The Octagon and the Non-Supersymmetric String Landscape

We present an orientifold of a toric singularity allowing for a configuration of fractional branes which dynamically breaks supersymmetry. This vacuum has no instability such as the ones conjectured to always afflict this class of models.

We present an orientifold of a toric singularity allowing for a configuration of fractional branes which dynamically breaks supersymmetry. This vacuum has no instability such as the ones conjectured to always afflict this class of models.
Introduction-Our understanding of gauge theories was given a dramatically new perspective when it was realized that they appear ubiquitously in string theory. In particular, four-dimensional gauge theories with N = 1 supersymmetry can be engineered by considering D-branes at Calabi-Yau three-fold singularities in type IIB string theory, in the limit in which supergravity decouples from the open string degrees of freedom [1]. For those singularities which are toric, there is a specific algorithm that completely determines the gauge theory, modulo Seiberg dualities. The gauge theories one gets are usually of quiver type, with all their data (gauge groups, chiral matter fields and superpotential couplings) best encoded in a dimer model [2,3].
It is a question of interest to ask whether all kinds of N = 1 supersymmetric gauge theories can be engineered in this way, or at least if it is possible to engineer theories which reproduce all kinds of low-energy behavior. This question can be ascribed, at least remotely, to the swampland program (see [4][5][6]), where physics at the low-energy, effective level is deemed possible only if it is consistent with quantum gravity, with an embedding in string theory fulfilling such a consistency requirement. While confinement, generation of a mass gap and of a chiral condensate can be shown to arise in very simple models [7], as well as N = 2 Coulomb-like branches in others [8][9][10], the fascinating possibility that the vacuum of the gauge theory dynamically breaks supersymmetry requires more work. Supersymmetry can be broken in different ways. The gauge theory may have both supersymmetric vacua and meta-stable supersymmetry breaking vacua, which can be parametrically long-lived. This situation can be engineered with D-branes at singularities, see e.g. [11][12][13][14]. Another possibility is that there is simply no vacuum in the theory, leading to what is called a runaway. It turns out that such a situation is rather frequent in configurations of branes at singularities, see [15][16][17][18].
The last possibility that remains is that supersymme-try is dynamically broken in a fully stable vacuum. This has proven to be a harder problem to engineer with Dbranes at singularities. This is partly due to the nongenericity of known gauge theories that display such a non-supersymmetric vacuum. After attempts to turn the runaway into a stable vacuum proved unsuccessful [19], it was shown in [20] that introducing an orientifold projection it is possible to engineer configurations which at lowenergies reproduce the well-known dynamical supersymmetry breaking (DSB) model of [21], henceforth referred to as 'the SU (5) model.' The same model was argued to arise in a wider number of singularities in [22,23]. The existence of such models is important because, in principle, it could be in tension with recent conjectures such as [24][25][26]. Somewhat in a plot twist, the DSB configurations of [20,22] were more closely scrutinized in [26], where it was found that they are actually not fully stable. Indeed, when the DSB configuration is probed by N regular D3branes, an instability appears where the regular branes split along the Coulomb branch of so-called N = 2 fractional branes [16], eventually settling the configuration in a supersymmetric vacuum. This phenomenon was further investigated in [27], where many examples of brane configurations at orientifolded singularities with a DSB model were found, all with the same kind of instability. In fact, a no-go theorem was proven in [27] showing that for any singularity allowing for a DSB model, N = 2 fractional branes, if present, always destabilize the supersymmetry breaking vacuum and set, eventually, the vacuum energy to zero. All of this was mounting evidence for what could be interpreted as the impossibility of engineering stable DSB with D-branes at singularities. In more dramatic words, could stable DSB be in the swampland?
In this letter we argue that DSB is still in the landscape. We produce an orientifold of a toric singularity which allows for a brane configuration displaying a variant of the SU (5)  such as the one described in [26,27], simply because it does not admit N = 2 fractional branes. Of course we cannot rule out any other hitherto unforeseen instability, possibly involving subleading effects in 1/N . On the other hand, we have a counter-example to what could have been conjectured, namely that DSB models were possible only in singularities admitting N = 2 fractional branes, and hence, following the no-go theorem presented in [27], unstable towards supersymmetric vacua [28].
