Contents

On Classical Stability with Broken Supersymmetry




I. Basile, J. Mourad  and  A. Sagnotti

[15pt]

Scuola Normale Superiore and INFN

Piazza dei Cavalieri, 7

56126 Pisa  ITALY

e-mail: basile@sns.it, sagnotti@sns.it


APC, UMR 7164-CNRS, Université Paris Diderot – Paris 7

10 rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13  FRANCE

e-mail: mourad@apc.univ-paris7.fr






Abstract

We study the perturbative stability of four settings that arise in String Theory, when dilaton potentials accompany the breaking of Supersymmetry, in the tachyon–free and orientifold models, and also in the heterotic model. The first two settings are a family of vacua of the orientifold models and a family of vacua of the heterotic model, supported by form fluxes, with small world–sheet and string–loop corrections within wide ranges of parameters. In both cases we find some unstable scalar perturbations, as a result of mixings induced by fluxes, confirming for the first class of vacua a previous result. However, in the second class of vacua they only affect the modes, so that a projection induced by an overall parity in the internal space suffices to eliminate them, leading to perturbative stability. Moreover, the constant dilaton profiles of these vacua allow one to extend the analysis to generic potentials, thus exploring the possible effects of higher–order corrections, and we exhibit wide nearby regions of perturbative stability. The solutions in the third setting have nine–dimensional Poincaré symmetry. They include regions with large world–sheet or string–loop corrections, but we show that these vacua have no perturbative instabilities. Finally, the last setting concerns cosmological solutions in ten dimensions where the “climbing” phenomenon takes place: they have bounded string–loop corrections but large world–sheet ones close to the initial singularity. In this case we find that perturbations generally decay, but homogeneous tensor modes exhibit an interesting logarithmic growth that signals a breakdown of isotropy. If the Universe then proceeds to lower dimensions, milder potentials from other branes force all perturbations to remain bounded.

1 Introduction

When Supersymmetry is broken in String Theory [1], one is inevitably confronted with stability issues. These are particularly severe when tachyons emerge, but even in the few cases when this does not occur runaway dilaton potentials destabilize the original ten–dimensional Minkowski vacua. Supergravity [2] is the key tool to analyze these systems, within its limits of applicability.

There are three ten–dimensional string models with broken Supersymmetry and no tachyons, and two of them are orientifolds [3] of closed–string models. The first of these is the model [4] with “Brane Supersymmetry Breaking” (BSB) [5], a peculiar type–IIB orientifold which combines the presence of a gravitino in the low–energy Supergravity with a non–linearly realized Supersymmetry [6]. The second has no Supersymmetry at all, and is the model [7], a non–tachyonic orientifold of the tachyonic model [8]. In both cases, an exponential dilaton potential emerges at the (projective–)disk level, which reflects the residual tension of the branes and orientifolds that are present in ten dimensions. In the Einstein frame it reads

(1.1)

There is finally a heterotic model of this type [9], with an gauge group. Its torus amplitude gives a contribution that in the string frame is independent of the dilaton, but in the Einstein frame this translates into the dilaton potential [10]

(1.2)

In the first case there is a class of vacua [11], which are sustained by three–form (electric) fluxes that permeate , and where the dilaton has constant profiles. In the second case similar steps lead to a class of vacua [11], which are sustained by three–form (magnetic) fluxes that permeate the internal , and where the dilaton has again constant profiles 111There are also descriptions in terms of dual potentials, which involve seven–form fluxes in internal space for the orientifold models and in spacetime for the heterotic one.. In both cases there are wide corners of parameter space where large radii and small string couplings are possible.

In this paper we investigate perturbative stability in the presence of broken Supersymmetry in four types of settings. The first two concern these vacua for the orientifold and heterotic models, where fluxes counteract the effects of gravity. The third concerns the Dudas–Mourad solutions with nine–dimensional Poincaré symmetry [12], the first that were found for the low–energy equations of String Theory when the potentials (1.1) or (1.2) arise. No form fluxes are present in this case, and the resulting vacua include corners where the string coupling is large or curvature corrections are expected to be important. The fourth concerns cosmological solutions with nine–dimensional Euclidean symmetry, which were also first found in [12] and where the string coupling has an upper bound, so that string corrections are in principle under control, while curvature corrections are still large close to the initial singularity. Some generalizations of this solution were found later in [15], and were elaborated upon in detail in [16], where they were associated to the picture of a “climbing scalar”. Amusingly, the orientifold potential in eqs. (1.1) corresponds indeed to a “critical” logarithmic slope where the scalar (the dilaton in ten dimensions or, in lower dimensions, a mixing of it with the breathing mode of the internal space [17]) becomes compelled to climb up the potential when emerging from the initial singularity. In the subsequent descent the potential energy thus collected can give the initial impulse to start inflation [18]. This is the case if milder potentials, which could arise from lower–dimensional branes, are taken into account [19, 20]. The generic indication of these scenarios, if a short inflation happened to have left us some glimpses of its inception, resonates with the lack of power displayed by the first CMB multipoles [21]. Our discussion will make these considerations a bit more concrete from a top–down perspective.

