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.
A related project, which has partly been joint work with Prof. Werner Nahm, is based on results of my Ph.D. 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.
