Principal hierarchies of infinite-dimensional Frobenius manifolds: the extended 2D Toda lattice

TL;DR

This paper introduces an extended dispersionless 2D Toda hierarchy on a specific function space, constructs its Frobenius manifold connection, and proves their equivalence, advancing the understanding of integrable hierarchies and Frobenius manifolds.

Contribution

It defines a new extended hierarchy and establishes its equivalence with the principal hierarchy of a known Frobenius manifold, extending prior integrable systems theory.

Findings

The extended hierarchy generalizes the dispersionless 2D Toda hierarchy.

The deformed flat connection of the Frobenius manifold is explicitly constructed.

The extended hierarchy coincides with the principal hierarchy of the Frobenius manifold.

Abstract

We define a dispersionless tau-symmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0$/$\infty$ respectively, which extends the dispersionless 2D Toda hierarchy of Takasaki and Takebe. Then we construct the deformed flat connection of the infinite-dimen\-sional Frobenius manifold $M_0$ introduced by Carlet, Dubrovin and Mertens in Math. Ann. 349 (2011) 75--115 and, by explicitly solving the deformed flatness equations, we prove that the extended 2D Toda hierarchy coincides with principal hierarchy of $M_0$.