There is an extensive activity aimed at identifying and cutting out, within the vacua that arise in Supergravity, wide subsets that should not admit ultraviolet completions, and therefore should not pertain to String Theory proper [22]. There is also a widespread feeling that instabilities show up generically in non–supersymmetric contexts. When this is the case perturbations, even if initially small, grow generically in time, and a pathology of this type is precisely what the negative squared masses of tachyons signal in flat space. In vacua matters are a bit subtler, and the proper stability conditions, which are called Breitenlohner–Freedman (BF) bounds [23], allow for finite ranges of negative squared masses, in ways that depend on the dimension and on the nature of the fields. Hence, one can study perturbative stability in backgrounds analyzing the eigenvalues of (properly defined) squared mass matrices for the available modes. This will be a primary task of this paper, and in particular we shall confirm the instability found in [24] for the family of vacua, which were actually first considered there, and we shall also present a similar result for the heterotic vacua of [11]. The instabilities present in both cases comply with the general expectations in [22], but the relative simplicity of these settings allows us to move further.

In the vacua of [24, 11] the dilaton has constant profiles. This makes it possible to scan the behavior of perturbations in a wide class of systems simply modifying three constants, the vacuum value of the potential and those of its first derivative and of its second derivative . One can explore, in this fashion, whether perturbative instabilities are generic within the potentials that give rise to similar flux vacua. The lesson that we shall gather is clearcut: close to the actual values, and as soon as becomes negative, there are wide regions of perturbative stability. This interesting phenomenon occurs both in the orientifold vacua and in the heterotic ones, and the nearby regions could play a role when string corrections are taken into account, although we are unable to justify dynamically the emergence of such deformations via quantum corrections. Moreover, in the heterotic vacua the instability affects only the scalar modes, so that it can be eliminated by an antipodal projection in the internal sphere. We also explored similar, albeit more complicated, operations for the orientifold vacua. In particular, in the simpler case of an internal one could work with unit quaternions, removing all unwanted spherical harmonics with via projections based on the symmetry group of the cube in . This would leave no fixed sub–varieties, and therefore constitutes an alternative option for the heterotic case. Its direct counterpart in , however, appears much more complicated, since it would rest on octonions. We thus explored, as an alternative, a construction based on three quaternion pairs, which does eliminate all unwanted spherical harmonics but unfortunately fixes some sub–varieties.

The following sections deal with the nine–dimensional vacua with Poincaré symmetry and the cosmological solutions that were discovered in [12], and also with the linear–dilaton systems of [13, 14] and other generalizations. As we shall see, the nine–dimensional vacua are perturbatively stable solutions of Einstein’s equations although, from the vantage point of String Theory, they include regions of high curvature and strong coupling, where corrections to Supergravity are expected to play an important role. Some features of these higher–derivative couplings were explored in [20]. In the spirit of the preceding extensions and in view of their potential applications, we also analyze the cosmological solutions that arise, in lower dimensions, with similar potentials , for different values of . These potentials were first studied in [15], and this will also connect the present analysis to the work in [16]. The resulting indications are that perturbations are well-behaved, up to an intriguing behavior of the homogeneous, mode that manifests itself, in ten dimensions, with the potentials of eqs. (1.1) and (1.2) in the absence of milder terms, to which we shall return in Section 7.1.

