Metastable String Vacua

We argue that tachyon-free type I string vacua with supersymmetry breaking in the open sector at the string scale can be interpreted, via S and T-duality arguments, as metastable vacua of supersymmetric type I superstring. The dynamics of the process can be partially captured via nucleation of brane-antibrane pairs out of the non-supersymmetric vacuum and subsequent tachyon condensation.


Introduction and conclusions
It is a widespread belief that all perturbative string constructions with broken supersymmetry are unstable and that the dynamics universally drives them towards trivial configurations [1]. Typically, the simplest sign of instability of non-supersymmetric string vacua is the presence of tachyonic excitations, at least in some regions of moduli space. Although, in the past tachyon-free ten-dimensional vacua with broken supersymmetry have been proposed [2,3,4,5] possibly violating the standard lore, it was soon evident that most of these string vacua develop tachyonic instabilities once some dimensions are compactified. For instance, the O(16) × O(16) heterotic model [2] is continuously related to its tachyonic cousins after proper Wilson lines are introduced in nine dimensions [6], while for the circle reduction of the 0 ′ B model [3] either the winding or the momentum excitations of the closed-string tachyon are still present after the orientifold projection, and actually become tachyonic in the small or large radius region of moduli space, respectively.
The so-called type I vacua with brane supersymmetry breaking [4,5], however, seem to be non-tachyonic, and thus stable, in any space-time dimension and in any corner of moduli space, thus offering a notable counter example to this common belief [1]. These models are characterised by a supersymmetric closed-string sector, while supersymmetry is explicitly broken in the open-string sector at the string scale, where bosonic and fermionic excitations are assigned different representations with of the Chan-Paton gauge group. Although the presence of gauge singlet fermions hints to the fact that the vacuum is already in its broken phase, where supersymmetry is non-linearly realised [7], there is no obvious candidate for a supersymmetric vacuum configuration to which it could decay into.
Whether or not these models are quantum mechanically stable is an open issue that we shall try to elucidate in the present letter. Actually, the construction of metastable vacua in field theories with rigid supersymmetry [8] has acquired some interest, and it is believed that they are more natural than traditional models with dynamical supersymmetry breaking [9] (see [10] for earlier constructions of metastable vacua). Some proposals to extend the field theory constructions in [8] to string theory using D-branes at orbifold singularities have been suggested [11], while in [12] it was argued that metastable vacua could play an active role in attempts to stabilise moduli. Despite much progress in the field theory and/or string theory constructions with metastable phases, identifying a fullfledged string theory vacuum of this type is still an important unsolved problem. Clearly, around such a metastable vacuum the non-supersymmetric spectrum should be free of tachyonic excitations, precisely as in the case for orientifolds with brane supersymmetry breaking [4,5]. It is then natural to propose that these vacua actually represent metastable local minima in the moduli space, where the true global minimum would correspond to the supersymmetric type I superstring. The purpose of this note is to collect some evidence in favour of this conjecture. In fact, we shall show that the models in [4,5] are naturally driven towards strong coupling. A Montonen-Olive duality then leads to a natural perturbative description in terms of type I superstring with pairs of branes and anti-branes that are expected to decay to the SO(32) superstring after brane and anti-brane annihilation.
We shall also show how this dynamics could be partly captured by the condenstation of tachyons on the pairs of branes and anti-branes.

