PreprinttypesetinJHEPstyle-HYPERVERSION Fermions on the Anti-Brane: Higher Order Interactions and Spontaneously Broken Supersymmetry 6 1 0 2 Keshav Dasgupta, Maxim Emelin, and Evan McDonough c e Ernest Rutherford Physics Building, McGill University, D 3600 University Street, Montr´eal QC, Canada H3A 2T8 7 [email protected], [email protected] [email protected] ] h t - p e Abstract: It has been recently argued that inserting a probe D3-brane in a flux background breaks h supersymmetry spontaneously instead of explicitly, as previously thought. In this paper we argue [ that such spontaneous breaking of supersymmetry persists even when the probe D3-brane is kept in a 3 v curvedbackgroundwithaninternalspacethatdoesn’thavetobeaCalabi-Yaumanifold. Toshowthis 9 we take a specific curved background generated by fractional three-branes and fluxes on a non-K¨ahler 0 4 resolved conifold where supersymmetry breaking appears directly from certain world-volume fermions 3 becoming massive. In fact this turns out to be a generic property even if we change the dimensionality 0 . of the anti-brane, or allow higher order fermionic interactions on the anti-brane. We argue for the 1 0 former by taking a probe D7-brane in a flux background and demonstrate the spontaneous breaking 6 of supersymmetry using world-volume fermions. We argue for the latter by constructing an all order 1 : fermionic action for the D3-brane from which the spontaneous nature of supersymmetry breaking can v i be demonstrated by bringing it to a κ-symmetric form. X r a Contents 1. Introduction 1 1.1 Spontaneous vs. explicit supersymmetry breaking with anti-branes 2 1.2 Outline of the paper 3 2. D3-brane in a Resolved Conifold Background: Soft (and Spontaneous) Breaking of Supersymmetry 4 2.1 A SUSY perturbation of the resolved conifold 8 2.2 Bosonic action for a D3-brane 12 2.3 SUSY breaking and the fermionic action for a D3 13 2.4 Perturbing away from the probe limit 14 2.5 Moduli stabilization and de Sitter vacua 17 3. Probe D7 in a GKP Background 18 3.1 The fermionic action for a D7 in a flux background 18 3.2 Fermions in 4d and spontaneous SUSY breaking in a GKP background 21 3.3 Inclusion of F 23 3.4 Effect of more general F 24 5 4. Towards the κ-symmetric All-Order Fermionic Action for a D3-brane 25 4.1 Towards all-order θ expansion from dualities 27 4.2 κ-symmetry at all orders in θ 35 5. Conclusion and Discussion 41 1. Introduction It has recently been shown [1, 2] that a probe D3-brane in a flux background breaks supersymmetry spontaneously, and furthermore, if the D3 is placed on an orientifold plane, the only low-energy field content is a single massless fermion1. The implications of this are two-fold: (1) that SUSY breaking is spontaneous, as opposed to explicit, indicates that there is no perturbative instability in the D3-D3 systemfamouslyusedtoconstructtheKKLTdeSittersolution[6],and(2)astheonlyfour-dimensional field content is a single massless fermion, which can be expressed in the d = 4 N = 1 supergravity theory as the spinor component of a nilpotent multiplet, this provides a natural starting point for a string theory embedding of the inflation models proposed in [9, 8, 10] and other works. This result, and the connection to string cosmology, provides impetus to further investigate Dp- branesystems;inordertopopulatethelandscapeofstablenon-supersymmetriccompactificationswith Dp-branes, to better understand supersymmetry breaking in these models, and to perhaps stumble upon new string theory settings where de Sitter space and inflation naturally arise. It is with these goalsinmindthatwepresentthreeinterconnectedanalyses, whichgeneralizeandbuilduponthework of [3, 1, 2]. 