Weak mixing in interval exchange transformations of periodic type

Interval exchange transformations (IETs) are piecewise isometries of the interval, obtained permuting a certain number of subintervals. We give a condition on IETs in the special subclass of IETs with periodic Rauzy-Veech cocycle which guarantees weak mixing, i.e. the continuity of the spectrum. The proof involves the study of the associated spectral measures. The condition can be checked explicitly by computing a certain Galois group of a field related to the Ravzy-Veech cocycle. Explicit examples of weakly mixing IETs are constructed in the Appendix.