  We study the free energy of the Laughlin state on curved backgrounds,
starting from the free field representation. A simple argument, based on the
computation of the gravitational effective action from the transformation
properties of Green functions under the change of the metric, allows to compute
the first three terms of the expansion in large magnetic field. The leading and
subleading contributions are given by the Aubin-Yau and Mabuchi functionals
respectively, whereas the Liouville action appears at next-to-next-to-leading
order. We also derive a path integral representation for the remainder terms.
They correspond to a large mass expansion for a related interacting scalar
field theory and are thus given by local polynomials in curvature invariants.
