Calabi-Yau geometry and electrons on 2d lattices

Abstract

The B-model approach of topological string theory leads to difference equations by quantizing algebraic mirror curves. It is known that these quantum mechanical systems are solved by the refined topological strings. Recently, it was pointed out that the quantum eigenvalue problem for a particular Calabi--Yau manifold, known as local ${\mathbb{F}}_0$, is closely related to the Hofstadter problem for electrons on a two-dimensional square lattice. In this paper, we generalize this idea to a more complicated Calabi--Yau manifold. We find that the local ${\mathcal{B}}_3$ geometry, which is a three-point blow-up of local ${\mathbb{P}}^2$, is associated with electrons on a triangular lattice. This correspondence allows us to use known results in condensed matter physics to investigate the quantum geometry of the toric Calabi--Yau manifold.

arXiv comment38 pages, 11 Figures; v2: the title changed, the discussions added, published in Physical Review D

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Cited by in corpus (12)

• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Blowup Equations for Refined Topological Strings
• Hofstadter's Butterfly and Langlands Duality
• Spectral equations for the modular oscillator
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• New results in $\mathcal{N}=2$ theories from non-perturbative string
• Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
• Bloch electrons on honeycomb lattice and toric Calabi-Yau geometry
• Hofstadter-Toda spectral duality and quantum groups
• Lattice random walks and quantum A-period conjecture
• Recent Developments in Topological String Theory
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux