Model Building and SUSY Breaking on D-Branes

Model building and SUSY breaking on D-branes

Dmitry Malyshev

a dissertation presented to the faculty of princeton university in candidacy for the degree of doctor of philosophy

recommended for acceptance by the department of physics Adviser: Herman Verlinde

September 2008 c Copyright by Dmitry Malyshev, 2008.

In the recent years there has been an increase of the interest in applying the String theory to construct viable extensions of the Standard Model and to find stringy analogs of the known field theory models for supersymmetry breaking. In my dissertation I will focus on the constructions involving the D-branes at the Calabi-Yau singularities in type IIB string theory. The main motivations for this choice are: the decoupling of the D-brane field theory from the supergravity, the well established tools for deriving the field theories from the configurations of the D-branes, and the possibility of using the gauge-gravity correspondence in solving some non-perturbative aspects in the field theory. A construction of an extended supersymmetric Standard Model on a D3-brane at the del Pezzo 8 singularity is presented in the first part of the dissertation. In the second part we discuss the possible representations of the SUSY breaking models in String Theory and obtain the metastable SUSY breaking vacuum in a system of D-branes on the suspended pinch point singularity. The gauge mediation of this SUSY breaking model is described in details. In the simplest model the spectrum of particles in the visible sector has a split SUSY breaking, the sfermions are one order of magnitude heavier than the gauginos. More complicated models and non-generic deformations are expected to give better spectra of the superpartners.

iii Acknowledgments

I am very grateful to my advisor Herman Verlinde for his support in my research and help I’ve received during the years at the graduate school. I am also thankful to Igor Klebanov, Nikita Nekrasov, Alexander Polyakov, Chris Herzog, Nathan Seiberg, and Zohar Komargodsky for discussions. I would like to thank my friends and collaborators M. Buican, A. Solovyov, A. Dymarsky, V. Pestun, A. Murugan, S. Pufu, S. Franco, D. Rodrigues-Gomes, D. Shih, D. Baumann, and D. Melnikov for the work we’ve done together, for continuous stimulating discussions and for their interest in my research.

iv Contents

Abstract iii

Acknowledgments iv

1 Introduction 1 1.1 Model building in String Theory ...... 1 1.2 SupersymmetryinStringTheory ...... 6 1.2.1 Compactiﬁcation ...... 8 1.2.2 D-branes...... 13 1.3 Outline...... 18

2 Standard Model on a Calabi-Yau cone 20 2.1 Calabi-Yaucones ...... 21 2.1.1 Thecanonicalclass ...... 23 2.1.2 DelPezzosurfaces ...... 27 2.2 D-branesonaCalabi-Yausingularity ...... 31 2.2.1 D-branesandfractionalbranes ...... 31 2.3 Quivergaugetheories...... 37 2.3.1 D3-brane on a del Pezzo singularity ...... 41 2.4 Geometric Identiﬁcation of Couplings ...... 45 2.4.1 Superpotential ...... 46 2.4.2 Kahlerpotential...... 49

v 2.4.3 SpectrumofU(1)GaugeBosons ...... 52 2.4.4 Symmetrybreaking...... 55 2.5 Bottom-UpStringPhenomenology ...... 56 2.5.1 AStandardModelD-brane ...... 58 2.5.2 Identiﬁcationofhypercharge ...... 61

3 Singularities on compact CY manifolds 63 3.1 Compactiﬁcation of del Pezzo singularities ...... 64 3.2 Del Pezzo 6 and conifold singularities on the quintic CY ...... 68 3.2.1 QuinticCY ...... 71 3.2.2 Quintic CY with del Pezzo 6 singularity ...... 72 3.2.3 Quintic CY with 36 conifold singularities ...... 76 3.2.4 Quintic CY with del Pezzo 6 singularity and 32 conifold singularities...... 77 3.3 Compact CY manifolds with del Pezzo singularities ...... 80 3.3.1 Generalconstruction ...... 81 3.3.2 Adiscussionofdeformations ...... 84

4 SUSY breaking in String Theory 87 4.1 ISSmodel ...... 89 4.1.1 SUSYrestorationinISSmodel ...... 91 4.2 F-term SUSY breaking in IIB String Theory ...... 92 4.3 DeformedZ2orbifoldsingularity...... 96 4.4 ISS from the Suspended Pinch Point singularity ...... 99 4.4.1 D-branes at a deformed SPP singularity ...... 100 4.4.2 DynamicalSUSYbreaking ...... 103 4.4.3 SUSYrestoration ...... 105

vi 5 Conclusion 112 5.1 MediationofISSmodel...... 114

A Compact CY with del Pezzo singularities 121

B ISS quiver via an RG cascade 132

Bibliography 135

vii Chapter 1

Introduction

1.1 Model building in String Theory

Any science has a particular set of fundamental problems. These problems are usually easy to formulate and very difficult to solve. Every new step in the solution marks a significant development in the field. In string theory, one of the fundamental problems is the problem of unification of all particles and interactions under a single concept of the string [1][2]. This question was one of the main motivations for creating the string theory and until now it doesn’t have a completely satisfactory solution. Apart from solving the intrinsic fundamental problems, every successful science has a lot of integration with the other fields. String theory is no exception and it has numerous applications in other fields such as algebraic geometry [3] [4], condensed matter physics [5][6], heavy ion collisions [7][8][9], non-relativistic conformal field theories [10] etc. Along with the dramatic development in these fields, a lot of effort is still put in the unification of forces within the string theory. The main purpose of my dissertation is a description of the current state of Model Building in String Theory. In particular, I will be studying the type IIB D-branes at the Calabi-Yau singularities. In the introduction, I will give a sketch the history of Model building in order to motivate why the D-branes at singularities

1 1.1. Model building in String Theory 2 are one of the most promising ways for embedding the Standard Model. All approaches to string phenomenology can be divided into top-bottom and bottom-top approaches. The main motivation for the top-bottom approach comes from the highly restricted form of the string theory itself. For instance, perturbative superstring theory has no conformal anomalies only in 10 dimensions. Absence of conformal anomalies also fixes the low energy effective action which turns out to be the supersymmetric extension of Einstein gravity together with some extra fields: the dilaton, the B-field, and the Ramond-Ramond fields. The idea of the top-bottom approach is that various consistency constraints should single out just a few possible candidates one of which will have the Standard Model in the low energy limit. The top-bottom approach was developed during the “compactification era” of string theory (80’s and early 90’s). The compactification refers to the necessity to hide the extra 6 dimensions, complementary to the observable 4D Minkowski space. The idea is to make the extra dimensions small and compact, which renders them invisible for the current experiments. Then the four-dimensional effective action is obtained by Kaluza-Klein reduction from the 10 dimensional supergravity. The problem is to find the suitable compact manifold that will give the correct gauge group and the matter content of the Standard Model. In Kaluza-Klein reduction it is rather easy to get abelian gauge groups. The non- abelian gauge groups are harder to obtain. One of the ways is to introduce the open strings where the gauge group indices, the Chan-Paton indices, are carried at the ends of the string. Green and Schwarz have proved that the only possible anomaly free gauge groups are E E and SO(32) [11]. This result sparked the interest in 8 × 8 string theory from the point of view of Grand Unification since it singled out two possible candidates for the grand unified gauge group. The remaining problem was to find a plausible mechanism for symmetry breaking down to SU(3) SU(2) × × U(1). The starting point for the open string compactification is a 10D Supergravity together with the 2-form B-field and an SO(32) or E E gauge field. The problem 8 × 8 1.1. Model building in String Theory 3 is to find a compactification that will reduce these field content to the content of the Standard Model or its supersymmetric extension. The two main problems that appear in any approach to building the standard model in string theory are to find the embedding that contains the Standard Model and to eliminate the extra fields. In the top-down approach these two problems are highly intertwined. In order to embed the Standard Model one needs rather complicated internal manifolds, but the complicated manifolds tend to have a lot of moduli that reduce to massless fields after compactification. Among the most successful top-down approaches to model building we can point out the type IIA/M- theory compactifications [12] [13] [14] and the Heterotic string compactifications [15] [16] [17]. The bottom-up approach to string phenomenology [18][19] developed after the discovery of the D-branes [20]. The idea is to separate the problem of model building from the problem of compactification. The Standard Model is supposed to live on a particular combination of the D-branes. The internal manifold provides a background for the system of D-branes and it doesn’t give rise to the standard model fields, although the couplings in the Standard Model generally depend on the shape of the background manifold. The D-branes can be defined as the submanifolds where the open strings can end, i.e. the open strings have Dirichlet boundary condition in the directions perpendicular to the D-brane and Neuman boundary condition in the directions along the D-brane [20] [21]. The D-branes are the sources of the Ramond-Ramond fields, the higher form fields in the 10D SUGRA [1][2]. Since the open strings carry gauge indices at the ends, the open strings attached to a D-brane realize the gauge field on the world volume of the D-brane. A stack of D-branes corresponds to the non-abelian gauge field. Intersecting two stacks of D-branes, we get open strings from one stack to the other which give rise to the matter charged with respect to the two gauge groups at the ends. These simple 1.1. Model building in String Theory 4 rules can be used to realize field theories that come relatively close to the standard model. The main advantage in using the D-branes is that the information about the gauge theory is encoded in the local configuration of the branes and doesn’t depend on the global properties of the manifold (although the global structure of the manifold puts some constraints on the D-brane configurations). However in the intersecting D-branes models [22][23][24], one has to take additional care in order to avoid the massless fields in the low energy action. In fact, the problem gets even worse, since on top of the fields coming from the compactification, there are fields describing the fluctuations of the D-branes, the fields in the adjoint representation of the gauge groups. The next major advancement comes from the AdS/CFT correspondence [25][26] [27][28]. This correspondence is possible due to the decoupling limit where the gauge fields on the D-branes can be decoupled from the SUGRA fields living in the bulk. In model building this decoupling limit is extremely important in eliminating the extra SUGRA fields from the low energy action. The idea is that all the gauge theory fields are localized on the brane, consequently the internal manifold doesn’t need to be compact in order to be invisible and the SUGRA fields which are not localized in the vicinity of the D-brane will not be dynamical. In the decoupling limit, the massless SUGRA fields are still present somewhere in the geometry, but they decouple from the gauge theory fields. The massless fields corresponding to the fluctuation of the D-branes can be avoided by putting the D-branes at the singularities of the Calabi-Yau manifolds. A D-brane placed at the singularity splits into a collection of fractional branes [29][30]. Every fractional brane corresponds to a gauge field and the intersection of fractional branes gives rise to bifundamental fields. For some special set of singularities the fractional branes don’t have holomorphic deformations and the corresponding adjoint field is not present as a massless degree of freedom. There is a massless “center of mass” degree of freedom and the corresponding overall U(1) gauge symmetry but 1.1. Model building in String Theory 5 they decouple from the rest of the fields. The summary of the objectives for studying the IIB D-branes at the singularities in the Model Building

The singularities of Calabi-Yau manifolds have rather restricted form and are • well described in the math literature.

The structure of the singularities is rich enough to provide a lot of room for • embedding the Standard Model.

The SUGRA fields living in the bulk of the geometry are decoupled from the • gauge fields on the D-branes at the singularity, i.e. they are not present as massless fields in the low energy effective action.

Absence of adjoint ﬁelds. • Among the challenges of this approach one can point out

Abundance of possible singularities. • Calculational diﬃculties in ﬁnding some of the parameters in the gauge theory • from the geometrical data (for example, the Kahler potential).

Viable mechanism of SUSY breaking. • There are also some generic features of these constructions that may be viewed as unpleasant form the phenomenological point of view

Gauge groups are independent – no grand uniﬁcation. • Possibility of Landay pole / duality cascade before the Plank scale. • 1.2. Supersymmetry in String Theory 6

1.2 Supersymmetry in String Theory

In this subsection I will try to argue why supersymmetry can be viewed as a bridge between the abstract mathematics of string theory and the physics. Various symmetries, or constraints, play important role both in physics and in mathematics. In mathematics we put the constraints by hand and derive the consequences, for instance, if the function is harmonic, then it’s values in a region are defined by the boundary condition. In physics the symmetries can be obtained as a consequence of a dynamical principle. The harmonic functions minimize the action for a massless field. It is highly improbable that a random function turns out to be harmonic, but in order to minimize the energy the static field is forced to satisfy the Laplace equation. If we let a chunk of water in the space, then it takes the form of the sphere. It is highly improbable that a random shape will be a sphere, but the surface tension, in order to minimize the energy, forces the water to form the sphere. The Supersymmetry is very restrictive: it requires all the particles to assemble in multiplets with equal number of fermions and bosons in each multiplet, it relates the masses and couplings of the particles in one multiplet etc. It is highly improbable that a given lagrangian will be supersymmetric. Why should we expect then the world to be supersymmetric? Let us forget for a moment about the gravity and the problems with the cosmological constant. Then a reason for the world to by supersymmetric is the zero of the vacuum energy in the SUSY vacuum

Φ V Φ = 0 (1.1) h | | iSUSY where Φ denotes the set of the vacuum expectation values of the fields and the | i matrix element computes the energy in this vacuum. Moreover any non-SUSY vacuum has a positive vacuum energy. Thus if we think about the Big-Bang evolution, as the world cools down, the field theory may be forced to choose the vacuum with 1.2. Supersymmetry in String Theory 7 minimal vacuum energy and the absolute minimum is provided by SUSY theories. The next question is why then the SUSY is broken at the low energies? One simple argument is that there can exist classically stable vacua that break supersymmetry: suppose there is a set of vacuum expectation values (ϕ1, . . . , ϕn) of the fields such that the vacuum energy

V (ϕi) > 0 ∂V = 0, i = 1, . . . , n (1.2) ∂ϕi ∂2V is positive definite ∂ϕi∂ϕj then this vacuum breaks SUSY and is classically stable, i.e. small fluctuations of the fields increase the vacuum energy. If there exists a SUSY vacuum for some values of the fields, then this vacuum is a false vacuum and it can tunnel to the true SUSY vacuum. If the energy of the false vacuum is not too big and the barrier is high enough, then the life time can be bigger than the age of the universe and the metastable vacuum becomes phenomenologically acceptable. It is interesting to note that the process of tunneling from a vacuum with higher energy to the vacuum with lower energy until we reach the vacuum with small enough vacuum energy and long enough life-time is a viable scenario for dynamical relaxation of the cosmological constant [31]. In order to keep the energy of the metastable vacuum at the low level, we need the theory to be supersymmetric at the energies above the vacuum energy. This can be achieved if the lagrangian is supersymmetric but at the low energies the SUSY is broken spontaneously by the VEVs of the fields. So the question is why do we expect the metastable SUSY breaking vacua to exist in a supersymmetric theory? One possibility is that there exists another symmetry that prevents the vacuum energy from being zero. The most celebrated example is the R-symmetry that makes impossible to solve the F-terms term equations and breaks SUSY [32]. The argument is very simple, 1.2. Supersymmetry in String Theory 8

suppose that the superpotential W (ϕ1, . . . , ϕn) has an unbroken R-symmetry

R : ϕ eiqiαϕ i → i iq1α iqnα 2α W (e ϕ1, . . . , e ϕn) = e W (ϕ1, . . . , ϕn) (1.3) here we need the overall charge of 2 since the Grassman coordinates have the charge

α 2 1, θ e− θ, and L = d θW should have charge zero. For the canonical kinetic − → terms, the potential energyR is n ∂W 2 V = (1.4) ∂ϕ i=1 i X

In general, n equations ∂W/∂ϕi = 0 in n complex variables will have a solution which gives a configuration with V = 0. In the presence of the R-symmetry, only (n 1) variables are independent, consequently not all the equations ∂W/∂ϕ = 0 − i can be satisfied and V > 0. Why would the R-symmetry then introduce the SUSY breaking at low energies and not at the high energies? The reason is that the R-symmetry may arise as an accidental symmetry of the low energy action. For instance, suppose that there is a massive field that breaks R-symmetry. At the energies below its mass the field decouples and there appears an accidental R-symmetry. The accidental R-symmetry in the IR and the metastable SUSY breaking were nicely realized in the Intriligator, Seiberg, Shih construction [33]. We will find a string theory realization of ISS model in the 4th chapter of the thesis but for now let us proceed with the supersymmetric string theory and look for possible consequences.

1.2.1 Compactiﬁcation

The main motivation for the following discussion is to ﬁnd the consequences of the Supersummetry on the dynamics of the String Theory. At the moment it is not clear if there exists a dynamical reason for the nature to choose among the possible String Theories. We will simply choose one of them, the type IIB String Theory. 1.2. Supersymmetry in String Theory 9

The perturbative type IIB string theory requires the background manifold to be 10-dimensional. In general, the question of compactification reduces to finding the zero modes of the graviton and other fields in the internal 6-dimensional manifold. In practice, it is very difficult to find the zero modes for the Laplace operators acting on some tensor fields in a general 6 dimensional manifold. One of the first very restrictive consequences of Supersymmetry is that the 6 dimensional manifold is in fact a complex manifold with zero Ricci curvature, i.e. it is a Calabi-Yau 3-fold [1]. This is a very helpful restriction but the problem is that there are no compact CY 3-fold with explicitly known metric, i.e. we don’t even know how to write the equations we need to solve. What saves us is that we only want to know how many zero modes there are and we don’t need to know the explicit expressions. On a complex manifold, the number of zero modes for a (p, q) form ω is equal to the dimension of the corresponding cohomology group Hp, q, where p and q denote the number of holomorphic and anti-holomorphic indices of the form. This statement follows from the Hodge decomposition theorem [34] that any form can be decomposed as

[Any form] = [Harmonic] + [Exact] + [Co exact] (1.5) −

Thus any generator of Hp, q corresponds to exactly one harmonic (p, q)-form. In particular, for a connected manifold H0, 0 = 1 and the only holomorphic function on a compact complex manifold is a constant (which is also the only harmonic function). In many cases, the problem of finding the dimension of the cohomology group reduces to a simple algebraic equation, or even to a combinatorial problem, which are much easier to solve than to find the number of zero modes for a differential equation on a manifold with an unknown metric.

Φ The ﬁelds in the type IIB supergravity include the metric, gµν, the dilaton, e− , (0) (2) (4) NSNS 2-form, Bµν , and the Ramond-Ramond form ﬁelds C , Cµν , Cµνλρ [2]. 1.2. Supersymmetry in String Theory 10

The importance of the Hodge decomposition (1.5) is that for the known dimensions of the cohomology groups we can ﬁnd the corresponding zero modes for all the

SUGRA ﬁelds except for the graviton, gµν , which we are going to describe next. On a complex n-dimensional manifold the metric can be uniquely described by a two-form J H1,1 and the complex structure [34]. On the Calabi-Yau 3- ∈ fold the choice of the complex structure is equivalent to the choice of the maximal holomorphic form Ω H3,0 [1]. The beauty of the CY 3-folds is that the zero modes ∈ of the metric are given by independent deformations δJ H1,1 and δΩ H2,1 ∈ ∈ [35]. Thus the number of zero modes for the metric also follows from the Hodge decomposition (1.5). The dimensions of the cohomology groups of CY 3-folds are quite restrictive. In fact the only two cohomology groups that have diﬀerent dimensions are

b1,1 = dim H1,1 b2,1 = dim H2,1 (1.6) we also have b1,2 = b2,1 b2,2 = b1,1 (1.7) b0,0 = b3,0 = b0,3 = b3,3 = 1 All other dimensions are zero [1, 36]. The type IIB String Theory compactiﬁed on a Calabi-Yau 3-fold becomes = 2 N SUSY theory in Minkowski space. All the ﬁelds in = 2 theory are organized in N hypermultiplets, vector multiplets and the rank 2 tensor multiplet. On the CY there

1,1 (2) 1,1 2,2 are b two-cycles αi corresponding to the two-forms in H and b four-cycles (4) 2,2 1,1 βi corresponding to the four-forms in H . Thus we have b + 1 hypermultiplets with the bosonic components

(2) (4) 1,1 (2) J (2) C (2) B (4) C i = 1, . . . , b αi αi αi βi (1.8) Φ (0) (2) eR− C R Bµν RCµν R In the last hypermultiplet one can deﬁne the Hodge duals in 4 dimensions da = dC(2) and db = dB in order to get scalars out of the two-tensors. ∗ ∗ 1.2. Supersymmetry in String Theory 11

The variation of the metric corresponding to the deformations of the complex structure δΩ H2,1 and the compactiﬁcation of C(4) on b2,1 corresponding three- ∈ cycles give rise to the b2,1 vector multiplets.

(4) The metric gµν plus compactification of C on the last three-cycle corresponding to b3,0 gives the tensor multiplet. This derivation of the field contents in the compactification is rather remarkable: the requirement of supersymmetry plus some basic knowledge about the topology of the internal manifolds (i.e. the numbers of two-cycles and three-cycles) fixes completely the fields in the 4-dimensional low energy action. One can even find the action, using the = 2 supersymmetry in 4D. The N vector multiplets are characterized by the prepotential which can be derived from the Kähler potential on the moduli space of complex deformations of the Calabi-Yau manifold [35] i Ω Ω¯. (1.9) ∧ ZCY3 The Kähler potential for the hypermultiplets can be derived from the Kähler potential for the H1,1 deformations of the Calabi-Yau [35]

4 J J J. (1.10) 3 ∧ ∧ ZCY3 These formulas, in principle, give the = 2 action for the fields after compactifi- N cation, though the actual expressions can be difficult to find. The = 2 supersymmetry can be broken to = 1 by the background fluxes N N on the Calabi-Yau manifold. In particular, a non-trivial flux G(3) = dC(2) + τdB(2),

(0) Φ where τ = C + ie− , introduces the superpotential [37] [38] for the scalars in the vector multiplets W = G(3) Ω. (1.11) ∧ ZCY3 This superpotential can be used to stabilize the scalar moduli in the vector multiplets. In the context of gauge theory/gravity duality the role of this superpotential 1.2. Supersymmetry in String Theory 12 becomes even more significant as its simple geometrical form captures rather complicated non-perturbative properties on the gauge theory side [38] [39] [40]. To conclude the subsection, let us summarize the consequences of supersymmetry in string compactification. First of all, imposing the absence of Weyl anomalies in the string worldsheet one can fix the number of dimensions, the set of worldsheet vertex operators and the low energy Supergravity action. Then the target space supersymmetry requires the internal 6-dimensional manifold to be a Calabi-Yau manifold, i.e. complex Ricci flat manifold. Then the effective theory in 4D is obtained by the compactification of the 10D Supergravity on the CY manifold. The fields in the 4D action correspond to the zero modes of the SUGRA fields on the CY manifold that are in one to one correspondence with the cohomology classes of Calabi-Yau. If we turn on the fluxes on the CY manifold then a superpotential for the scalar components of the vector fields is generated and the = 2 symmetry is N broken to = 1. N The compactification is an example of the top-bottom approach to phenomenology, where we start with the complete string theory in 10D and use some simple principles such as supersymmetry and absence of anomalies to derive the action in the 4D Minkowski space. This actions depends only on the discrete choice of the internal manifold with fluxes. As we have mentioned in the introduction, the compactification paradigm by itself was not rich enough for the realistic model building. The compactification of SUGRA fields is one of the important ingredients in the D-brane story as well, but the role is different. Now the SUGRA fields are not the players by themselves, they provide a background for the fields on the D-branes. The values of the background fields parameterize the couplings in the D-brane field theory. By the gauge/gravity correspondence the SUGRA equations on the fields are transformed to the RG equations in a corresponding gauge theory (the most notable example is the Klebanov-Strassler solution for the warped deformed conifold [42]). Now let us discuss the consequences of supersymmetry on the effective field 1.2. Supersymmetry in String Theory 13 theory on the D-branes.

