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

TL;DR

The paper develops a quantum geometry framework for local Calabi–Yau manifolds built from del Pezzo surfaces in the Nekrasov–Shatshvili limit, where the refined topological string partition function acts as a brane wave function and quantum corrections are encoded in a deformed differential on the Seiberg–Witten curve. By modeling branes with β-ensembles and employing a quantum (NS) special geometry, the authors derive second-order differential operators that map classical periods to their quantum-corrected counterparts, yielding NS free energies across various toric geometries and moduli points, including large radius, orbifold, and conifold regimes. They provide explicit results for several geometries (e.g., local F0, local P^2, local F1, and E8 del Pezzo deformations), verify modular and elliptic-curve structures at genus one, and compare with known refined/topological-vertex data where possible, noting both agreements and subtle discrepancies tied to constants of integration or non-normalizable moduli. The work advances a systematic, operator-based approach to compute quantum deformations of periods and NS free energies, with potential extensions to open amplitudes and nonperturbative completions. The results emphasize a clear separation between Coulomb (normalizable) moduli, which receive quantum corrections, and mass (non-normalizable) parameters, which remain classical in the NS limit, and highlight the role of modular and elliptic structures in organizing the NS expansions.

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.

Figures (8)

• Figure 1: local $\mathcal{B}_2$. In the first column we denote the divisors and in the fourth column the moduli and parameters associated with them.
• Figure 2: Polyhedron 2 depicting the toric geometry $\mathds{F}_0$.
• Figure 3: Toric diagram of local ${\cal O}(-3)\rightarrow {\mathbb P}^2$.
• Figure 4: Toric diagram of local $\mathds{F}_1$.
• Figure 5: Toric diagram of $\mathcal{O}(-K_{\mathds{F}_2})\rightarrow \mathds{F}_2$.