Extended r-spin theory and the mirror symmetry for the Ar–1-singularity

By a famous result of K. Saito, the parameter space of the miniversal deformation of the Ar−1Ar−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 rr 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 rr 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 D4D4 and present conjectures for the singularities E6E6 and E8E8.