1.2.2 D-branes

The Dirichlet branes can be defined as subspaces where the open strings can end [2] [20]. The D-branes have scalar fields corresponding to the deformations of there world volume in the perpendicular directions and vector fields corresponding to deformations in the directions tangent to the brane. The classical action for the fields on the D-brane in the background of Supergravity fields is

S = SDBI + SCS (1.12) where the Dirak-Born-Infeld and the Chern-Simons parts are

Φ S = T e− det(G + ) (1.13) DBI − p − F Z p (n) SCS = iµp C eF (1.14) n Z X where = B + 2πα′F and T = µ [2]. The first term is the generalization of the F p p energy contained in the volume of the D-brane with the tension Tp. The second term is the interaction between the D-brane and the RR form fields C(n). In the Model Building part of the Introduction, we have discussed that the D-branes is a convenient tool for modeling the field theories. By putting several D-branes in a stack one gets the non-abelian gauge groups and the intersection of the D-branes gives rise to the matter fields. A general problem is that there usually exist some extra fields such as the scalar fields describing the fluctuation of the D-branes in the perpendicular directions. Such fields should be eliminated. Let us discuss how one can avoid the moduli fields for the D-branes by putting them on the singularities of the Calabi-Yau manifolds. At first I would like to point out the basic physical reason and then to give a more formal derivation of the absence of translational modes. 1.2. Supersymmetry in String Theory 14

Suppose that a D-brane wraps a zero-size cycle at a singularity of the CY manifold. The wrapping number is a conserved charge on the D-brane and if we try to move the brane away from the singularity, then it should also wrap this cycle. But now the cycle has a finite volume, consequently it costs some energy to move the brane. In terms of the effective action this cost of energy corresponds to the mass term for the scalar field, i.e. this field is no longer a zero mode.

Massless Mode

Massive Mode

Figure 1.1: Massless and massive modes for a brane with a non-trivial wrapping number over a zero size cycle at the singularity. If the singularity is isolated, then all the transverse modes are massive.

Mathematically the absence of the zero mode for a D-brane at a singularity can be described as follows. In order to preserve the supersymmetry, the D-brane should wrap holomorphic cycles within the internal CY manifold and the deformations of the D-brane worldvolume should be holomorphic. Now the zero-mode will be absent if there are no holomorphic deformations of the D-brane at the singularity within the CY manifold. In order to give an elementary example of such situation, consider the tangent 1.2. Supersymmetry in String Theory 15 vectors to the sphere S2. It is known that any continuous vector field tangent to the sphere should have at least two zeros, i.e. at two points on the sphere the length of the vectors should be zero (or there is a second order zero at one point). Thus we can continuously deform the sphere along the vector field but there will always be two fixed points.

Figure 1.2: The impossibility to comb the hairy sphere: there are at least two points where the vectors are perpendicular to the sphere. The projections of the vectors on the tangent directions provide an example of a section of (2) bundle over complex projective space P1 S2. Since there are at least two perpendicularO vectors, after ≃ the projection there will be two zero vectors.

The sphere is isomorphic to the complex projective space P1 and the tangent vector fields are sections of the so called (2) line bundle over P1. A general line bundle O over P1 is denoted as (n) where n is the difference between the number of zeros O and poles in the sections of the bundle. For example, any section of ( 2) bundle O − should have a least 2 poles. Thus there are no continuous holomorphic deformations of P1 along the directions of ( 2) line bundle. In fact the total space of ( 2) line O − O − 1 2 bundle over P is equivalent to the blowup of the C /Z2 orbifold singularity. The absence of holomorphic deformations of P1 inside the ( 2) bundle corresponds to O − 1.2. Supersymmetry in String Theory 16 the absence of the adjoint fields in the quiver that represent the motion of the branes away from the singularity, the adjoint fields in the corresponding quiver represent

2 the motion of the branes along the line of C /Z2 singularities where the size of the hidden P1 is zero [29]. This is a general phenomenon, the D-branes with non-trivial wrapping numbers over the zero size cycles in the singularities don’t have massless fields corresponding to moving away from the singularity. If the singularity is non-isolated, i.e. there is a line of singularities, then the branes can move along this line and there is a corresponding massless field. This property of D-branes at singularities is very helpful in building the field theory models without additional massless scalar fields. The D-branes on non-isolated singularities will show up in the description of supersymmetry breaking. Another important problem that can be solved by representing the Standard Model on the D-branes at the singularities is the decoupling from gravity. In particle physics one neglects the gravitational interactions because they are many orders of magnitude smaller than the electro-weak and strong interactions. This decoupling of field theory from the gravity is a highly non-trivial fact in string theory. It is easy to understand the decoupling of massive Kaluza-Klein modes, since their masses are too big for the experiments. But we don’t even see the massless compactification modes. The problem of decoupling from the gravity has a natural solution in the D- branes at the singularities framework. The idea is that all the Standard Model fields are localized near the D-branes and at low energies they don’t propagate away from the singularity to the bulk of Calabi-Yau. Consequently they can interact only with the modes localized near the singularity. Formally this decoupling can be represented via the decompactification limit, where the volume of the Calabi-Yau manifold is taken to infinity. In this case the metric for the zero modes given in (1.9) and (1.10) becomes divergent, i.e. the zero modes become non-normalizable 1.2. Supersymmetry in String Theory 17 and decouple from the rest of the theory.

Figure 1.3: The open strings on the Standard Model D-branes at the conical singularity of a Calabi-Yau manifold decouple from the closed strings living in the bulk.

The local geometry near the singularity has the form of a Calabi-Yau cone. In the second chapter of the thesis we provide an extensive mathematical description of geometry and give an example of building a Standard Model-like theory on the D-branes at the tip of the cone. We will show that some of the supergravity zero modes are localized near the tip of the cone and survive the decompactification limit. For the complex cone over a del Pezzo surface these modes correspond to the four-cycle of the del Pezzo itself and its Poincarédual two-cycle, the canonical class of del Pezzo. These modes will be responsible for giving a mass to anomalous U(1) gauge symmetries in the quiver gauge theory via the Green-Schwarz mechanism [11]. In fact the Green-Schwarz mechanism can be used to eliminate the unwanted U(1) symmetries in the quiver gauge theory. Similarly to absorbtion of goldstone 1.3. Outline 18 bosons in Higgs mechanism, the U(1) fields get masses by absorbing corresponding massless SUGRA modes. Every U(1) gauge symmetry corresponds to a cycle in the local geometry near the singularity. If this cycle is non-trivial also in the global CY geometry, then there is a corresponding zero mode and the U(1) field becomes massive. The U(1) fields corresponding to the local cycles that are trivial in the global geometry remain massless. We will need one massless U(1) symmetry to represent the hypercharge. Thus the decoupling of the gravity zero modes has two origins: the supergravity zero modes corresponding to the cycles localized away from the singularity decouple from fields near the D-branes because they correspond to different elements in the cohomology and the zero modes on the cycles localized at the singularity become massive via the interaction with the corresponding U(1) gauge fields. Although the SUGRA zero modes decouple, they may still have non-zero vacuum expectation values that enter in the low energy action as parameters in the Standard Model. The parametrization may depend on a particular model. In the cases we will encounter, the dilaton and the B-field parameterize the gauge couplings similar to [41][42], the Kähler deformations of the CY metric correspond to FI parameters similar to [29], the complex deformations of the shape of singularity parameterize the superpotential of the field theory [43] and the complex deformations that smooth out the singularity correspond to some VEVs in the field theory as in the Klebanov- Strassler solution [42].

1.3 Outline

In chapter 2, the necessary facts about the Calabi-Yau manifolds and their singularities are collected. Also we construct an extended MSSM on a D3-brane at the del Pezzo 8 singularity. In this model there is a problem of extra massless U(1) gauge symmetries. We show that it can be solved by embedding the singularity in 1.3. Outline 19 a compact CY manifold. The U(1) gauge bosons can acquire a mass through the generalized Green-Schwarz mechanism [11]. In chapter 3 we discuss the embedding of the cones over del Pezzo surfaces in compact Calabi-Yau manifolds. A general construction of compact CY manifolds for all del Pezzo singularities is presented. In chapter 4 we study an example of the string theory realization of a SUSY breaking vacuum. The ﬁeld theoretic model is the recent ISS construction of a metastable SUSY breaking vacuum. We show that the ISS model can be realized by a stack of fractional branes at the Suspended Pinch Point singularity. In chapter 5 we discuss the problem of R-symmtry breaking in the ISS model and the possible applications to the phenomenology of the SUSY breaking. Chapter 2

Standard Model on a Calabi-Yau cone

In this chapter we derive an extended MSSM (Minimal Supersymmetric Standard Model) on the D3-brane at a Calabi-Yau singularity. In the beginning we review the relevant facts about the Calabi-Yau (CY) manifolds and their singularities. In this chapter we focus on the non-compact singularities, such as the conifold and the cones over del Pezzo surfaces. Then we describe the quiver gauge theories living on the world volume of the D-branes at the singularities. In the end, as an example, we find a model that comes reasonably close to the Standard Model. This model has some extra U(1) gauge fields and extra Higgs fields. The Higgs fields are unpleasant but not as troublesome as the extra gauge fields. We postpone the question of breaking the extra gauge symmetries till the next chapter where we discuss the singularities on the compact CY manifolds. Most of the material in this chapter is based on the lecture notes [44] and the papers [45] [46]. An immediate question one can ask is why the cones over del Pezzo surfaces are so special? The answer has to do with the mathematical classification of possible singularities of CY manifolds that admit a resolution [47] [48] [49] [50]. The resolved CY manifold is equivalent to the initial manifold except for the singular points that are substituted by either a curve or a surface. The non-trivial point is that the

20 2.1. Calabi-Yau cones 21 process of resolution doesn’t destroy the CY condition. Locally the only primitive isolated singularities of CY manifolds are either the conifold or the cone over a del Pezzo surface [47] [48]. The singularity is called primitive if the manifold that we insert during the resolution is irreducible, the singularity is isolated if it is point- like (not a curve or a surface of singularities). Thus, on the one hand, the conifold and the cones over del Pezzo surfaces provide the most simple examples and, on the other hand, any isolated singularity can be split into irreducible components such that every component is either a conifold or the cone over a del Pezzo. The classiﬁcation of primitive non-isolated singularities is more complicated [50]. We will make use of the non-isolated singularities in chapter 4 for a construction of a SUSY breaking vacuum.

2.1 Calabi-Yau cones

Calabi-Yau manifolds are complex Ricci flat manifolds that provide a good background for string compactifications. The Ricci flatness is necessary for the absence of conformal anomalies. Calabi-Yau manifolds admit one covariantly constant spinor, and hence preserve at least one supersymmetry [1].1 More strictly, a Calabi-Yau manifold Y is a compact Kahler manifold with a vanishing first Chern class, c1(Y ) = 0 [36]. We will assume that the manifold has three complex dimensions. A complex manifold is called Kahler if its Kahler form

J = ig dxµ dxν¯ (2.1) µν¯ ∧ is closed [34]. One of the properties of an n-dimensional Calabi-Yau manifold is that it has a nowhere vanishing holomorphic n-form Ω. The form Ω has only holomorphic indices and depends on z1, . . . , zn (notz ¯1,..., z¯n)

¯ Ω = Ω1...ndz1 . . . dzn ; ∂Ω = 0 . (2.2) 1In the presence of fluxes or branes the Ricci flatness condition may be modified, also one can break the supersymmetry completely. 2.1. Calabi-Yau cones 22

For a general complex manifold M, Ω may have zeros and poles. The corresponding divisor is called the canonical class and is denoted by K(M). We will use the same notation K(M) for the line bundle associated to this divisor.2 The form Ω can be considered as a section of the line bundle K(M). For a general compact manifold M the class of the canonical bundle is minus the ﬁrst Chern class of the holomorphic tangent bundle

K(M) = c (TM). (2.3) − 1

The existence of non zero section of K(M) is equivalent to the triviality of the bundle, i.e. K = c = 0, which coincides with the Calabi-Yau condition. − 1 We will use the triviality of the canonical class for the definition of non compact Calabi-Yau manifolds. The motivation is that both the existence of the covariantly constant spinor η and the existence of the Ricci flat metric are related to the existence of everywhere non zero holomorphic n-form Ω [1]. Note, that the local Calabi-Yau condition is less restrictive than the global one.3 A singular manifold will be called a Calabi-Yau if it can be obtained by a complex or a Kahler deformation from a smooth Calabi-Yau. For example, the singularity of the conifold can be either deformed or resolved and in both cases the resulting manifold is a smooth CY [51]. A rich class of Calabi-Yau singularities is provided by complex cones over a base space X of complex dimension two. In order to get a CY cone we, first, take a line bundle over the base space X such that the canonical class of the total space of the bundle is trivial and, second, shrink the zero section of the bundle to a point. The line bundle over X is the normal bundle to X inside Y . We denote it by NX . The line bundle is not arbitrary: it is completely specified by the condition of vanishing

2In general, the category of line bundles is equivalent to the category of divisors, i.e. for every divisor there is a corresponding line bundle and vice versa, the sum of two divisors is the tensor product of the line bundles [34]. 3E.g. the complex projective space P1 is locally a complex line C1, the latter is evidently a non compact Calabi-Yau while the former is not a Calabi-Yau. 2.1. Calabi-Yau cones 23 canonical class. From the adjunction formula it follows that the divisor for the line bundle NX is equal to the canonical class of the base space. Indeed, the maximal holomorphic n-form on Y restricted to X can be decomposed in an (n 1)-form on − X and a one-form ”perpendicular” to X

K(Y ) = K(X) N ∗ , (2.4) |X ⊗ X and since from the Calabi-Yau condition it follows that the restriction of the canonical class to the base is trivial K(Y ) = 1, we have |X

NX = K(X). (2.5)

2 a ¯b To specify the geometry of the CY cone, let dsX = ha¯bdz dz be a K¨ahler-

Einstein metric over the base X with Ra¯b = 6ha¯b and ﬁrst Chern class ωa¯b = 6iRa¯b. 1 Introduce the one-form η = 3 dψ + σ where σ is deﬁned by dσ = 2ω and 0 < ψ < 2π is the angular coordinate for a circle bundle over the base X. Then the Calabi-Yau metric can then be written as follows

2 2 2 2 2 2 dsY = dr + r η + r dsX (2.6)

For the non-compact cone, the r-coordinate has inﬁnite range. Alternatively, we can think of the cone as a localized region within a compact CY manifold, with r being the local radial coordinate distance. In the following we will consider a particular class of Calabi-Yau singularities of this type, for which the base X is a del Pezzo surface. Before we describe the geometry of del Pezzo surfaces, let us study a few examples of complex manifolds and their canonical classes.

2.1.1 The canonical class

The canonical class of a complex manifold is an important geometric characteristic that will show up in many places in model building. First of all we will use the 2.1. Calabi-Yau cones 24 triviality of the canonical class as the deﬁnition of the non-compact Calabi-Yau manifold. Also it will be shown that the canonical class of the base of the Calabi- Yau cone is relevant for the identiﬁcation of anomalous U(1) gauge groups in the corresponding quiver gauge theories. In this subsection we give a few examples of complex manifolds and calculate their canonical classes. In the following Pn denotes the complex n-dimensional projective space.

Example 1. Canonical class of Pn.

1 The projective plane P has two coordinate charts parameterized by (1, z1) and 1 P1 (z0, 1). The two charts are glued together via z1 = z0− . Line bundles over are classiﬁed by their divisor. A section of (n) bundle for n > 0 may have n zeros and O no poles or n + 1 zeros and a single pole etc. Let λ = 1 be a section of (n) bundle in the chart (1, z ), then in the chart 0 O 1 n (z0, 1) the section is λ1 = z0 . The transition function is therefore

n f0 1 = λ1/λ0 = (z0/z1) . (2.7) →

The sections of the canonical bundle on P1 are holomorphic one forms. Consider 2 the one-form Ω= dz in the chart z = 0. On the intersection Ω = z− dz . The 1 0 6 − 0 0 one-form Ω thus has a double pole at z0 = 0, and the canonical class is therefore K(P1) = ( 2). O − For Pn, any section of the canonical bundle is a holomorphic n-form. Consider the chart z = 0 with coordinates (1, z , . . . , z ) and the form 0 6 1 n

Ω = dz1 . . . dzn. (2.8)

In the chart z = 0 introduce the coordinates 1 6

1 y2 yn (z1, z2, . . . , zn) = ( , ,..., ) (2.9) y1 y1 y1 2.1. Calabi-Yau cones 25 then the holomorphic form reads

1 Ω = n+1 dy1 . . . dyn. (2.10) −y1

Note that in homogeneous coordinates y1 =z0/z1. Consequently, the (n+1)-th order pole at y1 = 0 corresponds to the pole at the hyperplane z0 = 0. Thus the canonical bundle K(Pn) = (n + 1)H, (2.11) − where H is the hyperplane class of Pn. We can reach the same conclusion by using the total Chern class of Pn, given by

c(Pn) = (1 + H)n+1 (2.12) with H the hyperplane class. The canonical class is therefore

K(Pn)= c = (n + 1)H, (2.13) − 1 − as before.

Example 2. Line bundle (n) over P1. O Let λ be a section of the (n) line bundle over P1. Denote by λ and λ the O 0 1 restrictions of λ to the two charts (1, z1) and (z0, 1). Let us consider the section of the canonical bundle for the total space of the line bundle in the chart z1 = 1

Ω = dz0dλ1 (2.14)

In the intersection of the two charts, we have

n z0 z0 Ω = d( )d( n λ0) (2.15) z1 z1 so that in the chart z0 = 1 1 Ω = n+2 dz1dλ0 (2.16) −z1 2.1. Calabi-Yau cones 26

For n = 2, Ω has a non zero section, consequently the total space of the ( 2) − O − line bundle has a vanishing canonical class.4

Example 3. C2 blown up at the origin. The blowup of C2 at the origin is obtained by inserting a P1 instead of the point at the origin of C2. The blowup of C2 will be denoted by C2. The blown up P1 is called the exceptional divisor and is denoted by E. e Near the exceptional divisor, the coordinates on C2 can be written in the form

e z0 = λw0, z1 = λw1 (2.17)

1 where (w0, w1) are homogeneous coordinates on P and λ is the radial direction. In order for this parametrization to be compatible with the projective invariance of P1, λ should be a section of ( 1) bundle over P1 [34]. O − Before the blowup, C2 has the following holomorphic two-form

Ω = dz0dz1 (2.18)

After the blowup, this form is modiﬁed in the vicinity of the exceptional divisor.

Using the coordinates (2.17), we get in the chart w1 = 1

Ω = d(λ1w0)d(λ1w1) = λ1dw0dλ1 (2.19)

The form Ω has a zero at λ = 0, i.e. at the exceptional divisor E. Consequently the canonical class of C2 is K(C2) = E. (2.20) e We will use this result in the next subsection.e

4If we interpret the ( 2) bundle as the normal bundle to P1 inside the total space, then 1 O − NP1 = ( 2) = K(P ), in accordance with the general formula (2.5). O − 2.1. Calabi-Yau cones 27

2.1.2 Del Pezzo surfaces

The mathematical deﬁnition of del Pezzo surface is rather abstract: it is a two- dimensional complex manifold such that its anticanonical bundle is ample. We will simply use the fact that any dell Pezzo surface is either a P1 P1 or a P2 blown up × at n points, where 0 n 8 [53]. The P2 blown up at n points will be called the ≥ ≥ n-th del Pezzo surface and denoted by dPn. 1 Locally, every blown up point supports a P , called an exceptional divisor Ei. Together with the hyperplane class H of P2 the exceptional divisors form a basis for

H2(dPn, Z). Thus the dimension of H2(dPn, Z) is

b2(dPn) = n + 1. (2.21)

Also there is a class of a point, i.e. the zero cycle, and the class of the four-cycle:

b0(dPn) = b4(dPn) = 1. (2.22)

There are no other cycles thus the Euler characteristic of dPn is

χ(dPn) = n + 3 (2.23)

A del Pezzo surface is complex, so we can deﬁne complex cohomology classes hp,q. The corresponding dimensions are

h(0,0) = 1, h(1,1) = n + 1, h(2,2) = 1. (2.24)

The other cohomologies are trivial. These divisors have the following intersections

H H = 1 · H E = 0 (2.25) · i E E = δ i · j − ij Note, that the exceptional divisors have negative self intersection, E E = 1. The · − 1 surface near the exceptional P has the form of P1 ( 1) bundle. The self intersection O − 2.1. Calabi-Yau cones 28 of E is deﬁned as the intersection of E with a little perturbation of E along the normal directions, which is equal to the intersection of a section of P1 ( 1) with O − the zero section. The sign of the intersection depends on the relative orientation of the two curves. If a section has a zero, then the intersection is +1, if it has a pole, the intersection is 1. − In the examples above we have shown that the canonical class of P2 is 3H and − every exceptional divisor contributes Ei, consequently the canonical class of the dPn surface is K(dP ) = 3H + E + ... + E . (2.26) n − 1 n The self intersection of anti-canonical divisor is

( K) ( K) = 9 n. (2.27) − · − − A necessary condition for a surface to be dell Pezzo is that its anti-canonical divisor has positive self-intersection, i.e. n 8. ≤ For n 3 we will usually use the following basis of the two-cycles on dP ≥ n K = 3H + n E − i=1 i α = E E P, i = 1, . . . , n 1 (2.28) i i − i+1 − α = H E E E n − 1 − 2 − 3 The intersection matrix in this new basis takes the form

K α = 0 (2.29) · i α α = A (2.30) i · j − ij where A equals minus the Cartan matrix of the corresponding Lie groups E ij 3 ≡ A A , E A , E D , and the exceptional groups E , E and E . 2 × 1 4 ≡ 4 5 ≡ 5 6 7 8 The complex structure of del Pezzo surfaces depends on the position of the points

2 of P that we blow up, i.e. for dPn we have 2n complex parameters, since every point has two complex coordinates. Two surfaces are equivalent if they are related by an 2.1. Calabi-Yau cones 29 automorphism of P2, i.e. by a P Gl(3) transformation. This group has 32 1 = 8 − complex parameters, and so four generic points on P2 can be moved to given positions by P Gl(3). The remaining coordinates of the blown up points parameterize the non equivalent complex structures. For n > 4, the complex structure of dPn is parameterized by 2n 8 parameters. − Some equations that deﬁne the embedding of the dPn surfaces in the (products of) weighted projective spaces can be found in [36] [53].

Example: del Pezzo 8 surface As an example, let us prove that an equation of degree 6 in the weighted P3 projective space W 1123 is the dP8 surface. Recal that the weighted projective space W P3 consists of the points (u, v, x, y) = (0, 0, 0, 0) up to identiﬁcations 1123 6 (u, v, x, y) (λu, λv, λ2x, λ3y) with λ = 0. Let X denote the surface deﬁned by the ∼ 6 P3 homogeneous equation of degree 6 on W 1123

ay2 = bx3 + cv6 + du6 + ... (2.31)

Let us verify that this surface has the same cohomology and space of deformations as dP8. The total Chern class of the weighted projective space is

2 c(W P1123) = (1 + H) (1 + 2H)(1 + 3H) (2.32)

where H is the hyperplane class in W P1123. The class, Poincare dual to the surface deﬁned by an equation of degree 6 in W P1123, is 6H. Consequently the total Chern class of this surface is [36]

(1 + H)2(1 + 2H)(1 + 3H) c(X) = (2.33) 1 + 6H

Expanding the fraction and using the relation H3 = 0 on X we ﬁnd

c(X) = 1 + H + 11H2 (2.34) 2.1. Calabi-Yau cones 30

2 hence c1(X) = H and c2(X) = 11H . Since 6H is the Poincare dual two-form for X W P , we can extend the integration over X to the integration over W P . ⊂ 1123 1123 One obtains5

χ(X) = c2(X) = c2(X) 6H (2.35) P ∧ ZX ZW 1123 1 = 6 11=11 (2.36) 6 ·

The second cohomology for X is therefore

h1,1 = χ 2 = 9, (2.37) − in accord with the identiﬁcation of the surface as dP8. As we have shown the number of parameters describing the dP surfaces is 2n 8 − 8 = 8. The number of coeﬃcient in equation (2.31) is 23. Let us show that W P1123 has 15 coordinate reparameterizations. The coordinate y has degree 3, hence a generic change of coordinates involves addition of a polynomial of degree 3 in u,v and x to y. The transformations of y together with its rescaling give 7 parameters. The coordinate x has degree two, hence we can add a polynomial of degree 2 in u and v. The transformations of x give 4 parameters. Also there are arbitrary Gl(2) transformations of u and v, which give the last 4 reparameterizations. Consequently, the equations up to change of coordinates are described by 8 parameters and any

6 generic dP8 surface can be described by some equation. At special values of the complex structure moduli, the del Pezzo 8 surface may form ADE type singularities, at which one or more of the 2-cycles αi degenerate. We will make use of this possibility later on.

