Extended Affine Weyl Frobenius Manifolds

• Extended Affine Weyl Frobenius Manifolds are special Frobenius structures derived from orbit spaces of extended affine Weyl groups with quasi-homogeneous and semisimple properties.
• They feature a flat pencil of metrics, explicit prepotentials solving the WDVV equations, and symmetry realized through twist automorphisms and translations along fundamental weights.
• Their analytic and algebraic frameworks support applications in topological field theories, integrable hierarchies, and mirror symmetry, with extensions to multi-pole and infinite-dimensional settings.

An extended affine Weyl Frobenius manifold is a special class of (quasi-)homogeneous, semisimple Frobenius manifolds whose structure is governed by the orbit spaces of extended affine Weyl groups, including types A, D, E, B, C, and D. These manifolds encode deep connections between singularity theory, root system symmetries, isomonodromic deformations, and integrable hierarchies, and are central objects in the context of mirror symmetry for integrable systems of affine type. The construction proceeds via a root-theoretic extension of the original affine Weyl group, often via additional automorphisms (e.g., the twist automorphism), and yields a rich flat pencil geometry with explicit LG (Landau–Ginzburg) and Hurwitz-theoretic B-models. Their prepotentials solve the WDVV equations and manifestly realize the symmetries of the underlying extended affine Weyl groups, while their analytic and algebraic invariants are described in terms of both Lie-theoretic and Hurwitz-theoretic data (Otani, 2024, Dubrovin et al., 2015, Proserpio et al., 2024, Brini et al., 2021, Zuo, 2019, Jiang et al., 16 Jun 2025, Wu et al., 2013).

1. Extended Affine Weyl Groups and Their Automorphisms

The extended affine Weyl group arises from a root system $R$ (typically of affine ADE, B, C, D type), whose standard affine Weyl group $W_a$ is enlarged by additional automorphisms. In the simply-laced affine ADE case, the group is constructed from:

• A root lattice $L$ with nondegenerate bilinear “Cartan” form $I\colon L \times L \to \mathbb{Z}$ of signature $(\ell-1,1,0)$.
• Real roots $\Delta_\mathrm{re}$ generating $L$.
• A Coxeter transformation $c_A\in\operatorname{Aut}_{\mathbb{Z}}(L,I)$.

A key new automorphism, the twist automorphism $t_\delta$, is defined using the unique Euler form $\chi:L\times L\to\mathbb{Z}$ of Saito–Sekiguchi–Yano, by

$t_\delta(x) = x - \chi(\delta,x)\,\delta$

where $\delta$ spans the radical $\ker I$ (Otani, 2024). The full (modified) extended affine Weyl group is then

$$W_A^{\mathrm{ext}} := \langle\, W_A,\, t_\delta\, \rangle$$

yielding a split exact sequence

$$1 \longrightarrow W_A \longrightarrow W_A^{\mathrm{ext}} \overset{t_\delta}{\longrightarrow} \mathbb{Z} \longrightarrow 1.$$

For B, C, D types, the extension involves shifts along fundamental weights attached to marked Dynkin diagram vertices and yields families $W^{(k)}(R)$ parameterized by the marked vertex $k$ (Dubrovin et al., 2015).

In the A-case, and also for $BCD$-type, additional extensions by translation along coweights lead to more general extended affine Weyl groups, as in the introduction of $\widetilde{W}^{(k,k+1)}(A_l)$, defined by affine shifts along two distinct fundamental weights (Zuo, 2019). These extensions are essential to the structure of the associated Frobenius manifold and its symmetry properties.

2. Frobenius Manifold Structure on Orbit Spaces

The extended affine Weyl Frobenius manifolds are constructed on quotient spaces (orbit spaces) of the universal cover of the Cartan subalgebra or its exponential form, divided by $W^\mathrm{ext}$:

$$M = (\mathfrak{h} \oplus \mathbb{C}) / W^\mathrm{ext}$$

or, for the classical types,

$M = \mathbb{C}^{\ell+1}/\widetilde{W},$

where $\widetilde{W}$ is the appropriate extended affine Weyl group.

These manifolds support a Frobenius structure, characterized by:

• Existence of a flat pencil of metrics $(g, \eta)$, with $g$ from push-forward of the standard bilinear form and $\eta$ as its Lie derivative along a specified unity vector field $e$.
• Associative, commutative multiplication on tangent (or cotangent) spaces, defined via the Christoffel data of $g$ and the flat coordinates of $\eta$.
• Semisimple prepotential $F$ solving the WDVV equations (associativity equations), explicit in flat coordinates.
• Quasi-homogeneous Euler vector field $E$ encoding weight gradings, with charge $d=1$ in the semisimple case (Dubrovin et al., 2015, Proserpio et al., 2024, Zuo, 2019, Brini et al., 2021).

In exceptional and classical types (ADE, $B_l$, $C_l$, $D_l$), the invariant theory yields explicit polynomial (or exponential polynomial) bases for $\mathbb{C}[\mathfrak{h}]^{W^\mathrm{ext}}$, and the flat structure is effectively computed in terms of these variables (Dubrovin et al., 2015, Brini et al., 2021, Proserpio et al., 2024).

