Semi-Classical Mechanics in Phase Space: The Quantum Target of Minimal Strings

TL;DR

This paper explores the semi-classical phase space mechanics of minimal strings, revealing how quantum effects smooth classical singularities and deriving quantization rules for backgrounds with ZZ branes.

Contribution

It introduces a phase space approach to minimal strings, connecting matrix models to quantum mechanics via the Wigner function and deriving new quantization conditions.

Findings

Classical target space is a fold catastrophe smoothed by quantum effects

Quantum effects are modeled using the Wigner function in phase space

Quantization rules for backgrounds with ZZ branes are established

Abstract

The target space $M_{p,q}$ of $(p,q)$ minimal strings is embedded into the phase space of an associated integrable classical mechanical model. This map is derived from the matrix model representation of minimal strings. Quantum effects on the target space are obtained from the semiclassical mechanics in phase space as described by the Wigner function. In the classical limit the target space is a fold catastrophe of the Wigner function that is smoothed out by quantum effects. Double scaling limit is obtained by resolving the singularity of the Wigner function. The quantization rules for backgrounds with ZZ branes are also derived.