Non-supersymmetric branes
Niccolò Cribiori, Christoph Roupec, Magnus Tournoy, Antoine Van Proeyen, Timm Wrase
NNon-supersymmetric branes
Niccol`o Cribiori, a Christoph Roupec, a Magnus Tournoy, b Antoine Van Proeyen, b andTimm Wrase c,a a Institute for Theoretical Physics, TU Wien,Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria b Instituut voor Theoretische Fysica, KU Leuven,Celestijnenlaan 200D, B-3001 Leuven, Belgium c Department of Physics, Lehigh University,16 Memorial Drive East, Bethlehem, PA, 18018
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We discuss how to incorporate non-supersymmetric branes in compactifications of type IIstring theories. We particularly focus on flux compactifications on SU (3) × SU (3) structuremanifolds to four dimensions, so that a linear N = 1 supersymmetry is spontaneously brokenby spacetime filling Dp-branes. Anti-Dp-branes are a very special subset of such branes butour analysis is generic. We show that the backreaction of non-supersymmetric branes canbe incorporated into the standard 4d N = 1 supergravity by including a nilpotent chiralmultiplet. Supersymmetry in such setups is always spontaneously broken and non-linearlyrealized. In particular this means that, contrary to what was previously thought, branesupersymmetry breaking cannot be simply described by a D-term in 4d N = 1 supergravitytheories. a r X i v : . [ h e p - t h ] A p r Introduction
The fact that our universe is not supersymmetric clearly indicates the importance of study-ing compactifications of string theory where supersymmetry is broken in four dimensions.Beside fluxes, also local sources in string theory, like Dp-branes and Op-planes, genericallybreak some amount of the original supersymmetry. By choosing appropriately the micro-scopic ingredients, it is possible to construct string compactifications in which the breakingof supersymmetry in four dimensions is complete, but spontaneous. When supersymmetryis spontaneously broken, it can then be non-linearly realized. In this large class of model,particularly interesting are those in which there is no order parameter capable to restore lin-ear supersymmetry, namely supersymmetry is intrinsically non-linear. These models involvenon-supersymmetric Dp-branes, as we will show in the present work.In view of its connections to string theory, the study of non-linear supersymmetry hasbeen revived recently, building on the early work [1], and especially in relation with thegoldstino on an anti-D3-brane [2–8]. Indeed, the proper framework in which to insert allof these constructions is that of brane supersymmetry breaking [9–15] (see [16] for a recentreview), in which supersymmetry is broken at the string scale by the presence of non-mutuallyBPS objects in the vacuum and it cannot be restored below that scale.In this work, we study non-supersymmetric Dp-branes in string compactifications. Inparticular, we concentrate on the very broad class of flux compactifications of type II stringtheory on SU (3) × SU (3) structure manifolds to four-dimensional theories preserving a linear N = 1 supersymmetry. This is then spontaneously broken and non-linearly realized due tothe presence of spacetime filling Dp-branes. It is known that building a correct dictionarybetween the ten-dimensional and the four-dimensional setup can be very subtle. In this regard,we correct a long-standing misconception in the literature where such non-supersymmetricDp-branes in four-dimensional N = 1 supergravity have been described using D-terms. Wehere provide consistent four-dimensional supersymmetric actions which reproduce preciselytheir ten-dimensional counterparts. In order to obtain such general expressions, we have toset the model-dependent world volume fields to zero and include only the universal Goldstinofield that lives on the Dp-brane world volume. Then we can give a precise expression for thescalar potential that contains the couplings of all closed string moduli to the Dp-branes andthat is manifestly invariant under non-linear supersymmetry.We start by considering spacetime filling Dp-branes in flat space. This setup can beconsidered as a simple toy model in which the main differences between four-dimensional Note, that this does not mean that any of the related older papers contains wrong results. Usually theD-brane action is properly reduced and used in the analysis. It is, however, often incorrectly stated that in thesetting of 4d N = 1 supergravity the terms that result from the reduction of the non-supersymmetric D-braneaction can be written in terms of a D-term. – 2 –inear and non-linear supersymmetry can be understood from the brane perspective. Insection 2 we show how Dp-branes at angles with respect to an Op-plane (or equivalently Dp-branes with world volume fluxes) lead to a spontaneous breaking and non-linear realizationof supersymmetry, governed by the string scale. For this reason, we generically call theseobjects non-supersymmetric branes .Then, we show that, once supersymmetry is in the non-linear phase and the order pa-rameter contains a constant piece that is not moduli dependent, then there does not exist anyfield redefinition that absorbs the order parameter and brings you to linear supersymmetry.This means that string compactifications in which supersymmetry is broken by local sources,such that the supersymmetry breaking scale is related to the string scale, cannot be describedusing the language of standard linear supersymmetry or supergravity. This corrects a miscon-ception in the literature where people tried to use the familiar language of linear supergravityand in particular D-terms to describe such non-supersymmetric branes.In section 3 we consider non-supersymmetric Dp-branes in generic backgrounds with SU (3) × SU (3)-structure, following [17]. While the case of a D3 brane is somehow pecu-liar, since it cannot be placed at arbitrary angles, we provide a unified description of the p = 5 , , , N = 1 four-dimensional supersymmetric language, specifying a K¨ahler potential and asuperpotential. In other words, we build a precise dictionary between ten-dimensional andfour-dimensional quantities. The formalism of constrained multiplets is conveniently adoptedfor this purpose. The anti-D3-brane and intersecting D6-branes are discussed in more detailin specific examples before we give the general answer. In this section we recall some basic facts about Dp-branes and how they spontaneously breakhalf of the supersymmetry in flat space [18]. We discuss branes at angles and the differencebetween anti-Dp-branes and Dp-branes. We also review supersymmetry breaking by worldvolume fluxes, which is T-dual to branes intersecting at angles [19]. Then we show that SUSYbreaking by branes necessarily requires a description in terms of non-linear supersymmetry.Lastly, we discuss the restoration of supersymmetry at high energies, exemplified by someexplicit string theory setups.
