.$$

Dirac zero modes from intersecting D-branes
-------------------------------------------

When the configuration of $Y_a$ takes the block-diagonal form (\[eq: correspondence\_MY\]), the eigenvalue problem associated with the Dirac operator on the left-hand side of (\[eq: chirality\_in\_6d\_psi\]) reduces to that for each block matrix of $\psi^\mathrm{(6d)}$. Note that the diagonal blocks of $Y_a$ can be interpreted as D-branes, whereas an off-diagonal block of $\psi^\mathrm{(6d)}$ can be interpreted as the open string ending on the corresponding two D-branes. When the two D-branes intersect with each other at a point, the length of the open string can become zero as it comes close to the intersecting point, and therefore a chiral zero mode appears there. In what follows, we confirm that this indeed happens for the typical solutions.

For that purpose, it suffices to consider the solutions of with a block-diagonal structure $$\begin{aligned}
\label{eq: form_Ya}
Y_a &=
\begin{pmatrix}
Y_a^{(1)} & \\
 & Y_a^{(2)}
\end{pmatrix} \ ,\end{aligned}$$ where the block matrices $Y_a^{(1)}$ and $Y_a^{(2)}$ have to satisfy with the same $\zeta$, which we set to $\zeta=1$ without loss of generality. We denote the size of the block matrices $Y_a^{(1)}$ and $Y_a^{(2)}$ as $N_{\rm Y}^{(1)}$ and $N_{\rm Y}^{(2)}$, respectively.

Corresponding to the block-diagonal structure (\[eq: form\_Ya\]) of 