Formulas for A_n and B_n-solutions of WDVV equations

TL;DR

This paper derives recurrence relations to compute A_n and B_n solutions of WDVV equations, extending known polynomial solutions for singularities and connecting to dispersionless KP hierarchy coefficients.

Contribution

It introduces recurrence formulas enabling the calculation of all A_n and B_n potentials, expanding beyond previously known solutions for n ≤ 4.

Findings

Recurrence relations for A_n and B_n potentials

Recurrence formulas for dispersionless KP hierarchy coefficients

Extension of polynomial solutions to all n

Abstract

The simplest non-trivial solutions of WDVV equations are A_n and B_n-potentials, which describe metrics of K.Saito on spaces of versal deformation of A_n and B_n-singularities. These are some polynomials, which were known for $n\leqslant$ 4. We find some recurrence relations, which give a possibility to find all A_n and B_n-potentials. In passing we give recurrence formulas for coefficients of dispersionless KP hierarchy.