# Del Pezzo Singularities and SUSY Breaking

Authors: Dmitry Malyshev
Journal: Advances in High Energy Physics
 (2011)

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## Abstract

An analytic construction of compact Calabi-Yau manifolds with del Pezzo singularities
is found. We present complete intersection CY manifolds for all del Pezzo singularities
and study the complex deformations of these singularities. An example of the quintic CY
manifold with del Pezzo 6 singularity and some number of conifold singularities is studied
in detail. The possibilities for the ‘‘geometric’’ and ISS mechanisms of dynamical SUSY
breaking are discussed. As an example, we construct the ISS vacuum for the del Pezzo 6
singularity.


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## Body

## 1. Motivation

Recently, there has been a substantial progress in Model building involving the D-branes at the singularities of noncompact Calabi-Yau manifolds. On the one hand, the singularities provide enough flexibility to find phenomenologically acceptable extensions of the Standard Model [1, 2] and solve some problems such as finding metastable susy breaking vacua [3, 4]. On the other hand, the presence of the singularity eliminates certain massless moduli, such as the adjoint fields on the branes wrapping rigid cycles [1, 5].The main purpose of this paper is to study the del Pezzo and conifold singularities on compact CY manifolds that may be useful for the compactifications of dynamical SUSY breaking mechanisms. The stringy reallizations of metastable SUSY breaking vacua have been known for some time [6, 7]. We will focus on the two recent approaches to the dynamical SUSY breaking: on the “geometrical” approach of [8, 9] and on the ISS construction [10]. One of the main goals will be to study the topological conditions for the compactification of the above constructions.An important topological property of “geometrical” mechanism is the presence of several homologous rigid two-cycles. This is not difficult to achieve in the case of conifold singularities. For example, in the geometric transitions on compact CY manifolds [11, 12], several conifolds may be resolved by a single Kahler modulus, that is, the two-cycles at the tip of these conifolds are homologous to each other. However, this is not always true for the del Pezzo singularities, that is, the two-cycles in the resolution of del Pezzo singularity may have no homologous rigid two-cycles on the compact CY. In the paper, we explicitly construct a compact CY manifold with del Pezzo 6 singularity and a number of conifolds such that some two-cycles on the del Pezzo are homologous to the two-cycles of the conifolds. This construction opens up the road for the generalization of geometrical SUSY breaking in the case of del Pezzo singularities, where one may hope to use the richness of deformations of these singularity for phenomenological applications.A more direct way towards phenomenology is provided by the ISS mechanism. The realization of ISS construction for del Pezzo 5 and 8 singularities was considered in [4]. As an example, we find an ISS vacuum for the del Pezzo 6 singularity. The del Pezzo 6 surface can be embedded in ℙ3 by a degree 3 polynomial. This is one of the most simple analytical representations of del Pezzo surfaces, which enables us to find an analytical embedding of the corresponding del Pezzo 6 singularity in a compact Calabi Yau manifold, the quintic CY embedded in ℙ4 by a degree 5 polynomial.A nice feature of the del Pezzo singularities is that they are isolated. Thus, the fractional branes, that one typically introduces in these models, are naturally stabilized against moving away from the singularity. But, for example, in the models involving quotients of conifolds [3, 13], the singularities are not isolated and one needs to pay special attention to stabilize the fractional branes against moving along the singular curves.Apart from the application to SUSY breaking, the construction of compact CY manifolds with del Pezzo singularities may be useful for the study of deformations of these singularities. In particular, we will be interested in the D-brane interpretation of deformations.In general, a singularity can be smoothed out in two different ways, it can be either deformed or resolved (blown up). The former corresponds to the deformations of the complex structure, described by the elements ofH2,1; the latter corresponds to Kähler deformations given by the elements of H1,1 [14–16]. In terms of the cycles, the resolution corresponds to blowing up some two-cycles (four-cycles), while the complex deformations correspond to the deformations of the three-cycles. For example, the conifold can be either deformed by placing an S3 at the tip of the conifold or resolved by placing an S2 [17]. The process where some three-cycles shrink to form a singularity and after that the singularity is blown up is called the geometric transition [11, 12]. For the conifold, the geometric transition has a nice interpretation in terms of the branes. The deformation of the conifold is induced by wrapping the D5-branes around the vanishing S2 at the tip [18]. The resolution of the conifold corresponds to giving a vev to a baryonic operator, that can be interpreted in terms of the D3-branes wrapping the vanishing S3 at the tip of the conifold [19].The example of the conifold encourages to conjecture that any geometric transition can be interpreted in terms of the branes. The nonanomalous (fractional) branes produce the fluxes that deform the three-cycles. The massless/tensionless branes correspond to baryonic operators whose vevs are interpreted as the blow-up modes.However, there are a few puzzles with the above interpretation. In some cases, there are less deformations than nonanomalous fractional branes; in the other cases there are deformations but no fractional branes, The quiver gauge theory on the del Pezzo 1 singularity has a nonanomalous fractional brane; moreover, it has a cascading behavior [20] similar to the conifold cascade. But it is known that there are no complex deformations of the cone over dP1 [21–23]. The relevant observation [24] is that there are no geometric transitions for the cone over dP1. From the point of view of gauge theory, there is a runaway behavior at the bottom of the cascade and no finite vacuum [25].On the other side of the puzzle, there are more complex deformations of higher del Pezzo singularities, than there are possible fractional branes. It is known that the cone over del Pezzon surface has c∨(En)-1 complex deformations [24], where c∨(En) is the dual Coxeter number of the corresponding Lie group. For instance, the cone over dP8 has 29 deformations. But there are only 8 nonanomalous combinations of fractional branes [1].We believe that these puzzles can be managed more effectively if there were more examples of compact CY manifolds with local del Pezzo singularities. The advantage of working with compact manifolds is that they have finite a number of deformations and well-defined cohomology (there are no noncompact cycles).The organization of the paper is as follows. In Section2, we study the singularities on compact CY manifolds using the quintic CY manifold as an example. We restrict our attention to isolated singularities that admit crepant resolution, that is, their resolution does not affect the CY condition. There are two types of primitive isolated singularities on CY 3-folds: small contractions or conifold singularities, and del Pezzo singularities [26, 27]. We will study the example of del Pezzo 6 singularity and some number of conifolds on the quintic. The presence of conifold singularities is important if we want to put fractional branes at the del Pezzo singularity. Without conifolds, the nonanomalous two-cycles on del Pezzo (i.e., the ones that do not intersect the canonical class) are trivial within the CY manifold. It is impossible to put the fractional branes on such “cycles”, because the corresponding RR fluxes have “nowhere to go.” In the presence of conifolds, some of the two-cycles on del Pezzo may become homologous to the two-cycles of the conifolds (this will be the case in our example). Then we can put some number of D5-branes on the two-cycles of del Pezzo and some number of anti-D5-branes on the two-cycles of the conifolds. Such configuration of branes and antibranes is a first step in the geometrical SUSY breaking [8, 28]. Also the possibility to introduce the fractional branes will be crucial for the D-brane realizations of ISS construction.In Section3, we discuss the compactification of the geometrical SUSY breaking and the ISS model and find an ISS SUSY breaking vacuum in a quiver gauge theory for the dP6 singularity.In Section4, we formulate the general construction of compact CY manifolds with del Pezzo singularities and discuss the complex deformations of these singularities. We observe that the number of deformations depends on the global properties of the two-cycles on del Pezzo that do not intersect the canonical class and have self-intersection (−2). Suppose all such cycles are trivial within the CY, then the singularity has the maximal number of deformations. This will be the case for our embeddings of del Pezzo 5, 6, 7, and 8 singularities and for the cone over ℙ1×ℙ1. In the case of dP0=ℙ2 and dP1 singularities, we do not expect to find any deformations. In the case of del Pezzo 2, 3, and 4, our embedding leaves some of the (−2) two-cycles nontrivial within the CY; accordingly, we find less complex deformations. This result can be expected, since it is known that the del Pezzo singularities for n≤4 in general cannot be represented as complete intersections [27, 29]. In our case, the del Pezzo singularities are complete intersections but they are not generic. Specific equations for embedding of del Pezzo singularities and their deformations are provided in the appendix.Section5 contains discussion and conclusions.


## 2. Del Pezzo 6 and Conifold Singularities on the Quintic CY

The CY manifolds can have two types of primitive isolated singularities: conifold singularities and del Pezzo singularities [26, 27]. Correspondingly, we will have two types of geometric transitions.  (1)
Type I, or conifold transitions: severalℙ1’s shrink to form conifold singularities and then these singularities are deformed.(2)
Type II, or del Pezzo transition: a del Pezzo shrinks to a point and the corresponding singularity is deformed.In order to illustrate the geometric transitions, we will study a particular example of transitions on the quintic CY. The example is summarized in the diagram in Figure1. The type I transitions are horizontal, whereas the type II transitions are vertical. It is known [24] that the maximal number of deformations of a cone over dP6 is c∨(E6)-1=11, where c∨(E6)=12 is the dual Coxeter number of E6. Going along the left vertical arrow we recover all complex deformations of the cone over dP6. In this case, all the two-cycles that do not intersect the canonical class on dP6 are trivial within the CY.Figure 1
Possible geometric transitions of quintic CY. The numbers in parentheses denote the dimensions(h1,1,h2,1).For the CY with both del Pezzo and conifold singularities, the deformation of the del Pezzo singularity has only 7 parameters (right vertical arrow). The del Pezzo surface is not generic in this case. It has a two-cycle that is nontrivial within the full CY and does not intersect the canonical class inside del Pezzo. As a general rule, the existence of nontrivial two-cycles reduces the number of possible complex deformations.The horizontal arrows represent the conifold transitions. In our example, we have 36 conifold singularities on the quintic CY. These singularities have 35 complex deformations. In the presence ofdP6 singularity, there will be only 32 conifolds that have, respectively, 31 complex deformations. (It may seem puzzling that we need exactly 36 or 32 conifolds. One can easily find the examples of quintic CY with fewer conifold singularities. But it is impossible to blow up these singularities unless we have a specific number of them at specific locations. In the example considered in [11, 12], the quintic CY has 16 conifolds placed at a ℙ2 inside the CY.)In general, the del Pezzo singularity and the conifold singularities are away from each other but they still affect the number of complex deformations, that is, the presence of conifolds reduces the number of deformations of del Pezzo singularity and vice versa. The diagram in Figure1 is commutative, and the total number of complex deformations of the CY with the del Pezzo singularity and 32 conifold singularities is 42. But the interpretation of these deformations changes whether we first deform the del Pezzo singularity or we first deform the conifold singularities.Before we go to the calculations, let us clarify what we mean by the deformations of the del Pezzo singularity. We will distinguish three kinds of deformations. The deformations of the shape of the cone, the deformations of the blown up del Pezzo with fixed canonical class and deformations that smooth out the singularity.The first kind of deformations corresponds to the general deformations of del Pezzo surface at the base of the cone. Recall that thedPn surface for n>4 has 2n-8 deformations that parameterize the superpotential of the corresponding quiver gauge theory [5].The second kind of deformations is obtained by blowing up the singularity and fixing the canonical class on the del Pezzo. In this case, the deformations of del Pezzon surface can be described as the deformations of En singularity on the del Pezzo [30]. The deformations of this singularity have n parameters, corresponding to the n two-cycles that do not intersect the canonical class. Note that the intersection matrix of these two-cycles is (minus) the Cartan matrix of En. The En singularity on the del Pezzo is an example of du Val surface singularity [31] (also known as an ADE singularity or a Kleinian singularity). A three-dimensional singularity that has a du Val singularity in a hyperplane section is called compound du Val (cDV) [26, 31]. The conifold is an example of cDV singularity since it has the A1 singularity in a hyperplane section. The generalized conifolds [32, 33] also have an ADE singularity in a hyperplane section, that is, from the 3-dimensional point of view they correspond to some cDV singularities. In terms of the large N gauge/string duality, the deformation of the En generalized conifold singularity corresponds to putting some combination of fractional branes on the zero size two-cycles at the singularity. Hence, the deformtion of cDV singularity that restricts to En singularity on the del Pezzo can be considered as a generalized type I transition.We will be mainly interested in the the third type of deformations that correspond to smoothing of del Pezzo singularities. These deformations make the canonical class of del Pezzo surface trivial within the CY. If we put some number of nonanomalous fractional D-branes at the singularity, then the corresponding geometric transition smooths the singularity [24]. But not all the deformations can be described in this way.In order to get some intuition about possible interpretations of these deformations, we will consider the del Pezzo 6 singularity. It is known that thedP6 singularity has 11 complex deformations [21, 34] but there are only 6 nonanomalous fractional branes in the corresponding quiver gauge theory and there are only 6 two-cycles that do not intersect the canonical class [24]. It will prove helpful to start with a quintic CY that has 36 conifold singularities. The del Pezzo 6 singularity can be obtained by merging four conifolds at one point. There are 7 deformations of del Pezzo 6 singularity that separate these four conifolds (right vertical arrow). The remaining 4 deformations of dP6 cone correspond to 4 deformations of the four “hidden” conifolds at the singularity. Note that the total number of deformations is 11 (left vertical arrow).


### 2.1. Quintic CY

The description of the quintic CY is well known [16]. Here, we repeat it in order to recall the methods [16] of finding the topology and deformations that we use later in more difficult situations.The quintic CY manifoldY3 is given by a degree five equation in ℙ4(2.1)Q5(zi)=0,
where (z0,z1,z2,z3,z4)∈ℙ4. The total Chern class of this manifold is
(2.2)c(Y3)=(1+H)51+5H=1+10H2-40H3
and the first Chern class c1(Y3)=0.Let us calculate the number of complex deformations. The complex structures are parameterized by the coefficients in (2.1) up to the change of coordinates in ℙ4. The number of coefficients in a homogeneous polynomial of degree n in k variables is
(2.3)(n+k-1n)=(n+k-1)!n!(k-1)!.
In the case of the quintic in ℙ4, the number of coefficients is
(2.4)(95)=9!5!4!=126.
The number of reparametrizations of ℙ4 is equal to dimGl(5)=25. Thus, the dimension of the space of complex deformations is 101.The number of complex deformations of CY threefolds is equal to the dimension ofH2,1 cohomology group
(2.5)h2,1=h1,1-χ2,
where h1,1 can be found via the Lefschetz hyperplane theorem [16, 35]
(2.6)h1,1(Y3)=h1,1(P4)=1
and the Euler characteristic is given by the integral of the highest Chern class over Y3(2.7)χ=∫Y3c3=∫P4-40H3∧5H=-200,
here, we have used that 5H is the Poincare dual class to Y3 inside ℙ4. Consequently, h2,1=101 which is consistent with the number of complex deformations found before.


### 2.2. Quintic CY withdP6 Singularity

Suppose that the quintic equation is not generic but has a degree three zero at the point(w0,w1,w2,w3,w4)=(0,0,0,0,1),
(2.8)P3(w0,…,w3)w42+P4(w0,…,w3)w4+P5(w0,…,w3)=0,
where Pn’s denote degree n polynomials. The shape of the singularity is determined by P3(w0,…,w3) (we will see that this polynomial defines the del Pezzo at the tip of the cone). The deformations that smooth out the singularity correspond to adding less singular terms to (2.8), that is, the terms that have bigger powers of w4.The resolution of the singularity in (2.8) can be obtained by blowing up the point (0,0,0,0,1)∈ℙ4. Away from the blowup, we can use the following coordinates on ℙ4:
(2.9)(w0,…,w3,w4)=(tz0,…,tz3,s),
where (s,t)∈ℙ1 and (z0,…,z3)∈ℙ3. The blowup of the point at t=0 corresponds to inserting the ℙ3 instead of this point. Hence, the points on the blown up ℙ4 can be parameterized globally by (z0,…,z3)∈ℙ3 and (s,t)∈ℙ1. The projective invariance (s,t)~(λs,λt) corresponds to the projective invariance in the original ℙ4. In order to compensate for the projective invariance of ℙ3, we need to assume that locally the coordinates on ℙ1 belong to the following line bundles over ℙ3, s∈𝒪 and t∈𝒪(-H). Thus, the blowup of ℙ4 at a point is a ℙ1 bundle over ℙ3 obtained by projectivization of the direct sum of 𝒪ℙ3 and 𝒪ℙ3(-H) bundles, ℙ̃4=P(𝒪ℙ3⊕𝒪ℙ3(-H)) (for more details on projective bundles see, e.g., [36, 37]). In working with projective bundles, we will use the technics similar to [37].Using parametrization (2.9), we can write the equation on the blown up ℙ4 as
(2.10)P3(z0,…,z3)s2+P4(z0,…,z3)st+P5(z0,…,z3)t2=0.
This equation is homogeneous of degree two in the coordinates on ℙ1 and degree three in the zi’s. Note that t∈𝒪(-H), that is, it has degree (-1) in the zi’s and s∈𝒪 has degree zero.Let us prove that the manifold defined by (2.10) has vanishing first Chern class, that is, it is a CY manifold. Let H be the hyperplane class in ℙ3 and G the hyperplane class on the ℙ1 fibers. Let M=P(𝒪ℙ3⊕𝒪ℙ3(-H)) denote the ℙ1 bundle over ℙ3. The total Chern class of M is
(2.11)c(M)=(1+H)4(1+G)(1+G-H),
where (1+H)4 is the total Chern class of ℙ3, (1+G) corresponds to s∈𝒪ℙ3, and (1+G-H) corresponds to t∈𝒪ℙ3(-H). Note that G(G-H)=0 on this ℙ1 bundle and, as usual, H4=0 on the ℙ3.LetY3 denote the surface embedded in M by (2.10). Since the equation has degree 3 in zi and degree two in (s,t), the class Poincare dual to Y3⊂M is 3H+2G and the total Chern class is
(2.12)c(Y3)=(1+H)4(1+G)(1+G-H)1+3H+2G.
Expanding c(Y3), it is easy to check that c1(Y3)=0.The intersection ofY3 with the blown up ℙ3 at t=0 is given by the degree three equation P3(z0,…,z3)=0 in ℙ3. The surface B defined by this equation is the del Pezzo 6 surface [16, 35]. The total Chern class and the Euler character of B(2.13)c(B)=(1+H)41+3H=1+H+3H2,(2.14)χ(B)=∫Bc2(B)=∫P33H2∧3H=9.
In the calculation of χ(B), we have used that 3H is the Poincare dual class to B inside ℙ3.It is known that the normal bundle to contractable del Pezzo in a CY manifold is the canonical bundle on del Pezzo [38]. Let us check this statement in our example. The canonical class is minus the first Chern class that can be found from (2.13) (Slightly abusing the notations, we denote by H both the class of ℙ3 and the restriction of this class to B∈ℙ3.)
(2.15)K(B)=-H.
The coordinate t describes the normal direction to B inside Y3. Since t∈𝒪ℙ3(-H), restricting to B we find that t belongs to the canonical bundle over B. Hence locally, near t=0, the CY threefold Y3 has the structure of the CY cone over the del Pezzo 6 surface.The smoothing of the singularity corresponds to adding less singular terms in (2.8). These terms have 15 parameters, but also we get back 4 reparametrizations (now, we can add w4 to the other coordinates). Hence, smoothing of the singularity corresponds to 11 complex structure deformations that is the maximal expected number of deformations of dP6 singularity.In view of applications in Section4, let us describe the geometric transition between the CY with the resolved dP6 singularity and a smooth quintic CY in more details. As we have shown above, the CY with the blown up dP6 singularity can be described by the following equation in the ℙ1 bundle over ℙ3:
(2.16)P3(z0,…,z3)s2+P4(z0,…,z3)st+P5(z0,…,z3)t2=0.
This equation can be rewritten as
(2.17)P3(tz0,…,tz3)s2+P4(tz0,…,tz3)s+P5(tz0,…,tz3)=0.
Next, we note that, being a projective bundle, M is equivalent [35, 36] to P(𝒪ℙ3(H)⊕𝒪ℙ3), where locally s and t are sections of 𝒪ℙ3(H) and 𝒪ℙ3, respectively. We further observe that tzi, i=0⋯3 are also sections of 𝒪ℙ3(H) and the equivalence (t,s)~(λt,λs) induces the equivalence (tz0,…,tzi,s)~(λtz0,…,λtzi,λs). Consequently, if we blow down the section t=0 of M, then (tz0,…,tzi,s)∈ℙ4. Now, we define (w0,…,w3,w4)=(tz0,…,tz3,s) and rewrite (2.17) as
(2.18)P3(w0,…,w3)w42+P4(w0,…,w3)w4+P5(w0,…,w3)=0.
Not surprisingly, we get back (2.8).Above we have found that there are 11 complex deformations of thedP6 singularity embedded in the quintic CY manifold. In the view of further applications, let us rederive the number of complex deformations by calculating the dimension of H2,1.Expanding (2.12), we get the third Chern class
(2.19)c3(Y3)=-2G3-13HG2-17H2G-8H3.
The Poincare dual class to Y3∈M is 3H+2G and
(2.20)χ(Y3)=∫Y3c3(Y3)=∫Mc3(Y3)∧(3H+2G).
In calculating this integral, one needs to take into account that G(G-H)=0 on M. Finally, we get
(2.21)χ(Y3)=-176,h2,1=h1,1-χ2=90.
The number of complex deformations of the del Pezzo singularity is 101-90=11, which is consistent with the number found above.


