Dissertations / Theses on the topic 'Gauge theories'

Includes bibliographical references.<br>The thesis is organized as follows. Part I is a general introduction to LGT. The theory is discussed from first principles, so that for the interested reader no previous knowledge is required, although it is assumed that he/she will be familiar with the rudiments of relativistic quantum mechanics. Part II is a review of QCD on the lattice at finite temperature and density. Monte Carlo results and analytical methods are discussed. An attempt has been made to include most relevant data up to the end of 1987, and to update some earlier reviews existing on the subject. To facilitate an understanding of the techniques used in LGT, provision has been made in the form of a separate Chapter on Group Theory and Integration, as well as four Appendices, one of which deals with Grassmann variables and integration.

Die vorliegende Dissertationarbeit ist dem Problem der analytischen Beschreibung des Confinement-Mechanismus in der QCD und in anderen Eichtheorien gewidment. Als Leitprinzip der Arbeit wurde das sogenannte Wilsonsche-Confinement-Kriterium gewählt, gemäss welchem diese Erscheinung durch eine effektive Stringtheorie beschrieben werden kann. Die entstehenden Strings des Eichfeldes verbinden farbige-Objekte (Quarks, Gluonen) miteinander und hindern ihr Auseinandergehen auf makroskopische Abstände. Es werden verschiedene Verfahren der Ableitung dieser Stringstheorien aus unterschiedlichen Eichtheorien, einschliesslich der QCD, vorgestellt. Kapitel 2 enthält die Untersuchung der nichtlokalen effektiven Stringwirkung, die im Rahmen des sogenannten stochastischen Vakuum-Modells der QCD entsteht, wobei die Wechselwirkung zwischen den Elementen der String-Weltfläche durch den phänomenologischen Background-Gluon-Propagator vermittelt wird. Durch Entwicklung dieser Wirkung nach Ableitungen wurden die ersten Terme niedrigster Ordnung bestimmt. Die ersten beiden Terme dieser Entwicklung sind die Nambu-Goto- und Rigidity-Terme mit Kopplungskonstanten, die sich durch das Gluon-Kondensat und die Korrelationlänge des QCD-Vakuums ausdrücken lassen. Die Vorzeichen dieser Konstanten zeigen, dass die durch dieses Verfahren erhaltenen Strings stabil sind. Danach wurde eine mögliche Lösung des ``Crumpling'' Problems auf der Basis eines zusätzlichen topologischen Stringtermes im Instantongas-Modell des QCD-Vakuums vorgestellt. Mittels Störungstheorie im nicht-störungstheoretischen QCD-Background berechneten wir zusätzliche-Korrekturen zur ursprünglichen nicht-störungstheoretischen Stringwirkung. Diese Korrekturen führen zu neuen Formen der nichtlokalen effektiven Stringwirkung, die den störungstheoretischen Gluon-Propagator im Backgroundfeld zwischen den Elementen der Weltfläche enthalten. Durch Ableitungsentwicklung dieser Wirkung bekommen wir eine Korrektur zur Kopplungskonstante des Rigidity-Terms; die Stringsspannung des Nambu-Goto-Terms jedoch bleibt unverändert. Am Ende dieses Kapitels wurde der Hamilton-Operator des QCD-Strings mit spinlosen Quarks hergeleitet, der der effektiven Stringwirkung mit Rigidity-Term entspricht. Dieser Hamilton-Operator liefert einen Korrekturterm zur Wechselwirkung im relativistischen Quarkmodell-Operator. Im Kapitel 3 untersuchten wir das Problem der Stringdarstellung von Abelsch-projezierten Eichtheorien. Als erstes wurde die Herleitung der Stringdarstellung der erzeugenden Funktion für das einfachste Modell dieser Art, d.h. die Abelsch-projezierte SU(2)-QCD gegeben, die einem dualen Abelschen Higgs-Modell mit äusseren elektrisch geladendenen Teilchen äquivalent ist. Der Vorteil dieses Stringszuganges im Vergleich zum Zugang des stochastischen Vakuum-Modells der QCD besteht in der Berücksichtigung der Integration über String-Weltflächen, die auf Grund der Integration über den Singulärteil der Higgsfeld-Phase entsteht. Zusätzlich zur Stringdarstellung der erzeugenden Funktion wurde im London-Limes die Stringdarstellung für die erzeugenden Funktionale der Feldstärke- und Monopolstromkorrelatoren hergeleitet. Dies gab uns die Möglichkeit, die entsprechenden bilokalen Kumulanten zu finden und zu zeigen, dass die bilokalen Kumulanten der Feldstärke für grosse Abstände das gleiche Verhalten wie die entsprechenden eichinvarianten Kumulanten der QCD zeigen. Das Letztere wurde durch das stochastische Vakuum-Modell vorhergesagt und durch Gitterexperimente berechnet. Dieses Ergebnis unterstützt einerseits die Methode der Abelschen Projektion und gibt anderseits dem stochastischen Vakuum-Modell der QCD einen neuen feldtheoretischen Status. Danach erweiterten wir unsere Analyse über den Rahmen des London-Limes hinaus untersuchten den Zusammenhang von quartischen Kumulanten und bilokalen Kumulanten. Anschliessend wurde die Stringdarstellung der SU(3)-Gluodynamik hergeleitet. Dabei wurde die Stringdarstellung für ein entsprechendes duales Modell formuliert, das drei Arten des magnetischen Higgs-Feldes enthält. Infolgedessen liefert das Modell drei Strings, von denen nur zwei wirklich unabhängig sind. Alle diese Strings wechselwirken untereinander durch Austausch zweier massiver dualer Eichbosonen. Ausserdem erhielten wir die bilokalen Kumulanten des effektiven dualen Modells der SU(3)-Gluodynamik. Die entsprechenden bilokalen Kumulanten zeigen für grosse Abstände ein Verhalten wie es durch das stochastische Vakuum-Modell vorhergesagt wurde. Zum Schluss dieses Kapitels geben wir eine nützliche Darstellung für erzeugende Funktionen von Abelsch-projezierten Theorien in Form von Integralen über Monopolströme an. Im Kapitel 4 wurde ein weiteres Modell untersucht, das eine analytische Beschreibung des Confinement-Mechanismus zulässt, nämlich die 3D kompakte QED. Für den Wilson-Loop der entsprechenden Theorie mit Monopoldichten wurde die Äquivalenz zur sogenannten Confining-Stringtheorie demonstriert. Ausserdem wurde das Verhalten der bilokalen Kumulante der Feldstärke im Limes schwacher Felder untersucht. Dieses Verhalten befindet sich ebenfalls in Übereinstimmung mit den Voraussagen des stochastischen Vakuum-Modells. Erwartungsgemäss sind die Stringdarstellungen der erzeugenden Funktionen der 3D kompakten QED im Limes schwacher Felder und der dualen Abelschen Higgs-Modelle sehr ähnlich. Wir zeigten ausserdem, dass diese Entsprechung nicht zufällig ist. Die 3D kompakte QED ergibt sich nämlich im Limes verschwindender Eichbosonmasse aus dem 3D Abelschen Higgs-Modell mit äusseren Monopolen. Zum Schluss wurde ein allgemeines Verfahren der Beschreibung der Anregungen der Stringweltfläche in Abelsch-projezierten Theorien (kompakte QED und QCD) ausgearbeitet. Es ist auf der Methode der nicht-linearen Sigma-Modelle gegründet und gibt eine Möglichkeit, die in diesen Fluktuationen quadratische Effektive Wirkung zu erhalten. In der Dissertation wurden analytische nicht-störungstheoretische Verfahren ausgearbeitet, die neue Informationen über den Confinement-Mechanismus in der QCD und anderen Eichtheorien liefern und zum besseren Verständnis der Vakuumstruktur dieser Theorien beitragen können. Sie sind insbesondere relevant für die Herleitung effektiver Stringtheorien aus Eichtheorien.