Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups I
Lingrui Jiang, Si-Qi Liu, Yingchao Tian, Youjin Zhang
TL;DR
This work develops a framework to construct generalized Frobenius manifold structures on the orbit spaces of affine Weyl groups by introducing λ-dependent invariant Fourier polynomials that yield a metric g_λ linear in λ. It proves that the resulting monodromy groups are parabolic subgroups of the affine Weyl group and that a flat pencil of metrics g + λη exists when the pencil generators satisfy specific conditions, giving a Frobenius-algebra structure on the dense open locus M_D. The authors establish the polynomiality of pencil generators in flat coordinates, develop periods and extended flat structures, and determine the monodromy group as Stab_W(ω) ⋉ Z^ℓ, with explicit examples illustrating various phenomena. The paper also outlines a program (JLTZ-2) to construct pencil generators for several affine types and to connect these generalized Frobenius manifolds to relatedT-theoretic and integrable-system structures. Overall, the work links orbit-space geometry of affine Weyl groups to generalized Frobenius manifolds and their monodromy, providing a pathway to new connections with topological field theory and integrable hierarchies.
Abstract
We present an approach to construct a class of generalized Frobenius manifold structures on the orbit spaces of affine Weyl groups, and prove that their monodromy groups are parabolic subgroups of the associated affine Weyl groups.