Rozprawy doktorskie na temat „Gauge group SU(N)”

In dieser Arbeit praesentieren wir Resultate der topologischen Suszeptibilitaet “chi” und untersuchen die Faktorisierung der reinen SU(N) Yang-Mills Eichtheorie im 't Hooft'schen Grenzwert grosser N. Ein entscheidender Teil der Berechnung von chi in der Gittereichtheorie ist die Abschaetzung des topologischen Ladungsdichtekorrelators, die durch ein schlechtes Signal-Rausch- Verhaeltnis beeintraechtigt ist. Um dieses Problem abzuschwaechen, fuehren wir einen neuen, auf einem mehrstufigen Vorgehen beruhenden Algorithmus ein, um die Korrelationsfunktion von Observablen zu berechnen, die mit dem Yang-Mills Gradientenfluss geglaettet wurden. Angewandt auf unsere Observablen, erhalten wir Ergebnisse, deren Fehlerskalierung besser ist, als die von herkoemmlichen Monte-Carlo Simulationen. Wir bestimmen die topologische Suszeptibilitaet in der reinen Yang-Mills Eichtheorie fuer Eichgruppen mit N = 4,5,6 und drei verschiedenen Gitterabstaenden. Um das Einfrieren der Topologie zu umgehen, wenden wir offene Randbedingungen an. Zusaetzlich wenden wir die korrekte Definition der topologischen Ladungsdichte durch den Gradientenfluss an. Unser Endresultat im des Grenzfalls von grossen N repraesentiert eine neue Qualitaet in der Verifikation der Witten-Veneziano Formel. Schliesslich benutzen wir die Gitterformulierung, um die Erwartungswertfaktorisierung des Produkts eichinvarianter Operatoren im Grenzwert grosser N zu verifizieren. Wir arbeiten mit durch den Yang-Mills Grandientenfluss geglaetteten Wilsonschleifen und Simulationen bis zur Eichgruppe SU(8). Die Extrapolationen zu grossen N sind in Ueberstimmung mit der Faktorisierung sowohl fuer endlichen Gitterabstand als auch in Kontinnumslimes. Unsere Daten erlauben uns nicht nur die Verifizierung der Faktorisierung, sondern auch einen hochpraezisen Test des 1/N Skalierungsverhaltens. Hier konnten wir das quadratische Skalierungsverhalten in 1/N finden, welches von 't Hooft vorhergesagt wurde.
In this thesis we present results for the topological susceptibility “chi”, and investigate the property of factorization in the 't Hooft large N limit of SU(N) pure Yang-Mills gauge theory. A key component in the lattice gauge theory computation of chi is the estimation of the topological charge density correlator, which is affected by a severe signal to noise problem. To alleviate this problem, we introduce a novel algorithm that uses a multilevel type approach to compute the correlation function of observables smoothed with the Yang-Mills gradient flow. When applied to our observables, the results show an scaling of the error which is better than the one of standard Monte-Carlo simulations. We compute the topological susceptibility in the pure Yang-Mills gauge theory for the gauge groups with N = 4, 5, 6 and three different lattice spacings. In order to deal with the freezing of topology, we use open boundary conditions. In addition, we employ the theoretically sound definition of the topological charge density through the gradient flow. Our final result in the limit N to infinity, represents a new quality in the verification of the Witten-Veneziano formula. Lastly, we use the lattice formulation to verify the factorization of the expectation value of the product of gauge invariant operators in the large N limit. We work with Wilson loops smoothed with the Yang-Mills gradient flow and simulations up to the gauge group SU(8). The large N extrapolations at finite lattice spacing and in the continuum are compatible with factorization. Our data allow us not only to verify factorization, but also to test the 1/N scaling up to very high precision, where we find it to agree very well with a quadratic series in 1/N as predicted originally by 't Hooft for the case of the pure Yang-Mills gauge theory.

In questa tesi discutiamo un limite di particolari teorie di gauge 3d, le teorie T^sigma_rho[SU(N)], che ammetono una descrizione in termini di da quiver. Nel limite considerato, il numero di nodi è grande e i ranghi scalano quadraticamente con la lunghezza del quiver. Le energie libere sulla sfera e il topologically twisted index sono ottenuti usando la procedura della localizzazione supersimmetrica. Entrambi scalano quarticamente con la lunghezza del quiver e quadraticamente con N, con funzioni trilogaritmiche dipendenti dai dati del quiver come coefficienti. Le teorie precedentemente discusse con scaling N^2 \ln N sorgono come casi limite. Le SCFTs IR hanno duali di supergravità ben definiti in Tipo IIB: le energie libere corrispondono esattamente ai risultati olografici e gli indici, nel caso di un twist universale, riproducono correttamente l'entropia di un buco nero universale che può essere immerso nelle soluzioni olograficamente duali. Ogni teoria 3d descritta da un quiver bilanciato è legata ad una 5d, il cui modello matriciale è dominato dallo stesso punto di sella. Ciò conduce a strette relazioni tra osservabili BPS. In particolare, calcoliamo il valore di aspettazione dei Wilson loop in rappresentazioni antisimmetriche, trovando un perfetto accordo con il lato di gravità in un particolare esempio.
