In this work we review the basic principles of the theory of the relativistic bosonic string through the study of the action functionals of Nambu-Goto and Polyakov and the techniques required for their canonical, light-cone, and path-integral quantisation. For this purpose, we briefly review the main properties of the gauge symmetries and conformal field theory involved in the techniques studied. / Neste trabalho fazemos uma revisão dos princípios básicos da teoria da corda bosônica relativística através do estudo dos funcionais ação de Nambu-Goto e de Polyakov e das técnicas necessárias para sua quantização canônica, no cone de luz e usando integrais de trajetória. Para tanto apresentamos uma pequena revisão das principais propriedades das simetrias de calibre a da teoria de campos conforme envolvidas nas técnicas estudadas.

<p>The main topic of this doctoral thesis is D-branes in string theory, expressed in the language of conformal field theory. The purpose of string theory is to describe the elementary particles and the fundamental interactions of nature, including gravitation as a quantum theory. String theory has not yet reached the status to make falsifiable predictions, thus it is not certain that string theory has any direct relevance to physics. On the other hand, string theory related research has led to progress in mathematics.</p><p>We begin with a short introduction to conformal field theory and some of its applications to string theory. We also introduce vertex algebras and discuss their relevance to conformal field theory. Some classes of conformal field theories are introduced, and we discuss the relevant vertex algebras, as well as their interpretation in terms of string theory.</p><p>In string theory, a D-brane specifies where the endpoint of the string lives. Many aspects of string theory can be described in terms of a conformal field theory, which is a field theory that lives on a two-dimensional space. The conformal field theory counterpart of a D-brane is a boundary state, which in some cases has a natural interpretation as constraining the string end point. The main focus of this thesis is on the interpretation of boundary states in terms of D-branes in curved target spaces.</p>

<p>Conformal field theories (CFT) constitute an interesting class of twodimensionalquantum field theories, with applications in string theoryas well as condensed matter physics. The symmetries of a CFT can beencoded in the mathematical structure of a conformal vertex algebra.The rational CFT’s are distinguished by the property that the categoryof representations of the vertex algebra is a modular tensor category.The solution of a rational CFT can be split off into two separate tasks, apurely complex analytic and a purely algebraic part.</p><p>The TFT-construction gives a solution to the second part of the problem.This construction gets its name from one of the crucial ingredients,a three-dimensional topological field theory (TFT). The correlators obtainedby the TFT-construction satisfy all consistency conditions of thetheory. Among them are the factorization constraints, whose implicationsfor boundary conditions are the main topic of this thesis.</p><p>The main result reviewed in this thesis is that the factorization constraintsgive rise to a semisimple commutative associative complex algebrawhose irreducible representations are the so-called reflection coefficients.The reflection coefficients capture essential information aboutboundary conditions, such as ground-state degeneracies and Ramond-Ramond charges of string compactifications. We also show that the annuluspartition function can be derived fromthis classifying algebra andits representation theory.</p>

Conformal field theories (CFT) constitute an interesting class of twodimensionalquantum field theories, with applications in string theoryas well as condensed matter physics. The symmetries of a CFT can beencoded in the mathematical structure of a conformal vertex algebra.The rational CFT’s are distinguished by the property that the categoryof representations of the vertex algebra is a modular tensor category.The solution of a rational CFT can be split off into two separate tasks, apurely complex analytic and a purely algebraic part. The TFT-construction gives a solution to the second part of the problem.This construction gets its name from one of the crucial ingredients,a three-dimensional topological field theory (TFT). The correlators obtainedby the TFT-construction satisfy all consistency conditions of thetheory. Among them are the factorization constraints, whose implicationsfor boundary conditions are the main topic of this thesis. The main result reviewed in this thesis is that the factorization constraintsgive rise to a semisimple commutative associative complex algebrawhose irreducible representations are the so-called reflection coefficients.The reflection coefficients capture essential information aboutboundary conditions, such as ground-state degeneracies and Ramond-Ramond charges of string compactifications. We also show that the annuluspartition function can be derived fromthis classifying algebra andits representation theory.

The main topic of this doctoral thesis is D-branes in string theory, expressed in the language of conformal field theory. The purpose of string theory is to describe the elementary particles and the fundamental interactions of nature, including gravitation as a quantum theory. String theory has not yet reached the status to make falsifiable predictions, thus it is not certain that string theory has any direct relevance to physics. On the other hand, string theory related research has led to progress in mathematics. We begin with a short introduction to conformal field theory and some of its applications to string theory. We also introduce vertex algebras and discuss their relevance to conformal field theory. Some classes of conformal field theories are introduced, and we discuss the relevant vertex algebras, as well as their interpretation in terms of string theory. In string theory, a D-brane specifies where the endpoint of the string lives. Many aspects of string theory can be described in terms of a conformal field theory, which is a field theory that lives on a two-dimensional space. The conformal field theory counterpart of a D-brane is a boundary state, which in some cases has a natural interpretation as constraining the string end point. The main focus of this thesis is on the interpretation of boundary states in terms of D-branes in curved target spaces.

The sewing operation is an integral component of both Geometric Function Theory and Conformal Field Theory and in this thesis we explore the interplay between the two fields. We will first generalize Huang's Geometric Sewing Equation to the quasi-symmetric case. That is, given specific maps g(z) and f^{-1}(z), we show the existence of the sewing maps F_{1}(z) and F_{2}(z). Second, we display an algebraic procedure using convergent matrix operations showing that the coefficients of the Conformal Welding Theorem maps F(z) and G(z) are dependent on the coefficients of the map phi(z). We do this for both the analytic and quasi-symmetric cases, and it is done using a special block/vector decomposition of a matrix representation called the power matrix. Lastly, we provide a partial result: given specific maps g(z) and f^{-1}(z) with analytic extensions, as well as a particular analytic map phi(z), it is possible to provide a method of determining the coefficients of the complementary maps.

The sewing operation is an integral component of both Geometric Function Theory and Conformal Field Theory and in this thesis we explore the interplay between the two fields. We will first generalize Huang's Geometric Sewing Equation to the quasi-symmetric case. That is, given specific maps g(z) and f^{-1}(z), we show the existence of the sewing maps F_{1}(z) and F_{2}(z). Second, we display an algebraic procedure using convergent matrix operations showing that the coefficients of the Conformal Welding Theorem maps F(z) and G(z) are dependent on the coefficients of the map phi(z). We do this for both the analytic and quasi-symmetric cases, and it is done using a special block/vector decomposition of a matrix representation called the power matrix. Lastly, we provide a partial result: given specific maps g(z) and f^{-1}(z) with analytic extensions, as well as a particular analytic map phi(z), it is possible to provide a method of determining the coefficients of the complementary maps.

In this thesis, we investigate sigma models and algebraic structures emerging from a Hamiltonian description of their dynamics, both in a classical and in a quantum setup. More specifically, we derive the phase space structures together with the Hamiltonians for the bosonic two-dimensional non-linear sigma model, and also for the N=1 and N=2 supersymmetric models. A convenient framework for describing these structures are Lie conformal algebras and Poisson vertex algebras. We review these concepts, and show that a Lie conformal algebra gives a weak Courant–Dorfman algebra. We further show that a Poisson vertex algebra generated by fields of conformal weight one and zero are in a one-to-one relationship with Courant–Dorfman algebras. Vertex algebras are shown to be appropriate for describing the quantum dynamics of supersymmetric sigma models. We give two definitions of a vertex algebra, and we show that these definitions are equivalent. The second definition is given in terms of a λ-bracket and a normal ordered product, which makes computations straightforward. We also review the manifestly supersymmetric N=1 SUSY vertex algebra. We also construct sheaves of N=1 and N=2 vertex algebras. We are specifically interested in the sheaf of N=1 vertex algebras referred to as the chiral de Rham complex. We argue that this sheaf can be interpreted as a formal quantization of the N=1 supersymmetric non-linear sigma model. We review different algebras of the chiral de Rham complex that one can associate to different manifolds. In particular, we investigate the case when the manifold is a six-dimensional Calabi–Yau manifold. The chiral de Rham complex then carries two commuting copies of the N=2 superconformal algebra with central charge c=9, as well as the Odake algebra, associated to the holomorphic volume form.

