Infrared Duality in Unoriented Pseudo del Pezzo

We study the orientifold projections of the $\mathcal{N}=1$ superconformal field theories describing D3-branes probing the Pseudo del Pezzo singularities PdP$_{3b}$ and PdP$_{3c}$. The PdP$_{3c}$ parent theory admits two inequivalent orientifolds. Exploiting $a$ maximization, we find that one of the two has an $a$-charge smaller than what one would expect from the orientifold projection, which suggests that the theory flows to the fixed point in the infrared. Surprisingly, the value of $a$ coincides with the charge of the unoriented PdP$_{3b}$ and we interpret this as the sign of an infrared duality.


INTRODUCTION
The AdS/CFT correspondence, in its original formulation, states that the 4 dimensional N = 4 SU (N ) SYM gauge theory living on the worldvolume of a stack of N D3-branes in flat space is dual to type-IIB supergravity on an AdS 5 × S 5 background in the large N limit [1][2][3]. More generally, for a system of regular D3-branes probing the tip of a Calabi-Yau (CY) cone, the worldvolume conformal field theory is dual to IIB supergravity on AdS 5 × H 5 , where the horizon H 5 is a 5-dimensional Sasaki-Einstein manifold [4,5] and represents the base of the CY cone. The correspondence relates the central charge a of the conformal field theory a = 3 32 to the volume of the Sasaki-Einstein horizon H 5 by the relation [6] Vol(H 5 ) = π 3 4 where N is the number of units of 5-form flux. In general, the presence of U (1) flavour symmetries implies that the R-charges can not be unambiguously assigned a priori. When this happens, the superconformal R-charges are uniquely determined as those that maximize a [7,8]. While in the simpler cases of cones that are abelian orbifolds of C 3 the field content and superpotential of the gauge theory can be read directly from the N = 4 ones, systematic techniques have been developed to determine the gauge theories for D3-branes at the tip of more general toric cones [9]. A toric cone has at least a U (1) 3 isometry and its base H 5 is a T 3 fibration over a convex polygon known as the toric diagram. In particular, one can study the blow-up of a singularity adding points to the toric diagram and this is dual to the unhiggsing mechanism in the gauge theory [4,[9][10][11][12]. If applied to the dP 2 theory, corresponding to the complex cone over the del Pezzo surface obtained by the blow-up of 2 generic points of P 2 , this gives rise to either dP 3 or the Pseudo del Pezzo geometries PdP 3b or PdP 3c if the blow-up is non-generic [11,12]. The latter two theories, whose toric diagrams are drawn in Fig. 1, are the ones we are interested in.
From the toric diagrams, there is a systematic procedure to determine, up to Seiberg dualities, the field content and the superpotential of the gauge theory. Specifically, this information is encoded in a bipartite graph representing a web of NS5-branes, which is called dimer [13][14][15][16][17]. This web is drawn on a T 2 wrapped by D5-branes, and by performing two T-dualities along the torus the whole configuration is mapped back to the original system of D3-branes sitting at the singularity of the cone. We draw in Figs. 3 and 4 the dimers of the PdP 3b and PdP 3c theories. By looking at the nodes in the diagrams one can see that the two theories have different superpotentials but share the same quiver, given in Fig. 2.
In this setup, more general gauge theories can be constructed by introducing orientifold projections Ω. In the brane configuration, this corresponds to adding an orientifold plane, which induces a Z 2 involution on the space and flips the world-sheet parity of strings, making the theory unoriented [18][19][20][21] (for a review see [22]). The Z 2 involution of the orientifold can be represented on the dimer by the introduction of either fixed points or fixed lines [23], from which one can read the field content and the superpotential of the unoriented theory. As far as conformal invariance is concerned, the following scenarios are possible: either the unoriented theory does not have a fixed point at which a is maximized, or there is a fixed point and the orientifold yields O(1/N ) corrections to physical observables. Note that in the first scenario one could have a duality cascade [24] or conformal symmetry can be restored by the addition of flavour branes [25].
