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Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups II

Lingrui Jiang, Si-qi Liu, Yingchao Tian, Youjin Zhang

Abstract

This is a sequel to arXiv:2506.13656, in which an approach to construct a class of generalized Frobenius manifold structures on the orbit spaces of affine Weyl groups is presented. In this paper we apply this construction to the affine Weyl groups of type $A_\ell, B_\ell, C_\ell$ and $D_\ell$.

Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups II

Abstract

This is a sequel to arXiv:2506.13656, in which an approach to construct a class of generalized Frobenius manifold structures on the orbit spaces of affine Weyl groups is presented. In this paper we apply this construction to the affine Weyl groups of type and .
Paper Structure (19 sections, 26 theorems, 273 equations)

This paper contains 19 sections, 26 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. The Case of $(A_\ell, \omega_\ell)$
  3. The invariant $\lambda$-Fourier polynomial ring
  4. The pencil generators
  5. Flat coordinates of the metric $\eta$
  6. The generalized Frobenius manifold sturcture on $\mathcal{M}_D({A_\ell},\omega_\ell)$ and monodromy group
  7. Examples
  8. The Landau-Ginzburg superpotential
  9. The Case of $(C_\ell, \omega_1)$
  10. The $W_a(R)$-invariant $\lambda$-Fourier polynomial ring
  11. The pencil generators
  12. Flat coordinates of the metric $\eta$
  13. The generalized Frobenius manifold structures
  14. Examples
  15. Landau-Ginzburg superpotential
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose $\{z^1,\dots,z^\ell\}$ be a set of pencil generators associated with an irreducible reduced root system $R$ and a fixed weight $\omega$, and the vector field $E$ given by Euler is diagonalizable, then there exists a generalized Frobenius manifold structure of charge $d=1$ on $\mathcal{M}_D$,

Theorems & Definitions (52)

  • Theorem 1.1: JTZ2025-1
  • Theorem 1.2: Main Theorem
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • ...and 42 more