Refined Invariants and Quantum Curves from Supersymmetric Localization

We study an Aganagic-Vafa brane supported on a special Lagrangian submanifold $\mathcal{L}$ in a non-compact toric Calabi-Yau threefold $\mathcal{X}$. From the perspective of geometric engineering, the Aganagic-Vafa branes give rise to a special class of half-BPS codimension-two defects in 5d $\mathcal{N}=1$ supersymmetric field theories in the presence of $\Omega$-background. We propose that the defect partition functions give generating functions of refined, non-negative, integral open BPS invariants of the pair $(\mathcal{X},\mathcal{L})$, across different K\"{a}hler moduli chambers they are expanded in. In the Nekrasov-Shatashvili limit, the partition function provides a partially resummed solution to a $q$-difference equation that quantizes the mirror curve of $\mathcal{X}$ in an unambiguous fashion, in a polarization determined by the discrete labels of the Aganagic-Vafa brane. We demonstrate our method at examples of $\mathbb{C}^3$, resolved conifold, resolved $A_1$-singularity, local $F_0$, and local $F_1$.