$Y_a$, we focus on the off-diagonal block $\varphi$ in $$\begin{aligned}
\psi^\mathrm{(6d)}&=
\begin{pmatrix}
 & \varphi \\
 & 
\end{pmatrix} \ ,
\label{eq: off-diagonal_block_psi}\end{aligned}$$ for which the eigenvalue problem reduces to $$\label{eq: Diraceq_6d}
\gamma^a_{\alpha \beta} {\left[ Y_a^{(1)} \varphi_\beta
- \varphi_\beta Y_a^{(2)} \right]} =\lambda \varphi_\alpha \ ,$$ where $\varphi_\alpha \ (\alpha=1,\ldots,8)$ are the eigenvectors. The eigenvalue $\lambda$ corresponds to the mass in the (3+1) dimensions as one can see from Eq. (\[eq:ac-decompose\]).

We obtain numerically the eigenvalue spectrum of the Dirac operator for various pairs of classical solutions $Y_a^{(1)}$ and $Y_a^{(2)}$ to see in which case we obtain zero modes in the limit of infinite matrix size. Also, we look into the structure of the eigenvectors $\varphi_\alpha$, which we call the “wave functions” in what follows, and see whether it is localized as is expected from the picture of intersecting D-branes.

Since the Dirac operator under consideration anti-commutes with $\gamma_\chi$, the eigenvalues $\lambda$ and $- \lambda$ appear in pairs. If $\varphi$ is an eigenvector corresponding to the eigenvalue $\lambda$, then $\gamma_{\chi}\varphi$ is an eigenvector corresponding to the eigenvalue $- \lambda$. For the zero modes corresponding to $\lambda=0$, one can choose them to have definite chirality $\gamma_\chi \varphi = \pm \varphi$. In order to see 