Superembedding methods for 4d N=1 SCFTs

TL;DR

The paper develops a manifestly covariant embedding-space framework for ${\cal N}=1$ superconformal field theories by extending the six-dimensional lightcone construction to a superspace with coordinates $X_{AB}$ and ${\bar X}^{AB}$. The $SU(2,2|1)$ group acts linearly on these coordinates, automatically enforcing superconformal Ward identities and yielding compact, invariant expressions for correlators of chiral and anti-chiral multiplets, including the four-point function parameterized by two cross-ratios $u$ and $v$. It establishes the chiral sector correspondence between holomorphic embedding-space fields ${\Phi}(X,\bar X)$ and four-dimensional chiral multiplets, with the lowest component being a chiral primary and the scaling dimension tied to the $R$-charge. The formalism applies in any conformally flat background (e.g., $AdS_4$, ${\mathbb R}\times S^3$) and sets the stage for extensions to extended SUSY, curved spaces, and non-holomorphic multiplets.

Abstract

We extend SO(4,2) covariant lightcone embedding methods of four-dimensional CFTs to N=1 superconformal field theory (SCFT). Manifest superconformal SU(2,2|1) invariance is achieved by realizing 4D superconformal space as a surface embedded in the projective superspace spanned by certain complex chiral supermatrices. Because SU(2,2|1) acts linearly on the ambient space, the constraints on correlators implied by superconformal Ward identities are automatically solved in this formalism. Applications include new, compact expressions for correlation functions containing one anti-chiral superfield and arbitrary chiral superfield insertions, and manifestly invariant expressions for the superconformal cross-ratios that parametrize the four-point function of two chiral and two anti-chiral fields. Superconformal expressions for the leading singularities in the OPE of chiral and anti-chiral operators are also given. Because of covariance, our expressions are valid in any superconformally flat background, e.g., AdS_4 or R times S^3.