Non-BPS string vacua and strong coupling
Orientifold models are the subject of an intense activity, since their perturbative definition offers interesting new possibilities for low-energy phenomenology. These models have a very interesting geometrical description in terms of D-branes and orientifold planes, extended objects that carry a charge with respect to appropriate R-R potentials and have a tension proportional to the charge itself. Typically, tensions and charges of D-brane and O-planes saturate a BPS bound, so that individually they preserve a certain half of the original supersymmetries of the closed-string theory, depending on the relative sign of their tension and charge. For D-branes tension and charge are both positive, while two types of O-planes can be present in perturbative string vacua: those with negative tension and charge, here denoted Op − -planes, and those with positive tension and positive charge, here denoted Op + -planes 1 . In addition, there are of course anti-D-branes and anti-O-planes, with identical tension and opposite R-R charges. Moreover, using non-perturbative string dualities, a rich zoo of similar extended objects emerges [14] that will be used in the following sections to support our conjecture.
The consistency of orientifold constructions and a number of their most amusing features may be traced to the relation to suitable parent models of oriented closed strings, from which their spectra can be derived [13]. In this procedure, a special role is played by tadpole conditions for R-R and NS-NS states. Although space-time supersymmetry relates the two tadpole conditions, they are completely different in nature. In fact, while the former are to be regarded as global neutrality conditions for R-R charges, and are usually linked to gauge and gravitational anomalies, the latter simply force the configuration of D-branes and O-planes to be globally massless. As a result, while the R-R tadpoles have always to be cancelled in a consistent vacuum configuration, in principle NS-NS ones can be relaxed, thus calling for a background redefinition [15,16] whose proper implementation in string theory, however, is not fully understood.
This difference between R-R and NS-NS tadpoles turns out to play an important role in a class of models with broken supersymmetry. In these constructions [4,5], the closedstring sector is classically supersymmetric, whereas supersymmetry is broken at the string scale on some stack of D-branes. Geometrically, these models always involve Op + planes together with an appropriate number of anti-branes, termed Dp-branes in the following, whose negative R-R charge compensates that of the Op + planes. In the simplest known example [4], the ten-dimensional closed-string sector encoded in the torus and Klein-bottle is as in the supersymmetric type I superstring, while in the open-string sector encoded in the annulus and Möbius-strip amplitudes 2) a crucial sign difference in front the of NS sector in M yields a D-brane spectrum with broken supersymmetry. In fact, the orientifold projection is in this case Ω ′ = −Ω(−1) F , where (−1) F is the space-time fermion number, so that the massless gauge bosons have symmetric Chan-Paton matrices, As a result, after setting N = 32 as required by the cancellation of the R-R tadpole, the open-string spectrum has gauge group USp(32) and fermions in the reducible 496 = 495 +1 anti-symmetric representation, consistently with the cancellation of ten-dimensional gauge and gravitational irreducible anomalies.
As usual, the transverse-channel Möbius-strip amplitudẽ clearly spells out the nature of O-planes and D-branes involved in the construction that, as anticipated, are O9 + -planes and D9-branes. This non-BPS configuration breaks explicitly all supersymmetries directly at the string scale, and seems not continuously connected to any supersymmetric vacuum. Notice that no tachyonic excitations are present in the openstring sector, thus suggesting that this vacuum configuration is locally, classically, stable.
The quantum dynamics of this and related systems is, to the best of our knowledge, still an open question.
The impossibility of cancelling the NS-NS tadpole in these non-BPS configurations induces a tree-level potential in the low-energy effective action While crucial in order to couple consistently a non-supersymmetric open-string spectrum to a supersymmetric bulk 3 , this potential is incompatible with a maximally symmetric Minkowski space-time, and in fact leads to a "spontaneous compactification" to nine dimensions, with a manifest SO (1,8) Poincaré symmetry. More specifically, the metric and the dilaton field read [17] e φ = e φ 0 |u| 2/3 e 3u 2 /4 , 5) in the string frame, where u is the "internal" coordinate. Notice that in the Einstein frame the dilaton tadpole is proportional to e 3φ/2 , and hence one would naively expect the theory to be driven towards zero string coupling, with g s = e φ . Actually, this is not the case, and inspection of the solution (2.5) shows that this vacuum configuration necessarily enters a strong coupling regime for large u. This clearly suggests that the perturbative description is at best incomplete. Another hint pointing towards the inevitable presence of a strongly coupled phase comes from the analysis of the gauge theory on the D-branes. After a suitable reduction to four dimensions, the light excitations comprise gauge bosons and six scalars in the adjoint of USp(32) together with four Weyl fermions in the 496-dimensional anti-symmetric representation. This gauge theory is clearly asymptotically free, and its coupling becomes strong at low energies. To summarise, these non-BPS orientifolds are naturally driven towards a phase of strong coupling, and, as we shall see in the following sections, our conjecture is that non-perturbatively these vacua are metastable states of the supersymmetric type I superstring.

