  We use mirror symmetry, quantum geometry and modularity properties of
elliptic curves to calculate the refined free energies in the
Nekrasov-Shatashvili limit on non-compact toric Calabi-Yau manifolds, based on
del Pezzo surfaces. Quantum geometry here is to be understood as a quantum
deformed version of rigid special geometry, which has its origin in the quantum
mechanical behaviour of branes in the topological string B-model. We will argue
that, in the Seiberg-Witten picture, only the Coulomb parameters lead to
quantum corrections, while the mass parameters remain uncorrected. In certain
cases we will also compute the expansion of the free energies at the orbifold
point and the conifold locus. We will compute the quantum corrections order by
order on $\hbar$, by deriving second order differential operators, which act on
the classical periods.
