Integrable hierarchies associated to infinite families of Frobenius manifolds

Abstract

We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to A N , D N and B N singularities. In the case of A N Frobenius manifolds our hierarchy turns out to coincide with the dispersionless KP hierarchy; for B N Frobenius manifolds it coincides with the dispersionless BKP hierarchy; and for D N hierarchy it is a certain reduction of the dispersionless 2-component BKP hierarchy. As a side product to these results we illustrate the enumerative meaning of certain coefficients of A N , D N and B N Frobenius potentials.