The contents of this paper are as follows. In Section 2 we define the low–energy Lagrangians of interest and we explain our conventions. In Section 3 we review the orientifold vacua of [24, 11] and extend them in the most general way that is of interest with constant dilaton profiles, allowing for generic values of the potential , its first derivative and its second derivative . In Section 3.1 we linearize the field equations and set up the perturbative analysis, which is carried out in Section 3.2 for tensor and vector perturbations. No instabilities are found in these sectors. The lowest tensor modes arise from metric perturbations, while the lowest vector modes are the expected massless Kaluza–Klein vectors for the internal , which here arise from mixings of metric and form contributions. In Section 3.3 we analyze scalar perturbations. Here we confirm the results in [24]: there are no instabilities in the sector, where the available modes arise from the metric tensor and the dilaton, while there are instabilities for , where also the form field comes into play. However, we display a wide corner of modified potentials, which lie close to the lowest–order one in eq. (1.1) but have a negative , where no unstable modes are present. At the end of the section we point out how the instabilities could be removed via projections in the internal sphere for the original potential (1.1), although all examples that we have constructed feature fixed sub–varieties. In Section 4 we review the heterotic vacua of [11] and extend them in the most general way that is of interest with constant dilaton profiles, allowing again for generic values of the potential , its first derivative and its second derivative . In Section 4.1 we linearize the field equations and set up the perturbative analysis, which is carried out in Section 4.2 for tensor and vector perturbations. Again, no instabilities are found in these sectors. The lowest tensor and vector modes are massless spin–two excitations and the expected massless Kaluza–Klein vectors for the internal , which here arise, again, from mixed metric and form contributions. In Section 4.3 we analyze scalar perturbations and show that there are no instabilities in the sector, where only the metric tensor and the dilaton enter, while there is again an instability when the form field comes into play. Here it only affects the modes, and could be removed by an antipodal orbifold projection in the internal manifold, along with all odd– harmonics. As for the orientifold vacua, we display however a wide corner of modified potentials that lie close to the lowest–order one in eq. (1.1) and have a negative , where again no unstable modes are present. In Section 5 we review the vacua with nine–dimensional Poincaré symmetry of [12], and in Section 5.1 we set up the perturbative analysis. Then in Section 5.2 we show that there are no perturbative instabilities in the scalar sector for the potentials in eqs. (1.1) and (1.2), and in Section 5.3 we show that the same is true for vector and tensor perturbations. This background is therefore a perturbatively stable solution of Einstein’s equations, although from the vantage point of String Theory it includes strong–coupling regions. In Section 6 we analyze the stability of the linear–dilaton vacua that originate, below the critical dimension of String Theory, from [25, 13]. In Section 7 we review the salient features of the climbing scalar cosmologies that arise with the potentials of eqs. (1.1) and (1.2), and in Sections 7.1 and 7.2 we analyze the corresponding perturbations. We show that there is a logarithmic instability for the homogeneous tensor mode, but there are none for other modes. In Section 8 we consider the behavior of a class of milder exponential potentials, which can arise in lower dimensions, in the presence of branes, and we show that there are no instabilities for them. Finally, in Section 9 we analyze a special class of potentials related to the linear–dilaton cosmologies of [14], while Section 10 contains a short summary of our results and some indications on possible further developments. There are also four Appendices. In Appendix A we discuss some differential equations that are encountered in Sections 3.3 and 4.3, while in Appendix B we review some properties of tensor spherical harmonics that are used extensively in Sections 3.1 and 4.1. In Appendix C we collect some useful results on BF bounds, along with a sketchy derivation of them, and finally in Appendix D we collect some observations on cosmic and conformal time that are relevant for Section 8.

2 The Models

In this paper we explore the low–energy dynamics of the three ten–dimensional string models with broken Supersymmetry and no tachyons in their spectra. The first two models are the orientifold with BSB of [5] and the orientifold of [7], whose low–energy effective field theories are identical, insofar as the vacua addressed here are concerned, while the third is the heterotic model of [9]. In all cases we shall account for their exponential dilaton potentials, whose origin is however different in the orientifold and heterotic examples. In the former it reflects residual brane/orientifold tensions, and is thus an open–string (projective)disk–level effect, while in the latter it arises from the torus and is therefore a standard first quantum correction. As we have shown in [11], these potentials allow flux vacua in the two orientifold models and flux vacua in the heterotic model. In both classes of vacua the dilaton acquires constant values, which can be tuned into regions of small string coupling.

We use the “mostly plus” signature and the following definition of the Riemann curvature in terms of the Christoffel connection :

(2.1)

Moreover

(2.2)

is the Ricci tensor. Notice that with this definition the curvature of a sphere is negative, and that the relation to , the curvature in the frame formalism, is

(2.3)

Here we refer to the curvature defined in terms of , and in our conventions

(2.4)

The low–energy dynamics of the systems of interest is captured by a class of Lagrangians involving the metric tensor, a scalar field and a two–form gauge field of field strength . In the Einstein frame, these read

(2.5)

where in the orientifold examples, which we shall study with reference to the model with BSB and to the model, and

(2.6)

On the other hand, in the heterotic model and

(2.7)

The equations of motion read, in general,

(2.8)

Equivalently, one can combine the Lagrangian Einstein equation with its trace and work with

(2.9)

The orientifold vacua of interest are supported by a flux of the three–form field strength in the factor, while their heterotic counterparts are supported by a flux of the three–form field strength in the factor.

Actually, it proved very instructive to allow generic values, in the vacuum, of the potential and of its first and second derivatives and . As we shall see, while instabilities are present, among the available scalar perturbations, in the uncorrected vacua corresponding to eqs. (2.6) and (2.7), nearby interesting regions of stability exist with . We also extended the discussion to more general systems, allowing generic values of , but this did not add significant novelties to the picture.

The ensuing sections are devoted to the study of perturbations of the two classes of backgrounds described in the Introduction. In the following, we shall thus refer all covariant derivatives to the corresponding background metrics. Moreover, we shall distinguish space–time d’Alembertians, denoted as usual by , and internal Laplace operators, here denoted by , since we shall decompose all perturbations in the proper sets of internal spherical harmonics, on which has the eigenvalues discussed in Appendix B.

3 The (Generalized) Orientifold Flux Vacua

The background manifold is in this case , with metric

(3.1)

Here and are and metrics of unit radius, while and denote the actual and radii. Greek indices run from 0 to 2, while Latin indices run from 1 to 7, the dilaton has a constant vacuum value , and finally there is three–form flux in , with

(3.2)

Here is a constant and , while all other components of vanish in the vacuum.

For maximally symmetric spaces, in terms of ,

(3.3)

where and are the radii of the sphere and factors. Moreover, the preceding conventions imply that

(3.4)

where, here and in the following Section 3.3, covariant derivatives are computed referring the background. In addition, the zeroth–order dilaton equation gives

(3.5)

which links the three–form flux, sized by , to the derivative of the scalar potential. Notice that the allowed signs of and must coincide, a condition that holds for the perturbative orientifold vacuum, where . The Einstein equations translate into

(3.6)
(3.7)

and it is convenient to define the two variables

(3.8)

which will often appear in the next section. Notice that and

(3.9)

so that the value separates negative and positive values of for these generalized vacua, and for the (projective)disk–level orientifold potential

(3.10)

3.1 Perturbations of the Generalized Orientifold Flux Vacua

We can now discuss perturbations in these vacua, letting

(3.11)

and linearizing the resulting equations of motion. The perturbed tensor equations are

where, here and in the following,

(3.13)

in terms of the and sphere contributions. In a similar fashion, the perturbed dilaton equation is

(3.14)

Finally, the perturbed metric equations that follow from eq. (2.9) rest on the linearized Ricci tensor

(3.15)

and read

(3.16)

In all cases, perturbations depend on the coordinates and on the sphere coordinates , and will be expanded in corresponding spherical harmonics, whose structure is briefly reviewed in Appendix B. For instance, for internal scalars this will always result in expressions of the type

(3.17)

where and is totally symmetric and traceless in the Euclidean labels. However, the eigenvalues of the internal Laplace operator will only depend on . Hence, for the sake of brevity, and at the cost of being somewhat sketchy, we shall leave the internal labels implicit, although in some cases we shall refer to their ranges when counting multiplicities.

For tensors in internal space there are some additional complications. For example, for mixed metric components the expansion reads

(3.18)

where corresponds to a “hooked” Young tableau of mixed symmetry and , as explained in Appendix B. Here the are vector spherical harmonics, and we shall drop all internal labels, for brevity, also for the internal tensors that we shall consider.

3.2 Tensor and Vector Perturbations

Following standard practice, we classify perturbations referring to their behavior under the isometry group of the background. In this fashion, the possible unstable modes violate the bounds, which depend on the nature of the fields involved and correspond, in general, to finite negative values for (properly defined) squared masses. Indeed, as reviewed in Appendix C with reference to forms, care must be exercised to identify the proper masses to which the bound applies, since in general they differ from the eigenvalues of the corresponding d’Alembertian. In particular, aside from the case of scalars, massless field equations always exhibit gauge symmetries.