5 3 The factor 1/6 comes from the fact that P is a six-fold cover of W P1123, since the weighted projective space has a Z2 orbifold singularity near (0, 0, 1, 0): (u, v, 1, y) ( u, v, 1, y); and a 2 ∼3 − − − Z3 orbifold singularity at (0, 0, 0, 1): (u, v, x, 1) (ωu, ωv, ω x, 1) where ω = 1. 6In this respect the del Pezzo surfaces are diﬀerent∼ from the K3 surface, because not all the K3 surfaces are algebraic [54]. 2.2. D-branes on a Calabi-Yau singularity 31

2.2 D-branes on a Calabi-Yau singularity

The del Pezzo surface X forms a four-cycle within the full three-manifold Y , and itself supports several non-trivial two-cycles. Now, if we consider IIB string theory on a del Pezzo singularity, we should expect to ﬁnd a basis of D-branes that spans the complete homology of X: the del Pezzo 4-cycle itself may be wrapped by any number of D7-branes, any 2-cycle within X may be wrapped by one or more D5 branes, and the point-like D3-branes occupy the 0-cycle within X. We will now summarize some of their properties.

2.2.1 D-branes and fractional branes

The D-branes are charged with respect to the Ramond-Ramond (RR) fields [20]. In flat space the interaction between the Dp-brane and the RR fields is summarized by the Chern-Simons terms in the action [21]

F B (n) SCS = µp e − C . (2.38) n Z X Here the integral goes over the world volume of the brane, the index n runs over even integers for type IIB and over odd integers for the type IIA. The fields C(n) are the RR n-form fields. F is the gauge field on the world volume of the brane and B is the restriction of the B-field to the brane. In a more general settings, one may consider a stack of N D-branes wrapping some cycle M. It is convenient to introduce an N-dimensional vector bundle V over

M. The coordinates of the vectors v = (v1, . . . , vN ) in V are labeled by the the Chan-Paton index i = 1 ...N. The gauge ﬁeld F SU(N) living on the world ∈ volume of the stack of branes is interpreted as the curvature form for V . (It is a two-form on the tangent bundle TM that takes values in the operators acting on V .) 2.2. D-branes on a Calabi-Yau singularity 32

A priori, a D-brane can be placed on any subspace of the 10-dimensional space- time on which the string theory lives. But in general, such a configuration will not be stable. The stable objects are the BPS branes. The BPS branes have a minimum energy in a given topological class and preserve some supersymmetry. One of the main questions is to find the set of BPS branes for a given CY geometry. If one changes the geometry or moves a brane around the manifold, then the BPS brane can become non stable and decay to a combination of new BPS branes. For example, a D3 brane is stable at a smooth point in CY but it decays to a combination of so-called fractional branes near a singularity.7 In terms of the world-sheet CFT, that describes the string propagation on the singularity, they are in one-to-one correspondence with the allowed conformally invariant open string boundary conditions. Alternatively, by extrapolating to a large volume perspective, fractional branes may be represented in geometrical language as particular well- chosen collections of sheaves, supported on corresponding submanifolds within the local Calabi-Yau singularity. A geometric way of classifying the space of possible fractional branes at a Calabi-Yau singularity is in terms of the ”hidden” cycles that the branes can wrap. The hidden cycles can be seen by resolving the singularity. The fractional branes can be defined as appropriate stable bound states of branes wrapping these cycles. These bound states, in turn, can be thought of as a D- brane with some quantized magnetic flux F supported on cycles wrapped by the worldvolume of the maximal dimension brane. The form F is the Poincare dual form to the cycles on which the sub-branes are wrapped.8

7 3 The terminology ‘fractional brane’ is motivated as follows. Near the C /Z3 singularity, D- branes must form a representation of Z3, e.g. the Z3 can act by interchanging the branes placed at 3 image points. The BPS branes at the singularity may be called fractional because a D3-brane splits into three branes, each carrying a 1/3 of the D3-brane mass. 8Let us recall the deﬁnition of the Poincare dual form. Let M be a compact complex manifold and A be a cycle inside M. The form wA is said to be the Poincare dual to A inside M if for any form α

α = wA α (2.39) ∧ ZA ZM For example, the hyperplane class H in P2 is the two-form Poincare dual to P1 P2. ⊂ 2.2. D-branes on a Calabi-Yau singularity 33

Topologically non trivial conﬁgurations of F are interpreted as lower dimensional branes bound to the stack of N D-branes. The charges of these lower dimensional branes can be determined from the interaction with the RR ﬁelds. Let us introduce the notation for the charge vector

Aˆ(T ) = ch(V ) (2.40) Q sAˆ(N) where ch(V ) = Tr(eF ) (2.41) is the Chern character of the vector bundle V . The ﬁrst three characters are the rank, the ﬁrst Chern class and the ”instanton number”

ch0(V ) = rk(V ) = N

ch1(V ) = c1(V ) = Tr(F ) (2.42)

ch (V ) = 1 Tr(F F ) 2 2 ∧ The term in the square root in (2.40) is related to the curvature of the D-brane. It is necessary for the cancelation of gravitational anomalies [55]. The charge vector can be expanded in terms of the cohomology classes. In the case of del Pezzo surfaces there are several 2-cycles αA, consequently the D2-brane charges will come with an index A, labeling the cycles that the brane wraps. In terms of their Poincare dual forms the charge vector is expanded as

= Q + QAω + Q H H (2.43) Q 7 5 A 3 ∧ where H is the hyperplane class on P2. The ﬁrst number represents the D7-brane charge (as always, we include the four non compact dimensions in the world volume of the branes), the second element is the Poincare dual class to the cycle that the D5- brane wraps, and the last number is the D3-brane charge. The linear Chern-Simons coupling takes the form

(8) A (6) (4) LCS = Q7 C + Q5 C + Q3 C (2.44) P2 Z ZαA 2.2. D-branes on a Calabi-Yau singularity 34

in accord with the identifications of Qp with the wrapped Dp-brane charge. A D3-brane placed at the tip of the cone can split into fractional branes. This splitting is possible since it doesn’t violate the conservation of the charges. Whether the splitting will actually happen is a more difficult question. The answer to this question involves some knowledge about the masses of the branes. The masses of the BPS objects are proportional to the absolute value of their central charges. The Born-Infeld action for the D-branes in the BPS limit reduces to the absolute value of the central charge, which in the large volume limit has the form [56] ˆ B iJ A(T ) Z(V ) = e− − ch(V ) (2.45) Aˆ(N) Z s here B + iJ is the complexified Kahler form which is determined by the background geometry. The central charge then takes the form

A A Z(V ) = Π0Q7 + Π2 Q5 + Π4Q3 (2.46)

A The charges (Q7,Q5 ,Q3) can be found by expanding the charge vector (2.40) in terms of the forms as in (2.43). In the large volume limit the periods are found by

B iJ expansion of e− −

Π0 = 1

Π2 = (B + iJ) (2.47) − P1 Z 1 Π4 = (B + iJ) (B + iJ). 2 P2 ∧ Z These expressions for the periods in terms of the background ﬁelds B and J receive non perturbative corrections away from the large volume limit. The central charge is an important characteristic of the D-branes, as it tells what supersymmetry is preserved (broken) by the branes. The couplings and the FI terms of the gauge theory in dimensional reduction also depend on the central charges.

Example 1: Branes on a CY cone over P2. In this example we ﬁnd the charge vectors for the fractional branes on the CY cone 2.2. D-branes on a Calabi-Yau singularity 35

2 3 over P , this cone is equivalent to a C /Z3 singularity [56]. The charge of the brane at the tip of the cone can be expressed in terms of the cohomologies of P2

= Q + Q H + Q H H. (2.48) Q 7 5 3 ∧ where H is the hyperplane class of P2. If we have N D7-branes wrapping the P2

3 in the blowup of C /Z3, then depending on the ﬂuxes of the gauge ﬁeld F , i.e. depending on the character of the bundle V , the components of the charge vector are [57]

Q7 = rk(V ) = N

Q5 = P1 c1(V ) (2.49)

R 1 Q3 = P2 ch2(V ) + 8 rk(V )

The last term for Q3 comes from [55][57]R

Aˆ(T ) χ(P2) = 1 + H H (2.50) sAˆ(N) 24 ∧ where the Euler character χ(P2) = 3. This term can be interpreted as the D3 brane charge induced by the curvature of the branes. The calculation of the periods in the small volume limit is a non trivial problem since they receive non perturbative corrections. The periods and the central charges

3 for the branes on C /Z3 singularity can be found e.g. in [57].

Example 2: D3 at the tip of the cone over P1 P1. × In this example we ﬁnd the central charges of the fractional branes on the cone over S = P1 P1. We denote by H and H the 2-cycles Poincar´edual to the two × 1 2 P1’s. They have intersections

H2 = H2 = 0,H H = 1 . (2.51) 1 2 1 · 2

The canonical class is K(S) = 2(H + H ). (2.52) − 1 2 2.2. D-branes on a Calabi-Yau singularity 36

Line bundles over the base S are of the general form

(n, m) = (nH + mH ) . (2.53) O O 1 2

1 In other words, if we choose coordinates on the ﬁrst P as zα, α = 1, 2 and on the second P1 as w , β = 1, 2, then sections of H ( (n, m)) are polynomials P (z, w) of β 0 O total degree n in z and total degree m in w (assuming n, m 0). ≥ The basis of fractional branes on S is given by appropriate sheaves. In our example the sheaves are particularly simple: they are given by the following set of line bundles

F = (0) , F = (H ) , (2.54) 1 O 3 O 2 F = (H ) , F = (H + H ) , 2 O 1 4 O 1 2 and carry the following set of charge vectors

F1 = (1, 0, 0) , F3 = (1,H2, 0) , (2.55)

F2 = (1,H1, 0) , F4 = (1,H1 + H2, 1) .

Each charge vector indicates a corresponding bound state of wrapped D-branes. Charge conservation lets the D3-brane split into four fractional branes via

(0, 0, 1) = F F F + F (2.56) 1 − 2 − 3 4

Let us ﬁnd the central charges of these fractional branes and verify that the splitting is possible from the point of view of the masses. For the cone over P1 P1 the central × charge is F B Z = e − (2.57) ± ZS where the plus sign is for the branes and minus is for the antibranes. If we blow up one of the P1’s, then the geometry near the second P1 will be C C2/Z . The P1 in the blow up of the singularity is the second P1 in S. It is × 2 2.3. Quiver gauge theories 37

1 known [64] that as the P shrinks to form the Z2 singularity the value of the B field 1 period is 1/2, i.e. at the Z2 orbifold point P1 (B + iJ) = 2 . We will assume that changing the size of the first P1 doesn’t affectR the periods over the second one, then in the limit vol(S) = 0 the value of the B field is

1 B = (H + H ). (2.58) 2 1 2

As an example of the calculation, let us ﬁnd the central charge for the antibrane

H F¯2 = (1,H , 0) = e 1 − 1 −

1 H1 B (H1 H2) Z(F¯2) = e − = e 2 − (2.59) − − ZS ZS 1 1 = H1 H2 = (2.60) 4 P1 P1 ∧ 4 Z × It’s easy to check that the central charges of the other three conﬁgurations are also 1 equal to 4 . The sum of the masses is equal to 1 which is the mass of the D3-brane. The phases of the central charges are the same, i.e. the corresponding fractional branes break/preserve the same supersymmetry generators.

2.3 Quiver gauge theories

A collection of fractional branes gives rise to a quiver gauge theory. In the absence of orientifold planes the quiver is a graph with oriented edges. Every fractional brane corresponds to a vertex in the quiver. If there are N fractional branes of the same sort, then they correspond to the U(N) gauge group. An edge in the graph starting on U(N1) and ending on U(N2) corresponds to the bifundamental ﬁeld Φ (N¯ ,N ), where N¯ denotes the antifundamental representation of U(N ) ∈ 1 2 1 1 and N2 is the fundamental represenation of U(N2). Every edge corresponds to a massless mode of open strings stretching between fractional branes. The lightest open string modes are massless whenever the two branes intersect with each other. 2.3. Quiver gauge theories 38

The orientation of the open string is translated in the orientation of the edge. Note that there can be several edges between two vertices, also an edge can begin and end on the same vertex, in this case the field is in adjoint representation of the corresponding gauge group. If there are orientifold planes, then some of the open strings become unoriented. The corresponding gauge groups are SO(N) or Sp(N) and the fields are in real representations of these gauge groups. The orientation of the edge also corresponds to the chirality of the bifundamental field. At every vertex, the number of incoming and outgoing edges is the same. This property ensures that there are no cubic anomalies, i.e. anomalies with three SU(N) currents. But if there is a net number of chiral fields between two vertices, then the U(1) parts of the corresponding U(N) gauge groups have mixed anomalies. Note that some combinations of the anomalous U(1)’s can be non anomalous.9 Before we go to some practical details on finding the quiver gauge theory, let us mention the question of stability [58]. The conservation of the mass and the charge is a necessary but not a sufficient condition for a splitting of a brane into fractional branes to exist. The problem is that some fractional branes may further split into sub-branes or may form a new bound state. The collection of fractional branes should be stable against further reductions. For a mathematical description of various stability condition see for example [59][58]. From the point of view of the corresponding quiver gauge theory stability means that there are no adjoint fields and that for any two vertices all the edges between them have the same orientation (if there are any). The last condition is, in fact, more strict: there should exist an order

9For the cones over del Pezzo surfaces the combination of U(1)’s is not anomalous if the corresponding sum of fractional branes has no D7-brane charge and the class of the D5-brane doesn’t intersect the canonical class of the del Pezzo. In short the argument goes as follows. The divisor for the normal bundle over X is the canonical class. Thus the normal bundle is non trivial over X and over all cycles that intersect the canonical class K. If a cycle doesn’t intersect K, then the normal bundle over this cycle is trivial: it doesn’t intersect with any other cycle. If the two fractional branes don’t intersect, then there is no chiral matter between them. Hence the corresponding U(1) is non anomalous. 2.3. Quiver gauge theories 39 of the fractional branes (let the order be from left to right) such that the orientation of the edge is from the left fractional brane to the right one. The collection of fractional branes that satisfy the stability conditions is called exceptional.

Example: D3 near an orbifold singularity A useful illustration of how quiver gauge theories arise is provided by the example of a D3-brane near a general orbifold singularity [29][60]. Let G be some ﬁnite group of order G , that acts on C3. G can be abelian or non-abelian. In case G is a sub- | | group of SU(3), the world-volume theory is = 1 supersymmetric. To ﬁnd states N invariant under the orbifold projection, we have to consider the D3-brane and all of its images, making a total of G D3-branes. From now on let us denote | |

N = G (2.61) | |

Before performing the orbifold projection, the world-volume theory on the N D- branes is a U(N) gauge theory with a vector multiplet V and three chiral multiplets

3 Φi, that parametrize the transverse positions of the D3-branes along C . All ﬁelds are N N matrices. Projecting onto G invariant states amounts to imposing the × conditions

1 RregVRreg− , = V (2.62)

j 1 i (R3)ijRregΦ Rreg− = Φ (2.63) where R is the N N regular representation of G acting on the Chan-Paton index, reg × and R3 is the 3-d deﬁning representation. The regular representation is deﬁned as the group G acting on itself.

The regular representation Rreg is not irreducible; instead it decomposes into irreducible representations as

r a Rreg = naR (2.64) a=1 M 2.3. Quiver gauge theories 40 where r denotes the total number of irreducible represenations and

a na = dimR . (2.65)

a In other words, each irreducible representation R occurs na times in the regular representation. In explicit matrix notation, we have

1 R 1n1 0 ... 0 ⊗ 2  0 R 1n2 ... 0  Rreg = ⊗ (2.66)  . . .. .   . . . .     r   0 0 ... R 1nr   ⊗    where Ra 1 is the n2 n2 matrix ⊗ na a × a Ra 0 ... 0  a  a 0 R ... 0 R 1n = (2.67) a  . . .. .  ⊗  . . . .     a   0 0 ... R      From this form of Rreg we read oﬀ that the (2.63) breaks the U(N) gauge symmetry to r

U(na) (2.68) a=1 Y Translated into geometric language, we conclude that a D3-brane near an orbifold singularity splits up into fractional branes Fa, where a labels an irreducible representation Ra, and that each fractional brane occurs with multiplicity na = dimRa. The worldvolume theory of each fractional brane Fa contains a vector multiplet Va which in particular is an n n matrix. This result is a reﬂection of the decomposition a × a of the group algebra as a direct sum of n n matrices a × a r C[G] ∼= Mat(na) , (2.69) a=1 M which for us states that the vector multiplets Va, when all combined together, can be thought of as an element of C[G]. 2.3. Quiver gauge theories 41

From the condition (2.63), we learn that we can obtain the number of chiral

3 fields nab between two fractional branes Fa and Fb, transforming in the (na, nb) bifundamental representation, by decomposing the product of the defining and each irreducible representation into irreducible representations in the following way: r R Ra = n3 Rb. (2.70) 3 ⊗ ab Mb=1 Using that the multiplication of group characters reflects the representation algebra

3 of the group, we can compute these coeﬃcients nab as

3 1 3 a b n = χ (g)χ (g)χ (g)∗, (2.71) ab G g G | | X∈ where we used the orthogonality condition of group characters

1 a b χ (g)χ (g)∗ = δ . (2.72) G ab g G | | X∈ Eqns (2.68) and (2.71) provide the complete quiver data of the D3-brane gauge theory.

2.3.1 D3-brane on a del Pezzo singularity

Consider a D3-brane placed at the tip of the complex cone over a dell Pezzo surface.

The D3-brane will split in an exceptional collection of fractional branes. Let Fi denote the charge vector of the ith fractional brane in the collection, i = 1 . . . m.

Recall that a fractional brane correponds to a vector bundle Vi such that the charge vector (2.40) for the interaction with the RR ﬁelds is

= (rk(V ), c (V ), ch (V ) + rk(V )c) (2.73) Qi i 1 i 2 i i where c is a certain constant induced by the curvature of del Pezzo. The decomposition of the D3-brane charge introduces a set of multiplicities Ni via m (0, 0, 1) = N . (2.74) i Qi i=1 X 2.3. Quiver gauge theories 42

Note, that the total D7-brane charge is zero i Ni rk(Vi) = 0. This means that F¯ some of the Ni’s are negative: this correspondsP to taking Ni antibranes i. Since | | there are m fractional branes in the decomposition, the quiver gauge theory consists of a product of m gauge groups m

G = U(Ni). (2.75) i=1 Y The next step is to ﬁnd the matter ﬁelds. Consider two fractional branes Fi and

Fj. For the exceptional collection, if there are some ﬁelds Φij in the representation

(N¯i,Nj), then there are no fields in the opposite representation (N¯j,Ni) and the total number of chiral fields between Fi and Fj is given by the intersection of these fractional branes. In order to define the intersection number let us introduce the notation for the intersection of the 2-cycle that the Fi wraps with the canonical class of the del Pezzo surface deg(F ) = c (V ) K. (2.76) i 1 i · Then the matrix of intersections between the fractional branes is given by (here rk(Fi) = rk(Vi)) N = rk(F ) deg(F ) rk(F ) deg(F ). (2.77) ij j i − i j The absolute value N is the number of edges between F and F . The direction | ij| i j of the edges is determined by the sign of Nij. The geometrical motivation of the formula (2.77) is as follows. Let us find the intersection of Fi with Fj by deforming the cycles in Fi along the normal directions to the del Pezzo surface X. The surface X intersects itself via the canonical class.

Consequently, the D7-brane component of Fi intersects Fj via its component along the canonical class. Since this component is wrapped deg(Fj) times, and the D7- brane charge of Fi equals rk(Fj), the intersection number receives a contribution equal to the product deg(Fj) rk(Fi). The same logic works for the intersection of D5 component of Fi with D7 component of Fj. 2.3. Quiver gauge theories 43

Using the langauge of algebraic geometry, one can give a more invariant way of

finding the chiral matter in the collections of type IIB branes. In general, let V1 and V2 be two vector bundles associated to two fractional branes F1 and F2. Open strings from F1 to F2 have one Chan-Paton index in V1 and the other in V2. They correspond to homomorphisms from V1 to V2, i.e. a chiral matter field associated with edge between the quiver nodes F1 and F2 is an element of Hom(V1,V2). More generally, the fields that live on the intersections of two fractional branes F1 and F2 p are represented by the so-called extension groups Ext (F1, F2). The zeroth extension group is spanned by the homomorphisms

0 Ext (F1, F2) = Hom(F1, F2). (2.78)

For the exceptional collections of branes, all higher extension groups are trivial.

U(1) U(1) A 1,2 2 1

B1,2 C1,2 E i j

4 3 D U(1) 1,2 U(1)

Figure 2.1: Quiver theory of the cone over P1 P1. ×

Example: D3 at the tip of the cone over P1 P1. × Let us return to our example of a D3-brane at the tip of a cone over P1 P1. × The basis of fractional branes is speciﬁed via the charge vectors given in (2.55). The degree is deﬁned by the intersection of D5-brane charge with the canonical class, 2.3. Quiver gauge theories 44

K = 2(H + H ). So we have10 − 1 2

rk(F1) = 1 deg(F1) = 0

rk(F2) = 1 deg(F2) = 2 − (2.79) rk(F ) = 1 deg(F ) = 2 3 − 3 rk(F ) = 1 deg(F ) = 4 4 4 − Every fractional brane corresponds to a gauge group. Since in the decomposition (2.56) of a single D3 brane, all fractional branes occur with multiplicity 1, the ± gauge group of the gauge theory on the D3-brane is U(1)4. The number of chiral ﬁelds between two fractional branes is given by their oriented intersection number. Via the general formula (2.77), we ﬁnd

0 2 2 4 − −  2 0 0 2  F F − #( i, j) =   (2.80)    2 0 0 2   −     4 2 2 0   −    The resulting quiver gauge theory is given in figure 2.1. The number of chiral matter fields between the fractional branes is equal to the dimension of the corresponding space of homomorphisms. Recall the definitions (2.54) of the basis of fractional branes, and that, if we choose coordinates on the first P1 as z , α = 1, 2 and on the second P1 as w , β = 1, 2, sections of H ( (n, m)) α β 0 O are polynomials P (z, w) of total degree n in z and total degree m in w (assuming n, m 0). Let us denote the operators dual to z , z by z∗, z∗, etc. That is, z∗ is ≥ 1 2 1 2 1 10 Here we have flipped the sign of F2 and F3, since these occur with negative multiplicity in (2.56). 2.4. Geometric Identification of Couplings 45

the linear operator that maps z1 to 1, and all other coordinates to 0, etc. Then

F F Hom( 2, 1) = B1w1∗ + B2w2∗ F F Hom( 4, 2) = A1z1∗ + A2z2∗ F F Hom( 3, 1) = D1z1∗ + D2z2∗ (2.81) F F Hom( 4, 3) = C1w1∗ + C2w2∗ F F Hom( 1, 4) = i,jEijziwj P We will use these expressions later to compute the superpotential of the quiver gauge theory of Fig 2. Note, that the only non anomalous combination of the U(1) groups is the difference between the U(1) at the nodes 3 and 2. The fields A and D have the charge +1, the fields B and C have the charge 1, and the field E is neutral under this − U(1). The corresponding combination of the fractional branes is

F = F F = (0,E E , 0). (2.82) α 3 − 2 2 − 1

This combination is the only combination of fractional branes that has rk(Fα) = 0 and deg(Fα) = 0. Consequently it has no chiral intersection with any other fractional brane in the collection. This is an example of a more general statement, that the combination of fractional branes with no D7-brane charge and with the D5- brane charge that has no intersection with the canonical class corresponds to a non anomalous combination of the U(1) gauge groups.

