Superconformal Symmetry and Correlation Functions

TL;DR

This paper derives superconformal symmetry constraints on four-point functions of 1/2-BPS operators in N=2,4 theories, solving Ward identities, analyzing crossing symmetry, and computing conformal partial waves and anomalous dimensions.

Contribution

It introduces a simplified method using null vectors to solve superconformal Ward identities and provides explicit solutions for four-point functions with detailed symmetry and unitarity analysis.

Findings

Superconformal Ward identities reduce to simple differential equations.

Crossing symmetry fixes the form of four-point functions to free field form.

Explicit large N amplitude expressions for four identical 1/2-BPS operators are provided.

Abstract

The N = 2, 4 superconformal symmetry constraints in d = 4 for four point functions of chiral primary 1/2-BPS operators are derived. The operators are described by symmetric traceless tensors of the internal R-symmetry group. A substantial simplification is achieved by introduction of null vectors. Two variable polynomials corresponding to different R-symmetry representations are constructed. The Ward identities for superconformal symmetry are obtained as simple differential equations. The general solution is presented in terms of a constant, a single variable function and a two variable function. An interpretation in terms of the operator product expansion is given for the case of fields of equal dimension and for the so called (next-to)extremal cases. The result is shown to accommodate long multiplets, semishort and short multiplets with protected dimension. Generically also non-unitary multiplets can appear. It is shown how to remove them to obtain a unitary theory. Implications of crossing symmetry for the four point functions studied are derived and discussed. It is shown that crossing symmetry fixes the single variable function in the general solution to be of free field form using singularity arguments. For a restricted set of next-to-extremal correlation functions with S3 symmetry amongst the first three fields it is shown that the amplitude is fixed up to normalization to free field form. We compute the conformal partial wave expansion of all representations in this amplitude and compute an averaged value of the anomalous dimensions for long multiplets given spin and twist in each relevant representation at first order in 1/N. Finally assuming the universal singularity structure we derive the general large N amplitude of four identical 1/2-BPS operators in the [0, p, 0] representation in terms of D functions. Explicit expressions for all coefficients are given.