Let us begin by considering tensor perturbations, which result from transverse traceless , with all other perturbations vanishing. The corresponding dynamical equation

(3.19)

where we have replaced the internal radius with the radius using eq. (3.8), obtains when the first of (3.16) is combined with the results summarized in Appendix B on general spherical harmonics. These harmonics are eigenvectors of the internal Laplace operator in eq. (3.13), whose eigenvalues on scalars are , with an integer number. In order to properly interpret this result, however, it is crucial to notice that the massless tensor equation in is the one determined by gauge symmetry. In fact, the linearized Ricci tensor determined by eq. (3.15) is not gauge invariant under linearized diffeomorphisms of the background, but

(3.20)

However, the fluxes present in eq. (2.9) endow, consistently, its r.h.s. with a similar behavior, and in eq. (3.19) corresponds precisely to massless modes. Thus, as expected from Kaluza–Klein theory, eq. (3.19) describes a massless field for , and an infinity of massive ones for . These perturbations are all consistent with the bound, and therefore no instabilities are present in this sector.

There are also scalar excitations resulting from the traceless part of that is also divergence–free, which are tensors with respect to the internal rotation group. They satisfy (see Appendix B)

(3.21)

so that their squared masses are all positive. Finally, there are massive perturbations, which are divergence–free and satisfy

(3.22)

Vector perturbations are a bit more involved, due to mixings between and induced by fluxes. The relevant equations are

(3.23)

where and are divergence–free in both indices. It is now possible to write

(3.24)

but this does not determine uniquely, since the redefinitions

(3.25)

do not affect . The divergence–free of interest obtain, in particular, provided is divergence–free in its internal index , and divergence–free do not affect this condition.

One is thus led to the system 222In all these expressions that refer to vector perturbations , as described in Appendix B.

(3.26)

and the first of eqs. (3.26) could in principle accommodate a source term of the type . However, its contribution can be absorbed by a gauge redefinition of the type  (3.25), and from now on we shall ignore it. Similar arguments apply to the ensuing analysis of scalar perturbations in this section and to some cases in Section 4.

In terms of the combination of eq. (3.8) this system becomes

(3.27)

The eigenvalues of the mass matrix are thus

(3.28)

In order to refer to the BF bound in Appendix C, one should add 2 to these expressions and compare the result with zero. All in all, there are no modes below the BF bound in this sector. The vector modes lie above it for in the region , while they are massless for and all allowed values of , and also, for all , in the singular limit , which would translate into a seven–sphere of infinite radius. For there are 28 massless vectors arising from one of the eigenvalues above. Indeed, according to Appendix B they build up a second–rank antisymmetric tensor in internal vector indices, and therefore an adjoint multiplet of vectors. This counting is consistent with Kaluza–Klein theory and reflects the internal symmetry of , although the massless vectors originate here from mixed contributions of the metric and the two–form field.

3.3 Scalar Perturbations

Let us now focus on scalar perturbations of the complete system. To begin with, contributes to scalar perturbations, as can be seen letting

(3.29)

an expression that satisfies identically

(3.30)

On the other hand, they do not arise from and , since the corresponding contributions would be pure gauge. On the other hand, scalar metric perturbations can be parametrized as

(3.31)

up to a gauge transformation with independent parameters along and directions. The linearized tensor equation yields

(3.32)

where we use the decomposition of eq. (3.13), so that denotes the internal background Laplacian. After expanding in internal spherical harmonics, so that , eq. (3.32) becomes finally (an derivative of)

(3.33)

Notice that a redefinition , where depends only on internal coordinates, would not affect in eq. (3.29). As a result, while eqs. (3.32) and (3.33) could in principle contain a source term, this can be eliminated taking this fact into account. Similar considerations will apply for the field in Section 4.3.

In a similar fashion, the dilaton equation becomes

(3.34)

or, after combining it with eq. (3.33),

(3.35)

Finally, after eliminating with eq. (3.33), the metric equations (3.16) read

(3.36)

These equations have an unfamiliar form, and for this reason in Appendix A we prove that the terms involving gradients must vanish separately. One ought nonetheless to distinguish two cases.

For nothing depends on internal coordinates, the terms involving and become empty and also disappears. According to Appendix A, in this case one is thus led to the reduced system

(3.37)

to be supplemented by the linear relation

(3.38)

Using the definitions of and , eqs. (3.37) can be recast in the form

(3.39)

and the last column of the resulting mass matrix vanishes, so that there is a vanishing eigenvalue whose eigenvector is proportional to . This perturbation is however pure gauge, since eq. (3.29) implies that the corresponding field strength vanishes identically. Leaving it aside, one can work with the reduced mass matrix (in our convention) determined by the other two equations,

(3.40)

whose eigenvalues are