Ooguri–Vafa integrality product formula for framed geometric-seed wavefunctions

Let $\underline{\bf i}\in\mathbb{G}_{\mathrm{ad}}$ be a framed seed with wavefunction $\Psi_{\underline{\bf i}}$. Let $A$ be its framing and $L$ the Lagrangian of the deformed foam. For $d\in\mathbb{Z}_{\geq 0}^g$, write $X^d=\prod_iX_i^{d_i}$, and define

$$\Phi(z)=\prod_{n\geq 0}(1+q^{2n+1}z)^{-1}.$$

Ooguri–Vafa integrality conjecture. The wavefunction has the product expansion

$$\Psi_{\underline{\bf i}}=\prod_{d\in\mathbb{Z}_{\geq 0}^g\setminus\{0\}}\prod_{s\in\mathbb{Z}}\Phi\bigl(X^d(-q)^s\bigr)^{n_{d,s}^{(A)}},$$

with $n_{d,s}^{(A)}\in\mathbb{Z}$ the Ooguri–Vafa invariants. This gives the explicit all-genus integral refinement of the preceding wavefunction conjecture for framed geometric seeds; the underlying open Gromov–Witten theory is not yet rigorously defined.