Let us start by studying Dp-branes in type II supergravity in flat space R , . This allows usto easily review a few simple facts [20–24]. The type II supergravity background preserves32 supercharges that can be conveniently packaged into two Majorana-Weyl spinors (cid:15) and– 3 – . Adding a Dp-brane along the directions x , x , . . . , x p breaks half of the supersymmetryspontaneously [18]. The 16 supercharges that act linearly on the Dp-brane world volumefields satisfy (cid:15) = Γ ...p (cid:15) ≡ Γ Dp (cid:15) , (2.1)where Γ ...p = Γ Γ . . . Γ p with Γ M the flat Γ-matrices of D = 10. The 16 supercharges thatsatisfy (cid:15) = − Γ Dp (cid:15) are spontaneously broken and non-linearly realized on the world volumefields of the Dp-brane.For an anti-Dp-brane things are exactly opposite and the linearly realized and unbroken16 supercharges satisfy (cid:15) = − Γ ...p (cid:15) = − Γ Dp (cid:15) ≡ Γ Dp (cid:15) , (2.2)while the 16 non-linearly realized supercharges satisfy (cid:15) = Γ Dp (cid:15) ≡ − Γ Dp (cid:15) .More generically we find for a Dp-brane that extends along x , x , . . . , x p − and intersectsthe x p -axis with angle ϕ in the ( x p , x p +1 )-plane, that the 16 linearly realized superchargesare solutions to (cid:15) = (cid:0) cos( ϕ )Γ ...p + sin( ϕ )Γ ... ( p − p +1) (cid:1) (cid:15) ≡ Γ Dp ( ϕ ) (cid:15) . (2.3)We then find trivially Γ Dp (0) = Γ Dp and Γ Dp ( π ) = Γ Dp . This simply means that an anti-Dp-brane is the same as a Dp-brane with opposite orientation.Note that in the above discussion the entire separation into branes, anti-branes and branesat angles is rather artificial and not physically meaningful since we can always change (rotate)our coordinate system. So, the three separate cases above are all physically equivalent sincewe started with a maximally supersymmetric background. However, if the background breakssome of the supersymmetry, then these cases are not necessarily equivalent anymore.Let us consider again the flat space case and do an orientifold projection that projects outhalf of the 16 supersymmetries. In particular, we do a projection that gives us an Op-planethat extends along the directions 01 . . . p . This means that only supercharges that satisfy (cid:15) = Γ ...p (cid:15) ≡ Γ Op (cid:15) (2.4)survive and the other 16 supercharges are projected out. Now it is meaningful to talk abouta Dp-brane as the object that preserves the same 16 linear supercharges as the Op-plane,i.e. the brane that has Γ Dp = Γ Op . The anti-Dp-brane is the object for which the 16 linearsupercharges of the background are non-linearly realized on the world volume fields.The case of a brane at an angle relative to the Op-plane is the most interesting one.The 16 linearly realized supersymmetries of the background correspond to a combination of These spinors have opposite (the same) chirality in type IIA (IIB). This distinction will not be relevantfor us and we will treat type IIA and type IIB at the same time whenever possible. – 4 –inear and non-linear transformations for the world volume fields on the Dp-brane. Such acombination is always non-linear, as we will show in the next subsection.Next let us discuss branes with non-vanishing world volume flux F = B + F , where B isthe pull-back of the background Kalb–Ramond field and F = dA is the field strength on thebrane. The above Dp-brane with projection condition given in equation (2.1) together withthe Dp-brane at an angle ϕ and projection condition given in equation (2.3) are T-dual toan D(p+1)-brane with F = tan( ϕ ) dx p ∧ dx p +1 [19]. Thus, we expect that generically braneswith world volume fluxes will break the supersymmetry that is preserved by the correspondingorientifold projection. Concretely, the linearly realized supersymmetries that are preservedby a Dp-brane with world volume flux in flat space are given by (see for example section 5.3.in [25] and note that in our conventions ε ...p = − (cid:15) = − √ g + F (cid:88) n + (cid:96) = p +1 n ! (cid:96) !2 n ε a ...a n b ...b (cid:96) F a a . . . F a n − a n Γ b ...b (cid:96) (cid:15) ≡ Γ F Dp (cid:15) , (2.5)with Γ a = ∂ a y M Γ M being the pull-back of the Gamma matrices to the Dp-brane worldvolume. Since generically Γ Op (cid:54) = Γ F Dp the 16 linearly realized supercharges of the backgroundare non-linearly realized on the brane with world volume flux. We will show that all suchsetups cannot be described using standard linear supersymmetry.The above examples can be straightforwardly extended to compactifications of type IIsupergravity on spaces that partially preserve supersymmetry, like for example Calabi–Yaumanifolds. In this case one has to ensure that Gauss law is satisfied and the charges carriedby the branes, orientifolds and potentially background fluxes have to cancel. This leads to alarge sets of possibilities: for example in compactifications to four dimensions we can use asinternal space T , T × K manifold and get N = 8, N = 4 or N = 2 supergravities.We can then do orientifold projections that break half or more of the supersymmetry. AddedD-branes will now have world volume fields that transform linearly and/or non-linearly underthe remaining supercharges.The above branes at angles, anti-branes and branes with world volume fluxes are themain examples that we have in mind and we will discuss them in this paper. However,there are other supersymmetry breaking sources in type II like for example NS5-branes,(p,q)-branes, and KK-monopoles that can likewise be included in a supergravity action usingthe general idea of non-linear supersymmetry outlined below in this paper. Similarly, ourdiscussion applies to supersymmetric M-theory or heterotic string theory compactificationswith supersymmetry breaking sources in arbitrary dimensions. In the previous section we have seen that D-branes in string theory spontaneously break partor all of the supersymmetry preserved by a given background. Concretely, we have a given– 5 –ackground with fields that transform linearly under a certain number of supercharges { Q A } , A = 1 , . . . , N . We add to that D-branes or other localized sources that transform linearlyunder a (potentially empty) subset { Q a } , a = 1 , . . . , n < N . What does this mean for theremaining supercharges { Q i } , i = n + 1 , . . . , N ?Above we have seen that for a D-brane at angles in the presence of an Op-plane, the 16linearly realized supersymmetries of the background will combine with the non-linear super-symmetry of the world volume fields. A combination of linear and non-linear supersymmetrytransformations is generically non-linear, as it can be easily shown. Consider indeed a genericsituation in which a set of scalars φ A are mapped by supersymmetry to spin-1/2 fields λ A .The transformation of the λ A will be schematically of the form δλ A = S A (cid:15) + /∂φ A (cid:15) + . . . , (2.6)where S A are complex constants and dots stand for other model-dependent terms, which arenot relevant for the present argument. We can split S A = { S a , S i } , where S a = 0, while S i (cid:54) = 0 are the fermionic shifts, which break supersymmetry spontaneously. Let us focus onthe non-linear part of the supersymmetry transformations given by the shift, namely δ shift λ a = 0 ,δ shift λ i = S i (cid:15). (2.7)This part is rather universal and is generically present in any model with spontaneously brokensupersymmetry. We would like to show that, as long as at the set { S i } is not empty, thenit is not possible to simultaneously set to zero all the non-linear parts of the supersymmetrytransformations, even if we allow for field redefinitions. In other words, we show that acombination of linear and non-linear supersymmetry transformation is always non-linear.Since the set { λ a } already transforms linearly, we focus on the complementary set { λ i } . Wecan then define a spin-1/2 field v = g i ¯ λ i S ¯ ≡ λ T g ¯ S, (2.8)where g i ¯ is the K¨ahler metric on the scalar manifold, such that δ shift v = g i ¯ S i S ¯ (cid:15) ≡ ( S T g ¯ S ) (cid:15) . (2.9)We can redefine now all the other fields in the set { λ i } as˜ λ i = λ i − S i vS T g ¯ S , ˜ λ i g i ¯ S ¯ = 0 , (2.10)such that δ shift ˜ λ i = 0 . (2.11)– 6 –owever, for this to be consistent we have to require ( S T g ¯ S ) (cid:15) (cid:54) = 0 and therefore δ shift v (cid:54) = 0.Notice that the same requirement follows from the fact that the metric g i ¯ is positive definite,which actually implies ( S T g ¯ S ) = || S || >
0. This means that, even if by means of fieldredefinitions we can restrict the non-linearity to be present along just one direction in fieldspace, we can never really eliminate it, since the field space metric g i ¯ is positive definite. Westress that this argument is generic and can be easily adapted to any specific setup.This concludes a simple proof that shows that string compactifications in which super-symmetry is broken by local sources cannot be described using the language of standardlinear supersymmetry or supergravity. The reason is that the combined actions for the closedstring background fields and the local sources can never be invariant under linear supersym-metry, as proven above. The linearly realized supersymmetries of the background actionare non-linearly realized on the world volume fields of the sources and vice versa. Therefore,clearly such compactifications require that supersymmetry is non-linearly realized for somefields and for example for compactifications to four dimensions with four non-linearly realizedsupercharges this leads to the so-called dS supergravity theory [27–30] coupled to differentkinds of matter and gauge fields [31, 32].
In the previous subsection we have proven that the supersymmetry that is spontaneouslybroken by branes at angles or by branes with world volume fluxes, cannot be describedusing standard linear supersymmetry. When supersymmetry (or any other symmetry) isspontaneously broken at a certain scale F , then we expect that above that scale new degrees offreedom come in and restore the linear symmetry. The necessity for such new states manifestsitself in unitarity violations in for example the original Volkov-Akulov theory [33, 34], as wellas for example for a chiral superfield with very heavy scalar component coupled to supergravity[35]. So, the expectation is that non-linear supergravity theories will violate unitarity abovethe supersymmetry breaking scale and new degrees of freedom need to come in. One notableexception to this is inflation in non-linear supergravity, where a large Hubble scale H modifiesthe simple argument above. In particular, the SUSY breaking scale F ∼ m / M P l getsreplaced with F ∼ (cid:113) m / + H M P l [36], see [37–40] for a recent discussion of this point.Here we are restricting ourselves to models with negligible Hubble scale and are asking how isunitarity preserved in string theory models that give rise to non-linear supergravity theoriesat low energies? This statement is not in contradiction with the analysis in [26]. Indeed, there the attention was on the non-linear and model dependent interactions in the supersymmetry transformations, rather than on the constantshift, which was always assumed to be present. – 7 –elow we will study a few examples and find generically massive states with massesaround the SUSY breaking scale F , as is required for a unitary and UV complete theorylike string theory. However, generically we do not find a single set of such states but infinitetowers of states whose mass scale is set by the SUSY breaking scale. Such infinite towersof states invalidate the use of the original low energy effective theory at energies above theSUSY breaking scale but in some examples one can integrate in the entire new tower of statesand describe the resulting new theory. Given this observation it seems likely that any genericlow-energy effective supergravity theory that arises from string compactifications in whichsources (like Dp-branes) break SUSY, will only be valid below the supersymmetry breakingscale. This means that one would expect a substantial change in such a generic low energyeffective theory near the SUSY breaking scale. Here we mean by a generic theory one forwhich local sources are neither absent nor specially aligned such that they preserve (almost)linear supersymmetry.The above statement might seem fairly strong with substantial implications for our uni-verse, if it is generic in the above sense. So, let us add some words of caution. Firstly, it isnot impossible that a low energy effective theory undergoes substantial changes that do notaffect all of its sectors. For example, in the KKLT construction [41] an anti-D3-brane at thebottom of a warped throat breaks supersymmetry spontaneously. The SUSY breaking scaleis the warped down string scale, which in controlled settings should be above the warpedKK scale. So, below the SUSY breaking scale new massive KK states will appear and atthe SUSY breaking scale massive open string states appear. However, all these states arelocalized at the bottom of the throat and this might not substantially affect another sectorlocated in the bulk or even in a different throat. Secondly, local sources minimize their en-ergy if they preserve mutual linear supersymmetry. So, it seems possible that supersymmetrypreserving sources, albeit being non generic in the above sense, are very abundant since theyare dynamically favored. If all sources preserve linear supersymmetry and it is only brokenby for example background fluxes, then we are not aware of any reason why one would expectan infinite tower of states to appear near or above the SUSY breaking scale. So, the SUSYbreaking by generic branes seems somewhat different. Here we want to give a more intuitive physical explanation for why one expects an infinitetower of light states to appear near the SUSY breaking scale if SUSY is broken by local sourceslike D-branes. The reason is simply that the SUSY breaking scale for these objects is thestring scale that sets the scale for the tower of massive string states. Even if we manage It would be interesting to quantify this. One can however probably argue that generic string compactifications even without branes should nothave huge hierarchies nor preserve supersymmetry. So, in that sense the closed string sector is special, if itgives rise to an effective theory for which the SUSY breaking scale is well below the KK-scale and the stringscale, both of which are scales at which infinite towers of states appear. – 8 –o lower this scale, which is usually taken to be fairly high in string compactifications, thenwe also lower the scale for the tower of light strings on the D-brane. Concretely, let usconsider a D-brane (or anti-D-brane) in 10d flat space. It spontaneously breaks half of the32 supercharges preserved by the background. The associated SUSY breaking scale is thestring scale. We can project out the remaining linearly realized supercharges by combiningan Op-plane with an anti-Dp-brane. The resulting theory has no linear supersymmetry andas we have proven above cannot be possibly rewritten such that one gets a theory with linearsupersymmetry. Therefore, like for the original Volkov–Akulov theory, one expects that fourgoldstino interaction terms, that have to be present in order for the action to be invariantunder non-linear supersymmetry, lead to divergent cross-sections and unitarity violations atenergies well above the SUSY breaking scale. String theory cures this via the infinite tower ofmassive string states on the D-brane. Here notably the first level of massive open string statesdoes not contain the right number of states to allow for linear supersymmetry restoration,so it seems likely that one needs the entire tower of massive open string states. It wouldbe interesting to precisely show how linear supersymmetry is restored by including the entiretower of massive open string states. We hope to come back to this in the future.Next, we present a few examples of string compactifications with D-branes that break allthe linear supersymmetry.Our first example is the KKLT construction of dS vacua in string theory [41]. Here wehave a four-dimensional theory in AdS that preserves linear N = 1 supersymmetry. To thisan anti-D3-brane is added at the bottom of a warped throat in the compactification manifold.This spontaneously breaks supersymmetry and uplifts the AdS vacuum. The uplift energythat is the SUSY breaking scale is the warped down string scale. Since the anti-D3-braneis sitting at the bottom of a warped throat the masses of the entire open string tower arewarped down. Therefore, at or near the SUSY breaking scale we have the tower of massiveopen string states coming in, which should lead to a substantial modification of the effectivelow energy theory. However, here things can be more interesting. In particular for theKlebanov–Strassler throat [42], we have an S , i.e. a non-trivial 3-cycle, at the bottom ofthe throat. This means that we have a warped down KK scale that in controlled setups isbelow the warped string scale. So, there is an infinite tower of KK states coming in beforethe infinite tower of massive string states on an anti-D3-brane. This was studied in greatdetail in [8]. The authors study the situation with many anti-D3-branes that polarize into anNS5-brane that can wrap a metastable cycle on the S [1]. It turns out that it is possible torestore linear supersymmetry by including a particular tower of KK modes at the bottom of For example, one could consider an anti-D3-brane in flat space on an O3 − -plane. It is known that at theground