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Recent Aspects of Seiberg Duality: Metastable Vacua, Stringy Instantons and M2 Branes

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Recent Aspects of Seiberg Duality: Metastable Vacua, Stringy Instantons and M2 Branes UNIVERSITA' DEGLI STUDI DI MILANO BICOCCA SCUOLA DI DOTTORATO DI SCIENZE DOTTORATO DI RICERCA IN FISICA E ASTRONIMIA XXII CICLO Recent aspects of Seiberg duality: metastable vacua, stringy instantons and M2 branes Supervisor: Prof. Luciano Girardello Coordinator of the PhD school: Prof. Claudio Destri PhD dissertation of Antonio Amariti Matr. N. 042321 PACS: 11.17, 11.30.P,12.40.H, 14.80.L 23 October 2009 Contents Introduction 3 1 Seiberg duality 8 1.1 Generalities of Seiberg duality . 9 1.2 KSS duality . 16 1.3 Seiberg duality from intersecting brane . 19 1.4 Seiberg duality on a quiver . 21 1.5 Seiberg duality on the brane tiling . 23 2 Metastable Vacua 25 2.1 Spontaneous supersymmetry breaking . 25 2.2 SQCD with adjoint matter . 42 2.3 An gauge theories . 55 2.4 Geometric deformations . 71 2.5 Metastable vacua at two loop . 102 3 Stringy instantons 110 3.1 D-instantons . 111 3.2 A general action for instanton in toric quiver gauge theories . 114 3.3 Stringy Instanton and Seiberg Duality . 118 4 Three dimensions 133 4.1 The ABJM model . 133 4.2 N = 2 toric CS matter theory . 138 4.3 M2 branes and Seiberg duality . 140 4.4 Supersymmetry breaking . 163 Conclusions 174 A Analysis of = 1 supersymmetric theories 176 A.1 Quiver gaugeN theories . 176 A.2 Orientifold projection from dimers . 183 A.3 Hierarchy of scales . 185 A.4 Geometric transition and the superpotential . 187 1 A.5 Stability and UV completion . 188 A.6 Quantum analysis . 190 A.7 Bounce action for a triangular barrier in 4 D . 194 B Details of the instantonic calculations 198 B.1 Clifford Algebra and Spinors . 198 B.2 General result for U(1) instanton . 199 B.3 Bosonic integration . 200 B.4 Relation between our results and known models . 203 C Aspects of field theories in 3D 205 C.1 Seiberg duality in three dimensional CS SQCD from branes . 205 C.2 Parity anomaly . 206 C.3 Bounce action for a triangular barrier in 3 D . 208 C.4 Coleman-Weinberg formula in various dimensions . 210 2 Introduction Nowadays, the model which describes the fundamental interactions, in the particle physics, is the Standard Model. It is a quantum field theory, where the local gauge symmetry generates the gauge fields. These fields are the physical candidates to mediate the strong and electroweak interactions between the elementary particles. In the first case the local gauge symmetry is exact, while in the second one it is broken by the Higgs mechanism. The original Standard Model has to face the problem of the quadratic divergences of QFT which are at the origin of the so called hierarchy problem. Furthermore it seems natural to try to unify the above theories with gravity but it seems very difficult to combine QFT with general relativity, and the most promising attempt to overcome this problem is string theory. Now both the hierarchy issue as well as string theory require an additional global symmetry which ties bosons and fermions together, called supersymmetry. One can construct QFT with such a symmetry. It turns out that the vacuum state of such theories must have zero energy. From the phenomenological point of view supersymmetry has to be broken at low energy and can be restored at very high energy. Another important aspect of supersymmetry is its relation with confining gauge theo- ries. In many cases strongly coupled models admit a weakly coupled dual version. The use of duality has often helped the comprehension of physical phenomena. In supersymmetric gauge theories there exists a natural electric/magnetic type duality. The first historical example was the Montonen-Olive duality in = 4. Many extensions have been studied in N the following years. In this thesis we concentrate on the = 1 case, namely the Seiberg duality [1]. This is an electric/magnetic duality relatingNthe correlation functions of the the long distance physics of two different theories. The dual description is usually weakly coupled at low energy if the electric theory were free in the UV. Non perturbative prop- erties of the electric theory at low energy are perturbatively calculated in the magnetic theory. Many test of validity of Seiberg duality have been performed. For example the global symmetries are the same in the dual theories, while the gauge symmetries, which are unphysical are not the same. Moreover the number of supersymmetric vacua, Witten index and superconformal indexes coincide for dual theories. Also the a-maximization has checked the consistence of duality. Although the original duality matched two SQCD models, each one having a gauge group, Seiberg duality can be extended to theories with many gauge groups. These are the quiver gauge theories, which are associated, by the gauge/gravity correspondence, to the gauge theories living on D3 branes wrapped on Calabi-Yau (CY) singularity. In