Non-supersymmetric infrared perturbations to the warped deformed conifold
aa r X i v : . [ h e p - t h ] S e p Preprint typeset in JHEP style - HYPER VERSION
MAD-TH-09-08
Non-supersymmetric infrared perturbations tothe warped deformed conifold
Paul McGuirk, Gary Shiu, , and Yoske Sumitomo , Department of Physics, University of Wisconsin, Madison, WI 53706, USA Institute for Advanced Study, Hong Kong University of Science and Technology,Hong Kong, People’s Republic of China [email protected], [email protected], [email protected]
Abstract:
We analyze properties of non-supersymmetric isometry-preserving pertur-bations to the infrared region of the warped deformed conifold, i.e. the Klebanov-Strassler solution. We discuss both perturbations that “squash” the geometry, sothat the internal space is no longer conformally Calabi-Yau, and perturbations thatdo not squash the geometry. Among the perturbations that we discuss is the so-lution that describes the linearized near-tip backreaction of a smeared collection ofD3-branes positioned in the deep infrared. Such a configuration is a candidate gravitydual of a non-supersymmetric state in a large-rank cascading gauge theory. AlthoughD3-branes do not directly couple to the 3-form flux, we argue that, due to the pres-ence of the background imaginary self-dual flux, D3-branes in the Klebanov-Strasslergeometry necessarily produce singular non-imaginary self-dual flux. Moreover, sinceconformally Calabi-Yau geometries cannot be supported by non-imaginary self-dualflux, the D3-branes squash the geometry as our explicit solution shows. We also brieflydiscuss supersymmetry-breaking perturbations at large radii and the effect of the non-supersymmetric perturbations on the gravitino mass.
Keywords:
D-branes, supergravity, supersymmetry breaking. ontents
1. Introduction 12. Supergravity ansatz 5
3. Non-SUSY deformations from localized sources 7
4. Regular non-supersymmetric perturbations 18
5. Non-SUSY solutions in the KT region 21
6. Gravitino mass 257. Discussion 28A. Conventions 31B. Complex Coordinates 33
1. Introduction
Among the challenges in connecting string theory to our observable universe is thedifficulty in constructing controllable supersymmetry-breaking backgrounds. Whilesupersymmetry (SUSY) breaking is a prerequisite in any phenomenological study of– 1 –our-dimensional supersymmetric theories, the myriad of string theory moduli makesthis a formidable task. Unless all moduli are stabilized at a hierarchically higher scalethan the scale of SUSY breaking, one generically finds runaway directions that desta-bilize the vacuum, taking us away from the controllable background which describesthe original supersymmetric state.On top of this challenge, the observational evidence of an accelerating universeadds yet another layer of complication: in addition to the requirement that the SUSY-breaking background be (meta)stable, viable vacua must also have positive energydensity. Motivated by this cosmological consideration, several mechanisms to “uplift”the vacuum energy of string vacua have since been suggested, e.g., by adding D3-branes [1], by introducing D-terms from gauge fluxes [2], or by considering negativelycurved internal spaces [3, 4, 5, 6] (see also [7, 8, 9]). Though these mechanisms areoften discussed in terms of 4D effective field theories, it is of interest for a varietyof reasons discussed below to find backreacted supergravity solutions including suchuplifting sources as full 10D backgrounds.In this paper, we report on some properties of non-supersymmetric perturbationsto the Klebanov-Strassler (KS) solution [10], a prototypical warped supersymmetricbackground which is dual to a cascading SU ( N + M ) × SU ( N ) gauge theory in thestrong ’t Hooft limit, and is ubiquitous in flux compactifications and in describing mod-uli stabilization. The backreaction of a collection of D3-branes placed at the tip of thedeformed conifold should be describable by such perturbations. Such a configurationis known to be metastable against brane/flux annihilation provided that the numberof D3-branes is sufficiently small in comparison to the background flux [11]. Thoughfurther instabilities generically arise upon compactification when the closed string de-grees of freedom become dynamical and further stabilization mechanisms (e.g., fluxes,non-perturbative effects, etc) are needed, this local construction represents progresstowards a genuine metastable SUSY-breaking background. Other than being an es-sential feature in [1] for vacuum uplifting to de Sitter space, the warped D3 tensionintroduces an exponentially small supersymmetry breaking scale which can be usefulfor describing hidden sector dynamics (both in dimensionally reduced theories and intheir holographic descriptions).Although we are interested especially in modes related to D3-branes, the analysiswith more general modes brings us interesting features for the classification of near-tipperturbations. We analyze perturbations that are either singular or regular and thosethat either do or do not “squash” the geometry (i.e. those that do or do not leavethe internal geometry as conformally the deformed conifold) in accordance with theequations of motion. We also identify which modes can break supersymmetry. Themode related to D3s at the tip should have singular behavior, at least in the warp– 2 –actor, in order to capture the localized tension. We show below however that the onlysingular, non-squashed, non-SUSY mode corresponds to a point source for the dilaton,and thus cannot be identified as an D3-brane. Furthermore, the squashed backreactionof an D3-brane is supported by a 3-form flux that is no longer imaginary self-dual (ISD).The fact that an D3-brane squashes the geometry was observed in [12] where the D3-brane backreaction was studied in the Klebanov-Tseytlin (KT) region [13]. However,due to the decreased complexity of the geometry, the squashing of the geometry in [12]is less dramatic than the squashing in the near-tip region. Likewise, the resulting non-ISD flux near the tip is more complex than the non-ISD flux supporting the solutionof [12]. We also discuss these issues in the KT region.Other than the consideration of D3-branes, the existence of non-SUSY fluxes isinteresting to consider for many reasons. It is well known (see for example [14, 15,16, 17]) that such non-SUSY fluxes can give rise to soft SUSY-breaking terms in 4Deffective theories. Additionally, non-SUSY fluxes can play an important role in thecontext of D-brane inflation . While the deformed conifold can support certain non-SUSY fluxes (at least to the level of approximation at which we work), we show belowthat in order to have any non-ISD flux, the geometry must be squashed so that it is nolonger Calabi-Yau.Perturbations to the KS solution appear in many other places in the literature(and indeed most of our solutions have appeared elsewhere though previously nonehad been identified as describing the presence of D3-branes). Using an alternativeparametrization [24, 25] of the ansatz that we present below, the linearized equationsof motion for perturbations to the KS geometry have been written elsewhere as a systemof coupled first order equations, solutions for which can be written formally in termsof integrals [26, 27]. Although writing the equations of motion in this way can beconvenient, we choose to work directly with the linearized second order equations. Thesecond derivative equations were also directly solved in [28, 29], though we relax someof the assumptions made in those references. Analysis of perturbations to KS also arisein studies of the glueball spectrum of the dual theory [30, 31, 32].There are several other reasons why we are interested in analyzing non-SUSY per-turbations to the near-tip region of KS. First of all, being closest to the source of SUSYbreaking, this is the region where the supergravity fields are most affected. More-over, as is common in warped compactification, the wavefunctions of non-zero modes(e.g. the gravitino after SUSY breaking) tend to peak in the tip region. Thus, ourperturbative solutions are useful in determining the low energy couplings (includingsoft masses) in the 4D effective action involving these infrared localized fields. Addi- For recent reviews, see, e.g., [18, 19, 20, 21, 22, 23]. – 3 –ionally, as discussed in a companion paper [33], the backreacted D3 solution in thenear tip region provides a holographic dual of gauge mediation in a different parameterspace regime from that of [34]. As a result, strongly coupled messengers (and not onlyweakly coupled mesonic bound states) of the hidden sector gauge group can contributesignificantly to visible sector soft terms. Given the aforementioned applications, it isof importance for us to consider warped geometries which are infrared smooth beforeperturbations. Since we are focusing on the near tip region, our starting point is the KSsolution which provides a more accurate description at small radius than KT. Althoughthe KT background correctly reproduces the cascading behavior of the field theory, itbecomes singular in the IR where the effective D3 charge (which is dual to the scaledependent effective ’t Hooft coupling) becomes negative and the cascade must end. Theappropriate IR modification is the KS solution which is built on the deformed conifoldso that the solution is smooth even in the IR.This paper is organized as follows. In Section 2, we discuss