<br>The main problem addressed in the present Dissertation was an attempt of an analytical description of confinement in QCD and other gauge theories. As a guiding principle for our investigations served the so-called Wilson's picture of confinement, according to which this phenomenon can be described in terms of some effective theory of strings, joining coloured objects to each other and preventing them from moving apart to macroscopic distances. In this Dissertation, we have proceeded with a derivation of such string theories corresponding to various gauge ones, including QCD, i.e. with the solution of the problem of string representation of gauge theories. We have started our analysis with the nonlocal string effective action, arising within the so-called Stochastic Vacuum Model of QCD, where the interaction between the string world-sheet elements is mediated by the phenomenological background gluon propagator. By performing the derivative expansion of this action, we have derived the first few terms of a string Lagrangian. The first two nontrivial of them turned out to be the Nambu-Goto and rigidity terms with the coupling constants expressed completely via the gluonic condensate and correlation length of the QCD vacuum. The signs of these constants ensure the stability of strings in the so-obtained effective string theory. After that, we have investigated the problem of crumpling for the string world-sheets by derivation of the topological string term in the instanton gas model of the gluodynamics vacuum. Next, by making use of perturbation theory in the nonperturbative QCD vacuum, we have calculated perturbative corrections to the obtained string effective action. Those lead to a new form of the nonlocal string effective action with the propagator between the elements of the world-sheet being the one of a perturbative gluon in the confining background. By the derivative expansion of this action, we got a correction to the rigidity term coupling constant, whereas the string tension of the Nambu-Goto term occurs to get no corrections due to perturbative gluonic exchanges. Finally, we have derived the Hamiltonian of QCD string with spinless quarks at the ends, associated with the obtained string effective action including the rigidity term. In the particular case of vanishing orbital momentum of the system, this Hamiltonian reduces to that of the so-called relativistic quark model, albeit with some modifications due to the rigidity term, which might have some influence on the dynamics of the QCD string with quarks. All these topics have been elaborated on in Section 2, and form the essence of the string representation of QCD within the Stochastic Vacuum Model. In Section 3, we have addressed the problem of string representation of Abelian-projected theories. In this way, we have started with the string representation for the partition function of the simplest model of this kind, namely the Abelian-projected SU(2)-QCD, which is argued to be the dual Abelian Higgs Model with external electrically charged particles. The advantage of this approach to the string representation of QCD w.r.t. the one based on the Stochastic Vacuum Model is a possibility to get an integration over the string world-sheets, resulting from the integration over the singular part of the phase of the Higgs field. After the string representation of the partition function in the London limit, we have proceeded with the string representation for the generating functionals of the field strength and monopole current correlators. Those enabled us to find the corresponding bilocal cumulants and demonstrate that the large-distance asymptotic behaviour of the bilocal field strength cumulant matches the one of the corresponding gauge-invariant cumulant in QCD, predicted by the Stochastic Vacuum Model and measured in the lattice experiments. This result supports the method of Abelian projection on the one hand and gives a new field-theoretical status to the Stochastic Vacuum Model on the other hand. After that, we have extended our analysis beyond the London limit, and studied the relation of the quartic cumulant, which appears there, with the bilocal one in the London limit. Next, by making use of the Abelian projection method, we have addressed the problem of string representation of the SU(3)-gluodynamics. Namely, we have casted the related dual model, containing three types of magnetic Higgs fields, into the string form. Consequently, the latter one turned out to contain three types of strings, among which only two ones were actually independent. As a result, we have found, that both the ensemble of strings as a whole and individual strings display confining properties in a sense that all types of strings (self)interact via the exchanges of the massive dual gauge bosons. We have also derived bilocal cumulants in the effective dual model of confinement, corresponding to the SU(3)-gluodynamics, and they turned out to be also in line with the ones predicted by the Stochastic Vacuum Model. In conclusion of this topic, we have obtained another useful representation for the partition functions of the Abelian-projected theories in the form of an integral over the monopole currents. In Section 4, we have studied another model, allowing for an analytical description of confinement, which is 3D compact QED. In this way, by virtue of the integral over the monopole densities, we have derived string representation for the Wilson loop in this theory and demonstrated the correspondence of this representation to another recently found one, the so-called confining string theory. After that, we have calculated the bilocal cumulant of the field strength tensors in the weak-field limit of the model under study. It also turned out to be in line with the general concepts of the Stochastic Vacuum Model and therefore matches the corresponding results known from the lattice measurements in QCD and found analytically for the effective Abelian-projected theories in the previous Section. Besides that, string representations of the partition functions of the weak-field limit of 3D compact QED and of the dual Abelian Higgs Model turned out to be also quite similar. We have illustrated later on that this correspondence is not accidental. Namely, we have shown that 3D compact QED is nothing else, but the limiting case of 3D Abelian Higgs Model with external monopoles, corresponding to the vanishing gauge boson mass. Finally, we have elaborated on a unified method of description of the string world-sheet excitations in the Abelian-projected theories, compact QED, and QCD, based on the techniques of nonlinear sigma-models, and obtained the effective action, quadratic in the world-sheet fluctuations. In conclusion, the proposed nonperturbative techniques provide us with some new information on the mechanisms of confinement in QCD and other gauge theories and shed some light on the vacuum structure of these theories. They also show the relevance of string theory to the description of this phenomenon and yield several prescriptions for the construction of the adequate string theories from the corresponding gauge ones.