The Octagon-The toric singularity we start with is the following. We dub it the 'Octagon' because of its toric diagram, that we reproduce in Fig. 1. It has 8 edges and it is of area 14, where the unit of area is an elementary triangle with which one performs a triangulation of the diagram. Before any orientifold projection, D-branes probing this singularity lead to a gauge theory with 14 gauge groups.
The four-dimensional N = 1 gauge theories living on the worldvolume of (fractional) D3-branes probing toric Calabi-Yau three-fold singularities are fully encoded by bipartite graphs on a two-torus known as dimer models or brane tilings [2,3]. A simple dictionary connects dimers to the corresponding gauge theories. Faces, edges and nodes in the dimer correspond to gauge group factors, bi-fundamental or adjoint chiral fields and superpotential terms, respectively. Dimers significantly simplify the connection between the geometry of the singularity and the corresponding gauge theory. Moreover, dimers efficiently encode orientifolds, which translate into Z 2 involutions of the graph. We will focus on the class of involutions studied in [20], which have either fixed points or fixed lines [29].
According to the rules stated in [23], the Octagon does not admit any orientifold represented as a point projection on the dimer. On the other hand, the highly symmetric toric diagram shows that orientifolds represented by line projections are possible, both diagonal and vertical/horizontal (the latter two possibilities are obviously equivalent). In Fig. 2 we show the dimer of the Octagon, together with its two orientifold vertical lines. More precisely, this is the dimer corresponding to a particular toric phase where the vertical fixed lines are manifest. Other toric phases, obtained by Seiberg dualities, obviously exist, but in general do not display the symmetry required to perform the (vertical) line projection. That the dimer of Fig. 2 indeed corresponds to the toric singularity of Fig. 1 can be checked using standard techniques [2,3].
In an orientifold with vertical fixed lines, each line carries an independent sign, which controls the projections of gauge groups and matter fields. The orientifold lines identify the 6 faces (1-6) to the 6 faces (14-9), respectively, each corresponding to an SU gauge group. Faces 7 and 8 are self-identified. By assigning the sign + to the line on the edge of the unit cell, and the sign − to the line in the middle of the cell, faces 7 and 8 inherit an SO and an U Sp gauge group, respectively. Moreover, the SU groups of faces 1 and 3 have a matter field in the antisymmetric representation, while the SU groups of faces 5 and 6 have a matter field in the symmetric representation.
Before performing the orientifold projection, it is straightforward to see that the following rank assignment is anomaly free: faces 1, 2, 3, 7, 12, 13 and 14 have gauge group SU (N + M ), and all the others have gauge group SU (N ). Setting N = 0, one has only seven SU (M ) gauge groups: one isolated Super-Yang-Mills (SYM) on face 7 and the six others forming a loop whose links are bifundamentals, together with a sextic superpotential proportional to the only gauge invariant (it is represented by the white dot in the center of the unit cell). This rank assignment corresponds to a so-called deformation fractional brane [16]. One can easily see that such a gauge theory eventually leads to a confining behavior just like SYM. This can be naturally UV completed starting from a system of N regular and M fractional D3-branes which trigger a RG-flow that can be described by a duality cascade, similar to [7] and many other examples that were found since then. The effective number of regular branes diminishes along the flow and the deep IR dynamics is described by fractional branes only.
In the presence of an orientifold projection, it is no longer granted that an anomaly free rank assignment exists at all. For instance, in the present case it can be shown that it is not possible to find one if the signs of the two lines are the same. However, choosing opposite signs as in Fig. 2, one can see that there is a rank assignment which is anomaly free: SU (N + M + 4) for faces 1 and 3, SU (N + M ) for face 2, SO(N + M + 4) for face 7, SU (N ) for faces 4, 5 and 6, and U Sp(N ) for face 8. Setting N = 0 we obtain a gauge theory with an isolated SO(M + 4) 7 SYM theory, which confines on its own, together with a quiver gauge theory based on the group SU (M + 4) 1 × SU (M ) 2 × SU (M + 4) 3 with matter fields and a superpotential that we proceed to analyze.