### 2.3. Quintic CY with 36 Conifold Singularities

In this subsection, we use the methods of geometric transitions [11, 12, 16] to find the quintic CY with conifold singularities, that is, we describe the upper horizontal arrow in Figure 1. Consider the system of two equations in ℙ4×ℙ1(2.22)P3u+R3v=0,P2u+R2v=0,
where (u,v)∈ℙ1 and Pn, Rn denote polynomials of degree n in ℙ4.Suppose that at least one of the polynomialsP3, R3, P2, and R2 is nonzero, then we can solve for u, v and substitute in the second equation, where we get
(2.23)P3R2-R3P2=0,
a nongeneric quintic in ℙ4. The points where P3=R3=P2=R2=0 (but otherwise generic) have conifold singularities. There are 3·3·2·2=36 such points. The system (2.22) describes the blowup of the singularities, since every singular point is replaced by the ℙ1 and the resulting manifold is non singular.LetH be the hyperplane class of ℙ4 and G by the hyperplane class of ℙ1, then the total Chern class of Y3 is
(2.24)c=(1+H)5(1+G)2(1+3H+G)(1+2H+G),
since c1=0, Y3 is a CY.By Lefschetz hyperplane theoremh1,1(Y3)=h1,1(ℙ4×ℙ1)=2, there are only two independent Kahler deformations in Y3. One of them is the overall size of Y3 and the other is the size of the blown up ℙ1’s. Thus, the 36 ℙ1’s are not independent but homologous to each other and represent only one class in H2(Y3). If we shrink the size of blown up ℙ1’s to zero, then we can deform the singularities of (2.23) to get a generic quintic CY. In this case, the 35 three chains that where connecting the 36 ℙ1’s become independent three cycles. Thus, we expect the general quintic CY to have 35 more complex deformations than the quintic with 36 conifold singularities.Calculating the Euler character similarly to the previous subsections, we find(2.25)h2,1=66.
Recall that the smooth quintic has 101 complex deformations. Thus, the quintic with 36 conifold singularities has 101-66=35 less complex deformations than the generic one.


### 2.4. Quintic CY withdP6 Singularity and 32 Conifold Singularities

The equation for the quintic CY manifold with the blown updP6 singularity was found in (2.10). Here, we reproduce it for convenience
(2.26)P3(zi)s2+P4(zi)st+P5(zi)t2=0.
This equation describes an embedding of the CY manifold in the ℙ1 bundle M=P(𝒪ℙ3⊕𝒪ℙ3(-H)). As before, (z0,…,z3)∈ℙ3 and (s,t) are the coordinates on the ℙ1 fibers over ℙ3.In order to have more Kahler deformations, we need to embed (2.26) in a space with more independent two-cycles. For example, we can consider a system of two equations in the product (ℙ1 bundle over ℙ3) × ℙ1(2.27)(P1s+P2t)u+(Q1s+Q2t)v=0,(R2s+R3t)u+(S2s+S3t)v=0,
where (u,v) are the coordinates on the additional ℙ1. Let G, H, and K be the hyperplane classes on the ℙ1 fibers, on the ℙ3, and on the additional ℙ1, respectively. Then, the total Chern class of Y3 is
(2.28)c=(1+H)4(1+G)(1+G-H)(1+K)2(1+H+G+K)(1+2H+G+K),
and it is easy to see that the first Chern class is zero.For generic points on theℙ1 bundle over ℙ3, at least one of the functions in front of u or v is nonzero. Thus, we can find a point (u,v) and substitute it in the second equation, which becomes a nongeneric equation similar to (2.26)
(2.29)(P1S2-Q1R2)s2+(P1S3+P2S2-Q1R3-Q2R2)st+(P2S3-Q2R3)t2=0.The CY manifold defined in (2.27) has the following characteristics:
(2.30)χ=∫Y3c3=-112,h1,1=3,h2,1=h1,1-χ2=59.
Recall that the number of complex deformations on the quintic with the del Pezzo 6 singularity is 90. Since we lose 31 complex deformations, we expect that the corresponding three-cycles become the three chains that connect 32 ℙ1’s at the blowups of the singularities in (2.29). These singularities occur when all four equations in (2.27) vanish
(2.31)R2s+R3t=0,S2s+S3t=0,P1s+P2t=0,Q1s+Q2t=0.
The number of solutions equals the number of intersections of the corresponding classes ∫M(2H+G)2(H+G)=32, where M is the ℙ1 bundle over ℙ3 and G(G-H)=0.The right vertical arrow corresponds to smoothing of del Pezzo singularity in the presence of conifold singularities. Before the transition, the CY hash2,1=59 deformations and after the transition it has h2,1=66 deformations. Hence, the number of complex deformations of dP6 singularity is 66-59=7 which is less than c∨(E6)-1=11. This is related to the fact that the del Pezzo at the tip of the cone is not generic. The equation of the del Pezzo can be found by restricting (2.27) to t=0, s=1 section
(2.32)P1u+Q1v=0,R2u+S2v=0.
This del Pezzo contains a two-cycle α that is nontrivial within the full CY and does not intersect the canonical class inside dP6.In the rest of this subsection, we will argue thatα is homologous to four ℙ1’s at the tip of the conifolds. The heuristic argument is the following. The formation of dP6 singularity on the CY manifold with 36 conifolds reduces the number of conifolds to 32. Let us show that the deformation of the del Pezzo singularity that preserves the conifold singularities corresponds to separating 4 conifolds hidden in the del Pezzo singularity. The CY that has a dP6 singularity and 32 resolved conifolds can be found from (2.27) by the following coordinate redefinition (w0,…,w3,w4)=(tz0,…,tz3,s) (compare to the discussion after (2.17)):(2.33)(P1w4+P2)u+(Q1w4+Q2)v=0,(R2w4+R3)u+(S2w4+S3)v=0.
If we blow down the ℙ1, then we get the quintic CY with 32 conifold singularities and a dP6 singularity. For a finite size ℙ1, the conifold singularities and one of the two-cycles in the dP6 are blown up. The deformations of dP6 singularity correspond to adding terms with higher power of w4. After the deformation, the degree two zeros of R2 and S2 will split into four degree one zeros that correspond to the four conifolds “hidden” in the dP6 singularity. The blown up two-cycle of dP6 is homologous to the two-cycles on the four conifolds. (Formally, we can prove this by calculating the corresponding Poincaré dual classes. The Poincaré dual of ℙ1 on the blown up conifold is H3G—this is the ℙ1 parameterized by (u,v). The Poincaré dual of the canonical class on dP6 is (G-H)(H+K)(2H+K)(-H), where (G-H) restricts to t=0 section of the ℙ1 bundle, (H+K)(2H+K) restricts to dP6 in (2.32), while the restriction of (-H) is the canonical class on dP6 (see (2.15)). The class that does not intersect (-H) inside dP6 is dual to (G-H)(H+K)(2H+K)(2H-3G)=4H3G, q.e.d.)