our solution ansatzand express the KS solutions in accordance with this ansatz. In Section 3, we examinesingular perturbations to the warped deformed conifold and describe how we obtain theperturbative solution corresponding to placing a D3-brane point source in the warpeddeformed conifold. We also clarify that solutions where the internal space is unsquashedshould satisfy the ISD condition and cannot describe the backreaction of an D3-brane.In Section 4, we present regular solutions which also break supersymmetry, but do notcorrespond to the backreaction of a localized source. In Section 5, we present solutionsin the KT region, both with and without the ISD condition imposed. We calculatethe gravitino mass in these SUSY-breaking warped backgrounds in Section 6 and endwith some discussions in Section 7. Some useful details about our conventions and thecomplex coordinates of KS are relegated to the appendices.We note that after the completion of this paper, another preprint [37] that addressesthe question of adding D3-branes to the geometry was made available. Our treatmentof the D3-brane differs from [37] by the boundary conditions imposed in the IR aselaborated in Sec 3.4. By this we mean that, at least before the addition of 3-branes, the warp factor approaches aconstant, or, equivalently, the (minimal surface) dual Wilson loop [35, 36] has a finite tension. Additionally, the equations of motion in [37] are formally solved for all radii and would thus beuseful for further analysis connecting the IR and UV regions. – 4 – . Supergravity ansatz
In this section we give the ansatz that we will use for the metric and other fields,working in the Einstein frame of IIB supergravity. Our conventions are presentedin Appendix A. Since we are considering perturbations to the KS solution [10] (seealso [38]), our ansatz will be based on that solution. In particular, since we are lookingfor perturbations that preserve the isometry of KS, we take the metric ds = h − ( τ ) dx µ + h ( τ ) d ˜ s , (2.1a) d ˜ s = p ( τ ) dτ + b ( τ ) g + q ( τ )( g + g ) + s ( τ )( g + g ) , (2.1b)where τ is the radial coordinate and where the angular one-forms g i are reviewed inAppendix A. This metric ansatz includes the warped deformed conifold as a special caseby a certain choice of p , b , q , and s presented in the next section. For the axiodilatonwe take Φ =Φ( τ ) C = 0 , (2.2a)while for the fluxes, B = g s M α ′ f ( τ ) g ∧ g + k ( τ ) g ∧ g ] ,F = M α ′ − F ( τ )) g ∧ g ∧ g + F ( τ ) g ∧ g ∧ g + F ′ ( τ ) dτ ∧ ( g ∧ g + g ∧ g )] ,F = (1 + ∗ ) F , F = g s M α ′ ℓ ( τ ) g ∧ g ∧ g ∧ g ∧ g . (2.2b)These choices of fluxes respect the isometries of the deformed conifold and satisfy theBianchi identities dF = 0 , dH = 0 . (2.3) The KS solution [10] corresponds to placing M fractional D3-branes at a deformed con-fiold point (i.e. wrapping M D5-branes around the collapsing two-cycle) and smearing– 5 –hese branes over the finite S . It is recovered by the choice f KS ( τ ) = τ cosh τ − sinh τ ( τ / , k KS ( τ ) = τ cosh τ − sinh τ ( τ / , F KS ( τ ) = sinh τ − τ τ ,ℓ KS ( τ ) = f KS (1 − F KS ) + k KS F KS ,p KS ( τ ) = b KS ( τ ) = ε / K ( τ ) , q KS ( τ ) = ε / K ( τ ) cosh τ , (2.4) s KS ( τ ) = ε / K ( τ ) sinh τ , Φ ( τ ) = log g s , h KS ( τ ) = ( g s M α ′ ) ε − I ( τ ) ,K ( τ ) = (sinh(2 τ ) − τ ) / / sinh τ , I ( τ ) = Z ∞ τ dx x coth x − x (sinh 2 x − x ) , (2.5)where ℓ KS is determined by requiring F = B ∧ F , which ensures that the solutionis regular. However, if a regular D3-brane is added to the geometry, then ℓ must havean additional constant part. This introduces a τ − part to the warp factor and thesolution becomes singular. On the field theory side, this corresponds to the loss ofconfinement.We ar[e interested in solving for perturbations to the KS background. Since thegeometry is already relatively complicated, we will consider perturbations to the ge-ometry as a power expansion about τ = 0. It is therefore useful to to note the seriessolutions for the KS solution near the tip, f KS = τ − τ
80 + 17 τ − τ τ · · · ,k KS = τ τ − τ τ − τ · · · ,F KS = τ − τ
720 + 31 τ − τ τ · · · , (2.6a)for the three-form fields, and p KS = b KS = ε / / / ε / τ / /
10 + ε / τ / / (cid:0) (cid:1) / ε / τ − ε / τ / / · · · ,q KS = ε / / / + (cid:0) (cid:1) / ε / τ + 17 ε / τ / / ε / τ / / − ε / τ / / · · · , – 6 – KS = ε / τ / / − ε / τ / /
240 + 59 ε / τ / / − ε / τ / / ε / τ / / · · · ,h KS = ( g s M α ′ ) ε − (cid:20) a − τ / + τ / − τ / / τ / · · · (cid:21) , (2.6b)for the squashing functions and warp factor. Note that each of these functions containspowers of τ with only one parity (e.g. h KS contains only terms of the form τ k ).The given expansions satisfy the dilaton equation of motion (A.3b) up to O ( τ ), theEinstein equation (A.3a) up to O ( τ ), the H equations (A.4a) and (A.4b) up to O ( τ )and O ( τ ), and the F equation (A.5) up to O ( τ ). The leading constant of h can becaluculated numerically, a = I (0) ≈ .
3. Non-SUSY deformations from localized sources
In this section, we present perturbations to the KS geometry that are solutions to thesupergravity equations of motion with singular behavior. One of these solutions corre-sponds to adding D3-branes to the tip of the geometry. We first argue why a singularsolution is necessary to describe the point source behavior of a D3. We then discusstwo solutions, the first in which the internal space remains the deformed conifold, andthe second in which the geometry is “squashed” away from this geometry. We alsomatch the parameters in the latter solution to the tension and charge of the 3-branes.
For simplicity, we seek solutions that retain the isometry of the KS solution. Becausethe S remains finite as τ = 0, in general placing a point source into the internalgeometry will break the angular isometry even at τ = 0. Therefore, in order to retainthe KS isometry, these point sources must be smeared over the finite S . Alternatively,we can consider a collection of point sources that to good approximation are uniformlydistributed over the S .The effect of such a localized source on the geometry can be estimated by con-sidering the Green’s function in the unperturbed background. Using the metric (2.1),an S -wave solution (i.e. dependent only on τ ) to Laplace’s equation ∇ H = 0 can bewritten with integration constants P and φH = P − Z ∞ τ dx φq ( x ) s ( x ) s p ( x ) b ( x ) , (3.1)– 7 –f the warp factor h is obtained by solving the Killing spinor equations (A.6) (moreprecisely (A.9)), then the form of h is similar to that of the Green’s function, h = P + ( g s M α ′ ) Z ∞ τ dx ℓ ( x ) q ( x ) s ( x ) s p ( x ) b ( x ) . (3.2)A localized source of D3 charge gives a constant piece to ℓ , and this constant pieceindeed generates a Green’s function in h . The ansatz (2.1) admits solutions that areeither asymptotically flat or AdS (up to possible log corrections) and setting P = 0corresponds to demanding the latter.The integral (3.1) cannot be performed exactly for the KS solution. However, usingthe expansion about τ = 0 given above, one can write for small τ , H = P + 2 φ ε − (cid:18) − τ − τ
15 + τ
315 + · · · (cid:19) . (3.3)Thus if we include a point source at the tip that respects the same supersymmetry asKS, then the geometry should become singular as τ →
0. In particular, the warp factorwill depend as τ − .Dropping the constant term, in the large radius region the Green’s function takesthe form (in terms of r = 2 / ε / e τ/ ) H = − / ε − φ e − τ + · · · = − φr + · · · . (3.4)If the object that is added to geometry breaks supersymmetry, then it needs notperturb the warp factor by simply adding a Green’s function piece. Indeed, in the largeradius region, where a Green’s function behaves as r − , adding D3-D3 pairs perturbsthe warp factor by r − [12, 39] (though there are also log corrections). Heuristically, thepresence of the non-supersymmetric source adds a perturbation that scales as δh ∼ h H where h is the unperturbed warp factor (if the charge of the non-BPS source is non-vanishing, then there will be a Green’s function contribution as well). Since the KSwarp factor approaches a constant, this suggests that even for non-supersymmetricsources, we should look for perturbations to the warp factor that behave as τ − . Thisargument is only very heuristic, though we are able to check using boundary conditionsthat this behavior is indeed correct. We consider perturbing the KS solution by taking the ansatz (2.1), (2.2) and writing f = f KS + f p ( τ ) , k = k KS + k p ( τ ) , F = F KS + F p ( τ ) , ℓ = f (1 − F ) + kF, Φ = log g s + Φ p ( τ ) , h = h KS + h p ( τ ) ,p = p KS , b = b KS , q = q KS , s = s KS , (3.5)– 8 –here the KS solution is (2.4) and the subscript p indicates a perturbation to KS. Suchan ansatz changes the warp factor and the fluxes, but leaves the internal unwarpedgeometry as the deformed conifold. We do not attempt to solve for the perturbationsexactly, but write them as a power series about τ = 0. We then solve the equationsof motion (A.3) to first order in the perturbations and order-by-order in τ . To linearorder in perturbations the coefficients for the odd powers of τ in h p decouple from thecoefficients for the even powers. As argued above, to capture the behavior of a pointsource, the warp factor ought to behave as τ − , implying that we should focus on theodd powers in τ in h p . We find the solutionΦ p = φ (cid:18) τ + 2 τ − τ
315 + 2 τ (cid:19) ,F p = φ (cid:18) − − τ
