A flat Riemann manifold is called a Frobenius manifold if there is an associative symmetric 3-tensor with a local potential. For example, one could consider an associative quantum cup product in quantum cohomology of projective varieties, pencils of flat pairings on the base of the mini-versal deformations of hypersurface singularities, and Givental’s mirror construction relating Gromov-Witten invariants (quantum cohomology) to variation of Hodge structures of the mirror (Picard-Fuchs operators).
The authors provide a Lie-theoretic mirror symmetry construction for the orbit spaces \(\mathcal{M}^{DZ}_{\mathcal{R}}\) of extended affine Weyl groups as Frobenius manifolds. Mirror Frobenius manifolds are constructed as certain strata of a Hurewicz space, that is a space of isomorphism classes of covers of the complex lines by a genus g Riemann surface with ramification profile at infinity described by tuples of non-negative integers.
Roughly, the construction is the following: consider a complex simple Lie algebra and the highest weight of a non-trivial irreducible representation of minimal dimension. The characteristic polynomial in the irreducible representation for a pencil of group elements gives rise to spectral curves associated to affine relativistic Toda chains for arbitrary Dynkin types. It defines an embedding of the orbit spaces of extended affine Weyl groups into the Hurewicz space, where the image of the orbit spaces become sub-Frobenius manifolds.
The authors also provide three applications: first, they provide flat coordinates for the Saito metric of the Dubrovin-Zhang pencil and closed-form prepotentials for \(\mathcal{M}^{DZ}_{\mathcal{R}}\). Secondly, they compute the Lyashko-Looijenga multiplicity of the image of \(\mathcal{M}^{DZ}_{\mathcal{R}}\) inside the Hurewicz space. Lastly, they construct a bi-Hamiltonian dispersionless hierarchy on the loop space on \(\mathcal{M}^{DZ}_{\mathcal{R}}\) in Hamiltonian form for the canonical Poisson pencil associated to \(\mathcal{M}^{DZ}_{\mathcal{R}}\).