Large Order Behaviour of 2D Gravity Coupled to $d<1$ Matter

TL;DR

This paper analyzes the large order behavior and Borel summability of the topological expansion in 2D gravity models coupled to $(p,q)$ matter, deriving equations for key constants and showing non-Borel summability for unitary models.

Contribution

It provides a set of equations for the large order constant in general $(p,q)$ models and explicitly determines the behavior for models with known semiclassical limits.

Findings

Large order behavior follows a specific gamma function pattern.

The constant 'a' is determined by the form of differential operators in semiclassical limits.

Unitary models' topological expansions are not Borel summable.

Abstract

We discuss the large order behaviour and Borel summability of the topological expansion of models of 2D gravity coupled to general $(p,q)$ conformal matter. In a previous work it was proven that at large order $k$ the string susceptibility had a generic $a^k\Gamma(2k-\ud)$ behaviour. Moreover the constant $a$, relevant for the problem of Borel summability, was determined for all one-matrix models. We here obtain a set of equations for this constant in the general $(p,q)$ model. String equations can be derived from the construction of two differential operators $P,Q$ satisfying canonical commutation relations $[P,Q]=1$. We show that the equation for $a$ is determined by the form of the operators $P,Q$ in the spherical or semiclassical limits. The results for the general one-matrix models are then easily recovered. Moreover, since for the $(p,q)$ string models such $p=(2m+1)q\pm1$ the semiclassical forms of $P,Q$ are explicitly known, the large order behaviour is completely determined. This class contains all unitary $(q+1,q)$ models for which the answer is specially simple. As expected we find that the topological expansion for unitary models is not Borel summable. \preprint{SPhT/92-163}, Plain-TeX, macro harvmac