Correlators in higher-spin AdS₃ holography from Wilson lines with loop corrections

We study the correlators of the 2D W${}_N$ minimal model in the semiclassical regime with large central charge from the bulk viewpoint by utilizing open Wilson lines in $\text{sl}\left(N\right)$ Chern-Simons gauge theory. We extend previous works for the tree level of bulk theory to incorporate loop corrections in this paper. We offer a way to regularize divergences associated with loop diagrams such that three-point functions with two scalars and a higher-spin current agree with the values fixed by the boundary W${}_N$ symmetry. With the prescription, we reproduce the conformal weight of the operator corresponding to a bulk scalar up to two-loop order for explicit examples with $N=2,3$.