) ,X PX = 7 τ 2.24 itself, given in ( + 3 ) + + 4 3 ( ′ i + 3 τ j X 8 1 τ g ˜ − ξ λ 3 ,X ) and ( X )) sinh 3 − 2 2 7 X j h,i 5 into ( 4 g X X are functions of a single variable ,X 1 + 2 X 3 2.22 (1 + 2 − + (3 1 ˜ ξ ) contains 5 j ( − − + 6 τ X X τ (2( 5 X (( ( 4 . ( X , UV and IR expansions
= P 2.22 τ τ τ X + 4 τ g, A, b i )( 2 3 τ 2 g 4 τ − 3 6= 0 e X − ) cosh 3 P τ 1 cosh 2 7 τ (2  X P 2 sinh 2 sinh X (1 + 3  , and − 6 1 + + P ) − τ 5 + 2 = sinh 2 e of the homogeneous equations (again 8 X 2 ( , . . . , n ˜ 3 ξ X − P ( j h,i X g is nonzero, one can find integral expressions for the solutio + − = 1 1 ˜ i = (1 = 2 ξ 8 3 ˜ ˜ We then directly integrate ( So far we have introduced 8 integration constants ξ ξ ’s get an extra contribution proportional to ˜ where solutions the ansatz for thesolutions inhomogeneous where the solution coefficients byone are a has promoted linear to combin functions, a will in fact be used to solve the equations for contribution, is to recall the following method to solve a sy appears on the right hand side in all of the ξ but the right hand side of ( however the the zero-energy condition is a constraint among cannot be performed explicitly since Finally we insert the result for This leaves 7 independentsince constants, this but will for provide the a time clearer being picture w of4.2 the UV asymptoti Solution for When JHEP09(2010)087 ≡ + i 1 b X (4.18) (4.17) (4.15) (4.16) (4.19) showing → τ i ated. On  (typically and b ) ′ n τ ) ( . UV r 1 f 1 j ′ d the integrals X λ and that purely due dτ (log . 1 j h,i ˜ τ ξ , g ∞ ∼ ′ nsion is in powers of B.1 Z 1 n dτ  . X τ overall integration con- ) ′ ) unt of the homogeneous ) ′ + n τ ! 3 τ amounts to expanding the i ( exp ) ( g h 4 τ/ b τ  g τ rbation theory (they are both first ) 10 ( ′ . τ → 0 , and the integral can be done − 0 τ τ elation between i O e τ τ ( Z ( f + ′ O R there are negative powers of = . ) amounts to the change 2 ’s in appendix 1 i dτ ξ τ λ 2 ) + om the lower limit of the integral, namely we ∞ , g ) for small τ 2.24 0 P , λ 1 i Z 0 3 λ 12 A / h 4.1 )–(  1 1 R 3 −
e ’s can be obtained similarly by expanding the X This implies 2 3 ˜ ξ exp 2.18 (3 = + 7 – 18 – 3 . h 0 i 1 6= 1). IR 16 1 τ/ ξ λ 4 n X − − = 1, that will appear repeatedly in the following, we ≡ e , g ) are split into 1 = , doing the integral explicitly and then expanding the  1 j n ) (for ) b τ

we have chosen to absorb the ′ τ UV 1 j  4.1 ( τ i ) +1 ′ +1 λ ( ) h X IR n 1 n 1 g τ contains f − ′ ) ( − t − h X = 1 i τ f g n b ′ ( dτ ( f = ′ λ τ UV on the rest of the 1 dτ = ∞ ˜ ξ dτ are used to distinguish between the expression without 1 IR τ 1 Z n ˜ ˜ ξ τ 0 ξ b − ) =  0 t Z where τ n Z ( This gives  f 1 i 1 g = 1 appear). Fortunately all these terms can easily be integr b τ 8 exp 1 X R n . We give the IR behavior of all the exp X on the right hand side of ( a 1 + UV 1 → 1 i λ 1 . For the simple case + n X X ˜ i λ ξ and fixed the limits of the integrand in − , where at each power there are terms proportional to 0 i g t b 0 i ≡ = n → /r λ f 1 ≡ 1 i τ ˜ 0 ξ = 0 and λ ) R ∼ τ can in principle be done numerically. This gives n ( 3 The integrands can always be expanded around the IR and UV, an Adding The UV behavior is slightly more tricky. On one hand, the expa . They should not be confused with zero and first order in pertu i In general, if for a given expression to be integrated in the I The indices 0 and 1 in 1 f ˜ 8 7 ξ τ/ IR 1 1 − stants in can be performed in these limits. Expanding ( up in the expansion,replace we subtract the infinite contribution fr to order). where the integrand cananalytically. now The be integral expanded from zero around to large infinity giving the r integrands in The contribution from solution to The integration constant here corresponds to adding any amo have e integrand in a seriesexponential of again. powers of where the integration constant is related to the choice of Note that in the expression for only X X the other hand, expressions like ( JHEP09(2010)087 ). , = 0. . We olved 4.10 (4.26) (4.23) 1 i (4.25) (4.21) (4.22) (4.24) (4.20) 1  λ . 1 j X b j i IR 3 ) . . Y 1 − h + g    IR 1 ( 5 6 7 ′ he remaining in- ! Y λ λ λ is given in ( 4 dτ IR    + ˜ 2 2 X τ ˜ ξ ), assuming Y ∞ 3 !    / + ) are given by Z 1 1 0 3 2 + / 3 ) , 2 + 2.33 3 ˜ τ 2) ) X / i 2) IR τ 2 k 8 τ τ/ − ) )5 )–( ( τ/ Y ( f, k, F 2  ) 1 1 τ τ − 2 1 2 τ 1 2 ) 4 sinh + ˜ − ξ τ . We therefore define 2.26 ˜ X − X 3 ) 0 i − τ 2 sinh . / ) = ( ) τ λ 2 cosh . 2 = 7 τ − ) ˜ φ a sinh(2 τ B.2  , k sinh sinh(2 1 2 1 j 2) 6 2) 1 − b ( ˜ φ 1 2 τ/ j τ/ sinh(2 ) X , ( ( i ( 8 5 2 ) τ 1 2 2 τ ˜ τ ξ (sinh(2 ˜ 1 + ( φ 9 equations ( 32 − h and performs the integral analytically. The 3 ), which implies 16 i g dτ IR a + ( 2 cosh 2 sinh τ φ ′ − X sinh(2 τ – 19 – Z − τ 4.15 dτ 2 1 2 τ ≡ ( : ˜ τ φ 32 2 2) 2) + 1 ∞ ’s in appendix ˜ φ sinh ˜ − Z ξ φ dτ τ/ τ/ cosh UV a sinh + = + X Z −

1 j 2 τ ) b τ ˜ λ 2 64 j τ coth( 3 tanh( i τ 2 ˜ ) − φ − , which is given by − sinh 1 − , τ 3 ) − − h , = ˜ τ 2 ) τ φ τ τ ) g 8 τ τ ( τ is solved using ( ′ ( ˜ 2 φ ). 2 2 sinh equations is given by the following integral, which however cannot be s ˜ sinh λ dτ sinh ˜ − ) φ 4.8 i 8 sinh(2 ) ), we find that the fluxes ( τ + sinh ∞ − φ ˜ ( = shift the constants of integration in φ ’s, one uses the same expansions to obtain the UV behavior of τ 0 3 8 3 τ ˜ τ ξ (sinh(2 i 2 Z is the solution to the homogeneous equation, and 2 ˜ λ 2