2.4 Geometric Identiﬁcation of Couplings

Consider a D-brane placed at a singularity of a compact Calabi-Yau manifold. We can take a formal low energy limit, in which the distances that can be probed by the open strings are much smaller than the size of CY. In this decoupling limit, we 2.4. Geometric Identification of Couplings 46 may focus our attention to the local region of the singularity, which for simplicity we can take to be non-compact. In this limit the kinetic terms of the closed string modes propagating on the full CY become non normalizable, and the corresponding fields enter in the action as true non dynamical parameters. If the kinetic term of the closed string mode is localized near the singularity, then the corresponding field remains dynamical in the effective theory.

2.4.1 Superpotential

In our discussion thus far, we have concentrated our attention on the topological properties of D-branes on Calabi-Yau singularities. This restriction is partly by choice and partly by necessity: non-topological data are much harder to control and compute. There is one more valuable piece of gauge theory data, however, that can be extracted with precision from this geometric perspective, namely the holomorphic superpotential W . The superpotential in the quiver gauge theories for the type IIB D-branes is a holomorphic quantity, and does not depend on the Kähler moduli. Hence one can go to the large volume limit and find it from the topological B-model. Some superpotentials for the quiver gauge theories on the cones over dell Pezzo surfaces can be found e.g. in [43]. For quiver gauge theories, the superpotential is a sum of gauge invariant traces over ordered products of bi-fundamental chiral fields. There is one such term for each oriented closed loop on the quiver. For a given D3-brane configuration on a Calabi-Yau singularity, one can compute W as follows. For simplicity, let us assume that the closed loops in the quiver are all triangles, so that W is a purely cubic function11 a b c W = CabcTr(φ φ φ ) (2.83)

11This is true for the three block exceptional collections that we discuss below. 2.4. Geometric Identiﬁcation of Couplings 47

Now suppose we want to compute the cubic coupling of the chiral multiplets running between the fractional branes Fi, Fj and Fk. Geometrically, the chiral fields are elements of the Ext groups between the respective sheaves. If we give ourselves the freedom to choose any basis of chiral fields, we can pick any favorite set of generators of these Ext-groups and compute their so-called Yoneda pairings by taking the product of the first two sets of generators

l m 3 n Ext (F , F ) Ext (F , F ) Ext − (F , F ). (2.84) i j × j k → i k and decompose the result in terms of the third basis of generators. Here we used that the cubic pairing between three Ext groups is non-vanishing only if n + l + m = 3,

3 n n and that Ext − (Fi,Fk) is the natural dual space to Ext (Fk, Fi). This calculation was done for del Pezzo singularities with n 5 in [43]. ≤ The superpotential resulting from this calculation is a holomorphic function of the space of chiral bi-fundamental ﬁelds, as well as on the space of complex structure deformations of the del Pezzo singularity. For the n-th del Pezzo surface this amounts to 2(n 4) complex parameters, corresponding to the positions of the − n 4 blow up points that can not be held ﬁxed by using the SL(3, C) isometry − group of the underlying P2. More generally the superpotential depends also on the non-commutative deformations and ’gerbe’ deformations [61].

Example: D3 at the tip of the cone over P1 P1. × The classical superpotential depends on the complex deformations of the cone. Since there are no deformations of P1 P1 the potential will have no free parameters. × In order to find the terms in the superpotential, one, first, takes the fields around a loop in the quiver, this is necessary for the term to be gauge invariant, then finds the decomposition of the corresponding homomorphisms in a direct sum and picks up the terms proportional to identity. For example, the composition of the 2.4. Geometric Identification of Couplings 48 homomorphisms for the upper left triangular in figure 2 reads

Hom(F , F ) Hom(F , F ) Hom(F , F ) = (B w∗) (A z∗) (E z w ) = E A B +... 2 1 ◦ 4 2 ◦ 1 4 k k ◦ l l ◦ ij i j ij i j (2.85) where we assume the summation over all repeated indices and the dots denote the terms not proportional to identity operator. Taking into account the lower left triangular we get the superpotential

W = (A B C D )E (2.86) i j − i j ij i,j=1,2 X In order to get some intuition one can also compare this calculation with the orbifold calculation, where the terms in the superpotential consist of the fields around loops [60]. The classical complex deformations of the geometry are not the only source of deformations of the superpotential. Non commutative deformation can also play a role. Let us count the number of possible parameters in the superpotential for the quiver in figure 2. There are 32 gauge invariant combinations of the fields and the superpotential may have the form

W = λijklAiBjEkl + λ˜ijklCiDjEkl (2.87)

We may allow any Gl(2) transformations of the fields Ai, Bi, Ci, Di and any Gl(2) Gl(2) = Gl(4) transformations of E . The resulting number of repa- ⊗ ij rameterizations of the fields is 4 4 + 16 = 32, but there are three rescalings · 1 that don’t change the superpotential: the first one is A α1A and B B; → → α1 1 2 the second one is C α2C and D D; the third one is E α E and → → α2 → 3 (A, B, C, D) 1 (A, B, C, D). The superpotential thus has three deformation pa- → α3 rameters. And, indeed, there are non commutative deformations of the geometry given by the inverse B-field with holomorphic indices that give the three-parametric deformations of the superpotential [61]. 2.4. Geometric Identification of Couplings 49

2.4.2 Kahler potential

The Lagrangian for the quiver gauge theory can be deduced from the Born-Infeld and Chern-Simons parts of the action for the branes

∗ i φ s = e− s det(i G ) + i∗C eF , (2.88) S s∗ − Fs s p p Z p Z X

= F i∗B. (2.89) Fs s − s The parameters in this Lagrangian depend on the expectation values of the background fields corresponding to the closed string modes, such as the metric on the internal space, the NSNS B-field, and the RR fields. Here is∗ denotes the pull-back of the various fields to the world-volume of the branes.

Massless ﬁelds arising from type IIB superstrings compactiﬁed on a CY3 fold are organized in = 2 multiplets. In general, there are [62] N h1,1 + 1 hypermultiplets • h2,1 vector multiplets • 1 tensor multiplet • If we introduce a D-brane, then the = 2 supersymmetry gets broken to = 1. N N The = 2 hypermultiplets split into two sets of chiral multiplets. One set of these N multiplets corresponds to the holomorphic couplings for the gauge groups

θ 4π τ = + i (2.90) 2π g2 which enter the kinetic terms for the gauge ﬁelds on the branes

τ 1 θ Im d2θ W W α = F F + F F (2.91) 8π α −4g2 ∧ ∗ 32π2 ∧ Z The other ﬁeld is a holomorphic extension of the FI parameter

S = ρ + iζ. (2.92) 2.4. Geometric Identiﬁcation of Couplings 50

If the corresponding closed string modes have normalizable kinetic terms, then S is a dynamical ﬁeld and it is possible to write a gauge invariant mass term for the gauge ﬁeld on the brane [29][63]

1 d4θ (Im(S S¯ 2V ))2 (2.93) 4 − − Z 1 = (A dρ) (A dρ) ζD 2 − ∧ ∗ − − consequently ρ is interpreted as the Stuckelberg field and ζ is indeed the FI ‘parameter’. In the non compact geometry the normalizable modes correspond to Poincare dual cycles that are both compact. For the cone over del Pezzo surfaces, such cycles are the four cycle and the canonical class. The branes wrapping these compact Poincare dual cycles have chiral matter in their intersections, i.e. the corresponding U(1) gauge theories on their world volume have mixed anomalies. These anomalous U(1) gauge groups become massive due to the interaction with the normalizable modes of the closed strings via interaction (2.93). If the string modes corresponding to S are non normalizable, then the fields ρ and ζ become parameters and the only gauge invariant combination is ζD, i.e. the usual FI parameter of the gauge theory. Although the U(1) fields interacting with S don’t have anomalies in this case, they can still get a mass through the Higgs mechanism induced by non zero FI parameter ζ. Expanding the DBI and the Chern-Simons actions one finds the following gauge couplings for the D3 and D5 branes12

ϕ τ0 = C0 + ie− (2.94)

τ2 = (C2 + τ0B2) (2.95) P1 Z Stuckelberg ﬁeld and FI parameter for the D5-brane are

dρ = 4d C4 , ζ = J. (2.96) ∗ P1 P1 Z Z 12The expressions for the D7 brane can be derived in a similar way. 2.4. Geometric Identiﬁcation of Couplings 51

In general, the stable branes are represented by the bound states that have D3, D5 and D7 charges. Hence the coupling on these fractional branes will be a combination of the above couplings. It can be shown that the gauge coupling associated to a fractional BPS brane with the charge vector Q can be obtained from the central charge vector via 4π ϕ = e− Z(Q) (2.97) g2 | | while the FI parameter for the gauge theory on the fractional brane is proportional to the phase of the central charge [59]

1 ζ = Im log Z(Q) (2.98) π

The intuitive motivation for this identiﬁcation is that two D-branes with diﬀerent phases of their central charges break the supersymmetry in a similar way as FI parameters do. A formal derivation of this correspondence in terms of the SCFT on the branes can be found e.g. in [59].

Example: D3 near a Z2 orbifold point.

In this example we ﬁnd the gauge couplings for the fractional branes on the Z2 orbifold singularity. The fractional branes near a resolved Z2 orbifold have the central charges

Z = Q3 + Q5 (B + iJ) (2.99) P1 Z In the orbifold limit J = 0. The charges (Q ,Q ) of the fractional branes are ( 1, 1) 5 3 − and (1, 0). The corresponding couplings are

4π ϕ 4π ϕ = e− 1 B = e− B (2.100) g2 − g2 1 Z 2 Z For the orbifold B = 1/2 [64] and the couplings are equal. It is interesting to note that the more generalR expressions (2.100) work also for the conifold [42]. This is not too surprising because the conifold can be obtained from the Z2 orbifold by giving certain masses to the adjoint ﬁelds [65]. 2.4. Geometric Identiﬁcation of Couplings 52

The blow up of the two-cycle at the Z singularity corresponds to J = 0. In this 2 6 case the absolute value of the central charges (2.99) is bigger than 1/2, consequently it becomes more favorable for the fractional branes to recombine in D3-branes. From the quiver gauge theory point of view, the resolution of the singularity can be reproduced by the non zero FI parameters [29][66]. In the presence of FI parameters, some of the bifundamental ﬁelds get VEVs and break the gauge group U(N) U(N) U(N) . This corresponds to the recombination of two stacks × −→ diag of fractional branes in one stack of D3-branes. The massless ﬁelds that solve the F and D-term equations represent the motion of these D3-branes on the resolved manifold.

The coupling of the gauge fields to the RR-fields follows from expanding the CS-term of the action. In this way we derive that the θ-angle has the geometric expression (4) A (2) (0) θ = Q7 C + Q5 C + Q3 C (2.101) P2 Z ZαA In addition, each fractional brane may support a Stückelberg field, which arises by dualizing the RR 2-form potential that couples linearly to the gauge field strength C via F (2.102) C ∧ From the CS-term we read off that

(6) A (4) (2) = Q4 C + Q2 C + Q0 C (2.103) C P2 Z ZαA The linear coupling (2.102) determines the spectrum of U(1) gauge bosons, as we will now show.

2.4.3 Spectrum of U(1) Gauge Bosons

On dPn there are n+3 different fractional branes, with a priori as many independent gauge couplings and FI parameters. However, the above expression for the central 2.4. Geometric Identification of Couplings 53 charge contain only 2n + 4 independent continuous parameters: the dilaton, the (dualized) B-field, and a pair of periods (of B and of J) for every of the n + 1

2-cycles in dPn. Hence there must be two relations restricting the couplings. The interpretation of these relations is that quiver gauge theory always contains two anomalous U(1) factors. FI-parameters associated with anomalous U(1)’s are not freely tunable, but dynamically adjusted so that the associated D-term equations are automatically satisﬁed. This adjustment relates the anomalous FI variables and gauge couplings.

The non-compact cone Y0 supports two compact cycles for which the dual cycle is also compact, namely, the canonical class and the del Pezzo surface X. Corre- spondingly, we expect to find normalizable 2-form and 4-form on Y0. Their presence implies that two closed string modes survive as dynamical 4-d fields with normalizable kinetic terms; these are the two axions associated with the two anomalous U(1) factors. The two U(1)’s are dual to each other: a U(1) gauge rotation of one generates an additive shift in the θ-angle of the other. This naturally identifies the respective θ-angles and Stückelberg fields. The geometric origin of this identification is that the corresponding branes wrap dual intersecting cycles. The interaction between the anomalous U(1) fields and the Stückelberg fields produces the mass term for the U(1)’s. Let us show in a simple example how the U(1) mass term is generated. Consider the U(1) field corresponding to a two-cycle α with the normalizable two form ωα in the internal directions. The Kaluza-Klein decomposition of the RR four-form on the harmonic forms in the internal directions will contain the term C(4) = c ω 2 ∧ α 3,1 where c2 is a two-form in Minkowski space R . Then the interaction between the U(1) field and the zero-mode of C(4) supported on α takes the form

(4) V = F C = F c2 ωα (2.104) R3,1 α ∧ R3,1 ∧ α Z × Z Z Now let us introduce the scalar ﬁeld ρ such that dρ = dc then we can extend ∗ 2 2.4. Geometric Identiﬁcation of Couplings 54 interaction (2.104) to the gauge invariant mass term

dA c = A dρ (A ∂ ρ)2 (2.105) ∧ 2 ∧ ∗ −→ i − i Z Z Z The mechanism of getting the mass for a U(1) field via the interaction with the RR zero modes is quite general. It can be used to eliminate the unwanted U(1) gauge groups and keep the necessary hypercharge U(1)Y massless. On the non-compact cone, most of the U(1) gauge groups remain massless. The reason is that the corresponding harmonic forms are non-normalizable. These forms are simply obtained by extending the harmonic forms on the base of the cone to the forms that don’t depend on the radial coordinate. The corresponding 4-d RR- modes are non-dynamical: any space-time variation would carry infinite kinetic energy. The only normalizable modes correspond to the anomalous U(1) groups, all the non-anomalous U(1)’s remain massless. The story changes if we embed the cone in a compact Calabi-Yau manifold. Now all the forms are normalizable, but some of the forms that were harmonic in the local geometry may not extend to global harmonic forms. In geometric terms, such forms correspond to non-trivial cycles in the local geometry that turn out to be trivial globally. For example, if we intersect a two-sphere with a plane, we get a non-trivial 1-cycle in the intersection (the circle), however, in the global geometry of S2, this cycle is trivial, since it can be continuously contracted to a point. But if we intersect a two-torus with the plane, then the local two-cycle remains non-trivial in the global geometry. The U(1) gauge factors corresponding to the globally non-trivial two-cycles become massive due to the interaction with the RR zero modes on these two-cycles. If the U(1) field lives on the globally trivial cycle, then there is no RR zero mode and the U(1) field remains massless. As a special consequence, it may be possible to form linear combinations of 2.4. Geometric Identification of Couplings 55 fractional branes, such that the charge adds up to that of a D5-brane wrapping a 2- cycle within X that is trivial within the total space Y . As a result, the corresponding U(1) vector boson decouples from the massless RR-modes and remains an unbroken gauge symmetry after compactification. We will use this observation in the next section to identify the geometric condition for preserving a massless hypercharge

U(1)Y .

2.4.4 Symmetry breaking

The quiver gauge theory on a D3-brane at a CY singularity contains a number of U(1) factors, one for each type of fractional branes. For each non anomalous U(1), one can turn on an FI-parameter ζi. The FI-parameters typically correspond to blow-up modes that govern the size of two-cycles within the CY manifold. A more precise correspondence can be extracted by studying the D and F flatness equations that select the supersymmetric classical vacua of the = 1 gauge theory. N This space of vacua can be thought of as the configuration space of the D3 brane within the CY singularity. The F-flatness conditions follow from extremizing the holomorphic superpotential, ∂W/∂φa = 0, for all chiral matter fields φa. The D- term equations further restrict this space of solutions. There is one real D-flatness condition for each node i on the quiver

+ 2 2 µ φ φ− ζ = 0, (2.106) i ≡ | ia| − | ib| − i a X Xb + where φa and φb− denote the bi-fundamentals on each side of the corresponding node. The conditions (2.106) are implemented via a symplectic quotient: it identifies field configurations on a gauge orbit generated by δφ± = ǫφ and sets µ = 0. Each D- ia ± ia i constraint thus eliminates two dimensions from the solution space of the F-flatness equations. 2.5. Bottom-Up String Phenomenology 56

For a given quiver gauge theory associated to a D3-brane on a Calabi-Yau singularity, the moduli space of vacua, the space of solutions to the F- and D-term equations, reconstructs the geometry of the CY singularity. This correspondence may provide an interesting route towards reverse engineering the appropriate CY geometry associated to a given quiver gauge theory. In general, however, quiver gauge theories do not lead to simple commutative geometries. By varying the ambient Calabi-Yau geometry, fractional branes can become un- stable: they may decay into two or more components or form bound states. In the large volume theory, this happens because the central charges of the fractional branes may re-orient themselves such that the mutual triangle inequalities, that ensure their stability, get violated. From the perspective of the quiver gauge theory, the formation of a bound state is described by condensation of one or more bi-fundamental scalar fields. This generically happens as soon as some of the FI- parameters are non-zero. The D-term equations (2.106) then dictate that some scalar fields φc must acquire a non-vanishing vacuum expectation value φc . In the h i new vacuum, part of the gauge symmetry gets broken. In addition, several matter fields acquire a mass proportional to φc and get lifted from the moduli space of h i supersymmetric vacua via the F-flatness equation

∂W = C φb φc = 0 (2.107) ∂φa abc h i

Hence the number of ﬁelds that become massive is determined by the number of

c non-zero Yukawa couplings Cabc of the ﬁeld φ that acquires the vev.

2.5 Bottom-Up String Phenomenology

A practical way to find a realization of the Standard Model at the D-branes is to look for a configuration of D-branes that contain the MSSM and then to find a gauge symmetry breaking that reduces it to MSSM. In [45] a semi-realistic theory from a 2.5. Bottom-Up String Phenomenology 57 single D3-brane on a partially resolved del Pezzo 8 singularity was cunstructed. The final model, however, still had several extra U(1) factors besides the hypercharge symmetry. Such extra U(1)’s are characteristic of D-brane constructions: typically, one obtains one such factor for every fractional brane. Which of these U(1) factors remains massless depends on the embedding of the CY singularity inside a full compact CY geometry. As we have seen, the massless U(1) gauge bosons are in one-to-one correspondence with non-trivial 2-cycles within the local CY singularity that lift to trivial cycles within the full CY three-fold. This insight can be used to ensure that, among all U(1) factors, only the hypercharge survives as a massless gauge symmetry. The interrelation between the 2-cohomology of the del Pezzo base of the singularity, and the full CY thee-fold has other relevant consequences. Locally, all gauge invariant couplings of the D-brane theory can be varied via corresponding deformations of the local geometry. This local tunability is one of the central motivations for the bottom-up approach to string phenomenology. The embedding into a full string compactification, however, typically introduces a topological obstruction against varying all local couplings: only those couplings that descend from moduli of the full CY survive. Their value will need to be fixed via a dynamical moduli stabilisation mechanism. Let us summarize our general strategy:

Choose a non-compact CY singularity, Y , and ﬁnd a suitable basis of fractional • 0 branes Fi on Y0. Assign multiplicities ni to each Fi and enumerate the resulting quiver gauge theories.

Look for quiver theories that, after symmetry breaking, produce an SM-like • theory. Use the geometric dictionary to identify the corresponding resolved CY singularity. 2.5. Bottom-Up String Phenomenology 58

Identify the topological condition that isolates hypercharge as the only mass- • less U(1). Look for a compact CY 3-fold Y , with the right topological proper-

ties, that contains Y0. Use ﬂuxes and other ingredients to stabilize the moduli of Y at the desired values.

2.5.1 A Standard Model D-brane

The construction of [45] starts from a single D3-brane at the del Pezzo 8 singularity.

2 The dP8 surface is obtained by blowing up 8 points on P . That the cohomology of dP8 is spanned by the zero-cycle, the four-cycle and 9 two-cycles: the canonical class

K and 8 two-cycles αi, i = 1 ... 8 that don’t intersect with K and the intersection among αi is given by minus the Cartan matrix for exceptional Lie group E8. The two-cycles can be represented in terms of the hyperplane class H on P2 and 8

2 exceptional divisors Ei for the 8 blowups on P .

8 K = 3H + E − i i=1 X α = E E , i = 1 ... 7 (2.108) i i − i+1 α = H E E E 8 − 1 − 2 − 3

The D3-brane at the dP8 singularity splits into a collection of fractional branes F . Every fractional brane is characterized by three charges = (Q7,Q5,Q3) where i Qi i i i 5 F Qi is a linear combination of the two-cycle that i wraps. Since the total charge of the collection of fractional branes is the D3-brane charge, the multiplicities ni of the fractional branes are constrained by the condition (2.74). In particular there exists a collection of fractional branes corresponding to the quiver gauge theory with the gauge group = U(6) U(3) U(1)9. This particular quiver theory is related via G0 × × 3 a single Seiberg duality to the world volume theory of a D3-brane near a C /∆27 orbifold singularity – the model considered earlier in [19] as a possible starting point for a string realization of a Standard Model-like gauge theory. As shown in [45], by 2.5. Bottom-Up String Phenomenology 59

U(2)

H H U(3)d u U(1)

U ν D E

6 Figure 2.2: The MSSM-like quiver gaugeU(1) theory obtained in [45]. Each line represents three generations of bi-fundamentals. adding the FI parameters the gauge group can be broken to G0

= U(3) U(2) U(1)7. (2.109) G × ×

The resulting quiver diagram is drawn in fig 2.2. Each line represents three generations of bi-fundamental fields. The D-brane model thus has the same non-abelian gauge symmetries, and the same quark and lepton content as the Standard Model. It has an excess of Higgs fields – two pairs per generation – and several extra U(1)-factors. Our plan is to use the ideas discussed in section 2.4.3 in order to eliminate all the extra U(1) gauge symmetries except the hypercharge from the low energy theory. The gauge theory in fig 2.2 can be obtained by turning on some FI parameters for the non-anomalous U(1) symmetries corresponding to the α1 and α2 two-cycle.

