Topological strings, quiver varieties and Rogers-Ramanujan identities

TL;DR

This paper explores a conjectured link between string invariants in toric Calabi-Yau manifolds and the cohomology of quiver varieties, revealing connections to Rogers-Ramanujan identities through a toy model.

Contribution

It introduces a toy model demonstrating the correspondence between Ooguri-Vafa invariants and quiver variety Betti numbers, and relates these to classical identities.

Findings

Ooguri-Vafa invariants match Betti numbers of specific quiver varieties for certain parameters.

An infinite product formula emerges from the invariants, connecting to Rogers-Ramanujan identities.

The case τ=1 illustrates a deep link between string invariants and classical q-series identities.

Abstract

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.