IFT-UAM/CSIC-04-36

hep-th/yymmnn

Flux-induced SUSY-breaking soft terms on D7-D3 brane systems

[10mm] P. G. Cámara, L. E. Ibáñez and A. M. Uranga

[2mm] * Departamento de Física Teórica C-XI and Instituto de Física Teórica C-XVI,

[-0.3em] Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain.

[4mm] ** Theory Division, CERN, 1211 Geneva 23, Switzerland.

[9mm] Abstract

[7mm]

We study the effect of RR and NSNS 3-form fluxes on the effective action of the worldvolume fields of Type IIB D7/D3-brane configurations. The D7-branes wrap 4-cycles on a local Calabi-Yau geometry. This is an extension of previous work on hep-th/0311241, where a similar analysis was applied to the case of D3-branes. Our present analysis is based on the D7- and D3-brane Dirac-Born-Infeld and Chern-Simons actions, and makes full use of the R-symmetries of the system, which allow us to compute explicitly results for the fields lying at the D3-D7 intersections. A number of interesting new properties appear as compared to the simpler case of configurations with only D3-branes. As a general result one finds that fluxes stabilize some or all of the D7-brane moduli. We argue that this is important for the problem of stabilizing Kähler moduli through non-perturbative effects in KKLT-like vacua. We also show that (0,3) imaginary self-dual fluxes, which lead to compactifications with zero vacuum energy, give rise to SUSY-breaking soft terms including gaugino and scalar masses, and trilinear terms. Particular examples of chiral MSSM-like models of this class of vacua, based on D3-D7 brane systems at orbifold singularities are presented.

## 1 Introduction

In the search for realistic string compactifications two important problems appear. On one hand, in e.g. familiar Calabi-Yau compactifications, one typically has a large number of perturbatively massless fields, namely the complex dilaton, Kähler and complex structure moduli, and possibly others (e.g. gauge bundle or brane moduli). Thus the string coupling as well as geometric data of the compactification remain undetermined at the perturbative level. A second problem, possibly related to the first one, is how to break supersymmetry in a controled manner if we start with e.g. a supersymmetric compactification.

In the last few years it has been realized that a general compactification allows for an additional ingredient, not considered previously, namely field strength fluxes for the internal components of -form supergravity fields [1, 2, 3, 4, 5, 6, 7, 8, 9]. Interestingly, this new ingredient may have an important bearing on those two problems. The case of NSNS/RR 3-form field strength fluxes in Type IIB CY orientifold compactifications (or more generally, F/M-theory on CY fourfolds with 4-form flux) has been studied in particular detail (see e.g. [2, 4, 6]). The presence of the fluxes induces a non-trivial warp factor in the geometry, as well as a non-trivial source for the RR 4-form potential. However it has been shown that, due to the presence of O3-planes in the IIB construction (or more generally, the F/M-theory tadpole from the Euler characteristic of the fourfold [10]), the Type IIB 10-dimensional equations of motion can be solved consistently with 4d Minkowski space, if the internal 3-form flux is imaginary self-dual, ISD (analogously, if the F/M-theory 4-form flux is self-dual).

The fact that the (quantised) fluxes should be ISD to solve the equations of motion generically determines dynamically the value of the complex dilaton as well as all complex structure moduli, while Kähler moduli are not stabilized. For adequate choices of fluxes, the complex dilaton may be fixed at a perturbative value, so that one expects the description of the compactifications in terms of classical supergravity to be a reliable approximation (in the large volume limit in Kähler moduli space). It has further been argued that non-perturbative effects depending on the Kähler moduli, which are generically present in this kind of compactification, like non-perturbative superpotentials from strong infrared dynamics of gauge sectors on the D7-branes or from euclidean D3-brane instantons, may determine also all the Kähler moduli [11] (possibly at large volume). This has been further explored in explicit models in [12, 13], with positive results in a sizeable number of examples [12]. This setting thus provides the first examples of tractable string compactifications with all dilaton, Kähler and complex structure moduli determined dynamically.

