From Adler-Gelfand-Dickey Brackets to Logarithmic Dubrovin-Frobenius manifolds

TL;DR

This paper introduces a new Poisson bracket compatible with the Adler-Gelfand-Dickey structure, leading to a logarithmic Dubrovin-Frobenius manifold connected to permutation group orbits.

Contribution

It constructs a novel local Poisson bracket and demonstrates its relation to logarithmic Dubrovin-Frobenius manifolds and permutation group representations.

Findings

New compatible Poisson bracket with Adler-Gelfand-Dickey bracket

Bihamiltonian structure admits a dispersionless limit

Dubrovvin-Frobenius manifold constructed on permutation group orbits

Abstract

We construct a new local Poisson bracket compatible with the second unconstrained Adler-Gelfand-Dickey bracket. The resulting bihamiltonian structure admits a dispersionless limit and the leading term defines a logarithmic Dubrovin-Frobenius manifold. Furthermore, we show that this Dubrovin-Frobenius manifold can be constructed on the orbits space of the standard representation of the permutation group.