Calabi-Yau geometry and electrons on 2d lattices

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.

Figure 1

Left: The toric diagram of local ${\mathbb{F}}_0$. Right: The two-dimensional lattice corresponding to the Hofstadter problem. An electron on the lattice can hop to any nearest neighbors.

Figure 2

Left: The toric diagram of local ${\mathcal{B}}_3$. Right: The two-dimensional triangular lattice. An electron on the lattice can hop to any nearest neighbors as well as to two next-nearest neighbors connected by the solid lines. One can see that the toric diagram of local ${\mathcal{B}}_3$ is embedded in this lattice system.

Figure 3

We show the region ${\mathrm{e}}^x+{\mathrm{e}}^y+{\mathrm{e}}^{-x-y}+m_1{\mathrm{e}}^{-x}+m_2{\mathrm{e}}^{-y}+m_3{\mathrm{e}}^{x+y}\le \mathcal{E}$ in the phase plane $(x,y)$. The energy is set to be $\log \mathcal{E}=10$, and the mass parameters are (a) $(m_1,m_2,m_3)=(1,1,1)$ and (b) $(m_1,m_2,m_3)=(10,1,1/2)$, respectively.

Figure 4

The branch cut structure of the Kähler modulus $t(\mathcal{E},\hslash )$ in the quantum local ${\mathcal{B}}_3$ geometry with the symmetric masses $m_1=m_2=m_3=1$. The horizontal direction is the Planck constant $\hslash$, and the vertical direction is the energy $\mathcal{E}$. The figure is identical to the band spectrum of electrons on the 2d triangular lattice in the presence of the magnetic field [23]. In this context, the horizontal direction corresponds to the magnetic flux $\varphi$.

Figure 5

We show the density of states for the triangular lattice with ${\lambda }_1={\lambda }_2=1$. The left figure is the graph for no flux ($\varphi =0$), and the right is for $\varphi =4\pi /3$. The DOS is not vanishing only on the energy subbands, and it has singularities at $(-1{)}^{ab}{F}_{a/b}(\mathcal{E})=-2$.

Figure 6

The branch cut structure of the Kähler modulus $t^{\prime }({\mathcal{E}}^{\prime },{\lambda }_i,\hslash )$ for (a) $({\lambda }_1,{\lambda }_2)=(2,1/2)$ and (b) $({\lambda }_1,{\lambda }_2)=(1,1/8)$.

Figure 7

Left: The 2d lattice considered in [43]. An electron can hop to eight possible sites connected by the solid lines. Right: The toric diagram of the corresponding CY threefold.

Figure 8

The cut structure for local ${\mathbb{P}}^2$.

Figure 9

The pictorial description of the refined topological vertex. The red line is the preferred direction.

Figure 10

Left: The web diagram of local ${\mathcal{B}}_3$. Right: We can embed it into the web diagram of local ${\mathbb{P}}^2$.

Figure 11

Two equivalent toric diagrams for the mass deformed $E_8$ del Pezzo geometry.