1See also [3, 5], and especially the key papers [4], that motivated the research on spontaneous susy breaking in the presence of a D3-brane. – 1 – 1.1 Spontaneous vs. explicit supersymmetry breaking with anti-branes Before we proceed with our analysis, let us start with a discussion of spontaneous supersymmetry breaking. Spontaneous supersymmetry breaking is a crucial element of string theory model building. This is because a consistent study of four dimensional physics requires that all or almost all moduli be stabilized, and all known mechanisms of moduli stabilization2 are understood in terms of a super- symmetric four dimensional theory, e.g. the complex structure moduli are fixed via the flux induxed superpotential as in [12]. Without an underlying supersymmetric theory, i.e. in the case that super- symmetryisexplicitlybroken, itisnotcleartowhatextenttheknownmethodsofmodulistabilization are applicable. Spontaneous symmetry breaking occurs when the ground state of a theory does not respect the symmetries of the action. This is an essential part of model building in particle physics, supergravity, and string theory, as it gives theoretical control over corrections to the action. The situation in string theoryisslightlymorecomplicatedthaninparticlephysics, sinceproposeddeSittersolutionsinstring theory(forexampleKKLT[54])rarelyexistasthegroundstateofthetheory,butratherasmetastable minima. Given this, we will drop the phrase ‘ground state’ from our definition, and instead refer to non-supersymmetricstatesinasupersymmetrictheoryasspontaneouslybreakingthesupersymmetry. In simple cases, for example [2], there is a smoking gun of spontaneous supersymmetry breaking by antibranes: a worldvolume fermion remains massless, which one can identify with the goldstino of SUSY breaking. However, as discussed in [3], it will not in general be true that a worldvolume fermion remains massless. Instead, the goldstino of SUSY breaking will be some combination of open and closed string modes. Thus a more general diagnostic of spontaneous breaking is needed, which we will now develop. We will see that even in the absence of a massless fermion on the brane, supersymmetry breaking can still be shown to be spontaneous. Our diagnostic for spontaneous supersymmetry breaking by a probe Dp brane is the following: a solution breaks supersymmetry spontaneously if it is a solution of the theory with action: S = S +S , (1.1) IIB Dp where S is action of type IIB supergravity. The above action is explicitly supersymmetric, since IIB an anti-brane is 1/2 BPS, and thus negates the requirement to ‘find’ the goldstino in order to deduce thatsupersymmetrybreakingisspontaneous. AprobeD3inanon-compactGKPbackgroundwithout sources can be studied in this way. This reasoning applies directly to our second example: an D7 in a warped bosonic background without sources, which we will study in Section 3. However,thisdiagnosticislimitedinitsapplicability,asmanyinterestingbackgroundshaveexplicit brane or orientifold content in addition to the probe Dp. Fortunately, the condition (1.1) can in fact be extended to apply to a subset of these cases, by making use of string dualities to relate a flux background with branes to a background without branes. Again, this makes no recourse to the goldstino being a pure open-string mode, i.e. a worldvolume fermion. Our first example in this paper, a D3 in a resolved conifold background with wrapped five-branes, which we study in Section 2, is an example where dualities must be used to make sense of (1.1). One way to arrive at the resolved conifold with wrapped five-branes background is as a solution to S = S +S , in which case the addition of a D3 would break supersymmetry explicitly, since the IIB D5 D5 and D3 are invariant under different κ-symmetries. However, the resolved conifold background can alternatively be found as the dual to the deformed conifold with fluxes and no branes3, see for 2with the exception of ‘string gas’ moduli stabilization, see e.g. [74] 3The dual is succinctly described in supergravity when the number of wrapped D5-branes is very large [13, 14]. – 2 – example [14, 19]. In this dual frame the underlying action is source-free, and the addition of an D3 (again in the dual deformed conifold) will break SUSY spontaneously. The deformed conifold with D3 can then dualized back to a resolved conifold with wrapped D5 along with a D3, but the spontaneous (as opposed to explicit) nature of SUSY breaking is only manifest in the dual frame. As we will see, backreaction of the D3 on the resolved conifold induces masses for all the fermions, so there is no obvious candidate for the goldstino; this further indicates that the resolved conifold with wrapped D5 and a D3 system exhibits explicit breaking of supersymmetry. This is consistent with our discussion above: the spontaneous nature of SUSY breaking is only manifest in the dual deformed conifold description. In terms of moduli stabilization, a dual description in terms of spontaneous breaking allows one to consistently define a superpotential for both the K¨ahler and complex structure moduli, which is precisely the feature of ‘spontaneous breaking’ that is useful for studying 4d physics from string theory. 1.2 Outline of the paper Our first analysis, studied in section 2, considers a probe D3-brane, not in a Calabi-Yau background [11, 12] as studied in [2], but in a non-K¨ahler resolved conifold background with integer and fractional three-branes. We will construct a supersymmetric deformation to the Calabi-Yau resolved conifold that converts it to a non-K¨ahler resolved conifold, provides a non-zero curvature to the internal space, and which induces a non-zero amount of ISD fluxes. Once a probe D3 is introduced, supersymmetry is spontaneously broken by the coupling of ISD fluxes to the worldvolume fermions, giving masses to the world-volume fermions. This breaking is in fact ‘soft’ as the fluxes and fermion masses are set by the non-Kahlerity of the internal space, which is in turn a tune-able parameter. The picture is somewhat similar to the case with Calabi-Yau internal space as studied in [2] but the analysis differs in terms of fluxes and backreaction. In particular, the analysis in the probe approximation now yields two massless fermions, as opposed to one in [2]. This result is modified upon considering backreaction of the D3 on the bulk fluxes, which generates both (2, 1) and (1, 2) three-form fluxes, inducing masses for all the worldvolume fermions, i.e. there are zero massless fermions remaining in the spectrum. We also study certain aspects of de Sitter vacua from our analysis. It interesting to note that a curved internal space appears to be a requirement for de Sitter solutions in string theory, at least in many contexts, especially negatively curved internal spaces (see for example [15] and references therein). With this in mind, we consider moduli stabilization in this background, and the connection to de Sitter space in this model. The physics discussed above remains largely unchanged even if we change the dimensionality of the anti-brane. In section 3, we consider a second application of anti-brane fermionic actions and take a probe4 D7-brane, this time working with a Calabi-Yau background. Supersymmetry is again broken spontaneously via flux-induced fermion masses, and the masses are proportional to the piece of the three-form flux which is ISD in the space transverse to the brane. In the D3 case, where the transverse space is the entire internal space, this flux is precisely the flux of the GKP background5. However, in the D7 case, the fermion masses are now sourced by the subset of these fluxes which are ISD in the two-directions transverse to the brane. In other words, the fermion masses are now determined solely by fluxes that have two legs on the brane, and one leg off. We show that for a special class of flux background there can be many