We discuss several rigidity results connected to topological quantum field theories. We show that there are countably many rational TQFT's, and that the invariants of manifolds computed with them are algebraic numbers. We describe all the theories which have a generator of quantum dimension < 2, and show that they correspond to the subgroups of quantum SU(2) at roots of 1. For instance, the quantum binary icosahedral group has 32 representations and appears at the 30th root of 1. These subgroups arise naturally as symmetries of paths on the Coxeter ADE graphs. Their modular invariants explain the ADE classification of conformal invariants of Itzykson, Capelli and Zuber.