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We develop a first quantization description of fractional Chern insulatorsthat is the dual of the conventional fractional quantum Hall (FQH) problem,with the roles of position and momentum interchanged. In this picture, FQHstates are described by anisotropic FQH liquids forming in momentum-spaceLandau levels in a fluctuating magnetic field. The fundamental quantum geometryof the problem emerges from the interplay of single-body and interactionmetrics, both of which act as momentum-space duals of the geometrical pictureof the anisotropic FQH effect. We then present a novel broad class of idealChern insulator lattice models that act as duals of the isotropic FQH effect.The interacting problem is well-captured by Haldane pseudopotentials andaffords a detailed microscopic understanding of the interplay of interactionsand non-trivial quantum geometry.

Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators