Emergent Conformal Symmetry and Geometric Transport Properties of Quantum Hall States on Singular Surfaces

We study quantum Hall states on surfaces with conical singularities. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum, which is due solely to the gravitational anomaly. We also show that quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. Finally, we argue that the gravitational anomaly and conformal dimension determine the fine structure of the electronic density at the conical point. The singularities emerge as quasiparticles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.