˜ k φ 4.14  1 dτ 1    1 − ) dτ X ) sinh(2 Z X = τ Z = = =    4.16 are some constants that can be obtained numerically, and in t is given in ( 7 6 5 = 1 i 3 3 ˜ ˜ ˜ i φ φ φ 8 ˜ λ 1 ˜ λ φ ˜ k ξ ˜ 1 φ    For the other The equation for X tegral one expands thecontributions integrand for large get from ( where (sinh The same trick is used for 4.3 Solving the 4.3.1 The spaceHere of we solutions show how to solve the system of We obtain an integral expression for and then using ( We give the UV expansion of all the where First we note that analytically where JHEP09(2010)087 . ns IR 2 ! X    (4.27) (4.29) (4.28) 3 5 6 7 / b b b 1    3) . 3 matrix is /    ) × ) except that  τ )4(2 1) ) ) τ 4.15 IR τ τ 6 − ( i.e. the integration 7 sinh 2 X O − τ sinh τ cosh s, and the derivatives τ τ + + ) τ (4 τ 2 2 an integral expression. τ any constraints on our 3 τ ) to substitute for ( ! / IR cosh 5 2
τ O cosh ) 1 + X 2 sinh 8 sinh τ ( IR 6 4.32 − + cosh IR )+4 − 4 ( P IR − 1 τ ) τ Y − IR τ PX − τ X 8 1 ( τ + + Y 2 ).  + sinh
O 7 h 2 , P 4 0 0 (sinh(3 4 6 sinh b 3 , h ˜ + IR P φ 5 sinh / 5 − ) respectively, and the 3 τ /h )) + to zero) and 2 3  Y 2 τ / ) 4 Z 1 3) IR 7 τ )) P PX ˜ φ 0 / 6 2.32 3) τ X = h ) + / (cosh X 6 ) + 5 τ sinh 128 4 τ τ 3 τ 64(2 + − τ IR / 3 4 − 2 , λ sinh IR X 1 1 τ – 20 – IR IR ) and ( 5 6 3) 2 . We write the divergent terms and the constant + 16(2 − i X / 4 sinh X X ) (setting φ cosh(3 ) sinh − ( 1 3 (1 + τ 2.31 − P 2 1 ( − h  (cosh 4 h ) 4 1 2.33 + ˜ + IR − IR φ ), ( 3 + (2 5 3 τ 4 is solved using the same trick, namely ( − / 20 Y

X 3 λ IR 3 4 4 3 ) / / ˜ (5 1 2.30 τ φ 1 X = ). We will call the constants of integration in these equatio − ) 2 P τ 3) , but a combination of 4 sinh(3 τ 3) 5 / ˜ − sinh φ τ / τ + τ Y IR ( 2 2 4.26 3 1 sinh − Y 8(2 / dτ IR 8 1 τ 16(2

4 sinh X 3) Z τ IR 1 1 τ − / are given by 2 (6 (1 + τ X 3 1 2 a P + log / (cosh − 8(2 λ 1 8 1 4  are the rhs of ( is not just +  IR 3) 3 τ 4 7    / 5 Y ˜ τ IR φ , b 2 3 (2 = 6 log Y 3 is the right hand side of ( 1 τ , even though the three functions depend on the three of them ( 1 τ τ 16 = exp 3 matrix contains the solutions to the homogeneous equation , b +    7 4 5 ′ 5 ′ 6 ′ 7 h b b × λ λ λ 4 = = = ,Y ˜ 6 φ Finally, the equations for    3 8 2 ˜ ˜ ˜ φ φ φ ,Y 5 where the inverse of the one in ( This 3 4.3.2 IR behavior We now give the IR expansions of the the solution to theFormally, it homogeneous is equation can be found only as Y of the functions where constant in terms since termssolution which space. are Here regular we in have used the the IR zero energy will condition not ( provide JHEP09(2010)087 ) and ! τ 3 ) ( τ/ O IR 7 4 − + X ! e ) P IR 3 τ 3 3 + ( / ! Y 1 / 2 2 O 3) P P IR 6 / + 3 / X 1 0 ( 128 IR 2 0 1 h 3 3) 16(2 / h 3 Y P / 1 3 3 IR 4 / + ! 3 1 /
 X 2 ! ) 16(2 ) 3) 2  / −  IR 8 IR 3 IR 6 ! − 6 / 4 + ! )
1 X X ) 3 P 3 2 0 / PX / 3) IR h 6 + 1 4 + 3 IR / 6 3 / + 32(2 X ! 3) 1 3) X IR 4096 IR 5 / 3 5 / ! + 3) IR 6 32(2 + X P 2 X (2 / − ( ( 3 X P IR 0 / 0 5 IR 6 − 10(2 P IR 2 5 h  including terms which fall off as 0 6 X 2 X /h 2

IR 32(2 ( X 3) ) i h 7 512 3) 2 ( 3 ˜ 0 / P 3 − τ P φ 3 / + 2 / − ( 3 IR / P 3 h 1 4 / 2 PY / 3 P (4 – 21 – 2 O IR 1 4 / X 8 IR 0 P 5 0 3) IR 5 , certain modes have leading behavior in the UV 2 3) 3 h / − − 3) h X 3) 1 / / + 512(2 X + 16 X / 3) 2 / 3  / 3 (2 10  IR / IR  4 1 + (1 3 ) 1 IR 2 / − 3 10 + X 3 1 3 + 8(2 16(2 Y P 3) P IR / 32 3 6 2 2 3 1 PX / 3 / 2 /

1 / X 32 + 4 + 1 3) − + 64(2 (2 P 32 3 P 3) / / P 3 3 + 0 3) 3 9 3 3) 3 1 / / / IR IR / / 4 / 9 h 2 / 1 1 IR 16 1 1 6 2 3) Y IR 0 5 X (1 2) 3) / 2 3) 3 3) h  IR / / 1 X 64(2 + 4(2 / / PY + 16(2 3 − (2 ( 64 3 − X 0 IR /

0 )

4 3

1 IR 8(2 h τ h + 7 1 τ + + (2 X ( IR 32(2  2 IR Y 256(3 3) 1 0 IR 1 + / Y O + + 9 h IR IR + X 7 + 6 X IR IR 2 4 1 P 3 + Y IR Y

6

2 IR  X X 2 2 3 ) τ IR 16(2 Y τ 6 + 6 Y IR τ − 4 − − + + τ Y − ( − 3  Y 1  IR τ log 2

O

6 2 1 + 1 1 τ τ τ Y + log − + + + = = = = = 4 1 6 5 7 ˜ ˜ ˜ ˜ ˜ φ φ φ φ φ 4.3.3 UV behavior Here we provide the UVslower. asymptotics for However the as shown in the table JHEP09(2010)087 3 ) τ/ 4 3 − τ/ e τ τ 7 
τ τ ) − − − e τ ( e e UV 7 3 −

O τ/ ) ) (3 3 + 3 − UV UV PX 8 8 e τ τ τ/ τ/ 3 3
UV 7 7 UV 10 189 PY PY 2 3 3 − τ/ τ/ − Y e UV − − 7 τ/ τ/ − t e ( PX e e 4 4 ( + 2 + 2 − O − − e O 36 e e UV PX 3 UV UV 3 3 + )

UV UV τ τ − 5 5 ) ) X τ 3 3 + 2 τ ) + Y Y 3 3 4 − 
τ/ τ/ 4 4 PX PX ) e UV UV 4 4 6 6 τ − ) ) UV − − 8 − − τ τ  X X e e + 176 3 144 144 UV 2 2 7 X ) 2 2 137 1 1 τ − − + +   τ τ UV 2 e e + − 8 ) ) ( ( UV 4 − + − − + (10 PX Y 3 3 e UV UV 7 7 UV UV ) + 4 5 5 τ τ O O X ( UV UV τ/ τ/ 1) − 8 8 − − X X UV − − 3 O + + 112 e e + 3 ( ( UV 7 − e e PX PX / 2 2

 X PY PY

P P + τ X ) τ τ + + ( UV 3 3 2 τ − − UV  UV UV 3 − 8 8 4 UV UV UV P 3 7 7 τ/ τ/ + 2 + 2 X X 4 4 UV UV X X X 3 3 − UV UV UV – 22 – + 7 3 3 − − 5 5 PY PY 2 2 UV 5 e e 2 UV UV X X 108 2 2 3 3 Y Y )(2 + (9 P P 2 2 X τ   X X PX − − 4 PY )( − 5 5 3 − − UV 4 τ + 7 + 7 27 / UV UV 7 7 + 1 − − X UV UV ) + (2 ) + (2 2 2 UV 3 + 2 2 − UV UV 5 5 UV UV 3 3 − X PX PX UV + 2 UV UV UV UV 8 8 6 + 2 ( 7 7 PY PY 6 UV (3 + 4 3 3 Y Y Y X X PX PX − − PX PX 2 P 2 2 + 4 + 4 τ/ X UV 2 ( + 2 + 2 + + 2 + + 3 − e 3 − − P 79 X UV + 7 + 7 2 UV UV τ/ 3 3 3 UV UV UV UV 3 3 UV UV  4 2 2 6 6 5 5 36 36 X UV 3 UV UV UV X X 5 τ/ 3 2 2 − 2 Y Y Y Y UV UV 2 2  X X 2 / 8 8 X e Y 16 1 4 4 e X X − − 3 3 3 / 4 4 − − / 1) 20 24 + 4 + 4 − − 1 PX PX 3 3 τ/     10 UV ) 3 + 2 + ( + ( 2 / / 2 2 − − 2 τ 1 1 τ τ τ 2 τ 3 P P P P X e P P UV UV 2 τ 5 5 e e e 3 − τ/ Y Y 72 72 72 72 10 48 36 36 72 72 e (4 − + 2 4 τ/ 5 5 ( 3 3 3 3 3 3 3 3 3 3 4 2 1 UV UV UV − 2 2 2 / / / / / / / / / / 5 5 5 1 1 1 1 1 1 1 1 1 1 − O e  Y 1 1 Y Y 2 2 UV UV e 2 2 2 2 2 2 2 1 8 + − − − Y +2 +2 − + + − Y − − − − − +2 − − = = = = + = = = 5 1 7 6 8 3 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ φ φ φ φ φ φ φ which is more convergent than this. JHEP09(2010)087 and i ) modes , 0 UV 8 | τ i i 2 ] relates ξ X − . One key |O i 28 e 72 ( 0 φ of quantum , h O − i 27 O + UV 4 3 X τ/ ) 4 τ symmetric deformation − rator 8 malizable. These two e 2 Z ! ctionary [ ,X rms, it is expected that ich integration constants ) 2 4 × 5 + 8 3 is very interesting to note s τ ,Y − ′ ,X 6 3 1 6 on value (VEV) 5 /Y sists of one BPS mode and one /Y SU(2) /X /X + 2( 7 3 /X 4 5 1 × Y Y − − − − − − X modes it is useful to tabulate the ,X + 14900 UV 2 ,Y 2 i 8 2 ˜ int. constant X φ X τ ) 3 ,Y τ 7 τ/ 3 are integration constants for the Y 2 . For backgrounds like Klebanov-Strassler 1) i − τ/ 3 20785 e 2 X field theory deformations of the action field theory VEV ) − e τ/ : 2 − e UV 7 i τ τ – 23 – τ ↔ ↔ X O = 3 2 ) 8 7 6 4 + 1) UV x 4 5 + 72(1 + 8 5 − − 2 ∆ r are integration constants for the modes d − − − − − 2 − τ τ Y − + 25 i d ) 375(4 /r /r ∆ r 1) /r /r /r /r Y 1 τ 1 2 (2 P τ UV r 5 R 0 4 3 2 4 r/r − − − r r r r ∼ X 85 r r + 80 − ∼ ∼ UV 3 ) − τ τ τ − X (4 δS UV non-norm/norm 56 8 16 3279 + 17012 Y 3 (12 − ) / − 1 ( τ 1125(1 3 2 UV (11 + 8 1) 4 (95 5 4 3 2 8 7 6 X P + − Y P 3 64 dim ∆ 18 τ 32 τ/ − (4 + 9 − 4 + (3 + 4 e +