In this letter we show that a third scenario exists, in which the orientifold breaks the conformal symmetry of the parent theory but develops a new superconformal fixed point. We provide an example of such a scenario by studying the orientifold projection of the gauge theories arising from the toric CY cones over PdP 3b and PdP 3c . While the projection PdP Ω 3b is unique, PdP 3c admits two inequivalent orientifolds, which we call Ω 1 and Ω 2 . For each of the orientifolds, we determine the ranks of the gauge groups and the R-charges of the chiral fields using a maximization. Although one would always naively expect that the a central charge of the unoriented theory is a half of that of the parent theory, we find that this happens only for PdP Ω 3b and PdP Ω1 3c , while for PdP Ω2 3c one gets a smaller central charge a. To the best of our knowledge, this is the first time that such a mechanism occurs in the context of unoriented theories. Interestingly, both the Rcharges and the a central charge of PdP Ω2 3c are identical to the ones of PdP Ω 3b . To see this effect it is crucial that the computation is performed keeping N finite, while neglecting O(1/N ) corrections naively gives the same a central charge of PdP Ω1 3c . The fact that the central charge a is smaller than expected suggests that there is an RG flow from a conformal fixed point in the UV to another conformal fixed point in the IR, in which the theory coincides with PdP Ω 3b . This IR duality could be inherited by the relation between the parent theories, which are connected by a web of dualities involving specular [26,27] and Seiberg duality [28,29].

PdP 3b AND PdP3c
Let us introduce the parent gauge theories of interest. We begin with PdP 3b [11,12,26], whose quiver and dimer are drawn in Fig. 2 and 3. There are 6 gauge groups 6 a=1 U (N a ) and the matter fields are bifundamentals X ab corresponding to the edges in the dimer. For instance, X 12 transforms in the fundamental representation of U (N 1 ) and in the anti-fundamental of U (N 2 ) and is the edge between faces 1 and 2 in the dimer. The global symmetries of this model are U (1) 2 × U (1) R as mesonic symmetry and U (1) 5 as baryonic one, of which U (1) 2 is anomalous. The superpotential of the theory reads The gauge anomalies vanish imposing the following relation between the ranks of the gauge groups: We find the superconformal fixed point and the corresponding R-charges R ab for the fields X ab maximizing the a-charge. Requiring that the β-functions vanish (which is equivalent to non-anomalous R-symmetry) we have where the sum is over gauge groups a connected to b by a bifundamental field X ab . Together with the condition that the R-charge of the superpotential is R(W ) = 2, we have a system of equations with a priori eight independent R-charges. This can be seen also from the quiver, which enjoys a Z 2 symmetry. The a-charge in Eq. (1) is a two-variable function, namely, a flavour symmetry where N a = N ∀a = 1, . . . , 6 since this condition gives the only solution that respects the unitarity bound. Note that the expression of the a-charge is given at leading order in N .
The second theory we study is PdP 3c [11,12,26], whose dimer is drawn in Fig. 4. As in the previous case, the gauge group is 6 a=1 U (N a ) and the global symmetries are U (1) 2 × U (1) R as mesonic symmetry and U (1) 5 as baryonic one, of which U (1) 2 is anomalous. The matter fields are also the same of the PdP 3b theory and indeed the two models share the same quiver in Fig. 2. Nonetheless, they have a different dimer and therefore interact differently. In fact, the superpotential reads The gauge anomalies vanish imposing the condition in Eq. (4) as before. Computing the R-charges which max-imize the a-charge we find [26] R c where again N a = N ∀a = 1, . . . , 6 is the only solution that respects the unitarity bound. Note that the difference in the R-charges arises from the condition R(W ) = 2. It is crucial to note that in this case the a-charge is a three-variable function, namely, the non-R symmetry which mixes with the R-charge is U (1) 3 .

UNORIENTED PdP 3b AND PdP3c
The orientifold projection of a quiver gauge theory is represented as a line which identifies the two sides of the quiver and projects the groups and the fields that are mapped onto themselves [25,30,31]. For the PdP 3b and PdP 3c theories, the involution is the Ω line in Fig. 2. The gauge groups intersected by the line are projected onto either SO or Sp, while bifundamental fields charged under two groups that are identified by Ω are projected onto a symmetric or antisymmetric representation. Besides, if U (N a ) is mapped to U (N b ), a fundamental representation of U (N a ) is identified with the antifundamental of U (N b ).
