Quiver Diagonalization and Open BPS States

We show that motivic Donaldson–Thomas invariants of a symmetric quiver Q, captured by the generating function PQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_Q$$\end{document}, can be encoded in another quiver Q(∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q^{(\infty )}$$\end{document} of (almost always) infinite size, whose only arrows are loops, and whose generating function PQ(∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{Q^{(\infty )}}$$\end{document} is equal to PQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_Q$$\end{document} upon appropriate identification of generating parameters. Consequences of this statement include a generalization of the proof of integrality of Donaldson–Thomas and Labastida–Mariño–Ooguri–Vafa invariants that count open BPS states, as well as expressing motivic Donaldson–Thomas invariants of an arbitrary symmetric quiver in terms of invariants of m-loop quivers. In particular, this means that the already known combinatorial interpretation of invariants of m-loop quivers extends to arbitrary symmetric quivers.