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arXiv 1701.01561hep-th
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Calabi-Yau几何与二维晶格上的电子

Calabi-Yau geometry and electrons on 2d lattices

Yasuyuki Hatsuda, Yuji Sugimoto, Zhaojie Xu

备注：
38 pages, 11 Figures; v2: the title changed, the discussions added, published in Physical Review D
发表信息：
Phys. Rev. D 95, 086004 (2017)
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摘要：

拓扑弦论的B模型方法通过量子化代数镜像曲线导出差分方程。我们知道，这些量子力学系统都是用精化拓扑弦来求解的。最近，有人指出，一个特定的Calabi-Yau流形的量子本征值问题，称为局部$\mathbb{F}_0$，是密切相关的二维正方形晶格上的电子的Hofstadter问题。在本文中，我们将这一思想推广到一个更复杂的Calabi-Yau流形。我们发现，局部$\mathcal{B}_3$几何，这是一个三点爆破的局部$\mathbb{P}^2$，是与电子上的三角晶格。这种对应关系使我们能够利用凝聚态物理中的已知结果来研究复曲面卡拉比-丘流形的量子几何。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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