In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) proposals allow us to find the same quantities using calculations in general relativity. We will introduce main concepts of holography, quantum information and conformal field theory that will be used to derive the results of this thesis. In the first part of the thesis, we explicitly compute entanglement entropy of the rotating BTZ black hole by directly applying HRT proposal and finding lengths of spacelike geodesics. Rényi entropy of thermal state perturbed by a local quantum quench is computed by mapping correlators on two glued cylinders to the plane for field theory containing a single free boson and for 2d CFTs in the large c limit. We consider Thermofield Double State (TFD) which is an entangled state in direct product of two 2D CFTs. It is conjectured to be holographically equivalent to the eternal BTZ black hole. TFD state is perturbed by a local quench in one CFT and mutual information between two intervals in two CFTs is computed. We find when mutual information vanishes and interpret this as scrambling time, i.e. time scale required for the system to thermalize. This field theory result is modelled with a massive free falling particle in the BTZ black hole. We have computed the back-reaction of the particle on the metric of BTZ and used RT proposal to find holographic entanglement entropy. Finally, we generalize this calculation to the case of rotating BTZ with inner and outer horizons. It is dual to the CFT with different temperatures for left and right moving modes. We calculate mutual information and scrambling time and find exact agreement between results in the gravity and those in the CFT.

Made available in DSpace on 2014-06-11T19:32:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2001-06Bitstream added on 2014-06-13T19:47:10Z : No. of bitstreams: 1 echevarria_cat_me_ift.pdf: 657711 bytes, checksum: d365934122a9e19d3c69fab3a408cda4 (MD5) / Primeiramente apresenta-se uma breve descrição das teorias de primeira quantização covariante da corda bosônica, supercorda RNS e GS. Também apresentamos a descrição topológica da supercorda. Em seguida , fazemos uma introdução à teoria clássica de campos da corda bosônica aberta formulado por Witten. Depois, consideramos a toeria clássica de campo de supercorda aberta que usa as variáveis RNS. Discutimos o problema da divergência que surge em termos de contato relacionados com picture. O formalismo topológico permite construir uma ação para teoria de campos de supercorda que é do tipo WZW. Esta ação resolve o problema da divergência. Este fato é verificado com cálculo explícito da amplitude de espalhamento para quatro estados externos na camada da massa. Seguidamente, discutimos as branas não BPS, condensação do táquion e construimos os vértices com projeção GSO negativo para o táquion e o férmion não massivo usando variáveis híbridas. Finalmente incluimos o setor GSO (-) da supercorda numa ação generalizada do tipo WZW. / Firstly we present a brief description of the covariant first quantized theories of string, RNS and GS superstring. Also we present a topological description of the superstring. Then we give an introduction to the classical open string field theory formulated by Witten. After that we cosnider the classical open superstring field theory using the RNS variables, we discuss the problem of the divergence that appears in contact terms that is related with picture. The topological formalism allow us to construct an action for the open superstring field theory which is WZW-like. This action solves the divergence problem, and we verify this fact by explicit computation of the scttering amplitude for four external on-shell states. Next, we discuss nonBPS D-branes, tachyon condensation and construct the GSO negative projected vertices for the tachyona and the massless fermion using hybrid variables. Finally we include the GSO (-) sector of the superstring in a generalized WZW-like action.