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

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.