Coxeter elements for vanishing cycles of type $A_{1/2\infty}$ and $D_{1/2\infty}$

TL;DR

This paper constructs and analyzes entire functions with specific critical values, exploring their topological fibrations, vanishing cycles, and Coxeter elements, revealing spectral properties related to infinite-type quivers.

Contribution

It introduces new entire functions associated with infinite-type $A_{1/2infty}$ and $D_{1/2infty}$ quivers, and studies their Coxeter elements and spectral properties.

Findings

Critical points are ordinary double points.

Vanishing cycles span the middle homology group.

Spectra of Coxeter elements are continuous on (-1/2,1/2) except at 0 for type D.

Abstract

We introduce two entire functions $f_{A_{1/2\infty}}$and $f_{D_{1/2\infty}}$ in two variables. Both of them have only two critical values 0 and 1, and the associated maps $\C^2 to \C$ define topologically locally trivial fibrations over $\C\setminus{0,1\}$. All critical points are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed quivers of type $A_{1/2\infty}$ and $D_{1/2\infty}$, respectively. Coxeter elements of type $A_{1/2\infty}$ and $D_{1/2\infty}$, acting on the middle homology group, are introduced as the product of the monodromies around 0 and 1. We describe the spectra of the Coxeter elements by embedding the middle homology group into a Hilbert space. The spectra turn out to be strongly continuous on the interval $(-1/2,1/2)$ except at 0 for type $\DD$.