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

2025/06/13 by Dinar, Yassir
#13A50 #17B08 #17B68 #17B80 #37K25 #37k30 #53D17 #53D45 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2506.11569

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.

Citations

• Algebraic classical W-algebras and Frobenius manifolds
• Frobenius manifolds from subregular classical W-algebras
• Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures
• A Simple Way of Making a Hamiltonian System Into a Bi-Hamiltonian One
• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Flat pencils of metrics and Frobenius manifolds
• Geometry of 2d topological field theories
• Differential geometry of the space of orbits of a Coxeter group

Related

• On B type family of Dubrovin-Frobenius manifolds and their integrable systems
• Algebraic classical W-algebras and Frobenius manifolds
• Poisson bivectors on infinite dimensional manifolds
• Frobenius manifolds and quantum groups
• Analytic theory of Legendre-type transformations for a Frobenius manifold