This blowup removes the branes supported on α1 and α2 from the original brane spectrum, and replaces other branes in the spectrum by bound states which are independent of α1 and α2. The remaining bound state basis of the fractional branes 2.5. Bottom-Up String Phenomenology 60 is speciﬁed by the following set of charge vectors

8 1 ch(F ) = (3, 2K + E E , ) 1 − i − 4 2 i=5 X 8 ch(F ) = (3, E , 2) 2 i − i=5 X 4 1 ch(F ) = (3, 3H E , ) (2.110) 3 − i −2 i=1 X ch(F ) = (1,H E , 0) 4 − 4

ch(Fi) = (1, K + Ei, 1 ) i = 5,., 8 − 4 ch(F ) = (1, 2H E , 0) 9 − i i=1 X Here the first and third entry indicate the D7 and D3 charges; the second entry gives the 2-cycle wrapped by the D5-brane component of Fi. As shown in [45], the above collection of fractional branes is rigid, in the sense that the branes have the minimum number of self-intersections and the corresponding gauge theory is free of adjoint matter besides the gauge multiplet. From the collection of charge vectors, one easily obtains the matrix of intersection products via the formula (2.77), which gives the quiver diagram drawn in fig 2.2. The rank of each gauge group corresponds to the (absolute value of the) multiplicity of the corresponding fractional brane, and has been chosen such that weighted sum of charge vectors adds up to the charge of a single D3-brane. In other words, the gauge theory of fig 2.2 arises from a single D3-brane placed at the del Pezzo 8 singularity. Note that, as expected, all fractional branes in the basis (2.110) have vanishing D5 wrapping numbers around the two 2-cycles corresponding to the first two roots

α1 and α2 of E8. After eliminating the two 2-cycles α1 and α2, the remaining 2- cohomology of the del Pezzo singularity is spanned by the roots αi with i = 3, .., 8 and the canoncial class K. 2.5. Bottom-Up String Phenomenology 61

α 8

α α α α α α α 1 2 3 4 5 6 7

Figure 2.3: Our proposed D3-brane realization of the MSSM involves a dP8 singularity embedded inside a CY manifold, such that two of its 2-cycles, α1 and α2 are blown up, and all 2-cycles except α4 are non-trivial within the full CY.

2.5.2 Identiﬁcation of hypercharge

Let us turn to discuss the U(1) factors in the quiver of ﬁg 2.2, and identify the linear combination that deﬁnes hypercharge. We denote the node on the right by U(1)1, and the overall U(1)-factors of the U(2) and U(3) nodes by U(1)2 and U(1)3, resp. 6 3 3 The U(1) node at the bottom divides into two nodes U(1)u and U(1)d, where each

U(1)u and U(1)d acts on the matter ﬁelds of the corresponding generation only. We denote the nine U(1) generators by Q ,Q ,Q ,Qi ,Qi , . The total charge { 1 2 3 u d }

Qtot = Qs (2.111) s X decouples: none of the bi-fundamental fields is charged under Qtot. Of the remaining eight generators, two have mixed U(1) anomalies. As discussed, these are associated to fractional branes that intersect compact cycles within the del Pezzo singularity. In other words, any linear combination of charges such that the corresponding fractional brane has zero rank and zero degree is free of anomalies. The hypercharge is identified with the non-anomalous combination 1 1 1 3 3 Q = Q Q Qi Qi (2.112) Y 2 1 − 6 3 − 2 d − u i=1 i=1 X X The other non-anomalous U(1) charges are 1 1 Q Q = B L, (2.113) 3 3 − 2 1 − together with four independent abelian flavor symmetries of the form

Qij = Qi Qj ,Qij = Qi Qj. (2.114) u,d u − u b b − b 2.5. Bottom-Up String Phenomenology 62

We would like to ensure that, among all these charges, only the hypercharge survives as a low energy gauge symmetry. From our study of the stringy St¨uckelberg mechanism, we now know that this can be achieved if we ﬁnd a CY embedding of the dP8 geometry such that only the particular 2-cycle associated with QY represents a trivial homology class within the full CY three-fold. We will compute this 2-cycle momentarily.

The linear sum (2.112) of U(1) charges that deﬁnes QY , selects a corresponding linear sum of fractional branes, which we may choose as follows

1 F = F F F + F (2.115) Y 2 3 − 0 − i i i=4,5,9 i=6,7,8 X X A simple calculation gives that, at the level of the charge vectors

1 ch(F ) = ( 0 , α , ) α = E E (2.116) Y − 4 2 4 5 − 4

We read oﬀ that the 2-cycle associated with the hypercharge generator QY is the one represented by the simple root α4. Chapter 3

Singularities on compact CY manifolds

In the last chapter we have shown that the knowledge about the global properties of the Calabi-Yau manifold is essential for giving masses to U(1) gauge fields and preserving the massless hypercharge U(1)Y . Every cycle in the local singularity corresponds to a U(1) gauge symmetry. If this cycle is non-trivial in the global geometry, then the corresponding U(1) field acquires a mass via the interaction with a RR zero mode. If the cycle is globally trivial, then the zero mode is absent and the U(1) is massless. Another question, where the compactification may play a role, is the description of the superpotential. There are two types of deformations: the deformations that change the shape of the singularity but preserve the singularity itself, these deformations parameterize the terms in the superpotential; the deformations of the second type smooth out the singularity, partially or completely, these deformations correspond to the VEVs of some operators. For example, the deformation of the conifold is of the second type: it smooths out the singularity and corresponds to the VEV of gaugino bilinear. For the conifold there are no deformations of the first type and the superpotential doesn’t have free parameters (apart from the overall coefficient). For the cone over a del Pezzo surface, the deformations of the first type

63 3.1. Compactiﬁcation of del Pezzo singularities 64 correspond to the deformation of del Pezzo: the deformations of the base of the cone change the shape of the singularity and, in some special points, the degree of the singularity may increase, but generically the type of the singularity is unchanged. In this chapter we will mainly study the deformations of the second type, i.e., the deformations that smooth out the singularity. The compactiﬁcation of the singularity may impose additional constraints on the possible deformations. In the end of this chapter we will present the embeddings for all del Pezzo singularities in compact Calabi-Yau manifolds as complete intersections in the (weighted) projective spaces. There is a mathematical fact [48][49], that del Pezzo n singularities for n 4 in general cannot be represented via a complete intersection. Thus our ≤ method will not give all possible complex deformations of del Pezzo singularities for n 4 (for n = 0 and n = 1 there are no complex deformations at all). However ≤ we will reproduce all local deformations in the compact manifolds for del Pezzo n singularities with n 5 and for the cone over P1 P1. ≥ ×

3.1 Compactiﬁcation of del Pezzo singularities

One of the original motivations to study the deformations and the embeddings of the singularities in the compact CY manifolds came from the search for new dynamical mechanisms of SUSY breaking in String Theory. In particular, one can try to embed the “geometrical” approach of [67][68] in a compact manifold or try to represent the

ISS construction [33] in a string theory setup. In the case of dP6 singularity [69], the details of the dP6 geometry obscure the SUSY breaking sector and it will proof useful to study a simpler geometry that exhibits the ISS vacuum. This study will be done in great details in the next chapter where we will present the ISS vacuum for a particular combination of D-branes on the Suspended Pinch Point singularity. 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 3.1. Compactification of del Pezzo singularities 65 of conifold singularities. For example, in the geometric transitions on compact CY manifolds [70][71], 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. An example of an ISS vacuum for the del Pezzo 6 singularity was found in [69]. In the next section we will describe the ISS vacuum for the suspended pinch point singularity. 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 H2,1; the latter corresponds to Kähler deformations given by the elements of H1,1 [35][1] [36]. 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 [51]. The process where some three-cycles shrink to form a singularity and after that the singularity is blown up is called the geometric transition [70][71]. For the conifold, the geometric transition has a nice 3.1. Compactification of del Pezzo singularities 66 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 [42]. 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 [72]. 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 [73] similar to the conifold cascade. But it is known that there are no complex deformations of the cone over dP1 [74][75][76]. The relevant observation

[77] 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 [78]. 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 ) 1 complex deformations [77], where c∨(E ) n − n is the dual Coxeter number of the corresponding Lie group. For instance, the cone over dP8 has 29 deformations. But there are only 8 non anomalous combinations of fractional branes [45]. I 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 3.1. Compactification of del Pezzo singularities 67 and well defined cohomology (there are no non compact cycles). In this chapter we will first 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, i.e. their resolution doesn’t 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 [47][48]. 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 non anomalous two-cycles on del Pezzo (i.e. the ones that don’t 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 [67][79]. Also the possibility to introduce the fractional branes will be crucial for the D-brane realizations of ISS construction. Finally 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 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 P1 P1. In the case of × 2 dP0 = P and dP1 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 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 68 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 [48][49]. In our case the del Pezzo singularities are complete intersections but they are not generic.

3.2 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 [47][48]. Correspondingly we will have two types of geometric transitions

1. Type I, or conifold transitions: several P1’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.

∆h2,1 = 35 Smooth quintic CY Y3 (1, 101) Y3 (2, 66) with 36 conifolds

6 6

∆h2,1 = 11 ∆h2,1 = 7

Y3 (2, 90) with dP6 singularity Y3 (3, 59) with dP6 singularity ∆h2,1 = 31 and 32 conifolds

Figure 3.1: Possible geometric transitions of quintic CY. The numbers in parentheses denote the dimensions (h1,1, h2,1).

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 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 69

ﬁgure 3.1. The type I transitions are horizontal, the type II transitions are vertical.

It is known [77] that the maximal number of deformations of a cone over dP6 is c∨(E ) 1 = 11, where c∨(E ) = 12 is the dual Coxeter number of E . Going along 6 − 6 6 the left vertical arrow we recover all complex deformations of the cone over dP6. In this case all the two-cycles that don’t intersect the canonical class on dP6 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 dP6 singularity there will be only 32 conifolds that have, respectively, 31 complex deformations.1 In general, the del Pezzo singularity and the conifold singularities are away from each other but they still affect the number of complex deformations, i.e. the presence of conifolds reduces the number of deformations of del Pezzo singularity and vice versa. The diagram in figure 3.1 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

1It 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 [70][71], the quintic CY has 16 conifolds placed at a P2 inside the CY. 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 70 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 dPn surface for n > 4 has 2n 8 deformations that parameterize the superpotential of the corresponding quiver − gauge theory [43]. 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 En singularity on the del Pezzo [53]. 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 En. The En singularity on the del Pezzo is an example of du Val surface singularity [52] (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) [47][52]. The conifold is an example of cDV singularity since it has the A1 singularity in a hyperplane section. The generalized conifolds [99][102] 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 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 non anomalous fractional D-branes at the singularity, then the corresponding geometric transition smooths the singularity [77]. But not all the deformations can be 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 71 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 dP6 singularity has 11 complex deformations [74][80] 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 [77]. 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).

3.2.1 Quintic CY

The description of the quintic CY is well known [36]. Here we repeat it in order to recall the methods of ﬁnding the topology and deformations that we use later in more diﬃcult situations.

4 The quintic CY manifold Y3 is given by a degree ﬁve equation in P

Q5(zi) = 0 (3.1) where (z , z , z , z , z ) P4. The total Chern class of this manifold is 0 1 2 3 4 ∈ (1 + H)5 c(Y ) = = 1 + 10H2 40H3 (3.2) 3 1 + 5H − the first Chern class c1(Y3) = 0. Let us calculate the number of complex deformations. The complex structures are parameterized by the coefficients in (3.1) up to the change of coordinates in P4. The number of coefficients in a homogeneous polynomial of degree n in k variables 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 72 is n+k 1 (n + k 1)! ( − ) = − (3.3) n n!(k 1)! − In the case of the quintic in P4 the number of coefficients is

9! (9) = = 126 (3.4) 5 5!4!

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 of H2,1 cohomology group h2,1 = h1,1 χ/2 (3.5) − where h1,1 can be found via the Lefschetz hyperplane theorem [36][34]

1,1 1,1 4 h (Y3) = h (P ) = 1 (3.6) and the Euler characteristic is given by the integral of the highest Chern class over

Y3 3 χ = c3 = 40H 5H = 200, (3.7) P4 − ∧ − ZY3 Z 4 here we have used that 5H is the Poincare dual class to Y3 inside P . Consequently h2,1 = 101 which is consistent with the number of complex deformations found before.

3.2.2 Quintic CY with del Pezzo 6 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 P3(w0, . . . , w3)w4 + P4(w0, . . . , w3)w4 + P5(w0, . . . , w3) = 0 (3.8)

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 6 surface 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 73

[34] at the tip of the cone). The deformations that smooth out the singularity correspond to adding less singular terms to (3.8), i.e. the terms that have bigger powers of w4. The resolution of the singularity in (3.8) 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

(w0, . . . , w3, w4) = (tz0, . . . , tz3, s) (3.9) where (s, t) P1 and (z , . . . , z ) P3. The blowup of the point at t = 0 corresponds ∈ 0 3 ∈ to inserting the P3 instead of this point. Hence the points on the blown up P4 can be parameterized globally by (z , . . . , z ) P3 and (s, t) P1. The projective 0 3 ∈ ∈ 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 and t ( H). Thus the blowup of P4 at a point is a P1 bundle over ∈ O ∈ O − 3 P obtained by projectivization of the direct sum of P3 and P3 ( H) bundles, O O − 4 P = P ( P3 P3 ( H)) (for more details on projective bundles see, e.g. [81][82]). O ⊕ O − In working with projective bundles, we will use the technics similar to [82]. e Using parametrization (3.9), we can write the equation on the blown up P4 as

2 2 P3(z0, . . . , z3)s + P4(z0, . . . , z3)st + P5(z0, . . . , z3)t = 0 (3.10)

This equation is homogeneous of degree two in the coordinates on P1 and degree three in the z ’s. Note, that t ( H), i.e. it has degree ( 1) in the z ’s, and i ∈ O − − i s has degree zero. ∈ O Let us prove that the manifold deﬁned by (3.10) has vanishing ﬁrst Chern class, i.e. it is a CY manifold. Let H be the hyperplane class in P3 and G be the hyperplane

1 1 3 class on the P ﬁbers. Let M = P ( P3 P3 ( H)) denote the P bundle over P . O ⊕ O − The total Chern class of M is

c(M) = (1 + H)4(1 + G)(1 + G H) (3.11) − 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 74

4 3 where (1 + H) is the total Chern class of P , (1 + G) corresponds to s P3 and ∈ O 1 (1 + G H) corresponds to t P3 ( H). Note, that G(G H) = 0 on this P − ∈ O − − bundle and, as usual, H4 = 0 on the P3.

Let Y3 denote the surface embedded in M by (3.10). Since the equation has degree 3 in z and degree two in (s, t), the class Poincare dual to Y M is 3H +2G i 3 ⊂ and the total Chern class is

(1 + H)4(1 + G)(1 + G H) c(Y ) = − . (3.12) 3 1 + 3H + 2G

Expanding c(Y3), it is easy to check that c1(Y3) = 0. 3 The intersection of Y3 with the blown up P at t = 0 is given by the degree three 3 equation P3(z0, . . . , z3) = 0 in P . The surface B deﬁned by this equation is the del Pezzo 6 surface [36][34]. The total Chern class and the Euler character of B

(1 + H)4 c(B) = = 1 + H + 3H2 (3.13) 1 + 3H 2 χ(B) = c2(B) = 3H 3H = 9 (3.14) P3 ∧ ZB Z 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 [46]. Let us check this statement in our example. The canonical class is minus the ﬁrst Chern class that can be found from (3.13)2

K(B) = H. (3.15) −

The coordinate t describes the normal direction to B inside Y . Since t P3 ( H), 3 ∈ O − 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. 2Slightly abusing the notations, we denote by H both the class of P3 and the restriction of this class to B P3. ∈ 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 75

The smoothing of the singularity corresponds to adding less singular terms in (3.8). 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 section 4 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 2 P3(z0, . . . , z3)s + P4(z0, . . . , z3)st + P5(z0, . . . , z3)t = 0 (3.16)

This equation can be rewritten as

2 P3(tz0, . . . , tz3)s + P4(tz0, . . . , tz3)s + P5(tz0, . . . , tz3) = 0 (3.17)

Next we note that, being a projective bundle, M is equivalent [34][81] to P ( P3 (H) O ⊕ P3 ), where locally s and t are sections of P3 (H) and P3 respectively. We further O O O observe that tz , i = 0 ... 3 are also sections of P3 (H) and the equivalence (t, s) i O ∼ (λt, λs) induces the equivalence (tz , . . . , tz , s) (λtz , . . . , λtz , λs). Consequently, 0 i ∼ 0 i if we blow down the section t = 0 of M, then (tz , . . . , tz , s) P4. Now we deﬁne 0 i ∈ (w0, . . . , w3, w4) = (tz0, . . . , tz3, s) and rewrite (3.17) as

2 P3(w0, . . . , w3)w4 + P4(w0, . . . , w3)w4 + P5(w0, . . . , w3) = 0 (3.18)

Not surprisingly, we get back equation (3.8).

Above we have found that there are 11 complex deformations of the dP6 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 (3.12), we get the third Chern class

c (Y ) = 2G3 13HG2 17H2G 8H3. (3.19) 3 3 − − − − 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 76

The Poincare dual class to Y M is 3H + 2G and 3 ∈

χ(Y ) = c (Y ) = c (Y ) (3H + 2G). (3.20) 3 3 3 3 3 ∧ ZY3 ZM In calculating this integral one needs to take into account that G(G H) = 0 on − M. Finally we get χ(Y ) = 176 (3.21) 3 − and h2,1 = h1,1 χ/2 = 90 (3.22) − the number of complex deformations of the del Pezzo singularity is 101 90 = 11, − which is consistent with the number found above.

3.2.3 Quintic CY with 36 conifold singularities

In this subsection we use the methods of geometric transitions [70][71][36] to ﬁnd the quintic CY with conifold singularities, i.e. we describe the upper horizontal arrow in ﬁgure 3.1. Consider the system of two equations in P4 P1 ×

P3u + R3v = 0 (3.23)  P u + R v = 0  2 2 where (u, v) P1 and P , R denote polynomials of degree n in P4. ∈ n n Suppose that at least one of the polynomials P3,R3,P2 and R2 is non zero, then we can solve for u, v and substitute in the second equation, where we get

P R R P = 0 (3.24) 3 2 − 3 2

4 a non generic quintic in P . 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 (3.23) describes the blowup of the singularities, since every singular point is replaced by the P1 and the resulting manifold is non singular. 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 77

Let H be the hyperplane class of P4 and G by the hyperplane class of P1, then the total Chern class of Y3 is (1 + H)5(1 + G)2 c = , (3.25) (1 + 3H + G)(1 + 2H + G) since c1 = 0, Y3 is a CY. By Lefschetz hyperplane theorem h1,1(Y ) = h1,1(P4 P1) = 2, there are only 3 × 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 (3.24) 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

h2,1 = 66. (3.26)

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.

3.2.4 Quintic CY with del Pezzo 6 singularity and 32 conifold singularities

The equation for the quintic CY manifold with the blown up dP6 singularity was found in (3.10). Here we reproduce it for convenience

2 2 P3(zi)s + P4(zi)st + P5(zi)t = 0 (3.27)

This equation describes an embedding of the CY manifold in the P1 bundle M =

3 P ( P3 P3 ( H)). As before, (z , . . . , z ) P and (s, t) are the coordinates on O ⊕ O − 0 3 ∈ the P1 ﬁbers over P3. 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 78

In order to have more Kahler deformations we need to embed (3.27) 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 ×

(P1s + P2t)u + (Q1s + Q2t)v = 0 (3.28)  (R s + R t)u + (S s + S t)v = 0  2 3 2 3 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

(1 + H)4(1 + G)(1 + G H)(1 + K)2 c = − (3.29) (1 + H + G + K)(1 + 2H + G + K) and it’s easy to see that the ﬁrst Chern class is zero. For generic points on the P1 bundle over P3 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 (3.27)

(P S Q R )s2 + (P S + P S Q R Q R )st + (P S Q R )t2 = 0 (3.30) 1 2 − 1 2 1 3 2 2 − 1 3 − 2 2 2 3 − 2 3

The CY manifold deﬁned in (3.28) has the following characteristics

χ = c = 112; 3 − ZY3 h1,1 = 3; (3.31)

h2,1 = h1,1 χ/2 = 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 (3.30). These singularities occur when all four equations in 3.2. Del Pezzo 6 and conifold singularities on the quintic CY 79

(3.28) vanish

R2s + R3t = 0

S2s + S3t = 0

P1s + P2t = 0 (3.32)

Q1s + Q2t = 0

(3.33)

The number of solutions equals the number of intersections of the corresponding classes (2H+G)2(H+G) = 32, where M is the P1 bundle over P3 and G(G H) = M − 0. R The right vertical arrow corresponds to smoothing of del Pezzo singularity in the presence of conifold singularities. Before the transition the CY has h2,1 = 59 deformations and after the transition it has h2,1 = 66 deformations. Hence the number of complex deformations of dP singularity is 66 59 = 7 which is less than 6 − c∨(E ) 1 = 11. This is related to the fact that the del Pezzo at the tip of the cone 6 − is not generic. The equation of the del Pezzo can be found by restricting (3.28) to t = 0, s = 1 section

P1u + Q1v = 0 (3.34)  R u + S v = 0  2 2 This del Pezzo contains a two-cycle α that is non trivial within the full CY and  doesn’t intersect the canonical class inside dP6. In the rest of this subsection we will argue that α 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 (3.28) by the following coordinate redeﬁnition (w0, . . . , w3, w4) = 3.3. Compact CY manifolds with del Pezzo singularities 80

(tz0, . . . , tz3, s) (compare to the discussion after equation (3.17))

(P1w4 + P2)u + (Q1w4 + Q2)v = 0 (3.35)  (R w + R )u + (S w + S )v = 0  2 4 3 2 4 3 If we blow down the P1, then we get the quintic CY with 32 conifold singularities

1 and a dP6 singularity. For a ﬁnite size P , 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.3

3.3 Compact CY manifolds with del Pezzo singularities

The non compact CY manifolds with del Pezzo singularities are known [48][49]. The dP singularities for 5 n 8 and for the cone over P1 P1 can be represented n ≤ ≤ × as complete intersections.4 The CY cones over P2 and dP for 1 n 4 are n ≤ ≤ not complete intersections. The compact CY manifolds for complete intersection singularities where presented in [80]. Our construction is different from [80]. It enables one to 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 ≤ 3Formally we can prove this by calculating the corresponding Poincarédual classes. The Poincarédual of P1 on the blown up conifold is H3G – this is the P1 parameterized by (u, v). The Poincarédual of the canonical class on dP6 is (G H)(H + K)(2H + K)( H), where (G H) 1 − − − restricts to t = 0 section of the P bundle, (H + K)(2H + K) restricts to dP6 in (3.34), while the restriction of ( H) is the canonical class on dP6 (see Eq. (3.15)). The class that doesn’t intersect − 3 ( H) inside dP6 is dual to (G H)(H + K)(2H + K)(2H 3G) = 4H G, q.e.d. − 4Note, that in 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. − 3.3. Compact CY manifolds with del Pezzo singularities 81 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 P1 P1 we will represent ≥ × all complex deformations in our construction.

3.3.1 General construction

At ﬁrst we present the construction in the case of dP6 singularity, and then give a more general formulation.

3 The input data is the embedding of dP6 surface in P 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

3 1. Find the canonical class on B = dP6 in terms of a restriction of a class on P . Let us denote this class as K H1,1(P3). K can be found from expanding the ∈ total Chern class of B

(1 + H)4 c(B) = = 1 + H + ... (3.36) 1 + 3H

thus K = c (B) = H. − 1 −

1 3 2. Construct the P ﬁber bundle over P as the projectivisation M = P ( P3 O ⊕ P3 (K)). O

3 3. The Calabi-Yau Y3 is given by an equation of degree 3 in P and degree 2 in

the coordinates on the ﬁber. The total Chern class of Y3 is

(1 + H)4(1 + G H)(1 + G) c(Y ) = − (3.37) 3 1 + 3H + 2G

this has a vanishing ﬁrst Chern class. By construction, this Calabi-Yau has a del Pezzo singularity at t = 0. 3.3. Compact CY manifolds with del Pezzo singularities 82

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 [36]. 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 surface B X, deﬁned in terms of two ⊂ equations with the corresponding classes L ,L H1,1(X), 1 2 ∈ c(X) c(B) = = 1 + c1(X) L1 L2 + ... (3.38) (1 + L1)(1 + L2) − − thus the canonical class of X is obtained by the restriction of K = L + L 1 2 − c1(X).