massless fermions in the low energy spectrum, while in a general flux 4By assuming such a heavy object as probe simply means that the logarithmic backreactions of the D7-brane on geometry and fluxes are suppressed by powers of g . s 5Henceforth by GKP background we will always mean the background proposed in [11, 12]. – 3 – background there may be none. This provides yet another instance of a string theory realization of nilpotent goldstinos6, and a possible starting point for inflation and de Sitter solutions. Ourfinalapplicationisactuallyclosertoaderivation;westudythefermionicD3actionatall orders in the fermionic expansion. To do this, we promote the bosonic fields to superfields, and discuss the physics at the self-dual point. At the self-dual point we can use U-dualities to relate various pieces of the multiplet and consequently determine the fermionic completions of the different fields. Once we move away from the self-dual point, we can determine the fermionic completions of all the bosonic fields in a compact form. As an added bonus, we find that the all-order fermionic action can be written in a manifestly κ-symmetric form, even without precise details of the form of the terms in the action. The orientifolding action can then be easily incorporated in the action. This indicates that the spontaneous nature of supersymmetry breaking by anti-branes, both in the presence and in the absence of an orientifold plane, is not a leading order effect, but in fact continues to be true to all orders. This puts the conclusions of [1, 2], and its implications for KKLT, on solid footing. We conclude with a short discussion of the implications of our work and directions for future research. 2. D3-brane in a Resolved Conifold Background: Soft (and Spontaneous) Breaking of Supersymmetry The breaking of supersymmetry by a probe D3-brane in a warped bosonic background was studied recently in [2]. They studied a D3-brane in a GKP background, and found that supersymmetry was spontaneously broken by the coupling of ISD fluxes to the worldvolume fermions. In this section we perform a similar analysis, focusing instead on a probe D3-brane in a resolved conifold background. We will consider a deformation to the Calabi-Yau resolved conifold which maintains supersymmetry but provides a non-zero curvature to the internal space, and which induces ISD three-form fluxes from a set of integer and fractional D3-branes. Once a probe D3 is introduced, supersymmetry is again spontaneously (and softly) broken by the coupling of ISD fluxes to the worldvolume fermions, and the fermion masses can be straightforwardly computed. As we will see, the ‘soft’ nature of supersymmetry breaking is due to the tune-able nature of the non-K¨ahlerity of the internal manifold. The key details of the fermionic action for a D3-brane in a warped bosonic background are given in [2]. These will be the starting point of our analysis, so here we merely quote them. The worldvolume action is given, in a convenient κ-symmetry gauge, by (cid:20) (cid:21) LD3 = T e4A0θ¯1 2e−φΓµ∇ − i (cid:0)GISD −G¯ISD(cid:1)Γmnp θ1. (2.1) f 3 µ 12 mnp mnp where θ1 is a 16-component7 10d Majorana-Weyl spinor8, and we have defined the three from flux G 3 as G = F −τH . The 16-component spinor θ1 can be decomposed into four 4d Dirac spinors λ0, (3) (3) (3) λi with i = 1,2,3. On a Calabi-Yau manifold, the λ0 is a singlet under the SU(3) holonomy group of the internal Calabi-Yau manifold while the λi transform as a triplet. We can now rewrite the D3 brane action (2.1) using the 4d decomposition of the θ1 spinor in the following way: (cid:104) LD3 = 2T e4A0−φ λ¯¯0γµ∇ λ0 +λ¯¯γµ∇ λi δ (2.2) f 3 − µ + − µ + i¯ 6See [16, 17, 18] for even more examples. 716 complex components, or 32 real components. 