The Compactified D-Brane Cylinder Amplitude and T Duality

In this paper, we address how to implement T-duality to the closed string tree cylinder amplitude between a Dp brane and a Dp′ brane with p−p′=2n. To achieve this, we compute the closed string tree cylinder amplitude, for the first time, between these two D-branes with common longitudinal and transverse circle compactifications. We then show explicitly how to perform a T-duality for this amplitude along either a longitudinal or a transverse compactified direction to both branes. At the decompactification limit, we show that either the compactified cylinder amplitude or the T dual compactified cylinder gives the known non-compactified one as expected.