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${}^{\mathrm{\prime }}$ brane with $p-p^{\mathrm{\prime }}=2n$. For this, we first compute the closed string tree cylinder amplitude between these two D branes with common longitudinal and transverse circle compactifications. We then show explicitly how to perform a T duality to this amplitude along either a longitudinal or a transverse compactified direction to both branes. In the decompactification limit, we show that either the compactified cylinder amplitude or the T dual compactified cylinder one gives the known non-compactified one as expected.