After the previous chapter's brief introduction to perturbative string theory and T duality we are going to discuss how one arrives at the low-energy string effective (field theory) actions, what their meaning is, and what their limits of validity are. We are going to start exploiting them to study T duality and find Buscher's T-duality rules, that relate different curved backgrounds that are equivalent from the string-theory point of view. These rules are some of the most powerful tools of string theory.

Effective actions and background fields

The low-energy string effective action describes the low-energy dynamics of a given string theory. Here low energy means energies lower than the relevant energy scale: the string mass ms. Thus, the low-energy limit is the ά→ 0 limit, heuristically, the limit in which the string length can be ignored and a theory of particles (a field theory) is recovered.

On the other hand, at low energies only the massless modes are relevant and their dynamics is described by a theory of the corresponding massless fields. The obvious way to find this field theory is to compute string amplitudes for the massless modes, take the ά→ 0 limit and then construct a field theory that reproduces these amplitudes. In principle, the effective field theory has an expansion in powers of ά, although usually only the lowestorder terms are considered. The terms of higher order in ά are also of higher order in derivatives. Also, string amplitudes are calculated order by order in string perturbation theory and the effective action can also be expanded in powers of the string coupling constant, which here is the exponential of the dilaton field.