Thesis (Ph. D.)--University of California, San Diego, 2006.<br>Title from first page of PDF file (viewed June 27, 2006). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 159-165).

Cette thèse traite de plusieurs façons de construire une théorie quantiques des champs en quatre dimensions à partir de la théorie des cordes.<br /><br />Dans une première partie nous étudions la construction d'une théorie Yang-Mills supersymétrique, couplée à un superchamp chiral dans la représentation adjointe, à partir de la théorie des cordes de type IIB sur une variété Calabi-Yau non compacte, avec des D-branes qui enroulent certaines sousvariétés. Les propriétés de<br />la théorie de jauge sont alors reflétées dans la structure<br />géométrique de la variété Calabi-Yau. En particulier, on peut calculer en principe le superpotentiel effectif de basse énergie qui décrit la structure des vides de la théorie de jauge en utilisant la théorie des cordes (topologiques). Malheureusement, en pratique, ceci n'est pas faisable. Il est remarquable qu'on puisse cependant montrer que la dynamique de basse énergie de la<br />théorie de jauge est codée par la géométrie d'une autre variété Calabi-Yau non compacte, reliée à la première par une transition géométrique. La théorie des cordes de type IIB sur cette deuxième variété, dans laquelle sont allumés des flux de fond appropriés, génère une théorie de jauge en quatre dimensions, qui n'est d'autre que la théorie effective de basse énergie de la théorie de<br />jauge originale. Ainsi, pour obtenir le superpotentiel effectif de basse énergie il suffit simplement de calculer certaines intégrales dans la deuxième géométrie Calabi-Yau, ce qui est faisable, au moins perturbativement. On trouve alors que le problème extrêmement difficile d'étudier la dynamique de basse<br />énergie d'une théorie de jauge non Abelienne a été réduit à celui de calculer certaines intégrales dans une géométrie connue. On peut démontrer que ces intégrales sont intimement reliées à certaines quantités dans un modèle de matrices holomorphes, et on peut alors calculer le superpotentiel effectif comme fonction de<br />certaines expressions du model de matrices. Il est remarquable que la série perturbative du modèle de matrices calcule alors le superpotentiel effectif non-perturbatif.<br /><br />Ces relations étonnantes ont été découvertes et élaborée par plusieurs auteurs au cours des dernières années. Les résultats originaux de cette thèse comprennent la forme précise des relations de la ``géométrie spéciale" sur une variété Calabi-Yau<br />non compacte. Nous étudions en détail comment ces intégrales géométriques dépendent du cut-off, et leur relation à l'énergie libre du modèle de matrices. En particulier, sur une variété Calabi-Yau non compacte nous proposons une forme bilineaire sur le<br />produit direct de l'espace des formes avec l'espace des cycles, qui élimine toutes les divergences, sauf la divergence logarithmique. Notre analyse détaillée du modèle de matrices holomorphes clarifie aussi plusieurs aspects reliés à la méthode du col de ce modèle de matrices. Nous montrons en particulier qu'exiger une densité spectrale réelle restreint la forme de la<br />courbe Riemannienne qui apparaît dans la limite planaire du modèle de matrices. Çela nous donne des contraintes sur la forme du contour sur lequel les valeurs propres sont intégrées. Tous ces<br />résultats sont utilisés pour calculer explicitement l'énergie libre planaire d'un modèle de matrices avec un potentiel cubique.<br /><br />La deuxième partie de cette thèse concerne la génération de théories de jauge supersymétriques en quatre dimensions comportant des aspects caractéristiques du modèle standard à partir de<br />compactifications de la supergravité en onze dimensions sur une variété G_2. Si cette variété contient une singularité conique, des fermions chiraux apparaissent dans la théorie de jauge en quatre dimensions ce qui conduit potentiellement à des anomalies. Nous montrons que, localement à chaque singularité, les anomalies<br />correspondantes sont annulées par une non-invariance de l'action classique au singularités (``anomaly inflow"). Malheureusement, aucune métrique d'une variété G_2 compacte n'est connue explicitement. Nous construisons ici des familles de métriques sur des variétés compactes faiblement G_2, qui contiennent deux singularités coniques. Les variétés faiblement G_2 ont des propriétés semblables aux propriétés des variétés G_2, et alors ces exemples explicites pourraient être utiles pour mieux comprendre la situation générique. Finalement, nous regardons la<br />relation entre la supergravité en onze dimensions et la théorie des cordes hétérotiques E_8\times E_8. Nous étudions en détail les anomalies qui apparaissent si la supergravité est formulée sur le produit d'un espace de dix dimensions et un intervalle. Encore une fois nous trouvons que les anomalies s'annulent localement sur<br />chaque bord de l'intervalle si on modifie l'action classique d'une façon appropriée.