state we have 8 fermions, while the first excited level has 128 bosons. On the other hand, the N = 4representation of linear supersymmetry in four dimensions has 8+8 degrees of freedom. – 9 –he Klebanov–Strassler throat [8].Next, let us consider a D-brane that is rotated by a very small angle ϕ (cid:28) ϕ/ √ α (cid:48) , where1 / (2 πα (cid:48) ) is the string tension. By making ϕ arbitrarily small, we can have an arbitrarilysmall SUSY breaking scale that can certainly be well below the compactification scale andthe string scale set by simply 1 / √ α (cid:48) . So, it might seem natural that one could describe thisalso in terms of a standard effective low energy supergravity theory. However, this cannotbe the case as we have proven in the previous subsection. In the past such cases were oftendescribed as D-terms in standard linear 4d N = 1 supergravity for very small angles and wewill correct this in the next section. Here we are more interested in the breakdown of such alow energy effective theory near the SUSY breaking scale. One can actually show that ϕ/ √ α (cid:48) sets the scale of a tower of massive open string states that stretch between the two branes, seefor example [43–45]. Therefore, there will be a full tower of excited open string modes withmasses being multiples of the SUSY breaking scale. So, the effective low energy theory breaksdown and it is only when one includes an infinite tower of new states that one can potentiallyobtain a new theory that describes this string compactification and that has linearly realizedsupersymmetry. In this particular setup it is of course not too difficult to guess that sucha theory should be an SU (2) gauge theory arising from two coincident D-branes. In thattheory one can turn on a ϕ dependent vev for one of the scalar fields on one of the D-branes.This rotates one of the branes and breaks the gauge group to U (1) × U (1) (see [43] for somerelated discussions).The above example is somewhat peculiar in the sense that we fine-tuned the intersectionangle to get a very small supersymmetry breaking scale but nevertheless we ended up withan infinite tower of states. This tower is associated with strings stretching between two D-branes and therefore different from the tower of massive open string states that arise on anysingle D-brane and that we discussed before. Therefore, one can modify it such that one canfind examples without such infinite towers of states near the SUSY breaking scale. Let usconsider two intersecting Dp-branes, for example two D6-branes that intersect perpendicularon a toroidal orbifold, such that they preserve some supersymmetry. Now if we change theangle slightly away from this supersymmetric angle then this again leads to a very smallSUSY breaking scale that can be well below the KK scale and the string scale. This timethe intersection angle is not small and therefore there is no infinite tower of massive stringstates appearing near the SUSY breaking scale. This can be understood as follows. The finetuning of the angle to be very close to the supersymmetric angle leads to a world volumetheory on the slightly rotated Dp-brane that is almost invariant under linear supersymmetry.This allows the world volume fields to be almost in regular multiplets and to almost canceleach other in divergent cross-sections. So, no infinite tower of states needs to come in at the– 10 –USY breaking scale. However, there is no way to rewrite this in terms of linearly realizedsupersymmetry as proven above. Therefore, there is no way to write down the exact action inthis setup using standard linear supergravity. In this paper we are laying out the first stepstowards a correct low energy description of such setups in the next section.We believe that the above examples are describing a generic feature of string compactifi-cations in which all supersymmetry is broken by D-branes, namely the appearance of a towerof states whose mass is set by the SUSY breaking scale. This means that the resulting lowenergy effective theories that describe generic compactifications will be substantially modifiedat or near the SUSY breaking scale. However, there are ways of avoiding such modificationsin concrete setups by fine tuning as in the last example above.In the next section we will discuss how to explicitly describe particular setups below theSUSY breaking scale. We will use non-linearly realized supersymmetry and see that thereis no smooth limit in which we can obtain a theory with linear supersymmetry. The SUSYbreaking scale will appear with inverse powers and sending it to zero will lead to singularities.Alternatively, if one redefines the field so that there are no singularities one finds that theentire action of the goldstino field is proportionally to the SUSY breaking scale and hencedisappears once the SUSY breaking scale is send to zero. However, the goldstino is a fermionicfield that lives on the branes and it does not disappear at higher energies. On the contrary,generically new light fields will appear for energies near or above the SUSY breaking scale.So, it does not seem possible to capture this transition from non-linear supersymmetry tolinear supersymmetry in a simple low energy effective theory. N = 1 theories In this section we describe how to incorporate non-supersymmetric
Dp-branes in compact-ifications of type II string theory to four-dimensional theories with N = 1 supersymmetry.We have in particular flux compactifications and cosmological applications in mind [46–48]and we will neglect the model dependent world volume fields on the Dp-branes. However,our results are also relevant for the existing literature on intersecting D-brane model building[48, 49] and it is possible but technically challenging to include all world volume fields in anygiven model, as was done in [6, 7, 50–52] for the anti-D3-brane in the KKLT setup [41]. The case of the D3-brane is somehow peculiar since, in order to preserve Lorentz invariancein the non-compact space, the brane cannot be at arbitrary angles. Indeed, let us considera generic type IIB compactification on a Calabi-Yau 3-fold. In order to break 4d N = 2supersymmetry to N = 1 we introduce also an orientifold projection, implying (cid:15) = Γ (cid:15) ≡ Γ O3 (cid:15) . (3.1)– 11 –n the other hand, the presence of a D3-brane at some angle ϕ in the ( x , x )-plane requires (cid:15) = (cos( ϕ )Γ + sin( ϕ )Γ ) (cid:15) ≡ Γ D3 ( ϕ ) (cid:15) . (3.2)Since the term proportional to Γ would break Lorentz invariance in the non-compactspace, we have to require sin( ϕ ) = 0. Therefore the only possible angles for a D3-brane are ϕ = { , π } . In the first case, the brane preserves the same supersymmetries as the orientifoldprojection, while in the second case all the supersymmetries are spontaneously broken andthe brane is an anti-D3-brane. Notice that this also means that, in the D3-brane case withan orientifold projection, the world volume flux F has to vanish for both the allowed valuesof ϕ . For a compactification to four dimensions we can include Dp-branes with p > p − S Dp = S DBI , p + S CS , p = T Dp (cid:18)(cid:90) M , × Σ d x d p − y e − φ | Σ (cid:112) det ( − g | Σ + F | Σ ) − (cid:90) M , × Σ C | Σ ∧ e F| Σ (cid:19) , (3.3)where T Dp is the brane tension, φ the dilaton, g the metric and C is sum over the RR-fields.We also denoted the pull-back onto the world volume of the Dp-brane by | Σ . For the easeof the notation we will not spell out the pull-back anymore and we restrict in this paper tocompactifications that preserve N = 2 supersymmetry in four dimensions, broken to N = 1after doing an orientifold projection. In the case with vanishing fluxes the internal spaceis therefore a CY manifold and otherwise it is a more general SU(3)-structure manifold.These spaces (in the strict limit) have no non-trivial 1- and 5-cycles and are equipped witha K¨ahler (1,1)-form and a holomorphic (3,0)-form Ω. Supersymmetric Dp-branes in suchcompactifications have been reviewed in great detail in [48]. It turns out that one can expressthe DBI part of the action for supersymmetric branes in terms of the K¨ahler form J , theholomorphic 3-form Ω and the Kalb-Ramond 2-form B as follows S DBI , = T D5 (cid:90) M , × Σ d x e − φ J , Generic SU(3)-structure manifolds are not K¨ahler and the K¨ahler form J and holomorphic three-form Ωare not closed so we are slightly abusing the language here. The existence of the real 2-form J and the complex3-form Ω are a defining property of SU(3)-structure manifolds as explained for example in section 3.2 of [46].In the special case of Calabi-Yau manifolds they reduce to the familiar K¨ahler and holomorphic 3-form. We will not discuss the CS-part of the action much further here. It encodes the charge of the Dp-braneand enters in the tadpole cancelation condition that ensures that the total charge in the internal compactspace vanishes, if one combines the contribution from fluxes, D-branes and orientifold planes. – 12 –
