The new states of matter and concomitant quantum critical phenomena re- vealed by the quantum Hall effect appear to be accompanied by an emergent modular symmetry. The extreme rigidity of this infinite symmetry makes it easy to falsify, but two decades of experiments have failed to do so, and the location of quantum critical points predicted by the symmetry is in increasingly accurate agreement with scaling ex- periments. The symmetry severely constrains the structure of the effective quantum field theory that encodes the low energy limit of quantum electrodynamics of 10 10 charges in two dirty dimensions. If this is a non-linear σ-model the target space is a torus, rather than the more familiar sphere. One of the simplest toroidal models gives a critical (cor- relation length) exponent that agrees with the value obtained from numerical simulations of the quantum Hall effect.