  Motivated by some recent works on BPS invariants of open strings/knot
invariants, we guess there may be a general correspondence between the
Ooguri-Vafa invariants of toric Calabi-Yau 3-folds and cohomologies of Nakajima
quiver varieties. In this short note, we provide a toy model to explain this
correspondence. More precisely, we study the topological open string model of
$\mathbb{C}^3$ with one Aganagic-Vafa brane $\mathcal{D}_\tau$, and we show
that, when $\tau\leq 0$, its Ooguri-Vafa invariants are given by the Betti
numbers of certain quiver variety. Moreover, the existence of Ooguri-Vafa
invariants implies an infinite product formula. In particular, we find that the
$\tau=1$ case of such infinite product formula is closely related to the
celebrated Rogers-Ramanujan identities.
