Abstract
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.
Highlights
The idea of quantizing geometrical structures originated in topological string theory from an interpretation of background independence [1] of the string partition function Z
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
After Nekrasov [8] introduced the Ω-background with two deformation parameters 1 and 2 to regularize the moduli space of instantons in N = 2 Super-Yang-Mills theories, it became quickly clear [8], how to interpret these two parameters in the topological string partition function on local Calabi-Yau spaces, which are related to the gauge theories by geometric engineering
Summary
The idea of quantizing geometrical structures originated in topological string theory from an interpretation of background independence [1] of the string partition function Z. There are some advantages in using these operators, one is that we are able to compute the free energies in different regions of the moduli space, another is that we do not have to solve the period integrals This method of computing the higher order corrections clears up their structure. We will compute the free energies in the Nekrasov-Shatashvili limit of the topological string on local Calabi-Yau geometries with del Pezzo surfaces or mass deformations thereof as the base.
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