DBI , = T D6 (cid:90) M , × Σ d x e − φ Re(Ω) ,S DBI , = T D7 (cid:90) M , × Σ d x e − φ
12 ( J ∧ J − B ∧ B ) ,S DBI , = T D9 (cid:90) M , × Σ d x e − φ (cid:18) J ∧ J ∧ J − J ∧ B ∧ B (cid:19) , (3.4)where we used string frame and have set the world volume gauge flux F = 0, as well as allother world volume fields on the Dp-branes.The above calibration conditions have already been generalized to supersymmetric braneswith non-zero world volume flux F , see for example [53–55]. Such supersymmetric Dp-branes satisfy generalized calibration conditions and it is possible to use them to rewrite theDBI-action in equation (3.3) in terms of expressions that generalize the expressions in (3.4).In particular, for any Dp-brane wrapping a ( p − V DBI = (cid:90) Σ e A − φ Re ˆΨ ∧ e F . (3.5)Here e A denotes the warp factor and ˆΨ is a pure spinor (which is a polyform). The definitionof the pure spinor is different in type IIA and type IIB so that the above expression reducesto equation (3.4) for vanishing world volume flux and no warping.In this paper we are interested in non-supersymmetric Dp-branes and will describe thegeneral procedure of how to incorporate these into an effective low energy supergravity actionin the next subsection. Then we will discuss several examples where we apply this procedureto obtain proper dS supergravities that incorporate the universal contributions from thesenon-supersymmetric Dp-branes.The particular case of so-called pseudo-calibrated anti-Dp-branes in flux compactifica-tions was recently studied in [17]. There the contribution to the action from these anti-Dp-branes is essentially the same as above in equation (3.4). The only difference is an overallminus sign in the CS-term since the anti-Dp-branes have the opposite orientation, so theirvolume, as measured by the DBI-action is the same as for Dp-branes. Crucially however,for the supersymmetric Dp-branes the cancelation of the Dp-brane charge ensures that theabove DBI-action nicely fits into the standard linear supergravity formalism, while for anti-Dp-branes this is not the case and one finds new terms of the form given in equation (3.4)that can only be incorporated into the scalar potential when using non-linear supergravity,see [17] for details. Here we are interested in more general non-supersymmetric Dp-branesand we will generalize the results of [17]. – 13 – .3 The new supergravity action In this subsection we show how one can generically incorporate the new contributions to thescalar potential that arise from the non-supersymmetric Dp-branes. The world volume fieldsthat arise on a particular Dp-brane are model dependent and cannot be spelled out in fullgenerality. However, all the non-supersymmetric branes that we study have in common thatthey break supersymmetry spontaneously and lead to a non-linear realization. This meansthat the world volume fields are not appearing in standard multiplets anymore. In particular,there is one special so called nilpotent chiral multiplet S , that satisfies S = 0. This multiplethas only one fermionic degree of freedom ψ because the nilpotency condition fixes the scalarin the multiplet S = φ + √ ψ θ + F θ in terms of the fermion as φ = ψ / (2 F ). Here θ represents the superspace coordinates and F is an auxiliary field. Note, that this solutionfor φ necessarily requires that F (cid:54) = 0 and we cannot take the limit of sending F to zero. So, we see explicitly in this example that one cannot really take the limit that restores linearsupersymmetry.Nilpotent superfields were first used in [61] in order to linearize the Volkov-Akulov (VA)model [34]. The description of the VA-model in terms of nilpotent superfields serves as anillustration of how these fields realize non-linear supersymmetry. The action of the VA-modelreads S V A = − M (cid:90) E ∧ E ∧ E ∧ E with E µ = dx µ + ¯ λγ µ dλ (3.6)and is invariant under the non-linear symmetry transformation δ (cid:15) λ = (cid:15) + (cid:0) ¯ λγ µ (cid:15) (cid:1) ∂ µ λ . (3.7)Using the nilpotent field S this action can be written as S = (cid:90) d x (cid:90) d θ (cid:90) d ¯ θS ¯ S + M (cid:18)(cid:90) d x (cid:90) d θS + h.c. (cid:19) , (3.8)where we now also have to consider the superspace integral over the θ coordinates. One canshow that the two actions are equivalent, if one imposes the nilpotent condition S = 0. Inthis description ψ is the goldstino and the only degree of freedom. Note that this descriptionof the VA action using a nilpotent chiral field is not a unique choice and one can also use aconstrained vector multiplet [63, 64]. In addition to the nilpotent field we consider here there They have been studied for supersymmetric branes in for example [56–60] and even there they are notfully understood in all setups. If one rescales the fermions such that ψ → √ F ψ (cid:48) then the entire action for the rescaled fermion ψ (cid:48) vanishesin the limit of F →
0. However, ψ (cid:48) is a world volume field and cannot simply disappear, if we for examplerotate the D-brane. Note that the fermion λ in the VA-action and the fermion ψ in the nilpotent chiral superfield are not thesame but are rather related via a field redefinition, see [3, 62]. – 14 –re many more constrained supermultiplets that can be used to describe non-linearly realizedsupersymmetry [6, 62, 64–70]. A description on how to use these fields in supergravity can befound in [71]. Constrained multiplets have been used to great success in order to describe thecomplete action of the anti-D3-brane in the KKLT background [51, 52] and general anti-Dp-branes in [17]. In [72] the constrained multiplet formalism has been used in order to facilitatethe first uplift in type IIA, using anti-D6-branes.After this short review of constrained multiplets, let us return to describing the lowenergy contribution of supersymmetry breaking D-branes in four-dimensional N = 1 super-gravities. If the background preserves linear N = 1 supersymmetry, then the added non-supersymmetric branes would be the sole source of supersymmetry breaking. That meansthat non-supersymmetric branes would have to provide the Goldstino, i.e. one of the worldvolume fermions or a linear combination thereof is the Goldstino. We can incorporate thisGoldstino into our general action via the nilpotent chiral multiplet S . This allows us toincorporate the new contribution from the non-supersymmetric branes into the bosonic su-pergravity action as follows:We start with any four dimensional theory with linear N = 1 supersymmetry with aK¨ahler potential K before , superpotential W before , as well as potentially gauge kinetic functionsand D-terms. These are obtained from an explicit string theory compactification and dependon chiral multiplets Φ a . We now add non-supersymmetric Dp-branes and we want to incor-porate their backreaction on the Φ a fields. This backreaction can be obtained by explicitlyreducing the non-supersymmetric Dp-brane actions in a particular