In this chapter we shall study the spectrum of strings propagating in a spacetime that has a compact direction. The theory has all of the properties we might expect from the knowledge that at low energy we are placing gravity and field theory on a compact space. Indeed, as the compact direction becomes small, the parts of the spectrum resulting from momentum in that direction become heavy, and hence less important, but there is much more. The spectrum has additional sectors coming from the fact that closed strings can wind around the compact direction, contributing states whose mass is proportional to the radius. Thus, they become light as the circle shrinks. This will lead us to T-duality, relating a string propagating on a large circle to a string propagating on a small circle. This is just the first of the remarkable symmetries relating two string theories in different situations that we shall encounter here. It is a crucial consequence of the fact that strings are extended objects. Studying its consequences for open strings will lead us to D-branes, since T-duality will relate the Neumann boundary conditions we have already encountered to Dirichlet ones, corresponding to open strings ending on special hypersurfaces in spacetime.

Fields and strings on a circle

Let us remind ourselves of what happens in field theory, for the case of placing gravity on a spacetime with a compact direction.