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. All rights reserved. Abstract 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 sin- gularities.............................. 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 per- pendicular 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].