# Moscow Mathematical Journal

Volume 20, Issue 3, July–September 2020 pp. 475–493.

Extended $r$-Spin Theory and the Mirror Symmetry for the $A_{r-1}$-Singularity

**Authors**:
Alexandr Buryak (1)

**Author institution:**(1) School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

**Summary: **

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 $E_8$.

2010 Math. Subj. Class. 14H10, 53D45

**Keywords:**Moduli space of curves, Frobenius manifold, singularity, mirror symmetry.

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