720 + τ (cid:19) + U (cid:18) τ − τ τ − τ (cid:19) ,f p = φ (cid:18) τ − τ
80 + 61 τ (cid:19) + U (cid:18) − − τ τ − τ (cid:19) + H ,k p = φ (cid:18) τ + 94 − τ − τ (cid:19) + U (cid:18) − τ − − τ
120 + τ (cid:19) + H ,h p = ( g s M α ′ ) / ε − / / (cid:20) φ (cid:18) τ + 206 τ − τ (cid:19) + U (cid:18) − τ − τ
25 + 208 τ (cid:19) + H (cid:18) τ + 2 τ − τ
315 + 2 τ (cid:19)(cid:21) . (3.6)This solution is valid to linear order in the parameters φ , U , and H . It can be extendedto higher order in τ by expressing the higher order coefficients in terms of φ , U , and H so that no additional parameters need to be introduced. These perturbations satisfythe dilaton equation (A.3b) up to O ( τ ), the Einstein equation (A.3a) up to O ( τ ), andthe gauge equations (A.4a) up to O ( τ ), (A.4b) up to O ( τ ), and (A.5) up to O ( τ ).It is worth noting that even if we allow a perturbation to b ( τ ), which describeslimited squashing of the internal space (more general squashing is considered below),the solution (3.6) does not change and b remains unperturbed ( b p = 0). The squashingof this direction was considered in [12] to obtain a non-SUSY deformation of KT space,but in the KS region, there is no solution in which only this direction is squashed.To this order in the perturbations and in τ , the solution (3.6) respects the ISDcondition (A.8) of the 3-form flux as well as the first derivative SUSY condition forthe warp factor (A.9), even though the solution follows from solving second derivativeequations. However, we expect (and indeed we have checked to several higher ordersin τ ) that the flux remains ISD to all orders in τ since the dilaton takes the form of aGreen’s function (3.3). If the flux had an IASD component as well, then in general the– 9 –uxes would provide a potential for the dilaton and Φ would no longer satisfy ∇ Φ = 0.Indeed, since Φ does have the same form as (3.1), we can identify φ as correspondingto some point source for the dilaton smeared over the finite S at τ = 0.Some non-SUSY perturbations to the KS solutions, found by solving the first orderdifferential equations given in [24], were analyzed in [26, 27]. For φ = 0 (i.e. constantdilaton), the solution (3.6) is a small τ expansion of the exact solution appearing in [27],the flux part of which is F (cid:0) τ (cid:1) = 12 (cid:16) − τ sinh τ (cid:17) + U sinh τ + 5 T (cid:16) cosh τ − τ sinh τ (cid:17) ,f ( τ ) = τ cosh τ − sinh τ ( τ / − U ( τ /
2) (5 + 8 cosh τ ) + H
6+ 5 T
128 cosh ( τ /
2) (2 τ + 4 τ cosh τ − τ − sinh 2 τ ) ,k ( τ ) = τ cosh τ − sinh τ ( τ / − U ( τ /
2) ( − τ ) + H
6+ 5 T
128 sinh ( τ /
2) ( − τ + 4 τ cosh τ − τ + sinh 2 τ ) . (3.7)The solution is singular for non-vanishing U or H which are essentially the same pa-rameters that appear in (3.6), though (3.7) is an exact solution of (A.3) to all ordersin U , H , and T . The remaining parameter T appears in another solution (4.1), and ofthe parameters of (3.7), only a non-vanishing T leads to supersymmetry breaking. Theadditional parameter φ appearing in (3.6) comes from relaxing the condition that thedilaton Φ is constant. Note also that the parameter H is related to the gauge symmetry B → B + d Λ .To check if supersymmetry is preserved, we consider the SUSY variations of thegravitino and dilatino (A.6), taking into account the non-trivial dilaton profile. Since G is ISD, the last term of the dilatino variation (A.6b) vanishes. However, the termsinvolving the derivative of the dilaton do not. Indeed for small τ , δλ ∼ − i / φ / a / ( g s M α ′ ) / τ ˆΓ τ ǫ + · · · , (3.8)where ˆΓ indicates an unwarped Γ-matrix. Since this variation is non-vanishing, thesolution (3.6) breaks supersymmetry.The variation for the gravitino is also non-vanishing since the solution includes a(0 ,
3) part of G . From (B.8), we see that for the solution (3.6), G (0 , = φ (cid:18) τ + 115 τ − τ · · · (cid:19) ( z i d ¯ z i ) ∧ ( ǫ ijkl z i ¯ z j d ¯ z k ∧ d ¯ z l ) . (3.9)– 10 –s shown in Sec. 4.1, the exact solution (3.7), for which φ = 0, has an additionalcontribution to the (0 ,
3) part from T . For the perturbation (3.6), the (3 ,
0) and (1 , ,
1) and (0 ,
3) components.Both the variation of the dilatino and gravitino involve only φ . Therefore, eventhough the singular behavior seems to imply that U and H can be associated with apoint source, they do not break supersymmetry (though (3.7) breaks supersymmetryfor non-vanishing T ) and only φ is a possible candidate to describe the presence of alocalized SUSY-breaking source. The parameter φ characterizes a localized source forthe dilaton and therefore cannot correspond to the presence of D3-branes since D3sdo not directly couple to the dilaton. Furthermore, it was shown in [12] that an D3squashes the geometry so that it is no longer conformally Calabi-Yau. Extrapolatingthis result to short distances, the source associated with φ , which does not squash thegeometry, should therefore not be identified with an D3-brane. Indeed, this mode isthe small radius analogue of the r − mode for the dilaton that appeared in [12] (aswell as the flat space analysis of non-BPS branes in [39]) which could be turned offindependently of the existence of D3-branes as it does not contribute to the total massof the solution.Note that this solution possesses a curvature singularity at τ = 0; at small τ theRicci scalar behaves as R = −
45 2 / H + (45 2 / − / a )(12 U − φ )303 / a / g s M α ′ τ + · · · . (3.10)The presence of the curvature singularity indicates a breakdown of the supergravityapproximation, and so our solution is only expected to be valid for 1 / ( g s M α ′ ) ≪ τ < τ expansionand the lower bound comes from assuming that R is small in string units. We can generalize by considering solutions that “squash” the internal geometry so thatthe unwarped geometry is no longer that of the deformed conifold. At large distanceswhere the DKM solution [12] is valid, the only non-trivial squashing that occurs dueto the presence of a D3-brane is in the direction on which the U (1) isometry acts .However, as discussed in the previous section, at small radius the equations of motiondo not admit a solution in which the only squashing is in this direction. Thus, we In actuality, this U (1) isometry is broken to a discrete subgroup by the fluxes and deformation ofthe conifold singularity. – 11 –onsiderΦ = log g s + Φ p ( τ ) , h = h KS + h p ( τ ) ,f = f KS + f p ( τ ) , k = k KS + k p ( τ ) , F = F KS + F p ( τ ) , ℓ = f (1 − F ) + kF,b = b KS (1 + b p ( τ )) , q = q KS (1 + q p ( τ )) , s = s KS (1 + s p ( τ )) , p = p KS , (3.11)where we have used the freedom to redefine τ to keep p unperturbed but have allowed b , q , and s to be general so that the ansatz is the most general ansatz consistent withthe isometry of KS. This more general ansatz will allow G to have both ISD and IASDcomponents. We are again interested in describing the effect of a localized source andsince the even and odd powers of τ in the warp factor decouple from each other, wefocus on odd powers of τ in h p . We find a power series solution to (A.3) where thedilaton obtains a non-trivial profile that is regular at small τ Φ p = S τ + Y τ . (3.12a)However, the squashing functions for the solution are singular b p = S (cid:18) τ − τ (cid:19) + Y (cid:18) τ − τ (cid:19) + B (cid:18) τ − τ (cid:19) ,q p = S (cid:18) τ + 103 τ − τ (cid:19) + Y (cid:18) τ + 70 τ − τ (cid:19) + B (cid:18) τ − τ (cid:19) ,s p = S (cid:18) τ − τ
720 + 29999 τ (cid:19) + Y (cid:18) τ − τ τ (cid:19) + B (cid:18) τ − τ
12 + 529 τ (cid:19) . (3.12b)Similarly the fluxes are F p = S (cid:18) τ + 31235 τ / a + (cid:18) − −
299 3 / a
70 2 / (cid:19) τ (cid:19) + Y (cid:18) τ + 72 6 / a τ + (cid:18) − −
69 3 / a / (cid:19) τ (cid:19) + B (cid:18) τ − τ (cid:19) , – 12 – p = S (cid:18) − − / a
700 2 / + 13 3 / a / + (cid:18) − −
321 3 / a
112 2 / (cid:19) τ + (cid:18) / a / (cid:19) τ + (cid:18) − a / (cid:19) τ (cid:19) + Y (cid:18) − − / a / + 21 3 / a / + (cid:18) − −
165 3 / a / (cid:19) τ + (cid:18) / a
40 2 / (cid:19) τ + (cid:18) − a
640 6 / (cid:19) τ (cid:19) + B (cid:18) − − τ − τ
672 + 307 τ (cid:19) ,k p = S (cid:18) − τ −
513 3 / a
35 2 / τ − − / a
700 2 / + 13 3 / a / + (cid:18) − / a / (cid:19) τ (cid:19) + Y (cid:18) − τ −
39 6 / a τ − − / a / + 21 3 / a / + (cid:18) − / a
40 2 / (cid:19) τ (cid:19) + B (cid:18) − τ − − τ (cid:19) . (3.12c)The warp factor resulting from the fluxes exhibits the desired singular behavior h p =( g s M α ′ ) ε − (cid:20) S (cid:18) − /
105 3 / τ − a τ + (cid:18) − / − a (cid:19) τ (cid:19) + Y (cid:18) − / τ − a τ + (cid:18) − /
45 3 / − a (cid:19) τ (cid:19) + B (cid:18) − / τ −
29 2 / τ / (cid:19)(cid:21) . (3.12d)Perturbations that respect the ISD condition and were presented in the previous sectionhave been omitted. Again, S , Y , and B are treated as perturbations and so the solutionis valid to linear order in these parameters and can be extended to higher order in τ without introducing any new independent parameters.Since the dilaton does not exhibit a τ − behavior, the nontrivial profile cannot beinterpreted as resulting from a localized source for the dilaton. Instead it comes from– 13 –he lift of the vacuum energy due to the presence of both ISD and IASD componentsof G (A.3b), H − e F = 48 6 / S√ a g s M α ′ τ + 40 S −
