Topological strings, quiver varieties and Rogers-Ramanujan identities

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.