P 1 2 1) 3 UV 7 / UV − normalizable modes 1 5 1 − Y normalizable modes UV 2 τ Y 4 4 τ 4 2 − Y − + − . The UV behavior of all sixteen modes in the SU(2) 3(4 + non = To understand the holographic physics of the It perhaps bears repeating here that the 4 ˜ φ dimension ∆ in the field theory, the well known holographic di of the Klebanov-Strassler solution. Table 1 leading UV behavior coming from each mode. For each local ope two modes in dualgravity modes AdS are space, dual one respectively to normalizable the and vacuum one expectati non-nor and we recall that we have in the UV and break supersymmetry, while the the deformation of the action correspond to normalizablethat and in non-normalizable all modes. casesnon-BPS a mode. It normalizable/non-normalizable pair con which are asymptoticallythis AdS dictionary only still up holds. to certain In logarithm table te 1 we have summarized wh JHEP09(2010)087 . n − then (4.31) 5 cancels X = 6 (5) d or F 4 X ) in the UV this 4 ˜ φ , and for describing s fluxes. This is rel- 3 + S 1 ˜ φ t space has a warp factor in the UV. ) can be used to eliminate − . 3 ( es of flux, the IR boundary 1 3 he energy from , x s with ∆ = 4 in table 1. m of equations to correspond ter fixing the gauge freedom 2 xpansions we have provided, is arizing the PT ansatz around on of smeared anti-D3 branes. = 0 (4.32) = a we are interested in finding the d boundary conditions. 7 ξ , x A 4 1 X 9 ith the expectation from the AdS- ) (4.30) ) there is a constraint, coming from t, x φ − ( 5 2.5 and which is the only mode responsible V (since . X 1 = A X -dimensions, the warp factor will scale as b = 0 + 3 n 4 dτ a 0 dφ X a dτ – 24 – dφ + a dτ dφ ξ 3 ), we find that the zero energy condition is a linear ab X G 2.8 1 2 + 4 0 the full solution is smooth due to the infinite throat. 2 = 0 and smeared on the finite size X 6 → τ ) and ( ). Hence the linearized solution depends on an even number of − r 2.7 UV 1 Y and as , whose integration constant is 4 1 does not enter in the condition). Using this to eliminate 0 since it is now the solution to a wave equation in dimension ξ 1 Q/r → X r component of Einstein’s equation: as ττ n On our solution to ( The gravity solution for a stack of localized D3-branes in fla In addition we must impose boundary conditions on the variou R ) = 1 + 4+ r ( − that the mode result from this table which cannot be gleaned from the field e for the force on a probe D3-brane,4.4 is the most convergent The mode zeroIn energy addition condition to and the gauge first-order freedom equations of motion ( modes which might result fromThese the backreaction branes of are a placed collecti at algebraic relation amongst the integration constants the associated with redefining thethe radial KS coordinate. background this When becomes line a constraint on just the which must be enforced into order for solutions the of solutions of supergravity. our syste This constraint is imposed af the integration constant in the metric function leaves just two normalizable and two non-normalizable mode 5 Boundary conditions forWithin three-branes the space of solutions we have derived in section (note that In addition, the common rescaling of the coordinates ( integration constant is r However when these branes are smeared in physically-relevant parameters (fourteen), consistentCFT w correspondence. them it is necessary to carefully impose the correcth infrare against that from the curvature. In the presence of other typ This is the IR boundary condition we will imposeatively on straightforward the for metric D3-branes in flat space, where t JHEP09(2010)087 4 0 ˜ k φ and (5.2) (5.1) 3 Y and and ), whose 0 6 9 f ˜ τ φ has a very . ( , 8 h 5 Y ˜ φ  , and requiring flux. (1) UV first order, should 2 F Y (5) ∧ , by a large gauge trans- F = anes are added at the P (2) rane sources by shifting rm as in the original KS IR 2 B PS KS background. First, singularity in it is interesting to see how Y ∧ energy from the curvature: 3-branes should require the has no D3 Page charge. The ′ that this shifts on-shell, contributions to the ). The constant /τ branes (whose harmonic func- ltiple of (2) rom the 1 dτ o loose a criterion to signal the ˜ φ B 0 1 2 F . This shifts the D3 Page charge, 0 ) 1 , ′ are zero, k + 1 τ i P. ( ξ T N 2 (3) = ) +  of the warped deformed conifold. 0 F + K 5 ′ F 3 3 D τ (1) ∧ S D 2 − Q F or the warp factor, and hence will leave the physics (2) ∧ P , ) the D3 Page charge shifts by (5) B by a constant corresponds to shifting sinh – 25 – → Q (2 P (2) 0 − ∗F 3 = 0 f (2 (5) B D 3 τ (5) F Q − D N/ F Z Q sets these to zero. The same thing happens with  (3) 5 ∼ 2 F ˜ φ ) M  τ Z 3 ( S = 0. Furthermore, since 2 h ) Z . Shifting i 2 3 1 2 X / π 1 1 π ) τ (4 4 τ near the sources) on the 2 − = sinh = 4 τ 3 / /r 1 3 5 2 , in such a way that the warp factor becomes singular at the tip P age D P age D (sinh 2 . However, this will not affect (5) Q Q 2 remains invariant. This introduces in the IR a 1 B ∗F ) = fields, that correspond to modes that break supersymmetry at 5 τ i D ( ξ Q K by a constant multiple of the volume form on The integral that gives the warp factor remains of the same fo In order to “calibrate” our perturbation-theory machinery It is trivial to obtain the BPS solution in which smeared D3 br Note that one can also shift the D3 Page charge by an integer mu (for the latter we require no constant term in the UV of 9 (5) 1 invariant. The Klebanov-Strassler background is aD3 smooth and solution D5 and Page charges are 5.1 BPS three-branes by equal amounts. Under a shift of formation of stress tensor from allthis types is of the basic flux naturepresence taken of of together Einstein’s D3 cancel equation branes. but the dominant The this contribution correct is to boundary to the conditions stress-energy for tensor D to come f conditions are slightly more subtle. When the background is solution, Y clear meaning: it corresponds to a shift in the dilaton. Note this BPS solution can beall obtained the in an expansion around the B while and evaluated on the KS background these are be set to zero, and hence no divergent terms in the IR of tip of the deformedF conifold. In the PT ansatz one can add D3-b tion diverges as 1 physical interpretation is obvious: we have smeared BPS D3- with as well as JHEP09(2010)087 i 3 ˜ ct to φ (5) S by a 7 F , 4 6 , ˜ her φ 5 λ . and are equal S solution, N k then satisfy (3) to diverge in ). Similarly the 7 H ˜ 4 φ s coming from and ˜ φ ilar fashion, one 2.13 and has the same f and (3) d theory by adding ) by setting 6 ..., divergencies in the IR ˜ H φ flux, puting D3 branes , + (and subsequently that 5 3 4.26 ˜ φ (3) τ/ (5) F 4 . ing the sign of ing from the same Ricci-flat . It is rather straightforward F rves different Killing spinors − om ( N 3-branes in anti-KS. e ) P cal in the BPS solution changes 4 BPS D3 branes is obtained by − P ˜ φ s = 7 3 g 2 = Y anti-D3 branes smeared on the N 4 ... − 5 , and the functions N 3 P age D + τ P age D N Q flux and putting anti-D3 branes in; this 1)(1 1) + Q 1) = 0. The fields ; shifting in the expression for 2 s − − − (3) 7 g , which as argued above shift by 0 τ τ UV τ Y F 6 8 f ˜ 2 φ We consider perturbation where the dilaton is Y (4 (4 (4 3 3 – 26 – P = term, as one expects for pure (non-fractional) D3 τ/ τ/ 10 and 3 of opposite orientation compared to the solution 4 4 and 4 0 − − − IR 5 k 8 e e (3) /r ˜ φ 2 2 Y = which we have set for simplicity to 1 in eq. ( H P P 4 s ˜ φ g = = = 0. The only non-zero integration constant one can have and x 6 2 multiplies a non-normalizable mode corresponding to the fa Y ) contains a factor of − (5) 7 p Y 1), exactly as in our UV expansions. F +4 2.14 multiples a 1 − A 4 7 τ e Y (4 , exactly as one expects from the full BPS solution. All the ot . Note that this perturbation causes the warp factor = 0 and = . The D3 brane page charge is τ/ again diverges in the infrared as in eq. ( N 5 h Y h N h N/τ ∼ . This flips the signs of − . 7 . Hence, for one orientation one obtains BPS D3 branes in the K 7 Y (3) Y 5.1 F ) is determined by the orientation of these D3 branes. In a sim . Requiring no exponentially divergent terms in the UV and no (3) The KS solution with D3 branes at the bottom can be thought of a Hence, the perturbation corresponding to adding Note that in the UV 7 , The NSNS B-field has a factor of H 6 , 10 5 the warp factor and approach as expected from the exact KS solution. while for the opposite orientation one obtains the anti-BPS integral that gives not shifted at infinity and hence set Hence the new warp factor is Y the homogeneous equation, and their solution is obtained fr term proportional to first taking the Ricci-flat deformed conifold with a topologi set respectively in, and backreacting this configuration. The orientation of can obtain the anti-KSdeformed solution conifold with with anti-D3 the branes same by topological start 5.2 Anti-BPS 3-branes inThe anti-KS anti-Klebanov-Strassler solution is obtained from the KS solution by flipp branes. of is therefore and we can see that D3-branes: that we have changed the rank of the gauge group of the dual fiel just setting the infrared as backreacted system has change by subleading terms, except in section to see that thisthan solution the is original KS also solution. supersymmetric With but the it addition prese of but not of D5 brane Page charge as the KS solution, namely JHEP09(2010)087 ]. 