Semi-classical mechanics in phase space: the quantum target of minimal strings

The target space $M_{p,q}$ of $(p,q)$ minimal strings is embedded into thephase space of an associated integrable classical mechanical model. This map isderived from the matrix model representation of minimal strings. Quantumeffects on the target space are obtained from the semiclassical mechanics inphase space as described by the Wigner function. In the classical limit thetarget space is a fold catastrophe of the Wigner function that is smoothed outby quantum effects. Double scaling limit is obtained by resolving thesingularity of the Wigner function. The quantization rules for backgrounds withZZ branes are also derived.Comment: 16 pages, 6 figure