Constrained superfields on metastable anti-D3-branes

Journal of High Energy Physics, May 2017

We study the effect of brane polarization on the supersymmetry transformations of probe anti-D3-branes at the tip of a Klebanov-Strassler throat geometry. As is well known, the probe branes can polarize into NS5-branes and decay to a supersymmetric state by brane-flux annihilation. The effective potential has a metastable minimum as long as the number of anti-D3-branes is small compared to the number of flux quanta. We study the reduced four-dimensional effective NS5-brane theory and show that in the metastable minimum supersymmetry is non-linearly realized to leading order, as expected for spontaneously broken supersymmetry. However, a strict decoupling limit of the higher order corrections in terms of a standard nilpotent superfield does not seem to exist. We comment on the possible implications of these results for more general low-energy effective descriptions of inflation or de Sitter vacua.

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Constrained superfields on metastable anti-D3-branes

Received: April Constrained super elds on metastable anti-D3-branes Lars Aalsma 0 1 3 Jan Pieter van der Schaar 0 1 3 Bert Vercnocke 0 1 2 Open Access 0 1 c The Authors. 0 1 0 Celestijnenlaan 200D , bus 2415, 3001 Leuven , Belgium 1 University of Amsterdam , Science Park 904, 1098 XH Amsterdam , The Netherlands 2 Institute for Theoretical Physics , KU Keuven 3 Institute for Theoretical Physics Amsterdam, Delta Institute for Theoretical Physics We study the e ect of brane polarization on the supersymmetry transformations of probe anti-D3-branes at the tip of a Klebanov-Strassler throat geometry. As is well known, the probe branes can polarize into NS5-branes and decay to a supersymmetric state by brane- ux annihilation. The e ective potential has a metastable minimum as long as the number of anti-D3-branes is small compared to the number of ux quanta. We study the reduced four-dimensional e ective NS5-brane theory and show that in the metastable minimum supersymmetry is non-linearly realized to leading order, as expected for spontaneously broken supersymmetry. However, a strict decoupling limit of the higher order corrections in terms of a standard nilpotent super eld does not seem to exist. We comment on the possible implications of these results for more general low-energy e ective descriptions of in ation or de Sitter vacua. D-branes; Superstring Vacua; Supersymmetry Breaking 1 Introduction The bosonic KPV potential The fermionic KPV potential Supersymmetry transformations At the south pole At the north pole At the metastable minimum Comments and conclusions A Details on fermions A.1 Projection matrix A.2 Fermionic action The fermionic action up to second order Reduction to four dimensions Mass matrix in four dimensions Mass matrix at the poles Mass matrix at the metastable minimum Fermionic action: orientifold compatible gauge choice A.3 Supersymmetry transformations Introduction De Sitter vacua are at the heart of any cosmological model as both the early and late universe are well-approximated by a de Sitter phase. It is therefore of great importance to understand the construction of de Sitter vacua in string theory and supergravity. However, such constructions have proven to be a tremendous challenge. Kachru, Kallosh, Linde and Trivedi (KKLT) provided a generic mechanism of moduli stabilization in Anti-de Sitter and an uplift to de Sitter vacua in ten-dimensional string theory already in 2003 [1] and by now, many di erent approaches for de Sitter compacti cations have been uncovered. In contrast, the equivalent mechanism for de Sitter vacua in an e ective foursuper elds [2{6]. By imposing constraints on super elds it is not only possible to describe elds transforming non-linearly under the broken supersymmetry, but also to eliminate unwanted degrees of freedom. General prescriptions for constrained super elds from linearly transforming ones in a supergravity context were given in [7, 8]. For some recent reviews of constrained super elds and their applications to cosmology, see [9, 10]. Constrained super elds are often e ective descriptions of the low-energy excitations. For example, in the context of four-dimensional spontaneous supersymmetry breaking the massless goldstino can be packaged in a chiral super eld that satis es a nilpotency constraint. This constraint arises after the bosonic superpartner of the goldstino (the sgoldstino) becomes heavy enough to be integrated out [11{13]. As argued in [11] this can be extended to multiple elds. In general, integrating out additional heavy degrees of freedom results in extra constraints which describe the universal low-energy dynamics of the theory, see also [7, 13]. It is of crucial importance to understand the embedding of constrained super elds in a putative UV-complete description. Can we indeed realize large mass splittings such that the constrained super elds correspond to a good approximation of the relevant low-energy physics? This question is especially important when considering cosmological in ation. As one typically accesses high energy scales during in ation it is necessary to ensure that elds eliminated by the constraints have large enough masses to be integrated out. Otherwise, a constrained super eld description will be invalid. An important condition for obtaining universal (UV insensitive) couplings to the goldstino, and standard constrained