of $N$ attachment points for the string. The inner degree of freedom modelized by the Hilbert space $\mathbb C^2$ (with canonical basis $(|\uparrow \rangle, |\downarrow \rangle)$ is the spin of the fermionic string.\
The total system is represented by the matrices: $$\mathbf X_i = \left(\begin{array}{cc} X_i & |\delta x_i\rangle \\ \langle \delta x_i| & x_i \end{array} \right) ; \qquad \Psi = \left(\begin{array}{cc} 0 & |\psi{\rangle \hspace{-0.2em} \rangle}\\ 0 & 0 \end{array} \right)$$ where $|\delta x_i \rangle \in \mathbb C^N$ represents quantum fluctuations of the vacuum creating and annihilating bosonic strings linking the stack (the D2-brane) to the probe D0-brane. $|\delta x_i(t=0)\rangle$ is randomly choosen following a gaussian law.\
The dynamics of the system is governed by the Lagrangian densities: $$\mathscr L_{boson} = \frac{1}{2} \dot {\mathbf X}^i \dot {\mathbf X}_i + \frac{1}{4} [\mathbf X_j,\mathbf X^i][\mathbf X^j,\mathbf X_i]$$ $$\mathscr L_{fermion} = -\imath \Psi^\dagger \dot \Psi + \Psi^\dagger \sigma^i \otimes [\mathbf X_i,\Psi]$$ involving the following equation of motion: $$\label{equX}
  \ddot {\mathbf X}_i - [\mathbf X_j,[\mathbf X_i,\mathbf X^j]] = 0$$ with the Gauss constraint $[\dot {\mathbf X}_i,\mathbf X^i] = 0$, $$\label{equSchro}
 \imath |\dot \psi {\rangle \hspace{-0.2em} \rangle}= \sigma^i \otimes(X_i-x_i) |\psi {\rangle \hspace{-0.2em} \rangle}$$ $\{\mathbf X_i\}_i$ obeys then to a matrix classical wave equation ($[\mathbf 