Extended $r$-spin theory and the mirror symmetry for the $A_{r-1}$-singularity

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In this paper we show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using the extended $r$-spin theory, first considered by T. J. Jarvis, T. Kimura and A. Vaintrob, and studied in a recent paper of E. Clader, R. J. Tessler and the author. We prove a similar result for the singularity $D_4$ and present conjectures for the singularities $E_6$ and $E_8$.