Non-linear sigma models on K3 and Umbral Moonshine

In this thesis we study some aspects of non-linear sigma models on K3 surfaces, a particular case of N=(4,4) superconformal field theories which are widely studied in the context of string compactification. We focused in particular on the class of moonshine phenomena arising in the elliptic genus of such models known as umbral moonshine. Umbral moonshine consists of surprising mathematical relationships between mock modular forms, Niemeier lattices and finite group representations which are related, in a way still not completely understood, to the models mentioned above. As the original part of this work, we investigated the decomposition of the twining genera in terms of the irreducible representations of some umbral groups and we specialized it to a particular subgroup which is a symmetry group of some non-linear sigma models on K3 surfaces. With this technique we were able to extract some information on these models which are generally not accessible with standard methods. In particular we found evidences that their chiral algebra can be extended beyond the N=4 superconformal algebra. Subsequently we investigated a simple orbifold model which seemed to constitute a good candidate to possess the same extended chiral algebra. However, with an analysis similar to the previous one, we were able to conclude that this was not the correct extension of the chiral algebra for the models considered. With this work we underline the importance of the search and development of non-standard methods to extract information when the standard techniques fail.