Field strength fluxes may also be important for the second problem
mentioned above, the breaking of supersymmetry, since the generic choice
of fluxes in a compactification is in fact non-supersymmetric (although
many examples with supersymmetry preserving fluxes have been constructed).
In particular the field theories inside D-branes located in the CY
compactification, which are supersymmetric in the absence of fluxes, may
get SUSY-breaking soft terms in the presence of fluxes. This is
particularly interesting since it is possible to embed chiral gauge
sectors relatively close to the (MS)SM in D-brane configurations in the
presence of 3-form fluxes [14, 15, 16]. In the case of
gauge sectors localized on D3- (or anti-D3-) branes, such terms have been
recently computed in [17, 18, 19]. In particular it was found in
[18] (see also [19]) that only imaginary anti-selfdual (IASD)
fluxes gives rise to SUSY-breaking soft terms on the worldvolume of
D3-branes ^{1}^{1}1On the other hand ISD do give rise to soft terms on
the worldvolume of anti-D3-branes. The presence of these however
break supersymmetry in a less controlled fashion.. This is unfortunate
because, as we mentioned, only ISD fluxes solve the equations of motion.
Thus, in order to have non-vanishing soft-terms, we either add IASD
fluxes, and hence do not solve the equations of motion at this level
(hoping that other effects would perhaps stabilize the compactification),
or else we include anti-D3-branes in the compactification, thus
breaking supersymmetry in a less controlled fashion. The latter is
certainly an interesting possibility, as advocated in [18], since
anti-D3-branes turn out to be an important ingredient in the proposal in
[11] to obtain deSitter vacua in string theory.

In any event, it would be interesting to have Type IIB CY orientifold compactifications (or more generally F-theory compactifications on CY 4-folds) which have supersymmetry in the absence of fluxes, and yield SUSY-breaking soft terms upon turning them on, and still obeying the Type IIB equations of motion. One of the motivations of the present paper is to provide examples of this class. Specifically we analyze the effect of 3-form field strength fluxes on the world-volume fields of D7/D3-brane configurations, extending the results for D3-branes presented in [18, 19]. We show that, in contrast with the D3-brane case, ISD fluxes do give rise to non-vanishing soft terms for the fields on the worldvolume of D7-branes.

This result is interesting for a number of reasons. In particular this shows that Type IIB CY orientifolds in the presence of ISD fluxes provide us with (to our knowledge) the first known class of string compactifications to 4d Minkowski space which solve the classical equations of motion and lead to non-trivial SUSY-breaking soft terms, in a controled manner. Apart from its theoretical interest, the structure of soft terms, if applied to realistic models with the SM living on D7-branes, may be of phenomenological relevance (see [20]).

Another application of our results concerns the proposal in [11],
attempting to fix dynamically the Kähler moduli in type IIB orientifolds
of this class, mentioned above. A potential problem in generating
non-perturbative superpotentials from e.g. strong infrared dynamics on the
gauge theory on D7-branes, is the possible presence of too much
massless charged matter in the latter. Our results for soft terms
involving D7-brane matter fields show that the presence of ISD fluxes
generally give masses to all the D7-brane geometric moduli. This indicates
that generically D7-brane vector-like matter is massive, thus allowing
such non-perturbative superpotentials to appear ^{2}^{2}2An important
exception is D7-branes containing charged chiral fermions. This suggests
that the D7-branes responsible for Kähler moduli stabilization should
not correspond to the D7-branes on which we plan to embed the SM,
but to some non-chiral D7-brane sector.. This
is also in agreement with recent results derived from the F-theory
perspective in [21], and from generalized calibrations
[22].