3. Mirror Symmetry, LG Superpotentials, and Hurwitz Moduli

Mirror symmetry provides a B-model realization in terms of LG (Landau–Ginzburg) superpotentials or Hurwitz spaces. For each extended affine Weyl Frobenius manifold, one constructs a one-dimensional superpotential

$W(\mu;w) = \lambda(\mu;w),$

determined by the spectral curves of the corresponding affine Toda chain or via explicit character polynomials in the case of exceptional Lie type (Brini et al., 2021). In classical (e.g., type A) cases, the associated Hurwitz space is

$\mathcal{H}_{0,d}^\omega(a_1,a_2,\dots)$

with prepotential constructed via explicit residue formulas and dependent on the critical and pole data of rational functions $\lambda(w)$ (Proserpio et al., 2024, Wu et al., 2013).

For $BCD$-type, the LG potential is realized as a (cosine)-Laurent polynomial with residue pairings giving the Frobenius metric and structure constants. The superpotential construction is not merely a formal analogy; it provides an isomorphism of the resulting Frobenius manifolds with the orbit-space quotients (Dubrovin et al., 2015, Proserpio et al., 2024, Brini et al., 2021, Zuo, 2019).

4. Structure of the Prepotential, Diagonal Invariants, and Flat Coordinates

The genus-zero prepotential $F$ for extended affine Weyl Frobenius manifolds decomposes as a sum of polynomial, logarithmic, and polynomial–exponential terms, integrating all three-point functions and recovering the invariants of the underlying root system. In the Hurwitz-cover realization, $F$ is assembled as

$$F = F_{\mathrm{bulk}}(\bm t) + F_{\mathrm{tail}}(\alpha, \beta) + F_{\mathrm{int}}(\bm t, \alpha, \beta),$$

with

• $F_{\mathrm{bulk}}$ the Dubrovin–Zhang potential,
• $F_{\mathrm{tail}}$ describing log-diagonal expansion among pole residues,
• $F_{\mathrm{int}}$ built from constrained cubic and higher-degree diagonal invariants of the form $\widetilde{\Theta}_k$ (Proserpio et al., 2024).

Flat coordinates for both the Frobenius metric and the intersection form are constructed as explicit algebraic (often symmetric polynomial or exponential) functions of critical value parameters, with precise transformation laws depending on the root and weight data (Brini et al., 2021, Zuo, 2019, Wu et al., 2013). The intersection form and compatibility with the algebra structure are governed by Armstrong–Brieskorn–Saito–type results.

5. Monodromy, Symmetry Groups, and the Lyashko–Looijenga Map

The action of the extended affine Weyl group, including the twist automorphism $t_\delta$ when present, gives rise to the full symmetry group of the Frobenius manifold, including monodromy around the discriminant locus and at infinity. The twist realises the monodromy corresponding to translation along the additional affine direction, and its orbits classify root bases modulo Coxeter transformations (Otani, 2024).

Associated is the Lyashko–Looijenga (LL) map, sending a point in the Frobenius manifold to the set of coefficients in the characteristic polynomial of the multiplication operator by the Euler field. The degree of the LL map equals the number of root bases modulo twist automorphism, expressible as

$$e(\mathcal{B}_A) = \frac{\mu_A!}{a_1! a_2! a_3!}\left(\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}\right),$$

where $a_i$ enumerate the labelled branch numbers (Otani, 2024). The group-theoretic and Hurwitz-theoretic degrees match, confirming the normalization of the manifold and the enumeration of distinct Frobenius–semisimple points.

6. Generalizations and Infinite-Dimensional Limits

Extended affine Weyl Frobenius manifolds admit generalizations:

• For more general weights in the extended group (e.g., sums of fundamental weights), one obtains new Frobenius structures with monodromy subgroups of the affine Weyl group; these do not necessarily possess flat unity fields but solve Dubrovin’s axioms otherwise (Jiang et al., 16 Jun 2025).
• Infinite-dimensional Frobenius manifolds arise in connection with integrable hierarchies (e.g., dispersionless Toda), whose finite-dimensional slices reduce to the above finite-type orbit-space models (Wu et al., 2013).
• Additional pole data in the Hurwitz construction leads to a hierarchy of “multi-pole” extended Frobenius manifolds, with explicitly computed diagonal invariants parametrizing interaction terms (Proserpio et al., 2024).

7. Applications and Further Directions

Extended affine Weyl Frobenius manifolds underpin:

• Classification of 2D topological field theories with root-theoretic symmetry.
• Mirror symmetry for integrable hierarchies and moduli space geometry (Brini et al., 2021).
• Connections to representation theory (Artin, Seidel–Thomas braid groups) and perverse sheaves.
• Explicit construction of dispersionless integrable hierarchies (extended Toda and generalizations).
• A bridge between Coxeter group Frobenius manifolds and those arising from nontrivial monodromy extensions (Otani, 2024, Jiang et al., 16 Jun 2025, Wu et al., 2013).

Their algebraic, geometric, and analytical structures continue to provide a robust framework for modern interactions between singularity theory, algebraic geometry, integrable systems, and mathematical physics.