Del Pezzo singularities and SUSY breaking

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 details. 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.


Motivation
Recently there has been a substantial progress in Model building involving the D-branes at the singularities of non compact 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 meta-stable 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 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, i.e. 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, i.e. 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. We find an example of an ISS vacuum for the del Pezzo 6 singularity. 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 of H 2,1 ; the latter corresponds to Kähler deformations given by the elements of H 1,1 [14][15] [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 S 3 at the tip of the conifold or resolved by placing an S 2 [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 S 2 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 S 3 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 non anomalous (fractional) branes produce the fluxes that deform the three-cycles. The massless/tensionless branes correspond to baryonic operators whose vevs are interpreted as the blowup modes.
However, there are a few puzzles with the above interpretation. In some cases there are less deformations than non anomalous 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 non anomalous 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 dP 1 [21][22] [23] [24] [25]. The relevant observation [26] is that there are no geometric transitions for the cone over dP 1 . From the point of view of gauge theory, there is a runaway behavior at the bottom of the cascade and no finite vacuum [27].
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 Pezzo n surface has c ∨ (E n ) − 1 complex deformations [26], where c ∨ (E n ) is the dual Coxeter number of the corresponding Lie group. For instance, the cone over dP 8 has 29 deformations. But there are only 8 non anomalous 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 number of deformations and well defined cohomology (there are no non compact cycles).
The organization of the paper is as follows. In section 2 we construct an example of quintic CY manifold that has both the del Pezzo and conifold singularities. The compactness of CY manifold puts additional constrains on the possible configurations of branes and fluxes [28]. We would like to point out that the presence of conifolds may be necessary if we want to put fractional branes at a del Pezzo singularity. In our example, if the del Pezzo singularity is the only singularity on the quintic, then all non anomalous two-cycles on del Pezzo (i.e. the ones that don't intersect the canonical class) turn out to be trivial within the CY manifold. In the absence of orientifold planes we cannot put fractional branes on such 'cycles', because on a compact manifold the RR flux from these branes has 'nowhere to go'. But if there are some other singularities, such as conifolds, then it is possible that some non anomalous two-cycles on del Pezzo are homologous to the vanishing cycles on 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 anti-branes is a first step in the geometrical SUSY breaking [8] [29]. Also the possibility to introduce the fractional branes will be crucial for the D-brane realizations of ISS construction.
In section 3 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 dP 6 singularity.
In section 4 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 twocycles on del Pezzo that don't 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 P 1 ×P 1 . In the case of dP 0 = P 2 and dP 1 singularities we don't expect to find any deformations. In the case of del Pezzo 2,3, and 4, our embedding leaves some of the (-2) two-cycles non trivial 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 [22] [30].
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.
Section 5 contains discussion and conclusions. 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 figure 1.
The type I transitions are horizontal, the type II transitions are vertical. It is known [26] that the maximal number of deformations of a cone over dP 6 is c ∨ (E 6 ) − 1 = 11, where c ∨ (E 6 ) = 12 is the dual Coxeter number of E 6 . Going along the left vertical arrow we recover all complex deformations of the cone over dP 6 . In this case all the two-cycles that don't intersect the canonical class on dP 6 are trivial within the CY.
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 non trivial within the full CY and doesn't intersect the canonical class inside del Pezzo. As a general rule the existence of non trivial 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 of dP 6 singularity there will be only 32 conifolds that have, respectively, 31 complex deformations. 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 the dP n 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 Pezzo n surface can be described as the deformations of E n singularity on the del Pezzo [32]. The deformations of this singularity have n parameters corresponding to the n two-cycles that don't intersect the canonical class. Note, that the intersection matrix of these two-cycles is (minus) the Cartan matrix of E n . The E n singularity on the del Pezzo is an example of du Val surface singularity [33] (also known as an ADE singularity or a Kleinian singularity). A three dimensional singularity that has a du Val singularity in a hyperplane section 2 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's impossible to blow up these singularities unless we have a specific number of them at specific locations. In example considered in [11] [12], the quintic CY has 16 conifolds placed at a P 2 inside the CY.
is called compound du Val (cDV) [31] [33]. The conifold is an example of cDV singularity since it has the A 1 singularity in a hyperplane section. The generalized conifolds [34] [35] also have an ADE singularity in a hyperplane section, i.e. 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 E n 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 E n 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 non anomalous fractional D-branes at the singularity, then the corresponding geometric transition smooths the singularity [26]. 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 the dP 6 singularity has 11 complex deformations [21][36] but there are only 6 non anomalous fractional branes in the corresponding quiver gauge theory and there are only 6 two-cycles that don't intersect the canonical class [26]. 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 dP 6 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).

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 manifold Y 3 is given by a degree five equation in P 4 where (z 0 , z 1 , z 2 , z 3 , z 4 ) ∈ P 4 . The total Chern class of this manifold is Let us calculate the number of complex deformations. The complex structures are parameterized by the coefficients in (1) up to the change of coordinates in P 4 . The number of coefficients in a homogeneous polynomial of degree n in k variables is In the case of the quintic in P 4 the number of coefficients is The number of reparametrizations of P 4 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 of H 2,1 cohomology group where h 1,1 can be found via the Lefschetz hyperplane theorem [16][38] and the Euler characteristic is given by the integral of the highest Chern class over Y 3 here we have used that 5H is the Poincare dual class to Y 3 inside P 4 . Consequently h 2,1 = 101 which is consistent with the number of complex deformations found before.

Quintic CY with dP 6 singularity
Suppose that the quintic polynomial is not generic but has a degree three zero at the point (w 0 , w 1 , w 2 , w 3 , w 4 ) = (0, 0, 0, 0, 1) where P n 's denote degree n polynomials. The shape of the singularity is determined by P 3 (w 0 , . . . , w 3 ), (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 (8), i.e. the terms that have bigger powers of w 4 .
The resolution of the singularity in (8) can be obtained by blowing up the point (0, 0, 0, 0, 1) ∈ P 4 . Away from the blowup we can use the following coordinates on P 4 where (s, t) ∈ P 1 and (z 0 , . . . , z 3 ) ∈ P 3 . The blowup of the point at t = 0 corresponds to inserting the P 3 instead of this point. Hence the points on the blown up P 4 can be parameterized globally by (z 0 , . . . , z 3 ) ∈ P 3 and (s, t) ∈ P 1 . The projective invariance (s, t) ∼ (λs, λt) corresponds to the projective invariance in the original P 4 . In order to compensate for the projective invariance of P 3 we need to assume that locally the coordinates on P 1 belong to the following line bundles over P 3 , s ∈ O and t ∈ O(−H).
Thus the blowup of P 4 at a point is a P 1 bundle over P 3 obtained by projectivization of ) (for more details on projective bundles see, e.g. [37][39]). In working with projective bundles, we will use the technics similar to [39].
Using parametrization (9), we can write the equation on the blown up P 4 as This equation is homogeneous of degree two in the coordinates on P 1 and degree three in the z i 's. Note, that t ∈ O(−H), i.e. it has degree (−1) in the z i 's, and s ∈ O has degree zero.
Let us prove that the manifold defined by (10) has vanishing first Chern class, i.e. it is a CY manifold. Let H be the hyperplane class in P 3 and G be the hyperplane class Chern class of M is where (1 + H) 4 is the total Chern class of P 3 , (1 + G) corresponds to s ∈ O P 3 and (1 + G − H) corresponds to t ∈ O P 3 (−H). Note, that G(G − H) = 0 on this P 1 bundle and, as usual, H 4 = 0 on the P 3 .
Let Y 3 denote the surface embedded in M by (10). Since the equation has degree 3 in z i and degree two in (s, t), the class Poincare dual to Y 3 ⊂ M is 3H + 2G and the total Chern class is Expanding The intersection of Y 3 with the blown up P 3 at t = 0 is given by the degree three In the calculation of χ(B) we have used that 3H is the Poincare dual class to B inside It is known that the normal bundle to contractable del Pezzo in a CY manifold is the canonical bundle on del Pezzo [40]. Let us check this statement in our example. The canonical class is minus the first Chern class that can be found from (13) 3 The coordinate t describes the normal direction to B inside Y 3 . Since t ∈ O P 3 (−H), restricting to B we find that t belongs to the canonical bundle over B. Hence locally, near t = 0, the CY threefold Y 3 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 (8).
