Infinite-dimensional Frobenius Manifolds Underlying the Toda Lattice Hierarchy

TL;DR

This paper constructs a class of infinite-dimensional Frobenius manifolds associated with the Toda lattice hierarchy, linking them to finite-dimensional Frobenius manifolds on affine Weyl group orbit spaces, expanding the geometric understanding of integrable systems.

Contribution

It introduces a new class of infinite-dimensional Frobenius manifolds related to the Toda lattice hierarchy, extending previous finite-dimensional models and establishing their connection to affine Weyl group structures.

Findings

Constructed infinite-dimensional Frobenius manifolds for Toda hierarchy

Established links between these manifolds and affine Weyl group orbit spaces

Extended the geometric framework of integrable systems

Abstract

Following the approach of Carlet et al.(2011)\cite{CDM}, we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type $A$ defined by Dubrovin and Zhang.