Discrete symmetries of N = 4 String models

In this thesis we study 2-dimensional N=(4,4) superconformal field theories and their application to superstring theories. More specifically, we would like to analyze the behaviour of the fields under a certain set of discrete symmetries of a non-linear sigma model whose target space is a K3 surface, in particular, the T^4/Z_2 orbifold. A good way to understand this is to compute the Elliptic Genus and some Twining Genera obtained by inserting in the Elliptic Genus one of the allowed discrete symmetries of the model. More in detail, the set of discrete symmetries we are going to consider are the ones that respect the OPEs of the fields and fix the N=4 superconformal algebra. The peculiarity of K3 surfaces is that, when a d=10 IIA or IIB superstring theory with 32 supercharges is compactified on them, only 16 supercharges are preserved and this gives birth to 6-dimensional low-energy theories of supergravity. If we compactify on (K3 x T^2) surfaces, what we obtain are low-energy 4-dimensional theories of supergravity with 16 preserved supercharges. The Twining Genera we are going to compute, have been conjectured for a certain set of K3 surfaces with different metrics and different discrete symmetries, therefore, since they should not depend on the choice of the metric as long as the symmetry remains a symmetry of the orbifold, we expect to confirm the conjectured value. In conclusion, the study of non-linear sigma models living on K3 surfaces is useful to understand 6-dimensional and 4-dimensional supergravity theories with, at least, 16 supercharges.