In this thesis we discuss a limit of 3d T^sigma _rho[SU(N)] quiver gauge theories in which the number of nodes is large and the ranks scale quadratically with the length of the quiver. The sphere free energies and topologically twisted indices are obtained using supersymmetric localization. Both scale quartically with the length of the quiver and quadratically with $N$, with trilogarithm functions depending on the quiver data as coefficients. Previously discussed theories with $N^2 \ln N$ scaling arise as limiting cases. The IR SCFTs have well-behaved supergravity duals in Type IIB: the free energies match precisely with holographic results and the indices, in case of a universal twist, correctly reproduce the entropy of an universal black hole which can be embedded in the holographically dual solutions. Each balanced 3d quiver theory is linked to a 5d parent, whose matrix model is related and dominated by the same saddle point, leading to close relations between BPS observables. In particular, we compute the expectation value of Wilson loops in antisymmetric representations, finding perfect agreement with the gravity side in a particular example.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-graduação em Física
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Monopolos magnéticos têm sido objetos de grande interesse nos últimos anos, principalmente por serem previstos em algumas teorias de grande unificação e por, possivelmente, serem relevantes no fenômeno do confinamento em QCD. Consideramos uma teoria de Yang-Mills-Higgs com simetria de gauge SU(n) quebrada espontaneamente em SO(n) que apresenta condições topológicas necessárias para a existência de monopolos Z2. Construímos as formas assintóticas desses monopolos, considerando duas quebras distintas do SU(n) em SO(n), e verificamos que os monopolos fundamentais estão associados aos pesos da representação definidora da álgebra so(n)v.

Les modèles SU(2)xSU(2)xU(1) représentent une étape intermédiaire motivée par l'unification des groupes de jauge du Modèle Standard (MS). Un groupe de jauge étendu, par rapport a celui du MS, implique l'existence de nouveaux bosons de jauge, neutres et charges. Ces bosons dénotés Z' et W' sont recherches activement au Large Hadron Collider (LHC). Sur la base d'une analyse globale récente des contraintes sur ces modèles, provenant des expériences a basse énergie et du LEP, nous effectuons une analyse numérique au Leading Order (LO) des différentes signatures au LHC. Nous montrons que les sections efficaces totales pour les leptons et les paires de quarks de troisième génération, expérimentalement facilement accessibles, fournissent individuellement qu'uneinformation partielles sur le modèle réalise dans la nature. En revanche, les corrélations de ces mêmes sections efficaces pourraient bien conduire a une identification unique. Par la suite, nous étudions la production électrofaible d'une paire de quarks top au Next-to-Leading Order dans les extensions du MS prédisant un boson Z' supplémentaire et en supposant des couplages génériqueset diagonaux dans la base des saveurs. Nous calculons les corrections virtuelles et réelles a l'ordre de O(alS*alW^2) et les implémentons dans le générateur d'événements Monte Carlo POWHEG BOX qui permet de réaliser de manière cohérente la fusion du calcul QCD NLO avec les parton showers. Nousconstatons que les corrections QCD NLO peuvent être très importantes, mais que les K-facteurs restent modestes dans la région de masse invariante centrée autour de la masse de la résonance
General SU(2)x SU(2)x U(1) models represent a well-motivated intermediate step towards the unication of the Standard Model (SM) gauge groups. Extended gauge group sector, as compared to that of the SM, leads to additional neutral and charged gauge bosons. These so-called Z' and W' bosons are actively searched for at the Large Hadron Collider (LHC). Based on a recent global analysis of low-energy and LEP constraints of these models, we perform numerical scans of their various signals at the LHC at Leading Order accuracy. We show that total cross sections for lepton and third-generation quark pairs, while experimentally easily accessible, provide individually only partial information about the model realized in Nature. In contrast, correlations of these cross sections in the neutral and charged current channels may well lead to a unique identification. Subsequently we study the electroweak top-pair production at Next-to-leading Order (NLO) accuracy in the SM extensions with an additional Z' boson assuming general flavour-diagonal couplings. We calculate the virtual and real corrections at order O(alS*alW^2) and implement them in the POWHEG BOX framework which allows for consistent matching of NLO QCD calculations with parton showers. We find that the NLO corrections can be very important but the K-factors in the invariant mass region around the resonance mass are modest