5 3 Max D (5.7) (5.6) (5.5) (5.3) Q ]. 31 . In other ) by parts – 3 29 5.3 D Max 11 throughout the Q (5) F s not make sense to zero. te to the Maxwell charge. Integrating ( of freedom. However we are spectively. Note that ler solution, one expects the al charge as ) (5.4) urb the KS solution, and try es. The only quantity whose es, and in addition to have an -flux annihilation process [ sures the running of the rank i e sign of arily quantized. x gets contributions from branes nto “fractional” anti-D3 branes. slice at infinity. For the zero-th − , (3) 11 x 3 F (5) ( ng theory can be found in [ Max D T δ ∧ F , Q f i q 11 , ∞ (5) (3) X T + F Z (5) H + b 6 2 F q 11 0 ) T M (3) 2 1 Z Z = on the 3-cycle. Hence, from this perspective, π F 2 2 – 27 – (4 ) ) ∧ 3 2 2 (3) 1 1 Max D π π = F (3) Q (4 (4 Maxwell charge which we will add to the background H 3 Max D = = = Q b f q q (5) F , we have two strategies to look for the “anti-D3 brane in KS” d additional 1.1 Note that if one thinks about the solutions in this way, it doe As is clear from the Bianchi identity for A very nice description of the various charges in type II stri 11 are the charges from the explicit D3-brane sources and flux re where the second term comes from explicit D3-brane sources. of dual gauge groups and thusconcerned is a only rough with measure of the theand degrees thus we will abuse terminology and refer to this addition order KS solution, the Maxwell charge is not constant, it mea 6 Constructing the anti-D3As brane advertized solution in section where where we have indicated that the integral is over the solution. The first is to impose UV boundary conditions, pert yields If one now inserts anti-D3physics branes at the in tip the to BPS beextra dominated Klebanov-Strass D3-charge by the dissolved anti-D3 in branes flux, sourc As as a expected result, from we the needRecall brane that to the consider Maxwell how charge branes for and D3-branes fluxes is contribu defined as a-priori fix the signorientation of the is charge fixed of is the “fractional” the D3 integral bran of solution, and hence transforms the “fractional” D3 branes i 5.3 Anti-BPS 3-branes in Klebanov-Strassler and flux combined words we have normalized the background Maxwell D3 charge to is gauge invariant (unlike the Page charge) but is not necess changing the orientation of the D3 brane sources reverses th JHEP09(2010)087 . b q (see (6.5) (6.1) (6.2) (6.3) (6.4) (6.8) (6.6) UV 1 X In our ansatz ). 0 ,F . 0 ) 3 , k 0 . τ/ f demonstrates that the − ) ciple all the other non- icit UV and IR behavior e τ oundary conditions as a 3 ( ( ive new solutions are that he second is to impose the ) (6.7) O O is proportional to ble modes to be absent, and τ be that of anti-BPS 3-branes ( + + we find that b IR 1 11 q 3 T X ∼ O . Max D ) ∼ 0 Q . 5 4.3.3 UV 6 . ˜ φ ) ˜ )) vol 6= 0 = φ , k τ . P 5 ) F ( + = 0 3 ˜ . Since IR φ 1 O τ − + 2 UV ( P ), the perturbations to ( X 7 UV 1 UV 5 1 + UV 7 P 4 ˜ φ ˜ b φ ) we find the constraint X The total Maxwell charge is evaluated in the ,F Y ⇒ ( + 2 q (2 ) 7 PY 1 ∼ O 0 ˜ f 7 P φ – 28 – = f , k ∼ 0 ˜ , 12 φ ) we see that 1 + ) 6 5 f = 2 = − 6= 0 0 ˜ φ ˜ φ UV f 5 0 , , f ) 5 2.13 P k kF Y 3 2 − ˜ ) = ( UV 1 2 φ Max D 7 τ 0 ˜ ( φ X + ( = ( Q k , 0 6 5 ˜ to contain no divergent terms except those coming from φ F F , + ( ) i ∼ O 5 5 0 ˜ ˜ φ φ 0 ˜ φ F F ) 5 − ˜ φ 6 is non zero ˜ φ − ( ) we immediately find that, 1 6 X ˜ φ ) and the UV expansions in section ( 2.10 6.3 IR, and that there should be a non-zero force on a probe D3-brane. ), ( → ) and ( N )), demanding a non-zero force in the UV will imply 2.13 5.6 = 4.18 In the IR, we want the Now we need to extract from this abstract discussion the expl Recall that in our notation ( 3 12 Max D eq. ( From ( the first boundary condition requires a careful analysis of so that disallowing divergences in ( IR boundary conditions (which makein the near-tip anti-Klebanov-Strassler), solution and to tonon-normalizable perturbation try of to the obtain anti-KS solution. the UV6.1 KS b UV to match the expected boundary conditions in the infrared. T The UV boundary conditions weQ wish to enforce on our prospect in the UV. In addition the brane probe calculation of section UV by demanding force on a probe D3 brane depends only on In addition, we also demandhence very set divergent UV non-normaliza for the modes in our deformation space. These are the onlynormalizable UV modes conditions could be that turned we on. will impose so in prin Using ( From the zero-th order expressions ( JHEP09(2010)087 : 2 ˜ φ IR 1 N X (6.9) (6.12) (6.14) (6.13) (6.16) (6.10) ) put into must be set , contrary to N 3 ) (6.15) . IR ) 2 τ /r 3 ( Y , which is highly i τ O ( and the other vari- X O the equations for + 8 depends only on X IR 4 4 + + ˜ ity in the Einstein tensor φ Y ith the singularity in the 2 τ + . . )  IR 1 the anti-D3 brane sources. in the UV as 1 constants differ the UV and IR IR IR 1 7 IR 1 becomes PX term in X X ) (6.11) 3 5 6 4 combined with the equations above PX / 1 ˜ τ . φ 1 3 ) IR 6 6 P − 0 2 0 / . 0 ˜ 1 φ 3) . h h X τ / . Hence, the fact that 3 ( IR 6 3) IR + 1 / 6 4096 O = 0 X 4.2 X ) have already been fixed in terms of + IR − 5 − IR + 4(2 , whose values in the UV and IR expansions − 3 16(2 1 X IR 1 Y /τ ( IR 1 IR 5 | = IR X 1 0 – 29 – b implies 3 = X ,X X q h is the warp factor): ) = X 3 3 6 IR 4 3 imposes a relation between / / (5 4 ˜ IR φ / IR are zero and we have a BPS solution. Hence, any 7 4 1 IR = 6 further implies that ˜ ∼ | 2 2 X φ 8 6 2 3 P Y i Y ˜  4 7 φ X ˜ IR ˜ − φ 3 2 φ X 6 + + Y  = 3 IR 7 ) one can see that the IR 5 0 P Y IR 8 h X 3 ( P X 6.12 3 256 will be a very nontrivial function of all the / (recall that 2 in terms of the number of initial anti-D3 branes ( 0 + 3 h -divergences in 3 UV (5) 2 / IR 7 Y F UV /τ 10 7 Y 2 Y = − 5  ˜ φ 1 τ ) fixes = 4 6.5 ˜ that this implies φ IR 3 7 ˜ Y φ The absence of divergences in However we must eliminate other divergences. We can see from unlikely to be zero unless all the firstly, ( to zero implies that (due to the singularityenergy density in from the warp factor) is commensurate w Then on physical grounds we demand that in the IR the singular Furthermore the absence of a log-divergent mode in prospective anti-D3 solution will source modes that behave what one might expect based on codimension-counting. Furthermore, upon using ( and The absence of aables: divergent term in also implies and the absence of 1 So this divergence is allowed since it is clearly coming from Note however that the values for ( Upon using these conditions above, the expansions of and by an additive constant, as reviewed in section Note that unlike theare integration proportional constant to each other, all the other 13 integration JHEP09(2010)087 b ) q 6.9 ng (6.18) (6.20) (6.23) (6.21) )) 4 τ ( , IR expan- ) ) to obtain O 4 , since their es in the KS 1 g + b , . ) ∧ 6.16 2 4 τ 2  g ) g 2 2 ∼ ∧ g + 3 G 3 g + ) and ( ′ g √ k 2 1 ∧ g 6.8 ( + 1 ity analysis is more subtle 2 g 2 4 ( τ g ∧ ∧ + 1 ) (6.19)  g dτ 3 ′ ′ 2 5 τ ation with us. f ) (6.22) g . ( (recall that F ( ) (6.17) 2 2 1 O τ τ ∞ + 13 ∧ ( ( . + 5 + 2 1 2 2 2 2 < O O g 2 c 4 τ b τ dτ 2 . | g 2 τ 3 ∧ 2 + + 2 0 2 + + a τ 4 0 + b τ ) + g 4 a 0 2 4 1 2 1 2 3 4 by an additive constant. Secondly, eqs. ( GH c . One can then use ( c τ b ∼ + τ g g b ∧ ∼ 2 √ q 3 τ – 30 –  + 3 ∧ IR ∼ ) ∼ 1 x 7 g 0 H b 2 f 2 Y + 2 ) 3 3 c 10 g 2 F d + and F − H = + 0 ). From this we observe that if k dτ − b 7 Z 3 ( 2 IR 2 1 | ˜ 7 g φ P are somewhat more benign than that from ∼ Y , c  ∧ 1 2 2 f c 1 b α g + (2 ( 5 + ∧ g and 2 4 5 ∧ 0 g ds ) a 2 1 f g . At this point we have computed all the IR quantities includi in terms of α ) however would result in a divergent action. Similarly, the ∧ − IR ∼ 7 1 IR 1 k Y ( 6.19 2 10 X 1 2 F g and one might declare this to be the solution for anti-D3 bran ds = = N 3 3 : ) give F 3 H term in ( F 1 6.15 in terms of b We are very grateful to Igor Klebanov for sharing this observ IR 1 13 divergent energy density integrates to a finite action gives a singularity in the energy density of order then there is a singularity in the energy density of the form for some numerical constants ( the system, which is in turn related to The singularities from Furthermore, if sion of 6.1.1 IR singularities Recall that the NS and RR three forms in our ansatz are and ( X in terms of background that we havethan been that seeking. presented above. However, the singular The and in the IR the unperturbed metric is regular and is given by then the energy density has a term that divergences as JHEP09(2010)087 , and has a (6.25) (6.24) f 5 φ ) necessary in 6= 0), . However, both 6.2 ], and the brane- 2 0 5 − a n ( τ 6= 0), all of which are have potentially inad- 1 = 0) and thus there are c is singularity, and given violet is consistent with 0 -D3 branes, while in our lution [ ave any direct connection c ion, the amount of charge , e-form field strengths. The . R three-form