compactification, whichgives rise to a new scalar potential term V new (Φ a , ¯Φ ¯ a ). We now define the full K¨ahler potentialand superpotential as K = K before + e K before S ¯ SV new ,W = W before + S . (3.9)This leads to the following scalar potential with a generic new contribution from the non-supersymmetric branes V = V F + V D = e K (cid:16) K I ¯ J D I W D J W − | W | (cid:17)(cid:12)(cid:12)(cid:12) S =0 + V D = V before + V new . (3.10)Here I, J run over all fields including the constrained nilpotent chiral multiplet S . Since S only contains the Goldstino we have to set it to zero in the end to obtain the bosonic scalarpotential. We thus have D S W | S =0 = 1 and K S ¯ S (cid:12)(cid:12) S =0 = (cid:0) K S ¯ S (cid:12)(cid:12) S =0 (cid:1) − = e − K before V new . (3.11)The above actually works independently of whether the background has supersymmetrypreserving solutions or not and it will give us always the correct result. It is however not– 15 –lways immediately obvious that V new is a real function of the Φ a . We will argue that this isalways the case and this might also be intuitively clear because V new is a real function of theclosed string degrees of freedom that have been package into the Φ a . So, we conclude thatwe can always incorporate the new contributions from non-supersymmetric branes by using anilpotent chiral multiplet. Generically, the background fields will also break supersymmetry,in particular once we include the backreaction of the non-supersymmetric branes. This meansthat the Goldstino is not simply the fermion contained in S but rather a linear combinationlike F S λ S + F α λ α , where the index α runs over the chiral multiplets that arise from the closedstring sector.Our prescription above seems rather simple and ad hoc. However, it was shown, andwe will review this below, that it gives the correct description in all examples. This mightseem trivial since V new can be any real function of the other moduli. However, the changesto K and W become highly non-trivial, if one wants to include all world volume fields on aDp-brane and reproduce all couplings between the bosonic and fermionic world volume fieldsand the closed string background fields. This has only been worked out in full detail foran anti-D3-brane in the KKLT setup [51, 52]. The rather general elaborate answer in thiscase, see section 5 of [52], reduces exactly to the expression above, if we set all world volumefields but the Goldstino to zero. The simplicity and elegance of the equations (3.9) and(3.10) can also be understood from the fact that they use the universal feature of all SUSYbreaking Dp-branes, namely the presence of a Goldstino contained in a nilpotent multiplet S , and package everything else, like information about the compactification background, thedimension of the brane and so on, into the unspecified function V new . Below we will seewhat V new is in several concrete examples and show that the correct scalar potential can beobtained as described above. The anti-D3-brane is the simplest and most studied case of supersymmetry breaking by Dp-branes, since the anti-D3-brane plays an important role in the KKLT scenario [41], where itwas used to uplift a supersymmetric AdS vacuum to a dS vacuum. In the original paper theanti-D3-brane contribution was taken to be V D3 = µ ( − i( T − ¯ T )) , (3.12) It would be interesting and probably not too difficult to include the scalar degrees of freedom from theworld volume fields on the non-supersymmetric branes. These appear already in V new . One would have to addmore constrained multiplets to the original theory and include them in K such that one obtains the correctkinetic terms for the world volume scalars. – 16 –hich is the correct result for a generic anti-D3-brane in a compactification with a singleK¨ahler modulus T . Here µ is related to the tension of the anti-D3-brane. In KKLMMT [73]it was shown that the correct result for an anti-D3-brane in a warped throat is V w D3 = µ ( − i( T − ¯ T )) . (3.13)Both of these results can be reproduced as described in the previous subsection by settingeither V new = V D3 or V new = V w D3 as first noticed in [74], leading respectively to K = − (cid:2) − i( T − ¯ T ) (cid:3) + µ − S ¯ S ,K = − (cid:2) − i( T − ¯ T ) (cid:3) + µ − S ¯ S − i( T − ¯ T ) = − (cid:20) − i( T − ¯ T ) − µ S ¯ S (cid:21) , (3.14)where we used S = 0 in the last equality.The connection between the above description and an actual anti-D3-brane was firstestablished in [2, 3] and this has ultimately led to an explicit reduction of the full anti-D3-brane action, coupled to all background fields in [51, 52]. This gives us confidence that with asufficient amount of effort one can likewise extend any other explicit example to a full modelthat contains all world volume fields on the Dp-brane. Furthermore, it should be clear fromthe above example that our procedure is guaranteed to give the correct answer in the limitwhere we set all world volume fields on the Dp-brane to zero. In that limit one just has toequate the model-dependent new contribution with our V new . The first new example to which we apply our above procedure are intersecting D6-branes intype IIA string compactifications to four dimensions. Such intersecting branes have led tostandard like models, as for example reviewed in [49], and they have been included in fluxcompactifications, as for example reviewed in [48]. Here we are particularly interested inbrane setups that do not preserve the linear N = 1 supersymmetry of the background. Suchsetups have been argued to lead to D-term breaking potentially going back more than twentyyears ago and we would like to clarify here the following:1. The papers we have been looking at are studying such setups using 10d supergravity oreven string theory and are using the correct DBI and worldsheet (WS) action for theD6-branes. So, the obtained results are as far as we checked all correct.2. The repackaging or interpretation of the four-dimensional scalar potential arising fromnon-supersymmetric D-branes as D-terms is not correct.– 17 –he reason, as argued above, is that we cannot use the language of linear supersymmetry sincethe world volume fields on the D-branes transform non-linearly and this cannot be changedor undone by any field redefinition.A beautiful paper that discusses the contribution of D6-branes in type IIA flux compact-ifications is [75]. The authors focus in particular on D6-branes wrapping an arbitrary 3-cycleΣ on T / Z × Z . Defining Ω Σ = (cid:90) Σ e − φ Ω , (3.15)they find that the scalar potential contribution from the DBI action takes the form V DBI = T D ˆ s − (cid:113) (Re Ω Σ ) + (Im Ω Σ ) , (3.16)where ˆ s = e − φ vol is a real combination of the dilaton and geometric moduli. For supersym-metric D6-branes Im Ω Σ = (cid:82) Σ e − φ Im Ω = 0, so that V DBI = T D ˆ s − Re Ω Σ , where we usedthat Re Ω Σ > (cid:90) d x (cid:113) − g s (10) e − φ R (10) = (cid:90) d x (cid:113) − g s (4) e − φ vol R (4) + . . . . (3.17)We now only rescale the 4d metric g sµν → ˆ s − g Eµν , with again ˆ s = e − φ vol . This takes us tothe 4d Einstein frame. For the Dp-brane actions given in equation (3.4) this means that theyall pick up an extra factor of ˆ s − = e φ / ( vol ) from rescaling the 4d part of the metric only,which leads to the above result for the scalar potential.Following [76] the authors of [75] decompose the above into a putative F-term and D-termcontribution V DBI = V F + V D ,V F = T D ˆ s − Re Ω Σ ,V D = T D ˆ s − (cid:18)(cid:113) (Re Ω Σ ) + (Im Ω Σ ) − Re Ω Σ (cid:19) . (3.18)This decomposition is such that V D = 0 for supersymmetric D6-branes (Im Ω Σ = 0), inaccordance with the fact that D-terms cannot uplift the vacuum energy without breakingsupersymmetry. Such an idea, that we can write the SUSY breaking contributions from non-supersymmetric D-branes as a D-term in four-dimensional N = 1 supergravity, can be tracedback twenty years to papers like [77, 78]. However, given our improved understanding of We are following the conventions of for example [17, 48]. These differ from the conventions used byVilladoro and Zwirner (VZ) in [75], so that our Ω satisfies Ω = iΩ VZ , and our Ω Σ is ˜Ω π in [75]. In [75], N D6 branes were considered, which just changes all T D below to NT D . – 18 –on-linear supergravity and its connection to D-branes, we can correct this and give a properdescription of the low energy four-dimensional effective theory for such brane setups.As was noticed in [75], one can attempt to describe the above D-term scalar potential interms of a gauge kinetic function f and a D-term D . The U(1) symmetry is the axion shiftsymmetry as deduced from the vector coupling in the Chern-Simons term of the D6 brane,normalized with µ = T D . It turns out that its moment map is related to the imaginary partof (3.15): T D Im Ω Σ = ˆ s P . (3.19)The DBI action leads to a kinetic term for the vector that would identify in the N = 1formulation Re f = T D (cid:113) (Re Ω Σ ) + (Im Ω Σ ) = T D Re Ω Σ + O (cid:18) Im Ω Σ Re Ω Σ (cid:19) . (3.20)By omitting the second part, they identify a holomorphic function of the moduli, whose realpart agrees with the leading part in (3.20) and whose imaginary part agrees with the F ∧ F terms in the Chern–Simons term. Using that holomorphic f allows them to rewrite the lastline of (3.18) as (adapted from equation (2.23) in [75]) V D = T D ˆ s − (Im Ω Σ ) Re Ω Σ
11 + (cid:114) (cid:16)
Im Ω Σ Re Ω Σ (cid:17) = 12 Re f P
21 + (cid:114) (cid:16)
Im Ω Σ Re Ω Σ (cid:17) . (3.21)The authors of [75] note that this is only “compatible with the standard formula of N = 1supergravity”, if | Im Ω Σ | / | Re Ω Σ | (cid:28)
1. However, the mathematically rigorous statement isthat the expression above takes the form of a standard D-term, iff Im Ω Σ = 0. The latter istrue for supersymmetric D6-branes in which case the D-term contribution in equation (3.18)vanishes. Another interesting case are anti-D6-branes, which are wrapping supersymmetriccycles with the opposite orientation and therefore also have Im Ω Σ = 0. Their contributionto the scalar potential for general flux compactifications was derived, using the language ofnon-linear supergravity, in [17]. Here we generalize the results of [17] to arbitrary D6-branesby noting that the above DBI action in equation (3.16) can be added to any existing scalar Interestingly, the authors of [75] point out that the discrepancy should be cured by including higher deriva-tives interactions into the DBI action. In this respect, non-linear supersymmetry is providing precisely therequired higher-derivative interactions to make the whole description consistent. This supports the argumentin [75]. – 19 –otential by simply modifying the K¨ahler and superpotential as K = K before + e K before S ¯ ST D ˆ s − (cid:113) (Re Ω Σ ) + (Im Ω Σ ) ,W = W before + S . (3.22)Here S is a nilpotent chiral multiplet satisfying S = 0. The only dynamical field in S isthe goldstino that necessarily has to live on the D6-brane world volume since the D6-braneis breaking supersymmetry. Other world volume fields like the gauge field or scalar fieldsliving on the D6-brane are model dependent. One should be able to explicitly include themfor any given setup using additional constrained N = 1 multiplets. Here we are not pursuingthis road but would like to quickly discuss the generalization to multiple D6-branes. In thiscase one expects the following contribution to the scalar potential, generalizing the expressionabove in equation (3.16), V DBI = T D ˆ s − (cid:88) i (cid:113) (Re Ω Σ i ) + (Im Ω Σ i ) , (3.23)where i denotes the sum over the set of non-supersymmetric D6-branes. This can likewise beobtained by modifying the K¨ahler and superpotential as K = K before + e K before S ¯ ST D ˆ s − (cid:80) i (cid:113) (Re Ω Σ i ) + (Im Ω Σ i ) , (3.24) W = W before + S . (3.25)Now the goldstino contained in the nilpotent chiral superfield S is a linear combinationof world volume fermions from all the D6-branes that break the four-dimensional N = 1supersymmetry preserved by the background.There is a point that we have omitted so far, namely we have to ensure that V new , which is V DBI in equations (3.16) or (3.23), is actually a function of the closed string moduli. That thisis the case in the concrete model studied in [75] can be seen for example from their equation(2.9). We can also show that this is generically the case for non-supersymmetric D6-branes:Using the conventions of appendix A.1 in [17] and identifying their e − φ with our √ ˆ s , thecomplex structure moduli in type IIA are given by Z N = (cid:82) Σ N (cid:16) C + i √ ˆ s Re Ω (cid:17) , where theΣ N are a basis of the orientifold odd 3-homology. This means that Im Z N = (cid:82) Σ N √ ˆ s Re Ω. Itis easy to convince oneself that the overall volume and the dilaton are always real functions ofthe moduli, so ˆ s = e − φ vol is also a real combination of the closed string moduli. However,the complex structure moduli Z N do not involve the imaginary part of Ω. So, how canwe write Im Ω Σ and Im Ω Σ i in equations (3.16) or (3.23) as a real function of the complexstructure moduli? The answer lies in the involution in the O6-orientifold projection. It– 20 –ctually acts on Ω as σ : Ω → Ω. Due to the complex conjugation in the action of σ we canwrite Im Ω Σ N as a real function of Re Ω Σ N , as explained for example in subsection 3.1 of [79],where we set their phase θ = 0. So, we conclude that the contribution V new arising from theDBI actions of non-supersymmetric D6-branes is indeed a real function of the closed stringmoduli. Here we show how our result applies to a generic Dp-brane that spontaneously breaks thefour-dimensional linear supersymmetry preserved by flux compactifications of type IIA or typeIIB string theory. In particular, we will use results from [53, 54] and follow their notation.The Ansatz for the ten dimensional metric is ds = e A ( y ) dx µ dx µ + g mn ( y ) dy m dy n . (3.26)The internal space is assumed to have an SU (3) × SU (3) structure, which allows one todefine two pure spinors ˆΨ and ˆΨ that differ depending on whether we are studying typeIIA or type IIB (see [53] for details). When considering a Dp-brane extending along the fournon-compact directions and wrapping a ( p − W m dσ ∧ . . . ∧ dσ p − = ( − p (cid:104) e A − φ ( i m + g mk dy k ∧ ) ˆΨ (cid:105) Σ ∧ e F (cid:12)(cid:12)(cid:12) p − , D dσ ∧ . . . ∧ dσ p − = (cid:104) e A − φ Im ˆΨ (cid:105) Σ ∧ e F (cid:12)(cid:12)(cid:12) p − , Θ dσ ∧ . . . ∧ dσ p − = (cid:104) e A − φ Re ˆΨ (cid:105) Σ ∧ e F (cid:12)(cid:12)(cid:12) p − , (3.27)where the subscripts Σ denotes the pullback to the world volume and the subscript ( p − p − S DBI , p = T Dp (cid:90) M , × Σ d x d p − y e − φ | Σ (cid:112) det ( − g | Σ + F | Σ )= T Dp (cid:90) M , × Σ d x d p − σ (cid:112) Θ + e A D + 2 e A g mn W m W n . (3.28)The above expression reduces to the previous result for D6-branes given in equation (3.16), if W m = 0 and after going to 4d Einstein frame as discussed below equation (3.16). Similarly,it reproduces the result for non-supersymmetric D7-branes in type IIB O3/O7 flux compact-ifications as given for example in equations (5) and (6) of [80], if we set W m = 0. So, let ustry to understand how our approach is compatible with the general result for any Dp-branein an SU (3) × SU (3) manifold, i.e. why is W m = 0 for us and why is Θ + e A D a functionof the closed string moduli? – 21 –e have seen above that we can in principle get pretty much any new contribution to thescalar potential in equation (3.10) by adding the nilpotent field S and changing the K¨ahlerand superpotential as in equation (3.9). However, the field S contains only the goldstino fieldout of all world volume fields on the Dp-brane. So, our approach needs to be extended if wewant to include all open string fields. This seems like a daunting task, if