Quantization condition from exact WKB for difference equations

Abstract

Abstract: A well-motivated conjecture states that the open topological string partition function on toric geometries in the Nekrasov-Shatashvili limit is annihilated by a difference operator called the quantum mirror curve. Recently, the complex structure variables parameterizing the curve, which play the role of eigenvalues for related operators, were conjectured to satisfy a quantization condition non-perturbative in the NS parameter ħ. Here, we argue that this quantization condition arises from requiring single-valuedness of the partition function, combined with the requirement of smoothness in the parameter ħ. To determine the monodromy of the partition function, we study the underlying difference equation in the framework of exact WKB.