  We investigate the topological string correspondence of the five-dimensional
half-BPS Wilson loops on $S^1$. First, we propose the refined holomorphic
anomaly equations for the BPS sectors of the Wilson loop expectation values. We
then solve these equations and obtain many non-trivial novel integral refined
BPS invariants for rank-one models. By studying the Wilson loop expectation
values around the conifold point, we obtain the quantum spectra of the quantum
Hamiltonians of the associated integrable systems. Lastly, as an application,
the study of this paper leads to a generalization of the blowup equations for
arbitrary magnetic fluxes that satisfy the flux quantization condition.