8We have already fixed κ-symmetry. – 4 – (cid:105) + 1m λ¯0λ0 + 1m λ¯¯0λ¯0 +m λ¯0λi +m λ¯¯0λ¯ı + 1m λ¯i λj + 1m λ¯¯ı λ¯ , 2 0 + + 2 0 − − i + + ¯ı − − 2 ij + + 2 ¯ı¯ − − where we use ± subscripts to denote 4d Dirac spinors that satisfy λ± = 12(1±iΓ(cid:101)0123)λ, and the masses are defined as √ 2 m = ieφΩ¯uvwG¯ISD , from (0,3) flux, (2.3) 0 12 uvw √ 2 m = − eφeuG¯ISDJvw¯, from non-primitive (1,2) flux, (2.4) i 4 i uvw¯ √ m = 2ieφ(cid:0)ewet +ewet(cid:1)Ω guu¯gvv¯G¯ISD, from primitive (2,1) flux, (2.5) ij 8 i j j i uvw tu¯v¯ where J and Ω are the K¨ahler form and holomorphic 3-form respectively. We are interested in a more general background, where the SU(3) holonomy will be broken by a perturbation to the geometry. Compactifications on manifolds with SU(3) structure but not SU(3) holonomy have been studied in, for example, [20] and [21]. These are non-K¨ahler manifolds, which in general may or may not have an integrable complex structure, and are classified by five torsion classes W [26, 27, 28]. The simplest case, where all five torsion classes vanish, is a Calabi-Yau manifold that i supports no fluxes. We are looking for the case with fluxes, so that we can make use of equations (2.3), (2.5), and (2.4), and therefore some of the torsion classes must be non-zero. Moreover, the non-K¨ahler manifold that we need has to be a complex manifold, otherwise the flux decomposition in terms of (2, 1), (1, 2) or (0, 3) forms would not make any sense. In addition, the manifold should to be non-compact, so as to avoid any tension with Gauss’ law. The simplest internal manifoldthatsatisfies ourrequirementsis theresolved conifold withanon-K¨ahlermetric whichallows an integrable complex structure (and by definition doesn’t have a conifold singularity). The goal of this section will be to study the action (2.1) or (2.2) in a resolved conifold with an arbitrary amount of D3 branes and delocalized five branes (see [23] and [22] for more details on delocalized sources). More precisely, we will put a D3-brane in a supersymmetric background with metric given by: 1 (cid:112) ds2 = ds2 +e2φ/3 e2φ/3+∆ ds2, (2.6) (cid:112) 0123 6 e2φ/3 e2φ/3+∆ where eφ is related to type IIB dilaton eφ as φ = −φ and the factor ∆ encodes the backreaction of B B the 3-branes. It is defined using a parameter β as: (cid:16) (cid:17) ∆ = sinh2β e2φ/3−e−4φ/3 . (2.7) The other piece appearing in (2.6) is ds2, which is the metric of the internal six-dimensional non- 6 K¨ahler resolved conifold. This is expressed in terms of the coordinates (r,ψ,θ ,φ ) in the following i i way: 2 (cid:88) ds2 = F dr2+F (dψ+cos θ dφ +cos θ dφ )2+ F (dθ2+sin2θ dφ2), (2.8) 6 1 2 1 1 2 2 2+i i i i i=1 where the resolution parameter is proportional to F −F . 3 4 We will start by making an ansatze for the warp-factors F (r) appearing in (2.8) which will allow i us to see how to go from a Ricci-flat Calabi-Yau metric to a non-K¨ahler metric on a resolved conifold. – 5 – A more generic class of solutions for the warp-factors exists and has been discussed in [22], but we will only consider a subset given by: 1 r2F r2 r2 F = +δF, F = , F = +a2(r), F = +a2(r), φ = φ(r), (2.9) 1 F 2 9 3 6 1 4 6 2 where F, δF(r), a (r), and a (r), are functions of the radial coordinate only. From the above ansatze, 1 2 it is easy to see where the Calabi-Yau case fits in. It is given by: (cid:18)r2+9a2(cid:19) F(r) ≡ F = , δF(r) = 0, a (r) = a, a (r) = 0, φ = 0. (2.10) CY r2+6a2 1 2 The Calabi-Yau case is fluxless (with the vanishing of the flux enforced by supersymmetry), and has a constant dilaton. Once we switch on fluxes, we can no longer assume that the other pieces of the warp-factors appearing in (2.9) vanish. As a cautionary tale, let us first consider whether we can perturb away from Calabi-Yau resolved conifold simply by allowing for a small perturbation in F(r) and φ(r). We will see that this in fact does not lead to useful results, and thus we will need to be more careful in constructing our