Finite Group Reduction of the DR/DZ Hierarchies

Abstract

We show that the finite group reduction of the Dubrovin-Zhang/Double Ramification hierarchy associated with a semisimple CohFT preserves its bihamiltonian structure and its tau structure. By applying this result to the orbifold Gromov-Witten theory with the target ${\mathbb{P}}_{2,2,2,2}^1$, we obtain two non-equivalent integrable hierarchies which can be associated with the Frobenius manifold structure on the Hurwitz space $M_{1;1}$, and we show that one of them is equivalent to the genus one topological recursion.