Coxeter elements for vanishing cycles of types ${\mathrm{A}}_{\frac{1}{2}\infty }$ and ${\mathrm{D}}_{\frac{1}{2}\infty }$

Saito, Kyoji. "Coxeter elements for vanishing cycles of types $\mathrm{A}_{\frac{1}{2}\infty }$ and $\mathrm{D}_{\frac{1}{2}\infty }$." Annales de l’institut Fourier 61.7 (2011): 2959-2984. <http://eudml.org/doc/275581>.

@article{Saito2011,
abstract = {We introduce two entire functions $f_\{\mathrm\{A\}_\{\frac\{1\}\{2\}\infty \}\}$ and $f_\{\mathrm\{D\}_\{\frac\{1\}\{2\}\infty \}\}$ in two variables. Both of them have only two critical values $0$ and $1$, and the associated maps $\mathbf\{C\}^2\!\rightarrow \! \mathbf\{C\}$ define topologically locally trivial fibrations over $\mathbf\{C\}\!\setminus \!\lbrace 0,1\rbrace $. All critical points in the singular fibers over $0$ and $1$ 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 infinite quivers of type $\mathrm\{A\}_\{\frac\{1\}\{2\}\infty \}$ and $\mathrm\{D\}_\{\frac\{1\}\{2\}\infty \}$, respectively. Coxeter elements of type $\mathrm\{A\}_\{\frac\{1\}\{2\}\infty \}$ and $\mathrm\{D\}_\{\frac\{1\}\{2\}\infty \}$, 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 $(-\frac\{1\}\{2\},\frac\{1\}\{2\})$ except at $0$ for type $\mathrm\{D\}_\{\frac\{1\}\{2\}\infty \}$.},
affiliation = {Institute for the Physics and Mathematics of the Universe, University of Tokyo,},
author = {Saito, Kyoji},
journal = {Annales de l’institut Fourier},
keywords = {vanishing cycle; spectra; Coxeter element; transcendental function; spectrum},
language = {eng},
number = {7},
pages = {2959-2984},
publisher = {Association des Annales de l’institut Fourier},
title = {Coxeter elements for vanishing cycles of types $\mathrm\{A\}_\{\frac\{1\}\{2\}\infty \}$ and $\mathrm\{D\}_\{\frac\{1\}\{2\}\infty \}$},
url = {http://eudml.org/doc/275581},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Saito, Kyoji
TI - Coxeter elements for vanishing cycles of types $\mathrm{A}_{\frac{1}{2}\infty }$ and $\mathrm{D}_{\frac{1}{2}\infty }$
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 7
SP - 2959
EP - 2984
AB - We introduce two entire functions $f_{\mathrm{A}_{\frac{1}{2}\infty }}$ and $f_{\mathrm{D}_{\frac{1}{2}\infty }}$ in two variables. Both of them have only two critical values $0$ and $1$, and the associated maps $\mathbf{C}^2\!\rightarrow \! \mathbf{C}$ define topologically locally trivial fibrations over $\mathbf{C}\!\setminus \!\lbrace 0,1\rbrace $. All critical points in the singular fibers over $0$ and $1$ 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 infinite quivers of type $\mathrm{A}_{\frac{1}{2}\infty }$ and $\mathrm{D}_{\frac{1}{2}\infty }$, respectively. Coxeter elements of type $\mathrm{A}_{\frac{1}{2}\infty }$ and $\mathrm{D}_{\frac{1}{2}\infty }$, 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 $(-\frac{1}{2},\frac{1}{2})$ except at $0$ for type $\mathrm{D}_{\frac{1}{2}\infty }$.
LA - eng
KW - vanishing cycle; spectra; Coxeter element; transcendental function; spectrum
UR - http://eudml.org/doc/275581
ER -