Charting the space of 3D CFTs with a continuous global symmetry

We study correlation functions of a conserved spin-1 current J in three dimensional Conformal Field Theories (CFTs). We investigate the constraints imposed by permutation symmetry and current conservation on the form of three point functions and the four point function JJJJ(sigma) and identify the minimal set of independent crossing symmetry conditions. We obtain a recurrence relation for conformal blocks for generic spin-1 operators in three dimensions. In the process, we improve several technical points, facilitating the use of recurrence relations. By applying the machinery of the numerical conformal bootstrap we obtain universal bounds on the dimensions of certain light operators as well as the central charge. Highlights of our results include numerical evidence for the conformal collider bound and new constraints on the parameters of the critical O(2) model. The results obtained in this work apply to any unitary, three dimensional CFT.

Colliders and conformal interfaces

Marco Meineri, Antonin Tancrède Amaury Rousset

We set up a scattering experiment of matter against an impurity which separates two generic one-dimensional critical quantum systems. We compute the flux of reflected and transmitted energy, thus defining a precise measure of the transparency of the interface between the related two-dimensional conformal field theories. If the largest symmetry algebra is Virasoro, we find that the reflection and transmission coefficients are independent of the details of the initial state, and are fixed in terms of the central charges and of the two-point function of the displacement operator. The situation is more elaborate when extended symmetries are present. Positivity of the total energy flux at infinity imposes bounds on the coefficient of the two-point function of the displacement operator, which controls the free-energy cost of a small deformation of the interface. Finally, we study out-of-equilibrium steady states of a critical system connecting two reservoirs at different temperatures. In the absence of extended symmetries, our result implies that the energy flux across an impurity is proportional to the difference of the squared temperatures and controlled by the reflection coefficient.

SPRINGER2020

A New Method to Explore Conformal Field Theories in Any Dimension

Alessandro Vichi

The thesis represents an investigation into Conformal Field Theories (CFT's) in arbitrary dimensions. We propose an innovative method to extract informations about CFT's in a quantitative way. Studying the crossing symmetry of the four point function of scalar operators we derive consistency constraints on the CFT structure, in the form of functional sum rules. The technique we introduce allows to address the feasibility of the sum rule and translate it into restrictions on the CFT spectrum and interactions. Our analysis only assumes unitarity of the CFT, crossing symmetry of the four point function and existence of an Operator Product Expansion (OPE) for scalars. We demonstrate that a CFT satisfying the above hypothesis and containing a scalar operator is not compatible with arbitrary spectra of the operators nor with arbitrary large OPE coefficients. More specifically we prove two main results. First, the spectrum of the CFT must contain a second scalar operator with dimension smaller than a given value. Second, the value of the three point function of two scalars with equal dimension and a third arbitrary operator is bounded from above. As an application of the fist statement we present the bound on the smallest dimension operator entering the OPE of a real scalar with itself. We perform the analysis in two and four dimensions. The comparison of the two dimensional case with exactly solvable models shows a saturation of the bound. We repeat for CFT's with global symmetries and superconformal field theories in four dimensions. As a demonstration of the second result we provide a lower bound on the central charge for CFT in two and four dimensions without global symmetries and for superconformal field theories. We also discuss in the potentialities of the method and possible future research lines. Finally, we discuss possible implications for model building beyond the Standard Model of particle physics.

EPFL2011

Central charge bounds in 4D conformal field theory

Riccardo Rattazzi, Alessandro Vichi

We derive model-independent lower bounds on the stress tensor central charge C-T in terms of the operator content of a 4-dimensional conformal field theory. More precisely, C-T is bounded from below by a universal function of the dimensions of the lowest and second-lowest scalars present in the conformal field theory. The method uses the crossing symmetry constraint of the 4-point function, analyzed by means of the conformal block decomposition.

2011

A New Method to Explore Conformal Field Theories in Any Dimension

Alessandro Vichi

EPFL2011