## 2.1. Quintic CY

The description of the quintic CY is well known [16]. Here, we repeat it in order to recall the methods [16] of finding the topology and deformations that we use later in more difficult situations.The quintic CY manifoldY3 is given by a degree five equation in ℙ4(2.1)Q5(zi)=0,
where (z0,z1,z2,z3,z4)∈ℙ4. The total Chern class of this manifold is
(2.2)c(Y3)=(1+H)51+5H=1+10H2-40H3
and the first Chern class c1(Y3)=0.Let us calculate the number of complex deformations. The complex structures are parameterized by the coefficients in (2.1) up to the change of coordinates in ℙ4. The number of coefficients in a homogeneous polynomial of degree n in k variables is
(2.3)(n+k-1n)=(n+k-1)!n!(k-1)!.
In the case of the quintic in ℙ4, the number of coefficients is
(2.4)(95)=9!5!4!=126.
The number of reparametrizations of ℙ4 is equal to dimGl(5)=25. Thus, the dimension of the space of complex deformations is 101.The number of complex deformations of CY threefolds is equal to the dimension ofH2,1 cohomology group
(2.5)h2,1=h1,1-χ2,
where h1,1 can be found via the Lefschetz hyperplane theorem [16, 35]
(2.6)h1,1(Y3)=h1,1(P4)=1
and the Euler characteristic is given by the integral of the highest Chern class over Y3(2.7)χ=∫Y3c3=∫P4-40H3∧5H=-200,
here, we have used that 5H is the Poincare dual class to Y3 inside ℙ4. Consequently, h2,1=101 which is consistent with the number of complex deformations found before.


## 2.2. Quintic CY withdP6 Singularity

Suppose that the quintic equation is not generic but has a degree three zero at the point(w0,w1,w2,w3,w4)=(0,0,0,0,1),
(2.8)P3(w0,…,w3)w42+P4(w0,…,w3)w4+P5(w0,…,w3)=0,
where Pn’s denote degree n polynomials. The shape of the singularity is determined by P3(w0,…,w3) (we will see that this polynomial defines the del Pezzo at the tip of the cone). The deformations that smooth out the singularity correspond to adding less singular terms to (2.8), that is, the terms that have bigger powers of w4.The resolution of the singularity in (2.8) can be obtained by blowing up the point (0,0,0,0,1)∈ℙ4. Away from the blowup, we can use the following coordinates on ℙ4:
(2.9)(w0,…,w3,w4)=(tz0,…,tz3,s),
where (s,t)∈ℙ1 and (z0,…,z3)∈ℙ3. The blowup of the point at t=0 corresponds to inserting the ℙ3 instead of this point. Hence, the points on the blown up ℙ4 can be parameterized globally by (z0,…,z3)∈ℙ3 and (s,t)∈ℙ1. The projective invariance (s,t)~(λs,λt) corresponds to the projective invariance in the original ℙ4. In order to compensate for the projective invariance of ℙ3, we need to assume that locally the coordinates on ℙ1 belong to the following line bundles over ℙ3, s∈𝒪 and t∈𝒪(-H). Thus, the blowup of ℙ4 at a point is a ℙ1 bundle over ℙ3 obtained by projectivization of the direct sum of 𝒪ℙ3 and 𝒪ℙ3(-H) bundles, ℙ̃4=P(𝒪ℙ3⊕𝒪ℙ3(-H)) (for more details on projective bundles see, e.g., [36, 37]). In working with projective bundles, we will use the technics similar to [37].Using parametrization (2.9), we can write the equation on the blown up ℙ4 as
(2.10)P3(z0,…,z3)s2+P4(z0,…,z3)st+P5(z0,…,z3)t2=0.
This equation is homogeneous of degree two in the coordinates on ℙ1 and degree three in the zi’s. Note that t∈𝒪(-H), that is, it has degree (-1) in the zi’s and s∈𝒪 has degree zero.Let us prove that the manifold defined by (2.10) has vanishing first Chern class, that is, it is a CY manifold. Let H be the hyperplane class in ℙ3 and G the hyperplane class on the ℙ1 fibers. Let M=P(𝒪ℙ3⊕𝒪ℙ3(-H)) denote the ℙ1 bundle over ℙ3. The total Chern class of M is
(2.11)c(M)=(1+H)4(1+G)(1+G-H),
where (1+H)4 is the total Chern class of ℙ3, (1+G) corresponds to s∈𝒪ℙ3, and (1+G-H) corresponds to t∈𝒪ℙ3(-H). Note that G(G-H)=0 on this ℙ1 bundle and, as usual, H4=0 on the ℙ3.LetY3 denote the surface embedded in M by (2.10). Since the equation has degree 3 in zi and degree two in (s,t), the class Poincare dual to Y3⊂M is 3H+2G and the total Chern class is
(2.12)c(Y3)=(1+H)4(1+G)(1+G-H)1+3H+2G.
Expanding c(Y3), it is easy to check that c1(Y3)=0.The intersection ofY3 with the blown up ℙ3 at t=0 is given by the degree three equation P3(z0,…,z3)=0 in ℙ3. The surface B defined by this equation is the del Pezzo 6 surface [16, 35]. The total Chern class and the Euler character of B(2.13)c(B)=(1+H)41+3H=1+H+3H2,(2.14)χ(B)=∫Bc2(B)=∫P33H2∧3H=9.
In the calculation of χ(B), we have used that 3H is the Poincare dual class to B inside ℙ3.It is known that the normal bundle to contractable del Pezzo in a CY manifold is the canonical bundle on del Pezzo [38]. Let us check this statement in our example. The canonical class is minus the first Chern class that can be found from (2.13) (Slightly abusing the notations, we denote by H both the class of ℙ3 and the restriction of this class to B∈ℙ3.)
(2.15)K(B)=-H.
The coordinate t describes the normal direction to B inside Y3. Since t∈𝒪ℙ3(-H), restricting to B we find that t belongs to the canonical bundle over B. Hence locally, near t=0, the CY threefold Y3 has the structure of the CY cone over the del Pezzo 6 surface.The smoothing of the singularity corresponds to adding less singular terms in (2.8). These terms have 15 parameters, but also we get back 4 reparametrizations (now, we can add w4 to the other coordinates). Hence, smoothing of the singularity corresponds to 11 complex structure deformations that is the maximal expected number of deformations of dP6 singularity.In view of applications in Section4, let us describe the geometric transition between the CY with the resolved dP6 singularity and a smooth quintic CY in more details. As we have shown above, the CY with the blown up dP6 singularity can be described by the following equation in the ℙ1 bundle over ℙ3:
(2.16)P3(z0,…,z3)s2+P4(z0,…,z3)st+P5(z0,…,z3)t2=0.
This equation can be rewritten as
(2.17)P3(tz0,…,tz3)s2+P4(tz0,…,tz3)s+P5(tz0,…,tz3)=0.
Next, we note that, being a projective bundle, M is equivalent [35, 36] to P(𝒪ℙ3(H)⊕𝒪ℙ3), where locally s and t are sections of 𝒪ℙ3(H) and 𝒪ℙ3, respectively. We further observe that tzi, i=0⋯3 are also sections of 𝒪ℙ3(H) and the equivalence (t,s)~(λt,λs) induces the equivalence (tz0,…,tzi,s)~(λtz0,…,λtzi,λs). Consequently, if we blow down the section t=0 of M, then (tz0,…,tzi,s)∈ℙ4. Now, we define (w0,…,w3,w4)=(tz0,…,tz3,s) and rewrite (2.17) as
(2.18)P3(w0,…,w3)w42+P4(w0,…,w3)w4+P5(w0,…,w3)=0.
Not surprisingly, we get back (2.8).Above we have found that there are 11 complex deformations of thedP6 singularity embedded in the quintic CY manifold. In the view of further applications, let us rederive the number of complex deformations by calculating the dimension of H2,1.Expanding (2.12), we get the third Chern class
(2.19)c3(Y3)=-2G3-13HG2-17H2G-8H3.
The Poincare dual class to Y3∈M is 3H+2G and
(2.20)χ(Y3)=∫Y3c3(Y3)=∫Mc3(Y3)∧(3H+2G).
In calculating this integral, one needs to take into account that G(G-H)=0 on M. Finally, we get
(2.21)χ(Y3)=-176,h2,1=h1,1-χ2=90.
The number of complex deformations of the del Pezzo singularity is 101-90=11, which is consistent with the number found above.