Our results are based on an expansion of the relevant D7- and D3-brane Dirac-Born-Infeld plus Chern-Simons (DBI+CS) actions in the presence of fairly general type IIB closed string backgrounds. Since D-branes are mainly sensitive to the local geometry around them, our results are derived in an expansion around the location of D7- and D3-branes. Since D7-branes wrap a 4-cycle in the internal space, we center on the simple situation of D7-branes wrapping or in a local Calabi-Yau . The effect of fluxes on the fields living at D3-D7 (or ) intersections is more involved, but we explicitly derive it in full generality using (super)symmetry arguments. Finally, all results are compared, showing full agreement, with an analysis based on the use of the 4d effective action [23].

The paper is organized as follows. After reviewing the results of [18] in section 2, we compute the soft terms for the fields on the worldvolume of D7’s in section 3, checking also that our ansatz for the closed string backgrounds solves the equations of motion. In sections 4 (resp. 6) we make use of the local geometric symmetries of - (resp. ) configurations to carry out the computation of the soft terms for fields living at the -( -) intersections. Examples of soft terms obtained for different classes of ISD and IASD fluxes are described in section 5.

Some of the results obtained both for -brane systems may be understood from the effective low energy supergravity action [23, 24, 25]. In particular, we show in section 7 that fluxes correspond to non-vanishing expectation values for the auxiliary fields of the dilaton and Kähler moduli. We show that the soft terms obtained in section 3 for ISD fluxes may be understood as arising from a non-vanishing auxiliary field for the overall Kähler modulus . This is in agreement with previous results in [18, 19]. A noteworthy property is that the bosonic soft terms induced by ISD fluxes naturally combine with the SUSY scalar potential into a positive definite scalar potential. This fact, which appears already in section 3 from the DBI+CS action in the presence of fluxes, has a simple interpretation also from the effective supergravity low-energy action, as explained in section 7. We also show in that section that analogous soft terms are expected for the fields lying at the intersections of -branes wrapping different 4-cycles.

In section 8 we argue that many of the results obtained for D7-branes wrapping 4-tori are still valid when D7-branes wrap curved 4-cycles, in particular , and describe the explicit soft terms in this case. We also argue that similar patterns of soft terms are also obtained in more involved situations, where the normal bundle to the 4-cycle is non-trivial. For instance, in cases with multiple D7-brane geometric moduli, all of the latter are fixed in the presence of ISD backgrounds, in agreement with results in section 7.

Some particular examples and applications are described in section 9. In particular we present several examples of local D7/D3-brane configurations, with a semirealistic chiral gauge sector arising in the worldvolume of the D7-branes. The presence of ISD fluxes gives rise to phenomenologically interesting soft terms. Embedding this class of local configurations into a full F-theory compactification would give rise to semirealistic models in which calculable SUSY-breaking soft terms are induced. We also briefly discuss in section 9 the relevance of our results for the program in [11], as discussed above. We end up with some comments in section 10. Details of the computations are provided in an appendix.

## 2 Three-form fluxes and D3-branes

Let us first briefly review some of the main results in ref.[18] (see [17, 19] for related discussions). We consider D3-branes embedded in general Type IIB closed string backgrounds of the general form

(2.1) | |||||

where (with being the RR(NSNS) flux), and denotes the metric in transverse space, in the absence of flux backreaction, i.e. the Ricci-flat metric of the underlying Calabi-Yau. Eventually we will request these backgrounds to solve the Type IIB supergravity equations of motion. Since D3-branes are located at a point in the six dimensions parametrized by , it is natural to expand these backgrounds around the position of the D3-branes

(2.2) | |||||

where the coefficients , , , in the right hand side are constant, independent of . The piece of the 5-form background relevant for our purposes below is

(2.3) |

In the absence of fluxes, the massless fields on a stack of D3-branes are given by gauge bosons, six real adjoint scalars, and four Majorana adjoint fermions, the latter transforming in the representations and of the local symmetry. In supersymmetry language, they fill out a vector multiplet and three chiral multiplets. The effect of the above backgrounds on the effective action of the worldvolume fields can be obtained by plugging the above expansions in the DBI and CS action for the D3-branes.