field strength ) ) gravity is the negative-mass arity that must be excluded uld be acceptable while the = uch finite-action divergencies ane that mediates brane-flux τ 2 ergent terms are proportional ( e-action divergencies, but it is 1 τ m field strength, while one of olution must have finite-action nti-D3 branes into NS5 branes b ( = 0, which in turn implies that O O s fair to say that if this singularity + IR 1 + X IR 1 τ we find that X IR 1 3 are zero ( 6.1 / X 1 7 ˜ 3 , are hnce no infinite-action divergencies. φ  / 4 1 2 3 −  τ  3 2 – 31 – P  has a nonzero linear term ( 3 16 7 P 3 ˜ φ 8 − = = 6 7 : First, brane-flux annihilation results in the non-trivial ˜ ˜ φ φ 3 . As such, this singularity could not have been predicted as S − ) and this in turn is related to the IR and UV values of f 6= 0) and f 5 2 ˜ φ − , q b has a nonzero constant in its IR expansion ( b k q 6 14 ˜ and lead to an energy density which diverges as and the constant term in φ 5 − ˜ φ IR 1 inside the into ( 5 X ˜ φ 2 3 S Max D Q . = 0 and is thus in contradiction with the UV boundary conditio IR 1 Using the boundary conditions from section Furthermore, this singularity does not appear a-priori to h Clearly, finite-action divergencies are better than infinit Our analysis establishes that the only solution whose ultra Hence, given the absence of a microscopic explanation for th 6= 0). Third, in the probe analysis of brane-charge annihilat We are very grateful to Gary Horowitz for pointing this out. X UV 1 14 1 c missible singularities. and so we see that both the RR and NS three-form field strengths division of flux annihilation that occurswrapping via the an polarization of these a should be allowed. Thefrom best the example spectrum of of aSchwarzchild any solution. finite-action consistent singul theory of two derivative with the infrared physics of probe anti-D3 branes in the KS so order to have a nonzero force on a probenot D3 clear whether brane. in the context of the AdS-CFT duality even s the linear term in The difference annihilation couples magnetically withthe an fields NS-NS whose three-for energy( density diverges in our solution is a R a consequence of brane-flux annihilation. Second, the NS5 br proportional to only way to eliminate these divergencies is by setting that of would-be anti-D3 branesdivergences in in the the Klabanov-Strassler infrared, s both in the RR and the NS-NS thre no singular energy densities that scale like nonzero quadratic term ( not to the IR value of to dissolved in fluxessolution depends both nonlinearly the anti-D3 on charge and the the coefficient amount of of the div anti the absence of anynegative-mass Schwarzschild criterion singularity would under be not, which it this i singularity wo X JHEP09(2010)087 , ), 7 X 6.14 (6.26) to zero. and ) for the )–( = 0, and i modes in boundary 5 ξ 3 2.7 ˜ φ 6.10 /r ,X 4 = 2 ˜ ,X φ 3 is a constant, and X 8 zable 1 ˜ φ es, eqs. ( . These fields have an arity does not appear 4 rguing that one cannot − the UV expansion of the urned on. This problem τ te, as typical for D-branes, ties of the previous section, ies it should have been deemed g theory. Hence, it is possible e previous subsection we have rity may become benign. meared distribution of anti-D3 one must set all the d five-form fields whose energy es as ) to show that mply: ion of the singularity sourced by , , . . 6.26 as a function of ) . 7 = 0 = 0 2 to establish that ) this implies X 4 = 0 X = 0 IR 7 6 X − 4.3 6.16 Y in the UV and in the IR must be the same. X 5 = UV 1 i X = = 0 6 = ( X X 5 2 X P – 32 – IR = 3 X X − + Y 5 IR = 1 = = ) together with ( ) to express X 3 X 8 X IR X 2 4.25 4.12 Y we have already solved exactly the equations ( 1 X ) and ( 4.24 must be equal, and using ( ), ( ’s must vanish. Hence, if one tries to match all the correct IR 6 ˜ φ X , and thus the values of the 4.23 UV and 4.1 5 ˜ → φ However, as we have seen in the previous section, this singul One can now proceed exactly as in section Furthermore, since we do not want divergent and non-normali If we are to match the boundary conditions near the anti-bran in section i unacceptable. all by itself: thedensity infrared computed geometry in also first-order contains perturbation metric theory an diverg had appeared all by itself in a fully-backreacted solution, and therefore ξ the fields conditions and avoid un-physical non-normalizable modes, the UV, The infrared regularity equations in the previous section i use equations ( Note that in the absence of and therefore all and setting it to zero combined with the other equations impl However, in deriving the infraredused boundary the conditions zero in energy th condition ( We now start from aand solution argue that that eliminates theKS the UV IR solution expansion singulari of with this noneis solution but equivalent cannot to the match starting chargematch from non-normalizable the the mode anti-KS BPS solution t D3 in branes the in UV. the IR and a branes, the metric andand five-form we divergences expect are this singularitythat commensura to upon make taking perfect into sense accountthe in the smeared strin string-theoretic D3-branes, resolut the the weaker finite-action singula 6.2 IR obvious microscopic interpretation: they are sourced by a s JHEP09(2010)087 (6.28) (6.29) om the , , .   7 7 Y Y UV 7 he anti-D3 charge is + ) (6.27) ) exactly, and match Y , and have seen that + τ 6 6 8 ree form fluxes diverge Y = Y 6.1 rane charge throughout 4.26 ) + PY τ IR + n frared singularity, coming branes, we have computed o show that large numbers 7 8 ch solutions, with anti-D3 τ versa. ]) and there is a force on a τ Y − 5 uctions of de Sitter vacua in 5 -Strassler solution which re- ]. While this is all somewhat 5 singularity is better behaved 7 Y assler boundary conditions in PY n the infrared cannot match a Y = Y 13 ds that 7 − ergent energy density integrates + ) + we get 7 ) + 5 τ 5 Y Y ,Y Y 2 + 2 4.3.3 − τ UV sinh , 6 − 8 5 Y 8 Y = 0 τ . and = sinh PY PY + ( 6 5 ( τ Y τ Y IR τ 6 5 – 33 – )csch Y = Y − 4.3.2 6 , which is proportional to the effective D3-brane 5 τ Y (2 = 7 Y 5 τ Y 6 Y − + sinh + sinh τ (2 5 τ ,Y cosh Y 8 − 6 + (  cosh PY τ 2 τ  2 + 2 τ 2 IR cosh 5 just corresponds to a shift in the dilaton), cannot change fr csch 5 Y 1 2 Y 8 sech = Y 1 2 − − = = = UV 5 6 5 7 Y ˜ ˜ ˜ φ φ φ = 5 Y must be a constant. The next step is to solve the system ( 1 ˜ φ We have discussed this singularity in great detail in sectio We have found in our spectrum a single solution where some of t Comparing to the expansions in sections that from the fact thatquadratically the at energy the densities origin. of both the RRas and NS far th as first-orderthan other perturbation possible theory IR is singularities concerned, - this in particular its div probe D3 brane of theencouraging, same the form solution as the we one find computed must in [ necessarily have an in the infrared and ultraviolet integration constants. One fin Requiring no divergencies in the UV and IR sets Importantly, we see that the sign of charge (recall that string theory. Much work hasof been such done vacua using exist. this mechanism t dissolved in flux, supersymmetry is broken (as suggested in [ With a view towards identifyingthe the spectrum modes of sourced by linearizedspect anti-D3 fluctuations the around symmetries the ofbrane Klebanov boundary the conditions original in background. thethe infrared ultraviolet has and Finding been Klebanov-Str su a key unresolved step in many constr solution with anti-KS boundary conditions in the UV and vice 7 Discussion IR to the UV.Hence, the This solution implies that that preserves D3 this boundary solution conditions has i positive D3 b JHEP09(2010)087 . It is consequence th this solution and is singularity, to see tringy corrections could uments, the most direct ackground, since in that ]. If this is so, then this s one needs to relate the ng theory, or whether it is sis has found a first-order s quadratically. It may be ckground electric field as a e solution near the anti-D3 -brane or NS5-brane sources 32 tablish that this singularity e eliminated from the bulk e. Indeed, the breakdown of of this singularity is neither hat perturbation theory has of anti-D3 branes considered n, nor of the polarization of the IR diverges quartically in ctured metastable vacuum of e that the finite-action singu- n makes this singularity more uld be treated as other such s that in the ultraviolet can be 3 brane sources diverges. This ttom of the Klebanov-Strassler verging action but rather from n singularity is indeed resolved es in the RR and NS-NS fields solution also has a finite-action n for gravity does not provide a on, and it is very likely that this ies, or even their full supergravity ays dominates. The energy density ngularity we have found and render – 34 – ]. Our analysis does not fully establish that these 5 of the breakdown of general relativity but rather a signal On the other hand there is one reason that we should persist wi We have also tried to find a microscopic interpretation for th If the finite-action singularity is physical, than our analy It is clearly crucial to establish whether this finite-actio