Sheaves on Fibered Threefolds and Quiver Sheaves

This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer–Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. For the local Calabi–Yau case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\cong{\bf A}^1$$\end{document} and special choice of framing, one recovers the N =  1 ADE quiver studied by Cachazo–Katz–Vafa.