Arithmetic space-time geometry from string theory

An arithmetic framework to string compactification is described. The approach is exemplified by formulating a strategy that allows to construct geometric compactifications from exactly solvable theories at c = 3. It is shown that the conformal field theoretic characters can be derived from the geometry of space-time, and that the geometry is uniquely determined by the two-dimensional field theory on the worldsheet. The modular forms that appear in these constructions admit complex multiplication, and allow an interpretation as generalized McKay-Thompson series associated to the Mathieu and Conway groups. This leads to a string motivated notion of arithmetic moonshine.