Algebraic classical $W$-algebras and Frobenius manifolds

TL;DR

This paper constructs algebraic Frobenius manifolds from Drinfeld-Sokolov bihamiltonian structures linked to nilpotent elements, providing a uniform approach for certain Weyl group classes.

Contribution

It introduces a novel method to derive algebraic Frobenius manifolds from bihamiltonian structures associated with nilpotent elements, unifying their construction for specific Weyl group classes.

Findings

Constructed local bihamiltonian structures forming algebraic $W$-algebras.

Established dispersionless limits leading to Frobenius manifolds.

Unified the construction for regular cuspidal conjugacy classes in Weyl groups.

Abstract

We consider Drinfeld-Sokolov bihamiltonian structure associated to a distinguished nilpotent elements of semisimple type and the space of common equilibrium points defined by its leading term. On this space, we construct a local bihamiltonian structure which form an exact Poisson pencil, defines an algebraic classical $W$-algebra, admits a dispersionless limit, and its leading term defines an algebraic Frobenius manifold. This leads to a uniform construction of algebraic Frobenius manifolds corresponding to regular cuspidal conjugacy classes in irreducible Weyl groups.