2. Second, we construct the P1 fiber bundle over X as the projectivisation M = P ( (K)). OX ⊕ OX 3. In the case of two equations, the Calabi-Yau manifold Y M is not unique. 3 ⊂ 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, 5 the third one has L1 and L2 + 2G.

As an example, let us describe the ﬁrst system. The ﬁrst equation in this

system is given by L1 in X and has degree 2 in the coordinates on the ﬁbers.

The second equation is L2 in X. The total Chern class is c(X)(1 + G + K)(1 + G) c(Y3) = . (3.39) (1 + L1 + 2G)(1 + L2) Since K = L + L c (X), it is straightforward to check that the first Chern 1 2 − 1 class is trivial. 5 1,1 1,1 Here L1,L2 H (M) are defined via the pull back of the corresponding classes in H (X) with respect to the∈ projection of P1 the fibers π : M X. → 3.3. Compact CY manifolds with del Pezzo singularities 83

Let us show how this program works in an example of a CY cone over B = P1 P1. × The P1 P1 surface can be embedded in P3 by a generic degree two polynomial × equation [36][34]

P2(zi) = 0 (3.40) where (z , . . . , z ) P3. 6 0 3 ∈ The ﬁrst step of the program is to ﬁnd the canonical class of B in terms of a class in P3. Let H be the hyperplane class of P3. Then the total Chern class of B is

(1 + H)4 c(B) = = 1 + 2H + 2H2. (3.41) 1 + 2H

The canonical class is K(B) = c (B) = 2H (3.42) − 1 − 1 Next we construct the P bundle M = P ( P3 P3 (K)) with the coordinates O ⊕ O (s, t) along the ﬁbers, where locally s P3 and t P3 ( 2H). The equation that ∈ O ∈ O − describes the embedding of the CY manifold Y3 in M is

2 2 P2(zi)s + P4(zi)st + P6(zi)t = 0 (3.43)

This equation is homogeneous in z of degree two, since t has degree 2. i − The section of M at t = 0 is contractable and the intersection with the Y3 is P (z ) = 0, i.e. Y is the CY cone over P1 P1 near t = 0. 2 i 3 × The total Chern class of Y3 is

(1 + H)4(1 + G)(1 + G 2H) c(Y ) = − (3.44) 3 1 + 2H + 2G

It’s easy to check that c1(Y3) = 0.

6 3 By coordinate redeﬁnition in P one can represent the equation as z0z3 = z1z2. The solutions 1 1 of this equation can be parameterized by the points (x1, y1) (x2, y2) P P as (z0, z1, z2, z3) = 1 ×1 3 ∈ × (x1x2, x1y2, y1x2, y1y2). This is the Segre embedding P P P . × ⊂ 3.3. Compact CY manifolds with del Pezzo singularities 84

3.3.2 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 A.

The procedure is similar to the deformation of the dP6 singularity described in section 3.2. As before let Y M be an embedding of the CY threefold Y in M, a 3 ⊂ 3 P1 bundle over products of (weighted) projective spaces. If we blow down the section of the P1 bundle that contains the del Pezzo, then M becomes a toric variety that we denote by V . After the blow down, 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.7

Table 1. Some characteristics of del Pezzo surfaces.

del Pezzo # two-cycles # (-2) two-cycles Dynkin diagram c∨ 1 − P2 1 0 0 0 P1 P1 2 1 A 1 × 1 dP1 2 0 0 0

dP2 3 1 A1 1 dP 4 3 A A 3 3 2 × 1 dP4 5 4 A4 4

dP5 6 5 D5 7

dP6 7 6 E6 11

dP7 8 7 E7 17

dP8 9 8 E8 29

7 In the example of dP6 singularity on the quintic, the projective bundle is M = P ( P3 3 4 O ⊕ P3 ( H)), the manifold V, obtained by blowing down the exceptional P in M, is P , and the Osingular− equation is the singular quintic in P4. 3.3. Compact CY manifolds with del Pezzo singularities 85

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 − P2 0 0 0 0 P1 P1 1 1 1 1 × dP1 0 0 0 0

dP2 1 0 1 0

dP3 3 1 3 1

dP4 4 3 4 3

dP5 5 5 7 7

dP6 6 6 11 11

dP7 7 7 17 17

dP8 8 8 29 29

Let n 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 is c∨(E ) n − 1, where c∨(En) is the dual Coxeter number of En. These deformations can be per- formed 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, dP , dP , × 5 6 dP , dP ). If some of the ( 2) two-cycles become non trivial within the CY, then 7 8 − 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

[67], the generic deformations of the cones over dP2 and dP3 were constructed. The list compact embeddings of del Pezzo singularities and their deformations 3.3. Compact CY manifolds with del Pezzo singularities 86 can be found in the Appendix A. Chapter 4

SUSY breaking in String Theory

Supersymmetry is one of the most fascinating subjects in contemporary high energy physics. It combines exceptional theoretical beauty with exceptional resistance to experimental observations. Supersymmetry is the only possible extension of the Poincarésymmetry of S-matrix in the quantum field theory [83]. All other possible extensions, such as the flavor symmetries, are strictly internal, i.e. they commute with the Lorentz generators and translations in Minkowski space [84]. Supersymme- try is supposed to be necessary for the solution of the hierarchy problem, it cancels the quadratic divergences which otherwise would push the masses to the cutoff scale, the Planck mass. The lightest supersymmetric particle is stable and may account for the dark matter. Also, with higher degree of supersymmtry, it is possible to find exactly the effective potential for the interacting gauge theories [85]. But, in spite of all the theoretical successes, so far there has been no experimental evidence for supersymmetry and a lot of hope is know connected with the upcoming LHC experiment. One of the main problems with supersymmetry is the abundance of possible models for its breaking at low energies. The masses of the superpartners are constrained only by the observational lower bounds and by some general naturalness principles; in some models there are also bounds coming from the cosmology [86]. However the constraints usually leave a big window of possible masses of superpartners.

87 88

The simplest ways to break SUSY is to add the soft SUSY breaking terms. This method is useful if one wants to study some general properties of SUSY breaking without worrying about the particular mechanisms that generate the breaking. How- ever this mechanism is unsatisfactory if we want to study the unification of gauge theories with the gravity. In supergravity the supersymmetry transformations may depend on the coordinates, i.e. the supersymmetry becomes a local gauge symmetry. The gauged supersymmetry can be broken spontaneously in a non-supersymmetric vacuum. Thus theoretically more consistent approach would be to consider a supersymmetric theory and find a vacuum that breaks SUSY. The problem is that SUSY puts some stringent restrictions on the tree level masses for spontaneous SUSY breaking. In particular, the sums of the mass squared for the fermions and for the bosons are equal [87]. Usually this leads to the presence of tachyons in the spectrum. Thus it turned out to be quite difficult to find a spontaneous SUSY breaking in the observable sector. The standard approach now is to assume that SUSY is broken in a hidden sector and is mediated to the observable sector via some messenger fields. The SUSY breaking creates the mass splitings for the messenger fields that generate the masses for the superpartners in the visible sector at the loop level. In this chapter we will be concerned with the spontaneous SUSY breaking in the hidden sector. The possibilities for the mediation of SUSY breaking will be discussed in the next chapter. The global vacuum that breaks SUSY should have a positive energy density: if v is the vacuum state that is not annihilated by the SUSY generator Q v = | i | i v′ = 0, then from the SUSY algebra the translationally invariant vacuum satisfies | i 6 v H v v Q†Q v = v′ v′ > 0. By the same token any supersymmetric vacuum, h | | i ∼ h | | i h | i Q v = 0, has zero vacuum energy. The situation when any vacuum breaks SUSY | i spontaneously is quite rare. Usually there are some supersymmetric vacua and some vacua that break SUSY. In this case one has to make sure that SUSY breaking vacuum doesn’t have any classical instabilities, i.e. tachyonic directions towards a 4.1. ISS model 89 supersymmetric vacuum, and the life-time of this metastable vacuum is long enough. Recently the interest in metastable SUSY breaking has been revived after the construction of Intriligator, Seiberg and Shih [33]. We will review the ISS construction in the next section. Realizations of the ISS mechanism in string theory have since been found [88] [89] [90] [91]. Most of this chapter will be devoted to the description of the ISS model in a system of D-branes in type IIB string theory.

4.1 ISS model

In this section we review the Intriligator, Seiberg, and Shih model of meta-stable spontaneous SUSY breaking (the ISS model) [33]. The superpotential is

W = hTr(ϕ ˜Φϕ) hµ2TrΦ, (4.1) − where Φ is N N matrix, ϕ andϕ ˜ are N N and N N matrices respectively. f × f f × × f The global symmetries of the superpotential and the corresponding charges are

SU(Nf ) SU(N) U(1)B U(1)R

Φ Nf N¯f 1 0 2 ⊗ (4.2) ϕ Nf N¯ 1 0 ϕ˜ N¯ N 1 0 f − The F-term equation for the Φ ﬁeld is

ϕ ϕ˜ = µ2δ . (4.3) ik · kj ij

If Nf > N, then the matrix on the left has the rank at most N, whereas the matrix on the right has the rank Nf . Consequently, the F-term equations cannot be satisﬁed and SUSY is broken. The ﬁelds ϕ andϕ ˜ will get the VEVs in order to satisfy as many F-term equations as possible. Using the global symmetries of the 4.1. ISS model 90 superpotential, we can rotate the VEVs to take the form

µA 1 ϕ = ϕ˜ = µA− 0 (4.4)  0    where A is an N by N matrix. It can be represented as a product of the unitary and hermitian matrices U and R,

A = U R. (4.5) · The matrix U is the goldstone boson for the spontaneous breaking of the SU(N) × U(1)B global symmetry and R is a classical modulus. R is stabilized by the one-loop contributions at R = 1N N . Thus it is reasonable to consider the ﬂuctuations of × the ﬁelds near the vacuum A = µ1

Y Z˜ µ1 + χ Φ = ϕ = ϕ˜ = (µ1 +χ ˜ ρ˜) (4.6)  ZX   ρ      In this vacuum the superpotential takes the following form

W = hTr µY (χ +χ ˜) + µρZ˜ + µZρ˜ +ρXρ ˜ µ2X + ... (4.7) − The fields Z, Z,˜ ρ, ρ,˜ Y, (χ +χ ˜) are massive. The vacuum energy comes from the F-terms for the X field V = h2µ4(N N), (4.8) f − thus the fermion superpartner of X,ΨX , is the goldstino. The classical pseudomod- uli are X and Reχ Reχ˜ (this is the field corresponding to the infinitesimal part of − R in 4.5). These fields are massless at the tree level but get a positive mass squared at the one-loop level. The other massless bosonic fields are the goldstone bosons for the broken global symmetries. Since the global symmetry groups are compact, the goldstone bosons parameterize the points on the compact group manifolds and cannot lead to a runaway. Hence ISS have argued that the SUSY breaking vacuum (4.6) is metastable. 4.1. ISS model 91

4.1.1 SUSY restoration in ISS model

If all the symmetries in the model are global, then there are no supersymmetric vacua at any finite values of the fields. Let us show that supersymmetry can be restored at a finite point in the field space if we gauge the SU(N) group. Suppose we give a VEV to the Φ field in (4.1). Then the fields ϕ,ϕ ˜ become massive. Below Φ the SU(N) gauge group confines at some scale Λ and develops h i L the gaugino condensate. The value of the condensate depends on the VEV of Φ and generates the superpotential [92][33]

det Φ 1/N δW = NΛ3 = N hNf (4.9) L Nf 3N Λm − where Λm is the value of the Landau pole for the gauge group SU(N) at high energies. The scales ΛL and Λm are related by matching the gauge coupling at the energy E hΦ . The running of the gauge coupling at the low energy and at the ∼ h i high energy is 3N 8π +iθ ΛL e− g2(E) = E < hΦ (4.10) E h i 3N Nf 8π +iθ Λm − e− g2(E) = E > hΦ (4.11) E h i 3N N 3N − f thus ΛL = Λm det(hΦ). The total superpotential for Φ is det Φ 1/N W (Φ) = N hNf hµ2TrΦ. (4.12) Nf 3N − Λm − It has a supersymmetric minimum at

2N − µ Nf N hΦ = Λmǫ 1Nf ǫ (4.13) h i ≡ Λm

Above the Landau pole at E = Λm the theory has a Seiberg dual description[93][33] in terms of the gauge group SU(N ) = SU(N N) with N massive ﬂavors Q and c f − f Q˜ with W = µTrQQ.˜ (4.14) 4.2. F-term SUSY breaking in IIB String Theory 92

Below the conﬁnement scale of the SU(Nc), the theory is described in terms of the magnetic dual gauge group SU(N) and meson ﬁelds Φ = QQ˜ neutral under both

SU(Nc) and SU(N). One also has to add Nf ﬁelds charged under SU(N) [93] with the total superpotential (4.1). In the end of this chapter we will demonstrate that the ISS model in its magnetic phase below Λm can arise at the end of a duality cascade for some set of fractional branes at the Suspended Pinch Point singularity. The theory above Λm is then diﬀerent from the SU(Nc) theory in [33], i.e. there exist several consistent UV completions of the same low energy theory. Now we tern to the description of the ISS model or, more generally, the F-term SUSY breaking in terms of the type IIB D-branes.

4.2 F-term SUSY breaking in IIB String Theory

We will consider gauge theories on D-branes near singularities of Calabi-Yau manifolds in type IIB String theory. Our goal will be to identify a general geometric criterion for the existence of F-type SUSY breaking, and to use this insight to construct simple examples of D-brane systems that exhibit metastable SUSY breaking. F-type SUSY breaking corresponds to the unsolvability of F-term equations

∂W = 0 (4.15) ∂Φ 6 where W (Φ) is a superpotential depending on the chiral field Φ. The simplest example of this type is the Polonyi model, consisting of a single chiral field with superpotential W (Φ) = fΦ in which SUSY is broken by the non zero vacuum energy V f 2. ∼ | | For the gauge theory on the D-branes in type IIB string theory, deformations of the superpotential correspond to complex deformations in the local geometry. The deformed geometry still satisfies the Calabi-Yau condition and the D-brane 4.2. F-term SUSY breaking in IIB String Theory 93 lagrangian is fully supersymmetric but the vacuum configuration of the gauge theory breaks SUSY spontaneously. In the geometric setting, this corresponds to a D- brane configuration that, while submerged inside a supersymmetric background, gets trapped in a non-supersymmetric ground state.

2 As a simple illustrative example, consider type IIB string theory on a C /Z2 orbifold singularity [29][65], with N fractional D5-branes wrapped on the collapsed 2-cycle. The corresponding field theory consists of a U(N) gauge theory with a complex adjoint chiral field Φ. Since the Z2 orbifold locus defines a non-isolated singularity inside C3, the fractional D5-branes are free to move along a complex line. The location of the N branes along the non-isolated singularity is parameterized by the N diagonal entries of the complex field Φ. As we discuss in more detail in section 2, there exists a deformation of the singularity that corresponds to adding the F-term W = ζ TrΦ (4.16) to the superpotential. Geometrically, the parameter ζ is proportional to the period of the holomorphic two-form over the deformed 2-cycle. The fractional D-brane gauge theory then breaks SUSY in a similar way to the Polonyi model. This simple observation lies at the heart of many type IIB D-brane constructions of gauge theories that exhibit F-term SUSY breaking.1 SUSY breaking via D-terms can be described analogously.2 We wish to use this simple geometric insight to construct more interesting gauge theories with DSB, and in particular, with ISS-type SUSY breaking and restoration. When viewed as a quiver theory, the ISS model has two nodes, a “color” node with

1F-term SUSY breaking in type IIB D-brane constructions naturally involves deformed non- isolated singularities, that support finite size 2-cycles which D5-branes can wrap [29]. Deformations of isolated singularities correspond to 3-cycles that in type IIB cannot by wrapped by the space- time filling D-branes. 2Turning on the FI parameters of the type IIB D-brane gauge theory amounts to blowing up the collapsed two-cycles of a CY singularity. These blowup modes are Kähler deformations of the geometry, and are somewhat harder to control in a type IIB setup than the complex structure deformations that we use in our study. 4.2. F-term SUSY breaking in IIB String Theory 94

gauge group SU(N) and a “flavor” node with SU(Nf ) symmetry. The “flavor” node has an adjoint field. This suggests that the flavor node must be represented by a stack of Nf fractional branes on a non-isolated singularity. The “color” node, on the other hand, does not have an adjoint, and thus corresponds to branes that are bound to a fixed location. The natural representation for the color node is via a stack of N branes placed at an isolated singularity. Our geometric recipe for realizing an ISS model in IIB string theory is as follows:

1. Find a Calabi-Yau geometry with a non-isolated singularity passing through an isolated singularity such that there exists a deformation of the non-isolated singularity.

2. Put some number of D-branes on the isolated singularity and some number of fractional branes on the non-isolated singularity. By conservation of charge, the branes can not leave the non-isolated singularity.

3. When we deform the non-isolated singularity, an F-term gets generated that results in dynamical SUSY breaking. The fractional branes have a non-zero volume, and their tension lifts the vacuum energy above that of the SUSY vacuum.

4. There is a classical modulus corresponding to the motion of the fractional branes along the non-isolated singularity. This modulus can be ﬁxed in a way similar to ISS, by the interaction with the branes at the isolated singularity.

Following this recipe we will geometrically engineer, via an appropriate choice of the geometry and fractional branes, gauge theories that are known to exhibit metastable DSB. The eventual goal is to fully explain in geometric terms all field theoretic ingredients: the field content and couplings, the meta-stability of the SUSY-breaking vacuum, and the process of SUSY restoration. While in our examples we will be able identify all these ingredients, we will not have sufficient dynamical control over 4.2. F-term SUSY breaking in IIB String Theory 95 the D-brane set-up to prove the existence of a meta-stable state on the geometric side. Rather, by controlling the geometric engineering dictionary, we can rely on the field theory analysis to demonstrate that the system has the required properties. This wish to have geometrical control over the field theory parameters also mo- tivates why we prefer to work with local IIB D-brane constructions. Although we will work in a probe approximation, in principle we could extend our analysis to the case where the number of branes becomes large. In this AdS/CFT limit, there should exist a precise dictionary between the couplings in the field theory and the asymptotic boundary conditions on the supergravity fields [26] [27] . By changing these boundary conditions one can tune the UV couplings. This in principle allows full control over the IR couplings and dynamics. As a warmup example, we will use the F-term deformation of the D-branes gauge

2 theory on C /Z2 singularity. After that we present a realization of the meta-stable supersymmetry breaking via D-branes on the suspended pinch point singularity. We ﬁnd that supersymmetry restoration involves a geometric transition.

The material in this chapter is based on the publication [91] which has a lot of overlap with the results reported in [90].3 In agreement with our observations, in [90] the F-term SUSY breaking takes place due to the presence of fractional D5- branes on slightly deformed non-isolated singularities. One of the main points in [90] was to show that the deformation can be computed exactly in the framework of geometric transitions: this is an important step in ﬁnding calculable examples of SUSY breaking in string theory. The main point in my work will be to identify simple geometric criteria for the existence of SUSY breaking vacua that can have more direct applications in model building.

3The IIB string realizations of DSB found in [90] were motivated by the earlier related work [94] in type IIA theory. 4.3. Deformed Z2 orbifold singularity 96

4.3 Deformed Z2 orbifold singularity

2 The C /Z2 singularity, or A1 singularity, is described by the following complex equation in C3 cd = a2, (a, b, c) C3 . (4.17) ∈ 2 A D3-brane on C /Z2 has a single image brane. The brane and image brane recombine in two fractional branes. Correspondingly, the quiver gauge theory for N

D3-branes at the A1 singularity has two U(N) gauge groups. It also has two adjoint matter ﬁelds Φ1 and Φ2 (one for each gauge group), and two pairs of chiral ﬁelds Ai and Bj i, j = 1, 2 in the bifundamental representations (N, N¯) and (N,N¯ ) [29][65]. The superpotential reads

W = g TrΦ (A B B A ) + g TrΦ (A B B A ) (4.18) 1 1 2 − 1 2 2 2 1 − 2 1 A D3-brane has 3 transverse complex dimensions. The transverse space C2/Z C 2 × has a non-isolated A1 singularity. It is therefore possible to separate the fractional branes. This corresponds to giving different vevs to the two adjoint fields. In the limit of infinite separation one can consider a theory with only one type of fractional branes. This theory consists of a U(N) gauge field with one adjoint matter field and no fundamental matter. Let us add an F and a D-term

W = ζ Tr(Φ Φ ),V = ξ Tr(D D ). (4.19) F 2 − 1 D 2 − 1

The resulting F and D-term equations for A and B fields give Φ1 = Φ2, then the F-term equation for (Φ Φ ) and the D-term equation for (D D ) are 1 − 2 1 − 2 A B B A = ζ, (4.20) 1 2 − 1 2 A 2 + B 2 A 2 B 2 = ξ. | 1| | 1| − | 2| − | 2| For generic ζ and ξ, some of the A and B fields acquire vevs and break the U(N) × U(N) symmetry to a diagonal U(N). This corresponds to joining the 2N fractional 4.3. Deformed Z2 orbifold singularity 97 branes into N D3-branes. The space of solutions of the F and D-term equations is the space where the D3-brane moves, which turns out to be a deformed A1 singularity described by the equation4 cd = a(a ζ), (4.21) − where c = A1A2, d = B1B2 and a = A1B2 = B1A2 + ζ are the gauge invariant combinations of the fields (in the last definition we used the F-term equation for the Φ field). The F-term coefficient ζ deforms the singularity, the D-term coefficient, or FI parameter, ξ represents a resolution of the Z2 singularity. In two complex dimensions both the resolution and the deformation correspond to inserting a two-cycle, E P1, ∼ instead of the singular point. The parameters ξ and ζ are identified with the periods of the Kahler form and the holomorphic two-form on the blown up 2-cycle E

ξ = J (4.22) ZE ζ = Ω(2) . ZE The non-supersymmetric vacuum state arises in the regime where the vevs of the two adjoint fields Φ1 and Φ2 are both different. Geometrically, this amounts to separating the two stacks of fractional branes. The bifundamental fields (Ai,Bi), which arise as the ground states of open strings that stretch between the two fractional branes, then become massive. In the deformed theory, the F-term equations can not be satisfied and SUSY is broken. In the extreme case, where one of the two stacks of fractional branes has been moved off to infinity, so that e.g. Φ , the system h 2i → ∞ reduces to the Polonyi model: a single U(N) gauge theory with a complex adjoint Φ and superpotential W = ζ TrΦ . The vacuum energy V = N ζ 2 is interpreted 1 1 | | as the tension of the N fractional branes wrapped over the deformed two-cycle.

4The general deformations of orbifold singularities of C2 where found by Kronheimer [66] as some hyperkahler quotients. Douglas and Moore noticed [29] that these hyperkahler quotients are described by the F and D-term equations for D-branes at the corresponding orbifold singularities. 4.3. Deformed Z2 orbifold singularity 98

Strictly speaking the single stack of fractional branes on a deformed singularity is a supersymmetric configuration (one manifestation is that the spectrum of particles in Polonyi model is supersymmetric). In order to break SUSY we really need the second stack of different fractional branes on a large but finite distance. In this case, the SUSY breaking vacuum is not stable due to the attraction between the two stacks of branes.

