Conformal field theory

Summary

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that local scale in

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Conformal field theories (CFTs) play a very significant role in modern physics, appearing in such diverse fields as particle physics, condensed matter and statistical physics and in quantum gravity both as the string worldsheet theory and through the AdS/CFT correspondence. In recent years major breakthroughs have been made in solving these CFTs through a method called numerical conformal bootstrap. This method uses consistency conditions on the CFT data in order to find and constrain conformal field theories and obtain precise measurements of physical observables. In this thesis we apply the conformal bootstrap to study among others the O(2)- and the ARP^3- models in 3D. In the first chapter we extend the conventional scalar numerical conformal bootstrap to a mixed system of correlators involving a scalar field charged under a global U(1) symmetry and the associated conserved spin-1 current J. The inclusion of a conserved spinning operator is an important advance in the numerical bootstrap program. Using numerical bootstrap techniques we obtain bounds on new observables not accessible in the usual scalar bootstrap. Concentrating on the O(2) model we extract rigorous bounds on the three-point function coefficient of two currents and the unique relevant scalar singlet, as well as those of two currents and the stress tensor. Using these results, and comparing with a quantum Monte Carlo simulation of the O(2) model conductivity, we give estimates of the thermal one-point function of the relevant singlet and the stress tensor. We also obtain new bounds on operators in various sectors. In the second chapter we investigate the existence of a second-order phase transition in the ARP^3 model. This model has a global O(4) symmetry and a discrete Z_2 gauge symmetry. It was shown by a perturbative renormalization group analysis that its Landau-Ginzburg-Wilson effective description does not have any stable fixed point, thus disallowing a second-order phase transition. However, it was also shown that lattice simulations contradict this, finding strong evidence for the existence of a second-order phase transition. In this chapter we apply conformal bootstrap methods to the correlator of four scalars t transforming in the traceless symmetric representation of O(4) in order to investigate the existence of this second order phase transition. We find various features that stand out in the region predicted by the lattice data. Moreover, under reasonable assumptions a candidate island can be isolated. We also apply a mixed t-s bootstrap setup in which this island persists. In addition we study the kink-landscape for all representations appearing in the t times t OPE for general N. Among others, we find a new family of kinks in the upper-bound on the dimension of the first scalar operator in the "Box" and "Hook" representations.

EPFL2020

Thermal CFTs in momentum space

Andrea Manenti

We study some aspects of conformal field theories at finite temperature in momentum space. We provide a formula for the Fourier transform of a thermal conformal block and study its analytic properties. In particular we show that the Fourier transform vanishes when the conformal dimension and spin are those of a "double twist" operator = 2(phi) + l + 2n. By analytically continuing to Lorentzian signature we show that the spectral density at high spatial momenta has support on the spectrum condition |omega| > |k|. This leads to a series of sum rules. Finally, we explicitly match the thermal block expansion with the momentum space Green's function at finite temperature in several examples.

SPRINGER2020

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