geometry. Nonetheless, it is useful for establishing an algorithm for constructing solutions. Consider a small perturbation to (2.10) of the form: F(r) = F +σf(r), δF(r) = 0, a (r) = ae−φ, a (r) = 0, (2.11) CY 1 2 where σ is a dimensionless expansion parameter, that satisfies the the EOMs and takes the solution from the Calabi-Yau resolved conifold to the non-K¨ahler resolved conifold. We can narrow down our perturbation scheme by allowing the dilaton field to behave in the following way: (cid:18) (cid:19) 1 φ(r) = log , (2.12) rσ which would guarantee the existence of a small parameter σ that, while preserving supersymmetry, would be responsible in taking us away from the Calabi-Yau case. In the limit σ → 0, we go back to the fluxless Calabi-Yau case. This geometry is of course singular in the r → ∞ limit, but we will assume for this discussion that the geometry is capped off at some sufficiently large r. In any case, this issue will not be important, as this perturbation fails for other reasons. A way to construct such a background has already been discussed in [22], and therefore we will simply quote some of the steps. The best and probably the easiest way to analyze such a background is by using the torsion classes. For us the relevant torsion classes are W and W . They can be 4 5 expressed in terms of the warp-factors F (r) and the dilaton φ(r) in the following way: i √ √ F − F F F − F F 3r 1 2 4r 1 2 W = + +φ , 4 r 4F 4F 3 4 √ F F F −2 F F φ 3r 4r 2r 1 2 r ReW = + + + . (2.13) 5 12F 12F 12F 2 3 4 2 The other torsion classes take specific values, with W determining the torsion. This solution is 3 generated by following the duality chain described in [22], which generates both the RR and the NS three-forms F and H respectively. 3 3 Our aim then is to use these torsion classes to determine the functional form for the warp-factors F using the specific variation of the ansatze (2.9) i.e (2.11) and (2.12). The key relation, that allows i us to find the connection between F(r) and the dilaton φ(r), is the supersymmetry condition: 2W +ReW = 0. (2.14) 4 5 – 6 – Plugging in the ansatze (2.11) and (2.12) in (2.14) will allow us to determine f(r) completely in terms of the radial coordinate r and the resolution parameter a2. The functional form for f(r) turns out to be a non-trivial function of r: (cid:40) (cid:34) 3 (cid:35) 2 (cid:88) f(r) = 27a2(6a2+r2) Φ (r;a2)+r2log r (6a2+r2) i i=1 (cid:41) (cid:20) (cid:18) r2 (cid:19) r2log r (cid:21) − (9a2+r2)(6a2+r2) 3log +1 +2− , (2.15) 6a2 6a2+r2 which is defined for a2 > 0. For vanishing a2 the functional form for f(r) simplifies and has been studied earlier in [32]. The other variables appearing in (2.15) are defined in the following way: (cid:18) r2 (cid:19) Φ (r;a2) = F(0,0,1,0) −1,2,3,− , 1 2 1 6a2 (cid:18) r2 (cid:19) Φ (r;a2) = F(0,1,0,0) −1,2,3,− , 2 2 1 6a2 (cid:18) r2 (cid:19) Φ (r;a2) = F(1,0,0,0) −1,2,3,− , (2.16) 3 2 1 6a2 (0,1,0,0) (1,0,0,0) (0,0,1,0) wherethenotation F refersto∂ F [x;y;z;w],andsimilarlyfor F and F . This 2 1 y2 1 2 1 2 1 perturbation to F(r) corresponds to introducing a small Ricci scalar on the internal space. This could computed using the torsion classes ([56]), or computed directly using standard GR techniques. Using GR techniques, we find a simple expression emerges for small resolution parameter a2 and small value for the parameter σ: 72σ (cid:20) (cid:18)6a2(cid:19)(cid:21) δR = − 3−2log , (2.17) 6 r2 r2 whichisnegativeforr ≥ 1.2a. Furthermoreonecancheckthatforgeneral a,i.e. notsmalla,whilethe expression for δR is no longer simple, it is negative definite. It is interesting to note that negatively 6 curvedinternalspaceshavebeenwidelystudiedasamechanismforfindingdeSittersolutionsinstring theory, see the discussion and references in [15]. Theaboveanalysis, althoughinterestingbecauseofthecontrolwecanhaveonthenon-K¨ahlerityof the internal manifold, is