Unmixing the Wilson line defect CFT. Part I: spectrum and kinematics

This is the first of a series of two papers in which we study the one-dimensional defect CFT defined by insertions of local operators along a $\tfrac{1}{2}$-BPS Wilson line in $\mathcal{N}=4$ super Yang-Mills. In this first paper we focus on the kinematical implications of invariance under the $\mathfrak{osp}(4^*|4)$ superconformal algebra preserved by the line. We study correlation functions involving both protected and unprotected supermultiplets and derive the associated superconformal blocks, using two types of superspace for short and long representations. We also discuss the spectrum of defect theories defined by the Wilson line, focusing in particular on fundamental lines in the planar limit: in this case we provide a detailed analysis of the type and number of states both at weak 't Hooft coupling, via the free gauge theory description of the defect CFT, and at strong coupling, where there is a dual description via AdS/CFT. Focusing on the strongly-coupled regime, which will be subject to a detailed analysis using analytic bootstrap techniques in a companion paper, we also develop a strategy that allows to explicitly build superconformal primary operators and their superconformal descendants in terms of the elementary fields in the AdS Lagrangian description. The explicit results will be used in a companion paper to address the problem of operators mixing at strong coupling. This paper and its companion provide an extended version of the results presented in 2103.10440.