Emergent Conformal Symmetry of Quantum Hall States on Singular surfaces

We show that quantum Hall states on surfaces with conical singularities behave as conformal primaries near the singular points, with a conformal dimension controlled by the gravitational anomaly. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum equal to the conformal dimension, in units of the Planck constant. Finally, we argue that the gravitational anomaly also controls { the fine structure of electronic density at the tip, and } the exchange statistics of cones in the Laughlin states, 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.

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