## 2.3. Quintic CY with 36 Conifold Singularities

In this subsection, we use the methods of geometric transitions [11, 12, 16] to find the quintic CY with conifold singularities, that is, we describe the upper horizontal arrow in Figure 1. Consider the system of two equations in ℙ4×ℙ1(2.22)P3u+R3v=0,P2u+R2v=0,
where (u,v)∈ℙ1 and Pn, Rn denote polynomials of degree n in ℙ4.Suppose that at least one of the polynomialsP3, R3, P2, and R2 is nonzero, then we can solve for u, v and substitute in the second equation, where we get
(2.23)P3R2-R3P2=0,
a nongeneric quintic in ℙ4. The points where P3=R3=P2=R2=0 (but otherwise generic) have conifold singularities. There are 3·3·2·2=36 such points. The system (2.22) describes the blowup of the singularities, since every singular point is replaced by the ℙ1 and the resulting manifold is non singular.LetH be the hyperplane class of ℙ4 and G by the hyperplane class of ℙ1, then the total Chern class of Y3 is
(2.24)c=(1+H)5(1+G)2(1+3H+G)(1+2H+G),
since c1=0, Y3 is a CY.By Lefschetz hyperplane theoremh1,1(Y3)=h1,1(ℙ4×ℙ1)=2, there are only two independent Kahler deformations in Y3. One of them is the overall size of Y3 and the other is the size of the blown up ℙ1’s. Thus, the 36 ℙ1’s are not independent but homologous to each other and represent only one class in H2(Y3). If we shrink the size of blown up ℙ1’s to zero, then we can deform the singularities of (2.23) to get a generic quintic CY. In this case, the 35 three chains that where connecting the 36 ℙ1’s become independent three cycles. Thus, we expect the general quintic CY to have 35 more complex deformations than the quintic with 36 conifold singularities.Calculating the Euler character similarly to the previous subsections, we find(2.25)h2,1=66.
Recall that the smooth quintic has 101 complex deformations. Thus, the quintic with 36 conifold singularities has 101-66=35 less complex deformations than the generic one.


## 2.4. Quintic CY withdP6 Singularity and 32 Conifold Singularities

The equation for the quintic CY manifold with the blown updP6 singularity was found in (2.10). Here, we reproduce it for convenience
(2.26)P3(zi)s2+P4(zi)st+P5(zi)t2=0.
This equation describes an embedding of the CY manifold in the ℙ1 bundle M=P(𝒪ℙ3⊕𝒪ℙ3(-H)). As before, (z0,…,z3)∈ℙ3 and (s,t) are the coordinates on the ℙ1 fibers over ℙ3.In order to have more Kahler deformations, we need to embed (2.26) in a space with more independent two-cycles. For example, we can consider a system of two equations in the product (ℙ1 bundle over ℙ3) × ℙ1(2.27)(P1s+P2t)u+(Q1s+Q2t)v=0,(R2s+R3t)u+(S2s+S3t)v=0,
where (u,v) are the coordinates on the additional ℙ1. Let G, H, and K be the hyperplane classes on the ℙ1 fibers, on the ℙ3, and on the additional ℙ1, respectively. Then, the total Chern class of Y3 is
(2.28)c=(1+H)4(1+G)(1+G-H)(1+K)2(1+H+G+K)(1+2H+G+K),
and it is easy to see that the first Chern class is zero.For generic points on theℙ1 bundle over ℙ3, at least one of the functions in front of u or v is nonzero. Thus, we can find a point (u,v) and substitute it in the second equation, which becomes a nongeneric equation similar to (2.26)
(2.29)(P1S2-Q1R2)s2+(P1S3+P2S2-Q1R3-Q2R2)st+(P2S3-Q2R3)t2=0.The CY manifold defined in (2.27) has the following characteristics:
(2.30)χ=∫Y3c3=-112,h1,1=3,h2,1=h1,1-χ2=59.
Recall that the number of complex deformations on the quintic with the del Pezzo 6 singularity is 90. Since we lose 31 complex deformations, we expect that the corresponding three-cycles become the three chains that connect 32 ℙ1’s at the blowups of the singularities in (2.29). These singularities occur when all four equations in (2.27) vanish
(2.31)R2s+R3t=0,S2s+S3t=0,P1s+P2t=0,Q1s+Q2t=0.
The number of solutions equals the number of intersections of the corresponding classes ∫M(2H+G)2(H+G)=32, where M is the ℙ1 bundle over ℙ3 and G(G-H)=0.The right vertical arrow corresponds to smoothing of del Pezzo singularity in the presence of conifold singularities. Before the transition, the CY hash2,1=59 deformations and after the transition it has h2,1=66 deformations. Hence, the number of complex deformations of dP6 singularity is 66-59=7 which is less than c∨(E6)-1=11. This is related to the fact that the del Pezzo at the tip of the cone is not generic. The equation of the del Pezzo can be found by restricting (2.27) to t=0, s=1 section
(2.32)P1u+Q1v=0,R2u+S2v=0.
This del Pezzo contains a two-cycle α that is nontrivial within the full CY and does not intersect the canonical class inside dP6.In the rest of this subsection, we will argue thatα is homologous to four ℙ1’s at the tip of the conifolds. The heuristic argument is the following. The formation of dP6 singularity on the CY manifold with 36 conifolds reduces the number of conifolds to 32. Let us show that the deformation of the del Pezzo singularity that preserves the conifold singularities corresponds to separating 4 conifolds hidden in the del Pezzo singularity. The CY that has a dP6 singularity and 32 resolved conifolds can be found from (2.27) by the following coordinate redefinition (w0,…,w3,w4)=(tz0,…,tz3,s) (compare to the discussion after (2.17)):(2.33)(P1w4+P2)u+(Q1w4+Q2)v=0,(R2w4+R3)u+(S2w4+S3)v=0.
If we blow down the ℙ1, then we get the quintic CY with 32 conifold singularities and a dP6 singularity. For a finite size ℙ1, the conifold singularities and one of the two-cycles in the dP6 are blown up. The deformations of dP6 singularity correspond to adding terms with higher power of w4. After the deformation, the degree two zeros of R2 and S2 will split into four degree one zeros that correspond to the four conifolds “hidden” in the dP6 singularity. The blown up two-cycle of dP6 is homologous to the two-cycles on the four conifolds. (Formally, we can prove this by calculating the corresponding Poincaré dual classes. The Poincaré dual of ℙ1 on the blown up conifold is H3G—this is the ℙ1 parameterized by (u,v). The Poincaré dual of the canonical class on dP6 is (G-H)(H+K)(2H+K)(-H), where (G-H) restricts to t=0 section of the ℙ1 bundle, (H+K)(2H+K) restricts to dP6 in (2.32), while the restriction of (-H) is the canonical class on dP6 (see (2.15)). The class that does not intersect (-H) inside dP6 is dual to (G-H)(H+K)(2H+K)(2H-3G)=4H3G, q.e.d.)


## 3. SUSY Breaking

In this paper, we compare two mechanisms for dynamical SUSY breaking: the “geometrical” approach of Aganagic et al. [8] and a more “physical” approach of ISS [10].In both approaches, there is a confinement in the microscopic gauge theory leading to the SUSY breaking in the effective theory. But the particular mechanisms and the effective theories are quite different. In the “geometrical” approach the effective theory is a non-SUSY analog of Veneziano-Yankielowicz superpotential [39] for the gaugino bilinear field S. This potential has an interpretation as the GVW superpotential [40] for the complex structure moduli of the CY manifold. The original Veneziano-Yankielowicz potential [39] is derived for the pure YM theory without any flavors. It has a number of isolated vacua and no massless fields. This is a nice feature for the (meta) stability of the vacuum but, since all the fields are massive, the applications of this potential in the low-energy effective theories are limited (see, e.g., the discussion in [41]).In the ISS construction, the number of flavors is bigger than the number of colorsNc<Nf<3/2Nc (and probably Nf=Nc). After the confinement, the low-energy effective theory contains classically massless fields that get some masses only at 1 loop. Hence, this theory is a more genuine effective theory but the geometric interpretation is harder to achieve [3]. Moreover, the geometric constructions similar to [3] generally have D5-branes wrapping vanishing cycles. In any compactification of these models, one has to put the O-planes or anti D5-branes somewhere else in the geometry, that is, the analysis of [8, 9] becomes inevitable.In summary, it seems that the ISS construction is more useful for immediate applications to SUSY breaking in the low-energy effective theories, whereas more global geometric analysis of [8, 9] becomes inevitable in the compactifications.In the previous section, we constructed the compact CY with del Pezzo 6 singularity and some number of conifold singularities. We have shown that it is possible to make some two-cycles on del Pezzo homologous to the two-cycles on the conifolds. This is the first step in the geometric analysis of [8]. In the next subsection, we show how the ISS story can be represented in the del Pezzo 6 quiver gauge theories.