These terms have 15 parameters, but also we get back 4 reparametrizations (now we can add w 4 to the other coordinates). Hence smoothing of the singularity corresponds to 11 complex structure deformations that is the maximal expected number of deformations of dP 6 singularity.
In view of applications in section 4 let us describe the geometric transition between the CY with the resolved dP 6 singularity and a smooth quintic CY in more details. As 3 Slightly abusing the notations, we denote by H both the class of P 3 and the restriction of this class we have shown above, the CY with the blown up dP 6 singularity can be described by the following equation in the P 1 bundle over P 3 This equation can be rewritten as Next we note that, being a projective bundle, M is equivalent [38][37] to P ( where locally s and t are sections of O P 3 (H) and O P 3 respectively. We further observe  (17) as Not surprisingly, we get back equation (8).
Above we have found that there are 11 complex deformations of the dP 6 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 H 2,1 .
Expanding (12), we get the third Chern class The Poincare dual class to Y 3 ∈ M is 3H + 2G and In calculating this integral one needs to take into account that G(G − H) = 0 on M.
Finally we get and The number of complex deformations of the del Pezzo singularity is 101 − 90 = 11, which is consistent with the number found above.

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, i.e. we describe the upper horizontal arrow in figure 1. Consider the system of two equations in P 4 × P 1 where (u, v) ∈ P 1 and P n , R n denote polynomials of degree n in P 4 .
Suppose that at least one of the polynomials P 3 , R 3 , P 2 and R 2 is non zero, then we can solve for u, v and substitute in the second equation, where we get a non generic quintic in P 4 . The points where P 3 = R 3 = P 2 = R 2 = 0 (but otherwise generic) have conifold singularities. There are 3 · 3 · 2 · 2 = 36 such points. The system (23) describes the blowup of the singularities, since every singular point is replaced by the P 1 and the resulting manifold is non singular.
Let H be the hyperplane class of P 4 and G by the hyperplane class of P 1 , then the total Chern class of Y 3 is By Lefschetz hyperplane theorem h 1,1 (Y 3 ) = h 1,1 (P 4 × P 1 ) = 2, there are only two independent Kahler deformations in Y 3 . One of them is the overall size of Y 3 and the other is the size of the blown up P 1 's. Thus the 36 P 1 's are not independent but homologous to each other and represent only one class in H 2 (Y 3 ). If we shrink the size of blown up P 1 's to zero, then we can deform the singularities of (24) to get a generic quintic CY. In this case the 35 three chains that where connecting the 36 P 1 '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 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.

Quintic CY with dP 6 singularity and 32 conifold singularities
The equation for the quintic CY manifold with the blown up dP 6 singularity was found in (10). Here we reproduce it for convenience This equation describes an embedding of the CY manifold in the P 1 bundle M = P (O P 3 ⊕ O P 3 (−H)). As before, (z 0 , . . . , z 3 ) ∈ P 3 and (s, t) are the coordinates on the P 1 fibers over In order to have more Kahler deformations we need to embed (27) in a space with more independent two-cycles. For example, we can consider a system of two equations in the product (P 1 bundle over P 3 ) × P 1 where (u, v) are the coordinates on the additional P 1 . Let G, H, and K be the hyperplane classes on the P 1 fibers, on the P 3 , and on the additional P 1 respectively. Then the total Chern class of Y 3 is and it's easy to see that the first Chern class is zero.
For generic points on the P 1 bundle over P 3 at least one of the functions in front of u or v is non zero. Thus we can find a point (u, v) and substitute it in the second equation, which becomes a non generic equation similar to (27) ( The CY manifold defined in (28) has the following characteristics 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 P 1 's at the blowups of the singularities in (30). These singularities occur when all four equations in (28) vanish The of the del Pezzo can be found by restricting (28) to t = 0, s = 1 section This del Pezzo contains a two-cycle α that is non trivial within the full CY and doesn't intersect the canonical class inside dP 6 .
