# Calabi-Yau geometry and electrons on 2d lattices

## TL;DR

This paper explores the connection between Calabi-Yau geometries and electron behavior on 2D lattices, generalizing previous work to more complex geometries and lattice types, bridging string theory and condensed matter physics.

## Contribution

It introduces a new correspondence between the local _3 Calabi-Yau geometry and electrons on a triangular lattice, extending the geometric-electronic analogy to more intricate manifolds.

## Key findings

- Established a link between local _3 geometry and triangular lattice electrons
- Utilized condensed matter results to analyze quantum geometry of toric Calabi-Yau manifolds
- Generalized previous Calabi-Yau-electron correspondences to more complex geometries

## 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.