### 3.1. ISS Vacuum for thedP6 Singularity

Consider the quiver gauge theory for the cone overdP6 represented in Figure 2. This quiver can be found by the standard methods [1] from the three-block exceptional collection of sheaves [42]. But, in order to prove the existence of this quiver, it is easier to do the Seiberg dualities on the nodes 4, 5, 6, and 1 and reduce it to the known dP6 quiver [2].Figure 2
Quiver gauge theory for the cone overdP6.In the compact CY manifold, one can put the D5-branes only on cycles that are nontrivial globally. A deformation of thedP6 singularity in (2.33) leaves four conifold singularities. We will assume that after joining the 4 conifolds to form a dP6 singularity the two-cycles remain nontrivial. We also expect that these two-cycles are represented by the four two-cycles on del Pezzo that have self-intersection (−2) and do not intersect with each other. Note that the total number of nonanomalous fractional branes and the number of (−2) two-cycles is 6, but there are only 4 two-cycles that do not intersect with each other and with the canonical class. (It is interesting to note the similarity between the branes wrapping the non intersecting cycles on dP6 and the deformation D-branes in [3, 23].)LetAi denote the two-cycle corresponding to the D5-brane charge [1] of the bound state of branes at the ith node in Figure 2. Then, the four non intersecting (−2) two-cycles can be chosen as A2-A3, A4-A5, A6-A7, and A8-A9. Now, we would like to add K fractional branes to A4-A5 and N fractional branes to A6-A7 and to A8-A9. The corresponding quiver is depicted in Figure 3.Figure 3
Quiver gauge theory for the cone overdP6 after adding the fractional branes.The gauge groups at the nodes 6 and 8 haveNf=Nc. Consider the Seiberg duality in the strong coupling limit of these gauge groups. The moduli space consists of the mesonic and the baryonic branches [43, 44]. Suppose we are on the baryonic branch. For the generic Yukawa couplings, the two mesons Φ=BC couple linearly to the fields A and become massive together with two of the A fields.An important question is whether the baryons for the gauge groups in nodes 6 and 8 remain massless. The baryons are charged under the baryonicU(1)B symmetries. In the noncompact setting, these U(1)B symmetries are global [45]. If the baryons get vevs, then the symmetries are broken spontaneously and there are massless goldston bosons. But for the compact CY manifold the U(1)B symmetries are gauged and the goldstone bosons become massive [13, 45] through the Higgs mechanism. Integrating out the massive fields, we get the quiver in Figure 4.Figure 4
Quiver gauge theory for the cone overdP6 after confinement of nodes 6 and 8.Next, we assume that the strong coupling scale for the gauge groupSU(N+K) at node 4 is bigger than the scale for the SU(2N). This assumption does not include a lot of tuning especially if K≲N. The number of flavors for the gauge group SU(N+K) is Nf=2N>Nc=N+K. Consequently, we can assume that the mesons do not get VEVs after the confinement of SU(N+K) and remain massless. The corresponding quiver is shown in Figure 5. The subscripts of the bifundamental fields denote the gauge groups at the ends of the corresponding link. The subscript k=2,3 labels the two U(N) gauge groups on the left. For example, Ak1 denotes both the field A21 going from the node 2 to the node 1 and A31 going from 3 to 1.Figure 5
Quiver gauge theory for the cone overdP6 after Seiberg duality on node 4.The superpotential of the quiver gauge theory in Figure5 has the form
(3.1)W=Tr⁡(mA21M12+mA31M13)+Tr⁡(λM12C̃24B̃41+λA21B15C52+λM13C̃34B̃41+λA31B15C53).
In order to make the notations shorter, we do not write the subscripts of the couplings. (The couplings are different but have the same order of magnitude.)IfΛ1 for the SU(2N) gauge group at node 1 is close to Λ4 for SU(N+K) at node 4 in Figure 4, then it is natural to assume that for small values of corresponding Yukawa couplings the mass parameters m satisfy m≪Λ1. Now, we note that the SU(2N) gauge group has Nc=2N and Nf=3N-K, that is, Nc+1≤Nf<3/2Nc. This group is a good candidate for the the microscopic gauge group in the ISS construction. After the Seiberg duality, the magnetic gauge group has Ñc=N-K. The superpotential of the dual theory is
(3.2)W̃=Tr⁡(mM22+mM33)+Tr⁡(λM22M̃21Ã12+λM33M̃31Ã13)+Tr⁡(mM42C̃24+mM25C52+mM43C̃34+mM35C53)+Tr⁡(λM42M̃21B̃̃14+λM25B̃51Ã12+λM43M̃31B̃̃14+λM35B̃51Ã13).
The indices of the meson fields correspond to the two gauge groups under which they transform. In our case, this leads to unambiguous identifications, for example, M22=A21M12, M33=A31M13, M42=B̃41M12, and so forth. The mesons M22 and M33 are in adjoint representation of SU(N)2 and SU(N)3, and their F-term equations read
(3.3)m⋅1+λM̃21Ã12=0,m⋅1+λM̃31Ã13=0,
where 1 is the N×N identity matrix. The Seiberg dual gauge group at node 1 is SU(N-K); hence the rank of the matrices M̃21 and so forth, is at most N-K and the SUSY is broken by the rank condition of [10]. Classically, there are massless excitations around the vacua in (3.3). In order to prove that the vacuum is metastable, one has to check that these fields acquire a positive mass at 1 loop. Similarly to [10], we expect this to be true, but a more detailed study is necessary.As a summary, in this section we have found an example of dymanical SUSY breaking in the quiver gauge theory on del Pezzo singularity. An interesting property of this example is that there are massless chiral fields after the SUSY breaking. This behavoir seems to be quite generic, and we expect that similar constructions are possible for other del Pezzo singularities.


## 3.1. ISS Vacuum for thedP6 Singularity

Consider the quiver gauge theory for the cone overdP6 represented in Figure 2. This quiver can be found by the standard methods [1] from the three-block exceptional collection of sheaves [42]. But, in order to prove the existence of this quiver, it is easier to do the Seiberg dualities on the nodes 4, 5, 6, and 1 and reduce it to the known dP6 quiver [2].Figure 2
Quiver gauge theory for the cone overdP6.In the compact CY manifold, one can put the D5-branes only on cycles that are nontrivial globally. A deformation of thedP6 singularity in (2.33) leaves four conifold singularities. We will assume that after joining the 4 conifolds to form a dP6 singularity the two-cycles remain nontrivial. We also expect that these two-cycles are represented by the four two-cycles on del Pezzo that have self-intersection (−2) and do not intersect with each other. Note that the total number of nonanomalous fractional branes and the number of (−2) two-cycles is 6, but there are only 4 two-cycles that do not intersect with each other and with the canonical class. (It is interesting to note the similarity between the branes wrapping the non intersecting cycles on dP6 and the deformation D-branes in [3, 23].)LetAi denote the two-cycle corresponding to the D5-brane charge [1] of the bound state of branes at the ith node in Figure 2. Then, the four non intersecting (−2) two-cycles can be chosen as A2-A3, A4-A5, A6-A7, and A8-A9. Now, we would like to add K fractional branes to A4-A5 and N fractional branes to A6-A7 and to A8-A9. The corresponding quiver is depicted in Figure 3.Figure 3
Quiver gauge theory for the cone overdP6 after adding the fractional branes.The gauge groups at the nodes 6 and 8 haveNf=Nc. Consider the Seiberg duality in the strong coupling limit of these gauge groups. The moduli space consists of the mesonic and the baryonic branches [43, 44]. Suppose we are on the baryonic branch. For the generic Yukawa couplings, the two mesons Φ=BC couple linearly to the fields A and become massive together with two of the A fields.An important question is whether the baryons for the gauge groups in nodes 6 and 8 remain massless. The baryons are charged under the baryonicU(1)B symmetries. In the noncompact setting, these U(1)B symmetries are global [45]. If the baryons get vevs, then the symmetries are broken spontaneously and there are massless goldston bosons. But for the compact CY manifold the U(1)B symmetries are gauged and the goldstone bosons become massive [13, 45] through the Higgs mechanism. Integrating out the massive fields, we get the quiver in Figure 4.Figure 4
Quiver gauge theory for the cone overdP6 after confinement of nodes 6 and 8.Next, we assume that the strong coupling scale for the gauge groupSU(N+K) at node 4 is bigger than the scale for the SU(2N). This assumption does not include a lot of tuning especially if K≲N. The number of flavors for the gauge group SU(N+K) is Nf=2N>Nc=N+K. Consequently, we can assume that the mesons do not get VEVs after the confinement of SU(N+K) and remain massless. The corresponding quiver is shown in Figure 5. The subscripts of the bifundamental fields denote the gauge groups at the ends of the corresponding link. The subscript k=2,3 labels the two U(N) gauge groups on the left. For example, Ak1 denotes both the field A21 going from the node 2 to the node 1 and A31 going from 3 to 1.Figure 5
Quiver gauge theory for the cone overdP6 after Seiberg duality on node 4.The superpotential of the quiver gauge theory in Figure5 has the form
(3.1)W=Tr⁡(mA21M12+mA31M13)+Tr⁡(λM12C̃24B̃41+λA21B15C52+λM13C̃34B̃41+λA31B15C53).
In order to make the notations shorter, we do not write the subscripts of the couplings. (The couplings are different but have the same order of magnitude.)IfΛ1 for the SU(2N) gauge group at node 1 is close to Λ4 for SU(N+K) at node 4 in Figure 4, then it is natural to assume that for small values of corresponding Yukawa couplings the mass parameters m satisfy m≪Λ1. Now, we note that the SU(2N) gauge group has Nc=2N and Nf=3N-K, that is, Nc+1≤Nf<3/2Nc. This group is a good candidate for the the microscopic gauge group in the ISS construction. After the Seiberg duality, the magnetic gauge group has Ñc=N-K. The superpotential of the dual theory is
(3.2)W̃=Tr⁡(mM22+mM33)+Tr⁡(λM22M̃21Ã12+λM33M̃31Ã13)+Tr⁡(mM42C̃24+mM25C52+mM43C̃34+mM35C53)+Tr⁡(λM42M̃21B̃̃14+λM25B̃51Ã12+λM43M̃31B̃̃14+λM35B̃51Ã13).
The indices of the meson fields correspond to the two gauge groups under which they transform. In our case, this leads to unambiguous identifications, for example, M22=A21M12, M33=A31M13, M42=B̃41M12, and so forth. The mesons M22 and M33 are in adjoint representation of SU(N)2 and SU(N)3, and their F-term equations read
(3.3)m⋅1+λM̃21Ã12=0,m⋅1+λM̃31Ã13=0,
where 1 is the N×N identity matrix. The Seiberg dual gauge group at node 1 is SU(N-K); hence the rank of the matrices M̃21 and so forth, is at most N-K and the SUSY is broken by the rank condition of [10]. Classically, there are massless excitations around the vacua in (3.3). In order to prove that the vacuum is metastable, one has to check that these fields acquire a positive mass at 1 loop. Similarly to [10], we expect this to be true, but a more detailed study is necessary.As a summary, in this section we have found an example of dymanical SUSY breaking in the quiver gauge theory on del Pezzo singularity. An interesting property of this example is that there are massless chiral fields after the SUSY breaking. This behavoir seems to be quite generic, and we expect that similar constructions are possible for other del Pezzo singularities.