super eld descriptions, is that the masses of the heavy superpartners should be large compared to the supersymmetry breaking scale. If that condition is not ful lled, the constraints are higher-order and depend on the masses of the heavy elds [14]. This issue was recently reconsidered featuring global supersymmetry [15] and supergravity [16]. Those authors studied the emergence of the constraints by integrating out massive elds, instead of imposing the constraints by hand. In [16] the corrections to an in ationary model with two super elds were analyzed. One super eld was used to describe spontaneous supersymmetry breaking and a second one contained the in aton and its superpartner. The UV physics is described by a supergravity model with additional heavy super elds and supersymmetry is broken by an O'Raifeartaigh-mechanism. This particular UV model did not allow for an exact nilpotent super eld description, because the strict in nite mass limit of the sgoldstino that would decouple its uctuations as in [12] does not exist. Instead, corrections due to the nite sgoldstino mass during in ation signi cantly limit the range of parameters for which an e ective nilpotent description is available. It is not clear whether more generic UV models have similar restrictions on taking the large sgoldstino mass limit. In this paper we take a step back from in ation and study how universal the description of de Sitter vacua with a nilpotent super eld is, in the context of string theory. We build on supergravity and string theory. The uplift term of the KKLT mechanism is generated by anti-D3 branes in a Giddings-Kachru-Polchinski (GKP) background [17]. This uplift is an example of the generic string theory mechanism of supersymmetry breaking by branes in backgrounds with uxes. If the anti-D3-brane indeed breaks supersymmetry spontaneously [18{20] it should be possible to package a worldvolume fermion into a nilpotent super eld describing the goldstino. This expectation was con rmed explicitly by putting The e ective description for the rst constrained super eld models in the context of KKLT arises by explicitly putting the anti-D3-brane on top of the orientifold plane. To answer the question if a constrained super eld description of de Sitter vacua is still appropriate in a more general background, we remove the orientifold projection. One of us initiated this study with Kallosh and Wrase for a ten-dimensional at background [24]: the non-linear transformations for all massless worldvolume elds (vector, scalars, fermion) can indeed be described by constrained multiplets. The full understanding of anti-D3-branes in ux backgrounds should introduce corrections to the description in the at background of [24]. Anti-D3-branes at the bottom of a warped throat can polarize into NS5-branes under the in uence of background In this paper we show that one source of corrections is due to such polarization e ects.1 We write down the supersymmetric version of the action for the polarized brane and consider small uctuations around the metastable minimum from the four-dimensional point of view. This reveals that supersymmetry is indeed, to leading order in uctuations, nonlinearly realized at the minimum. The central question is at what scale the rst leading corrections to the standard four-dimensional non-linear description appear. We nd that this scale is not set by the mass of the scalar uctuations, but is instead smaller by a factor background. Interestingly, the strict limit that would decouple these corrections does not The rest of this paper is organized as follows. We review the potential for polarized anti-D3 branes from the perspective of the NS5 worldvolume theory in section 2, with special emphasis on the expected scale at which this description is valid. In section 3, we construct the supersymmetric completion of the polarized NS5-brane action. We analyze the four-dimensional supersymmetry transformations in section 4. Finally, in section 5 we comment on our ndings and the relation to the use of anti-branes in de Sitter uplifts. Appendix A contains a technical derivation of the fermionic terms in the action and the supersymmetry transformations, based on the S-dual D5-brane action in a ux background The bosonic KPV potential Let us start with a short review of some of the results of Kachru, Pearson and Verlinde (KPV) [18]. KPV added p anti-D3-branes to the warped deformed conifold geometry of Klebanov and Strassler [30]. The throat of this geometry is supported by M units of ux through the A-cycle and K units through the B-cycle. F3 = M H3 = The Klebanov-Strassler geometry is an example of a GKP background [17] that experiences a high degree of warping near the bottom of the throat in the six-dimensional geometry. 