The DSB model-The gauge theory has matter content (2) and superpotential The superpotential can be interpreted as follows. The gauge invariant X t 12 A 1 X 12 of group 1 and the gauge invariant X 23 A 3 X t 23 of group 3 are respectively in the 2 and 2 of gauge group 2, with W above providing a bilinear in these two invariants, thus akin to a mass term. It is obvious that the antisymmetrics of SU (M ) 2 can exist as such only if M ≥ 2. In this case, it is possible to argue that strongly coupled dynamics generates superpotential terms that, together with the tree level one, eventually lead to supersymmetric vacua. While the case M = 0 leads trivially to decoupled SYM groups at faces 1 and 3, and hence confinement, the case of interest is M = 1.
For M = 1 node 2 becomes trivial (SU (1) is empty) and, more importantly, the superpotential actually vanishes. Indeed, both nodes 1 and 3 are SU (5) gauge theories with matter in the ⊕ representations, and there is no chiral gauge invariant that can be written in this situation [21]. Hence the two gauge theories are effectively decoupled, and their IR behavior can be established independently. Both happen to be the SU (5) model for stable DSB. Since the SO(5) SYM on node 7 just confines, we thus determine that this configuration displays DSB in its vacuum. Quite interestingly, this DSB vacuum may then arise at the bottom of a duality cascade (possibly more complicated with respect to the simpler unorientifolded case, due to the orientifold projection which would modify it, see [30]), hence within a UV complete theory.
In principle, one could be concerned about stringy instantons, whose presence may affect the low energy dynamics. Indeed, the D-brane configuration giving rise to the DSB model, N = 0, M = 1, contains both a U Sp(0) and an SU (1) factor coupling to the SU (5) gauge groups. These are the two instances where contributions to the low-energy effective superpotential are allowed (see [31] and [32], respectively). However, no such contributions can be generated in our model simply because there are no chiral gauge invariants that can be written. We thus conclude that stringy instantons cannot alter the DSB dynamics.
Stability-Is this DSB vacuum stable? Though we cannot answer this question in the most general terms, we can check whether the instability discussed in [26,27] is present or not. This is very quickly done: as can be readily seen from the toric diagram of Fig. 1, this singularity does not admit N = 2 fractional branes. The latter arise when the singularity can be partially resolved to display, locally, a non-isolated C 2 /Z n singularity and a Coulomb-like branch associated to it. This translates into the presence of points inside some of the edges along the boundary of the toric diagram. The Octagon does not have this property.
Hence, without the presence of N = 2 fractional branes, there is no vacuum expectation value on which the energy of the DSB vacuum can depend on, or equivalently there is no Coulomb branch along which the energy can slide to zero value.
Finally, one could worry about the N = 4 Coulomb branch represented by regular D3-branes. In fact, as in the previously analyzed cases [26,27], this does not lead to any instability at leading order in N , essentially because of the conformality of the parent (nonorientifolded) gauge theory. 1/N corrections, which can get contributions both from the orientifold itself as well as from fractional branes [7,30], could spoil stability. Indeed, flat directions are usually not expected in a nonsupersymmetric vacuum. Such corrections are not easily calculable, particularly in a complicated singularity such as the Octagon, hence we cannot predict whether they will lead to a supersymmetric vacuum, or just lift the flat direction while preserving a DSB vacuum.
Conclusions-In this letter we have presented a model, the Octagon, which is the first instance, to our knowledge, of a DSB configuration of fractional branes free of any obvious instability. As an existence proof of such a stable configuration, this is enough. However, it is not by chance that this particular singularity has been found, rather one can be led to it by a series of arguments. This is reviewed in [33], where it is also shown that the Octagon is in fact the simplest of a family of singularities allowing for stable DSB. All of them realize DSB through the double SU (5) model, and this is the reason why the simplest occurrence of this phenomenon is a singularity corresponding to a quiver with no less than 14 gauge groups. More details on how to find such toric singularities, and subtleties regarding orientifold projections and anomaly cancellation conditions, will appear in [34].
With this example, we have shown that stable DSB can still be engineered by brane configurations at Calabi-Yau singularities. Stability can be ensured at the leading order in N , i.e. in the decoupling limit. 1/N corrections are possibly dangerous, but more work would be needed to verify stability at such next-to-leading order. We similarly cannot exclude other instabilities due to mechanisms that we are currently not aware of. Given the remarkable properties of this family of models, we consider it important to study them in further detail.