Before we get to our main example of the SPP singularity, let us make a few comments: 1. The gauge theory on N fractional branes on the C2/Z singularity is an = 2 2 N U(N) theory. If we deform the singularity, then SUSY is broken, whereas in general, = 2 theories are not assumed to have SUSY breaking vacua (see, e.g., Appendix N D of [33]). The point is that the SUSY breaking occurs in the U(1) part of U(N) that decouples from SU(N). Moreover the = 2 U(1) theory consists of two non- N interacting = 1 theories: a vector boson and a chiral field. Thus the chiral field N ϕ = TrΦ, responsible for SUSY breaking, is decoupled from the rest of the fields in = 2 U(N) and SUSY is broken in the same way as in the Polonyi model. N 2. In general, we consider = 1 theories on isolated singularities that intersect N non-isolated singularities. With appropriate tuning of the couplings, the fractional branes wrapping the non-isolated cycles provide an = 2 subsector in the = 1 N N quiver. Removing the D-branes along the non-isolated singularity reduces the field theory on their world volume to = 2 SYM. For this reason the fractional branes on N the non-isolated singularity can be called = 2 fractional branes [76][95]. Similarly N to C2/Z example, the presence of = 2 fractional branes on slightly deformed 2 N non-isolated singularity breaks SUSY. 3. The use of = 2 fractional branes is the distinguishing property of our con- N struction from SUSY breaking by obstructed geometry [76][96][97]. The presence of the non-isolated singularity enables the relevant RR-fluxes escape to infinity without creating a contradiction with the geometric deformations. In this way one can 4.4. ISS from the Suspended Pinch Point singularity 99 avoid the generic runaway behavior (see, e.g., [78] [98]) of obstructed geometries (in our case we still need to take the one loop corrections to the potential into account in order to stabilize the flat direction along the non-isolated singularity).

4.4 ISS from the Suspended Pinch Point singularity

In this section, we will show how to engineer a gauge theory with ISS-type SUSY breaking by placing fractional branes on the suspended pinch point (SPP) singularity. First, however, we summarize the arguments that lead us to consider this particular system. As we have seen in the previous section, several aspects of the ISS model are

2 quite similar to the C /Z2 = A1 quiver theory. The term linear in the adjoint in the ISS superpotential is the ζ deformation of the A1 singularity. Both models have two gauge groups (the global flavor symmetry SU(Nf ) in ISS can be thought of as a weakly coupled gauge symmetry). The flavor gauge group is bigger than the color gauge group – this can be achieved in the A1 quiver by introducing an excess of fractional branes of one type. The vevs of bifundamental fields break SU(N ) SU(N) SU(N) SU(N N). The breaking of SU(N) SU(N) f × → diag × f − × → SU(N)diag corresponds to recombination of N pairs of fractional branes into N (supersymmetric) D3-branes. The vacuum energy is proportional to the tension of the remaining N N fractional branes. f − There is however an important difference between the two systems. In ISS it is crucial that the color node SU(N) doesn’t have an adjoint field and that all the

2 classical moduli are lifted by one loop corrections. In the C /Z2 orbifold there is also 2 an adjoint in the “ﬂavor” node. Giving equal vevs to the two adjoints in the C /Z2 quiver corresponds to the “center of mass” motion of the system of branes along the 4.4. ISS from the Suspended Pinch Point singularity 100 non-isolated singularity. This mode doesn’t receive corrections and remains a ﬂat direction.

2 Thus, the key distinguishing feature of ISS relative to the C /Z2 model is that the color gauge group SU(N) has no adjoints. For constructing a geometric set-up, we need a mechanism that fixes the position of the N D3-branes. The gauge theories without adjoint fields are naturally engineered by placing D-branes on isolated singularities. Our strategy will be to find an example of a geometry that has a non-isolated

A1 singularity that at some point gets enhanced by an isolated singularity. The fractional branes on the A1 will provide the SU(Nf ) symmetry; they interact with N branes at the isolated singularity, that carry the SU(N) color gauge group. Such systems are easy to engineer. The most basic examples are provided by the generalized conifolds [99], the simplest of which is the suspended pinch point singularity.5 A similar mechanism of dynamical SUSY breaking for the SPP singularity was previously considered in [95]: SUSY is broken by the presence of D-branes on the deformed A1 singularity. The essential diﬀerence is that in our case the A1 singularity is deformed without the conifold transitions within the SPP geometry. In fact, we will show that the conifold transition is responsible for SUSY restoration.

4.4.1 D-branes at a deformed SPP singularity

The suspended pinch point (SPP) singularity may be obtained via a partial resolution of a Z Z singularity [28]. It is described by the following complex equation 2 × 2 in C4 cd = a2b, (a, b, c, d) C4 . (4.23) ∈ 5The relevance of generalized conifolds and, in particular, the suspended pinch point was stressed to us by Igor Klebanov. See also [88][100][101] for the earlier constructions of the metastable SUSY breaking vacua in the generalized conifolds. 4.4. ISS from the Suspended Pinch Point singularity 101

1 U(N)

X ~ Y ~ X Y

Z 2 3 ~ U(N) Z U(N)

Figure 4.1: Quiver gauge theory for N D3-branes at a suspended pinch point singularity.

There is a C2/Z singularity along b = 0. The quiver gauge theory for N D3-branes 2 6 at the SPP singularity is shown in ﬁgure 4.1. It was derived in [28] by turning on an FI parameter ξ in the Z Z quiver gauge theory, and working out the resulting 2 × 2 symmetry breaking pattern. The superpotential of the SPP quiver gauge theory reads W = Tr Φ(YY˜ XX˜ ) + h(ZZX˜ X˜ ZZY˜ Y˜ ) (4.24) − − 1/2 where h is a dimensionful parameter (related to the FI parameter via h = ξ− ). As a quick consistency check that this theory corresponds to a stack of D3-branes on the SPP singularity, consider the F-term equations for a single D3-brane. The gauge invariant combinations of the ﬁelds are

a = XX˜ = Y˜ Y c = XY˜ Z˜

b = ZZ˜ d = Y XZ˜ (4.25) where we used the F-term equation for Φ. These quantities (a, b, c, d) satisfy the constraint cd = a2b, which is the same as the equation for the SPP singularity. 4.4. ISS from the Suspended Pinch Point singularity 102

Following our recipe as outlined in the introduction, we now deform the non- isolated A1 singularity inside the SPP as follows

cd = a(a ζ)b. (4.26) −

This deformation removes the A1 singularity, replacing it by a ﬁnite size 2-cycle. The deformed SPP geometry has two conifold singularities, located at a = 0 and a = ζ, with all other coordinates equal to zero. In the ﬁeld theory, the above deformation corresponds to adding an F -term of the form

W = ζTr(Φ hZZ˜ ). (4.27) ζ − −

This extra superpotential term is chosen such that the F-term equations for Φ and Z XX˜ YY˜ ζ = 0, Z˜(YY˜ XX˜ + ζ) = 0, (4.28) − − − are compatible. The correspondence between (4.27) and (4.26) is easily veriﬁed. Again, consider the gauge theory on a single D3-brane. In view of the deformed F-term equation, the quantity a now needs to be deﬁned via

a = YY˜ = XX˜ + ζ. (4.29)

The constraint equation thus gets modified to cd = a(a ζ)b, which is the equation − for the deformed SPP singularity. As we increase ζ, the two conifold singularities at a = 0 and a = ζ become geometrically separated and the D-branes end up on either of the two conifolds. The field theory should thus contain two copies of the conifold quiver gauge theory. To verify this, consider the vacuum Y = Y˜ = √ζI, which solves both the F-term equations (4.28) and the D-term equations Y 2 Y˜ 2 = 0. These vevs break the | | − | | gauge group SU(N) SU(N) to SU(N) and give a mass to the Higgs-Goldstone 1× 3 diag field Y = 1 (Y Y˜ ). Substituting the remaining fields in the superpotential, one − √2 − 4.4. ISS from the Suspended Pinch Point singularity 103

finds that the fields Φ and Y = 1 (Y +Y˜ ) are also massive. The surviving massless + √2 fields with the superpotential

W = h(ZZX˜ X˜ ZZ˜ XX˜ ) (4.30) con − reproduce the conifold quiver gauge theory. In general, both X and Y have vevs and the D-branes split into two stacks

N1 + N2 = N that live on the two conifolds. Note, that the Z ﬁeld in (4.27) corresponds to strings stretching between the two conifolds. The mass of this ﬁeld is proportional to the length of the string given by the size of the deformed two-cycle.

4.4.2 Dynamical SUSY breaking

A straightforward way to generate dynamical SUSY breaking is to reproduce the ISS model by placing some fractional branes on the SPP singularity. Suppose that there are Nf = N + M fractional branes corresponding to node 1 in figure 4.1, N fractional branes corresponding to node 3, and no fractional branes at node 2. The reduced quiver diagram is shown in figure 4.2. The superpotential for this quiver gauge theory is W = h ζ Tr(Φ) h Tr ΦY Y˜ , (4.31) − which is the same as the ISS superpotential in the IR limit [33], with the SU(N) identified as the “color” group and SU(N + M) as the “flavor” symmetry. The only difference between our gauge theory and the ISS system is that the “flavor” symmetry is gauged. The corresponding gauge coupling is proportional to a certain period of the B-field. We can tune it to be small and treat the gauge group as a global symmetry in the analysis of stability of the vacuum.6

6In fact, the restriction on the coupling is not very strong, because the SUSY breaking field TrΦ couples only to the bifundamental fields Y , Y˜ through the superpotential (4.31) (see also figure 4.2). Since the stabilization of the SUSY vacuum comes from the masses of these bifundamental fields it is sufficient to require that the corrections to the masses due to the gauge interactions are small at the SUSY breaking scale. 4.4. ISS from the Suspended Pinch Point singularity 104

An empty node in the quiver introduces some subtleties, since there might be instabilities or ﬂat directions at the last step of duality cascade leading to this empty node. In Appendix B we show that this quiver can be obtained after one Seiberg duality from an SPP quiver without empty nodes. Recall that in the ﬁeld theory SUSY is broken since the F-term equations for Φ

Y Y˜ = ζ 1N+M (4.32) cannot be satisﬁed by the rank condition. In the vacuum where

Y Y˜ = ζ 1N , (4.33) the SU(N) SU(N + M) gauge symmetry is broken to SU(N) SU(M) . 1 × 3 diag × 3 The superpotential for the remaining M M part of the adjoint ﬁeld reduces to the × Polonyi form

W = h ζ TrM (Φ). (4.34)

The metastable ground state thus has a vacuum energy proportional to Mh2ζ2. We can interpret the SUSY breaking vacuum on the geometric side as follows. Our system contains N fractional branes that wrap one of the conifolds inside the deformed SPP singularity, and (N + M) fractional branes that wrap the 2-cycle of the deformed A1. The Φ = 0 vacuum corresponds to putting all the (N + M) fractional branes on top of the N branes at the conifold (see fig. 4.2). The Y modes represent the massless ground states of the open strings that connect the two types of branes. The non-zero expectation value (4.33) for Y Y˜ corresponds to a condensate of these massless strings between N branes wrapping α and N branes wrapping α = α α . As a result of condensation, these two 3 1 − 2 − 3 stacks of N fractional branes recombine into N fractional branes wrapping α at − 2 the second conifold. The remaining M fractional branes around the deformed A1 end up in a non-supersymmetric state. The diagonal entries of the M M block × in Φ parameterize the motion of the M branes along the deformed non-isolated singularity. The corresponding configuration of branes is represented in figure 4.3. 4.4. ISS from the Suspended Pinch Point singularity 105

Φ N ~ α α Y 3 2 1 3

SU(N+M)Y SU(N) N+M

α1

Figure 4.2: A particular combination of fractional branes on the SPP singularity and the corresponding quiver gauge theory that reproduce the ISS model. The cycle α1 is a non-isolated two-cycle of the deformed A1 singularity inside the SPP. The cycles α2 and α3 denote the isolated two-cycles on the two conifolds that remain after the deformation of the A1 singularity. The cycles satisfy α1 + α2 + α3 = 0. The N fractional branes wrapping α3 are supersymmetric. The N + M fractional branes wrapping α1 break SUSY. This combination of fractional branes corresponds to zero vevs of the bifundamental ﬁelds in the ISS.

The stability of the SUSY breaking vacuum is a quantum eﬀect in the ﬁeld theory— there are pseudo-moduli that acquire a stabilizing potential at one loop [33]. In the D-brane picture this should correspond to the back reaction of the branes that makes the two-cycle at the deformed A1 singularity grow as one moves away from the conifold. (Alternatively one can think about a weak attraction between the branes.) It would be interesting to derive this directly from SUGRA equations, since it would complete the geometric evidence for the existence of the SUSY breaking vacuum.

4.4.3 SUSY restoration

Let us discuss SUSY restoration in this setup. The SUSY vacuum is found by

2 separating the (N + M) fractional branes on the deformed C /Z2 singularity from the N fractional branes at the conifold. This separation amounts to giving a vev to Φ. Initially this costs energy. The ﬁelds Y and Y˜ become massive. Below their 4.4. ISS from the Suspended Pinch Point singularity 106

N α α 3 2

α1

Figure 4.3: In the metastable vacuum. N supersymmetric fractional branes wrap the α2 cycle of the second conifold. The remaining M fractional branes wrap the non-isolated− cycle α = α α and are weakly bound to the N branes at the 1 − 2 − 3 conifold. This configuration of fractional branes is obtained from the configuration in figure 2 by giving vevs to the bifundamental fields. mass scale, the theory on the N fractional branes at the conifold becomes strongly coupled and develops a gaugino condensate. This condensate deforms the conifold singularity, and generates an extra term in the superpotential for Φ that eventually restores SUSY. On the gauge theory side, the SUSY restoring superpotential term arises due to the fact that the value of the gaugino condensate depends on the masses of Y and Y˜ , and these in turn depend on the vev of Φ. As a result [33], the gaugino condensation modifies the superpotential for Φ to (here Nf = N + M)

1/N Nf (Nf 3N) W = N h Λ−m − det Φ hζ TrΦ. (4.35) low − Due to the extra term, the F-term equations

∂W low = 0 (4.36) ∂Φ can be solved. In fact there are N N =M SUSY vacua Φ=Φ , with k = 1, .. , M. f − k On the geometric side, the SUSY vacuum is interpreted as the ground state of N +M fractional branes in the presence of a deformed conifold singularity. Suppose 4.4. ISS from the Suspended Pinch Point singularity 107

α2

3 S N+M

α 1

Figure 4.4: To reach the supersymmetric ground state, the N + M fractional branes on the A1 2-cycle move away from the conifold. The N fractional branes on the conifold then drive the geometric transition: the two-cycle α3 is replaced by the three 3 sphere S . After the transition, the size of the A1 2-cycle reaches a zero minimum at a new conifold singularity (indicated by the position of α1). that the deformed conifold is the one located at a = ζ. One can describe the situation after the geometric transition by the following equation

cd = a((a ζ)b + ǫ) . (4.37) −

The original conifold singularity at a = ζ is now a smooth point in the geometry. However, a new singularity has appeared in the form of an undeformed conifold at a = c = d = 0 and b = ǫ/ζ. The D5-branes that were originally stretching between a = 0 and a = ζ can thus collapse to a supersymmetric state by wrapping the zero- size 2-cycle of the undeformed conifold. This process is the geometric manifestation of SUSY restoration in the underlying ISS gauge theory.7 Using the geometric dual description, it is possible to rederive the ﬁeld theory superpotential (4.35) and even compute higher-order corrections. The calculation

7A similar mechanism of SUSY restoration in the case of SPP singularity was anticipated in [76] 4.4. ISS from the Suspended Pinch Point singularity 108 goes as follows, [90]. Let us rewrite the geometry (4.37) as:

uv = (z x)((z + x)(z x ζ) + ǫ) , (4.38) − − − where z x = a, z + x = b. Also it is useful to introduce the following notation −

z1(x) = x

z˜ (x) = ζ/2 (x + ζ/2)2 ǫ 2 − − p z (x) = x (4.39) 2 − z˜ (x) = ζ/2 + (x + ζ/2)2 ǫ 3 −

p z3(x) = x + ζ

The conifold singularity is at z = x = ǫ/2ζ x . If initially the fractional D- ≡ ∗ branes on the deformed A1 were stretching between z1(x) and z3(x), then after the geometric transition, they stretch between z1(x) andz ˜3(x). They can minimize their energy by moving (or tunneling) to the conifold singularity at z = z1(x ) =z ˜3(x ). ∗ ∗ For the geometric derivation of the superpotential, we take the deformation parameter ǫ to be dynamical, and related to the gaugino condensate via ǫ = 2S.8 We also identify Φ with the location x of the D5-branes relative to the (deformed) conifold at a = ζ. The superpotential for the gaugino condensate together with the adjoint ﬁeld is [90]

S t ˜ W (S, Φ) = NS(log 3 1) + S + W (Φ,S) . (4.40) Λ − gs

The ﬁrst two terms comprise the familiar GVW superpotential [37] W = Ω G3 ∧ evaluated for the deformed conifold supported by N units of RR 3-form ﬂuxR [38] [39] [40]. The last term W˜ (Φ,S) has a closely related, and equally beautiful, geometric characterization in terms of the integral of holomorphic 3-form

W˜ (Φ,S) = Ω (4.41) ZΓ 8The constant 2 appears due to the consistency conditions between the geometric derivation of the superpotential and the KS superpotential for the conifold. 4.4. ISS from the Suspended Pinch Point singularity 109 4.4. ISS from the Suspended Pinch Point singularity 110 superpotential. In the appropriate limit, x >>ǫ, ζ, we ﬁnd from (4.42)

W˜ (S, Φ) = ζTrΦ S log Φ/Λ . (4.43) − m Here we identify (x, ǫ) with (Φ, 2S), and use the integration constant to introduce a scale Λm. Physically, Λm sets the scale of the Landau pole for the IR free theory with 3N < Nf . Minimizing (4.40) with respect to S we ﬁnd

1 3 N S = Λm det Φ/Λm (4.44) If we substitute S back in (4.40), we get exactly (4.35) (up to an overall sign and after the redefinition Φ hΦ). By expanding the full geometric expression (4.42) → to higher orders, one can similarly extract the multi-instanton corrections to the superpotential. Our system in fact has other supersymmetric vacua besides the one just exhib- ited. These arise because, unlike the ISS-system, the flavor symmetry is gauged. If we move M fractional branes away from the conifold singularity in figure 4.3, then the N fractional branes wrapping the conifold 2-cycle α2 may also induce a geometric transition. As in the above discussion, this transition also restores SUSY. For a suitable choice of couplings, the extra SUSY vacuum lies farther away than the one considered above. The ISS regime arises when the coupling of the initial “color” SU(N) gauge group is sufficiently bigger than the coupling of the gauged “flavor” group SU(N + M), g g . (Note that after the symmetry breaking, the coupling 3 ≫ 1 of SU(N) SU(N) SU(N + M) is of order g .) Here we were assuming that diag ⊂ × 1 we are in this ISS regime. In this chapter we have found a string theory representation of the ISS model. One of the reasons why the string theory models can be useful is the existence of the gauge/gravity duality that enables one in some cases to calculate non-perturbative quantities on the gauge theory side via perturbative expansion on the gravity side. An example of such calculation is the derivation of the non-perturbative correction to 4.4. ISS from the Suspended Pinch Point singularity 111 the superpotential (4.35) via the tree-level superpotential (4.40) for the geometrical moduli fields on the gravity side. Chapter 5

Conclusion

As a conclusion let us discuss a possible scenario for the mediation of the SUSY breaking to the visible sector. There are several mechanisms to mediate the SUSY breaking. The most common possibilities are the gravity mediation and the gauge mediation. In gravity mediation the SUSY breaking happens at the energies much higher than the electro-weak scale and is transmitted via the gravitational interactions [86][103]. In terms of the soft SUSY breaking parameters, the gravity mediation results in the higher dimensional operators suppressed by the powers of the Plank mass in the denominator. One of the biggest problems with gravity mediation is the flavor changing neutral currents (FCNC). In the standard model there are some neutral flavor changing processes, for example the K0 K˜ 0 oscillations due to the box diagrams. In gravity − mediation the induced soft SUSY breaking parameters are generally not flavor neutral. Consequently there is a danger of adding additional contributions to the mass matrix that will violate the experimental constrain on the flavor changing neutral currents. In gauge mediation, the mediator fields couple to the observable sector through the gauge groups, consequently the SUSY breaking parameters are flavor neutral. Also since the gauge interactions are much stronger than gravity, the scale of SUSY

112 113 breaking can be much smaller than in gravity mediation, i.e. the contribution of gravity modes mediating the SUSY breaking is negligible in this case and they don’t affect the flavor changing neutral currents. In this chapter I will describe a toy model for the gauge mediation of the ISS SUSY breaking mechanism. Although one of the biggest problems in gauge mediation is the µ/Bµ problem, I will not talk about it. The problem we will keep in mind when discussing this toy model is the split supersymmetry breaking, i.e. separation between the gaugino mass and the sfermion masses. In F-term SUSY breaking with gauge mediation, the split SUSY breaking may arise due to the R-symmetry necessary for the generice F-term SUSY breaking [32]. Even if the R-symmetry is broken spontaneously, the leading contribution to the gaugino masses is usually vanishing [106][107] and the subleading terms for the gaugino mass have an extra numerical factor of 1/10 on top of the bigger power of the SUSY breaking parameter, F . Thus in the F-term SUSY breaking with gauge mediation it is common to get the mass of the gauginos much smaller than the mass of the sfermions. The experimental searches at Tevatron put the lower bound for the gluino mass at (300-400)GeV. If the stop quark is much heavier than the gluino, e.g., 10 times heavier, then it’s mass should be at least (3-4)TeV. The leading contribution to renormalization of the Higgs mass comes from the difference of the stop and top quark loops. Thus if the stop mass is much bigger than the top mass, the difference will be proportional to the stop mass, i.e. the Higgs mass squared is the loop factor times the stop mass squared [104]. Without tuning, the Higgs turns out to be too heavy for the stop mass bigger than about 700GeV. There are several ways to deal with the heavy Higgs problem. First of all why is it a problem? Within the MSSM, the upper bound of 200GeV on the Higgs mass comes from the requirement that the quartic Higgs self-coupling doesn’t have a Landau pole before the GUT scale [104] also there are constraints from fitting 5.1. Mediation of ISS model 114 the electro-weak precision data that put the upper bound for the Higgs mass below 135GeV. On the other hand we can reduce the gap between the gluinos and the squarks by introducing more mediators. This is because in gauge mediation both the gaugino masses and sfermion masses squared are proportional to the number of messengers N. Consequently the the ratio of gaugino mass to the sfermion mass has an extra factor of √N. The problem is that for N & 5 the gauge couplings have Landau poles before the GUT scale [86]. Another possibility is to simply assume that there is some tuning of the parameters that fixes the Higgs mass problem. In any case, all these possibilities don’t seem to be very attractive from the point of view of phenomenology. However they don’t contradict any basic principles and the F-term SUSY breaking with gauge mediation has a very natural representation in String Theory.

