Extended r-spin theory and the mirror symmetry for the A-singularity

Alexandr Buryak, University of Leeds. Part of the integrable systems mathematics seminar series.

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. I will show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using an extension of the r-spin theory.  

Moreover, I will show that the Landau-Ginzburg mirror symmetry for the A-singularity can be simply derived from a certain WDVV type equation appearing in the extended r-spin theory.