1In recent years the literature has been divided on whether metastable anti-D3 probes are robust beyond probe level, for recent work see [25{28] and references. We want to discuss the appearance of non-linear supersymmetry and possible corrections rst at probe level and do not discuss back-reaction in this paper. e4A0 p 3=gs. At the north pole ( Q( ) at the metastable minimum ( min). This con guration can decay non-perturbatively to the supersymmetric minimum at the south pole ( ), where the p anti-D3-branes annihilated and the nal con guration contains (M p) D3-branes. Since probe anti-D3 branes in the Klebanov-Strassler background feel a net force towards the bottom of the throat we can describe the relevant physics by focusing on the region near the tip of the throat, with topology R4 S3. The metric near the tip is [18] ds2 = e2A0 dx dx + gsM b02(d 2 + sin2 rameter of the deformed conifold. Anti-branes carry opposite charge with respect to the supersymmetric background, breaking all supersymmetry. By brane polarization [31], the anti-branes can blow up to form an NS5-brane wrapping an S2 inside the S3. Depending on the S2, or shrinks all the way to the opposite south pole of the S3, brane- ux annihilation takes place and the nal con guration becomes supersymmetric, see gure 1. Since the non-supersymmetric and supersymmetric states are continuously connected by moving the NS5-brane from the north pole to the south pole on the S3, one expects the breaking of supersymmetry in the metastable vacuum to be spontaneous and supersymmetry to be realized non-linearly. We opt to describe the dynamics from the perspective of the e ective NS5-brane worldvolume theory. The bosonic action describing the NS5 worldvolume theory is given by2 SNS5 = We wrote the DBI term in terms of the metric components Gk spanned by the (anti-)D3 brane coordinates (Minkowski coordinates plus possibly motion in ) and G?, spanned by 2The e ective action on the NS5-brane is obtained by S-duality of the D5-brane DBI theory. Strictly speaking this description is therefore only valid for large gs, but some (supersymmetric) properties and structures are expected to be invariant. the coordinates on the S2. The form elds in the action are 2 F2 = 2 F2 dB6 = charge p carried by the NS5-brane: F2 = 2 p ; min = and F3 = dC2. becomes3 over the S2: 3Where we added a constant to the action such that the potential is zero at the supersymmetric minimum. where for later convenience we introduced the position-dependent angle ( ), which takes 1 at the poles of the S3: cos( ( )) We plot the potential in gure 1. It has a metastable minimum for relatively small values We are speci cally interested in the e ective dynamics in the angular direction the S3, which is transverse to the NS5-brane wrapped on an S2 inside the S3. The action S = F2 = ? = b20M Q describes the e ective D3-charge at position . From the action one can nd the potential (the Hamiltonian at zero momentum): V ( ) = 3 e4A0 pQ2 + P 2 (1 + cos( )) ; By expanding for small values of up to fourth order we nd Around the north pole of the S3 ( In a moment we will expand the action in around the minimum general expansion of an arbitrary potential in up to cubic order can be ) = V ( min) + Using the standard normalization in four dimensions, a scalar eld and the cubic coupling 3 have mass dimension 1. For < m2 = 3 the quadratic term is a good approximation of the relevant physics, but for larger uctuations the cubic term dominates, signaling a breakdown of the quadratic approximation. For natural couplings is some (high-energy) cut-o scale, this would just restrict the uctuations to values below m , but for `unnaturally' large cubic couplings the quadratic approximation would only be valid for uctuations signi cantly smaller than the mass scale m . The second situation is exactly what we observe in the Klebanov-Strassler throat. Expanding (2.11) up to cubic order around min we obtain ) = From this expression, we see that 3=m larger than the quadratic coupling. In the remainder of this article, we are interested in the quadratic approximation. We are then forced to restrict to uctuations that are not only small compared to the dimensionless mass parameter m = 2 2=b20, which is of order one, but small compared to a dimensionless parameter set by the eld value in the metastable minimum: The importance of this basic observation will become clear when we discuss the corrected supersymmetry transformations in the metastable vacuum. In the rest of this paper, we continue with the discussion of the fermions on the NS5 worldvolume. We will be concerned with the leading behaviour at a xed but small value of p=M . Then we consider the small expansion, and discuss small eld uctuations around a xed background position for small . We consider up to quadratic order in the scale From the action it is straightforward to obtain the potential for the canonically normalized eld. For small uctuations around a minimum at 1, the kinetic term gets a with the potential V~ ( ) = p 1 pQ( )2 + P ( )2 We will arrange the kinetic terms of the fermions to have the same constant prefactor (for small uctuations at least), such that we can consistently compare mass scales. SNS5 = Since we expect the metastable minimum to break supersymmetry spontaneously, there should exist an associated massless goldstino. For a single anti-D3-brane on top of an orientifold plane, the goldstino was identi ed as the 4d fermion on the worldvolume of the anti-brane, which is a singlet under the SU(3) holonomy of the 6d internal space [21, 22]. Removing the orientifold plane, we now want to revisit the situation for the polarized NS5brane. Based on the physical picture of the previous section, we expect the e ective 4d worldvolume description to reduce to the known results for p anti-D3-branes at the north pole and M p D3-branes at the south pole, both probing the GKP background. The fermionic action up to second order Just as for the bosonic action, we formally obtain the fermionic NS5-brane worldvolume action from S-duality of a D5-brane. The action up to quadratic order in fermions is given by [29] (notice that we have a background with a constant dilaton) det(g + 