16 6 / a ( S − Y ) τ a / g s M α ′ + · · · . (3.13)The non-vanishing potential for the dilaton implies the existence of an IASD componentsince, for C = 0, ∇ Φ = − g s e − Φ × (cid:20) H − e F (cid:21) ∝ Re (cid:0) G + mnp G − mnp (cid:1) (3.14)where G ± = iG ± ˜ ∗ G . One can also see directly from (A.8) that G is no longerpurely ISD. The parameters controlling the deviation from the ISD condition (A.8)are S and Y . Both of these are included in the squashing functions, implying thatthe squashing of the deformed conifold is needed to have non-vanishing (3 ,
0) or (1 , G .As was the case for (3.6), the geometry exhibits a curvature singularity at τ = 0.Indeed at small τ the Ricci scalar behaves as R ∼ O ( B , S , Y ) g s M α ′ τ , (3.15)where we have omitted numerical coefficients since the essential behavior is τ − . Thissingularity implies that the solution is valid only for S / ( g s M α ′ ) ≪ τ < r − and r − log r compared to the KT geometry itself (e.g. the KT warp factor included r − and r − log r while the perturbations behaved as r − and r − log r ). Similarly, the solu-tion (3.12) involves perturbations that behave as τ − relative to the KS solution (2.4).We note that τ − and r − are the small and large radius expansions of the Green’sfunction (3.1). This, together with the shared properties mentioned above, is a hintthat the solution (3.12) may describe the backreaction of D3-branes. We can confirmthat this is the case by checking boundary conditions. We now seek to match the parameters of this solution to the tension of the D3-branesthat are localized at τ = 0. Since the solution is singular at τ = 0, we expect the solution– 14 –o be modified by undetermined stringy corrections at distances τ . S / ( g s M α ′ ). Wewill therefore not try to obtain the coefficients exactly.Following from the behavior of the Green’s function (3.3), the O (1 /τ ) behavior ofthe warp factor is tied to the existence of a localized source of tension. Indeed if thereis a collection of D3-branes and D3-branes located at τ = 0 and angular positions Ω i ,then they contribute to the stress-energy tensor as T loc µν = − κ T δ ( τ ) √ p KS b KS s KS q KS X i δ (Ω − Ω i ) √ ˜ g η µν , (3.16)where ˜ g is the angular part of the determinant of the unwarped metric and the othercomponents of T loc MN vanish. We make the approximation that there are enough 3-branes that we can treat them as uniformly smeared over the finite S at the tip. Thenintegrating over the S gives Z S d vol S T loc µν = − κ T ( N D3 + N D3 ) δ ( τ ) √ p KS b KS s KS q KS δ (Ω) √ ˜ g η µν , (3.17)where δ (Ω) fixes the angular position on the vanishing 2-cycle, ˜ g is the unwarpedmetric for that 2-cycle, and N D3 and N D3 are the numbers of D3 and D3-branes addedto the tip. This localized source of tension should cause a 1 /τ behavior in the warpfactor. Tracing over the Einstein equation in the presence of the localized source, wehave − τ ∂ τ (cid:0) τ ∂ τ h p (cid:1) ∼ κ T ( N D3 + N D3 ) δ ( τ ) r p KS b KS q KS s KS V , (3.18)where we have integrated over the angular directions and defined V = R d x √ ˜ g .Integrating over τ , we find that near the tip of the deformed conifold, h p ∼ ( N D3 + N D3 ) κ T ε − / V τ , (3.19)where h KS = h + O ( τ ). That is, the τ − coefficient in the warp factor is proportionalto the total tension of the 3-branes added to the tip. Using (3.12), we can use thisrelation to match the parameters to this tension.Similarly, one can match to the total charge added to τ = 0 by considering theconstant part of ℓ , ℓ ( τ = 0) ∝ N D3 − N D3 . (3.20)Since the solution (3.12) involves ratios of relatively large numbers, we omit the detailedform of this expression, but by some choice of parameters, we can take the solution– 15 –o correspond to adding negative charge. Thus, for some choice of S , B , and Y (asdiscussed below, one must additionally include the U -mode of (3.6)), the solution (3.12)corresponds to adding D3-branes to the tip of KS. Another choice of parameters allowsus to describe the influence of D3-D3 pairs which adds tension, but no net charge tothe solution and so is the small radius analogue of the solution presented in [12].Alternatively, we might match the parameters to the tension of the 3-branes bycalculating the analogue of the ADM mass. For spaces which do not necessarily asymp-tote to either flat space or AdS, a generalization of the ADM mass was given in [40].However, this is applicable only at large distances (and indeed was used in the largeradius solution [12]) while the solution (3.11) is valid for small τ . Although analoguesof the ADM mass exist for arbitrary surfaces, and not just those at infinity [41, 42], itis more efficient to match to the localized tension discussed above.The behavior of the 3-form fluxes in (3.12) gives rise to divergent energy densities H and F . In particular, the leading order behavior F p ∼ S τ − (for the remainder ofthe section, S will be used as short hand for linear combinations of S , B , and Y ) leadsto F ∼ S τ − . The contribution to the action then diverges since √ gF ∼ S τ − .Similarly, the τ − behavior of k p − f p gives H ∼ S τ − which also gives a divergentaction. Since the D3-branes do not directly source these fields, one should imposethat these very singular behaviors should be absent from the solution describing thebackreaction of D3-branes. Since two of S , B , and Y are fixed by matching to thetension and charge of the 3-branes, there is not enough freedom to cancel both ofthese divergences using just the modes in (3.12). However, these divergences can becancelled by additionally including the U -modes given in (3.6). Imposing this additionalcondition on S , B , Y , and U gives the leading order behavior F p ∼ S τ, k p − f p ∼ S τ . (3.21)From these, F ∼ H ∼ S τ . (3.22)That is, even after imposing that the most singular parts of the 3-form flux vanish,the energy densities H and F are divergent. Furthermore, these divergences cannotbe removed by including any of the other modes discussed here without setting all ofthese constants to be zero. However, these do not lead to a divergent action since √ gF ∼ √ gH ∼ τ .The fact that F and H are divergent may be at first be surprising since theD3s do not directly couple to the 3-form flux and thus the singularities in H and F – 16 –ave no obvious physical interpretation . Here, however, we suggest that such singularbehavior might have been anticipated from the equations of motion and the boundaryconditions. Indeed, the coupling between the 3-form and 5-form flux can be written as(see for example [43]) d Λ + i Im ( τ ) dτ ∧ (cid:0) Λ + ¯Λ (cid:1) = 0 , (3.23)where the external part of C has been written C = αdx ∧ dx ∧ dx ∧ dx , (3.24)and Λ = Φ + G − + Φ − G + (3.25)where Φ ± = h − ± α and G ± = iG ± ˜ ∗ G . Since in the KS background, τ isconstant and both Φ − and G − vanish, the second term in (3.23) is higher order inthe perturbations and the remainder of the equation implies d Λ = 0. To leading orderin perturbations, this implies Φ + δG − = − δ Φ − G + . (3.26)Although the D3-branes do not directly couple to G ± , they do directly source Φ − .Since the KS geometry has both Φ + and G + non-vanishing, this direct coupling impliesthat G − must be non-vanishing when an D3-brane is added. Furthermore, since δ Φ − will have singular behavior at small τ while Φ + and G + are regular, δG − must havesmaller powers of τ than G + . For example, in the KS background f − k ∼ τ whileΦ + ∼ τ . Since δ Φ − ∼ τ − , one might expect f p − k p ∼ τ . Due to the presence of theHodge- ∗ which will introduce the squashing functions into this analysis this argumentalone is not conclusive and one must solve the equations of motion as we did above.Nevertheless, it provides a heuristic argument for why this singular behavior for H and F is present in this solution. The intuition that an D3-brane should not result insuch singular behavior comes partially from the flat space case where G + = 0 and thisargument fails. Similarly, it fails for the addition of D3-branes to KS since D3s sourceonly Φ + and not Φ − .The backreaction of D3 s was also addressed in [37]. In [37], the existence of theconstant part of k p − f p and the linear part of F p , after imposing that the more singularparts vanish, was deduced from a slightly different logic. The authors used the fact that For example, it was argued in [37] that the resulting H has the wrong orientation and dependenceon the D3-charge to be due to the NS5-branes that were described in [11]. We use notation which is slightly different than the remainder of the paper to match with thenotation in [43] and to present (3.23) simply. – 17 – probe D3-brane in the geometry will be attracted to D3-branes at the tip . Under theassumption that the backreaction of the D3-branes could be described as a linearizedperturbation to the Klebanov-Strassler geometry with at least some non-normalizablemodes absent, it was shown in [37] that the existence of this force implies such behavior.It was then argued that this may imply that treating the D3s-branes as a perturbationto the Klebanov-Strassler background is not a valid procedure because the D3-branesdo not directly couple to H and F and that therefore the resulting singular H and F are unphysical. The point of view that we adopt is that although it is true thatadding D3-branes to KS or D3-branes to flat space will not result in such behavior, inlight of (3.23) it is not surprising that such modes exist when adding D3-branes to KS.Therefore, unlike the possibility discussed in [37], we do not impose that H and F are non-singular.
