In this thesis we study the connection between conformal symmetry breaking and the the renormalization group. In the first chapter we review the main properties of conformal field theories (CFTs), Wilsonian RG and describe how renormalization induces a flow between different CFTs. The prominent role is given to the trace of energy-momentum tensor (TEMT) as a measure for conformal symmetry violation. Scaling properties of supersymmetric gauge theories are also reviewed . In the second chapter the quantum action principle is introduced as a scheme for renormalizing composite operators. The framework is then applied to derive conditions for UV finiteness of two-point correlators of composite operators with special emphasis on TEMT. We then proceed to discuss the application of the Feynman-Hellmann theorem to evaluate gluon condensates. In the third chapter the basic elements the Trace anomaly on curved space are examined. The finiteness results from Chapter 2 are given physical meaning in relation with the RG flow of the geometrical quantity ~ d (coefficient of □R in the anomaly). The last chapter is dedicated to the a-theorem. First we apply some of the results derived in Chapter 3 to extend the known perturbative calculation for the flow of the central charge βa for gauge theories with Banks-Zaks fixed point. In the last part we review the main ideas of the recent proof of the a-theorem by Komargodski and Schwimmer and apply their formalism to re-derive the known non-perturbative formula for ∆ βa of SUSY conformal window theories.

As ideias fundamentais sobre entropia de emaranhamento e fluxos de renormalização são expostas, assim como uma introdução a CFTs e sua ligacão com a estrutura do espaco de parâmetros. A anomalia de traço é calculada em uma abordagem semi-clássica usando o método de heat kernel\" e regularização por função zeta . Mostramos que os coeficientes de Seeley-DeWitt são responsáveis pela quebra de simetria conforme em um espaço-tempo curvo de dimensão par, com isso alcançamos uma definição geométrica para as cargas centrais. A inexistência de anomalias no caso de dimensões ímpares também e mostrado. O C-theorem\", que prova a monotonicidade das cargas centrais sob o fluxo de renormalização, é demonstrado como feito por Zamolodchikov por meio de uma abordagem euclideana assumindo unitariedade, positividade por reflexão e condições de renormalizabilidade. A análise feita por Cardy também e demonstrada, nela considera-se os mesmos ingredientes. Por fim, a prova tecida por Casini & Huerta é demonstrada com detalhes, essa prova utiliza das propriedades de strong subadditivity da entropia de emaranhamento, unitariedade e invariância sob o grupo de Poincaré. Com isso, uma conexão com informação quântica é feita naturalmente. No último capítulo generalizamos o conceito de carga central para dimensões ímpares as definindo como o termo universal na entropia de emarahamento de uma esfera. As considerações geométricas feitas para provar o C-theorem\" são estendidas para um espaço-tempo de Minkowski com três dimensões. Como consequência temos a prova do F-theorem\" que é o analogo em três dimensões do C-theorem\". / The fundamental ideas of entanglement entropy and RG flows are laid out, as well as the basics of CFTs and its connection to the framework of RG flows. The trace anomaly is calculated in a semi-classical fashion by using the heat kernel method and zeta-function regularization. It is shown that the Seeley-DeWitt coefficients are responsible for the breaking of conformal symmetry in a curved even-dimensional background, which also achieves a geometrical definition of a central charge. The absence of anomalies in odd space-time dimensions is also contemplated. The C-theorem, which proves the monotonicity of the two dimensional central charge under RG flows, is demonstrated as first done by Zamolodchikov in an euclidean approach assuming unitarity, reflection positivity, and renormalizability conditions. Cardy\'s analysis is also demonstrated by considering the same conditions as Zamolodchikovs . And at last the proof via entanglement entropy by Casini & Huerta which relies on the strong subadditivity property of EE, unitarity and Poincaré invariance is explained in detail, providing a quantum information approach to the problem. In the last chapter a generalization of central charges to odd dimensional space-times is given through the universal term of the EE of a sphere. We provide the extension of the geometrical setup considered in the proof of the C-theorem to a three dimensional Minkowski space-time, which ultimately yields the F-theorem, constituting the three dimensional analog of the C-theorem.