Recent efforts to classify off-shell representations of supersymmetry without a central charge have focused upon directed, supermultiplet graphs of hypercubic topology known as Adinkras. These encodings of Super Poincare algebras, depict every generator of a chosen supersymmetry as a node-pair transformtion between fermionic / bosonic component fields. This research thesis is a culmination of investigating novel diagrammatic sums of gauge quotients by supersymmetric images of other Adinkras, and the correlated building of field theoretic worldline Lagrangians to accommodate both classical and quantum venues. We find Ref [40], that such gauge quotients do not yield other stand alone or ”proper” Adinkras as afore sighted, nor can they be decomposed into supermultiplet sums, but are rather a connected ”Adinkraic network”. Their iteration, analogous to Weyl's construction for producing all finite-dimensional unitary representations in Lie algebras, sets off chains of algebraic paradigms in discrete-graph and continuous-field variables, the links of which feature distinct, supersymmetric Lagrangian templates. Collectively, these Adiankraic series air new symbolic genera for equation to phase moments in Feynman path integrals. Guided in this light, we proceed by constructing Lagrangians actions for the N = 3 supermultiplet YI /(iDI X) for I = 1, 2, 3, where YI and X are standard, Salam-Strathdee superfields: YI fermionic and X bosonic. The system, bilinear in the component fields exhibits a total of thirteen free parameters, seven of which specify Zeeman-like coupling to external background (magnetic) fluxes. All but special subsets of this parameter space describe aperiodic oscillatory responses, some of which are found to be surprisingly controlled by the golden ratio, ? ? 1.61803, Ref [52]. It is further determined that these Lagrangians allow an N = 3 ? 4 supersymmetric extension to the Chiral-Chiral and Chiral-twisted- Chiral multiplet, while a subset admits two inequivalent such extensions. In a natural progression, a continuum of observably and usefully inequivalent, finite-dimensional off-shell representations of worldline N = 4 extended supersymmetry are explored, that are variate from one another but in the value of a tuning parameter, Ref [53]. Their dynamics turns out to be nontrivial already when restricting to just bilinear Lagrangians. In particular, we find a 34-parameter family of bilinear Lagrangians that couple two differently tuned supermultiplets to each other and to external magnetic fluxes, where the explicit parameter dependence is unremovable by any field redefinition and is therefore observable. This offers the evaluation of X-phase sensitive, off-shell path integrals with promising correlations to group product decompositions and to deriving source emergences of higher-order background flux-forms on 2-dimensional manifolds, the stacks of which comprise space-time volumes. Application to nonlinear sigma models would naturally follow, having potential use in M- and F- string theories.
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A presente tese aborda dois problemas distintos e independentes: triangularização de sistemas não-lineares com duas entradas e controle de sistemas sem arrasto que evoluem no grupo especial unitário SU(n). Em relação ao primeiro, estabeleceu-se, através da generalização de resultados bem conhecidos, condições geométricas para que um sistema com duas entradas seja descrito por uma forma triangular específica após uma mudança de coordenadas e uma realimentação de estado estática regular. Para o segundo problema, desenvolveu-se uma estratégia de controle que força o estado do sistema a rastrear assintoticamente uma trajetória de referência periódica que passa por um estado objetivo arbitrário. O método de controle proposto utiliza os resultados de convergência de tipo- Lyapunov que foram estabelecidos pela presente pesquisa e que tiveram como inspiração uma versão periódica do princípio da invariância de LaSalle. Apresentou-se, ainda, os resultados de simulação obtidos com a aplicação da técnica de controle desenvolvida a um sistema quântico consistindo de duas partículas de spin-1/2, com o objetivo de gerar a porta lógica quântica C-NOT.
This thesis treats two distinct and independent problems: triangularization of nonlinear systems with two inputs and control of driftless systems which evolve on the special unitary group SU(n). Concerning the first, one has established, by means of the generalization of well-known results, geometric conditions for a system with two inputs to be described by a specific triangular form after a change of coordinates and a regular static state feedback. For the second problem, one has developed a control strategy that forces the state of the system to track in an asymptotic manner a periodic reference trajectory which passes by an arbitrary goal state. The proposed control method uses Lyapunovlike convergence results that were established in this research and which were inspired in a periodic version of LaSalles invariance principle. Furthermore, one has shown the simulation results obtained from the application of the developed control technique to a quantum system consisting of two spin-1/2 particles, with the aim of generating the C-NOT quantum logic gate.