Differential operators for superconformal correlation functions

We present a systematic method to expand in components four dimensional superconformal multiplets. The results cover all possible N = 1 multiplets and some cases of interest for N = 2. As an application of the formalism we prove that certain N = 2 spinning chiral operators (also known as \exotic" chiral primaries) do not admit a consistent three-point function with the stress tensor and therefore cannot be present in any local SCFT. This extends a previous proof in the literature which only applies to certain classes of theories. To each superdescendant we associate a superconformally covariant differential operator, which can then be applied to any correlator in superspace. In the case of threepoint functions, we introduce a convenient representation of the differential operators that considerably simplifies their action. As a consequence it is possible to efficiently obtain the linear relations between the OPE coefficients of the operators in the same superconformal multiplet and in turn streamline the computation of superconformal blocks. We also introduce a Mathematica package to work with four dimensional superspace.

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

R-current three-point functions in 4d N=1 superconformal theories

Andrea Manenti, Alessandro Vichi

In 4d N = 1 superconformal field theories (SCFTs) the R-symmetry current, the stress-energy tensor, and the supersymmetry currents are grouped into a single object, the Ferrara-Zumino multiplet. In this work we study the most general form of three-point functions involving two Ferrara-Zumino multiplets and a third generic multiplet. We solve the constraints imposed by conservation in superspace and show that non-trivial solutions can only be found if the third multiplet is R-neutral and transforms in suitable Lorentz representations. In the process we give a prescription for counting independent tensor structures in superconformal three-point functions. Finally, we set the Grassmann coordinates of the Ferrara-Zumino multiplets to zero and extract all three-point functions involving two R-currents and a third conformal primary. Our results pave the way for bootstrapping the correlation function of four R-currents in 4d N = 1 SCFTs.

SPRINGER2018

Bootstrapping superconformal field theories in four dimensions

Andrea Manenti

EPFL2020