An explicit local geometric Langlands for supercuspidal representations: the toral case

We formulate a conjecture on local geometric Langlands for supercuspidal representations using Yu's data and Feigin-Frenkel isomorphism. We refine our conjecture for a large family of regular supercuspidal representations defined by Kaletha, and then confirm the conjecture for toral supercuspidal representations of Adler whose Langlands parameters turn out to be exactly all the irreducible isoclinic connections. As an application, we establish the conjectural correspondence between global Airy connections for reductive groups and the family of Hecke eigensheaves constructed by Jakob-Kamgarpour-Yi.