ultimately not useful for finding the masses of the D3 world-volume fermions, as it in fact renders the internal manifold with a non-integrable complex structure. Thus, there exists an almost complex structure but the manifold itself may not be complex9. This means we cannot decompose our G flux in terms of (1, 2), (2, 1) or (0, 3) forms in a global sense, making the fermionic 3 mass decompositions given in (2.5), (2.4) and (2.3), not very practical in analyzing the fermions on the probe D3. This of course doesn’t mean that we cannot study the spontaneous susy breaking; we can, but the analysis will not be so straightforward as was with the complex decomposition of the three-form fluxes. The question then is: can we have a complex non-K¨ahler resolved conifold satisfying a more generic ansatze like (2.9) where we can use equations (2.3), (2.5), and (2.4), to study spontaneous susy breaking with a probe D3? The answer turns out to be in the affirmative, and in the following section we elaborate the story10. 9There might exist a non-trivial integrable complex structure, but we haven’t been able to find one. 10Note that there is some subtlety with the mapping to [55] at this stage, for example the possibility of a non-Ka¨hler special Hermitian solution with a constant dilaton that we get here demanding supersymmetry as opposed to a Calabi- Yau resolved conifold with a constant dilaton studied in [55]. This has been discussed in details in [22] so we will not dwell on this any further. – 7 – 2.1 A SUSY perturbation of the resolved conifold Let us start with a simple example of a D3-brane located at a point in an internal manifold specified by the metric ds2 where ds2 is given by: 6 6 ds2 = dr2+g dymdyn, (2.18) 6 mn where (r,ym) are the coordinates of the internal six-dimensional space. To avoid contradiction with Gauss’ law, the internal manifold has to be non-compact, although a compact example could be constructed by either inserting orientifold planes, or anti-branes. Details of this will be discussed later. The backreaction of the D3-brane converts the vacuum manifold: ds2 = ds2 +ds2, (2.19) vac 0123 6 with ds2 being the Minkowski metric along the space-time directions, to the following: 0123 1 √ ds2 = √ ds2 + hds2, (2.20) 10 0123 6 h where h is the warp-factor. The five-form flux in the background (2.20) is now given as: 1 F = (1+∗ )dh−1∧dx4. (2.21) 5 10 g s The above analysis is generic, but it is highly non-trivial to actually compute the warp-factor h. For a complicated internal space, the equation for h typically becomes an involved second-order PDE. Furthermore, in the presence of other type IIB fluxes, for example the three-form fluxes H and F , 3 3 the metric is more complicated than (2.20). Additionally, the string coupling constant generically will not be constant. There is, however, a way out of the above conundrum if we analyze the picture from a more general setting. Wecanusethepowerfulmachineryoftorsionalanalysis[27,28,29]towritethebackgroundof a D5-brane wrapped on some two-cycle, parametrized by (θ ,φ ), of a generic six-dimensional internal 1 1 space. Assuming that the size of the wrapped cycle is smaller than some chosen scale, any fluctuations along the (θ ,φ ) will take very high energy to excite. This means at low energies the theory will be 1 1 of an effective D3-brane11 and the source charge of the wrapped D5-brane C will decompose as: 6 (cid:18) (cid:19) C (→−x,θ ,φ ) = C (→−x)∧ eθ1√∧eφ1 , (2.22) 6 1 1 4 V where V is the volume of the two-cycle on which we have the wrapped D5-brane. Therefore using the criteria (2.22), the supergravity background for the configuration of the effective D3-brane is given by: ds2 = e−φds2 +eφds2, 0123 6 (cid:16) (cid:17) F = e2φ∗ d e−2φJ , (2.23) 3 6 whereφisthedilatonandtheHodgestarandthefundamentalformJ arewrttothedilatondeformed metric e2φds6. The five-brane charge