Fractional Dp-Branes with Transverse and Tangential Dynamics in the Presence of Background Fluxes

We shall construct two boundary states which are corresponding to a dynamical fractional D$p$-brane in the presence of the fluxes of the Kalb-Ramond field and a $U(1)$ gauge potential, in the partially orbifold spacetime ${\mathbb{R}}^{1,5}\times {\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_2$. These states accurately describe the D$p$-brane in the twisted and untwisted sectors under the orbifold projection. We use them to compute the interaction of two parallel fractional D$p$-branes with the transverse velocities, tangential rotations and tangential linear motions. Various properties of the interaction, such as its long-range force, will be discussed.