In the rest of this subsection we will argue that α is homologous to four P 1 's at the tip of the conifolds. The heuristic argument is the following. The formation of dP 6 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 dP 6 singularity and 32 resolved conifolds can be found from (28) by the following coordinate redefinition (w 0 , . . . , w 3 , w 4 ) = (tz 0 , . . . , tz 3 , s) (compare to the discussion after equation (17)) If we blow down the P 1 , then we get the quintic CY with 32 conifold singularities and a dP 6 singularity. For a finite size P 1 , the conifold singularities and one of the two-cycles in the dP 6 are blown up. The deformations of dP 6 singularity correspond to adding terms with higher power of w 4 . After the deformation, the degree two zeros of R 2 and S 2 will split into four degree one zeros that correspond to the four conifolds "hidden" in the dP 6 singularity. The blown up two-cycle of dP 6 is homologous to the two-cycles on the four conifolds. 4

SUSY breaking
In the 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 [41] for the gaugino bilinear field S. This potential has an interpretation as the GVW superpotential [42] for the complex structure moduli of the CY manifold. The original Veneziano-Yankielowicz potential [41] 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 [43]).
In the ISS construction the number of flavors is bigger than the number of colors N c < N f < 3/2N c (and probably N f = N c ). 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] [4]. Moreover the geometric constructions similar to [3][4] 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, i.e. 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's 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. Consider the quiver gauge theory for the cone over dP 6 represented in figure 2. This quiver can be found by the standard methods [1] from the three-block exceptional collection of sheaves [44]. 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 dP 6 quiver [2].
On compact CY manifolds, it is possible to have D5-branes only in the presence of specific orientifolds or anti branes wrapping homologous cycles somewhere else in the geometry. In the previous section we have found a non anomalous two-cycle α on del Pezzo 6 that is homologous to the two-cycles of the conifolds.
Let A i denote the two-cycle corresponding to the D5-brane charge [1] of the bound state of branes at the i-th node in figure 2. Note that the cycles A 4 − A 5 , A 6 − A 7 , and A 8 − A 9 correspond to non-anomalous U(1) symmetries. We will assume that it is possible to construct a compact CY manifold such that these cycles are homologous to some two-cycles on the conifolds (or some other singularities away from del Pezzo). Now we would like to add K fractional branes to A 4 − A 5 and N fractional branes to A 6 − A 7 and to A 8 − A 9 . The corresponding quiver is depicted in figure 3.
In order to make the notations shorter, we don't write the subscripts of the couplings. hence the rank of the matricesM 21 etc. 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 (35). 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.

Compact CY manifolds with del Pezzo singularities
The non compact CY manifolds with del Pezzo singularities are known [22] [30]. The dP n singularities for 5 ≤ n ≤ 8 and for the cone over P 1 × P 1 can be represented as complete intersections. 5 The CY cones over P 2 and dP n for 1 ≤ n ≤ 4 are not complete intersections. The compact CY manifolds for complete intersection singularities where presented in [36]. The construction of elliptically fibred compact CY manifolds with del Pezzo singularities can be found e.g. in [4].
In our construction we use both the methods of complete intersection CY manifolds [16] and the methods of spherical/elliptic fibrations similar to [4]. Recall the construction of compact CY manifolds with local del Pezzo singularities via elliptic fibrations [4]. The first step is to take a particular P 1 bundle over the del Pezzo. The resulting threefold B 3 can be viewed as a base for the F-theory CY fourfold. In the type IIB limit of F-theory the CY fourfold becomes a CY threefold that has the form of a double cover of B 3 . This double cover of the P 1 bundle is an elliptic fibration over the del Pezzo.
In our construction we first embed the del Pezzo surface B in a space X, where X is a projective space, a product of projective spaces, or a weighted projective space [16][32].
Then we consider a particular P 1 bundle over the space X (not only over the del Pezzo).
