Frobenius Manifolds Associated to Bl and Dl, Revisited

Frobenius manifolds on orbits spaces

“…1. The standard reflection representation of a finite irreducible Coxeter group: We prove there is a natural rational Frobenius structure different from the ones constructed in [4] and [25]. We give details in section 3.1.…”

“…This result led to the classification of irreducible semisimple polynomial Frobenius manifolds with positive degrees (see section 3.1 for more details). His method was used in [25] when W is a Coxeter group of type B r or D r to construct r Frobenius manifolds on O(ρ ref ). However, all above-mentioned constructions seem like they depend on the fact that the invariant rings are polynomial rings.…”

Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy

“…where ζ(z) and l(z) are certain even functions similar as (1.7), and F m,n is the potential of the Frobenius manifold defined on the orbit space of the Coxeter group of type B m+n , see [1,13,27] for details. Note that the replacement of the factor log(z 2 − z 1 ) in [26] (see (1.19)) by log z 2 −z 1 z 2…”

“…It turns out that the underlying Frobenius manifold is of infinite dimension. Following their approach, a class of infinite-dimensional Frobenius manifolds was constructed by Xu and one of the authors [26] for the dispersionless two-component BKP hierarchy, and these manifolds are related to finite-dimensional Frobenius manifolds corresponding to Coxeter groups of types B and D [1,13,27]. Along the same line, now we want to revise and generalize the construction in [7] with some skills developed in [26].…”

A Class of Infinite-dimensional Frobenius Manifolds and their Submanifolds

“…In particular, the manifold M m,1 is defined on the orbit space C m+1 /D m+1 of the Coxeter group D m+1 . The Frobenius manifolds M m,n were reconstructed by Zuo [29], whose method is based on the polynomial contravariant metric of the orbit space induced from the Euclidean metric (see for example Lemma 4.1 in [11]) and different choices of unity vector field.…”