# $n=0$ mode Fourier expansion on string theory

I am quite baffled at what the $n=0$ mean in compactification, why is this mode important?

I mean if $n=0$ was applied here we'd just be left with

I know that (39) holds from (37), but I can't figure out why (38) is of particular significance.

• Which textbook? – Qmechanic Dec 18 '15 at 14:55
• I'm note too confident in this context, but generally speaking $n=0$ is the center-of-mass motion – Bort Dec 18 '15 at 15:24

In the usual Kaluza-Klein reduction for scalar fields in $D$ dimension you got an infinite tower of scalar fields from the $D-1$ dimensional point of view. These scalars come with increasing mass labelled by an integer $n$.
The case $n=0$ is special, because is the massless case. For instance in String Theory, when you compactify on a torus, a consistent truncation of your theory is to consider only the massless field, that is looking at the Supergravity approximation.