Differential operators for superconformal correlation functions

TL;DR

The paper develops a systematic method using differential operators to analyze superconformal multiplets and correlators in 4D supersymmetric theories, proving certain operators cannot appear in local SCFTs.

Contribution

It introduces a formalism for expanding superconformal multiplets and constructs differential operators to simplify correlator computations, extending previous results to more general cases.

Findings

Proves certain $ abla$-chiral operators cannot appear in local SCFTs.

Provides a Mathematica package for superspace calculations.

Simplifies the derivation of superconformal blocks.

Abstract

We present a systematic method to expand in components four dimensional superconformal multiplets. The results cover all possible $\mathcal{N} = 1$ multiplets and some cases of interest for $\mathcal{N} = 2$. As an application of the formalism we prove that certain $\mathcal{N} = 2$ 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 three-point 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.