Universality at large transverse spin in defect CFT

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators. We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE, analogous to the Caron-Huot formula for the four-point function [1]. Analyticity of the formula in s implies that the scaling dimensions of the defect operators are aligned in Regge trajectories (Delta) over cap (s). These results require the correlator of two local operators and the defect to be bounded in a certain region, a condition that we do not prove in general. We check our conclusions against examples in perturbation theory and holography, and we make specific predictions concerning the spectrum of defect operators on Wilson lines. We also give an interpretation of the large s spectrum in the spirit of the work of Alday and Maldacena [2].

Perturbative and non-perturbative aspects of RG flows in quantum field theory

Lorenzo Vitale

This thesis explores two aspects of the renormalization group (RG) in quantum field theory (QFT). In the first part we study the structure of RG flows in general Poincaré-invariant, unitary QFTs, and in particular the irreversibility properties and the relation between scale and conformal invariance. Within the formalism of the local Callan--Symanzik equation, we derive a series of results in four and six-dimensional QFTs. Specifically, in the four dimensional case we revisit and complete existing proofs of the

a

-theorem and of the equivalence between scale and conformal invariance in perturbation theory. We then present an original derivation of similar results in six-dimensional QFTs. In the second part we present the Hamiltonian Truncation method and study its applicability to the numerical solution of non-perturbative RG flows. We test the method in the Phi^4 model in two dimensions and show how it can be used to make quantitative predictions for the low-energy observables. In particular, we calculate the numerical spectrum and estimate the critical coupling at which the theory becomes conformal. We also compare our results to previous estimates. The main original ingredient of our analysis is an analytic renormalization procedure used to improve the numerical convergence. We then adapt the method in order to treat the strongly-coupled regime of the model where the Z2 symmetry is spontaneously broken. We reproduce perturbative and non-perturbative observables and compare our results with analytical predictions.

EPFL2016

Bootstrapping superconformal field theories in four dimensions

Andrea Manenti

The conformal bootstrap is a non-perturbative technique designed to study conformal field theories using only first principles, such as unitarity, crossing symmetry and the existence of an Operator Product Expansion. In this thesis we discuss an application of the bootstrap method in four dimensional conformal field theories. We also consider in detail the special case where the theory is supersymmetric. In particular we focus on the case study of four abelian currents. The non-supersymmetric setup applies to all conformal field theories with a global abelian symmetry group. When we include the assumption of supersymmetry, the current is taken to be the generator of the R-symmetry, which is tied to the stress tensor due to the superconformal algebra. The supersymmetric setup therefore applies to all local superconformal field theories. We start by introducing all the necessary ingredients. In particular, we discuss the formalism of the embedding space and of the conformal frame to study conformal kinematics. We also give a supersymmetric generalization of the conformal frame formula to count three-point tensor structures. Then we address the important problem of expanding superspace correlators in their components. To this aim we introduce a set of differential operators that act in superspace. Using this formalism we are able to compute the linear relations among the operators in the same superconformal multiplet. This is a necessary step in the computation of superconformal blocks, but it will also be useful for other purposes that we discuss before passing to the bootstrap analysis. First we use it to impose the averaged null energy condition on arbitrary superconformal field theories. This will lead to interesting consequences on their local operator spectrum. Next we focus on the case of local superconformal field theories with eight supercharges and we prove that a certain class of operators termed ``exotic primaries'' cannot exist. Finally, after a pedagogical introduction to the notion of the conformal bootstrap, we carry out a detailed study of the correlator of four conserved currents. In particular, we compute the conformal and superconformal blocks and the crossing equations. We conclude by proposing several numerical studies and strategies and by showing some preliminary results for the non-supersymmetric case.

EPFL2020