The compactified D-brane cylinder amplitude and T duality

TL;DR

This paper explores how T duality transforms the closed string tree cylinder amplitude between D-branes with compactified dimensions, ensuring consistency with known non-compactified results in the decompactification limit.

Contribution

It provides a detailed method for implementing T duality on the D-brane cylinder amplitude with compactified dimensions, clarifying the relationship between compactified and non-compactified amplitudes.

Findings

Explicit computation of the cylinder amplitude between D-branes with compactified dimensions.

Demonstration of T duality transformation along compactified directions.

Recovery of known non-compactified amplitude in the decompactification limit.

Abstract

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 $p - p' = 2 \,n$. 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.