Various effects of finite fermionic densities in gauge theories are studied. The phase structure of SU(N) gauge theories with fermions as a function of imaginary chemical potential is related to the confining properties of the theory. This phase structure is controlled by a remnant of the Z[sub N] symmetry which is present in the absence of fermions. At high temperature the theory has a first-order phase transition as a function of imaginary chemical potential. This transition is expected to be absent in the low-temperature phase. It is shown that properties of the theory at nonzero fermion density can be deduced from its behaviour at finite imaginary chemical potential. Anomalies in gauge theories are introduced using various two-dimensional models. In particular, the chiral Schwinger model is shown to be consistent despite being anomalous. The effects of finite densities in anomalous gauge theories is investigated. It is found that, contrary to some recent claims, an effective Harniltonian (obtained by integrating out the fermions) cannot be obtained by the simple inclusion of a Chern-Simons term multiplying the fermionic chemical potential. The importance of dynamical effects is stressed and a mechanism for producing primordial magnetic field is suggested.<br>Science, Faculty of<br>Physics and Astronomy, Department of<br>Graduate

Lattice gauge theories, through Monte-Carlo simulations, provide the most powerful methods available for the non-perturbative study of many models. These techniques, however, become very inefficient as we approach the continuum limit, a problem known as Critical Slowing Down. Over recent years cluster methods have generated significant improvements over established techniques. In part I of this thesis we introduce such an algorithm for the <I>Z</I><SUB>2</SUB> Kalb-Ramond model in four dimensions, and find that we can improve the efficiency of the simulation by orders of magnitude. In the second part of this thesis we make preliminary investigations towards using duality transformations as an aid to improving numerical simulations, (where by duality we mean an extension of the technique of Kramers and Wannier). We study the compact <I>U</I>(1) lattice gauge theory in (2+1) dimensions as an example. The dual to this model is known to be a discrete Gaussian Solid On Solid model. We find the discrete nature of the model makes each update faster. However the structures which develop at high temperature, make naive simulations, overall, inefficient.

This thesis is concerned with so called chiral gauge theories, also known as self dual gauge theories. In particular, the aim of this thesis to investigate the role that chiral gauge theories play in duality symmetries in lower dimensions through dimensional reduction. Chapter one serves as an introduction to the notions of duality in field and string theory. The problems of formulating well defined actions for self-dual gauge theories are introduced as well as a brief presentation of the different approaches used to over come these problems. Chapter two introduces dimensional reduction and demonstrates how duality symmetries arise from the dimensional reduction of self-dual theories in a variety of dimensions and on different compact spaces. Examples are presented where the couplings of the resulting theories are calculated explicitly in terms of the geometrical data of the compact space. The duality generators acting on these couplings are also calculated explicitly and related to the geometry/topology of the compact space. Chapter three deals with the idea of duality manifest actions and their relation to the self-dual theories in higher dimensions. Non-linear Born-Infeld type actions are introduced and again dimensional reduction is shown to play a role in the duality of the Born-Infeld action. This leads to a duality manifest version of the Born-Infeld action. Chapter four describes perhaps the main application of this thesis. The effective action of the M-theory five brane wrapped on a torus is identified with the effective action of the IIB D-3 brane dimensionally reduced on a circle (after some appropriate world volume dualizations). The IIB S-duality then arises as a result of the modular symmetry of the torus. The final chapter contains a brief summary and a hint of further directions for research that were outside the scope of this thesis.

There has been great progress in recent years in the understanding of the mathematical structure of scattering amplitudes in Quantum Field Theory as well as the development of powerful methods for their calculation, particularly in the arena of N = 4 Super Yang-Mills where hidden and manifest symmetries lead to striking simplifications. In this thesis, we will discuss the extensions of such methods away from the case of on-shell amplitudes in conformal N = 4. After introducing the necessary mathematical background and physical setting, we consider in Chapter Three the form factors of BPS operators in N = 4 Super Yang- Mills. These objects have several physical applications, and share many properties with scattering amplitudes. However, they are off-shell, which makes them a natural starting point to set out in the direction of correlation functions. After demonstrating the computation of form factors by BCFW recursion and unitarity based methods, we go on to show how the scalar form factor can be supersymmetrised to encompass the full stress-tensor multiplet. In Chapter Four, we discuss the Sudakov form factor in ABJM Theory. This object, which first appears at two loops and controls the IR divergences of the theory, is computed by generalised unitarity. In particular, we note that the maximal transcendentality of three dimensional integrals is related to particular triple cuts. Finally, in Chapter Five we consider massive amplitudes on the Coulomb Branch of N = 4 at one loop. Here we find that vertex cut conditions inherited from the embedding of the theory in String Theory lead to a restricted class of massive integrals.

In this thesis we study partition functions of supersymmetric gauge theories on compact backgrounds in various dimensions, with particular focus on infinite dimensional symmetry algebras encoded in these observables. The compact space partition functions of the considered theories can be decomposed into products of holomorphic blocks which are identified with partition functions on elementary geometries. Partition functions on different compact spaces can be obtained by fusing the holomorphic blocks with pairings reflecting the geometric decomposition of the backgrounds. An example of this phenomenon is given by the S4 partition function of 4d N = 2 theories, which can be written as an integral of two copies of the R4 Nekrasov partition function. Remarkably, the AGT correspondence identifies the S4 partition function of class S theories with Liouville CFT correlators. The perturbative integrand is identified with the product of CFT 3-point functions, while each copy of the non-perturbative instanton partition function is identified with conformal blocks of the Virasoro algebra. In this work we define a class of q-deformed CFT correlators, where chiral blocks are controlled by the q-Virasoro algebra and are identified with R4xS1 instanton partition functions. We derive the 3-point functions for two different q-deformed CFTs, and we show that non-chiral correlators can be identified with S5 and S4xS1 partition functions of certain 5d N = 1 theories. Moreover, particular degenerate correlators are mapped to S3 and S2xS1 partition functions of 3d N = 2 theories. This fits the interpretation of the 3d theories as codimension two defects. We also study 4d N = 1 theories on T2 fibrations over S2. We prove that when anomalies are canceled, the compact space partition functions can be expressed through holomorphic blocks associated to R2xT 2. We argue that for particular theories these objects descend from R4xT 2 partition functions, which we identify with the chiral blocks of an elliptically deformed Virasoro algebra.

In this thesis we discuss supersymmetric gauge theories, focusing on exact results achieved using methods of integrability. For the larger portion of the thesis we study the N = 4 super Yang-Mills theory in the planar limit, a recurring topic being the Konishi anomalous dimension, which is roughly the analogue for the mass of the proton in quantum chromodynamics. The N = 4 supersymmetric Yang-Mills theory is known to be integrable in the planar limit, which opens up a wealth of techniques one can employ in order to nd results in this limit valid at any value of the coupling. We begin with perturbation theory where the integrability of the theory rst manifests itself. Here we showcase the rst exact result, the so-called slope function, which is the linear small spin expansion coecient of the generalized Konishi anomalous dimension. We then move on to exact results mainly achieved using the novel quantum spectral curve approach, the method allowing one to nd scaling dimensions of operators at arbitrary values of the coupling. As an example we nd the second coecient in the small spin expansion after the slope, which we call the curvature function. This allows us to extract non-trivial information about the Konishi operator. Methods of integrability are also applicable to other supersymmetric gauge theories such as ABJM, which in fact shares many similarities with N = 4 super Yang-Mills. We briefly review these parallel developments in the last chapter of the thesis.

This work presents recent developments on brane tilings and their vacuum moduli spaces. Brane tilings are bipartite periodic graphs on the torus and represent 4d N = 1 supersymmetric worldvolume theories living on D3-branes probing Calabi-Yau 3-fold singularities. The graph and combinatorial properties of brane tilings make the set of supersymmetric quiver theories represented by them one of the largest and richest known so far. The aim of this work is to give a concise pedagogical introduction to brane tilings and a summary on recent exciting advancement on their classification, dualities and construction. At first, particular focus is given on counting distinct Abelian orbifolds of the form C3/[gamma]. The presented counting of Abelian orbifolds of C3 and in more general of CD gives a first insight on the rich combinatorial nature of brane tilings. Following the classification theme, the work proceeds with the identification of all brane tilings whose mesonic moduli spaces as toric Calabi-Yau 3-folds are represented by reflexive polygons. There are 16 of these special convex lattice polygons. It is shown that 30 brane tilings are associated with them. Some of these brane tilings are related by a correspondence known as toric duality. The classification of brane tilings with reflexive toric diagrams led to the discovery of a new correspondence between brane tilings which we call specular duality. The new correspondence identifies brane tilings with the same master space - the combined mesonic and baryonic moduli space. As a by-product, the new correspondence paves the way for constructing brane tilings which are not confined to the torus but are on Riemann surfaces with arbitrary genus. We give the first classification of genus 2 brane tilings, illustrate the corresponding supersymmetric quiver theories and analyse their vacuum moduli spaces.

It is expected that the number of flavors in a gauge theory plays an important role in model building for physics beyond the standard model. We study the phase structure of the 12 flavor case through lattice simulations using a Rational Hybrid Monte Carlo (RHMC) algorithm for different masses, betas, and volumes, to investigate the question of conformality for this number of flavors. In particular, we analyze the Fisher's zeroes, in the vicinity of the endpoint of a line of first order phase transitions. This is motivated by previous studies that show how the complex renormalization group (RG) flows can be understood by looking at the zeros. The pinching of the imaginary part of these zeros with respect to increasing volume provides information about a possible unconventional continuum limit. We also study the mass spectrum of a multiflavor linear sigma model with a splitting of fermion masses. The single mass linear sigma model successfully described a light sigma in accordance to recent lattice results. The extension to two masses predicts an unusual ordering of scalar masses, providing incentive for further lattice simulations with split quark mass.

Taking advantage of the integrable structures emergent in the theory, non-local observables such as null polygonal Wilson loops are studied in 4d planar N = 4 Super Yang-Mills. Their duality with the 4d gluon scattering amplitudes makes the analysis even more interesting. The so-called Pentagon approach, an application of the Operator Product Expansion (OPE) method to the null polygonal Wilsol loops, makes possible a non-perturbative evaluation of these objects. They are recast as an OPE series over the 2d GKP flux-tube excitations, a description reminescent of the QCD flux-tube stretching between quarks. The integrability of the flux-tube allows us to evaluate the series, in principle, for any value of the coupling constant. From this analysis, several results have been obtained. In the strong coupling regime we reproduced the TBA-like equations expected from the minimal area problem in string theory, finding agreement with the AdS/CFT prediction. In this respect, of fundamental importance is the emergence of effective bound states between elementary fermionic excitations. Along the way, we encountered some intriguing analogies between these null polygonal Wilson loops and the Nekrasov instanton partition function Z for N = 2 theories. Furthermore, a new non-perturbative enhancement of the classical string argument has been confirmed, stemming from the dynamics of the string in the five sphere S5 and described by the non-linear σ-model O(6). Some properties of a fundamental building block of the OPE series, the SU(4) structure of the form factors of a specific twist operator P, have been analysed. This SU(4) matrix part is given a representation in terms of rational functions, organized in a Young tableaux pattern.