The CY threefold Y 3 is embedded in this P 1 bundle via a complete intersection of a system of equations. One of the sections of the P 1 bundle is contractible and intersects Y 3 by the del Pezzo surface. The contraction of this section corresponds to forming the del Pezzo singularity on the CY manifold. The description as a system of equations enables one to identify more easily the complex deformations of the singularity than in the case of elliptic fibrations. Also our construction is different from [36]. We construct the complete intersection compact CY manifolds for all del Pezzo singularities. This construction doesn't 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, i.e. they will not have the maximal number of complex deformations.
Whereas for the del Pezzo singularities with n ≥ 5 and for P 1 × P 1 we will represent all complex deformations.

General construction
At first we present the construction in the case of dP 6 singularity, and then give a more general formulation. The input data is the embedding of dP 6 surface in P 3 via a degree three equation. The problem is to find a CY threefold such that it has a local dP 6 singularity. The solution has several steps 1. Find the canonical class on B = dP 6 in terms of a restriction of a class on P 3 . Let us denote this class as K ∈ H 1,1 (P 3 ). K can be found from expanding the total thus K = −c 1 (B) = −H.
2. Construct the P 1 fiber bundle over P 3 as the projectivisation M = P (O P 3 ⊕O P 3 (K)). This construction has a generalization for the other del Pezzo surfaces. Let B 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 L 1 and L 2 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.

The Calabi-Yau
1. First we find the canonical class of surface B ⊂ X, defined in terms of two equations with the corresponding classes L 1 , L 2 ∈ H 1,1 (X), thus the canonical class of X is obtained by the restriction of K = L 1 + L 2 − c 1 (X).
2. Second, we construct the P 1 fiber bundle over X as the projectivisation M = P (O X ⊕ O X (K)).
3. In the case of two equations, the Calabi-Yau manifold Y 3 ⊂ 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 L 1 + 2G and L 2 , the second one has L 1 + G and L 2 + G, the third one has L 1 and L 2 + 2G. 6 As an example, let us describe the first system. The first equation in this system is given by L 1 in X and has degree 2 in the coordinates on the fibers. The second equation is L 2 in X. The total Chern class is Since K = L 1 + L 2 − c 1 (X), it is straightforward to check that the first Chern class of Y 3 is trivial.
Let us show how this program works in an example of a CY cone over B = P 1 × P 1 .
The P 1 × P 1 surface can be embedded in P 3 by a generic degree two polynomial equation [16][38] where (z 0 , . . . , z 3 ) ∈ P 3 . 7 The first step of the program is to find the canonical class of B in terms of a class in The canonical class is Next we construct the P 1 bundle M = P (O P 3 ⊕ O P 3 (K)) with the coordinates (s, t) along the fibers, where locally s ∈ O P 3 and t ∈ O P 3 (−2H). The equation that describes the embedding of the CY manifold Y 3 in M is This equation is homogeneous in z i of degree two, since t has degree −2. The section of M at t = 0 is contractable and the intersection with the Y 3 is P 2 (z i ) = 0, i.e. Y 3 is the CY cone over P 1 × P 1 near t = 0. The total Chern class of Y 3 is It's easy to check that c 1 (Y 3 ) = 0.

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 the dP 6 singularity described in section 2. As before let Y 3 ⊂ M be an embedding of the CY threefold Y 3 in M, a P 1 bundle over products of (weighted) projective spaces. If we blow down the section of the P 1 bundle that contains the del Pezzo, then M becomes a toric variety that we denote by V . After deformations is maximal (this will be the case for P 1 × P 1 , dP 5 , dP 6 , dP 7 , dP 8 ). If some of the (−2) two-cycles become non trivial within the CY, then the number of complex deformations of the corresponding cone is smaller. We will observe this for our embedding of dP 2 , dP 3 , and dP 4 . 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 dP 2 and dP 3 were constructed. The list of embeddings of del Pezzo singularities and their deformations can be found in the Appendix. The results on the number of complex deformations and the comparison with the maximal number of deformations (c ∨ − 1) are presented in the tables below. Table 1. Some characteristics of del Pezzo surfaces.
del Pezzo # two-cycles # (-2) two-cycles Dynkin diagram c ∨ − 1 Table 2. Complex deformations of del Pezzo singularities studied in the paper del Pezzo # (-2) two-cycles # trivial (-2) two-cycles c ∨ − 1 # complex deforms In this paper we have constructed a class of compact Calabi-Yau manifolds that have del Pezzo singularities. The construction is analytic, i.e. the CY manifolds are described by a system of equations in the P 1 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. Also we find an ISS vacuum in the quiver gauge theory for dP 6 singularity.
In the last section and in the Appendix, we describe the deformations of del Pezzo singularities. The del Pezzo n surface corresponds to the Lie group E n . The expected number of complex deformations for the cone over del Pezzo is c ∨ (E n ) − 1, where c ∨ is the dual Coxeter number for the Lie group E n . In the studied examples, the cones over P 1 × P 1 and over dP 5 , dP 6 , dP 7 , and dP 8 have generic deformations. But the cones over dP 2 , dP 3 and dP 4 have less deformations, i.e. these cones do not describe the most generic embedding of the corresponding del Pezzo singularities. 9 We propose that for the generic embedding the two-cycles on del Pezzo with self- 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 non trivial 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 3dimensional 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 9 It is known that the generic embeddings of del Pezzo n singularities for n ≤ 4 (or rank k = 9 − n ≥ 5) cannot be represented as complete intersections [22] [30], in our construction the del Pezzo singularities are non generic complete intersections.

Appendix. A list of compact CY with del Pezzo singularities
In the appendix we construct the embeddings of all del Pezzo singularities in compact CY manifolds and describe the complex deformations of these embeddings. This description follows the general construction in section 4.
In the following B denotes the two-dimensional del Pezzo surface and X denotes the space where we embed B. The space X will be either a product of projective spaces or a weighted projective space. For example, if B ⊂ X = P n × P m × P k , then the coordinates on the three projective spaces will be denoted as (z 0 , . . . , z n ), (u 0 , . . . , u m ), and (v 0 , . . . , v k ) respectively. The hyperplane classes of the three projective spaces will be denoted by H, K, R respectively.
A polynomial of degree q in z i , degree r in u j , and degree s in v l will be denoted by P q,r,s (z i ; u j ; v l ).
If there are only two or one projective space, then we will use the first two or the first one projective spaces in the above definitions.
For the weighted projective spaces, we will use the notations of [32]. For example, consider the space W P 3 11pq , where p, q ∈ N. The dimension of this space is 3, the subscripts (1, 1, p, q) denote the weights of the coordinates with respect to the projective identifications (z 0 , z 1 , z 2 , z 3 ) ∼ (λz 0 , λz 1 , λ p z 2 , λ q z 3 ).
The P 1 bundles over X will be denoted as M = P (O X ⊕ O X (K)), where K is the class on X that restricts to the canonical class on B. The coordinates on the fibers will be (s, t) so that locally s ∈ O X and t ∈ O X (K). The hyperplane class of the fibers will be denoted by G, it satisfies the property G(G + K) = 0 for M = P (O X ⊕ O X (K)). In the construction of the P 1 bundles, we will use the fact that K(B) = −c 1 (B) and will not calculate K(B) separately.
The deformations of some del Pezzo singularities will be described via embedding in particular toric varieties. We will call them generalized weighted projective spaces.
Consider, for example, the following notation GW P 511100002 00011001 00000111 (45) The number 5 is the dimension of the space. This space is obtained from C 8 * by taking the classes of equivalence with respect to three identifications. The numbers in the three rows correspond to the charges under these identifications.
The embedding space V = W P 4 11113 has the coordinates (z 0 , . . . , z 3 ; w) and the singular CY is The embedding space V = W P 4 11112 has the coordinates (z 0 , . . . , z 3 ; w) and the singular CY is This equation has one deformation kw 3 and the spaces M and V have the same number of coordinate redefinitions. Thus the space of complex deformations is onedimensional.
