  We obtain two boundary states corresponding to the two folds of a
fractional-wrapped D$p$-brane, i.e. the twisted version under the orbifold
$\mathbb{C}^{2}/\mathbb{Z}_{2}$ and the untwisted version. The brane has
rotation and linear motion, in the presence of the following background fields:
the Kalb-Ramond tensor, a $U(1)$ internal gauge potential and a tachyon field.
The rotation and linear motion are inside the volume of the brane. The brane
lives in the $d$-dimensional spacetime, with the orbifold-toroidal structure
$T^n \times \mathbb{R}^{1, d-n-5} \times\mathbb{C}^{2}/\mathbb{Z}_{2}$ in the
twisted sector. Using these boundary states we calculate the interaction
amplitude of two parallel fractional D$p$-branes with the foregoing setup.
Various properties of this amplitude such as the long-range behavior will be
analyzed.