2 gsF ) (1 = g = r We only included terms in the action that are non-zero at the tip of the throat, because we are not interested in dynamics taking us away from the tip (we dropped terms with form and one-form eld strengths). The indices m; n are ten-dimensional curved indices, ; indicate worldvolume indices. To avoid confusion with the equations below, we wrote the pullbacks of gamma matrices on the worldvolume with hats: ^ we underline tangent space indices (m; n : : :). The fermion is a doublet of Majorana-Weyl spinors with positive chirality. We now use the speci c embedding of the NS5-brane of the previous section and use the leg structure of the three-forms to simplify the expressions. The F3 ux is fully along the S3 spanned by ( ; ; ) while H3 is orthogonal to F3 in the internal space. This means we can drop H3 terms with legs along the worldvolume of the NS5-brane. Also we will drop the terms with @ coming from the pullbacks of gamma matrices, as those do not contribute to the mass matrix. We only highlight the main points of the calculation here. For more general expressions and more detailed information, see appendix A. The combination in right brackets of (3.1) gives: = (M~ 1 cos(2 )Fmnp 3 + (1 + sin2 )gs 1Hmnp 1 with the position-dependent angle de ned in (2.10). It is important for our calculations to note that NS5 is o -diagonal. As explained in appendix A, at the tip of the deformed conifold, this projector takes on a fairly simple form: NS5 = 45 sin( )) : We still need to gauge x the kappa-symmetry on the brane. We do this by taking the gauge xing condition on the doublet = ( 1; 2) 3 = 1 = 0 : Now we can express the action in terms of the spinor 2 only. This gauge xing condition is convenient due to its simplicity, but it is not suitable when one also wants to perform an orientifold projection. The calculation for the mass matrix can also be done in a gauge where we set (1 + that this choice of gauge does not change the mass matrix. We introduce the notation for the remaining spinor components Taking care of the o -diagonal matrix NS5 and using that for a 10d Majorana-Weyl spinor the only fermion bilinears that are non-zero have three or seven gamma matrices, we nd SNS5 = with d the volume element on the unit two-sphere. The only terms that contribute to the mass matrix M are M = cos(2 )Fmnp This is the mass matrix on the six-dimensional world volume. The reduction to four dimensions could also pick up extra mass terms coming from the reduction of the kinetic term [32]. To determine if these extra mass terms still allow for a massless fermion, we have to make sure the internal piece of the modi ed Dirac operator together with the mass matrix [(M~ 1 + M] has a zero mode. In the remainder of this section we show that this is indeed the case and the lowest Kaluza-Klein modes reveal the existence of a massless fermionic mode, which we will identify as the massless goldstino. In the previous section we obtained the action for the worldvolume fermions from the sixdimensional point of view. We now discuss the four-dimensional interpretation. we perform the reduction to four dimensions, we will write in terms of four fermions: a singlet 0 and a triplet i under the SU(3) holonomy of the six-dimensional transverse internal space. This decomposition can for instance be found in [22]. Let us rst focus on the reduction of the mass matrix. We observe that, up to angles that parameterize the position of the NS5 on the S3, it is completely determined by the ux of the background, which can be written in terms of the complexi ed three-form = mij i+ j+ + m{| { | ; G3 = F3 This immediately implies that the only relevant structure for the fermionic mass matrix we have to reduce to four dimensions is the real part of the complex three-form: M = (cos(2 ) + cos( )) (G3 + G3)mnp Up to the coordinate-dependent prefactor (cos(2 ) + cos( )), this is the known mass term for anti-D3 branes in a supersymmetric background with uxes that carry only D3-brane charges, as reviewed in [22]. The general discussion of our mass terms also carries through directly as in [22]. The background three-form is (2,1) and primitive, and therefore we nd that the only non-zero contributions to the mass matrix come from the triplet: where the mij are linear in the components of the background ux and subscripts denote 4d Weyl spinors massless, similar to a single anti-D3-brane that does not polarize [22]. The kinetic term of the fermions still contains a `modi ed Dirac operator' = ((g + 2 gs 3F ) 1) that could contribute to the mass matrix in four dimensions. We can ask whether there is a fermion that remains massless and signals the spontaneous breaking of supersymmetry. The ux is crucial. If we would reduce the Dirac operator on an S2 without worldvolume ux F , this would leave no fermion massless, as the 2-sphere admits no covariantly constant spinors. However, we have a non-zero worldvolume ux F on the S2 that induces the (anti-)D3 brane charge. This allows for the possibility that the gauge eld twists the Dirac operator on the S2 such that the modi ed Dirac operator can have a zero mode on the 2-sphere, along the lines of [33]. If that zero mode agrees with the 0 direction, we can 0 as the four-dimensional goldstino, as was suggested in [19]. Instead of explicitly solving (M~ 1 mode from the dual perspective of the non-abelian gauge theory on the anti-D3 branes. From this point of view the situation is more transparent because a reduction