4. Regular non-supersymmetric perturbations
Here we present solutions which do not include a singular O (1 /τ ) behavior in the warpfactor. In this case, the warp factor is a power series in τ consisting only of even powers. The equations of motion (A.3) admit a solution that is regular and unsquashed. Takingthe ansatz (3.5), we find for the dilaton and fluxesΦ p = P ,F p = P (cid:18) − τ − τ − τ (cid:19) + T (cid:18) τ
48 + τ
288 + 13 τ (cid:19) ,f p = P (cid:18) τ − τ
240 + τ (cid:19) + T (cid:18) − τ
192 + τ (cid:19) ,k p = P (cid:18) − τ − τ − τ (cid:19) + T (cid:18) τ
12 + τ
36 + 5 τ (cid:19) , (4.1a)while the warp factor is h p = 2 ( g s M α ′ ) ε − (cid:20) A + T (cid:18) − τ /
24 + 5 τ / (cid:19)(cid:21) . (4.1b)Again, these solutions are valid up to linear order in the parameters A , P , and T .The solution satisfies the dilaton equation (A.3b) up to O ( τ ), the gravitational equa-tions (A.3a) up to O ( τ ), the H equations (A.4a) and (A.4b) up to O ( τ ) and O ( τ ), Of course, requiring the solution to exhibit non-SUSY behavior and that the warp factor behaveas τ − will result in such a force. – 18 –nd the F equation (A.5) up to O ( τ ). It can be easily extended to higher order in τ without introducing additional independent parameters (i.e. higher order coefficientscan be expressed in terms of P , T , and A ). The solution related to the parameter T isthe same solution appeared in the exact solution (3.7) after expanding around τ = 0.As was the case for the singular unsquashed perturbation given in Sec. 3.2, thefluxes in this solution respect the ISD condition. The solution has a non-vanishing(0 , G (0 , = (8 P − T ) (cid:18) τ −
124 + τ − τ
756 + · · · (cid:19) ( z i d ¯ z i ) ∧ ( ǫ ijkl z i ¯ z j d ¯ z k ∧ d ¯ z l ) , (4.2)while the (3 ,
0) and (1 , ,
3) part impliesthat the gravitino variation is non-vanishing for general choices of P and T and thussupersymmetry is broken even though the flux is ISD. However, taking 8 P = 5 T results in an N = 1 supersymmetric solution. This special case is a generalization ofKS, corresponding to a constant shift of the string coupling and a canceling shift in H such that G is unchanged. Indeed in this case, F p = 0 while f p ∝ f KS and k p ∝ k KS . As was the case for the singular perturbations, it is possible to obtain solutions thatbreak the ISD condition by adopting the more general squashed ansatz (3.11). Weagain find such a solution to (A.3) as a power series in τ . The dilaton profile is againnon-trivial Φ p = ϕ (cid:18) − τ
16 + τ − τ τ (cid:19) . (4.3a)The metric squashing functions are b p = D (cid:18) − τ + 13 τ − τ (cid:19) + M (cid:18) − τ τ − τ (cid:19) + Q (cid:18) − τ + τ − τ (cid:19) + ϕ (cid:18) − τ
560 + 223 τ (cid:19) ,q p = D (cid:18) τ − τ (cid:19) + M (cid:18) τ − τ (cid:19) + Q (cid:18) τ − τ − τ (cid:19) + ϕ (cid:18) − τ
560 + 1669 τ (cid:19) ,s p = D (cid:18) τ − τ
300 + 13817 τ (cid:19) + M (cid:18) − τ
400 + 157 τ (cid:19) + Q (cid:18) τ
100 + 697 τ (cid:19) + ϕ (cid:18) τ − τ (cid:19) . (4.3b)– 19 –nd the warp factor is h p =( g s M α ′ ) ε − (cid:20) D (cid:18) − τ / + 317 2 / τ / (cid:19) + M (cid:18) − τ
40 6 / + 1163 τ / (cid:19) + Q (cid:18) − τ / + 13 τ / (cid:19) ϕ (cid:18) τ
40 6 / − τ / (cid:19)(cid:21) . (4.3c)The perturbed fluxes are F p = D (cid:18) τ
10 + 17 τ − τ τ (cid:19) + M (cid:18) τ
240 + 319 τ − τ τ (cid:19) + Q (cid:18) τ τ − τ τ (cid:19) + ϕ (cid:18)(cid:18) − − / a
16 2 / (cid:19) τ + (cid:18) − a
160 6 / (cid:19) τ + (cid:18) − − a / (cid:19) τ + (cid:18) a / (cid:19) τ (cid:19) ,f p = D (cid:18) − / a τ / + (cid:18) − / a
40 2 / (cid:19) τ + (cid:18) − − a
560 6 / (cid:19) τ + (cid:18) − a / (cid:19) τ (cid:19) + M (cid:18)(cid:18) −
18 + a / (cid:19) τ + (cid:18) − a
160 6 / (cid:19) τ + (cid:18) − a / (cid:19) τ + (cid:18) a / (cid:19) τ (cid:19) + Q (cid:18) − τ − τ
700 + 22003 τ (cid:19) + ϕ (cid:18) a τ
32 6 / + (cid:18) − − a
128 6 / (cid:19) τ + (cid:18) a / (cid:19) τ + (cid:18) − − a / (cid:19) τ (cid:19) , – 20 – p = D (cid:18)(cid:18)
265 + 3 6 / a (cid:19) τ + (cid:18) / a / (cid:19) τ + (cid:18) / a
40 2 / (cid:19) τ + (cid:18) a
560 6 / (cid:19) τ (cid:19) + M (cid:18)(cid:18) − / a / (cid:19) τ + (cid:18) − a / (cid:19) τ + (cid:18) − a
160 6 / (cid:19) τ + (cid:18) − a / (cid:19) τ (cid:19) + Q (cid:18) τ + 2 τ τ − τ (cid:19) + ϕ (cid:18)(cid:18) −
120 + 3 / a / (cid:19) τ + (cid:18) −
150 + 3 / a
160 2 / (cid:19) τ + (cid:18) − / a / (cid:19) τ + (cid:18) − a / (cid:19) τ (cid:19) , (4.3d)where we have again omitted terms presented in Sec. 4.1. This solutions is valid tolinear order in the parameters ϕ , M , Q , and D which characterize the perturbationand again one could extend this to higher orders in τ .The resulting G is no longer purely ISD since H − e F = 6 / ϕa / g s M α ′ + 3( − / a ) ϕτ a / g s M α ′ + · · · 6 = 0 . (4.4)This can also be checked more directly using (A.8). Although only the parameter ϕ appears in the potential for the dilaton, making any of these independent parametersnon-zero leads to a non-ISD flux .
5. Non-SUSY solutions in the KT region
It is also possible to find non-SUSY perturbations to the KT solution. We again willfind that non-ISD fluxes can be found only if the conifold is squashed. As before, weconsider solutions that are linear in the perturbations, though since we are working atlarge τ , we do not perform a power series expansion around τ = 0. The vanishing of the potential (3.14) merely implies Re (cid:0) G + mnp G − mnp (cid:1) = 0, not that G + = 0. – 21 – .1 Klebanov-Tseytlin solution The ansatz (2.1), (2.2) also includes the KT solution [13]. This solution correspondsto adding N D3-branes and M fractional D3-branes (i.e. M D5-branes wrapping acollapsing 2-cycle) to the undeformed conifold singularity and is valid at large distancesfrom the conifold point. It is recovered by f KT ( r ) = k KT ( r ) = 32 log rr , F KT ( r ) = 12 ,ℓ KT ( r ) = f KT (1 − F KT ) + k KT F KT + πNg s M ,p KT ( r ) =1 , b KT ( r ) = r , q KT ( r ) = s KT ( r ) = r , Φ KT ( r ) = log g s , h KT ( r ) = 27 π r (cid:18) g s N α ′ + 38 π ( g s M α ′ ) + 32 π ( g s M α ′ ) log rr (cid:19) , (5.1)where r = 2 / ε / e τ/ . In contrast to the KS solution, ℓ is chosen to satisfy F = 27 πα ′ N vol T , + B ∧ F , (5.2)where vol T , is the volume form of the angular space. This reflects the fact that theeffective D3 charge receives contributions from both the 3-form fluxes and the N regularD3-branes which provide a localized source for the charge. In analogy with the analyses of unsquashed perturbations of KS in Sections 3.2 and 4.1,we first consider perturbations for which the unwarped 6D space is still the unsquashedconifold and the flux is ISD. We take the ansatzΦ = log g s + Φ p ( r ) , h = h KT + h p ( r ) , ℓ = f (1 − F ) + kF + πNg s M ,f = f KT + f p ( r ) , k = k KT + k p ( r ) , F = F KT + F p ( r ) ,p = p KT , b = b KT , q = q KT , s = s KT . (5.3)Solving the ISD condition (A.8) and the first order equation (A.9) yieldsΦ p = P + φr ,F p = G r , – 22 – p = C + G r − φ r + 3 P rr ,k p = C − G r − φ r + 3 P rr ,h p =( g s M α ′ ) (cid:20) A + 27 C r + 81 P r (cid:18)
14 + log rr (cid:19) − φ r (cid:21) , (5.4)where we have retained only solutions that are regular as r → ∞ . The solution is validto linear order in the parameters P , φ , C , G , and A which characterize the perturbation.Note that some terms in the perturbation are sub-dominant to the corrections to theKT geometry coming from the full KS solution; however even if these corrections areincluded, the perturbations are not corrected until even higher order in 1 /r .The parameters P and φ are essentially the same parameters that appear in theperturbations to KS in Sections 4.1 and 3.2 respectively. That is, P is a constantshift of the string coupling and the part including φ is a solution to Laplace’s equation ∇ Φ = 0. The parameters G and U are related to those appearing in (3.6) and (3.7)as G = 2 ε U and C = H − U (the remaining parameter T appearing in (3.7) is notregular as r → ∞ ).The parameter φ is also the same parameter appearing in [12]. By calculating theHawking-Horowitz mass [40] (the generalization of ADM mass), which is valid at largeradius, the authors of [12] concluded that the relevant behavior of the perturbation tothe warp factor due to the D3-D3 pairs should include a term behaving as r − log r .However no such a term appears in (5.4). Moreover there is no squashing and theflux remains ISD. Therefore, even though SUSY is broken in this solution, it does notcorrespond to the presence of D3-branes. A perturbation of KT which is no longer ISD was found in [12]. Based on a similaranalysis of AdS × S , the authors of [12] assume the perturbations due to the D3-branesbehave as O ( r − , r − log r ) relative to the original KT solution and took an ansatzwhich squashes each of the SU (2)-isometry directions in the same way. However, it isinteresting to relax this condition and take the more general ansatz (2.1), (2.2) withΦ = log g s + Φ p ( r ) , h = h KT + h p ( r ) , ℓ = f (1 − F ) + kF + πNg s M ,f = f KT + f p ( r ) , k = k KT + k p ( r ) , F = F KT + F p ( r ) ,b = b KT (1 + b p ( r )) , q = q KT (1 + q p ( r )) , s = s KT (1 + s p ( r )) , p = p KT . (5.5)– 23 –uch an ansatz in general squashes the spheres in different ways. Assuming