Superconformal field theories have been widely explored over the last years, especially because of their large amount of symmetry which allows to derive exact results. The most prominent example within this class is the four-dimensional maximally supersymmetric N = 4 Super Yang-Mills theory, which played a pivotal role in the proposal of the AdS/CFT correspondence, a conjectured weak/strong duality relating superconformal gauge theories to string theories on curved backgrounds. Soon after the formulation of the correspondence the presence of integrable structures was discovered both in planar N = 4 SYM and in its string counterpart, and since then integrability in superconformal theories became one of the main research topics in theoretical physics. Interestingly, integrability in N = 4 was later shown to be intimately related to the presence of the so called dual conformal symmetry, a hidden symmetry of the planar amplitudes that puts even stronger constraints on their structure. This suggests that some crucial aspects of a theory can be investigated through the computation of its scattering amplitudes, which can thereby provide a powerful tool also for the study of less supersymmetric superconformal field theories. In particular it is essential to understand whether some of the beautiful properties of N = 4 survives when supersymmetry is not maximal. In this thesis we present computations of massless scattering amplitudes in two different not maximally supersymmetric conformal theories: N = 2 superconformal QCD in four dimensions and N = 6 Chern-Simons matter theory (ABJM) in three dimensions. In N = 2 SCQCD we compute all possible four-point amplitudes at one loop and the two-loop amplitude with fundamental fields as external legs. In ABJM we extend the two-loop computation of four-point scattering amplitudes and of the Sudakov form factor beyond the planar limit. We also discuss our results in relation to the corresponding ones in N = 4 SYM, paying particular attention to the possible presence of dual conformal invariance and to maximum transcendentality.

This thesis is dedicated to the study of Wilson loops in supersymmetric gauge theories, we will study their properties and how they are related to physical observables, such as the Bremsstrahlung function that controls to the energy radiated by an accelerating particle. We will give various examples of the computation of their expectations values using perturbative and non-perturbative techniques. For SU(N) superconformal QCD we perform a three-loop calculation of the generalized cusp anomalous dimension of the BPS Wilson loop operator using HQET formalism. We obtain an expression which is valid at generic geometric and internal angles and finite gauge group rank N. For equal and opposite angles this expression vanishes, proving that at these points the cusp becomes BPS. From its small angle expansion we derive the corresponding Bremsstrahlung function at three loops, matching the matrix model prediction given in terms of derivatives of the Wilson loop on the ellipsoid.<br>This thesis is dedicated to the study of Wilson loops in supersymmetric gauge theories, we will study their properties and how they are related to physical observables, such as the Bremsstrahlung function that controls to the energy radiated by an accelerating particle. We will give various examples of the computation of their expectations values using perturbative and non-perturbative techniques. For SU(N) superconformal QCD we perform a three-loop calculation of the generalized cusp anomalous dimension of the BPS Wilson loop operator using HQET formalism. We obtain an expression which is valid at generic geometric and internal angles and finite gauge group rank N. For equal and opposite angles this expression vanishes, proving that at these points the cusp becomes BPS. From its small angle expansion we derive the corresponding Bremsstrahlung function at three loops, matching the matrix model prediction given in terms of derivatives of the Wilson loop on the ellipsoid.

A written report of a paper titled Holographic dual of collimated radiation by Veronika E. Hubeny where a new and easier method is proposed to estimate the “radiation due to an accelerated quark in a strongly coupled medium”. The method is able to reproduce the results from an earlier paper without the need of solving the linearized Einstein equations but by way of calculating geodesics in AdS using the AdS/CFT correspondence and the gravitational dual of the quark being a string. A quick introduction to synchrotron radiation and general relativity is given after which the AdS/CFT correspondence is introduced along with the results and method of V. Hubeny.<br>A bachelor thesis in theoretical physics.

Un ordinateur quantique (OQ), i.e. utilisant les ressources de la physique Q, superposition et intrication, pourrait fournir un gain exponentiel de temps de calcul. Une simulation utilisant ces ressources est appelée simulation Q (SQ). L’avantage des SQs sur les simulations classiques est bien établi au niveau théorique, i.e. software. Leur avantage pratique requiert un hardware Q. L’OQ, sous-entendu universel (cf. plus bas), n’a pas encore vu le jour, mais les efforts en ce sens sont croissants et variés. Aussi la SQ a-t-elle déjà été illustrée par de nombreuses expériences de principe, grâce à des calculateurs ou simulateurs Qs de taille réduite. Les marches Qs (MQs) sont des schémas de SQ particulièrement étudiés, étant des briques élémentaires pour concevoir n’importe quel algorithme Q, i.e. pour le calcul Q universel. La présente thèse est un pas de plus vers une simulation des théories Qs des champs basée sur les MQs à temps discret (MQTD). En effet, il est montré, dans certains cas, comment les MQTD peuvent simuler, au continu, l'action d'un champ de jauge Yang-Mills sur de la matière fermionique, et la rétroaction de cette-dernière sur la dynamique du champ de jauge. Les schémas proposés préservent l’invariance de jauge au niveau de la grille d’espace-temps, i.e. pas seulement au continu. Il est proposé (i) des équations de Maxwell sur grille, compatibles avec la conservation du courant sur la grille, et (ii) une courbure non-abélienne définie sur la grille. De plus, il est montré comment cette matière fermionique à base de MQTD peut être couplée à des champs gravitationnels relativistes du continu, i.e. des espaces-temps courbes, en dimension 1+2<br>A quantum (Q) computer (QC), i.e. utilizing the resources of Q physics, superposition of states and entanglement, could fournish an exponential gain in computing time. A simulation using such resources is called a Q simulation (QS). The advantage of QSs over classical ones is well established at the theoretical, i.e. software level. Their practical benefit requires their implementation on a Q hardware. The QC, i.e. the universal one (see below), has not seen the light of day yet, but the efforts in this direction are both growing and diverse. Also, QS has already been illustrated by numerous experimental proofs of principle, thanks too small-size and specific-task Q computers or simulators. Q walks (QWs) are particularly-studied QS schemes, being elementary bricks to conceive any Q algorithm, i.e. to achieve so-called universal Q computation. The present thesis is a step more towards a simulation of Q field theories based on discrete-time QWs (DTQWs). Indeed, it is shown, in certain cases, how DTQWs can simulate, in the continuum, the action of Yang-Mills gauge fields on fermionic matter, and the retroaction of the latter on the gauge-field dynamics. The suggested schemes preserve gauge invariance on the spacetime lattice, i.e. not only in the continuum. In the (1+2)D Abelian case, consistent lattice equivalents to both Maxwell’s equations and the current conservation are suggested. In the (1+1)D non-Abelian case, a lattice version of the non-Abelian field strength is suggested. Moreover, it is shown how this fermionic matter based on DTQWs can be coupled to relativistic gravitational fields of the continuum, i.e. to curved spacetimes, in several spatial dimensions

