Title: Quantized Laplacian growth, III: On conformal field theories of Laplacian growth,
Abstract: A one-parametric stochastic dynamics of the interface in the quantized
Laplacian growth with zero surface tension is introduced. The quantization
procedure regularizes the growth by preventing the formation of cusps at the
interface, and makes the interface dynamics chaotic. In a long time asymptotic,
by coupling a conformal field theory to the stochastic growth process we
introduce a set of observables (the martingales), whose expectation values are
constant in time. The martingales are connected to degenerate representations
of the Virasoro algebra, and can be written in terms of conformal correlation
functions. A direct link between Laplacian growth and the conformal Liouville
field theory with the central charge $c\geq25$ is proposed.