perturba-tions that behave as O (1 , log r, r − , r − log r ) relative to the KT solution , the equationsof motion (A.3) admit a solutionΦ p = − S log( r/r ) r ,b p = J + S r , q p = s p = J k p = f p = S r (cid:18) N π g s M + 94 log rr (cid:19) , F p = 0 ,h p = − π J r (cid:18) g s N α ′ + 38 π ( g s M α ′ ) + 32 π ( g s M α ′ ) log rr (cid:19) , + S r (cid:18) π g s N α ′ + 1053256 ( g s M α ′ ) + 8116 ( g s M α ′ ) log rr (cid:19) , (5.6)where J and S parameterize the perturbation and we omit the parameters which haveappeared in the previous subsection. The parameter S is the same parameter appearingin [12] and breaks the ISD condition and thus breaks SUSY. It was shown in [12] that S contributes a finite amount to the ADM mass as one would expect from the additionof D3s or D3s but since it does not contribute to the net charge, S characterizes theinfluence of D3-D3 pairs.Similarly, while turning on the parameter J preserves the ISD condition (A.8)and the first derivative equation for warp factor (A.9) and does not introduce a (0 , G , it causes the unwarped 6D space to no longer be Ricci flat (andtherefore no longer Calabi-Yau) so that there is no spinor covariantly constant withrespect to the unwarped metric, implying that supersymmetry is broken. In orderfor the flux part of the Killing spinor equations to vanish, any Killing spinor of theperturbed geometry would have to satisfy the same chirality conditions as the Killingspinor of the unperturbed geometry (i.e. Γ z ǫ = 0 where z is any holomorphic coordinateof KT). Therefore, while the flux part of the SUSY variation of the gravitino vanishes,the spin connection part does not and SUSY is broken. A priori, one might expect acancellation between the flux and spin connection parts might be possible for a differentchoice of chirality, but one can show that this cannot occur (see e.g. [44] and referencestherein).Although a non-vanishing J breaks supersymmetry, it does not describe the pres-ence of D3-branes. Note that while taking J 6 = 0 does not add any charge to thebackground, it still might describe the presence of D3-D3-brane pairs. However, such a As was the case in the previous section, including the finite deformation corrections to KT willnot change the form of the perturbations. – 24 –onfiguration would still provide a localized source of tension. The constant shift J inthe squashing cannot be a result of a localized tension since such a source should causea functional form that is singular as r →
0. Similarly, the perturbed warp factor is nota result of additional localized sources of tension, but results as a solution of (A.9) withthe perturbed squashing functions. Thus, the large radius backreaction of D3-D3-pairsis found by setting J = 0, reproducing the result of [12] .
6. Gravitino mass
In this section we calculate the effective 4D gravitino mass that results from dimensionalreduction of the SUSY breaking solution. The gravitino can potentially obtain a massfrom interactions with the 5-form flux F and the 3-form flux G . This problem wasaddressed previously in [45], though in their analysis they considered a background forwhich the warp factor satisfied the condition (A.9). However, even when this conditionis not satisfied, their method can still be applied and we follow it closely here. Notethat although we are interested in the specific case of the warped deformed conifold,this discussion applies to any perturbation of a warped Calabi-Yau.Since we work in the Einstein frame, we relate the Einstein frame spinors to thosein the string frameΨ EM = g s e − φ Ψ sM − i g s e − φ Γ M λ s ∗ , λ E = g − s e φ λ s , Γ EM = g s e − φ Γ sM , ǫ E = g s e − φ ǫ s . (6.1)Up through bilinear terms, the action for the type IIB Einstein frame gravitino is S f = 1 κ Z d x √− g (cid:0) L + L (cid:1) , L = i ¯Ψ M Γ MNS (cid:18) D N Ψ S + i e Φ ∂ N C Ψ S + i g s
192 Γ R R R R F SR R R R Ψ N (cid:19) , L = − i g / s e Φ /
192 ¯Ψ M Γ MNS (cid:0) Γ R R R S G R R R − R R G SR R (cid:1) Ψ ∗ N + h . c ., (6.2)where D M is the covariant derivative which, when acting on Ψ µ , is given by D [ µ Ψ ν ] = ˆ D [ µ Ψ ν ] −
18 Γ [ µ / ∂ log h Ψ ν ] ,D [ m Ψ ν ] = ˆ D [ m Ψ ν ] + 18 Γ [ m / ∂ log h Ψ ν ] − ∂ [ m log h Ψ ν ] , (6.3) We thank S. Kachru and M. Mulligan for some useful comments related to this discussion. – 25 –here ˆ D µ and ˆ D m are the covariant derivatives built from the unwarped metrics ˆ g µν and ˆ g mn .The Ψ µ part of the 10D gravitino is decomposed as a product of a 4D gravitino ψ µ and a 6D spinor χ that is covariantly constant with respect to the unwarped metricΨ µ ( x µ , x m ) = ψ µ ( x µ ) ⊗ h − χ ( x m ) , (6.4)where χ is normalized such that χ † χ = 1. The h − / factor of the warp factor comesfrom requiring that the spinor is covariantly constant with respect to the warped metric,ˆ D m Ψ µ = 0 [46].The 4D kinetic term following from (6.2) can be evaluated by dimensional reduction1 κ Z d x √− g i ¯Ψ µ Γ µνρ ˆ D ν Ψ ρ = 1 κ Z d x p − ˆ g i ¯ ψ µ ˆΓ µνρ ˆ D ν ψ ρ , (6.5)where on the right hand the indices are contracted with the unwarped metric ˆ g µν andwhere the 4D gravitational constant and the warped volume are1 κ ≡ κ V w , V w ≡ Z d y p ˆ g h. (6.6)If the supersymmetry condition on the warp factor (A.9) is satisfied, then thecoupling to F is canceled by the spin connection. However in general this interactionterm could a priori contribute to the gravitino mass and we have1 κ Z d x √− g i ¯Ψ M Γ MNR (cid:18) − ω ABR ˆΓ AB + i g s
16 / F Γ R (cid:19) Ψ N ∋ − κ Z d x p − ˆ g ¯ ψ µ ˆΓ µν ψ ν Z d y p ˆ g i h / (cid:18) h ′ √ p + ( g s M α ′ )4 ℓ √ bqs (cid:19) χ † ˆΓ τ χ, (6.7)where ω ABM is the spin connection with letters from the beginning of the alphabetdenoting tangent space indices and where on the right hand side, terms involving theunwarped spin-connections have been omitted and indices are again contracted withthe unwarped metric. The gravitino mass resulting from the 5-form flux is then i V w Z d y p ˆ g h − / (cid:18) h ′ √ p + ( g s M α ′ )4 ℓ √ bqs (cid:19) χ † ˆΓ τ χ. (6.8)However, this term vanishes as a result of the 6D chirality of χ and thus F does notcontribute to the gravitino mass.The essential contribution to the gravitino mass comes from the 3-form flux. Di-mensional reduction gives1 κ Z d x p − ˆ g ¯ ψ µ ˆΓ µν ψ ∗ ν Z d y p ˆ g i √ g s e Φ / χ † ˆΓ mnp χ ∗ G mnp + h . c . ! . (6.9)– 26 –ince ˆΓ ¯ ı χ = 0, we can write χ † ˆΓ mnp χ ∗ = χ † ˆΓ ¯ ı ¯ ¯ k χ ∗ = Ω ¯ ı ¯ ¯ k , (6.10)where Ω is the holomorphic 3-form of the underlying Calabi-Yau whose explicit formfor the deformed conifold is given in (B.11). Thus only the (0 , G contributes to the gravitino mass . This has been shown previously [45], but here weargued that it holds even when (A.9) is not satisfied. The 4D gravitino mass resultingfrom the 3-form flux is then m / = 3 √ g s i V w Z e Φ / Ω ∧ G , (6.11)which is quite similar to what follows from the Gukov-Vafa-Witten superpotential [47].With the explicit formula for the of K¨ahler potential and restoring the K¨ahler modulus ρ , we can write the gravitino mass as [45] m / ∝ κ e K W GV W , (6.12)where W GV W is the GVW superpotential and K is the K¨ahler potential.If we apply these expressions for the gaugino mass to (3.12), we find m / ∼ κ ( S + 10 T ) ε / a ( g s M α ′ ) τ min , (6.13)In evaluating this, we have assumed that most of the contribution to the gravitinomass should come from small τ , close to where the source of SUSY breaking is located,and cut the integral at some lower bound τ min . The lower bound must be introducedbecause for sufficiently small τ , the supergravity approximation breaks down. For thesingular solutions of Sec. 3 where the warp factor behaves at small τ as O (1 /τ ), theRicci scalars of these backgrounds behave as R ∼ S / ( g s M α ′ τ ) where S stands forany of the parameters characterizing the perturbation (which we expect to be all ofthe same order for a given solution). Thus, the solutions are valid for τ satisfying1 / ( g s M α ′ ) ≪ τ <
1. If we na¨ıvely take τ min to be this lower bound then m / ∼ κ S ε / . (6.14) We are treating the background as a non-SUSY perturbation to a warped Calabi-Yau. Moregenerally, when the Calabi-Yau is squashed there will be additional potential contributions from termssuch as g ij g kl g ¯ mn G ik ¯ m Ω jln , but these are higher order in perturbations since the unperturbed metrichas g ij = g ¯ ı ¯ = 0. Here we continue to follow [45], but in the presence of strong warping, the K¨ahler potential shouldbe modified from the expression used there [48, 49, 50, 51, 52, 53, 54]. – 27 –his is a finite value even if g s M is large. A more precise calculation of the gravitinomass would require extending the integral to smaller τ where the stringy correctionsto the geometry become important.We also found solutions which behaves regularly at τ = 0. The result of thecalculation for the solution in Section 4.2 is m / ∼ κ ε / g s M α ′ (cid:2)(cid:0) −
318 + 20 6 / a (cid:1) M − Q− (cid:0)
624 + 240 6 / a (cid:1) D + (cid:0) / a (cid:1) ϕ (cid:3) (6.15)This is a finite value, but since S is taken to be perturbatively small, and g s M is large,the mass of the gravitino is highly suppressed.The solutions (3.6) and (4.1) yield values for the gravitino mass that are similarto (6.13) and (6.15) respectively.