Calabi-Yau manifolds are endowed with a complex structure, hence it is useful to rewrite the world-volume fields and the 3-form background in complex coordinates. Let us introduce local complex coordinates , , . We define the complex scalars , , , and use to denote . Denoting the four-plet of fermions by their weights, the fermion belongs to the vector multiplet, and is referred to as the gaugino, denoted . The fermions combining with the above complex scalars to give chiral supermultiplets are , , , corresponding to the weights , and , respectively.

In a preferred complex structure, only an (times ) subgroup of the symmetry is manifest, hence it is useful to decompose the 3-form flux background under it. The antisymmetric flux transforms as a 20-dimensional reducible representation, decomposing as . The irreducible representations , correspond to the imaginary self-dual (ISD) and imaginary anti self-dual (IASD) parts, respectively, defined as

(2.4) |

It is useful to classify the components of the ISD and IASD parts of according to their behaviour under , . For that purpose, we introduce the tensors [26]

defined in terms of the complex components of , and which transform in the representation and under . One similarly defines and . The properties of each component is displayed in table 1.

ISD | IASD | ||||
---|---|---|---|---|---|

rep. | Form | Tensor | rep. | Form | Tensor |

In [18] the soft term Lagrangian for the worldvolume fields of D3-branes was computed to be

(2.5) | |||||

where we have defined

(2.6) |

In the notation of the first appendix in [18] one thus has soft terms for the D3-brane worldvolume fields

(2.7) |

The supergravity equations of motion impose some constraints on the background, namely [18]

(2.8) |

(2.9) |

(2.10) |

They allow to relate the scalar masses to the flux background, as follows

(2.11) |

As emphasized in [18], the scalar mass matrix is not fully determined in terms of the fluxes, only its trace is.

Our purpose in the present paper is to carry out a similar analysis for the effect of fluxes on configurations including D7-branes.

## 3 D7-branes and RR, NSNS 3-form fluxes

### 3.1 D7-branes and the 3-form flux background

There are a number of differences between the case of D7 and that of
D3-branes reviewed in the previous section. First of all the D7-branes
in general wrap a 4-cycle on the internal space. Thus an
expansion of the closed string background on all 6 transverse coordinates
as we did in the D3-brane case is no longer the correct
procedure. Rather, we need to describe the local geometry around the
4-cycle wrapped by the D7-brane, namely a tubular neighbourhood around the
4-cycle, given by the normal bundle of the 4-cycle, i.e. the wrapped
4-cycle, on which we fiber the normal direction. Then we may expand the
background in a power series in
its dependence on the normal coordinate (while in principle keeping its
full dependence on ). For most purposes, we will center on
cases where the fibration is trivial, and the local geometry is
. ^{3}^{3}3This restricts us to backgrounds
or K3, which is a restricted set, but sufficiently
rich to have non-trivial effects. Moreover, the qualitative features can
be extrapolated to more involved situations, see section 8.
We will comment later on possible generalizations of this background.

The strategy then is to obtain the 8d action for the D7-brane, taking the effect of fluxes into account. An important ingredient is that the symmetry of the problem is reduced as compared with the D3-brane case. In fact, the supergravity background need only respect 4d Lorentz invariance. On the other hand, the local geometric symmetry in the internal space is only , where the first factor is the local euclidean rotation group on , and the second in the normal direction. Hence the local 8d flux-induced terms must respect that symmetry (taking into account its action on the world-volume fields as well).

A second step involves the Kaluza-Klein compactification of this 8d action on to 4d. In this process, the geometry of plays a crucial role even in the absence of fluxes, and determines the number and kind of the massless fields on the D7-brane gauge sector in 4d. Our purpose is to determine the effect of the 8d flux induced terms on these massless fields, hence to compute it we will need to center on concrete cases. Our analysis will mainly center on the case of (which can be subsequently employed to study orbifolds thereof), and K3, with trivial worldvolume gauge bundles.

