Ooguri–Vafa integrality conjecture for the open topological string

Let $X$ be a toric Calabi–Yau $3$-fold and let $\mathcal{D}$ be an Aganagic–Vafa A-brane. Write $Z^{(X,\mathcal{D})}(\mathbf{x}^1,\ldots,\mathbf{x}^n;q,a)$ for the open-string partition function, let $s_{\vec{\mu}}$ be the Schur function indexed by a tuple of partitions $\vec{\mu}$, and define

$$f_{\vec{\mu}}(q,a)=(q^{1/2}-q^{-1/2})\left\langle \operatorname{Log}\left(Z^{(X,\mathcal{D})}(\mathbf{x}^1,\ldots,\mathbf{x}^n;q,a)\right),s_{\vec{\mu}}(\vec{\mathbf{x}})\right\rangle.$$

Ooguri–Vafa integrality conjecture. One has

$$f_{\vec{\mu}}(q,a)=\sum_{i,j}N_{\vec{\mu},i,j}a^{i/2}q^{j/2}\in\mathbb{Z}[q^{\pm1/2},a^{\pm1/2}],$$

for integer invariants $N_{\vec{\mu},i,j}$. This predicts that the generally rational open Gromov–Witten theory is governed by integer BPS invariants; the paper applies the prediction to the framed $\mathbb{C}^3$ model and studies its consequences for Rogers–Ramanujan identities.