7. Discussion
In this paper, we analyze several solutions to type IIB supergravity, correspondingto non-supersymmetric perturbations to the warped deformed conifold. Of particularinterest are the solutions presented in Sec. 3.3 which capture some key properties of asolution describing the backreaction of D3-branes smeared over the finite S at τ = 0.In particular, we discussed the necessary boundary conditions in the IR for the solutionto describe a localized D H that was discussed in [37]. These solutions are thus relatedto a small τ expansion of a background whose large radius behavior was found in [12]and is dual to a metastable SUSY breaking state.For all of the above solutions, we have assumed the validity of a linearized ap-proximation. For a small number D3-branes, it is natural to expect that the linearizedapproximation is valid at least at large distances where the background flux largelydominates the effects of the D3, though an extrapolation to larger radii would benecessary to confirm this. For small distances, one can ensure that the linearized ap-proximation is good for τ above some particular value determined by the parametersof the solution. The linearized approximation requires, for example that F p ≪ F KS .Using the perturbations of Sec. 3.3 and taking S ∼ B ∼ Y , this gives the condition τ ≫ S / where S ∼ κ T ( N D3 + N D3 ) / ( V g s M α ′ ). Similar or less restrictive con-ditions follow by considering the other functions in the perturbation. As discussedabove, a similar constraint is imposed by demanding that the curvature (3.15) is smallin string units . Note that for large M , τ is allowed to be quite small. For the other The additional requirement that h p . h p can be satisfied if ε is not too small – 28 –olutions presented above for which there is not always an obvious boundary conditionto impose, the validity of the linearized approximation is more difficult to check.There are several remaining open lines of research. A particularly important re-maining open problem is to find a solution that interpolates between the small and largeradius regions . Such a solution would be important for many reasons. For example,all of the above solutions should admit a dual description as either deformations ofthe KS gauge theory or states in the (possibly deformed) KS gauge theory. Althoughfor some of the solutions the field theory interpretation has been studied (for example,the dual of the D3 solution was considered in [12]), analysis of the remaining solutionswould clearly require extrapolating them to the UV. Additionally, the boundary con-ditions discussed in Section 3.4 do not seem to be sufficient to fix all of the integrationconstants. Having a solution that is valid at all distances would allow for a calculationof quantities such as the Hawking-Horowitz mass or the asymptotic charge which couldprovide other conditions to fix the integration constants. Finally, an interpolating solu-tion would allow for a more precise calculation of the flux-induced gravitino mass andsimilar quantities. Unfortunately, even the linearized equations of motion are likelytoo complex to solve analytically in which case the solution could only be presentednumerically or formally in terms of integrals, an analysis that we leave for future work.The solutions could be improved in other ways. For example, the solutions pre-sented in Sections 3.2 and 3.3 exhibit curvature singularities as τ → S . One can also consider similar perturbations to the baryonicbranch solution [56, 57].Along similar lines, the solution (3.12), which, for some choice of parameters, woulddescribe the effect of D3-branes on the near tip geometry of KS, has been argued to bea metastable background [11]. However, it would be interesting to use the explicit solu-tion to analyze fluctuations about this geometry to confirm the perturbative stability,though this would require moving beyond the linearized approximation.Our solutions have potential applications to model building in warped compactifi-cations. For example, the addition of D3-branes into the warped deformed conifold wasan important step in the construction of stabilized de Sitter vacua [1] and in the mod-eling of inflation (see [22, 58, 19, 20, 21] and references therein). It would be interestingto understand the impact of the backreaction of the D3-branes on these scenarios. The As mentioned in the introduction, some progress was made in this direction after this paper wascompleted [37]. The localization of three-branes was considered in [55] in different context. – 29 –onstruction in [1] further inspired the scenario of mirage mediation [59] and one mightuse the solutions given here to provide a more string theoretical understanding of thisscheme.A related though conceptually distinct application is in the context of gauge-gravityduality. The large radius solution [12] was used in [34] as a holographic dual of ametastable SUSY breaking state. The large amount of isometry in this large radiusregion was found to suppress gaugino masses in their construction. However, the smallradius solution presented in Sec. 3.3, has reduced isometry, and should result in moresignificant contributions. Details of the application to holographic gauge mediation willbe discussed in a companion paper [33].
Acknowledgments
We have benefited from discussions with A. Dymarsky, A. Ishibashi, S. Kachru, I. R.Klebanov, F. Marchesano, L. McAllister, M. Mulligan, P. Ouyang, Y. Tachikawa, andH. L. Verlinde. PM and GS also thank T. Liu for collaboration and preliminary dis-cussions on this and related topics. We would like to thank the Institute for AdvancedStudy and the Hong Kong Institute for Advanced Study, Hong Kong University ofScience and Technology for hospitality and support. PM and GS also thank the Stand-ford Institute for Theoretical Physics and SLAC for hospitality while some preliminarydiscussions were held. YS appreciates the KEK Theory Center and the InternationalVisitor Program for hospitality and the opportunity to present this work. PM and GSwere supported in part of NSF CAREER Award No. Phy-0348093, DOE grant DE-FG-02-95ER40896, a Cottrell Scholar Award from Research Corporation, a Vilas AssociateAward from the University of Wisconsin, and a John Simon Guggenheim MemorialFoundation Fellowship. YS was supported by Nishina Memorial Foundation. GS alsowould like to acknowledge support from the Ambrose Monell Foundation during hisstay at the Institute for Advanced Study.– 30 – . Conventions
We work in the type IIB supergravity limit where the bosonic part of the Einsteinframe action is [60] S = 12 κ Z d x √− g (cid:20) R − ∂ M Φ ∂ M Φ − e ∂ M C∂ M C − g s × e − Φ H − g s × e Φ ˜ F − g s ×
