  Chiral conformal field theories are characterized by a ground-state current
at finite temperature, that could be observed, e.g. in the edge excitations of
the quantum Hall effect. We show that the corresponding thermal conductance is
directly proportional to the gravitational anomaly of the conformal theory,
upon extending the well-known relation between specific heat and conformal
anomaly. The thermal current could signal the elusive neutral edge modes that
are expected in the hierarchical Hall states. We then compute the thermal
conductance for the Abelian multi-component theory and the W-infinity minimal
model, two conformal theories that are good candidates for describing the
hierarchical states. Their conductances agree to leading order but differ in
the first, universal finite-size correction, that could be used as a selective
experimental signature.
