Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit

Understand

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.