Twisted Bundles on Noncommutative $T^4$ and D-brane Bound States

We construct twisted quantum bundles and adjoint sections on noncommutative $T^4$, and investigate relevant D-brane bound states with non-Abelian backgrounds. We also show that the noncommutative $T^4$ with non-Abelian backgrounds exhibits SO$(4,4\mathrm{\mid }Z)$ duality and via this duality we get a Morita equivalent $T^4$ on which only D0-branes exist. For a reducible non-Abelian background, the moduli space of D-brane bound states in Type II string theory takes the form ${\prod }_a(T^4{)}^{q_a}\mathrm{/}{S}_{q_a}$.