The $\mathcal{N}=1$ Super-Grassmannian for CFT$_3$ and a Foray on AdS and Cosmological Correlators

TL;DR

This paper develops a Super-Grassmannian integral framework for $ ext{SCFT}_3$ that simplifies correlator relations, facilitates boundary correlator construction in (A)dS$_4$, and confirms flat-space limits.

Contribution

It introduces a novel Super-Grassmannian formalism for $ ext{SCFT}_3$ correlators, enabling algebraic relations and boundary correlator construction from particle exchange contributions.

Findings

Component correlators are related through simple algebraic relations.

Constructed (A)dS$_4$ boundary correlators from particle exchange diagrams.

Confirmed the flat-space limit matches existing results.

Abstract

We construct a Super-Grassmannian integral representation for $n-$point functions in $\mathcal{N}=1$ SCFT$_3$. In this formalism, conformal invariance, supersymmetry, and special superconformal invariance are implemented manifestly through (operator-valued) delta function constraints. An important feature of this framework is the fact that we obtain simple algebraic relations among component correlators, which enable us to determine any component correlator in terms of just one of the component correlators. In particular, this formalism enables us to construct (A)dS$_4$ boundary correlators with contact diagrams from those that receive contributions purely from particle exchanges. We illustrate this by determining the (A)dS$_4$ Yang-Mills gluon four-point function from its gluino counterpart. Further, we establish the flat-space limit in super-space, finding a perfect agreement with existing flat-space results.