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There is plenty of evidence for the relevance of automorphic forms within string theory. Unfortunately, many of the statements in the literature remain well-supported conjectures, and a complete mathematical interpretation is lacking. To gain a better understanding of these issues is a longterm research goal of mine.

For SCFTs associated to K3 a "topological part" of all partition functions was given by Eguchi, Ooguri, Taormina and Yang and corrected in my thesis. However, an explicit geometric and SCFT interpretation for it is lacking. I am studying this in collaboration with Dr. Anne Taormina. The revelant function can be expressed in terms of Appell functions, whose quasiperiodic behavior under modular transforms is possibly linked to automorphic forms on the moduli space. We already have concrete results concerning the interpretation of symplectic automorphisms of certain K3 surfaces in terms of the Mathieu group M24, which are expected to relate to the role of the latter group in the context of the elliptic genus:
Anne Taormina, Katrin Wendland,
The symmetries of the tetrahedral Kummer surface in the Mathieu group M24;
preprint arXiv1008.0954 [hep-th]

A related project, which has partly been joint work with Prof. Werner Nahm, is based on results of my PhD. thesis on a connection between the condition of rationality for a conformal field theory and the condition of complex multiplication in an appropriate geometric interpretation of the theory. Such a connection had also been conjectured by Nahm and Kontsevich and was studied by Gukov and Vafa. For related issues arising in the construction of dyonic black holes in Calabi-Yau compactifications of IIB string theory see Gregory Moore's work on arithmetic and attractors.

 
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