Calabi-Yau geometry and electrons on 2d lattices
Calabi-Yau geometry and electrons on 2d lattices
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DOI：
10.1103/physrevd.95.086004
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发表时间：
2017-01
影响因子：
5
通讯作者：
Yasuyuki Hatsuda;Yuji Sugimoto;Zhaojie Xu
中科院分区：
文献类型：
--
作者：
Yasuyuki Hatsuda;Yuji Sugimoto;Zhaojie Xu
拓扑弦理论的B模型方法通过量化代数镜面曲线得到差分方程组。众所周知，这些量子力学系统是通过精化的拓扑弦来求解的。最近，有人指出，一种特殊的Calabi-Yau流形的量子本征值问题，即局域的$\mathbb{F}_0，与二维正方形晶格上电子的Hofstadter问题密切相关。在本文中，我们将这一思想推广到更复杂的Calabi-Yau流形上。我们发现，局域$数学{B}_3$几何是局域$\mathbb{P}^2$的三点爆破几何，它与三角形晶格上的电子有关。这种对应关系允许我们使用凝聚态物理中的已知结果来研究Toric Calabi-Yau流形的量子几何。
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.