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Deux essais sur les theories quantiques des champs conformes et fortement couplees en d > 2 dimensions

Matthijs Hogervorst

Published Jun 29, 2015• 2 citations
OpenAlex

Abstract

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.

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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.
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In this dissertation we describe progress towards carving out the space of non-integrable conformal field theories (CFTs) in 1+1 dimensions using conformal bootstrap techniques. We develop and apply numerical and analytical techniques to explore the non-perturbative consequences of conformal symmetry, unitarity and locality on the spectrum and dynamics of 2D CFTs. Throughout we focus on compact irrational CFTs, which have a discrete spectrum consisting of an infinite number of primary operators and exhibit chaotic dynamics. In the first part of the dissertation we report on numerical advances in charting the space of 2D CFTs. We use semidefinite programming to compute bounds on the gap in the spectrum of primary operators as a function of the central charge and on spectral functions that capture the spectral density of primary operators. We then develop recursion relations for arbitrary Virasoro conformal blocks, and explore the constraints of modular invariance of the genus-two partition function, leading to ``critical surfaces'' that bound the heavy structure constants of a CFT. In the second part of the dissertation we develop the analytic bootstrap program for two dimensional conformal field theories, focusing on universal aspects of irrational unitary CFTs and holography. We show that the large-spin sector of irrational CFTs is universally governed by a ``Virasoro mean field theory,'' incorporating exact stress tensor dynamics into the analytic bootstrap and leading to non-perturbative insights about the bound-state dynamics of three-dimensional quantum gravity. We then leverage a generalized notion of modular invariance to derive a universal asymptotic formula for the average value of the structure constants whenever one or more of the operators has large conformal dimension. The large central charge limit makes contact with quantum gravity in three dimensions, where the averaging over heavy states corresponds to coarse-graining over black hole microstates. Finally, we study the Euclidean gravitational path integral of pure AdS3 quantum gravity, and use a modular bootstrap analysis to suggest new saddle point configurations that can be consistently included in the gravitational path integral and whose inclusion renders the dual torus partition function unitary.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.
This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results.
This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.
This thesis explores the application of numerical bootstrap techniques to two-point functions of bulk insertions in the presence of a boundary in two-dimensional conformal field theories. The first part of this thesis is devoted to presenting a comprehensive introduction to essential concepts in the numerical bootstrap of two-dimensional CFTs. Namely, an extensive introduction to the basics of conformal field theories (CFTs), their properties arising from the algebraic structure of conformal transformations, consitency conditions on the observables in a CFT, as well as the effects of the presence of defects in two-dimensional CFTs, is given. This is followed by a quick recount of the numerical techniques employed in the literature to solve for the aforementioned consistency conditions. These chapters form an overview of the seminal works that set the backdrop of the contributions made in the second part of the thesis. In the contributions of the thesis, we discuss the progress made in the investigations in (Meineri and Radhakrishnan 2025) where two-point functions in BCFTs are bootstrapped by treating them as a part of a larger system of consistency conditions involving the partition function on an annulus, the two-point function in the presence of a boundary, and four-point functions on the infinite plane. This perspective of treating the two-point function as a part of a larger system makes accessible parts of the BCFT data that have so-far remained inaccessible, and we present results on the boundary gap in the two-point function, and improvements in existing numerical bounds on the boundary entropy.