WDVV solutions associated with the genus one holomorphic differential

Abstract

Consider the genus one Hurwitz space ${\mathcal{H}}_1(n_0,\dots ,n_m)$ of ramified covering of fixed degree with $m+1$ prescribed poles of order $n_0+1,\dots ,n_m+1$, respectively. Based on a recent formula proved in \cite{Rejeb23}, we derive an explicit solution to the WDVV equations associated with the genus one Dubrovin-Hurwitz-Frobenius manifold structure induced by the normalized holomorphic differential. The obtained solution is written in terms of Bell polynomials, Eisenstein series as well as the Weierstrass functions.