Holomorphic anomaly in the quantum Hall system

Experimental and theoretical evidence which is accumulating in favor of the existence of a global sub-modular symmetry in the quantum Hall system is reviewed. The scaling data suggest that the zeros of the beta-function are effectively anti-holomorphic, and it is explained how this leads to a superuniversal scaling function. This motivates the first construction of a candidate beta-function which agrees with all scaling data, as well as the restrictions coming from both the perturbative and dilute instanton gas analysis of the non-linear sigma-model of planar charge transport. The key mathematical concept allowing global analysis of this non-holomorphic function is quasi-holomorphic automorphy, and a firm mathematical foundation is given which exhibits the intimate relationship with the holomorphic anomaly that appears in supersymmetric string- and field-theories in high energy physics. It is therefore natural to conjecture that quasi-holomorphy is simply a consequence of the well-known existence of an effective supersymmetry in the low-energy field theory of a disordered medium.