To derive the phase transition of large N gauge theories, the calculation of the partition function is restricted to the singlet sector. In this thesis, we examined whether the singlet constraint of gauge theory at the deconfinement phase was a physical consequence. We found changing the integration variables from group elements to eigenvalues explains the discrepancy of the free energy between imposing and relaxing the single constraint. Gauge theory cannot simply relax the singlet constraint at high temperature without adding a normalization factor.

Understanding physical configurations and how these can emerge from the underlying gauge theory is a fundamental problem in modern particle physics. This thesis investigates the study of these configurations primarily focussing on the need for gauge invariance in constructing the gauge invariant fields for any physical theory. We consider Wu’s approach to gauge invariance by identifying the gauge symmetry preserving conditions in quantum electrodynamics and demonstrate how Wu’s conditions for one-loop order calculations (under various regularisation schemes) leads to the maintenance of gauge invariance. The need for gauge invariance is stressed and the consequences discussed in terms of the Ward identities for which various examples and proofs are presented in this thesis. We next consider Zwanziger’s description of a mass term in Yang-Mills theory, where an expansion is introduced in terms of the quadratic and cubic powers of the field strength. Although Zwanziger introduced this expansion there is, however, no derivation or discussion about how it arises and how it may be extended to higher orders. We show how Zwanziger’s expansion in terms of the inverse covariant Laplacian can be derived and extended to higher orders. An explicit derivation is presented, for the first time, for the next to next to leading order term. The role of dressings and their factorisation lies at the heart of this analysis.

In this thesis we develop semi-classical techniques in three-dimensional gauge theories and apply them to models with extended supersymmetry. We investigate extensively the correspondence between three-dimensional gauge thesis with N = 4 supersymmetry and monopole moduli spaces. In the case of SU(z) gauge theory, we prove that the Coulomb branch is equivalent to the centred moduli space of two BPS monopoles constructed by Atyah and Hitchin. For SU(z) gauge groups, the results give new information about the moduli space of n BPS monopoles. We further extend the correspondence to include monopole moduli spaces of all simply-laced Lie groups. We also consider instanton effect in N=2 and N=8 theories.

Quiver gauge theories are widely studied in the context of AdS/CFT, which establishes a correspondence between CFTs and string theories. CFTs in turn offer a map between quantum states and Gauge Invariant Operators (GIOs). This thesis presents results on the counting and correlators of holomorphic GIOs in quiver gauge theories with flavour symmetries, in the zero coupling limit. We first give a prescription to build a basis of holomorphic matrix invariants, labelled by representation theory data. A fi nite N counting function of these GIOs is then given in terms of Littlewood-Richardson coefficients. In the large N limit, the generating function simpli fies to an in finite product of determinants, which depend only on the weighted adjacency matrix associated with the quiver. The building block of this product has a counting interpretation by itself, expressed in terms of words formed by partially commuting letters associated with closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. We compute the free fi eld two and three point functions of the matrix invariants. These have a non-trivial dependence on the structure of the operators and on the ranks of the gauge and flavour symmetries: our results are exact in the ranks, and their expansions contain information beyond the planar limit. We introduce a class of permutation centraliser algebras, which give a precise characterisation of the minimal set of charges needed to distinguish arbitrary matrix invariants. For the two-matrix model, the relevant non-commutative algebra is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams.

There has been a resurgence of interest in theories of massive spin-2 fields, owing to the recent discovery of ghost-free self-interactions. In addition to reviewing the historical and recent progress in this subject, I discuss my contributions, including the derivation of the complete decoupling limit of dRGT ghost-free massive gravity, proving no-go theorems on ghost-free interactions for charged spin-2 fields, updating the method of Dimensional Deconstruction for fermions to obtain massive supersymmetric gauge theories, and my progress towards supergravity theories with non-zero graviton mass.

