Quantized mirror curves and resummed WKB

Understand

Based on previous insights, we present an ansatz to obtain quantization conditions and eigenfunctions for a family of difference equations which arise from quantized mirror curves in the context of local mirror symmetry of toric Calabi-Yau threefolds.

• It is a first principles construction, which yields closed expressions for the quantization conditions and the eigenfunctions when $\hbar/2\pi \in \mathbb Q$.
• The key ingredient is the modular duality structure of the underlying quantum integrable system.
• We use our ansatz to write down explicit results in some examples, which are successfully checked against purely numerical results for both the spectrum and the eigenfunctions.