Dimensional Reduction for Conformal Blocks

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients.Comment: 12 pages, 1 figur

Radial Coordinates for Conformal Blocks

We develop the theory of conformal blocks in CFT_d expressing them as power series with Gegenbauer polynomial coefficients. Such series have a clear physical meaning when the conformal block is analyzed in radial quantization: individual terms describe contributions of descendants of a given spin. Convergence of these series can be optimized by a judicious choice of the radial quantization origin. We argue that the best choice is to insert the operators symmetrically. We analyze in detail the resulting "rho-series" and show that it converges much more rapidly than for the commonly used variable z. We discuss how these conformal block representations can be used in the conformal bootstrap. In particular, we use them to derive analytically some bootstrap bounds whose existence was previously found numerically.Comment: 27 pages, 9 figures; v2: misprints correcte

The ABC (in any D) of Logarithmic CFT

Logarithmic conformal field theories have a vast range of applications, from critical percolation to systems with quenched disorder. In this paper we thoroughly examine the structure of these theories based on their symmetry properties. Our analysis is model-independent and holds for any spacetime dimension. Our results include a determination of the general form of correlation functions and conformal block decompositions, clearing the path for future bootstrap applications. Several examples are discussed in detail, including logarithmic generalized free fields, holographic models, self-avoiding random walks and critical percolation.Comment: 55 pages + appendice

Deux essais sur les théories quantiques des champs conformes et fortement couplées en d > 2 dimensions

This thesis investigates two aspects of Conformal Field Theories (CFTs) in D dimensions. Its first part is devoted to conformal blocks, special functions that arise in the partial wave expansion of CFT four-point functions. We prove that these conformal blocks admit an expansion in terms of polar coordinates and show that the expansion coefficients are determined by recursion relations. Conformal blocks are naturally defined on the complex plane: we study their restriction to the real line, and show that they obey a fourth-order differential equation there. This ODE can be used to efficiently compute conformal blocks and their derivatives in general D. Several applications to the conformal bootstrap program are mentioned.The second half of this thesis investigates RG flows that are defined by perturbing a CFT by a number of relevant operators. We study such flows using the Truncated Conformal Space Approach (TCSA) of Yurov and Zamolodchikov, a numerical method that allows for controlled computations in strongly coupled QFTs. Two different RG flows are considered: the free scalar field deformed by a mass term, and phi^4 theory. The former is used as a benchmark, in order to compare numerical TCSA results to exact predictions. TCSA results for phi^4 theory display spontaneous Z_2 symmetry breaking at strong coupling: we study the spectrum of this theory both in the Z_2-broken and preserved phase, and we compare the critical exponents governing the phase transition to known values. In a separate chapter, we show how truncation errors can be reduced by adding suitable counterterms to the bare TCSA action, following earlier work in two dimensions.Cette thèse examine deux aspects des théories conformes des champs (TCC) en D dimensions. Sa première partie est dédiée aux blocs conformes, des fonctions spéciales qui contribuent au développement en ondes partielles des fonctions à quatre points dans les TCC. On montre que ces blocs admettent un développement en coordonnées polaires dont les coefficients se calculent par une récurrence. Les blocs conformes sont naturellement définis sur le plan complexe: on considère alors leur restriction à l'axe réel, afin de montrer qu'ils obéissent à une équation différentielle sur ce domaine, ce qui mène à un algorithme efficace pour calculer les blocs conformes et leurs dérivées pour tout D. Quelques applications au programme de bootstrap sont développées.La seconde partie de cette thèse examine les perturbations d'une TCC par des opérateurs pertinents. On étudie de tels flots du groupe de renormalisation en utilisant la Méthode de Troncature Conforme (MTC) de Yurov et Zamolodchikov, une méthode numérique qui permet de faire des calculs non-perturbatifs en théorie quantique des champs. Deux théories différentes sont considérées: le boson libre avec un terme de masse, et la théorie phi^4. Pour le dernier cas, les résultats de la MTC mettent en évidence la brisure de symétrie Z_2. Finalement, on développe une méthode pour réduire les erreurs de troncature en ajoutant des contre-termes à l'action "nue" de la MTC, suivant des travaux antérieurs en deux dimensions

Finite size scaling and triviality of \phi^4 theory on an antiperiodic torus

Worm methods to simulate the Ising model in the Aizenman random current representation including a low noise estimator for the connected four point function are extended to allow for antiperiodic boundary conditions. In this setup several finite size renormalization schemes are formulated and studied with regard to the triviality of \phi^4 theory in four dimensions. With antiperiodicity eliminating the zero momentum Fourier mode a closer agreement with perturbation theory is found compared to the periodic torus.Comment: 20 pages, 3 double-figures, 6 table