This dissertation explores a 2015 conjecture of Codesido-Grassi-Marino in topologicalstring theory that relates the enumerative invariants of toric CY 3-folds to the spectra of operators attached to their mirror curves. In the maximally supersymmetric case, our first theorem relates zeroes of the higher normal function associated to an integral K2-class on the mirror curve to the spectra of the operators for curves of genus one, and suggests a new link between analysis and arithmetic geometry. On the other hand in the ’t Hooft limit, [KM, MZ] deduced from the [CGM] conjecture that the limiting values of the local mirror map at the maximal conifold point are given by values of the Bloch-Wigner dilogarithm at algebraic arguments. Our second theorem establishes these assertions by calculating regulator periods on the mirror curves attached to 3-term operators coming from triangles. As a consequence numerous series identities involving the Bloch-Wigner dilogarithm are demonstrated
