Exact tachyon condensation on non-commutative torus

We construct the exact non-commutative solutions on tori. This gives an exa ct description of tachyon condensation on bosonic D-branes, non-BPS D-brane s and brane-antibrane systems. We obtain various bound states of D-branes a fter the tachyon condensation. Our results show that these solutions can be generated by applying the gauge Morita equivalence between the constant cu rvature projective modules. We argue that there is a general framework of t he non-commutative geometry based on the notion of Morita equivalence which underlies this specific example.