S-duality, supersymmetry breaking and metastable states
In the previous section we have introduced two different types of O-planes that exist in perturbative string theory. We have called them Op ± planes where the suffix refers to the sign of their tension and charge. Actually, the difference between these two types of orientifold planes resides in a discrete B ab background, always allowed by the orientifold projection [18], that implies the possibility of having a non-trivial discrete holonomy for the NS-NS B field The holonomy contributes to a term e 2iπθ NS to the RP 2 amplitude and thus introduces and additional minus sign responsible for the exchange of Op + and Op − planes. Actually, it was realised that also R-R field could have a non-trivial discrete holonomy, that would in turn yield new variants of orientifold planes. For instance, in the case of O3 planes one could allow for the holonomy As a result, there are four different types of orientifold planes characterised by the values of the holonomies (θ NS , θ R ) and yield different types of gauge theories on stacks of D3-branes coincident with them, as summarised in table 1 [14].
Unlike the Op ± cases, however, O-planes carrying a non-vanishing θ R holonomy cannot be described in perturbation theory since they involve a non-trivial R-R background. In fact, the SL(2, Z) duality of the type IIB superstring exchanges θ NS and θ R , so that O3 − and O3 + planes are fixed, while O3 + and O3 − planes are interchanged. If we include D3 branes, the S-duality of type IIB becomes the Montonen-Olive duality for the N = 4 supersymmetric gauge theory living on their world-volume [14]. Notice, that an O3 − plane has the same charge and tension as an O3 − plane with a stuck D3 brane on it, and indeed it was argued in [14] that in the strong coupling limit the O3 + plane with positive tension and positive charge is naturally described in terms of an O3 − together with a stuck D3.
These are all the ingredients we need to describe the strong-coupling dynamics of the orientifold vacua introduced in the previous section.
For simplicity, let us consider a local configuration of an O3 + plane with a number m of D3 branes on it -together with their images under Ω. Clearly this configuration is not BPS, and indeed the gauge theory on the anti-branes has gauge bosons and six scalars in the adjoint representation of a USp(2m) group while the four Weyl fermions are in the anti-symmetric representation. This is a local version of the model described in the the previous section and introduced in [4,5]. Although, strictly speaking, Montonen-Olive duality does not apply to this configuration, if the D3 branes are moved a distance δ ≫ √ α ′ from the O-plane then supersymmetry is only mildly broken, and one can assume that Sduality is almost exact. Hence, the configuration of O3 + and D3 branes that is stable at weak coupling is naturally driven towards a strongly coupled regime where it is more conveniently described in terms of 4 O3 − and D3 or, better, in terms of a negatively charged O3 − plane plus m physical D3 and a stuck D3 brane.
The vacuum energy of the initial configuration receives contribution entirely from the Möbius-strip amplitude

3)
4 Notice that the D3 branes in the bulk have an N = 4 supersymmetric massless spectrum with gauge group U(m) that is self-dual.
where the leading contribution originates from the exchange of massless closed-string states in the tree-level channel, and in going from the first to the second line we have used the standard relations between the proper times t for the open-string propagation and ℓ for the closed-string propagation [13].
In the weakly coupled S-dual configuration, however, the D3 branes not only interact with the orientifold plane, but also with the stuck D3 brane, so that now both the annulus and Möbius-strip diagrams contribute to the vacuum energy