We are thus interested in determining first the 8d action, including the effects of fluxes, and second in discussing the compactification to 4d. We will concentrate on the most relevant terms, namely up to dimension three, hence scalar masses, scalar trilinears and fermion masses.

Still this is a rather involved problem in the general case, and we will have to make some restricting assumptions on the geometry. In particular, we assume certain restricted classes of closed string backgrounds. The bottomline is that we consider the flux to have the structure in eq. (3.6), and to be pure ISD or pure IASD; we will also center in situations where the background is constant over .

To specify our assumptions and also for practical purposes it is now useful to write the 3-form fluxes in terms of the local geometric symmetry . Thus now the decomposition of the ISD (IASD) pieces of in representations of the symmetry are as follows

(3.1) |

where the subindex gives the charge. Choosing our 4-cycle to be parametrized by , , and localized in the transverse direction , the triplets of are related with the fluxes in notation by

(3.2) |

Regarding the triplets of groups as vectors of , for future convenience we define the invariant scalar product

(3.3) |

On the other hand the components transforming like correspond to the SU(3) components . These fluxes are special in several respects. In particular, if contains 3-cycles , this multiplet contains fluxes such that

(3.4) |

This is the case for instance for , on which much of our analysis
centers. This is problematic because non-zero integrals of on a
D-brane cycle generate a world-volume tadpole for the dual world-volume
gauge potential , rendering the configuration
inconsistent [27] ^{4}^{4}4 This can be solved as for
the baryonic brane in [28], by introducing additional
branes ending on the D7-branes [29], but this completely
changes the kind of configurations here considered.. Moreover, in
general, such fluxes along world-volume directions are
quantized, and cannot be diluted away keeping the D7-brane physics
four-dimensional. Hence their presence can lead to qualitatively
large changes in the 4d physics, which may not be well described
with our techniques (which are implicitly perturbative in the flux
density).

For these reasons, we restrict our analysis to situations where fluxes in the multiplet are absent. Still, the remaining fluxes transforming like include the most interesting cases, and lead to interesting and non-trivial effects.

For the computation of soft terms up to the order of interest, it is
enough to consider the leading term in the expansion of in the
normal direction, namely the -independent term. Assuming in addition
that the background 3-form flux is independent of the coordinates on the
4-cycle, as mentioned above, one may obtain the NSNS and RR 2-form
potentials, in a particular gauge ^{5}^{5}5We
have chosen a gauge on which all the coordinates are on equal footing.
This is somehow a natural choice for a closed string background,
which is independent of the D-brane configuration under consideration.

The RR 2-forms may be obtained analogously, but we will not need its explicit expression.

Out of the above components, only those linear in ,
will actually be relevant in the computation of soft terms, see appendix
A. This suggests that the computation of the 2-form gauge
potential can be recast in a more compact way
^{6}^{6}6An additional advantage is that, as we argue in section
8, this derivation is valid even in situations with
3-form backgrounds non-constant over the 4-cycle., as follows. The
fluxes have always
two legs in . This allows us to associate to each of the above
representations a different 2-form in . In
fact, it is possible to decompose as

(3.6) |

where , and , are selfdual and anti selfdual two-forms in , namely

(3.7) |

(and similarly for the primed forms), corresponding respectively to the , , and pieces of .

For clarity we summarize in table 2 the main properties of these induced 2-forms.

Form | SD/ASD in | Corresponding rep. | ISD/IASD flux |
---|---|---|---|

SD | IASD | ||

SD | ISD | ||

ASD | ISD | ||

ASD | IASD |

The above scalar product between SU(2) triplets will induce a positive definite product in given by

(3.8) |

With all of this, the part of the B-field laying completely in is given by

(3.9) |

with , etc.

In particular, it is important to notice the explicit dependence of these components of the B-field on the transverse coordinates. As mentioned, these are the only components relevant in the computation of soft terms (see appendix A).

