A toroidal sigma-model of magneto-transport is analysed, in which integer and fractional quantum Hall effects are automatically unified by a holomorphic modular symmetry, whose group structure is determined by the spin-structure of the target space. Hall-quantisation is protected by the topology of stable holomorphic vector bundles on tori, and plateaux values of the Hall-conductivity are rational because such bundles are classified by a rational number called the slope. By exploiting a quantum equivalence called mirror symmetry, this model is mapped to a tractable mirror model, in which topological protection is offered by more familiar winding numbers. The phase diagram and partition function of the mirror model is compared to some of the experimental and numerical data accumulated over the past three decades. The geometry of scaling flows extracted from quantum Hall experiments is in good agreement with modular predictions, including the location of many quantum critical points. One toroidal model gives a value of the critical delocalisation exponent that agrees with the best available numerical value at the per mille level, suggesting that the toroidal model is in the same universality class as the Chalker-Coddington model. The real delocalisation exponent can be disentangled from other scaling exponents in finite-size scaling experiments, giving an experimental value that is smaller than the toroidal value. We explain why this is another success of the modular model.