Gauge-Invariant Variables Reveal the Quantum Geometry of Fractional Quantum Hall States

We introduce the framework of gauge invariant variables to describe fractional quantum Hall (FQH) states, and prove that the wavefunction can always be represented by a unique holomorphic multi-variable complex function. We use this representation to combine the quantum geometry of charged particles in a magnetic field with the theory of coherent states and provide an analytical route, hitherto elusive, for deriving the properties of fractional quantum Hall phases from experimentally relevant microscopic Hamiltonians.