singularities in gravity and must be removed. perform further tests onlarity it is despite not the a singularity. It may b that one may expectresolve to this find singularity. in thea We infrared, have direct and argued consequence hence that whether ofanti-D3 the s branes the existence into expected NS5 brane-fluxconclusion branes annihilatio would in be the that infrared. this From finite-energy such singularity arg sho the fact that the derivativegood expansion perturbation in the theory. effective Hence, actio are divergent as energy unphysical densiti asbroken a down. divergent Ricci scalar, and indicate t whether the divergent fields can be explained as coming from D acceptable than others, itis cannot physical. be For used example, the assingularity, negative-mass a and Schwarzschild criterion yet to thisspectrum es in solution order is forgeneral unphysical, the relativity gauge/gravity and at duality a must to singularity make b does sens not come from the di to a finite action. However, even if the finiteness of the actio anti-D3 branes source onlyUV and normalizable IR modes integrationsolution - constants, would to which represent prove the is thi supergravity quite dual nontrivial of [ the conje solution represents the backreactionby of Kachru the Pearson smeared and system Verlinde in [ this way and could benot acceptable acceptable and from must the be point removed. of view ofbackreacted stri solution that has anti-D3 branesolution. charge at The the objects bo with anti-D3treated brane as charge source a mode perturbation of the Klebanov-Strassler soluti the IR, whereas thethat energy the stringy density resolution of ofbackreaction, the the will anti-D3 three brane also singularit resolve forms the diverge milderthe finite-action whole si solution physical. branes by doing perturbationregion the theory energy around in the theis fields similar original sourced to by KS the the explicit b perturbation, fact anti-D since that near one the electron cannotof its treat electric the an field fields alw electron sourced in by the a explicit ba anti-D3 brane sources in clear that one can never completely capture the physics of th JHEP09(2010)087 this 15 able vacua one step s when one tries con- alysis establishes that hat diverge in the in- , this result would be nd changes sign as one ation around the BPS it is is fact not once one ot a perturbation of the mensional dynamics, and al direction, coming from amics, and a perturbation le modes (other than the ompatible solution, his is a first hint that the ued in the Introduction, all s from the BPS one by a pergravity solution dual to ull solution consistent with ing the vacuum expectation of the unperturbed solution. crease our understanding of suggest that the dual gauge wed but the anti-D3 branes nal KS solution. radius. It is not hard to see ple to those modes, and hence garithmically with the energy. as a perturbation of the latter. ity: the anti-D3 branes in the s confirmed by the subsequent d in flux in the KS background he solution whose infrared has at the bottom of the Klebanov- o characterize very precisely the with a running coupling constant solution for the back-reacted anti-D3 the ]. First, if one tries to obtain this vacuum by 4 – 35 – ]. 4 Furthermore, this behavior would mimic exactly what happen While this conclusion might seem at first glance unexpected, On the other hand, if this singularity is not physical, our an If one takes the analogy with the MQCD construction of metast If the finite-action singularity is excluded, or if it is allo Given by eq. (3.22) in [ 15 to source perturbations thatthat grow the energy logarithmically of with such logarithmic the modes is as strongstructing as that an MQCD metastable vacuum [ is such modes does notsupergravity decay solutions at infinity. dual Indeed, to as four-dimensionalmust we theories have have arg bulk modesthe that fact depend that logarithmically the couplingHence, on it constants the is of radi quite these natural theories to run expect the lo anti-D3 branes to cou the solution corresponding to backreactingStrassler anti-D3 branes background cannotvacuum. be treated Hence, incharge) these the whose anti-D3 energy UV branes is as at source least a non-normalizab as perturb strong asrealizes that that of some the of origi the modes may have two-dimensional dyn a metastable vacuum ofvalues a of four-dimensional various theory. operators, this Byphysics solution explor would of allow this one metastable t gauge vacuum, theory and metastable would vacua in quite general. likely in the Klebanov-Strassler field theory, and would be the first su anti-Klebanov-Strassler solution.solution Much differs like from the thelatter MQCD BPS and IR-c one is by BPS log-growing by modes, itself. it is If n this is indeed This is again aonce reflection excited of in the the infrared fact it that does this not mode decayfurther, at has one infinity. two-di can alsoanti-D3 branes identify in an the obvious infrared candidate and for no the finite-action singular f frared, and cannot matchfully-backreacted the metastable vacuum correct might boundary notbackreacted conditions. exist, calculation. and T i Furthermore, ifregular one boundary tries to conditions, findlogarithmically-growing one t mode, and finds that cannot a be solution obtained that differ perturbation theory around the BPS one, one also finds modes t changes the orientation of these branes. source modes that aretheory nonnormalizable, does this not would strongly have a metastable vacuum. Though unexpected branes, this would imply thatmust the always sign of have the the charge same dissolve sign of the D3 branes at the bottom, a JHEP09(2010)087 ] . 5 g 33 (A.2) = (A.1) 3 ]). Any , ǫ 34 . as defined ) ] 3 a ξ g 12 ich appear in [ by + . i ] by lnr 3 1 g RCS07-12 and by g d τ/ 20 ( . Gomez-Reino, S. 1 e p , 2 orted in part by the 1 = √ +4 17 o find any microscopic 0 r , rature definitively show x = + e have constructed is the 16 d in the Introduction, this em of the existence of KS 0 2 les used in other papers is a in string theory, and the p , -action singularity is physi- particularly H. Verlinde for 3 ly prove that such a vacuum na, J. Polchinski, N. Seiberg, e , e from a near-brane analysis of 12 rn may invalidate many string singular behavior in the fields 6 m structure of this theory [ . ) would like to thank Princeton he UV , 4 √ 3 g ] are related to rule out the existence of KS anti- ending the analysis of [ rity is physical would ideally come = + 16 ] 2 g 16 [ of [ ( 2 2 du 1 to indicate that we refer to ) √ , e 1 ] p 1 = + 20 [ , e 0 1 3 p a ξ 4( – 36 – , ǫ , e e 2 ) 3 , ǫ = 1 g ] ǫ 3 [ + 1 dτ g 1 p − 4 ( e 2 1 = √ ] 17 = [ 2 dτ is related to the radial variables in [ 1 p , ǫ τ 4 ) e 4 g = ]. This is to try to keep straight the different conventions wh ] + 20 2 20 [ g dτ − ( 2 = 1 √ dτ Our analysis crystalizes and reduces the complicated probl It is fair to say that neither our analysis, nor any in the lite = 1 ǫ in reference [ consistent with the fact that a careful analysis of the vacuu the various papers onthe the following. subject The . angular variables The relation to the variab the Marie Curie IRG 046430. A Conventions We use the somewhat unsightly notation Kachru, I. Klebanov, G. Horowitz,D. S. Shih, Kuperstein, F. J. Vernizzi Maldace andchallenging E. debates Witten regarding for theUniversity useful infra-red discussions for physics. and hospitality. N.H. DSM The CEA-Saclay, work by of the ANR IB, grants MG BLAN and 06-3-137168 NH and was JCJC E supp developments on these issues would clearly be very exciting Acknowledgments We would like thank O. Aharony, C. Charmousis, A. Dymarsky, M metastable vacua to thecally simpler acceptable problem or of not. whether Establishingfrom the whether this a finite singula microscopic understanding ofthe the fully infrared backreacted, localized physics or solution (for example ext that the finite-action singularity isD3 unphysical, and metastable hence vacua. Onarguments the or other singularity-resolution hand, arguments wemay of have be why not this acceptable, been able whichsupergravity t dual would of establish the that KSexists. metastable the vacuum, solution and implicit w has not found such awould vacuum. probably Furthermore, affect as we the haveexistence KKLT explaine of construction a of landscape deSitter ofphenomenology such vacu vacua and in string general, cosmology which constructions. in tu To zeroth order in perturbation this last relation reads in t Our radial variable JHEP09(2010)087 (A.4) (A.3) ) 4 = 0 and τ ( y ( ) (B.1) O 2 . τ + ( a, b ) 3 O 2 τ τ ) ( + lnr + a IR τ O 7 , ] and ] X ! + = 20 20 [ τ − IR Y y 6 ) IR 7 X IR = = 5 4 τ X + X ) 12 ( 4 , , ( 4 ] ] 3 used in [ O τ P IR − 5 ( 12 12 [ 9 2 [ + X O k k IR 5 τ + + − 2 X 2 = 6 = 6 3 2 τ P ] ] τ + . P 3 ] q, y, p, Y 2 ! 17 17 1 / [ [ 3 τ IR 3 ! 1 ,A 17 0 / k [ f
) 1 IR ) h 6 X ! F 4 3 2 0
4 IR 6 3 8 6)) τ X 2 3 h τ ) = 2 15 = 2 ( ( X IR 6 √ ] ] = 2 + + – 37 – 15 O IR IR 15 7 1 O X ] + 12 20 20 2 IR [ [ 1 , y X X + IR 20 τ + 2 5 + [ 6 X IR 5 2 2 P k P f X ! 16 ln6 + ln(3 − 2 2 τ X P IR 2 5 56 PF are related to b 2 + τ 3 + − − − X /

