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

TL;DR

This paper explores the extended r-spin theory's role in understanding the mirror symmetry of A_{r-1} singularities, linking deformation parameters to geometric interpretations within Frobenius manifolds.

Contribution

It establishes a geometric interpretation of miniversal deformation parameters via extended r-spin theory and extends results to D_4, with conjectures for E_6 and E_8.

Findings

Parameters of miniversal deformation relate to extended r-spin theory.

Proved results for A_{r-1} and D_4 singularities.

Conjectures proposed for E_6 and E_8 singularities.

Abstract

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 $E_8$.