Vertex algebra constructions for (analytic) geometric Langlands in genus zero

We employ two-dimensional chiral algebra techniques to produce solutions of certain differential and integral equations which occur in the context of the Analytic Geometric Langlands Program. In particular, we build “Multiplication Kernels” K₃(x, x′, x′′) which intertwine the action of $\mathfrak{s}{\mathfrak{l}}_2$ \( \mathfrak{s}{\mathfrak{l}}_2 \) Gaudin Hamiltonians on three sets of variables.