Rebooting quarter-BPS operators in $\mathcal{N}=4$ Super Yang-Mills

We start a systematic study of quarter-BPS operators in four-dimensional $\mathcal{N}=4$ Super Yang-Mills with gauge group $\mathrm{SU}(N)$ making use of recently developed tools in conformal field theory. We adapt the technology of embedding space tensor structures in four dimensions to the problem of computing R-symmetry tensor structures and we use the underlying chiral algebra to obtain the superconformal Ward identities. This allows us to fix the protected part of the four-point correlators, up to few ambiguities. As applications, we use the Lorentzian inversion formula to study the leading order OPE data in the large~$N$ supergravity limit and we make contact with the OPE limit of the five-point function of half-BPS operators.