to four dimensions is not needed. In the non-abelian theory all elds become matrix-valued. The transverse scalars i have a potential that describes the brane polarization. One nds that the local minimum of this potential occurs when the scalars take an irreducible representation of SU(2) [18], which agrees with the metastable minimum of the wrapped NS5-brane. In the non-abelian theory, the analogue of the abelian 2-sphere with coordinates ; is a non-commutative fuzzy 2-sphere. The non-abelian theory is studied in detail in [19], with a decomposition of the 10d worldvolume fermion (which is promoted to a matrix) to the 4d 0 (`gaugino') and i (`modulini'), analogous to the abelian theory. For supersymmetry preserving ISD couplings between i G3- ux, the gaugino mass terms vanish and only the modulini are massive, in agreement with our NS5 mass matrix M. An additional mass contribution might come from Yukawa The scalars i have a vacuum expectation value in the metastable minimum such that the Yukawa coupling can be viewed as an o -diagonal contribution to the mass matrix. To nd the massless goldstino, we expand the fermions in terms of eigenfunctions on the fuzzy sphere (the non-commutative analogue of spherical harmonics [34]). One nds that the leaving 0 massless. Higher (` > 0) modes correspond to a Kaluza-Klein tower [33] and can be ignored when the radius of the fuzzy sphere is su ciently small. Clearly, ignoring ` > 0 modes we are left with an abelian truncation of the non-abelian fermionic action where we can identify 0 as a goldstino. This veri es the idea that in this setup spontaneous supersymmetry breaking should come with a massless fermion. M = Mass matrix in four dimensions To facilitate comparison with similar treatments in the literature, we will now explicitly compute the mass matrix M at the three relevant positions: the two poles of the S3 and the metastable minimum at min. As mentioned before, we can rewrite the mass matrix in terms of the complexi ed three-form G3. The general form of the mass matrix then becomes (cos(2 ) + cos( )) (G3 + G3)mnp We now give the four-dimensional reduction and discuss the fermionic mass matrix on the positions of interest. Mass matrix at the poles The mass matrix M becomes At the North pole, we have M = 0) = = ) = 0 : From the small expansion of In terms of G3 ux we have the mass matrix ( min) = M = These match earlier results for anti-D3 branes or D3 branes on GKP backgrounds derived in [22, 35] (note that we are working in an S-dual frame compared to those references, so one should take G3 ! igs 1G3 for comparison to those references.) Mass matrix at the metastable minimum To obtain the mass matrix in the metastable minimum, we expand cos( ) to lowest nontrivial order and we evaluate this expression at the minimum: M = Supersymmetry transformations In the previous section we argued that in the metastable minimum supersymmetry is spontaneously broken by identifying the corresponding massless goldstino. This also suggests that the e ective low-energy dynamics can be described in terms of a nilpotent supereld [11]. In this section we analyze the supersymmetry transformations to verify this picture and identify the leading corrections. To begin we need the expressions for the supersymmetry transformations in non-trivial ux backgrounds, which can be found in short in appendix A, adapted from [29]. Supersymmetry of the background requires that (1 + i 2 0123) = 0 2 = With a slight abuse of notation, we will write the 32-component Majorana-Weyl spinor 2 2. We have the following supersymmetry transformations: (1 + ) and the operator de ned as 0123 +, see eq. (A.12): We do not write fermion terms in , as those result in transformations that take use beyond the quadratic fermion order in the action. More details on these transformations can be found in appendix A. From here, we can already see the general form of the transformations around the poles, since = 0 : = 1 + : : : ; = +1 + : : : ; where the ellipses denote terms with eld uctuations. So around = 0 we nd non-linear transformations and at linear ones. To obtain four-dimensional supersymmetry transformations, in the end we always decompose the spinor into the singlet 0 and the triplet i under the SU(3) holonomy. Moreorientation of the S2 inside the transverse S3, corresponding to the superpartner of the at the south pole where supersymmetry is restored. The other directions come along for the ride and we will ignore them throughout. We are also interested in the supersymmetry transformations with parameter 0, the SU(3) singlet component of the 32-component Majorana-Weyl spinor , as this is the supersymmetry preserved by the With all the relevant information in place, we present a summary of the fourdimensional fermionic, scalar and gauge eld supersymmetry transformations at the di erent locations of interest: both poles and most importantly the metastable minimum. At the south pole Let us rst analyze the south pole = 1 + and the reduction of the supersymmetry transformations to four dimensions gives 0 = 3 = p ~ = p where we rede ned the scalar as follows. and rescaled spinors as ~ = (gsM b02)1=2 ; ! p12 . We conclude that, as expected, at and ( 3; ) correspond to a chiral multiplet. If we would have included the other two directions on the S2 that we now have ignored, they would form two additional chiral At the north pole At the (unstable) north pole we expect the e ective description to formally reduce to