Curiosities at c-effective = 1

The moduli space of all rational conformal quantum field theories with effective central charge ceff = 1 is considered. Whereas the space of unitary theories essentially forms a manifold, the non unitary ones form a fractal which lies dense in the parameter plane. Moreover, the points of this set are shown to be in one-to-one correspondence with the elements of the modular group for which an action on this set is defined. (orig.)