(3.4)
This configuration is clearly unstable since the D3 branes are attracted by the O3 − plane and the stuck D3 brane. However, in contrast with the original non-BPS configuration, for δ < √ α ′ a tachyonic mode now appears in the open-string spectrum and the D3's and the stuck D3 tend to partially annihilate. In the next section we shall see how this local construction can be extended to vacuum configurations with brane supersymmetry breaking.
In the original non-BPS configuration the vacuum energy in the Einstein frame has a qualitative dependence on the string coupling constant of the form V ∼ T − g s /δ 4 that indeed drives the system towards a non-perturbative regime. However, as g s becomes strong, the non-BPS configuration has a natural weakly coupled description in terms of type I with pairs of branes and antibranes that is still characterised by a vacuum energy of is now very small and hence the corresponding vacuum energy is bigger, thus interposing an energy barrier between the original non-BPS configuration and the final type I superstring state. We are therefore led to conclude that the original non-BPS configuration, with O3 + plane and D3 branes, is a locally metastable vacuum of a type IIB orientifold with O3 − planes. Clearly, this argument is somewhat qualitative, and more detailed studies are needed in order prove that this non-BPS configuration is metastable.

Strong coupling limit of vacua with brane supersymmetry breaking
We can now use the strong coupling properties of the local model studied in the previous section to describe the dynamics of the non-BPS vacuum configuration of interest [4,5]. In fact, let us consider the four-dimensional orientifold obtained by projecting the T 6 reduction of type IIB superstring by Ω ′ = ΩI 6 (−1) F L , where Ω is the standard orientifold projection, I 6 reverts the coordinates of the internal six-torus, and (−1) F L is the lefthanded space-time fermion index. This orientifold introduces 64 O3 + planes at the 64 fixed points of the Ω ′ orientifold together with 32 D3 branes needed to cancel the R-R tadpole.
The presence of a non-vanishing dilaton tadpole or, in turn, an attractive force between the O3 + planes and the antibranes makes the configuration unstable and drives the model towards a strong coupling regime. If the D3 are placed in the bulk at a suitable distance from the O-planes, it is reasonable to assume that type IIB S-duality still holds, so that a weakly coupled description is in terms of 64 O3 − planes, or in terms of 64 O3 − planes with 64 stuck D3 branes. This configuration is indeed allowed since the six Wilson lines needed to distribute the D3 branes on the orientifold planes, have positive determinant and mutually commute when acting on spinors. One can then decompactify this configuration and at the same time undo these Wilson lines, so that the D3 branes can be brought together to yield an SO(64) gauge group.
Finally, the 32 pairs of branes and antibranes annihilate via open-string tachyon condensation [19] and one is left with the type I superstring with negatively charges O-planes and 32 D-branes with gauge group SO(32).
This strongly coupled dynamics of the USp(32) model and its connection with the type I superstring can be nicely captured to a large extent by tachyon condensation already in ten dimensions. Let us consider, in fact, the type I superstring with additional pairs of branes and antibranes. In the presence of the O9 − plane these have two possible ways to decay. Either they fully annihilate in pairs, or a pair of stuck D9-D9 branes is left with an O(1) gauge group on each world-volume. Taking into account also the N = 32 D9 branes of type I, one is altogether left with p = 1 stuck antibranes and 33 = N + q branes whose one-loop amplitudes read and The light spectrum now comprises 1 2 33 · 32 = 528 gauge bosons on the D9 branes, 32 + 1 tachyons, denoted T 32 and T 1 , 496 + 32 + 1 left-handed Majorana-Weyl fermions, denoted ψ L 496 , ψ L 32 and ψ L 1 , and 32 + 1 right-handed Majorana-Weyl fermions, denoted λ R 32 and λ R 1 . These massless excitations are compatible both with a SO(33) and a USp(32) gauge group. From the point of view of the former, tachyon condensation breaks it to its SO(32) subgroup and theory becomes the supersymmetric type I. However, we can interpret the end-point of tachyon condensation also from the viewpoint of USp(32) gauge group. In this case, condensing the singlet tachyon, T 1 = 0, yields mass terms for the 33 non-chiral fermions and for the 32 (N, p) tachyons T 32 through couplings of the form T 1 ψ L 32 λ R 32 , T 1 ψ L 1 λ R 1 and T 2 1 T 2 32 . As a result, the surviving massless modes are the 496 left-handed fermions ψ L 496 and 528 gauge bosons, precisely the massless content of the non-supersymmetric USp(32) gauge theory with chiral fermions in the anti-symmetric representation!

