at a point. This requires generically that the dimensionality of the branes adds up to six. If the sum of the dimensionality is less than six, the two branes do not intersect, and if the sum is more than six, the two branes intersect but not at a point. In order to specify the dimensionality of the brane, say to $d$, we set $6-d$ components of $Y_a$ to zero.

We generate 3d and 4d solutions numerically using the algorithm described in section \[sec:classicalsolution\]. The 2d solutions, on the other hand, can be constructed analytically by defining $Z$ as $$\label{eq: Z}
{{\left[ Y_1, Y_2 \right]}}=i Z \ ,$$ so that we can reduce with $\zeta=1$ to $$\label{eq: Y1Z_Y2Z}
{{\left[ Y_2, Z \right]}} = i  Y_1  \ , \quad
{{\left[ Z, Y_1 \right]}}= i  Y_2 \ .$$ Eqs.  and imply $$Y_1 = L_1 \ , \quad Y_2 = L_2 \ , \quad Z = L_3 \ ,
\label{2d-solution-generator}$$ where $L_i\ (i=1,2,3)$ are some representation of the SU(2) algebra. Here we restrict ourselves to the irreducible representation without loss of generality since the Dirac spectrum for the reducible representation simply becomes the sum of those for the irreducible representations which the reducible representation 