The θ vacuum reveals itself as the fundamental theory of the quantum Hall effect. I. Confronting controversies

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The θ dependence of the Grassmannian U(m+n)/U(m)× U(n) non-linear σ model is reexamined. This general theory provides an important laboratory for studying the quantum Hall effect, in the special limit m=n=0 (replica limit). We discover however that the quantum Hall effect is in fact independent of this limit and exists as a generic topological feature of the theory for all non-negative values of m and n. The results are in concflict with many of the historical ideas and expectations on the basis of the large N expansion of the CPⁿ⁻¹ or SU(N)/U(N-1) model