In the dimer, the Z 2 involution can be realized by the introduction of either fixed points of fixed lines on the torus [23]. In the first case, the orientifold projects both coordinates of the torus, preserving the mesonic flavour symmetry. There are four fixed points in the fundamental cell and each one carries a sign i = ±, i = 1, . . . , 4. The superpotential terms are halved since black nodes are mapped to white nodes. The four fixed points are physically four orientifold planes and their charges are the signs of the fixed points. When a fixed point with = +(−) lies on a face, the corresponding U (N ) gauge group is projected onto SO(Sp). When a fixed point with = +(−) lies on an edge it projects the corresponding field onto a symmetric (antisymmetric) representation and it identifies the gauge groups on the two sides of the edge. Supersymmetry requires that where N W is the number of terms in the superpotential of the parent theory. Besides, the anomaly cancellation condition may further constrain the orientifold signs. In the second case, the involution acts projecting only one coordinate of the torus, thus breaking the U (1) 2 flavour symmetry to a combination of the two. Nodes in the dimer are mapped to each other w.r.t. the fixed lines. Physically, a fixed line is an orientifold plane that intersects the torus and its charge is the sign of the line. There is no constraint in this case, other than the anomaly cancellation condition.

Unoriented PdP 3b
We consider the orientifold involution of PdP 3b with two fixed lines and we choose the configuration with signs (−, +), as in Fig. 3. As a consequence, the gauge group U (N 5 ) is identified with U (N 3 ) and U (N 6 ) with U (N 2 ), while U (N 1 ) becomes Sp(N 1 ) and U (N 4 ) becomes SO(N 4 ) since they lie on top of the fixed lines. The resulting theory has gauge groups Sp(N 1 )×U (N 2 )× U (N 3 ) × SO(N 4 ), where the fields X A 35 and X S 62 belong to the antisymmetric and symmetric representations of the gauge groups U (N 3 ) and U (N 2 ) respectively. The superpotential reads (11) and the anomaly cancellation condition is As in the parent theory, there are eight a priori independent R ab . The condition for the β-functions to vanish changes slightly due to the orientifold involution. In fact, one has if the group labelled by b is orthogonal (−) or symplectic (+). Likewise, for unitary gauge groups one has where R b is the R-charge of the symmetric (+) or the antisymmetric (−) field charged under U (N b ). The only consistent solution requires N 2 = N 3 = N 1 +2 = N 4 −2 = N , the flavour symmetry is U (1) 2 and the R-charges are the same as the ones of the parent theory in Eq. (6). The same result holds if we take the large N limit in the system of equations, because these reduce exactly to the ones of the parent theory. The value of the a-charge is Note that the central charge a is half the one of the parent theory. This is expected since the degrees of freedom have been halved. Besides, also the volume of the PdP 3b after the orientifold is half the volume of the parent space. To see this, consider that the orientifold acts as a Z 2 projection on the geometry. As a consequence, the number of units of 5-form flux is N/2 [32]. Thus, the proper ratio between the volumes reads This is similar to the case of the Z n orbifold of flat space, where the volume is a fraction n of the volume of the sphere S 5 .

Unoriented PdP3c
Let us turn to the orientifold of PdP 3c . As shown in Fig. 4, the dimer admits only the projection with fixed points, whose signs ( 1 , 2 , 3 , 4 ) project the group U (N 1 ), the group U (N 4 ), the field X 35 and the field X 62 , respectively. The parent theory has N W = 8, thus First, we focus on Ω 1 . The unoriented theory has gauge groups SO( 35 and X S 62 are antisymmetric and symmetric representations of U (N 3 ) and U (N 2 ) respectively. The superpotential reads W Ω1 3c = X 13 X A 35 X 56 X 61 − X 45 X 56 X S 62 X 24 + X 12 X 24 X 41 − X 13 X 34 X 41 (17) and the anomaly cancellation condition remains as Eq. (12). The superconformal fixed point of this unoriented model has the same R-charges of the parent theory in Eq. (9), with the flavour U (1) 3 symmetry mixing with R-symmetry. The a-charge is where N 1 = N 2 = N 3 − 2 = N 4 − 2 = N is the only consistent solution. Again, the a-charge is halved because of the orientifold projection, and the ratio between the volumes is 1/2 as before.