We investigate semiclassical contributions to correlation functions in N = 1 supersymmetric gauge theories. Our principal example is the gluino condensate, which signals the breaking of chiral symmetry, and should be exactly calculable, according to a persymmetric non-renormalisation theorem. However, the two calculational approaches previously employed, SCI and WCI methods, yield different values of the gluino condensate. We describe work undertaken to resolve this discrepancy, involving a new type of calculation in which the space is changed from R(^4) to the cylinder R(3) x S(1) This brings control over the coupling, and supersymmetry ensures that we are able to continue to large radii and extract answers relevant to R(^4). The dominant semiclassical configurations on the cylinder are all possible combinations of various types of fundamental monopoles. One specific combination is a periodic instanton, so monopoles are the analogue of the instanton partons that have been conjectured to be important at strong coupling. Other combinations provide significant contributions that are neglected in the SCI approach. Monopoles are shown to generate a superpotential that determines the quantum vacuum, where the theory is confining. The gluino condensate is calculated by summing the direct contributions from all fundamental monopoles. It is found to be in agreement with the WCI result for any classical gauge group, whereas the values for the exceptional groups have not been calculated before. The ADS superpotential, which describes the low energy dynamics of matter in a supersymmetric gauge theory, is derived using monopoles for all cases where instantons do not contribute. We report on progress made towards a two monopole calculation, in an attempt to quantify the missed contributions of the SCI method. Unfortunately, this eventually proved too complicated to be feasible.

Gauge theories is one of the most pervasive and important subject of modern theoretical physics, and there are still many things about them we do not understand. In particular dealing with strongly coupled theories where normal perturbative techniques do not apply is a fundamental open problem. In this thesis, we study a particular class of toy-models that have supersymmetry, which makes them much easier to deal with. We employ the mathematical technique of localization, which for supersymmetric theories lets us evaluate certain path integrals exactly and for any value of the coupling. This is used to study the 5d N=1 theories placed on toric Sasaki-Einstein manifolds and compute their partition functions, finding that they factorize into a product of contributions from each closed Reeb orbit of the manifold. This computation leads us to define two new hierarchies of special functions associated to these manifolds, and we study their properties. Finally, we use the 5d N=1 theories to construct new 4d N=2 theories on a large class of curved backgrounds. These theories have some interesting features, such as supporting both instantons and anti-instantons, and having a position-dependent complexified coupling constant.

In this thesis we present several results concerning effective potentials in gauge theories. Firstly we calculate the effective potential up to one loop of weakly coupled large N N = 4 super Yang-Mills theory with gauge group SU(N) on S[3] x S[1] at low temperature and near-critical chemical potentials. We show that the theory contains a metastable phase for chemical potentials below a certain temperature- dependent value. We then give a review of Vilkovisky's formulation of the effective action, and use this method to calculate the effective potential of scalar electrodynamics with background VEVs for the scalar field and gauge field. Finally we describe the classical vacuum structure of a class of N = 1 gauge theories, which may be used to calculate the exact effective potential using the Dijkgraaf-Vafa conjecture.

In the first chapter we present some general concepts related to theories with extra dimensions adopting the KK perspective. In particular we report the general form of the action for scalar, fermion and gauge fields on a 5D interval with generic warped metric. At the same time we briefly discuss the boundary conditions which are allowed in these theories. Afterwards we present the general features of orbifold compactifications including a review of Scherk-Schwarz boundary conditions and Wilson lines. In chapter 2 we give a comprehensive presentation of the holographic approach. In the first part of the chapter we discuss the general framework, focusing in particular on the non-trivial case of gauge fields. In the second part we present three interesting applications of the holographic procedure. We begin with the computation of the treelevel holographic action for a generic 5D model, showing how it can be used to compute the one-loop Higgs effective potential in GHU theories. Then we use holography to obtain the Chiral Perturbation Theory (xPT) Lagrangian in Holographic QCD. As a last, more theoretical, application, we study the consequences of the introduction of a Chern-Simons (CS) term into a 5D gauge theory. The last three chapters are devoted to the study of 5D GHU theories compactified on the orbifold 8 1 /Z2. We begin with a review of the simplest implementation of the GHU idea which gives rise to the semi-realistic model of [15]. In particular we will discuss the general framework of this model pointing out its appealing features and its shortcuts. In chapter 4 we show how the problems of the previously considered model can be solved by assuming a breaking of the Lorentz symmetry along the fifth dimension and introducing the Z2 mirror symmetry. Furthermore, we will present a detailed study of this theory from both a qualitative and a quantitative point of view, determining the constraints on the parameter space which arise from the EW precision measurements and giving an estimate of the amount of fine-tuning of the model. As side remarks, we briefly discuss the presence of a viable DM candidate in the theory and we suggest a possible mechanism which could give origin to a spontaneous breaking of the Lorentz invariance in a purely 5D context. Finally, in chapter 5, we study the properties of GHU models at finite temperature. First of all we discuss the general features of the finite-temperature Higgs effective potential at one loop, showing that an Electroweak Phase Transition (EWPT) is always present in such models at a temperature of order 1/(2nR). Then we specialize our analysis to the models presented in chapters 3 and 4, and we perform a detailed study of the phase transition using analytic and numerical techniques. As a last issue, in section 5.3, we give some evidence of the stability of the finite-temperature one-loop potential in GHU models showing that the leading higher-loop corrections are not very relevant up to temperatures well above the EWPT. We also include at the end some, quite technical, appendices. In the first one we present a more formal derivation of the gauge-fixing procedure employed in the holographic approach of chapter 2. In the next appendix, we show how it is possible to handle boundary terms and localized fields within the holographic description and we exemplify this possibility by treating the toy model of a scalar with boundary mass terms on a warped space. In appendix C we collect some useful formulae needed to compute the holographic action for the AdS5 and flat space cases. In appendix D we give the decomposition of the most relevant SU(3) representations in terms of multiplets of the SU(2) x U(l) subgroup. This decomposition results useful in the study of the GHU models presented in chapters 3, 4 and 5. In appendix E another application of the holographic approach is given, namely we show how one can compute the distortion of the ZbLbL vertex (see section 4.4.1) in a simplified toy model. In appendix F we analyze the localization of the fermion zero-modes in the model of chapter 4. In the last appendix the explicit form of the finite-temperature one-loop Higgs potential is reported. In particular we collect various equivalent formulae by which one express the finite-temperature contributions to the potential.