Spectral determinants as a non-perturbative completion of the topological string

We discuss possible non-perturbative completions of the topological
string and focus on one, which suggests an exact relation of the refined topological string to the spectral determinant of the trace  class operator associated to the quantum curve of a local Calabi-Yau geometry. This is checked to high precision for the local Calabi-Yau spaces associated to the anti canonial bundle over del Pezzo surfaces.