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 sl2$$ \mathfrak{s}{\mathfrak{l}}_2 $$ Gaudin Hamiltonians on three sets of variables.