(\[eq: IRcutoff\]) is crucial in obtaining nontrivial solutions.

In this paper we search for solutions with a quasi direct-product structure for the (3+1) dimensions and the extra six dimensions [@Nishimura:2013moa] given as $$\label{eq: ansatz of A}
A_\mu = X_\mu \otimes M \quad (\mu=0,1,2,3) \ , \quad
A_a = \mathbf{1} \otimes Y_a \quad (a=4, \ldots, 9) \ ,$$ where the $N_{\rm X} \times N_{\rm X}$ Hermitian matrices $X_\mu$ represent the structure in the (3+1) dimensions, and the $N_{\rm Y} \times N_{\rm Y}$ Hermitian matrices $Y_a$ represent the structure in the extra six dimensions with $N=N_{\rm X} N_{\rm Y}$. This structure is actually compatible with the SO(3,1) symmetry $$\label{eq: Lorentzsymmetry}
U A_\mu U^\dagger = {O_\mu}^\nu A_\nu\ ,\quad
U A_a U^\dagger = A_a  \ ,$$ where $U= g \otimes \mathbf{1} \in \mathrm{SU}(N)$, $g \in \mathrm{SU}(N_{\rm X})$ and ${O_\mu}^\nu \in \mathrm{SO}(3,1)$ if $X_\mu$ satisfy $g X_\mu g^\dagger = {O_\mu}^\nu X_\nu$. Note that the $N_{\rm Y} \times N_{\rm Y}$ Hermitian matrix $M$ does not have to be $M = \mathbf{1}$, which would correspond to the direct-product structure. We will see that the matrix $M\neq \mathbf{1}$ plays an important role in making $Y_a$ block-diagonal, which will be interpreted later as dynamical generation of intersecting D-branes.

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