decomposes to. The above construction suggests that the 2d brane is something like a “fuzzy disk”, which can be obtained by projecting a fuzzy sphere onto a plane. More precisely, it should be regarded as two coinciding fuzzy disks corresponding to the two hemispheres of the fuzzy sphere.

the cases with intersection at a point
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#### “3d-3d” ansatz {#subsec: 3d-3d}

For the first brane $Y_a^{(1)}$, we set $Y_4^{(1)} = Y_5^{(1)} = Y_6^{(1)} = 0$, and for the second brane $Y_a^{(2)}$, we set $Y_1^{(2)} = Y_2^{(2)} = Y_3^{(2)} = 0$. We use four 3d solutions for each of them. Thus we consider $4\times 4=16$ backgrounds with the matrix size $N_{\rm Y}^{(1)}=N_{\rm Y}^{(2)}=32$, $40$, $48$, $56$ and $64$, on which we solve the eigenvalue problem (\[eq: Diraceq\_6d\]).

![The ratio $\mu_0/\mu_1$ with the “3d-3d” ansatz is plotted against $1/N_{\rm Y}^{(1)}$ for $N_{\rm Y}^{(1)}=N_{\rm Y}^{(2)}=32$, $40$, $48$, $56$ and $64$. The dashed line represents a fit to the quadratic function $a+b/N_{\rm Y}^{(1)}+c/(N_{\rm Y}^{(1)})^2$ of $1/N_{\rm Y}^{(1)}$ with $a=-0.04(7)$, $b=34(6)$ and $c=-5(1) \times 10^2$. []{data-label="ratio_EV_3d"}](ratio_3d-3d.eps){width="10.0cm"}

![The wave functions $|(\varphi_{\mathrm{R}1})_{pq}|^2$ and $|(\varphi_{\mathrm{L}5})_{pq}|^2$ $(p=1,\ldots,N_{\rm Y}^{(1)},  \ q=1,\ldots,N_{\rm Y}^{(2)})$ with the “3d-3d” ansatz are plotted for $N_{\rm Y}^{(1)}=N_{\rm Y}^{(2)}=64$. []{data-label="wavefn_3d-3d"}](wavefn_3d-3d_2.eps){width="16.5cm"}

We obtain the smallest absolute value $|\lambda|$ 