the results for a supersymmetry breaking anti-D3-brane in a GKP background. We will write the transformations to at most quadratic order in uctuations. Since sin( ) = O( 2); cos( ) = 1 + O( 4), we set cos the results of [24]. order in the supersymmetry transformations. Then we indeed reproduce to quadratic order We will expand the supersymmetry transformations up to the rst non-trivial order in the elds. Then we only have to expand the operator to rst order: The supersymmetry transformations around = 0 are = 1 ~ = ~ = ~ = ~ = ~ = ~ = A the collection of all elds = f ; ; A g . We recognize the rst terms as the standard non-linear transformations. By requiring the elds to transform non-linearly under the supersymmetry we can perform appropriate eld rede nitions of the spinors, scalar and gauge eld, that x the transformations uniquely: and we have the standard-looking transformations With an additional rescaling of the spinors ~ 2 , we then nd the following supersymmetry transformations in terms of the appropriate four-dimensional elds ~0 = 0 + O( 2) ~3 = 0 + O( 2) ~ = (~0 ~ = (~0 We conclude that indeed, as anticipated by the physical interpretation in terms of braneux decay, this seems to describe an exact non-linear realization of (broken) supersymmetry when adding anti-D3-branes to the GKP background and ignoring the (higher order) dynamics describing the polarization in the transverse S3 directions. This matches the results for anti-D3 branes in supersymmetric backgrounds of [21{24]. Note that this (direct) expansion of the theory around the north pole is only a formal result: since the scalar eld sits at the maximum of its potential, this is an expansion around an unstable con guration. At the metastable minimum Now let us include the polarization dynamics and determine the transformations at the true metastable minimum the S3. We rst expand in min, which should include corrections due to the dynamics on and then in the uctuations around the metastable minimum. The expansion for around the metastable minimum is then ) = captured by expanding in powers of : j =0 ; where j =0 is given by (4.13). We nd that after the eld rede nition (4.15) and the spinor rescalings the transformations (4.16) are corrected by the -expansion (or equivalently ~ = ~ = ~ = j =0 j =0 A j =0 + The transformations in the metastable minimum become ~0 = 0 ~3 = 0 ~ = (~0 ~ = (~0 The rst terms correspond to the standard non-linear transformations. Remember that the expansion of min is given by (4.21). We identify two types of corrections. First terms are just proportional to (the square of) p=M and re ect the shift towards the metastable minimum. In fact, if we could ignore the eld (as well as the spinor 3), the probe limit would consistently reproduce a subset of the non-linear supersymmetry transformations at the north pole. In other words, if the elds were in nitely massive, the probe limit takes you to the north pole and a constrained super eld description of the goldstino and the gauge eld would be adequate. However, it can be seen from (2.14) that the mass of the scalar is always of the same order of the potential energy scale in the metastable vacuum, so uctuations in be decoupled. Interestingly the corrections that are proportional to 2 are all, except for large and one should include (all) higher order terms. This is in line with the discussion of section 2: at order are forced to conclude that a strict decoupling limit in which the e ective description in terms of non-linearly realized supersymmetry becomes UV independent does not exist. As a consequence the validity of a constrained super eld description is restricted. Just how restricted can be estimated by observing that the corrections become comparable to the shift term when the uctuation is of order p=M or equivalently not come as a complete surprise, since this is where the expansion in min. This should breaks down. We can translate this into a corresponding mass scale using the potential, giving a scale that is In other words, the description in terms of non-linearly realized supersymmetry seems to break down at scales far below the mass scale of relevance in the metastable vacuum. Closing this section, we would like to make a nal comment. It is important to realize that one should not perform an additional eld rede nition at min that would remove the leading corrections. For instance, an additional eld rede nition of 3 that removes the corrections at the same time modi es the form of the transformations at the north pole and also changes the fermionic mass matrix for 0. In this case, the rede ned spinor cannot be identi ed with the massless goldstino. Comments and conclusions Constrained super elds provide a powerful technique in the context of a universal (UV insensitive) low-energy description of spontaneously broken supersymmetry. A crucial requirement is a stable and large enough hierarchy between the scale of the elds that are projected out by the constraints and the relevant scale of the low energy e ective theory. In some cases such a hierarchy might not be achievable, precluding the existence of a standard constrained super eld description. In general however the appropriate constrained supereld description is valid up to some energy scale that should be identi ed and compared to the supersymmetry breaking scale. In this work we studied the leading corrections to the nilpotent goldstino super eld description of anti-D3-branes in the GKP background from polarization e ects. Our main observation is that the (non-linear) supersymmetry transformations in the metastable vacuum receive corrections that cannot be `decoupled' and actually become large in the probe limit p ! 