  We develop a collective field theory for fractional quantum Hall (FQH)
states. We show that in the leading approximation for a large number of
particles, the properties of Laughlin states are captured by a Gaussian free
field theory with a (filling fraction dependent) background charge. Gradient
corrections to the Gaussian field theory arise from ultraviolet regularization.
They are the origin of the gravitational anomaly and are described by the
Liouville theory of quantum gravity. The field theory simplifies the
computation of correlation functions in FQH states and makes manifest the
effect of quantum anomalies.