As ideias fundamentais sobre entropia de emaranhamento e fluxos de renormalização são expostas, assim como uma introdução a CFTs e sua ligacão com a estrutura do espaco de parâmetros. A anomalia de traço é calculada em uma abordagem semi-clássica usando o método de heat kernel\" e regularização por função zeta . Mostramos que os coeficientes de Seeley-DeWitt são responsáveis pela quebra de simetria conforme em um espaço-tempo curvo de dimensão par, com isso alcançamos uma definição geométrica para as cargas centrais. A inexistência de anomalias no caso de dimensões ímpares também e mostrado. O C-theorem\", que prova a monotonicidade das cargas centrais sob o fluxo de renormalização, é demonstrado como feito por Zamolodchikov por meio de uma abordagem euclideana assumindo unitariedade, positividade por reflexão e condições de renormalizabilidade. A análise feita por Cardy também e demonstrada, nela considera-se os mesmos ingredientes. Por fim, a prova tecida por Casini & Huerta é demonstrada com detalhes, essa prova utiliza das propriedades de strong subadditivity da entropia de emaranhamento, unitariedade e invariância sob o grupo de Poincaré. Com isso, uma conexão com informação quântica é feita naturalmente. No último capítulo generalizamos o conceito de carga central para dimensões ímpares as definindo como o termo universal na entropia de emarahamento de uma esfera. As considerações geométricas feitas para provar o C-theorem\" são estendidas para um espaço-tempo de Minkowski com três dimensões. Como consequência temos a prova do F-theorem\" que é o analogo em três dimensões do C-theorem\". / The fundamental ideas of entanglement entropy and RG flows are laid out, as well as the basics of CFTs and its connection to the framework of RG flows. The trace anomaly is calculated in a semi-classical fashion by using the heat kernel method and zeta-function regularization. It is shown that the Seeley-DeWitt coefficients are responsible for the breaking of conformal symmetry in a curved even-dimensional background, which also achieves a geometrical definition of a central charge. The absence of anomalies in odd space-time dimensions is also contemplated. The C-theorem, which proves the monotonicity of the two dimensional central charge under RG flows, is demonstrated as first done by Zamolodchikov in an euclidean approach assuming unitarity, reflection positivity, and renormalizability conditions. Cardy\'s analysis is also demonstrated by considering the same conditions as Zamolodchikovs . And at last the proof via entanglement entropy by Casini & Huerta which relies on the strong subadditivity property of EE, unitarity and Poincaré invariance is explained in detail, providing a quantum information approach to the problem. In the last chapter a generalization of central charges to odd dimensional space-times is given through the universal term of the EE of a sphere. We provide the extension of the geometrical setup considered in the proof of the C-theorem to a three dimensional Minkowski space-time, which ultimately yields the F-theorem, constituting the three dimensional analog of the C-theorem.

In dieser Dissertation analysieren wir drei Aspekte von Konforme Feldtheorien (CFTs). Erstens betrachten wir Korrelationsfunktionen von sekundären Zuständen (SZ) in zweidimensionalen CFTs. Wir diskutieren eine rekursive Formel zu ihrer Berechnung und erstellen eine Computerimplementierung dieser Formel. Damit können wir jede Korrelationsfunktion von SZ des Vakuums erhalten und für Nicht-Vakuum-SZ den Korrelator als Differentialoperator, der auf den jeweiligen primären Korrelator wirkt, ausdrücken. Mit diesem Code untersuchen wir dann einige Verschränkungs- und Unterscheidbarkeitsmaße zwischen SZ, i.e. die Rényi-Entropie, den Spurquadratabstand und die Sandwich-Rényi-Divergenz. Mit unseren Ergebnissen können wir die Rényi Quanten-Null-Energie-Bedingung testen und stellen neue Werkzeuge zur Analyse der holographischen Beschreibung von SZ bereit. Zweitens untersuchen wir vierdimensionale Weyl-Fermionen auf verschiedenen Hintergründen. Unser Interesse gilt ihrer Spuranomalie, und der Frage, ob die Pontryagin-Dichte auftritt. Deshalb berechnen wir die Anomalien von Dirac-Fermionen, die an vektorielle und axiale Eichfelder gekoppelt sind, und dann auf einem metrisch-axialen Tensor Hintergrund. Geeignete Grenzwerte der Hintergründe erlauben es dann, die Anomalien von Weyl-Fermionen, die an Eichfelder gekoppelt sind, und in einer gekrümmten Raumzeit zu berechnen. Wir bestätigen das Fehlen der Pontryagin-Dichte in den Spuranomalien. Drittens liefern wir die holographische Beschreibung einer vierdimensionalen CFT mit einem irrelevanten Operator. Wenn der Operator eine ganzzahlige konforme Dimension hat, modifiziert sein Vorhandensein in der CFT die Weyl-Transformation der Metrik, was wiederum die Spuranomalie ändert. Unter Ausnutzung der Äquivalenz zwischen Diffeomorphismen im Inneren und Weyl-Transformationen auf dem Rand, berechnen wir diese Modifikationen mithilfe der dualen Gravitationstheorie. Unsere Ergebnisse repräsentieren einen weiteren Test der AdS/CFT-Korrespondenz. / Conformal field theories (CFTs) are amongst the most studied field theories and they offer a remarkable playground in modern theoretical physics. In this thesis we analyse three aspects of CFTs in different dimensions. First, we consider correlation functions of descendant states in two-dimensional CFTs. We discuss a recursive formula to calculate them and provide a computer implementation of it. This allows us to obtain any correlation function of vacuum descendants, and for non-vacuum descendants to express the correlator as a differential operator acting on the respective primary correlator. With this code, we study some entanglement and distinguishability measures between descendant states, i.e. the Rényi entropy, trace square distance and sandwiched Rényi divergence. With our results we can test the Rényi Quantum Null Energy Condition and provide new tools to analyse the holographic description of descendant states. Second, we study four-dimensional Weyl fermions on different backgrounds. Our interest is in their trace anomaly, where the Pontryagin density has been claimed to appear. To ascertain this possibility, we compute the anomalies of Dirac fermions coupled to vector and axial non-abelian gauge fields and then in a metric-axial-tensor background. Appropriate limits of the backgrounds allow to recover the anomalies of Weyl fermions coupled to non-abelian gauge fields and in a curved spacetime. In both cases we confirm the absence of the Pontryagin density in the trace anomalies. Third, we provide the holographic description of a four-dimensional CFT with an irrelevant operator. When the operator has integer conformal dimension, its presence in the CFT modifies the Weyl transformation of the metric, which in turns modifies the trace anomaly. Exploiting the equivalence between bulk diffeomorphisms and boundary Weyl transformations, we compute these modifications from the dual gravity theory. Our results represent an additional test of the AdS/CFT conjecture.