in (2.23) decomposes as (2.22) once we express it as a seven-form 2 11Also known as a fractional D3-brane. There is yet another way to generate a fractional D3-brane which we don’t explorehere. ForexampleifwetakewrappedD5-D5-braneswith(n ,n )amountofgaugefluxesoneachofthem,then 1 2 wecanhaveboundD3-braneswithchargesn andn respectively. Ifn arefractional,thesegivefractionalthree-branes 1 2 i with vanishing global five-brane charges. See [30, 31] for more details. – 8 – F = ∗ F . The metric ds2 is in general a noncompact non-K¨ahler metric that may not even have 7 10 3 6 an integrable complex structure. If we allow for background three-forms F and H , the above background (2.23) changes. One 3 3 way to see the change would be to work out the precise EOMs. However there exists another way, using a series of duality transformations, to study the background in the presence of the three-form fluxes. The steps have been elaborated in [24, 25, 22]. The solutions we will study here are specific realizations of the general solutions found and analyzed in [22], where supersymmetry of the final ’dualized’ solution was explicitly confirmed12. The idea is to: • Compactify the spatial coordinates x1,2,3 and T-dualize three times along these directions. The resulting picture will now be in type IIA theory. • Lift the type IIA configuration to M-theory and make a boost along the eleventh direction using a boost parameter β. This boosting will create the necessary gauge charges. • Reduce this down to type IIA and T-dualize three times along the spatial coordinates to go to type IIB theory. The IIB background now automatically has the three-form fluxes, as well as a five-form flux. The result of this duality procedure is that the type IIB background (2.23) now converts to exactly what we expect in (2.20), namely13: 1 √ 1 (cid:112) ds2 = √ ds2 + h ds2 = ds2 +e2φ/3 e2φ/3+∆ ds2, (2.24) 0123 6 (cid:112) 0123 6 h e2φ/3 e2φ/3+∆ confirming the low-energy effective D3-brane behavior, and the following background for the three- and the five-form fluxes: (cid:16) (cid:17) (cid:16) (cid:17) F = cosh β e2φ∗ d e−2φJ , H = −sinh β d e−2φJ , 3 6 3 (cid:16) (cid:17) (cid:16) (cid:17) dF(cid:101)5 = −sinh β cosh β e2φ d e−2φJ ∧∗6d e−2φJ , (2.25) with the type IIB dilaton eφB = e−φ. One may verify that (2.24) and (2.25) together solve the type IIB EOMs. We will concentrate on a specific background given by a (generically non-K¨ahler) singular, resolved or deformed conifold. The typical internal metric ds2 in this class is given by a variant of (2.8) as: 6 2 ds2 = F dr2+F (dψ+cos θ dφ +cos θ dφ )2+(cid:88)F (cid:0)dθ2+sin2θ dφ2(cid:1) (2.26) 6 1 2 1 1 2 2 2+i i i i=1 + F sin ψ(dφ dθ sin θ +dφ dθ sin θ )+F cos ψ(dθ dθ −dφ dφ sin θ sin θ ), 5 1 2 1 2 1 2 6 1 2 1 2 1 2 where F (r) are warp factors that are functions of the radial coordinate r only14 and in the following, i unless mentioned otherwise, we will only consider the resolved conifold, i.e we take F = F = 0 5 6 12In addition, the fact that the T-duality transformations lead to solutions that solve explicitly the supergravity EOMs has been shown earlier in [44, 45, 47]. In [21] and [22], this was confirmed using torsion classes. The subtlety thatsuchtransformationsdo notleadtonon-trivialJacobiansfollowsfromthefactthatthesupergravityfieldshaveno dependenceontheT-dualitydirections. IfthesupergravityfieldsstarttodependontheT-dualitydirections,therewill arise non-trivial Jacobians as discussed in some details in [75]. We thank the referee for raising this question. 13There is some subtlety in interpreting the final background with fluxes or with sources. This has been discussed in [25] which the readers may refer to for details. 14OnemaygeneralizethistomakethewarpfactorsF functionsofallcoordinatesexcept(θ ,φ ),i.ethedirectionsof i 1 1 the wrapped brane. We will not discuss the generalization here. – 9 –