The unoriented theory obtained from Ω 2 = (−, +, −, +) has gauge groups Sp(N 1 )×U (N 2 )×U (N 3 )× SO(N 4 ), where fields X A 35 and X S 62 are unchanged w.r.t. the previous case. The superpotential W Ω2 3c is formally identical to W Ω1 3c in Eq. (17) and again the anomaly cancellation condition remains as Eq. (12). The amaximization in this case is more subtle. If one took naively the limit N → ∞ before solving the equation for vanishing β-functions and R(W ) = 2, one would obtain the R-charges and half the a-charge of the parent theory, with non-R flavour U (1) 3 . On the other hand, we find that for any finite value of N , the only consistent solution is N 2 = N 3 = N 1 + 2 = N 4 − 2 = N exactly as in PdP Ω 3b , with one flavour U (1) broken and the remaining U (1) 2 mixing with R-symmetry. This has the crucial effect of giving the value of the superconformal R-charges as which are different from the R-charges of the parent theory in Eq. (9), and the a Ω2 3c -charge takes the value which is smaller than the value of a Ω1 3c in Eq. (18). Consequently, the ratio between a Ω2 3c and that of the parent theory is while the ratio between the volumes is The fact that a Ω2 3c is less than halved w.r.t. the central charge a 3c of the parent theory can be taken as a sign of an RG flow towards the IR. Thus, a natural question is what is the endpoint of this RG flow. It is even more surprising that the R-charges and the a-charge in Eqs. (19) and (20) are exactly those of PdP Ω 3b given in Eqs. (6) and (15). Furthermore, one can show that this equivalence still holds taking into account the 1/N corrections to the a-charges. This suggests the two theories are dual at the conformal fixed point. In other words, the RG flow is going from PdP Ω2 3c in the UV to PdP Ω 3b in the IR. DISCUSSION We have shown that the value of the a-charge of the superconformal fixed point of the unoriented PdP Ω2 3c is smaller than expected and this gives strong evidence that there is an RG flow from the UV to the IR, along which a U (1) is broken. On the fixed point in the IR, the Rcharges and a-charge are exactly those of the unoriented PdP Ω 3b . As a consequence, 't Hooft anomalies match in the IR as well as the quantum numbers of the gauge invariant operators and thus we conjecture that in the IR the two unoriented theories describe the same physics.
The mechanisms known in the literature to produce RG flows in the context of holographic field theories do not seem to explain our result. In particular, the flow described above is not due to mass deformations [33,34], to a Higgs mechanism [11] or to the introduction of fractional branes [35]. We can expect some kind of kink solution interpolating between two asymptotic geometries like in [36], but the metrics of the CY complex cones over PdP 3b and PdP 3c are unknown and thus we do not have control of the bulk theory.
Since orientifolds realized in the dimer by fixed points do not break the U (1) 2 × U (1) R mesonic symmetry [14], the U (1) broken by the Ω 2 orientifold must be baryonic. The picture is thus that the configuration of branes and orientifold planes in the PdP Ω2 3c theory breaks, together with a baryonic U (1), also conformal symmetry in the UV, but makes the theory flow to a different IR fixed point that is the one of the PdP Ω 3b theory, with which PdP Ω2 3c shares all the symmetries. In the bulk, we can thus expect form fluxes that make the volume increase so that only asymptotically the metric is AdS. The scale associated to the flow can be identified with the size of the cycles wrapped by the branes that generate the fluxes. Solutions with broken baryonic U (1)'s associated to branes wrapping cycles have been discussed in the literature [37,38].
Another direction that can be explored is the possibility that the duality is a consequence of the PdP 3b and PdP 3c parent theories being connected by specular duality [26,27], which in general is a map between theories with the same master space. In the case of theories whose toric diagram has only one internal point, like the ones we are discussing, specular duality exchanges mesonic and anomalous baryonic symmetries. The chain of maps that relate the two parent theories is more precisely a specular duality followed by a Seiberg duality [28] and again another specular duality. One could even investigate the possibility that a chain of Seiberg dualities relates the two unoriented theories. Seiberg dualities in the case in which (anti-)symmetric fields are present have been considered in [39,40], where one describes them as mesons of a new confining symplectic or orthogonal gauge group. The problem of this approach is that one needs to add a gauge group going towards the UV and that can not describe the flow from PdP Ω 3b to PdP Ω2 3c . We expect to find other examples of pairs of orientifolds sharing the same features of the theories discussed in this letter. This would allow us to understand the physical origin of this infrared duality. However, as the number of gauge groups increases, it becomes computationally harder to find the exact local maximum of the acharge. To give more evidence that the unoriented PdP Ω2 3c flows to PdP Ω 3b in the IR, we plan to compute and compare the superconformal indices and study the moduli spaces of the two unoriented theories. Another possible line of investigation would be to check whether S-duality and strong coupling effects are involved, along the lines of [41][42][43][44].