Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau–Ginzburg superpotentials

Abstract

For the root systems of type B l , C l and D l , we generalize the result of [7] by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in [7]. It also depends on certain additional data. We also construct Landau-Ginzburg superpotentials for these Frobenius manifold structures. References 55Date: May 21, 2019.2010 Mathematics Subject Classification. Primary 53D45.

Frobenius manifolds and a new class of extended affine Weyl groups of A-type

“…According to Lemma D.1 in [10] (or see Lemma 3.3 in [19]) and using Lemma 3.7, Propsition 3.8 and Lemma 3.6, we have Theorem 3.9. g ij (y) and η ij (y) form a flat pencil of metrics, i.e., the metric…”

“…Conditions (2.8) and (2.9) are essential for this construction as did in [12,19]. For simplicity, we introduce a set of numbers d j,k := (ω j , ω k ) = j(l−k+1) l+1…”

“…A natural question is that " Whether the geometric structures that were revealed by Dubrovin-Zhang's construction also exist on the orbit spaces of the extended affine Weyl groups for an arbitrary choice of α k ?" Our recent work in [19] is to give an affirmative answer to this question for the root systems of type B l , C l and also for D l . We show, by fixing another integer 0 ≤ m ≤ l − k, that on the corresponding orbit spaces there also exist Frobenius manifold structures with potentials F (t) that are weighted homogeneous polynomials w.r.t t 1 , · · · , t l+1 , 1 t l−m , 1 t l , e t l+1 .…”

“…B.Dubrovin and Y.Zhang in [12] (also [19]) defined an extended affine Weyl group W (k) (R), which acts on the extended space V ⊕ R and is generated by the transformations…”

“…In this paper we will present a new extension of affine Weyl groups denoted by W (k,k+1) (R), which is different from those in [12,19], and study Frobenius manifold structures on the corresponding orbit spaces M (k,k+1) (R), where the new extended affine Weyl groups W (k,k+1) (R) act on the extended space V ⊕ R 2 generated by the transformations…”