## 4. Compact CY Manifolds with Del Pezzo Singularities

Noncompact CY singularities are useful in constructing local geometries that enable SUSY breaking configurations of D-branes. However, for a consistent embedding of these constructions in string theory, one needs to find compact CY manifolds that posses the corresponding singularities.The noncompact CY manifolds with del Pezzo singularities are known [27, 29]. The dPn singularities for 5≤n≤8 and for the cone over ℙ1×ℙ1 can be represented as complete intersections. (Note that in the mathematics literature, the del Pezzo surfaces are classified by their degree k=9-n, where n is the number of blown up points in ℙ2.) The CY cones over ℙ2 and dPn for 1≤n≤4 are not complete intersections. The compact CY manifolds for complete intersection singularities where presented in [34].Our construction is different from [34]. It enables one to construct the complete intersection compact CY manifolds for all del Pezzo singularities. This construction does not contradict the statement that for n≤4 the del Pezzo singularities are not complete intersections. The price we have to pay is that these singularities will not be generic, that is, they will not have the maximal number of complex deformations. Whereas for the del Pezzo singularities with n≥5 and for ℙ1×ℙ1 we will represent all complex deformations in our construction.


### 4.1. General Construction

At first, we present the construction in the case ofdP6 singularity and, then, give a more general formulation.The input data is the embedding ofdP6 surface in ℙ3 via a degree three equation. The problem is to find a CY threefold such that it has a local dP6 singularity. The solution has several steps. (1)
Find the canonical class onB=dP6 in terms of a restriction of a class on ℙ3. Let us denote this class as K∈H1,1(ℙ3). K can be found from expanding the total Chern class of B(4.1)c(B)=(1+H)41+3H=1+H+⋯.
Thus, K=-c1(B)=-H.(2)
Construct theℙ1 fiber bundle over ℙ3 as the projectivisation M=P(𝒪ℙ3⊕𝒪ℙ3(K)).(3)
The Calabi-YauY3 is given by an equation of degree 3 in ℙ3 and degree 2 in the coordinates on the fiber. The total Chern class of Y3 is
(4.2)c(Y3)=(1+H)4(1+G-H)(1+G)1+3H+2G.
This has a vanishing first Chern class. By construction, this Calabi-Yau has a del Pezzo singularity at t=0.This construction has a generalization for the other del Pezzo surfaces. LetB denote a del Pezzo surface embedded in X as a complete intersection of a system of equations [16]. Assume, for concreteness, that the system contains two equations and denote by L1 and L2 the classes corresponding to the divisors for these two equations in X. The case of other number of equations can be obtained as a straightforward generalization.(1)
First, we find the canonical class of surfaceB⊂X, defined in terms of two equations with the corresponding classes L1,L2∈H1,1(X),
(4.3)c(B)=c(X)(1+L1)(1+L2)=1+c1(X)-L1-L2+⋯.
Thus, the canonical class of X is obtained by the restriction of K=L1+L2-c1(X).(2)
Second, we construct theℙ1 fiber bundle over X as the projectivisation M=P(𝒪X⊕𝒪X(K)).(3)
In the case of two equations, the Calabi-Yau manifoldY3⊂M is not unique. Let G be the hyperplane class in the fibers, then we can write three different systems of equations that define a CY manifold: the classes for the equations in the first system are L1+2G and L2, the second one has L1+G and L2+G, and the third one has L1 and L2+2G (here L1,L2∈H1,1(M) are defined via the pull back of the corresponding classes in H1,1(X) with respect to the projection of ℙ1 the fibers π:M→X).As an example, let us describe the first system. The first equation in this system is given byL1 in X and has degree 2 in the coordinates on the fibers. The second equation is L2 in X. The total Chern class is
(4.4)c(Y3)=c(X)(1+G+K)(1+G)(1+L1+2G)(1+L2).SinceK=L1+L2-c1(X), it is straightforward to check that the first Chern class is trivial.Let us show how this program works in an example of a CY cone overB=ℙ1×ℙ1. The ℙ1×ℙ1 surface can be embedded in ℙ3 by a generic degree two polynomial equation [16, 35]
(4.5)P2(zi)=0,
where (z0,…,z3)∈ℙ3. (By coordinate redefinition in ℙ3 one can represent the equation as z0z3=z1z2. The solutions of this equation can be parameterized by the points (x1,y1)×(x2,y2)∈ℙ1×ℙ1 as (z0,z1,z2,z3)=(x1x2,x1y2,y1x2,y1y2). This is the Segre embedding ℙ1×ℙ1⊂ℙ3.)The first step of the program is to find the canonical class ofB in terms of a class in ℙ3. Let H be the hyperplane class of ℙ3. Then, the total Chern class of B is
(4.6)c(B)=(1+H)41+2H=1+2H+2H2.
The canonical class is
(4.7)K(B)=-c1(B)=-2H.Next, we construct theℙ1 bundle M=P(𝒪ℙ3⊕𝒪ℙ3(K)) with the coordinates (s,t) along the fibers, where locally s∈𝒪ℙ3 and t∈𝒪ℙ3(-2H). The equation that describes the embedding of the CY manifold Y3 in M is
(4.8)P2(zi)s2+P4(zi)st+P6(zi)t2=0.
This equation is homogeneous in zi of degree two, since t has degree −2.The section ofM at t=0 is contractable, and the intersection with the Y3 is P2(zi)=0, that is, Y3 is the CY cone over ℙ1×ℙ1 near t=0.The total Chern class ofY3 is
(4.9)c(Y3)=(1+H)4(1+G)(1+G-2H)1+2H+2G.
It is easy to check that c1(Y3)=0.


### 4.2. A Discussion of Deformations

In this subsection, we will discuss the deformations of the del Pezzo singularities in the compact CY spaces. The explicit description of the singularities and their deformations can be found in the appendix.The procedure is similar to the deformation of thedP6 singularity described in Section 2. As before, let Y3⊂M be an embedding of the CY threefold Y3 in M, a ℙ1 bundle over products of (weighted) projective spaces. If we blowdown the section of the ℙ1 bundle that contains the del Pezzo, then M becomes a toric variety that we denote by V. After the blowdown, equation for the CY in M becomes a singular equation for a CY embedded in V. The last step is to deform the equation in V to get a generic CY. (In the example of dP6 singularity on the quintic, the projective bundle is M=P(𝒪ℙ3⊕𝒪ℙ3(-H)), the manifold V, obtained by blowing down the exceptional ℙ3 in M, is ℙ4, and the singular equation is the singular quintic in ℙ4.)Letn denote the number of two-cycles on del Pezzo with self-intersection (−2). The intersection matrix of these cycles is minus the Cartan matrix of the corresponding Lie algebra En.The maximal number of complex deformations of del Pezzo singularity isc∨(En)-1, where c∨(En) is the dual Coxeter number of En. These deformations can be performed only if the del Pezzo has a zero size. As a result of these deformations, the canonical class on the del Pezzo becomes trivial within the CY and the del Pezzo singularity is partially or completely smoothed out. In the generic situation, we expect that all (−2) two-cycles on del Pezzo are trivial within the CY, then the number of complex deformations is maximal (this will be the case for ℙ1×ℙ1, dP5, dP6, dP7, dP8). If some of the (−2) two-cycles become nontrivial within the CY, then the number of complex deformations of the corresponding cone is smaller. We will observe this for our embedding of dP2, dP3, and dP4. This reduction of the number of complex deformations depends on the particular embedding of del Pezzo cone. In [8], the generic deformations of the cones over dP2 and dP3 were constructed (Tables 1 and 2).Table 1
Some characteristics of del Pezzo surfaces.
Del PezzoNo. two-cyclesNo. (−2) two-cyclesDynkin’s diagramc∨-1ℙ21000ℙ1×ℙ121A11dP12000dP231A11dP343A2×A13dP454A44dP565D57dP676E611dP787E717dP898E829Table 2
Complex deformations of del Pezzo singularities studied in the paper.
Del PezzoNo. (−2) two-cyclesNo. trivial (−2) two-cyclesc∨-1No. complex deformsℙ20000ℙ1×ℙ11111dP10000dP21010dP33131dP44343dP55577dP6661111dP7771717dP8882929