The equation defining B has degree one in z i and degree one in u j The total Chern class of B c(B) = (1 + H) 3 (1 + K) 2 1 + H + K = 1 + 2H + K + H 2 + 3HK (56) The P 1 bundle is M = P (O X ⊕ O X (−2H − K)). The equation for the Calabi-Yau threefold Y 3 is The embedding space V = GW P 4 111002 000111 has the coordinates (z 0 , z 1 , z 2 ; u 0 , u 1 ; w) and the singular CY is There are no complex deformations of this equation.
The del Pezzo surface is defined by a system of two equations. The first equation has degree one in z i and degree one in u k . The second equation has degree one in z i and degree one in v k .
The total Chern class of B The system of equations for the Calabi-Yau threefold Y 3 can be written as The space V = GW P 511100002 00011001 00000111 has the coordinates (z 0 , z 1 , z 2 ; u 0 , u 1 ; v 0 , v 1 ; w) and the singular CY is P 1,1,0 (z i ; u k ; v k )w 2 + P 3,2,1 (z i ; u k ; v k )w + P 5,3,2 (z i ; u k ; v k ) = 0 There are no complex deformations of this equation. This is in contradiction with the general expectation of one complex deformation, i.e. the embedding is not the most general. This is connected to the fact that all the two-cycles on the del Pezzo are non trivial within the CY.
The del Pezzo surface is defined by an equation of degree one in z i , degree one in u j and degree one in v k .
The total Chern class of B c(B) = (1 + H) 2 (1 + K) 2 (1 + R) 2 (1 + H + K + R) = 1 + (H + K + R) + 2(HK + HR + KR) (61) where H, K and R are the hyperplane classes on the three P 1 's. The P 1 bundle is ). The equation for the Calabi-Yau threefold Y 3 is The embedding space V = GW P 41100001 0011001 0000111 has the coordinates (z 0 , z 1 ; u 0 , u 1 ; v 0 , v 1 ; w) and the singular CY is This equation has one deformation kw 3 and the spaces M and V have the same number of reparameterizations. Consequently, there is one complex deformation of the cone. This is related to the fact that 3 out of 4 two-cycles on dP 3 are independent within the CY and there is only one (−2) two-cycle on dP 3 that is trivial within the CY.
6. B = dP 4 ⊂ X = P 2 × P 1 Equation defining B has degree two in z i and degree one in u j The total Chern class of B where H and K are the hyperplane classes on P 2 and P 1 respectively. The P 1 bundle ). The equation for the Calabi-Yau threefold Y 3 is P 2,1 (z i ; u j )s 2 + P 3,2 (z i ; u j )st + P 4,3 (z i ; u j )t 2 = 0 (66) The embedding space V = GW P 4 111001 000111 has the coordinates (z 0 , z 1 , z 3 ; u 0 , u 1 ; w) and the singular CY is P 2,1 (z i ; u j )w 2 + P 3,2 (z i ; u j )w + P 4,3 (z i ; u j ) = 0 (67) The deformations of the singularity have the form of degree one polynomial in Now we find the deformations of this cone over dP 5 . The P 1 bundle M is, in fact, the P 5 blown up at one point. By blowing down the t = 0 section of M we get P 5 .
The CY three-fold with the dP 5 singularity is embedded in P 5 by the system of two equations P 2 (z i )w 2 + P 3 (z i )w + P 4 (z i ) = 0; The deformations of the singularity correspond to taking a general degree four poly- The second CY with the dP 5 singularity is described by Using the same methods as for the first CY, one can show that this singularity also has 7 complex deformations.
The case of dP 6 was described in details section 2, here we just repeat the general results.
The equation defining dP 6 ⊂ P 3 The P 1 bundle is M = P (O X ⊕O X (−H)). The equation for the Calabi-Yau threefold Y 3 P 6 (z i )s 2 + P 7 (z i )st + P 8 (z i )t 2 = 0 (81) The problem with this CY is that for any polynomials P 6 , P 7 and P 8 it has a singularity at s = z 0 = z 1 = z 2 = z 3 = 0 and z 4 = 1. As a consequence the naive calculation of the Euler number gives a fractional number The good feature of this singularity is that it is away from the del Pezzo, thus one can argue that this singularity should not affect the deformation of the dP 8 cone.
In order to justify that we calculate the number of complex deformations of the