5! ˜ F (cid:21) − g s κ Z C ∧ H ∧ F , (A.1)where we use˜ F = dC − CH , ˜ F = dC + B ∧ F = (1 + ∗ ) F , κ = (2 π ) α ′ g s , (A.2)and the self-duality of ˜ F is imposed at the level of the equations of motion. The stringframe metric is related to the Einstein frame metric by g EMN = g s e − Φ2 g sMN .The equations of motion resulting from (A.1) are R MN − g MN R − ∂ M Φ ∂ N Φ − e ∂ M C∂ N C − g s × e − Φ H MR R H R R N − g s × e Φ ˜ F MR R ˜ F R R N − g s ×
4! ˜ F MR R R R ˜ F R R R R N + 12 g MN (cid:20)
12 ( ∂ Φ) + 12 e ( ∂C ) + g s × e − Φ H + g s × e Φ ˜ F + g s ×
5! ˜ F (cid:21) = 0 , (A.3a) ∇ Φ − e ( ∂ M C ) + g s e − Φ × h H − e ˜ F i = 0 , (A.3b) d ∗ (cid:16) e − Φ H − Ce Φ ˜ F (cid:17) + g s F ∧ F = 0 , (A.3c) d ∗ (cid:16) e Φ ˜ F (cid:17) − g s F ∧ H = 0 , (A.3d) d ∗ ˜ F − H ∧ F = 0 . (A.3e)Note that imposing the self-duality of ˜ F implies ˜ F = 0. With the ansatz (2.1), (2.2),and taking ℓ = f (1 − F ) + kF , the Bianchi identity for ˜ F is automatically satisfied.– 31 –ith this ansatz, the equations for H (A.3c) can be written ddτ e − Φ h − s bp qs f ′ ! + e − Φ h r pb ( k − f ) − g s M α ′ h r pb ℓ (1 − F ) qs = 0 , (A.4a) ddτ e − Φ h − s bp sq k ′ ! − e − Φ h r pb ( k − f ) − g s M α ′ h r pb ℓFqs = 0 , (A.4b)while the equation for F (A.3d) is ddτ e Φ h − s bp F ′ ! + e Φ h r pb (cid:20) (1 − F ) sq − F qs (cid:21) − g s M α ′ h r pb ℓ ( k − f ) qs = 0 . (A.5)The bosonic and fermionic actions together are invariant under the supersymmetrictransformations for the gravitino Ψ M and dilatino λ , δ Ψ M = D M ǫ + i √ g s e Φ / (cid:0) Γ R R R M G R R R − R R G MR R (cid:1) ǫ ∗ + i g s
192 Γ R R R R F MR R R R ǫ, (A.6a) δλ = i R (cid:0) ie Φ ∂ R C + ∂ R Φ (cid:1) ǫ ∗ − e Φ R ∂ R Cǫ + √ g s e Φ /
24 Γ R R R G R R R ǫ, (A.6b)together with accompanying bosonic transformations. Here, G ≡ F − τ ad H , F = dC , τ ad ≡ C + ie − Φ . (A.7)We can use these transformations to check if supersymmetry is respected by thesolution. Using the ansatz (2.1), (2.2), the supersymmetry conditions δ Ψ M = δλ = 0imply1 − F − g s e − Φ s bp qs f ′ =0 , F − g s e − Φ s bp sq k ′ = 0 , F ′ − g s e − Φ r pb ( k − f ) = 0 , (A.8)which impose that the flux G is imaginary-self-dual (ISD). One can further show thatsupersymmetry requires that the flux be a primitive (2 , F , h ′ = − ( g s M α ′ ) ℓqs r pb . (A.9)– 32 –his condition implies that the BPS condition equates the tension and charge of 3-branes added to the geometry.The conifold and its related geometries make use of the angular 1-forms e = − sin θ dφ , e = dθ , e = cos ψ sin θ dφ − sin ψdθ ,e = sin ψ sin θ dφ + cos ψdθ , e = dψ + cos θ dφ + cos θ dφ . (A.10)In terms of these it is also useful to define [61] g = e − e √ , g = e − e √ , g = e + e √ , g = e + e √ , g = e , (A.11)which satisfy d ( g ∧ g + g ∧ g ) = g ∧ ( g ∧ g − g ∧ g ) ,d ( g ∧ g − g ∧ g ) = − g ∧ ( g ∧ g + g ∧ g ) ,d ( g ∧ g + g ∧ g ) = 0 ,dg ∧ g ∧ g = dg ∧ g ∧ g = 0 ,d ( g ∧ g ∧ g ) = d ( g ∧ g ∧ g ) = 0 . (A.12) B. Complex Coordinates
The angular coordinates and radial coordinate of the deformed conifold are related tothe complex coordinates z i by [62] W = L · W · L † ≡ (cid:18) z + iz z − iz z + iz − z + iz (cid:19) , (B.1a) L j = cos θ j e i ( ψ j + φ j ) / − sin θ j e − i ( ψ j − φ j ) / sin θ j e i ( ψ j − φ j ) / cos θ j e − i ( ψ j + φ j ) / ! , W = (cid:18) εe τ/ εe − τ/ (cid:19) , (B.1b)and the z i satisfy X i =1 z i = ε . (B.2)The angles ψ i always appear in the combination ψ = ψ + ψ . For ε = 0, τ is definedby R = X i =1 z i ¯ z i = 12 Tr (cid:0) W · W † (cid:1) = ε cosh τ. (B.3)– 33 –he deformed conifold metric can be written as [62] ds = ∂ i ∂ ¯ F dz i d ¯ z j = 14 F ′′ ( R ) (cid:12)(cid:12) Tr (cid:0) W † dW (cid:1)(cid:12)(cid:12) + 12 F ′ ( R ) Tr (cid:0) dW † dW (cid:1) = − iJ i ¯ dz i d ¯ z ¯ , (B.4a)where J = j dc ( τ )( g ∧ g + g ∧ g ) + dj dc ( τ ) ∧ g , F ′ ( R ) = ε − K ( τ ) ,j dc ( τ ) = ε τ F ′ ( R ) , (B.4b)and where ′ indicates a derivative with respect to R and J is the almost complexstructure.It is convenient to write G in terms of these complex coordinates. Following [63],we consider the SO (4) invariant 1-forms and 2-forms ξ = ¯ z i dz i , ξ = z i d ¯ z i ,η = ǫ ijkl z i ¯ z j dz k ∧ d ¯ z l , η = ǫ ijkl z i ¯ z j dz k ∧ dz l , η = ǫ ijkl z i ¯ z j d ¯ z k ∧ d ¯ z l ,η = ( z i d ¯ z j ) ∧ (¯ z j dz j ) , η = dz i ∧ d ¯ z i . (B.5)In terms of these, dτ = 1 ε sinh τ (cid:0) z i d ¯ z i + ¯ z i dz i (cid:1) , g = iε sinh τ (cid:0) z i d ¯ z i − ¯ z i dz i (cid:1) , (B.6a) g ∧ g = i (cid:0) τ )2 ε sinh τ ǫ ijkl (cid:0) z i ¯ z j dz k ∧ d ¯ z l − z i ¯ z j dz k ∧ dz l − z i ¯ z j d ¯ z k ∧ d ¯ z l (cid:1) , (B.6b) g ∧ g = i tanh τ ε sinh τ ǫ ijkl (cid:0) z i ¯ z j dz k ∧ d ¯ z l + z i ¯ z j dz k ∧ dz l + z i ¯ z j d ¯ z k ∧ d ¯ z l (cid:1) , (B.6c) g ∧ g + g ∧ g = 1 ε sinh τ ǫ ijkl (cid:0) − z i ¯ z j dz k ∧ dz l + z i ¯ z j d ¯ z k ∧ d ¯ z l (cid:1) , (B.6d) g ∧ g + g ∧ g = − i cosh τε sinh τ (cid:0) ¯ z j dz j (cid:1) ∧ (cid:0) z i d ¯ z i (cid:1) + 2 iε sinh τ dz i ∧ d ¯ z i . (B.6e)– 34 –he other remaining 1-forms cannot be as easily written in terms of the complexcoordinates. However, we find g + g = − ε sinh ( τ /
2) sinh τ (cid:2) (¯ z · dz ) + ( z · d ¯ z ) + 2 cosh τ (¯ z · dz )( z · d ¯ z )+ ε sinh τ ( dz · dz + d ¯ z · d ¯ z − dz · d ¯ z ) (cid:3) ,g + g = 12 ε cosh ( τ /
2) sinh τ (cid:2) (¯ z · dz ) + ( z · d ¯ z ) − τ (¯ z · dz )( z · d ¯ z )+ ε sinh τ ( dz · dz + d ¯ z · d ¯ z + 2 dz · d ¯ z ) (cid:3) . (B.7a)In terms of these complex coordinates G (3 , = M α ′ ε (cid:20) (cid:26) (1 − F ) tanh τ τ − F τ τ − F ′ sinh τ (cid:27) + g s e − Φ (cid:26) − f ′ τ τ + k ′ tanh τ τ + k − f τ (cid:27)(cid:21) ξ ∧ η , (B.8a) G (0 , = M α ′ ε (cid:20) − (cid:26) (1 − F ) tanh τ τ − F τ τ − F ′ sinh τ (cid:27) + g s e − Φ (cid:26) − f ′ τ τ + k ′ tanh τ τ + k − f τ (cid:27)(cid:21) ξ ∧ η . (B.8b)For the KS solution (2.4), these components vanish since each of the terms in bracesvanishes independently. The remaining components of G are G (2 , = M α ′ ε (cid:26) (cid:0) a +1 + a +2 (cid:1) ξ ∧ η + (cid:0) a − − a − − a +3 (cid:1) ξ ∧ η (cid:27) (B.9a) G (1 , = M α ′ ε (cid:26) (cid:0) a − + a − (cid:1) ξ ∧ η + (cid:0) a +1 − a +2 − a − (cid:1) ξ ∧ η (cid:27) (B.9b)where we have defined a ± ( τ ) = tanh τ τ (cid:0) ± (1 − F ) + g s e − Φ k ′ (cid:1) , (B.10a) a ± ( τ ) = 1 + cosh τ τ (cid:0) ± F + g s e − Φ f ′ (cid:1) , (B.10b) a ± ( τ ) = 1sinh τ (cid:18) ± F ′ + g s e − Φ k − f (cid:19) . (B.10c)For the KS solution, the only non-vanishing term is the (2 , G ∧ J = 0.– 35 –n calculating the gravitino mass, we make use of the holomorphic (3 , ε √ − sinh τ ( g ∧ g + g ∧ g ) + i cosh τ ( g ∧ g − g ∧ g ) − i ( g ∧ g + g ∧ g )] ∧ ( dτ + ig )= 14 √ ε sinh τ ( ǫ ijkl z i ¯ z j dz k ∧ dz l ) ∧ (¯ z m dz m ) . (B.11)Ω is normalized so that Ω ∧ ¯Ω / || Ω || = i vol with || Ω || ≡ Ω ijk ¯Ω ijk /
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