We will further assume that the flux background is purely imaginary self-dual (ISD) or purely imaginary anti-selfdual (IASD). This assumption is not compulsory, but simplifies enormously the computations, since in this situation the equations of motion imply that the dilaton is constant. It is important to mention that consequently, although our expressions below include both ISD and IASD components, it is understood that they are valid just when only ISD or only IASD components are turned on. Namely, the expressions below do not contain possible interference terms that may arise when both are present.

Concerning the rest of the closed string backgrounds, we will consider a general form analogous to that considered above for D3-branes

(3.10) |

Expanding on the warp factor and 5-form flux, we have

(3.11) |

with . At this level, and may depend on the longitudinal components , . The conditions these backgrounds must satisfy in order to solve Type IIB supergravity equations of motion are discussed in section 3.4.

As a last comment, notice that the metric ansatz does not contain any component mixing the coordinate with the , . Hence we take the local geometry to factorize as , namely we work on local geometries where the normal bundle is trivial. This restricts us to and K3, although clearly the general technique may be applied to non-trivial normal bundles.

### 3.2 The action

Our starting point will be the DBI and CS actions for the D7-brane.
Myers’ action in the Einstein’s frame for a D7-brane is given by
^{7}^{7}7 In the CS action we only keep pieces giving possible
contributions to the soft terms.

where run over the six directions transverse to 4d Minkowski space, denotes the pullback of the 10d background onto the D7-brane worldvolume, and

(3.12) | |||||

We will label by and the longitudinal and transverse indices to the D7-brane worldvolume. Within the class of backgrounds considered, we have , thus

(3.13) |

where is given by

(3.14) |

Expanding the above determinants we arrive to the 8d action in terms of the NS and RR background. The details of the computation can be found in the appendix A. The contributions to the soft terms are given by

(3.15) | |||

with given in eq. (3.9). Here the dots refer to derivative couplings and other 8-dimensional contributions which are not explicitly needed in our computations below.

Writing now and in terms of the two forms induced in
by the flux, and plugging them, together with
(3.9), into the above expression we obtain the eight
dimensional soft lagrangian. We refer again to the appendix
A for details. The final
result for the bosonic action in terms of flux
components, including the kinetic and quartic terms is
^{8}^{8}8Notice that the derivative terms mentioned below are required to
make the expression local Lorentz invariant in 8d. However, the listed
terms are the only ones explicitly need for our discussion below.

where are the components of the 8-dimensional gauge boson along the directions longitudinal to the D7-brane and are (adjoint) scalar fields describing the transverse degrees of freedom of the D7-brane. To avoid long expressions we have not included here either the YM action of gauge bosons or couplings depending on derivatives of the dimensions longitudinal to the D7-brane. Those terms are understood to be included in the dots, but, as we commented before, they will not be relevant for our discussion below.

The explicit dependence on the antisymmetric backgrounds (in
notation) and the warp factor is shown. Concerning the latter, notice that
for pure ISD or pure IASD fluxes, the equations of motion require the warp
factor to correspond to a black 3-brane solution, with , and the
warp factor dependence drops from the above expression ^{9}^{9}9This is
similar to the cancellation of force in systems of a D7-brane in the
background of D3- or anti-D3-branes, whose supergravity solution is also
of the black 3-brane form.. We will consider this situation in what
follows.

Some of the terms in (3.2) correspond to the Yang-Mills action for the gauge boson and the kinetic terms for the scalars . In addition there are terms which depend on the 3-form fluxes. In particular note that the transverse scalars get mass terms for some non-vanishing fluxes. This is an important point since it means that, irrespective of the compactification space, in general 3-form fluxes stabilize the positions of the D7-branes. This was already discussed in section 5 in [18]. Note also that in addition there are SUSY-breaking scalar-(vector) couplings proportional to some of the fluxes. Upon further compactification to these terms will give rise to trilinear scalar couplings, as described below.

It may be useful for later purposes to display this result in terms of the flux components