of the eigenvalues for each of the 16 cases and denote the average as $\mu_0$. Similarly, we denote as $\mu_1$ the average of the second smallest absolute value $|\lambda|$. Since the normalization of the solutions is somewhat arbitrary for different matrix size, we take the ratio $\mu_0/\mu_1$ and plot it against $1/N_{\rm Y}^{(1)}$ in Fig. \[ratio\_EV\_3d\]. The fit to the quadratic function of $1/N_{\rm Y}^{(1)}$ suggests that zero modes appear in the $N_{\rm Y}^{(1)} \to \infty$ limit.

Next we consider the wave function corresponding to the lowest eigenvalue for one of the 16 cases with $N_{\rm Y}^{(1)}=N_{\rm Y}^{(2)}=64$. We calculate $|(\varphi_{\mathrm{R}\alpha})_{pq}|^2$ for each $\alpha$ and find that the wave function almost vanishes except for the spinor component $\alpha=1$. Similarly, we calculate $|(\varphi_{\mathrm{L} \alpha})_{pq}|^2$ for each $\alpha$, and find that the wave function almost vanishes except for the spinor component $\alpha=5$. In Fig. \[wavefn\_3d-3d\], we plot $|(\varphi_{\mathrm{R}1})_{pq}|^2$ and $[(\varphi_{\mathrm{L}5})_{pq}|^2$, the component of the wave functions that has nonzero elements. We find that the wave functions are localized at the $(1,1)$ element. These results are consistent with the picture that the left-handed and right-handed zero modes appear from the intersecting D-branes.

#### “2d-4d” ansatz {#subsec: 2d-4d}

For the first brane $Y_a^{(1)}$, we 