0. To arrive at that result we constructed, to leading order in the elds, the supersymmetric completion of the e ective theory on an NS5-brane wrapped on an S2 inside the transverse S3 at the tip of the KS throat geometry of [18]. We identi ed the massless goldstino of spontaneous supersymmetry breaking as well as the gauge eld and transverse that describes the position of the S2 inside the S3. In the absence of an orientifold plane that projects out the bosonic degrees of freedom, they should also transform non-linearly. In the metastable state we again identi ed 0, the singlet under the SU(3) holonomy of the `internal' space, as the massless goldstino associated with the spontaneously broken supersymmetry. We argued this from the non-abelian point of view, which 1). From the abelian perspective this should correspond to twisting the Dirac operator with a gauge the 2-sphere, as was done in [36]. A full treatment of the modi ed Dirac operator on the 4d reduced abelian NS5-brane should also reveal this zero mode at the position of the metastable minimum. We hope to come back to this question in future work. We found that uctuations of the scalar eld around the metastable minimum cannot be decoupled. Moreover, corrections to the non-linear supersymmetry transformations become large at a scale far below the mass scale set by the scalar uctuations in the brane number and the background ux. This limits a nite parameter window where an e ective low-energy description of the metastable vacuum in terms of a constrained super eld is appropriate. This might not come as a total surprise. When the source of spontaneous supersymmetry breaking is intrinsically higher-dimensional, it might not admit any low-energy description in terms of (simple) constrained super elds. This is clearest for more energetic uctuations around the metastable minimum, with p=M . Those uctuations are not localized around the metastable minimum, as they exceed the energy di erence between the metastable state and the north pole (left maximum in gure 1). However, they are still localized on the northern hemisphere of the S3, as they have less energy than the absolute maximum of the potential. Those uctuations describe full 6-dimensional uctuations around the nilpotent super eld description of anti-D3 branes, governing the non-linear transformations around the north pole further will invalidate the non-linear description altogether, and will lead to a restoration of the linear transformations by higher-dimensional excitations. The uctuations we study in this paper are of a di erent nature. They capture excitations very close to the metastable minimum and obey p=M . They can be captured in a four-dimensional language (albeit not with standard constrained super elds). Determining the relevant uctuations in the KK reduction to four dimensions is subtle, since we discussed two di erent descriptions with opposite regimes of validity. The polarized NS5brane point of view is only valid for a large S2 and is hence intrinsically 6-dimensional. In section 3.2, we argued however from the dual non-abelian anti-D3 point of view that the set of lowest mass states of the KK spectrum in four-dimensions contains the massless goldstino. It is straightforward to check that the requirement p=M is a direct consequence of the relevance of higher order terms in the DBI action around the metastable vacuum. The expansion of the polarization potential around the metastable minimum (2.14) shows that the higher order terms become important when p=M , as we also concluded from the supersymmetry transformations. From the low-energy e ective eld theory point of view the theory becomes strongly coupled as soon as Our observations appear to be in line with the discussion of [14]. The mass of the around the minimum of the potential is in fact of the same order as the supersymmetry breaking scale, as can easily be seen from (2.14) m2 = As explained in [14], integrating out massive elds with masses of the order of the supersymmetry breaking scale does not lead to universal couplings of the goldstino and instead give rise to generalized holomorphic constraints on super elds. The UV dependence in our setup becomes apparent at scales correct the supersymmetry transformations. Whether and how this can be described in terms of generalized (higher order) constrained super elds, or in another approach such as the `goldstino brane' [37, 38], is a question we hope to come back to in the future. Let us nally brie y elaborate on what the general consequences of our ndings might be in the context of string cosmology. Following the arguments of [14], to allow for a standard universal nilpotent super eld description one would require a stable hierarchy between the scale of supersymmetry breaking and the mass of the transverse scalar . In the original KKLT scenario, the scale of supersymmetry breaking is set by the uplift energy of the metastable anti-D3 brane and hence seems to remain of the order of the mass uctuations around the metastable vacuum. As a consequence the uplift with p metastable polarized branes might lead to a similar breakdown of a putative universal constrained super eld description at energies far below the supersymmetry breaking scale. An e ective description of the metastable minimum by nilpotent super elds all the way up to the supersymmetry breaking scale with polarized anti-branes would require a version of the KKLT mechanism where the supersymmetry breaking scale and the uplift energy can be decoupled. Broad classes of such models are available: for instance in [39], or antibrane uplifts of an AdS minimum where supersymmetry is already broken, as in the Large Volume Scenario and related work [40{42]. We hope to address some of these questions in future work. Acknowledgments We thank Daniel Baumann, Eric Bergshoe , Nikolay Bobev, Ben Freivogel, Thomas Hertog, Dan Roberts, Gary Shiu, Hagen Triendl, Thomas Van Riet for discussions; Riccardo Argurio, Luca Martucci and Timm Wrase for feedback on a draft version of the paper; and Renata Kallosh and Timm Wrase for collaboration on related work. BV thanks the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support during the completion of this work. BV was supported during the initial stages of work by: the European Commission through the Marie Curie Intra-European fellowship 328652-QM-sing and Starting Grant of the European Research Council (ERC-2011-SrG 279617 TOI). Currently, BV is supported in part by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy (P7/37), by the European Research Council grant no. ERC-2013-CoG 616732 HoloQosmos and the KU Leuven C1 grant Horizons in High-Energy Physics. This work is part of the Delta ITP consortium, a program of the Netherlands Organisation for Scienti c Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). The work of LA and JPvdS is also supported by the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scienti c Research (NWO). Details on fermions In this appendix we review and apply the relevant details of the fermionic action of a Dp-brane of [29, 43, 44], its supersymmetry transformations and gauge xing. We take the results for a D5 brane with worldvolume ux in the S-dual background to KlebanovStrassler. We follow the conventions of [29]. For easy comparison with the literature on gauge- xed fermionic D-brane actions, we keep this appendix wholly in that `D5-frame' and we adapt notation slightly to match as much as possible the related work for Dp-branes at space [45] used in the recent literature on non-linear supersymmetries on anti-D3 branes [22{24, 46]. To transform the results of this appendix (`app') to the expressions used in the text, one has to apply the following S-duality rules to the NS5-frame: H3app = = (gs 1)text ; Projection matrix We obtain the matrix D5 from [29]: We have + = 1 and the relation denote pull-backs on the worldvolume ^ = @ XM (F ) = +( F ). Note that hats on gamma matrices D5 = (D05) = We will split the eld and the metric in a four-dimensional part (along the D3 worldvolume) and a transverse part along the two-sphere as: F = F It is not hard to see that the matrix in the projector splits as: + = 2 F ? 8 F 1 2 F 3 4 The ellipses indicates terms higher order in elds and indices have been raised and lowered with the metric G k and XI are the transverse coordinates. This is the straightforward covariantization of the kappa-symmetry matrix for a D3-brane. +? = cos( ) +k = where Greek letters still refer to worldvolume indices, but we make a split: the middle of The calculation of the term +? follows straightforwardly from the discussion of secF ? = Q( )volS2 : The four-dimensional part of the projector parallels that of the projector dubbed appendix of [24]. Note that we only consider the bosonic terms, as fermionic terms in would take us beyond the quadratic fermionic order in the action. The result for +k is +k = +? = +k = det(G? + F ?) det(Gk + F k) ( ; : : : = 4; 5). tion 2, with Fermionic action and writing We split the terms not involving a covariant derivative along the four-dimensions and the two-sphere as k = [G M? = [(G? + 2 gs 3F ) 1] = M We nd (using ; for directions on the two sphere, and ; for four-dimensions) k = M? = sin2( ) Hmnp 3 + e Fmnp 1 cos( ) = det(G? + F ) sin( ) = det(G? + F ) The signs in these last two equations are chosen for later convenience. Now we use that the ux H3 is fully along S3 and F is along S2, while F3 is orthogonal. So the non-zero terms in Mk; M? are k = M? = sin2( ) Hmnp 3 + e Fmnp 1 which gives the result (3.3). (as they are higher order in the action), we get From (A.2) and (A.12) we nd that for vanishing F and neglecting the derivative terms D5 = Now we use that for Majorana-Weyl bilinears only terms with three or seven gamma matrices are non-zero. = 0 for n 2= f3; 7g We now see that the last term in M? will not contribute at all and we nd cos(2 )Hmnp + cos(2 )Hmnp + cos( )e Fmnp 0123 mnp = 0 mnp 3 = mnp 1 = 2 Hmnp 2 cos( ) Fmnp Hmnp + e cos( )Fmnp We then nd after some algebra that Where we again used (A.23) to eliminate some terms. The total mass matrix is then cos(2 )Hmnp + e cos( )Fmnp 0123 = (?6F )mnp ; with ?6 the Hodge star operator on the six-dimensional internal manifold. This yields the nal result (3.8): cos(2 )Hmnp + e cos( )Fmnp 0123 Fermionic action: orientifold compatible gauge choice For completeness, we show that taking the alternative gauge choice D5) = 0 1 = 0123(cos( ) + sin( ) 45) 2 ; to x the kappa-symmetry we obtain the same mass matrix. 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Lars Aalsma, Jan Pieter van der Schaar, Bert Vercnocke. Constrained superfields on metastable anti-D3-branes, Journal of High Energy Physics, 2017, 89, DOI: 10.1007/JHEP05(2017)089