The Wilson loop CFT: Insertion dimensions and structure constants from wavy lines

We study operator insertions into the $1/2$ BPS Wilson loop in ${\cal N}=4$ SYM theory and determine their two-point coefficients, anomalous dimensions and structure constants. The calculation is done for the first few lowest dimension insertions and relies on known results for the expectation value of a smooth Wilson loop. In addition to the particular coefficients that we calculate, our study elucidates the connection between deformations of the line and operator insertions and between the vacuum expectation value of the line and the CFT data of the insertions.