Integrable hierarchies associated to infinite families of Frobenius manifolds

TL;DR

This paper introduces a new method to construct integrable hierarchies from Frobenius manifolds, revealing connections to well-known hierarchies like KP, BKP, and their reductions, with applications to enumerative geometry.

Contribution

It provides a novel construction linking Frobenius manifolds to integrable hierarchies, including explicit identifications with KP, BKP, and reduced hierarchies for specific singularities.

Findings

Hierarchies for A_N, D_N, B_N Frobenius manifolds identified with KP, D_N reduction, and BKP hierarchies.

New connections between Frobenius potentials and enumerative geometry coefficients.

Construction applicable to infinite series of Frobenius manifolds satisfying stabilization.

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 KP hierarchy; for $B_N$ Frobenius manifolds it coincides with the BKP hierarchy; and for $D_N$ hierarchy it is a certain reduction of the 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.