+ IR 2 6 P 2 1 0 − a
2 = = = 2 3 IR P h + 5 2 X + P a / 3 2 0 3 P 3 1 3 k 2 − 0 f X / h F − (2 /
3 IR 1 h 5 1 ] (relations to the latter are only valid in the UV) by P / 3 3 2 3 1
+ = 0
1 3 IR 3 1 X 3 2 IR x, y, p, A in the 3-form fluxes are related by 0 5 ] 3 h 2
/ 12 = h 1 2 3 P X ˜ ξ 20 X + [
IR 1 3 3 45 q 0 is the following 2 3 p / 16 16 (5 h IR 1 X 1 a IR − 6 IR 1
˜ ˜ P ξ ξ + τ ] X − 3 2 + 2 f, k, F X 6 16 1 X IR 0 1 1 20 [ q 4 τ 32 h = 0) in [ IR 1 + X + + p 2

c −

8 IR 1 X = = 3 IR

5 IR IR IR 5 3 4 8 1 − P X p x X 8 − X + X X

+ ======6 5 7 3 4 8 1 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ξ ξ ξ ξ ξ ξ ξ Our metric functions The functions in the solution B.1 IR behavior of B Behavior of The IR behavior of JHEP09(2010)087 ) ] τ , 3 − e ( ommons O ) + ) (B.2) 3 τ 2 τ τ ) 3 ( − τ − e 3 O e  − ( hep-th/0007191 e + [ ) ( O UV 2 7 3 O τ + X τ/ τ + ! 10 + ) − τ − (1997) 658 e 6 e − IR ! ( 6 e X  mmercial use, distribution, ) r(s) and source are credited. O (2000) 052 X τ  + IR UV 6 7  + τ B 507 UV 08 4 UV ) X 7 ) + X IR 7 τ X 5 2 IR 5 X + 1 UV 3 7 1 2 6 X + ( PX X IR ) 5 + + JHEP 4 τ P 3 3 X , 5 3 UV 6 ( ]. + 3 − − UV 2 UV 6 X e 23 + 140 P + ( ) Nucl. Phys. X IR 3 X UV 3 − 6 + , − 3 O ( IR 4 X − τ/ 3 X + τ/ Dynamical SUSY breaking in meta-stable vacua 4 3 + 4 X τ IR 10 4 UV τ/ 5 SPIRES − – 38 – 4 τ − − τ  UV − X 5 e X + ][ − e e − ) ( e e UV 5 ) X 2 3 + −  UV IR 3 5 5 UV O 1 ! X P ) τ/ 2 ) UV 1 Supergravity and a confining gauge theory: Duality τ/ X 3 UV + 2 ) + 10 P / X 10 3 2 PX PX UV 7 1 3 − UV 7 3 X 4 0 − + (3
e / 3 UV 7 τ/ X − e 25 ( + 1 3 h 2 X 2 4 + ( −
2 0 X τ ]. τ O − − 3 2 + τ ) O ) 15 IR h 2 e − −
+ IR 3 1 ) UV  7 X τ UV UV UV τ 3 5 7 7 2 ˜ X IR ) + ξ 1 hep-th/0602239 UV [ 6 UV − UV 5 τ/ X X τ X 7 12 is the following + 2 PX X 4 ( e 56 X SPIRES X τ − a − 12 − − ( ˜ 16 − ξ 2 e + ][ IR 4 PX − τ + 5 − τ  2 (3 UV 5 X + SB-resolution of naked singularities UV 5 + This article is distributed under the terms of the Creative C 4 UV 1 IR  1 1 3 χ (3 UV + 5 X UV 18 3 X 3 X − UV X 1 IR ( τ 1 − Branes and the dynamics of QCD + 9 X 2( UV X 3 (2006) 021 e 2 τ τ/ ( P X P τ 

X 2 4 2 3 ]. X e − e P − − − UV + +

+ 2 3 7 − 04 e  + UV  7 X − P = UV UV UV UV UV 2 1 1 4 5 8 3

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