On localization in the theory of the quantized hall effect: A two-dimensional realization of the θ-vacuum

It is shown that the localization problem in the theory of the quantized Hall effect is governed by the zero-component grassmannian U(2m) non-linear σ-model with θ-term, a two-dimensional analogue of the θ-vacuum in Yang-Mills theory. In this case, θ is to be interpreted as the “bare” value for the Hall conductivity, determined by an underlying non-critical theory. A detailed derivation is presented starting from the replica method and a delta function distribution for the impurities.