Super-Grassmannians for $\mathcal{N}=2$ to $4$ SCFT$_3$: From AdS$_4$ Correlators to $\mathcal{N}=4$ SYM scattering Amplitudes

We construct a Super-Grassmannian for $n-$point functions in $\mathcal{N}=2$ to $4$ SCFT$_3$. The constraints imposed by super-conformal invariance and $R-$symmetry are completely manifest in this formalism through (operator-valued) delta functions. We test our formalism in $\mathcal{N}=2$ and $\mathcal{N}=4$ AdS$_4$ super Yang-Mills theories. In the $\mathcal{N}=2$ case, for instance, we reproduce the four-gluon correlator using the four-point scalar correlator as input. For $\mathcal{N}=4$, we construct the super-operator in two distinct ways. In one approach, the super-operator has a lowest component of spin zero and includes all states up to spin two. In the other approach, we build the super-operator in a CPT self-conjugate manner, which contains only operators with spin zero, spin half, and spin one mimicking flat space $\mathcal{N}=4$ SYM super-field constructions. The latter construction is particularly interesting, as it matches directly with the $\mathcal{N}=4$ SYM amplitudes in the flat space limit, thereby demonstrating the non-triviality and usefulness of our framework. It is interesting to note that the $R-$symmetry group enhances from $SO(\mathcal{N})$ to $SU(\mathcal{N})$ in the flat space limit.