## 4.1. General Construction

At first, we present the construction in the case ofdP6 singularity and, then, give a more general formulation.The input data is the embedding ofdP6 surface in ℙ3 via a degree three equation. The problem is to find a CY threefold such that it has a local dP6 singularity. The solution has several steps. (1)
Find the canonical class onB=dP6 in terms of a restriction of a class on ℙ3. Let us denote this class as K∈H1,1(ℙ3). K can be found from expanding the total Chern class of B(4.1)c(B)=(1+H)41+3H=1+H+⋯.
Thus, K=-c1(B)=-H.(2)
Construct theℙ1 fiber bundle over ℙ3 as the projectivisation M=P(𝒪ℙ3⊕𝒪ℙ3(K)).(3)
The Calabi-YauY3 is given by an equation of degree 3 in ℙ3 and degree 2 in the coordinates on the fiber. The total Chern class of Y3 is
(4.2)c(Y3)=(1+H)4(1+G-H)(1+G)1+3H+2G.
This has a vanishing first Chern class. By construction, this Calabi-Yau has a del Pezzo singularity at t=0.This construction has a generalization for the other del Pezzo surfaces. LetB denote a del Pezzo surface embedded in X as a complete intersection of a system of equations [16]. Assume, for concreteness, that the system contains two equations and denote by L1 and L2 the classes corresponding to the divisors for these two equations in X. The case of other number of equations can be obtained as a straightforward generalization.(1)
First, we find the canonical class of surfaceB⊂X, defined in terms of two equations with the corresponding classes L1,L2∈H1,1(X),
(4.3)c(B)=c(X)(1+L1)(1+L2)=1+c1(X)-L1-L2+⋯.
Thus, the canonical class of X is obtained by the restriction of K=L1+L2-c1(X).(2)
Second, we construct theℙ1 fiber bundle over X as the projectivisation M=P(𝒪X⊕𝒪X(K)).(3)
In the case of two equations, the Calabi-Yau manifoldY3⊂M is not unique. Let G be the hyperplane class in the fibers, then we can write three different systems of equations that define a CY manifold: the classes for the equations in the first system are L1+2G and L2, the second one has L1+G and L2+G, and the third one has L1 and L2+2G (here L1,L2∈H1,1(M) are defined via the pull back of the corresponding classes in H1,1(X) with respect to the projection of ℙ1 the fibers π:M→X).As an example, let us describe the first system. The first equation in this system is given byL1 in X and has degree 2 in the coordinates on the fibers. The second equation is L2 in X. The total Chern class is
(4.4)c(Y3)=c(X)(1+G+K)(1+G)(1+L1+2G)(1+L2).SinceK=L1+L2-c1(X), it is straightforward to check that the first Chern class is trivial.Let us show how this program works in an example of a CY cone overB=ℙ1×ℙ1. The ℙ1×ℙ1 surface can be embedded in ℙ3 by a generic degree two polynomial equation [16, 35]
(4.5)P2(zi)=0,
where (z0,…,z3)∈ℙ3. (By coordinate redefinition in ℙ3 one can represent the equation as z0z3=z1z2. The solutions of this equation can be parameterized by the points (x1,y1)×(x2,y2)∈ℙ1×ℙ1 as (z0,z1,z2,z3)=(x1x2,x1y2,y1x2,y1y2). This is the Segre embedding ℙ1×ℙ1⊂ℙ3.)The first step of the program is to find the canonical class ofB in terms of a class in ℙ3. Let H be the hyperplane class of ℙ3. Then, the total Chern class of B is
(4.6)c(B)=(1+H)41+2H=1+2H+2H2.
The canonical class is
(4.7)K(B)=-c1(B)=-2H.Next, we construct theℙ1 bundle M=P(𝒪ℙ3⊕𝒪ℙ3(K)) with the coordinates (s,t) along the fibers, where locally s∈𝒪ℙ3 and t∈𝒪ℙ3(-2H). The equation that describes the embedding of the CY manifold Y3 in M is
(4.8)P2(zi)s2+P4(zi)st+P6(zi)t2=0.
This equation is homogeneous in zi of degree two, since t has degree −2.The section ofM at t=0 is contractable, and the intersection with the Y3 is P2(zi)=0, that is, Y3 is the CY cone over ℙ1×ℙ1 near t=0.The total Chern class ofY3 is
(4.9)c(Y3)=(1+H)4(1+G)(1+G-2H)1+2H+2G.
It is easy to check that c1(Y3)=0.


## 4.2. A Discussion of Deformations

In this subsection, we will discuss the deformations of the del Pezzo singularities in the compact CY spaces. The explicit description of the singularities and their deformations can be found in the appendix.The procedure is similar to the deformation of thedP6 singularity described in Section 2. As before, let Y3⊂M be an embedding of the CY threefold Y3 in M, a ℙ1 bundle over products of (weighted) projective spaces. If we blowdown the section of the ℙ1 bundle that contains the del Pezzo, then M becomes a toric variety that we denote by V. After the blowdown, equation for the CY in M becomes a singular equation for a CY embedded in V. The last step is to deform the equation in V to get a generic CY. (In the example of dP6 singularity on the quintic, the projective bundle is M=P(𝒪ℙ3⊕𝒪ℙ3(-H)), the manifold V, obtained by blowing down the exceptional ℙ3 in M, is ℙ4, and the singular equation is the singular quintic in ℙ4.)Letn denote the number of two-cycles on del Pezzo with self-intersection (−2). The intersection matrix of these cycles is minus the Cartan matrix of the corresponding Lie algebra En.The maximal number of complex deformations of del Pezzo singularity isc∨(En)-1, where c∨(En) is the dual Coxeter number of En. These deformations can be performed only if the del Pezzo has a zero size. As a result of these deformations, the canonical class on the del Pezzo becomes trivial within the CY and the del Pezzo singularity is partially or completely smoothed out. In the generic situation, we expect that all (−2) two-cycles on del Pezzo are trivial within the CY, then the number of complex deformations is maximal (this will be the case for ℙ1×ℙ1, dP5, dP6, dP7, dP8). If some of the (−2) two-cycles become nontrivial within the CY, then the number of complex deformations of the corresponding cone is smaller. We will observe this for our embedding of dP2, dP3, and dP4. This reduction of the number of complex deformations depends on the particular embedding of del Pezzo cone. In [8], the generic deformations of the cones over dP2 and dP3 were constructed (Tables 1 and 2).Table 1
Some characteristics of del Pezzo surfaces.
Del PezzoNo. two-cyclesNo. (−2) two-cyclesDynkin’s diagramc∨-1ℙ21000ℙ1×ℙ121A11dP12000dP231A11dP343A2×A13dP454A44dP565D57dP676E611dP787E717dP898E829Table 2
Complex deformations of del Pezzo singularities studied in the paper.
Del PezzoNo. (−2) two-cyclesNo. trivial (−2) two-cyclesc∨-1No. complex deformsℙ20000ℙ1×ℙ11111dP10000dP21010dP33131dP44343dP55577dP6661111dP7771717dP8882929


## 5. Conclusions and Outlook

In this paper, we have constructed a class of compact Calabi-Yau manifolds that have del Pezzo singularities. The construction is analytic, that is, the CY manifolds are described by a system of equations in theℙ1 bundles over the projective spaces.We argue that this construction can be used for the geometrical SUSY breaking [8] as well as for the compactification of ISS [10]. As an example, we find a compact CY manifold with del Pezzo 6 singularity and some conifolds such that some 2-cycles on del Pezzo are homologous to the 2-cycles on the conifolds, that is, this manifold can be used for the geometrical SUSY breaking. Also we find an ISS vacuum in the quiver gauge theory for dP6 singularity.In order to have a consistent string theory representation of the SUSY breaking vacua, one needs to find compact CY manifolds that have the necessary local singularities. In the last section, we present embedding of del Pezzo singularities in complete intersection CY manifolds and study the complex deformations of the singularities. The del Pezzon surface corresponds to the Lie group En. The expected number of complex deformations for the cone over del Pezzo is c∨(En)-1, where c∨ is the dual Coxeter number for the Lie group En. In the studied examples, the cones over ℙ1×ℙ1 and over dP5, dP6, dP7, and dP8 have generic deformations. But the cones over dP2, dP3, and dP4 have less deformations, that is, these cones do not describe the most generic embedding of the corresponding del Pezzo singularities. (It is known that the generic embeddings of del Pezzon singularities for n≤4 (or rank k=9-n≥5) cannot be represented as complete intersections [27, 29], in our construction the del Pezzo singularities are nongeneric complete intersections.)We propose that for the generic embedding the two-cycles on del Pezzo with self-intersection (−2) are trivial within the full Calabi-Yau geometry. The nontrivial two cycles with self-intersection (−2) impose restrictions on the complex deformations. This proposal agrees with the above examples of the embeddings of del Pezzo singularities. Also we get a similar conclusion when the CY has some number of conifolds in addition to the del Pezzo singularity. Although the conifolds are away from the del Pezzo and the del Pezzo itself is not singular, it acquires a nontrivial two-cycle and the number of deformations is reduced.Sometimes the F-theory/orientifolds point of view has advantages compared to the type IIB theory. Our construction of CY threefolds can be generalized to find the 3-dimensional base spaces of elliptic fibrations in F-theory with the necessary del Pezzo singularities. Also we expect this construction to be useful as a first step in finding the warped deformations of the del Pezzo singularities and in the studies of the Landscape of string compactifications.



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