S-duality in freely acting orbifolds with brane supersymmetry breaking
Other non-BPS configurations similar to that discussed in section 2 have been proposed in the literature [5], and their fate is also an open question. Clearly, it would be nice if similar arguments based on S-duality could be applied also to these cases. Unfortunately, in most of the other models the non-supersymmetric branes are embedded in an N = 2 or N = 1 closed-string setting, and S duality is not fully under control. For this reason, we shall study here a new vacuum partially related to that in [5], but where the various ingredients -O-planes and D-branes -are fairly separated in the transverse directions, and therefore do not interact strongly. The model is based on a freely acting (T 4 × S 1 × S 1 )/Z 2 orbifold of the type IIB superstring, where the single Z 2 generator g reverts the sign of the four coordinates of the T 4 g : (X 6 , X 7 , X 8 , X 9 ) → −(X 6 , X 7 , X 8 , X 9 ) , (5.1) and simultaneously shifts the first S 1 coordinate g : X 5 → X 5 + πR , (5.2) by half of the length of the circle, while leaving untouched the X 4 coordinate of the second S 1 . At the level of the type IIB superstring it interpolates between N = 2 vacua and N = 4 vacua in the limit R → ∞. Here we have used our standard notation [13] for the Z 2 characters Γ (1,1) .

(5.5)
Standard supersymmetric orientifold projections of this interpolating type IIB configuration have already been studied in [20], however we are interested now in non-BPS configurations with Op + planes and for this reason, as in [5], we combine the world-sheet parity ΩI 45 (−1) F L , where I 45 denotes a simultaneous inversion along the X 4 and X 5 coordinates, with an automorphism σ that reverts the contribution of the twisted sector. The Klein-bottle amplitude is then where, as usual, P and W denote the truncation of the Narain lattice to pure momenta and to pure winding zero modes. After an S modular transformation to the tree-level channel, this amplitude clearly spells-out the geometry of O-planes: this interpolating orientifold contains two O7 − planes both with X 5 = 0, together with 32 O3 + planes all at X 5 = πR, and dislocated at the 32 fixed points of the T 4 and of the spectator S 1 .
As expected, the open-string sector needed to cancel R-R tadpoles involves N = 16 D7 and M = 16 D3 branes, whose spectra are encoded in the annulus and Möbius-strip Also in this case the configuration is unstable, although tachyon free, and is driven towards a strongly coupled regime. After the D3 are displaced in the bulk sufficiently far from the O-planes and from the D7 branes, one can use the same arguments based on S-duality and describe this model with g s ≫ 1 in terms of a weakly coupled configuration where the 32 O3 + planes are traded for 32 O3 − ones ∼ 32(O3 − planes + stuck D3 branes. The sixteen bulk D3 branes can annihilate half of the stuck D3 ones and yield a fully supersymmetric configuration. The resulting massless spectrum has N = 4 supersymmetry and gauge group SO(16) × SO (16) as in the model in [20], that was argued to be related to the heterotic M-theory of Horava and Witten [21].
It would be interesting to gain also some understanding of the strongly coupled regime of more general models with brane supersymmetry breaking, where the closed-string sector and presumably the final weakly coupled D-brane configuration have reduced supersymmetry. However, our arguments are based on the SL(2, Z) duality of type IIB, that is well established for N = 4 theories but not fully understood for non-maximally supersymmetric models. Although in principle it is not applicable to non-supersymmetric environments, in the models we have analysed in this letter S duality is only marginally broken since, if the antibranes are placed in the bulk, the configurations preserve to leading order sixteen supercharges, so that the strongly coupled regime is partly under control.