5.1 Mediation of ISS model

A number of attempts has been made to use the ISS model of SUSY breaking as the hidden sector in the gauge mediation scenario [106][107]. One of the biggest problem with the ISS approach is the non-broken R-symmetry in the SUSY breaking vacuum. Recall the ISS superpotential

W = hTr(ϕ ˜Φϕ) hµ2TrΦ. (5.1) −

The R-charge of W is 2, consequently the R-charges of the ﬁelds can be chosen as R(Φ) = 2 and R(ϕ) = R(ϕ ˜) = 0. In the SUSY vacuum (4.4) the vev of Φ ﬁeld is zero and the R-symmetry remains unbroken. This is a problem for generating the gaugino masses, since the gaugino mass term has R-charge 2. The F-term supersymmetry is compatible with the spontaneous breaking of the 5.1. Mediation of ISS model 115

R-symmetry [32]. For instance, one can shift the VEV of the pseudomodulus X in

Y Z˜ Φ = (5.2)  ZX  Since R(X) = 2, the R-symmetry is spontaneously broken. In general one could expect the mass of the gaugino to be proportional both to the SUSY breaking scale F = µ2 and the R-symmetry breaking scale X. g2 XF m N (5.3) 1/2 ∼ 16π2 µ2 and the sfermion masses to be proportional to the SUSY breaking scale g2 2 F 2 m2 N (5.4) 0 ∼ 16π2 µ2 If X µ, then m m and there is split SUSY breaking. ∼ 1/2 ∼ 0 However the bad news is that the leading contribution (5.3) to the gaugino mass is vanishing [106][108]. As a toy model let us gauge a part of the unbroken global symmetry in ISS. Recall that the global symmetry in the SUSY breaking vacuum is broken as SU(N ) f × SU(N) SU(N) SU(N N). To be concrete let us consider a toy model −→ diag × f − where we gauge the SU(N)diag part and treat it as a ”grand uniﬁed” gauge group. The ISS superpotential near the SUSY breaking vacuum has the form

W = hTr µY (χ +χ ˜) + µZρ + µZ˜ρ˜ +ρXρ ˜ µ2X + ... (5.5) − The goldstino superpartner X is the ”hidden” sector that breaks the SUSY, F = 0. X 6 The fields Z, Z˜, ρ, andρ ˜ charged under SU(N)diag and interacting with X are the ”messenger” fields. The general formula for the gaugino mass is [86] [105] [106] g2N ∂ m = F log det M (5.6) 1/2 (4π)2 X ∂X where M is the mass matrix for the messengers. As one can see from (5.5), the mass matrix X µ M = (5.7)  µ 0    5.1. Mediation of ISS model 116 has the determinant that doesn’t depend on X, consequently the leading contribution vanishes. This vanishing of the leading contribution in the gauge mediation of O’Raifeartaigh model was first noticed [109] where the first subleading contribution XF 3/µ6 was computed. In the ISS model F = µ2, consequently it is necessary to know the exact one loop answer for the gaugino mass as a function of X and F/µ2. Before we compute the exact one-loop gaugino mass, let me first show how one can get a vev of X without affecting the metastability. A natural deformation of ISS superpotential is to add a term 1/2hmTrΦ2. This term breaks the R-symmetry explicitly and creates a supersymmetric vacuum at

µ2 Φ = . (5.8) 0 m

If we want to preserve the metastability, then the correction due to m should be smaller than the one-loop Coleman-Weinberg potential for the pseudomodulus X. Thus we assume that h4µ2 h2m2 m2 (5.9) ≪ 1l ≈ 16π2 In this case the correction to the classically massive ﬁelds is negligible and we can

1 2 keep only the 2 hmX part of the deformation. The terms with the X ﬁeld and the messengers are 1 W = h(˜ρXρ + µρZ˜ + µZρ˜ FX + mX2) (5.10) − 2 This superpotential has the form of O’Raifeartaigh superpotential with mX2 deformation [110]. Let us integrate out the heavy ﬁelds ρ and Z. Then the tree level plus one-loop potential for X is

V (X) = h2 mX F 2 + m2 X 2. (5.11) | − | 1l| |

As a ﬁrst order approximation, we neglect higher order corrections in X for X µ, ≤ in particular we don’t take into account that m1l depends on X. The potential can be rewritten as (assume that all the constants are real and 5.1. Mediation of ISS model 117 take the real part of X)

h2mF 2 V (X) m2 X + const. (5.12) ≈ 1l − m2 1l we see that the new minimum is at h2mF 16π2 X = 2 2 m (5.13) m1l ≈ h Notably, the vacuum value of X is proportional to the R-symmetry breaking parameter m that should be smaller than the one-loop mass in order to preserve the metastability, but it also has the one-loop mass squared in the denominator that can pull X to higher values. Let us show that we can get X µ. The deformation parameter is ∼ h2 m X (5.14) ∼ 16π2 The SUSY restoring vacuum is far away from the region we are interested in 16π2 µ2 X µ (5.15) SUSY ∼ h2 X ≫ and the the one-loop mass is much bigger than the deformation parameter h2µ µ m 4π m m (5.16) 1l ≈ 4π ∼ X ≫ i.e., the vacuum is stable in this approximation. The exact one loop gaugino mass can be computed by the standard methods. The only diﬃculty is that the basis of the ﬁelds where the interactions are diagonal is not the same as the basis of the mass matrix eigenvectors, i.e. one needs to insert the analog of the CKM matrix in the vertices, i.e., the additional SU(N) matrix that transforms the interaction eigenstates to the mass matrix eigenstates. In our case the number of messengers is N N. The exact one-loop contribution f − to the gaugino mass is g2 m = (N N)µ f(X/µ, F/µ2) (5.17) 1/2 16π2 f − 5.1. Mediation of ISS model 118

Denote x = X/µ and y = F/µ2, the function f(x, y) has the following expansion

1 1 1 4 f(x, y) = x x3 + ... y3 + x3 + x + ... y5 + ... (5.18) 15 − 28 −36 105 Note, that the leading contribution in F vanishes and the ﬁrst non vanishing contribution is XF 3/15µ6 which is highly suppressed in usual gauge mediation. In ISS F = µ2, i.e. y = 1. In this case f has the maximum value f 0.117 at x = 1, i.e., ≈ X = µ. The graph of the function f with respect of x for y = 1 is shown in ﬁgure 5.1.

0.12

0.1

0.08

f 0.06

0.04

0.02

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x

Figure 5.1: Gaugino mass as a function of x = X/µ with F = µ2.

The sfermion mass is expected to be of order

g2 m2 2 (N N)C (5.19) 0 ≈ 16π2 f − N where C = (N 2 1)/2N N/2 is the quadratic Casimir for the fundamental N − ≈ representation of SU(N). Thus the ratio

m Nf N 1/2 − f(X/µ, 1). (5.20) m0 ≈ p √N 5.1. Mediation of ISS model 119

The ISS has a consistent description in the IR if Nf > 3N. In the minimal case for

N = 5, Nf > 15, the maximal value of the ratio is for X = µ

m 1/2 0.17 (5.21) m0 ≈

Consequently if m1/2 > 300GeV , then m0 > 2T eV and the contribution to the Higgs mass is δmh > 500GeV which is unfortunately 3-4 times bigger than the acceptable values. We see that the simplest string theory embedding of the gauge mediation is about one order of magnitude off from the expected ratio of the sfermion and gaugino masses. A little better results can be obtained if one includes non-generic deformations of ISS model [106][111]. Another way to overcome the R-symmetry problem in field theory is to extend [112] our notion of the gauge mediation and try to use more general models for the hidden sector [113]. One can also try to employ some of the stringy constructions in order to generate the metastable SUSY breaking and its mediation. A simple example of the metastable vacuum in string theory is the anti D3-brane at the tip of the conifold [114]. A possible scenario of SUSY breaking is to embed a system of D-branes in the conifold geometry that will model the MSSM. The anti D3-brane will represent the hidden sector that breaks the Supersymmetry and the strings stretching from the D3-brane correspond to the messenger fields that transmit the SUSY breaking to the MSSM branes. As a conclusion let me summarize the main results of the dissertation. We review the general framework for the model building on the D-branes at the singularities and use it to construct an extended MSSM on the D3-brane at del Pezzo 8 singularity. Some of the field theory properties depend on the embedding of the local CY singularity in a compact CY manifold. We present a general construction and examples of embedding the del Pezzo singularities in compact CY manifolds. We 5.1. Mediation of ISS model 120 also construct a D-brane representation of the metastable SUSY breaking vacuum and study its mediation to a sector representing the standard model. In spite of the partial success in various directions of the model building, the various pieces still need to be collected together in a model that reproduces the MSSM with a viable spectrum of the superpartners. We expect that there is still a lot of room for discoveries and improvement, both for the D-branes at the CY manifolds and in more general setups such as the F-theory [115][116]. Appendix A

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 = Pn Pm Pk, ⊂ × × then the coordinates on the three projective spaces will be denoted as (z0, . . . , zn),

(u0, . . . , um), and (v0, . . . , vk) respectively. The hyperplane classes of the three projective spaces will be denoted by H, K, R respectively.

A polynomial of degree q in zi, degree r in uj, and degree s in vl will be denoted by Pq,r,s(zi; uj; vl). 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 [53]. For example, consider the space W P3 , where p, q N. The dimension of this space is 3, 11pq ∈ the subscripts (1, 1, p, q) denote the weights of the coordinates with respect to the projective identifications (z , z , z , z ) (λz , λz , λpz , λqz ). 0 1 2 3 ∼ 0 1 2 3

121 122

The P1 bundles over X will be denoted as M = P ( (K)), where K OX ⊕ OX 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 and t (K). The hyperplane ∈ OX ∈ OX class of the fibers will be denoted by G, it satisfies the property G(G + K) = 0 for M = P ( (K)). In the construction of the P1 bundles, we will use the fact OX ⊕ OX that K(B) = c (B) and will not calculate K(B) separately. − 1 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

511100002 GW P 00011001 (A.1) 00000111

The number 5 is the dimension of the space. This space is obtained from C8∗ by taking the classes of equivalence with respect to three identiﬁcations. The numbers in the three rows correspond to the charges under these identiﬁcations.

(z , z , z , z , z , z , z , z ) (λ z , λ z , λ z , z , z , z , z , λ2z ) 1 2 3 4 5 6 7 8 ∼ 1 1 1 2 1 3 4 5 6 7 1 8 (z , z , z , z , z , z , z , z ) (z , z , z , λ z , λ z , z , z , λ z ) (A.2) 1 2 3 4 5 6 7 8 ∼ 1 2 3 2 4 2 5 6 7 2 8 (z , z , z , z , z , z , z , z ) (z , z , z , z , z , λ z , λ z , λ z ) 1 2 3 4 5 6 7 8 ∼ 1 2 3 4 5 3 6 3 7 3 8

1. B = P2 X = P3. ⊂ The equation for B

P1(zi) = 0. (A.3)

The total Chern class of B

c(B) = (1 + H)3 = 1 + 3H + 3H2 (A.4)

The P1 bundle is M = P ( ( 3H)). The equation for the Calabi-Yau OX ⊕ OX − threefold Y3 2 2 P1(zi)s + P4(zi)st + P7(zi)t = 0 (A.5) 123

P4 The embedding space V = W 11113 has the coordinates (z0, . . . , z3; w) and the singular CY is

2 P1(z0, . . . , z3)w + P4(z0, . . . , z3)w + P7(z0, . . . , z3) = 0 (A.6)

This is already the most general equation, i.e. there are no additional complex deformations.

2. B = P1 P1 X = P3. × ⊂ The equation for B

P2(zi) = 0. (A.7)

The total Chern class of B (1 + H)4 c(B) = = 1 + 2H + 2H2 (A.8) 1 + 2H The P1 bundle is M = P ( ( 2H)). The equation for the Calabi-Yau OX ⊕ OX − threefold Y3 2 2 P2(zi)s + P4(zi)st + P6(zi)t = 0 (A.9)

P4 The embedding space V = W 11112 has the coordinates (z0, . . . , z3; w) and the singular CY is 2 P2(zi)w + P4(zi)w + P6(zi) = 0 (A.10)

This equation has one deformation kw3 and the spaces M and V have the same number of coordinate redeﬁnitions. Thus the space of complex deformations is one-dimensional.

3. B = dP X = P2 P1 1 ⊂ × The equation deﬁning B has degree one in zi and degree one in uj

P1(zi)u0 + Q1(zi)u1 = 0. (A.11)

The total Chern class of B (1 + H)3(1 + K)2 c(B) = = 1 + 2H + K + H2 + 3HK (A.12) 1 + H + K 124

The P1 bundle is M = P ( ( 2H K)). The equation for the Calabi- OX ⊕ OX − − Yau threefold Y3 is

2 2 P1,1(zi; uj)s + P3,2(zi; uj)st + P5,3(zi; uj)t = 0 (A.13)

P4 The embedding space V = GW 111002 has the coordinates (z0, z1, z2; u0, u1; w) 000111 and the singular CY is

2 P1,1(zi; uj)w + P3,2(zi; uj)w + P5,3(zi; uj) = 0 (A.14)

There are no complex deformations of this equation.

4. B = dP X = P2 P1 P1 2 ⊂ × × The del Pezzo surface is deﬁned by a system of two equations. The ﬁrst

equation has degree one in zi and degree one in uk. The second equation has

degree one in zi and degree one in vk.

P1(zi)u0 + Q1(zi)u1 = 0 (A.15)

R1(zi)v0 + S1(zi)v1 = 0

The total Chern class of B (1 + H)3(1 + K)2(1 + R)2 c(B) = = 1 + 2H + K + R + 2H(K + R) + KR (1 + H + K)(1 + H + R) (A.16) The P1 bundle is M = P ( ( 2H K R)). The system of equations OX ⊕ OX − − − for the Calabi-Yau threefold Y3 can be written as

2 2 P1,1,0(zi; uk; vk)s + P3,2,1(zi; uk; vk)st + P5,3,2(zi; uk; vk)t = 0 (A.17)

Q1,0,1(zi; uk; vk) = 0

P511100002 The space V = GW 00011001 has the coordinates (z0, z1, z2; u0, u1; v0, v1; w) 00000111 and the singular CY is

2 P1,1,0(zi; uk; vk)w + P3,2,1(zi; uk; vk)w + P5,3,2(zi; uk; vk) = 0 (A.18)

Q1,0,1(zi; uk; vk) = 0 125

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.

5. B = dP X = P1 P1 P1 3 ⊂ × × The del Pezzo surface is deﬁned by an equation of degree one in zi, degree one

in uj and degree one in vk.

P1,1,1(zi; uj; vk) = 0 (A.19)

The total Chern class of B

(1 + H)2(1 + K)2(1 + R)2 c(B) = = 1 + (H + K + R) + 2(HK + HR + KR) (1 + H + K + R) (A.20) where H, K and R are the hyperplane classes on the three P1’s. The P1 bundle is M = P ( ( H K R)). The equation for the Calabi-Yau threefold OX ⊕OX − − − Y3 is

2 2 P1,1,1(zi; uj; vk)s + P2,2,2(zi; uj; vk)st + P3,3,3(zi; uj; vk)t = 0 (A.21)

P41100001 The embedding space V = GW 0011001 has the coordinates (z0, z1; u0, u1; v0, v1; w) 0000111 and the singular CY is

2 P1,1,1(zi; uj; vk)w + P2,2,2(zi; uj; vk)w + P3,3,3(zi; uj; vk) = 0 (A.22)

This equation has one deformation kw3 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 are independent within the CY and there is only one ( 2) two-cycle on 3 − dP3 that is trivial within the CY. 126

6. B = dP X = P2 P1 4 ⊂ × Equation deﬁning B has degree two in zi and degree one in uj

P2(zi)u0 + Q2(zi)u1 = 0. (A.23)

The total Chern class of B

(1 + H)3(1 + K)2 c(B) = = 1 + H + K + H2 + 3HK (A.24) 1 + 2H + K

where H and K are the hyperplane classes on P2 and P1 respectively. The P1 bundle is M = P ( ( H K)). The equation for the Calabi-Yau OX ⊕ OX − − threefold Y3 is

2 2 P2,1(zi; uj)s + P3,2(zi; uj)st + P4,3(zi; uj)t = 0 (A.25)

P4 The embedding space V = GW 111001 has the coordinates (z0, z1, z3; u0, u1; w) 000111 and the singular CY is

2 P2,1(zi; uj)w + P3,2(zi; uj)w + P4,3(zi; uj) = 0 (A.26)

The deformations of the singularity have the form of degree one polynomial in

3 z0, z1, z2 times w . Consequently, there are three deformation parameters and the spaces V and M have the same reparameterizations. In this case we have three complex deformations and three ( 2) two-cycles on dP that are trivial − 4 within CY.

7. B = dP X = P4. 5 ⊂ The del Pezzo surface is deﬁned by a system of two equations. Both equation

have degree 2 in zi.

P2(zi) = 0 (A.27)

R2(zi) = 0 127

The total Chern class of B

(1 + H)5 c(B) = = 1 + H + 2H2 (A.28) (1 + 2H)2

The P1 bundle is M = P ( ( H)). The system of equations for the OX ⊕ OX − ﬁrst possible Calabi-Yau threefold Y3 is

2 2 P2(zi)s + P3(zi)st + P4(zi)t = 0 (A.29)

R2(zi) = 0

It has the following characteristics

χ(Y ) = 160; 3 − 1,1 h (Y3) = 2; (A.30) h2,1 = 82.

1 Now we ﬁnd the deformations of this cone over dP5. The P bundle M is, in fact, the P5 blown up at one point. By blowing down the t = 0 section of M

5 5 we get P . The CY three-fold with the dP5 singularity is embedded in P by the system of two equations

2 P2(zi)w + P3(zi)w + P4(zi) = 0; (A.31) R2(zi) = 0.

The deformations of the singularity correspond to taking a general degree four polynomial in the ﬁrst equation. This general CY has

χ = 176; − 1,1 h (Y3) = 1; (A.32) h2,1 = 89.

Since the system (A.31) has only the dP5 singularity and the general CY manifold has 89 82 = 7 more complex deformations, we interpret these extra − 128

7 deformations as the deformations of the cone over dP5. This number is

consistent with the general expectation, since c∨(D5) 1 = 7, where c∨(D5) = − 8 is the dual Coxeter number for D5.

The second CY with the dP5 singularity is described by

P2(zi)s + P3(zi)t = 0 (A.33)

R2(zi)s + R3(zi)t = 0.

Using the same methods as for the ﬁrst CY, one can show that this singularity also has 7 complex deformations.

8. B = dP X = P3. 6 ⊂ The case of dP6 was described in details section 2, here we just repeat the general results.

The equation deﬁning dP P3 6 ⊂

P3(zi) = 0. (A.34)

The P1 bundle is M = P ( ( H)). OX ⊕ OX −

The equation for the Calabi-Yau threefold Y3

2 2 P3(zi)s + P4(zi)st + P5(zi)t = 0 (A.35)

The total Chern class of Y3 (1 + H)4(1 + G)(1 + G H) c(Y ) = − (A.36) 3 1 + 3H + 2G

The Euler number and the cohomologies for the CY with the dP6 singularity are

χ = 176 − h1,1 = 2 (A.37)

h2,1 = 90 129

The deformation of this singularity is a quintic in P4, that has

h2,1 = 101 (A.38)

complex deformations. The diﬀerence between the number of complex defor-

mations is 101 90 = 11, which is consistent with c∨(E6) 1 = 11. − − 9. B = dP X = W P3 . 7 ⊂ 1112 The equation deﬁning B is homogeneous of degree four in zi’s

P4(zi) = 0. (A.39)

The P1 bundle is M = P ( ( H)). The equation for the Calabi-Yau OX ⊕ OX − threefold Y3 2 2 P4(zi)s + P5(zi)st + P6(zi)t = 0 (A.40)

The total Chern class of Y3 (1 + H)3(1 + 2H)(1 + G)(1 + G H) c(Y ) = − (A.41) 3 1 + 4H + 2G

The Euler number and the cohomologies for the CY with the dP6 singularity are

χ = 168 − h1,1 = 2 (A.42)

h2,1 = 86

P4 Blowing down the t = 0 section of M we get V = W 11112. The general CY is given by the degree six equation in V . The total Chern class of this CY is

(1 + H)4(1 + 2H) c = (A.43) (1 + 6H)

And the number of complex deformations

h2,1 = 103 (A.44) 130

The diﬀerence 103 86 = 17 is equal to c∨(E7) 1 = 17, where c∨(E7) = 18 − − is the dual Coxeter number of E7. Consequently, we can represent all complex

deformations of dP7 singularity in this embedding.

10. B = dP X = W P3 . 8 ⊂ 1123 The equation deﬁning B has degree six

P6(zi) = 0. (A.45)

The P1 bundle is M = P ( ( H)). The equation for the Calabi-Yau OX ⊕ OX − threefold Y3 2 2 P6(zi)s + P7(zi)st + P8(zi)t = 0 (A.46)

The total Chern class of Y3

(1 + H)2(1 + 2H)(1 + 3H)(1 + G)(1 + G H) c(Y ) = − (A.47) 3 1 + 6H + 2G

The problem with this CY is that for any polynomials P6, P7 and P8 it has a

singularity at s = z0 = z1 = z2 = z3 = 0 and z4 = 1. As a consequence the naive calculation of the Euler number gives a fractional number

2 χ = 150 . (A.48) − 3

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

dP8 cone. In order to justify that we will calculate the number of complex de-

formations of the CY manifold with dP8 singulariy by calculating the number of coeﬃcients in the equation minus the number of reparamterizations of M. The result is h2,1 = 77. (A.49)

P4 Blowing down the t = 0 section of M we get V = W 11123. The general CY is given by the degree eight equation in V . The number of coeﬃcients minus 131

P4 the number of reparamterizations of V = W 11123 is

h2,1 = 106. (A.50)

The diﬀerence 106 77 = 29 is equal to c∨(E8) 1 = 29, where c∨(E8) = 30 − − is the dual Coxeter number of E8. Thus all complex deformations of dP8 singularity can be realized in this embedding. Appendix B

ISS quiver via an RG cascade

In this appendix we show that the ISS quiver in figure 4.2 can be obtained after one Seiberg duality from an SPP quiver in figure B.1. This quiver is obtained from the quiver in figure 4.1 by adding M fractional branes to node 3, and N + M fractional branes to node 2, so that the respective ranks of the gauge groups become N + M and 2N + M. Note, that this theory has an infinite duality cascade that increases the ranks of the gauge groups, i.e. we can suppose that we start in the UV with some big ranks of the gauge groups and after a number of duality steps arrive at quiver B.1. Let us show that after one more duality at node 2 we reproduce the ISS model. The theory has gauge group U(N) U(2N +M) U(N +M), one adjoint under × × U(N) and three vector-like pairs of bi-fundamentals. The superpotential is given by the sum of (4.24) and (4.27)

W = Tr ζΦ˜ + Φ(˜ YY˜ XX˜ ) + h(ZZX˜ X˜ ZZY˜ Y˜ + ζZZ˜ ) . (B.1) − − − The SU(2N + M) gauge group confines first. This gauge group has Nf = Nc and thus the gauge group after the Seiberg duality is U(N) U(N + M). The two × U(1) factors can be represented as the overall U(1) (that decouples) and the non- anomalous U(1)B gauge group. Denote the meson fields as Mxx = XX˜ , Mxz = XZ˜ ,

Mzx = ZX˜ , and Mzz = ZZ˜ . In addition there are two baryons A and B. After the

132 133

Φ~

1 U(N)

X ~ Y ~ X Y

Z 2 3 ~ U(2N+M)Z U(N+M)

Figure B.1: Quiver gauge theory of the fractional brane conﬁguration on the SPP singularity that reduces to ISS model after conﬁnement of the SU(2N + M) gauge group at node 2.

Seiberg duality, the superpotential is

W˜ = Tr ζΦ˜ + Φ(˜ YY˜ M ) + h(M M M Y Y˜ + ζM ) (B.2) − − xx xz zx − zz zz Mxx Mxz + λ det AB Λ4N+2M    − −  Mzx Mzz     Here λ is a lagrange multiplier ﬁeld. Its constraint equation is the quantum deformed relation between the baryon and meson ﬁelds, and dictates that either the baryons or mesons acquire a non-zero vev. We assume that we are on the baryonic branch

AB = Λ4N+2M . (B.3) −

The vevs of the baryons break the non-anomalous U(1)B. The D-term equations for U(1) fix A 2 = B 2. B | | | | The adjoint field Φ,˜ and the meson fields Mxx, Mxz, and Mzx are all massive. So we can integrate them out. The reduced gauge theory has gauge group SU(N) × SU(N + M), a pair of bi-fundamental fields (Y, Y˜ ), and a meson field

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