one explicitly worksout the component field action for a particular Dp-brane in a particular flux compactificationincluding all the fermionic terms, since the latter are not fixed by non-linear supersymmetry.However, it might be possible to find a more elegant way that as the expression above capturesall world volume fields. We are not trying this here but again restrict to the case where weset all world volume fields except the goldstino to zero. Then we use the fact that W m is thederivative of a holomorphic superpotential W that encodes the open string fields [54]. Thismeans W m = 0 for us, since we have set the world volume fields to zero. However, this is notsufficient for our method in equation (3.9) to work. We also need to show that V new is a (real)function of the closed string moduli since we included it in the K¨ahler potential. For thegeneric case above we see that this is the case as follows: The complex closed string moduli,universally denoted by Φ a , have in our conventions as real part combinations of the RR axionsand as imaginary part NSNS fields. These imaginary parts, Im Φ a , are directly obtained fromexpanding the polyform Re ˆΨ in cohomology. Similarly to the type IIA example above, wehave in full generality that Im ˆΨ and Re ˆΨ are directly related to each other as explainedat the end of Section 5.1 and in footnote 21 of [46], see also [81, 82]. This allows us to writeschematically S DBI , p = T Dp (cid:90) M , × Σ d x d p − σ (cid:112) Θ + e A D = T Dp (cid:90) M , × Σ d x d p − σ (cid:112) f (Im Φ a ) + g (Im Φ a ) , (3.29)where f (Im Φ a ) and g (Im Φ a ) are real functions and for supersymmetric Dp-branes one wouldhave g (ImΦ a ) = e A D = 0. So, this shows that our approach seems to be very broadly ap-plicable and to extend beyond the most well studied classes of Calabi–Yau compactificationor SU (3)-structure manifolds to the most general case of SU (3) × SU (3) structure compact-ifications of type IIA and type IIB string theory. The above general expression given in equation (3.28) has a non-vanishing contribution inthe supersymmetric limit where D = W m = 0. This means that it applies to branes thatcan in principle be supersymmetric in the particular background. Examples are D3- andD7-branes in type IIB compactification with an O3/O7 orientifold projection or D5- and D9-branes in type IIB with an O5/O9 orientifold projection. We have described above how to– 22 –ncorporate such branes in a four dimensional non-linear N = 1 supergravity action, evenif they do not wrap supersymmetric cycles and D (cid:54) = 0. However, our approach of includingnon-supersymmetric Dp-branes in a four dimensional non-linear N = 1 supergravity actionapplies more broadly.Let us look at one more example to understand this: If we study type IIB compactifi-cations with an O3/O7 orientifold projection, then any D5-brane will break supersymmetry.Nevertheless, we can include the backreaction of such non-supersymmetric D5-branes in thissetup by using our formalism. The four dimensional N = 1 closed string moduli are τ = C + i e − φ ,G a = (cid:90) Σ a (2) C + τ B ,T a = (cid:90) Σ (4) a (cid:20) C + C ∧ B + 12 C B ∧ B − i e − φ J ∧ J − B ∧ B ) (cid:21) . (3.30)A D5-brane could wrap any 2-cycle and have for example, see equation (3.4) above, S DBI , = T D5 (cid:90) M , × Σ d x e − φ J . (3.31)Such a D5-brane would be supersymmetric, if we had done an O5/O9-orientifold projectionbut it always breaks SUSY for the O3/O7-orientifold projection. The above action seemsnot immediately compatible with the way we identified the four dimensional N = 1 moduli:There is a J instead of a J ∧ J in the action. However, it was shown in [83] that the v a = (cid:82) Σ a (2) J defined in their equation (3.4) are functions of the moduli τ, G a , T a (see for exampletheir equation (3.51) and the text below). Thus, we find that our approach works evenfor non-supersymmetric D5-branes in type IIB compactifications with an O3/O7 orientifoldprojection. Likewise we expect it to work for any other non-supersymmetric source, also ifthe source is not even a Dp-brane. In this paper we have studied string (flux) compactifications on SU (3) × SU (3)-structuremanifolds in which supersymmetry is broken by Dp-branes. We have shown that such setups cannot be described using the standard language of linear, four-dimensional N = 1 super-gravity but require the more general, so-called dS supergravity [27–30]. We have reviewedin explicit examples as well as in full generality that the bosonic contribution from the DBIaction of non-supersymmetric Dp-branes to the closed string scalar potential takes a universalknown form. In the case where we set the world volume scalars on the non-supersymmetricDp-brane to zero we can easily include the new term in the scalar potential by modifying theK¨ahler and superpotential of the compactification in a universal way. All we needed for this– 23 –s the fact that supersymmetry breaking Dp-branes have among their world volume fields theGoldstino, which is guaranteed by Goldstone’s theorem [84–86]. This leads to the very simpleexpression in equation (3.9) that encodes the new contribution V new from non-supersymmetricDp-branes that includes the couplings of the non-supersymmetric brane to the closed stringmoduli.Our general results should have important applications to flux compactifications andmight help to improve our understanding of a tractable large class of non-supersymmetricstring compactifications. It essentially allows one to include manifestly supersymmetry break-ing terms to the scalar potential. For example, in the best studied class of type IIB flux com-pactifications with an O3/O7 orientifold projection, one can include supersymmetry breakingD5- and/or D7-branes. Both of these will lead to new terms in the scalar potential that arereal functions of the closed string moduli and in particular of the K¨ahler moduli that usuallydo not appear at tree-level.Extending our results to include all the open string world volume fields on a given non-supersymmetric Dp-brane in a particular compactification is an interesting exercise. So far,this has only been done explicitly for an anti-D3-brane in a warped throat in type IIB fluxcompactifications [51, 52]. However, while technically non-trivial there should be no concep-tual problem in working this out for other cases. It might also be possible to find a moregeneral way of including the open string moduli using the language of generalized geometry.While we have focused here on compactifications to four dimensions that preserve N = 1linear supersymmetry spontaneously broken by Dp-branes, many of our ideas apply muchmore broadly. For example, to compactifications to other dimensions and/or that preservedifferent amounts of supersymmetry. It should also not matter whether supersymmetry isspontaneously broken by Dp-branes or other sources like for example, M2-, M5-branes, NS5-branes or KK-monopoles. It would be interesting to study such related setups in more detailin the future. Acknowledgments
We are thankful to F. Farakos, R. Kallosh, J. Moritz and E. Palti for useful discussions andE. Palti for bringing this interesting problem to our attention. The work of NC, CR andTW is supported by an FWF grant with the number P 30265. CR is furthermore supportedby the Austrian Academy of Sciences and the Doctoral College Particles and Interactionswith project number W1252-N27. The work of MT and AVP is supported in part by theKU Leuven C1 grant ZKD1118 C16/16/005. The work of MT is supported by the FWOodysseus grant G.0.E52.14N. TW thanks the KITP for hospitality and the organizers of theworkshop “The String Swampland and Quantum Gravity Constraints on Effective Theories”– 24 –or providing a stimulating environment during part of this work. This research was supportedin part by the National Science Foundation under Grant No. NSF PHY-1748958.
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