Quiver Diagonalization and Open BPS States

We show that motivic Donaldson-Thomas invariants of a symmetric quiver Q, captured by the generating function P-Q, can be encoded in another quiver Q((8)) of (al-most always) infinite size, whose only arrows are loops, and whose generating func-tion P-Q(8) is equal to P-Q upon appropriate identification of generating parameters. Consequences of this statement include a generalization of the proof of integrality of Donaldson-Thomas and Labastida-Marino-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.