1 Coxeter Elements for Vanishing Cycles of Types

We introduce two real entire functions fA 1 2∞ and fD 1 2∞ in two variables, having only two critical values 0 and 1. Associated maps C → C define topologically locally trivial fibrations over C\{0, 1}. The critical points over 0 and 1 are ordinary double points, whose associated vanishing cycles in the generic fiber span its middle homology group and their intersection diagram forms the bi-partite decomposition of quivers of type A 1 2∞ and D 1 2∞ , respectively. Coxeter element of type A 1 2∞ and D 1 2∞ are introduced as the product of the monodromies of the fibrations around 0 and 1. We describe the spectra of the intersection form (normalized in the iterval [0, 4]) and the Coxeter elements (normalized in the interval (−12 , 1 2 )).