So far, we have considered strings propagating on a flat space-time . However, consistency demands dimension . The simplest example is to modify the geometry to , where is a circle of radius . This leads us to a surprising phenomenon called T-duality, which indicates that strings probe geometry in a fundamentally different way to particles.
First, some notation. We define and the coordinates . We also require that the final coordinate is periodic, i.e., . It is then convenient to index the remaining non-periodic coordinates by .