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We shall construct two boundary states which are corresponding to a dynamicalfractional D$p$-brane in the presence of the fluxes of the Kalb-Ramond fieldand a $U(1)$ gauge potential, in the partially orbifold spacetime$\mathbb{R}^{1, 5}\times\mathbb{C}^{2}/\mathbb{Z}_{2}$. These states accuratelydescribe the D$p$-brane in the twisted and untwisted sectors under the orbifoldprojection. We use them to compute the interaction of two parallel fractionalD$p$-branes with the transverse velocities, tangential rotations and tangentiallinear motions. Various properties of the interaction, such as its long-rangeforce, will be discussed.

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