these cases all the groups are gauge groups and the matter fields are two index tensors charged under 3 the adjoint or the bifundamental representations of the gauge groups. The rules of duality are the same as in the SQCD case, but the number of equivalent theories is enlarged. Recently new applications of Seiberg duality have received a large attention. Here we dis- cuss some of them like the ISS model of metastable supersymmetry breaking, the exotic contribution of a class of stringy instanton and the AdS4/CFT3 correspondence. The first application that we discuss is the existence of ISS [2] metastable vacua in SQCD. The phenomenological application of many models of supersymmetry breaking have been excluded by the constraints on the soft masses. Some of these models are re- covered by requiring an hidden sector of supersymmetry breaking. In the past the study of the properties of the supersymmetry breaking hidden sectors has been problematic be- cause many models were strongly coupled in the IR. This complication made the analysis only qualitative, and the spectra were often uncalculable. As we explained above duality is helpful in the analysis of the strongly coupled sectors. Indeed the ISS model uses Seiberg duality to build a hidden sector of supersymmetry breaking. This is a SQCD model with NF massive flavour of quarks. The choice of the values of NF and NC makes the electric theory free in the UV regime and the dual the- ory free in the IR. This theory has supersymmetric vacua and does not seems a viable candidate for an hidden sector. Nevertheless in the dual version of the theory some new vacua appear. These are supersymmetry breaking metastable vacua, emerging at weak coupling in the gauge theory. Seiberg duality guarantees that these vacua are also vacua of the electric theory, but the strong couplings made them too difficult to find there. Metastability can be regarded as a problem for these vacuum states. However, as long as they have long lifetime, they are phenomenologically acceptable vacua. Metastable vacua are rather generic when the supersymmetry breaking sector is coupled with the MSSM. Supersymmetric vacua usually arise from this coupling and the super- symmetry breaking vacuum becomes metastable, if tachyons are not generated. This implies that the only difference with the ISS model is that metastability here already is accepted in the hidden sector. In this way many calculable and natural examples of supersymmetry breaking are found. The ISS model leaves many phenomenological problems opened. It cannot be used for gauge mediating supersymmetry breaking, because R-symmetry is not broken. Here an approximate R-symmetry is preserved around the supersymmetry breaking vacuum. This symmetry forbids a gaugino mass term in the perturbative expansion of the Lagrangian, also at higher loops. For this reason it is important to find phenomenologically appealing deformations of the ISS model. One can try to add explicit R-symmetry deformations. This usually restores supersymmetry [3] but as we will show later it is not always the case. Another possibility is that the R-symmetry is broken at loop level by the scalar potential, as in [4]. Another interesting question is the relation of the ISS model with the gauge/gravity duality. Some progress has been made by studying the interactions among systems of intersecting branes. Here Seiberg duality is the statement that the moduli space of two system of branes, obtained after an exchanging of two NS branes, coincide. Many models of metastable vacua have been worked out from system of branes [5, 6, 7, 8, 9]. Another 4 promising possibility comes from theories of fractional branes wrapped on CY3 singu- larities. Usually the geometric deformations of the singularity add new terms to the superpotential of the underlying quiver gauge theories. Metastable vacua are ubiquitous in these models [10, 11, 12, 13]. A second application of duality that we discuss is the relation with non-perturbative exotic contribution that arise from stringy instantons. Usually instantons provide non perturbative contributions to the gauge theories. In super- symmetry only the action for the instantonic zero modes is non vanishing. It is obtained from the ADHM construction [14]. The non perturbative contribution due to the instan- tons is calculated by integrating the action over the bosonic and fermionic zero modes. This field theory construction has a natural interpretation in string theory, especially in theories of branes at a singularity [15]. The dimensional reduction from ten dimensional supergravity, in a flat background, of a system of D3 branes and D( 1) branes, recovers − the ADHM construction of the instantonic action of = 4 SYM. By orbifolding and higgsing the = 4. This is a dynamical phenomenonNdue to the cascading behaviour. N Indeed the addiction of fractional branes usually breaks the conformal invariance and the theory flows to lower energy through many steps of Seiberg dualities.
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